Handbook of Colloid and Interface Science: Volume 2 Basic Principles of Dispersions 9783110541953, 9783110539912

Citation preview

Tharwat F. Tadros Handbook of Colloid and Interface Science De Gruyter Reference

Also of Interest Handbook of Colloid and Interface Science. Volume 1: Basic Principles of Interface Science and Colloid Stability Tadros, 2017 ISBN 978-3-11-053990-5, e-ISBN 978-3-11-054089-5

Handbook of Colloid and Interface Science. Volume 3: Industrial Applications: Pharmaceuticals, Cosmetics and Personal Care Tadros, 2018 ISBN 978-3-11-055409-0, e-ISBN 978-3-11-055525-7 Polymeric Surfactants. Dispersion Stability and Industrial Applications Tadros, 2017 ISBN 978-3-11-048722-0, e-ISBN 978-3-11-048728-2

Suspension Concentrates. Preparation, Stability and Industrial Applications Tadros, 2017 ISBN 978-3-11-048678-0, e-ISBN 978-3-11-048687-2

Formulations. In Cosmetic and Personal Care Tadros, 2016 ISBN 978-3-11-045236-5, e-ISBN 978-3-11-045238-9

Emulsions. Formation, Stability, Industrial Applications Tadros, 2016 ISBN 978-3-11-045217-4, e-ISBN 978-3-11-045224-2

Tharwat F. Tadros

Handbook of Colloid and Interface Science

| Volume 2: Basic Principles of Dispersions

Author Prof. Tharwat F. Tadros 89 Nash Grove Lane Workingham RG40 4HE Berkshire, UK [email protected]

ISBN 978-3-11-053991-2 e-ISBN (PDF) 978-3-11-054195-3 e-ISBN (EPUB) 978-3-11-054019-2 Set-ISBN 978-3-11-054196-0

Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Cover image: satori13/iStock/Getty Images Plus Typesetting: PTP-Berlin, Protago-TEX-Production GmbH, Berlin Printing and binding: CPI Books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Preface A colloidal dispersion is a two-phase system in which one phase (the disperse phase) is dispersed in a second continuous phase (the dispersion medium). The disperse phase can be solid, liquid or gas and the same applies for the dispersion medium. On this basis one can distinguish several classes of colloidal dispersions: solid/liquid (suspension), liquid/liquid (emulsion), gas/liquid (foam), liquid/gas (aerosol), liquid/solid (gel), solid/gas (smoke) and solid/solid (composite). Another important class of colloids are produced by self-assembly of surfactants that form micelles with dimensions in the colloid range and these were described in Vol. 1. In all disperse systems such as suspensions, emulsions, foams, etc., the structure of the interfacial region determines its colloidal properties. For convenience, I will list the topics of colloid and interface science under two main headings: disperse systems and interfacial phenomena. This subdivision does not imply any separation for the following reasons. All disperse systems involve an interface. Many interfacial phenomena are precursors for formation of disperse systems, e.g. nucleation and growth, emulsification, etc. The main objective of the present handbook is to cover the following topics: the basic principles that are involved in the formation of colloidal dispersions and their stabilization. The field of colloid and interface science has no boundary since chemists, physicists, engineers, biologists, and mathematicians can all be engaged in the field. For successful applications in industry, multidisciplinary teams are required. Understanding the basic principles of colloid and interface science will enable industry to develop many complex systems in a shorter period of time. Most colloidal systems used in industry are multiphase and complex formulations. They may contain more than one disperse phase, e.g. suspension/emulsion systems (suspoemulsions). Chapter 1 describes the flow characteristics (rheology) of colloidal dispersions. It starts with a section on the basic principles of rheology that includes steady state (shear stress-shear rate measurements), constant stress (creep measurements), constant strain (stress relaxation measurements) and dynamic (oscillatory) techniques. The various rheological models that are used to describe each technique are described. This is followed by a section on very dilute and moderately concentrated colloidal dispersions (which takes hydrodynamic interaction into account). The rheology of concentrated colloidal dispersions is then described by considering four different systems. The first is represented by hard sphere interactions where both repulsion and attraction are screened. The semi-empirical model that can be applied to describe the rheology of hard sphere dispersions is described. The second system is that of “soft” or electrostatic interaction whereby the rheology of the system is determined by double layer repulsion. The third system is that of sterically stabilized dispersions. The importance of the ratio of adsorbed layer thickness to particle radius is described. The

https://doi.org/10.1515/9783110541953-001

vi | Preface

fourth system is that of flocculated dispersions and a distinction can be made between weak and strong flocculation. Chapter 2 describes the various processes of wetting, spreading and adhesion. It starts with the equilibrium thermodynamic treatment and Young’s equation. The calculation of surface tension and contact angle is described. This is followed by a description of the process of spreading of liquids on surfaces and the definition of the Harkins spreading coefficient. The process of contact angle hysteresis and effect of roughness and surface heterogeneity is described. This is followed by a section on the critical surface tension of wetting and the effect of surfactant adsorption. The dynamic process of adsorption and wetting is described. Chapter 3 deals with solid/liquid dispersions (suspensions). Both colloidal and non-colloidal organic and inorganic particles suspended in aqueous and nonaqueous media are described. Particular attention is given to nanosuspensions that cover the size range 10–100 nm. The general methods of preparation and stabilization of dispersions are described. The distinction between colloidal and physical stability is described. This is followed by a section on preparation of suspensions by bottom-up processes, namely by nucleation and growth. The thermodynamic theory of nucleation and growth is described at a fundamental level. This is followed by a section on precipitation kinetics and control of particle size distribution. The preparation of suspensions by bottom-down processes, namely by breaking of aggregates and agglomerates, dispersion and comminution (milling) is then described. The role of wetting and dispersing agents is described at a fundamental level. This is followed by a section on electrostatic and steric stabilization of suspensions. The Ostwald ripening (crystal growth) of suspensions that results from the difference in solubility between small and large particles is described. This is followed by a section on sedimentation of suspensions and prevention of formation of hard sediments (“clays”). Chapter 4 describes the liquid/liquid dispersions (emulsions). It starts with a section on the classification of emulsions based on the nature of the emulsifier and the structure of the system. Special attention is given to nanoemulsions covering the size range 10–100 nm. The general instability problems with emulsions are described. This is followed by a section on the thermodynamics of emulsion formation and breakdown with particular reference to the importance of having a repulsive energy barrier to ensure the kinetic stability of the system. The interaction forces between emulsion droplets are briefly described. This is followed by a description of the mechanism of emulsification and the role of the emulsifier. The various methods that can be applied for emulsification are described with particular reference to the use of high pressure homogenizers. The various methods for selection of emulsifiers are described. The process of creaming/sedimentation of emulsions and its prevention is described. This is followed by sections on flocculation of emulsions and its prevention, Ostwald ripening and its reduction, emulsion coalescence and its prevention and phase inversion. Chapter 5 deals with multiple emulsions. It starts with a definition and types of multiple emulsions. The breakdown processes of multiple emulsions are described.

Preface

| vii

This is followed by a section on preparation of multiple emulsions with particular reference to the role of the emulsifiers and the nature of the oil phase. The factors affecting stability of multiple emulsions are described at a fundamental level. This is followed by a section on the methods that can be applied for characterization of multiple emulsions and their long-term physical stability. Chapter 6 deals with gas (air)/liquid dispersions (foams). It starts with the methods of foam preparation and foam structure. The classification of foam stability is described. This is followed by a section on drainage and thinning of foam films. The various theories of foam stability are described. This is followed by a description of foam inhibitors. The physical properties of foams are described and this is followed by a section on the experimental techniques for studying foam structure and stability. Chapter 7 describes liquid/solid dispersions or gels. It starts with a definition of a gel and a description of the rheological behaviour of a gel. The various classifications of gels are described, starting with physical gels obtained by polymer chain overlap. Gels produced by associative thickeners are described. This is followed by a section on crosslinked (chemical) gels. The second part of this chapter deals with particulate gels formed by crosslinking of finely divided particles. Chapter 8 deals with the subject of polymer colloids (latexes). It starts with a section on the preparation by emulsion polymerization, with particular reference to the theories and the role of the emulsifier. The use of block and graft copolymers in emulsion polymerization is described. This is followed by a section on dispersion polymerization for preparation of nonaqueous polymer latexes. Particular attention is given to the importance of the presence of a “protective colloid” to prevent the flocculation of the resulting latex. Chapter 9 describes microemulsions. It starts with the definition of microemulsions, their spontaneous formation and thermodynamic stability. The various theories of microemulsion formation and stability are described, with particular reference to the importance of having an ultra-low interfacial tension. The use of mixed surfactants for preparation of microemulsions is described. The various techniques that can be applied for characterization of microemulsions are described. Chapter 10 deals with liposomes and vesicles and their important applications. The various lipids that are used for preparation of liposomes and vesicles are described. This is followed by a description of the driving force for formation of vesicles and the importance of the critical packing parameter concept. The enhancement of stability of vesicles by incorporation of block copolymers is described. Chapter 11 deals with the process of deposition of particles and their adhesion at interfaces, with particular reference to the role of interparticle interactions. The methods of measurement of particle deposition using rotating disc and cylinder techniques are described. The effect of polymers and polyelectrolytes on particle deposition at interfaces is described. This is followed by a section on the effect of nonionic, anionic and cationic polymers on particle deposition. The process of particle-surface adhesion is described with particular reference to the importance of short-range forces

viii | Preface

and calculation of the force of adhesion. The surface energy approach to adhesion is described. This is followed by the experimental methods for measuring particle deposition and adhesion. Chapter 12 deals with the various techniques that can be applied for characterization of colloidal dispersions. It starts with the definition of particle size distribution and polydispersity. Various sections are devoted to the application of optical microscopy, image analysis, confocal laser scanning microscopy, diffraction methods and scanning and electron microscopy. The various scattering techniques that can be applied for determining particle size and polydispersity index are described with particular reference to time-average and dynamic (quasi-elastic) (photon correlation spectroscopy) light scattering methods. The methods that can be applied to assess creaming or sedimentation, flocculation, Ostwald ripening and coalescence are described in this chapter. This handbook (Vol. 2) gives a comprehensive fundamental analysis of the processes involved in the preparation of various disperse systems. It would be valuable for graduate students requiring knowledge of the subject as well as research workers who are engaged in the field. It will also be extremely valuable for industrial chemists and chemical engineers who are engaged in the formulation of various disperse systems. It will also provide the formulation chemist with a fundamental approach on how to arrive at a required state of a colloidal dispersion and how to maintain the ease of its application. Tharwat Tadros April 2017

Contents Preface | v 1 1.1 1.2 1.2.1 1.2.2 1.2.3

1.4.2 1.4.3

Flow characteristics (rheology) of colloidal dispersions | 1 Introduction | 1 Rheological techniques | 1 Steady state shear stress σ–shear rate γ measurements | 1 Rheological models for analysis of flow curves | 3 Time effects during flow – thixotropy and negative (or anti-) thixotropy | 5 Strain relaxation after sudden application of stress – constant stress (creep) measurements | 7 Stress relaxation after sudden application of strain | 10 Dynamic (oscillatory) measurements | 12 Rheology of colloidal dispersions | 17 The Einstein equation | 18 The Bachelor equation | 18 Rheology of concentrated dispersions | 18 Examples of strongly flocculated (coagulated) suspension | 34 Coagulation of electrostatically stabilized suspensions by addition of electrolyte | 34 Strongly flocculated sterically stabilized systems | 36 Models for interpretation of rheological results | 40

2 2.1 2.2 2.3 2.4 2.5 2.6 2.6.1 2.6.2 2.7 2.7.1 2.7.2 2.7.3 2.7.4 2.7.5 2.8

Wetting and spreading | 45 Introduction | 45 The concept of contact angle | 47 Adhesion tension | 49 Work of adhesion Wa | 50 Work of cohesion | 51 Calculation of surface tension and contact angle | 51 Good and Girifalco approach | 52 Fowkes treatment | 53 The spreading of liquids on surfaces | 55 The spreading coefficient S | 55 Contact angle hysteresis | 56 Reasons for hysteresis | 57 Wenzel’s equation | 57 Surface heterogeneity | 58 The critical surface tension of wetting | 59

1.2.4 1.2.5 1.2.6 1.3 1.3.1 1.3.2 1.3.3 1.4 1.4.1

x | Contents

2.9 2.10 2.11 2.12 2.13 2.14 2.14.1 2.14.2 2.15 2.15.1 2.15.2 2.15.3 2.15.4 2.15.5 2.16 2.17 2.18 3 3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.5

4 4.1 4.2

Theoretical basis of the critical surface tension | 60 Effect of surfactant adsorption | 61 Wetting of powders by liquids | 62 Rate of penetration of liquids: The Rideal–Washburn equation | 63 Measurement of contact angles of liquids and surfactant solutions on powders | 64 Assessment of wettability of powders | 65 Sinking time, submersion or immersion test | 65 List of wetting agents for hydrophobic solids in water | 65 Measurement of contact angles on flat surfaces | 66 Sessile drop or adhering gas bubble method | 66 Wilhelmy plate method | 67 Capillary rise at a vertical plate | 68 Tilting plate method | 69 Capillary rise or depression method | 69 Wetting kinetics | 70 The dynamic contact angle | 70 Effect of viscosity and surface tension | 73 Solid/liquid dispersions (suspensions) | 75 Introduction | 75 Preparation of suspension concentrates by the bottom-up process | 83 Nucleation and growth | 85 Precipitation kinetics | 87 Seeded nucleation and growth | 92 Surface modification | 92 Other methods for preparation of suspensions by the bottom-up process | 93 Preparation of suspensions using the top-down process | 96 Wetting of the bulk powder | 97 Breaking of aggregates and agglomerates into individual units | 101 Wet milling or comminution | 107 Stabilization of the suspension during dispersion and milling and the resulting nanosuspension | 111 Prevention of Ostwald ripening (crystal growth) | 115 Sedimentation of suspensions and prevention of formation of hard sediments | 123 Liquid/liquid dispersions (emulsions) | 145 Introduction | 145 Thermodynamics of emulsion formation and breakdown | 148

Contents |

4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5 5.1 5.2 5.3 5.4

xi

Interaction forces between emulsion droplets and factors affecting their stability | 153 Mechanism of emulsification and the role of the emulsifier | 155 Methods of emulsification | 162 Selection of emulsifiers | 172 Creaming/sedimentation of emulsions and its prevention | 181 Flocculation of emulsions | 196 Ostwald ripening in emulsions and its prevention | 206 Emulsion coalescence and its prevention | 213 Phase inversion and its prevention | 223

5.7 5.8 5.8.1 5.8.2

Multiple emulsions | 235 Introduction | 235 Preparation of multiple emulsions | 236 Types of multiple emulsions and their breakdown processes | 237 Factors affecting stability of multiple emulsions and criteria for their stabilization | 239 Polymeric surfactants used for preparation of multiple emulsions | 240 Interaction between oil or water droplets containing an adsorbed polymeric surfactant – steric stabilization | 242 Examples of multiple emulsions using polymeric surfactants | 249 Characterization of multiple emulsions | 250 Droplet size measurements | 250 Rheological measurements | 251

6 6.1 6.2 6.3 6.4 6.5 6.6 6.6.1 6.6.2 6.6.3 6.6.4 6.6.5 6.6.6 6.7 6.7.1 6.7.2

Gas (air)/liquid dispersions (foams) | 255 Introduction | 255 Foam preparation | 255 Foam structure | 256 Classification of foam stability | 258 Drainage and thinning of foam films | 259 Theories of foam stability | 262 Surface viscosity and elasticity theory | 262 The Gibbs–Marangoni effect theory | 262 Surface forces theory (disjoining pressure π) | 263 Stabilization by micelles (high surfactant concentrations > cmc) | 266 Stabilization by lamellar liquid crystalline phases | 267 Stabilization of foam films by mixed surfactants | 267 Foam inhibitors | 267 Chemical inhibitors that lower viscosity and increase drainage | 267 Solubilized chemicals which cause antifoaming | 268

5.5 5.6

xii | Contents

6.7.3 6.7.4 6.7.5 6.7.6 6.8 6.8.1 6.8.2 6.8.3 6.8.4 6.8.5 6.9 6.9.1 6.9.2 6.9.3 6.9.4 7 7.1 7.2 7.3 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.5.2 7.5.3 7.6 7.6.1 7.6.2 7.6.3 7.7 7.8 8 8.1 8.2 8.3

Droplets and oil lenses which cause antifoaming and defoaming | 268 Surface tension gradients (induced by antifoamers) | 269 Hydrophobic particles as antifoamers | 270 Mixtures of hydrophobic particles and oils as antifoamers | 270 Physical properties of foams | 271 Mechanical properties | 271 Rheological properties | 271 Electrical properties | 272 Electrokinetic properties | 273 Optical properties | 273 Experimental techniques for studying foams | 273 Techniques for studying foam films | 273 Techniques for studying structural parameters of foams | 274 Measuring foam drainage | 275 Measuring foam collapse | 275 Liquid/solid dispersions (gels) | 277 Introduction | 277 Classification of gels | 277 Gel-forming materials | 278 Rheological behaviour of a “gel” | 279 Stress relaxation (after sudden application of strain) | 279 Constant stress (creep) measurements | 281 Dynamic (oscillatory) measurements | 282 Polymer gels | 282 Physical gels obtained by chain overlap | 282 Gels produced by associative thickeners | 284 Crosslinked gels (chemical gels) | 287 Particulate gels | 288 Aqueous clay gels | 288 Organo-clays (bentones) | 291 Oxide gels | 292 Gels produced by mixtures of polymers and finely divided particulate solids | 293 Gels based on surfactant systems | 294 Polymer colloids (latexes) | 297 Introduction | 297 Emulsion polymerization | 297 Polymeric surfactants for stabilizing preformed latex dispersions | 307

Contents |

8.4 8.5 9 9.1 9.2 9.3 9.3.1 9.3.2 9.4 9.4.1 9.4.2 9.4.3 9.5 9.5.1 9.5.2 9.5.3 9.5.4 9.5.5

xiii

Dispersion polymerization | 311 Particle formation in polar media | 316

9.6 9.7 9.8

Microemulsions | 319 Introduction | 319 Thermodynamic definition of microemulsions | 320 Mixed film and solubilization theories of microemulsions | 321 Mixed film theories | 321 Solubilization theories | 323 Thermodynamic theory of microemulsion formation | 325 Reason for combining two surfactants | 325 Free energy of formation of a microemulsion | 326 Factors determining W/O versus O/W microemulsions | 328 Characterization of microemulsions using scattering techniques | 330 Time-average (static) light scattering | 330 Calculating droplet size from interfacial area | 333 Dynamic light scattering (photon correlation spectroscopy, PCS) | 333 Neutron scattering | 335 Contrast matching for determining the structure of microemulsions | 335 Characterization of microemulsions using conductivity | 336 NMR measurements | 339 Formulation of microemulsions | 339

10 10.1 10.2 10.3

Liposomes and vesicles | 343 Introduction | 343 Nomenclature of liposomes and their classification | 344 Driving force for formation of vesicles | 345

11 11.1 11.2 11.3 11.4

Deposition of particles at interfaces and their adhesion | 351 Introduction | 351 Particle deposition | 352 Effect of polymers and polyelectrolytes on particle deposition | 357 Experimental methods for studying kinetics of particle deposition | 360 11.5 Linear deposition regime | 363 11.6 Nonlinear particle deposition (high coverage) | 370 11.7 Particle deposition on heterogeneous surfaces | 375 11.8 Particle–surface adhesion | 378 11.8.1 Fox and Zisman critical surface tension approach | 380 11.8.2 Neumann’s equation of state approach | 380

xiv | Contents

12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.9.1 12.9.2 12.9.3 12.9.4

Characterization, assessment and prediction of stability of colloidal dispersions | 385 Introduction | 385 Assessment of the structure of the solid/liquid interface | 386 Measurement of surfactant and polymer adsorption | 389 Assessment of creaming/sedimentation of dispersions | 391 Assessment of flocculation | 395 Measurement of Ostwald ripening | 409 Assessment of coalescence of emulsions | 410 Bulk properties of dispersions: Equilibrium sediment or cream volume (or height) and redispersion | 410 Application of rheological techniques for the assessment and prediction of the physical stability of dispersions | 411 Rheological techniques for predicting sedimentation or creaming and syneresis | 411 Prediction of emulsion creaming | 414 Assessment and prediction of flocculation using rheological techniques | 416 Assessment and prediction of emulsion coalescence using rheological techniques | 422

Index | 429

1 Flow characteristics (rheology) of colloidal dispersions 1.1 Introduction Investigations on the flow characteristics, or rheology, of colloidal dispersions are of considerable importance in a number of aspects. For example, the rheology of colloidal dispersions can be applied for evaluation of the stability/instability of the dispersion without any dilution (which can cause significant changes in the structure of the system) and this requires carefully designed techniques that should cause as little disturbance to the structure as possible [1–7]. These measurements provide accurate information on the state of the system such as its flocculation, coalescence (with emulsions) and creaming. These measurements are also applied for the prediction of the long-term physical stability of the dispersion. In this chapter, I will start with descriptions of the various rheological techniques that can be applied to study the state of the dispersion. This is followed by an analysis of the rheological behaviour of the different colloidal dispersions. Four main types of systems will be considered, namely hard-sphere dispersions (where both repulsion and attraction are screened), electrostatically stabilized dispersions (where the rheology is mainly determined by double layer repulsion), sterically stabilized dispersions (where the rheology is determined by steric repulsion) and flocculated dispersions. In the latter case a distinction will be made between weakly and strongly flocculated systems.

1.2 Rheological techniques Four different types of rheological techniques can be applied and these are summarized below.

1.2.1 Steady state shear stress σ–shear rate γ measurements In this case, the dispersion is placed in the gap between two concentric cylinders (Fig. 1.1), two parallel plates or a cone and plate geometry (Fig. 1.2). In the latter geometry, the cone is truncated to avoid damage to the tip of the cone and to have a sufficient gap in order to avoid grinding of the dispersion. The inner cylinder or the cone is subjected to an angular rotation Ω which allows one to calculate the shear rate γ.̇ The torque M on the outer cylinder or the plate is measured and this allows one to obtain the stress σ. This requires the use of a shear rate controlled instrument [1].

https://doi.org/10.1515/9783110541953-002

2 | 1 Flow characteristics (rheology) of colloidal dispersions

Fig. 1.1: Schematic picture of a concentric cylinder.

Ω R r

θ

(a) Cone and plate

R

R1 (b) Truncated cone and plate

Fig. 1.2: Schematic picture of the cone and plate.

∞ η (0) η /Pas

(d) Dilatant (b) Plastic Yield vaue + Pseudoplastic (c) Pseudoplastic (a) Newtonian (e)

. γ Fig. 1.3: Viscosity–shear rate relationship.

Most dispersions, particularly those with high volume fraction and/or containing rheology modifiers, do not obey Newton’s law. This can be clearly shown from plots of shear stress σ versus shear rate as illustrated in Fig. 1.3. Five different flow curves can be identified:

1.2 Rheological techniques |

(a) (b) (c) (d) (e)

3

Newtonian; Bingham plastic; pseudoplastic (shear thinning); dilatant (shear thickening); yield stress and shear thinning.

The variation of viscosity with shear rate for the above five systems is shown in Fig. 1.3. Apart from the Newtonian flow (a), all other systems show a change of viscosity with applied shear rate.

1.2.2 Rheological models for analysis of flow curves 1.2.2.1 Newtonian systems σ = η γ.̇

(1.1)

η is independent of the applied shear rate, e.g. simple liquids and very dilute dispersions.

1.2.2.2 Bingham plastic systems [8] σ = σ β + ηpl γ.̇

(1.2)

The system shows a (dynamic) yield stress σ β that can be obtained by extrapolation to zero shear rate. Clearly at and below σ β the viscosity η → ∞. The slope of the linear curve gives the plastic viscosity ηpl . Some systems like clay suspensions may show a yield stress above a certain clay concentration. The Bingham equation describes the shear stress/shear rate behaviour of many shear thinning materials at low shear rates. Unfortunately, the value of σ β obtained depends on the shear rate ranges used for the extrapolation procedure.

1.2.2.3 Pseudoplastic (shear thinning) system In this case the system does not show a yield value. It shows a limiting viscosity η(0) at low shear rates (that is referred to as residual or zero shear viscosity). The flow curve can be fitted to a power law fluid model (Ostwald de Waele) σ = k γ ̇n ,

(1.3)

where k is the consistency index and n is the shear thinning index (n < 1). By fitting the experimental data to equation (1.3) one can obtain k and n. The viscosity at a given shear rate can be calculated η=

σ k γ ̇n = = k γ̇n−1 . γ̇ γ̇

(1.4)

4 | 1 Flow characteristics (rheology) of colloidal dispersions

The power law model (equation (1.3)) fits the experimental results for many nonNewtonian systems over two or three decades of shear rate. Thus, this model is more versatile than the Bingham model, although one should be careful in applying this model outside the range of data used to define it. In addition, the power law fluid model fails at high shear rates, whereby the viscosity must ultimately reach a constant value, i.e. the value of n should approach unity.

1.2.2.4 Dilatant (shear thickening) system In some cases the very act of deforming a material can cause rearrangement of its microstructure such that the resistance to flow increases with increasing shear rate. In other words the viscosity increases with applied shear rate and the flow curve can be fitted with the power law, equation (1.3), but in this case n > 1. The shear thickening regime extends over only about a decade of shear rate. In almost all cases of shear thickening, there is a region of shear thinning at low shear rates. Several systems can show shear thickening such as wet sand, corn starch dispersed in milk and some polyvinyl chloride sols. Shear thickening can be illustrated when one walks on wet sand whereby some water is “squeezed out” and the sand appears dry. The deformation applied by one’s foot causes rearrangement of the closepacked structure produced by the water motion. This process is accompanied by volume increase (hence the term dilatancy) as a result of “sucking in” of the water. The process amounts to a rapid increase in the viscosity.

1.2.2.5 Herschel–Bulkley general model [9] Many systems show a dynamic yield value followed by a shear thinning behaviour. The flow curve can be analysed using the Herschel–Bulkley equation: σ = σ β + k γ̇n .

(1.5)

When σ β = 0, equation (1.5) reduces to the Power Fluid Model. When n = 1, equation (1.5) reduces to the Bingham model. When σ β = 0 and n = 1, equation (1.5) becomes the Newtonian equation. The Herschel–Bulkley equation fits most flow curves with a good correlation coefficient and hence it is the most widely used model.

1.2.2.6 The Casson model [10] This is a semi-empirical linear parameter model that has been applied to fit the flow curves of many paints and printing ink formulations, 1/2

σ1/2 = σC

1/2

+ ηC γ̇1/2 .

(1.6)

Thus a plot of σ1/2 versus γ̇1/2 should give a straight line from which σC and ηC can be calculated from the intercept and slope of the line. One should be careful in using the

1.2 Rheological techniques |

5

Casson equation since straight lines are only obtained from the results above a certain shear rate.

1.2.2.7 The Cross equation [11] This can be used to analyse the flow curve of shear thinning systems that show a limiting viscosity η(0) in the low shear rate regime and another limiting viscosity η(∞) in the high shear rate regime. These two regimes are separated by a shear thinning behaviour as schematically shown in Fig. 1.4. η − η(∞) 1 , = η(0) − η(∞) 1 + K γ̇m

(1.7)

where K is a constant parameter with dimension of time and m is a dimensionless constant. An equivalent equation to (1.7) is, η0 − η = (K γ̇m ). η − η∞

(1.8)

η(0) 20

Newtonian region Shear thinning

ηr 15

Newtonian region

10

η(∞) –4

10

–3

10

10

–2

–1

. γ

10

1

10

Fig. 1.4: Viscosity versus shear rate for shear thinning system.

1.2.3 Time effects during flow – thixotropy and negative (or anti-) thixotropy When a shear rate is applied to a non-Newtonian system, the resulting stress may not be achieved simultaneously. (i) The molecules or particles will undergo spatial rearrangement to follow the applied flow field.

6 | 1 Flow characteristics (rheology) of colloidal dispersions

(ii) The structure of the system may change: Breaking of weak bonds; aligning of irregularly shaped particles; collision of particles to form aggregates.

Time

Shear stress

Shear rate

Shear stress

The above changes are accompanied by decreasing or increasing viscosity with time at any given shear rate. These changes are referred to as thixotropy (if the viscosity decreases with time) or negative thixotropy or anti-thixotropy (if the viscosity increases with time). Thixotropy refers to the reversible time dependent decease of viscosity. When the system is sheared for some time, the viscosity decreases but when the shear is stopped (the system is left to rest) the viscosity of the system is restored. Practical examples for systems that show thixotropy are: paint formulations (sometimes referred to as thixotropic paints); tomato ketchup; some hand creams and lotions. Negative thixotropy or anti-thixotropy: when the system is sheared for some time, the viscosity increases, but when the shear is stopped (the system is left to rest) the viscosity decreases. A practical example of the above phenomenon is corn starch suspended in milk. Generally speaking, two methods can be applied to study thixotropy in a suspension. The first and the most commonly used procedure is the loop test whereby the shear rate is increased continuously and linearly in time from zero to some maximum value and then decreased to zero in the same way. This is illustrated in Fig. 1.5. The main problem with this procedure is the difficulty of interpreting the results. The nonlinear approach used is not ideal for developing loops because by decoupling the relaxation process from the strain one does not allow the recovery of the material. However, the loop test gives a qualitative behaviour of the suspension thixotropy.

Time

Shear rate

Fig. 1.5: Loop test for studying thixotropy.

An alternative method for studying thixotropy is to apply a step change test, whereby the suspension is suddenly subjected to a constant high shear rate and the stress is followed as a function of time whereby the structure breaks down and an equilibrium value is reached. The stress is further followed as a function of time to evaluate the rebuilding of the structure. A schematic representation of this procedure is shown in Fig. 1.6.

7

Shear rate

1.2 Rheological techniques |

Time Shear stress

Breakdown

Equilibrium level of stress

Rebuilding Time

Fig. 1.6: Step change for studying thixotropy.

1.2.4 Strain relaxation after sudden application of stress – constant stress (creep) measurements A constant stress σ is applied on the system (that may be placed in the gap between two concentric cylinders or a cone and plate geometry) and the strain (relative deformation) γ or compliance J (= γ/σ, Pa−1 ) is followed as a function of time for a period of t. At t = t, the stress is removed and the strain γ or compliance J is followed for another period t [1]. The above procedure is referred to as “creep measurement”. From the variation of J with t when the stress is applied and the change of J with t when the stress is removed (in this case J changes sign) one can distinguish between viscous, elastic and viscoelastic responses as illustrated in Fig. 1.7.

σ removed Viscous

σ applied σ J/Pa–1

Viscoelastic

Elastic

t=0

t

Fig. 1.7: Creep curves for viscous, elastic and viscoelastic responses.

8 | 1 Flow characteristics (rheology) of colloidal dispersions

Viscous response: In this case, the compliance J shows a linear increase with increasing time, reaching a certain value after time t. When the stress is removed after time t, J remains the same, i.e. in this case no creep recovery occurs. Elastic response: In this case, the compliance J shows a small increase at t = 0 and it remains almost constant for the whole period t. When the stress is removed, J changes sign and it reaches 0 after some time t, i.e. complete creep recovery occurs in this case. Viscoelastic response: At t = 0, J shows a sudden increase and this is followed by a slower increase for the time applied. When the stress is removed, J changes sign and J shows an exponential decrease with increasing time (creep recovery) but it does not reach 0 as in the case of an elastic response.

1.2.4.1 Analysis of creep curves (1) Viscous fluid: The linear curve of J versus t gives a slope that is equal to the reciprocal viscosity γ γṫ t J(t) = = = . (1.9) σ σ η(0) (2) Elastic solid: The increase in compliance at t = 0 (rapid elastic response) J(t) is equal to the reciprocal of the instantaneous modulus G(0) J(t) =

1 . G(0)

(1.10)

(3) Viscoelastic response (a) Viscoelastic liquid: Fig. 1.8 shows the case for a viscoelastic liquid whereby the compliance J(t) is given by two components: an elastic component Je that is given by the reciprocal of the instantaneous modulus, and a viscous component Jv that is given by t/η(0) J(t) =

t 1 + . G(0) η(0)

(1.11)

Fig. 1.8 also shows the recovery curve which gives σ0 Je0 and when this is subtracted from the total compliance gives σ0 t/η(0). The driving force for relaxation is spring and the viscosity controls the rate. The Maxwell relaxation time τM is given by, τM =

η(0) . G(0)

(1.12)

(b) Viscoelastic solid: In this case complete recovery occurs as illustrated in Fig. 1.9. The system is characterized by a Kelvin retardation time τk that is also given by the ratio of η(0)/G(0).

1.2 Rheological techniques |

9

Creep is the sum of a constant value Je σ0 (elastic part) and a viscous contribution σ0 t/η0

J

σ0 t/η σ0 Je0 σ0 Je0

σ0 t/η0 t=0

t

Fig. 1.8: Creep curve for a viscoelastic liquid.

σ

J/Pa–1

σ

t=0

t

t=t

Fig. 1.9: Creep curve for a viscoelastic solid.

1.2.4.2 Creep procedure In creep experiments one starts with a low applied stress (below the critical stress σcr , see below) at which the system behaves as a viscoelastic solid with complete recovery as illustrated in Fig. 1.9. The stress is gradually increased and several creep curves are obtained. Above σcr , the system behaves as a viscoelastic liquid showing only partial recovery, as illustrated in Fig. 1.8. Fig. 1.10 shows a schematic representation of the variation of compliance J with time t at increasing σ (above σcr ). From the slopes of the lines one can obtain the viscosity η σ at each applied stress. A plot of η σ versus σ is shown in Fig. 1.11. This shows a limiting viscosity η(0) below σcr and above σcr the viscosity shows a sharp decrease with a further increase in σ. η(0) is referred to as the residual or zero shear viscosity, which is an important parameter for predicting sedimentation. σcr is the critical stress above which the structure “breaks down”. It is sometimes referred to as the “true” yield stress.

10 | 1 Flow characteristics (rheology) of colloidal dispersions

Increasing σ J

Slope ∝

1 ησ

t Fig. 1.10: Creep curves at increasing applied stress.

σc

η(σ)

η(0) Residual or zero shear viscosity

σ Critical stress is a useful parameter (related to yield stress) as denotes the stress at which structure „breaks down“

Fig. 1.11: Variation of viscosity with applied stress.

1.2.5 Stress relaxation after sudden application of strain In this case a small strain is rapidly applied within a very short period of time (that must be smaller than the relaxation time of the system) and is kept at a constant value. The shear rate remains constant within this period [1]. This is illustrated in Fig. 1.12. The stress will follow the strain and increases to a maximum value σ(0). For a perfectly elastic material, σ(0) remains constant over time t. For a viscoelastic liquid, the stress decreases exponentially with time reaching 0 at infinite time. The stress required to maintain a constant strain decreases with time due to viscous flow. This is illustrated in Fig. 1.13. For a viscoelastic solid, the stress reaches a limiting value at infinite time. The variation of stress with time is similar to a kinetic process represented by first order equations.

1.2 Rheological techniques |

11

ξ

. γ

γ

t=0

σ/ Pa

Fig. 1.12: Schematic representation of a strain experiment.

t=0

t

Fig. 1.13: Stress relaxation after sudden application of strain.

The stress σ(t) is related to the initial maximum stress σ(0) by σ(t) = σ(0) exp(−

t ), τm

(1.13)

where τm is the Maxwell relaxation time that is given by the ratio of the viscosity η to the modulus G η τm = . (1.14) G If the shear stress in equation (1.13) is divided by the applied strain γ one obtains the shear modulus G(t) G(t) =

σ(t) σ(0) t t = exp(− ) = G(0) exp(− ). γ γ τm τm

(1.15)

Fig. 1.14 shows the variation of the modulus G with time for a viscoelastic liquid, whereas Fig. 1.15 shows the trend for a viscoelastic solid. For a viscoelastic solid, the modulus reaches a limiting value Ge at long time (sometimes referred to as the equilibrium modulus). In this case, equation (1.15) has to be modified to account for Ge G(t) = G(0) exp(−

t ) + Ge . τm

(1.16)

12 | 1 Flow characteristics (rheology) of colloidal dispersions

1

Go

0.8

G/Pa

0.6 τ = σo /e

0.4 0.2 0 20

40

60 t/s

80

100

Fig. 1.14: Variation of modulus with time for a viscoelastic liquid.

120 G = 80 Pa

100

G(t)/Pa

80 60

Ge

G η

40 η/G = τ = 1 s

20

Ge = 30 Pa

0 0.001

0.1

1

10

100

1000

t/s Fig. 1.15: Variation of G(t) with t for a viscoelastic solid.

Note that according to equations (1.8) and (1.10) that t = τm when σ(t) = σ(0)/e or when G(t) = G(0)/e. This shows that stress relaxation can be used to obtain the relaxation time for a viscoelastic liquid.

1.2.6 Dynamic (oscillatory) measurements [1] This is the response of the material to an oscillating stress or strain. When a sample is constrained in, say, a cone and plate or concentric cylinder assembly, an oscillating strain at a given frequency ω (rad s−1 ) (ω = 2νπ, where ν is the frequency in cycles s−1 or Hz) can be applied to the sample. After an initial start-up period, a stress develops

1.2 Rheological techniques |

13

in response to the applied strain, i.e. it oscillates with the same frequency. The change of the sine waves of the stress and strain with time can be analysed to distinguish between elastic, viscous and viscoelastic responses. Analysis of the resulting sine waves can be used to obtain the various viscoelastic parameters as discussed below Three cases can be considered: Elastic response: whereby the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case there is no time shift between stress and strain sine waves. Viscous response: whereby the maximum of the stress is at the point of maximum shear rate (i.e. the inflection point) where there is maximum energy dissipation. In this case the strain and stress sine waves are shifted by ωt = π/2 (referred to as the phase angle shift δ, which in this case is 90°). Viscoelastic response: in this case the phase angle shift δ is greater than 0 but less than 90°.

1.2.6.1 Analysis of oscillatory response for a viscoelastic system Let us consider the case of a viscoelastic system. The sine waves of strain and stress are shown in Fig. 1.16. The frequency ω is in rad s−1 and the time shift between strain and stress sine waves is ∆t. The phase angle shift δ is given by (in dimensionless units of radians) δ = ω∆t. (1.17) As discussed before: – Perfectly elastic solid: δ = 0 – Perfectly viscous liquid: δ = 90° – Viscoelastic system: 0 < δ < 90°

σo

γo Strain Stress

Δt

Δt = time shift for sine waves of stress and strain Δt ω = δ, phase angle shift ω = frequency in radian s–1 ω=2πν δ=0 Perfectly elasic solid Perfectly viscous liquid δ = 90° 0 < δ < 90° Viscoelastic system

Fig. 1.16: Strain and stress sine waves for a viscoelastic system.

14 | 1 Flow characteristics (rheology) of colloidal dispersions

The ratio of the maximum stress σ0 to the maximum strain γ0 gives the complex modulus |G∗ | σ0 |G∗ | = . (1.18) γ0 The complex modulus can be resolved into G󸀠 (the storage or elastic modulus) and G󸀠󸀠 (the loss or viscous modulus) using vector analysis and the phase angle shift δ as shown below.

1.2.6.2 Vector analysis of the complex modulus G󸀠 = |G∗ | cos δ, 󸀠󸀠

∗

G = |G | sin δ, tan δ =

G″

G󸀠󸀠

(1.19) (1.20)

.

(1.21)

G󸀠󸀠 . ω

(1.22)

G󸀠

|G*|

δ G′

Dynamic viscosity η󸀠 : η󸀠 =

Note that η → η(0) as ω → 0. Both G󸀠 and G󸀠󸀠 can be expressed in terms of frequency ω and Maxwell relaxation time τm by, (ωτm )2 , 1 + (ωτm )2 ωτm G󸀠󸀠 (ω) = G 1 + (ωτm )2 G󸀠 (ω) = G

(1.23) (1.24)

In oscillatory techniques one has to carry two types of experiments: Strain sweep: The frequency ω is kept constant and G∗ , G󸀠 and G󸀠󸀠 are measured as a function of strain amplitude. Frequency sweep: The strain is kept constant (in the linear viscoelastic region) and G∗ , G󸀠 and G󸀠󸀠 are measured as a function of frequency.

1.2 Rheological techniques |

15

1.2.6.3 Strain sweep The frequency is fixed, say at 1 Hz (or 6.28 rad s−1 ) and G∗ , G󸀠 and G󸀠󸀠 are measured as a function of strain amplitude γ0 . This is illustrated in Fig. 1.17. G∗ , G󸀠 and G󸀠󸀠 remain constant up to a critical strain γcr . This is the linear viscoelastic region where the moduli are independent of the applied strain. Above γcr , G∗ and G󸀠 start to decrease whereas G󸀠󸀠 starts to increase with any further increase in γ0 . This is the nonlinear region.

G* G′

G″

γcr

G* G′ Linear viscoelastic G″ γo

Linear viscoelastic region G*, G′ and G″ are independent of strain amplitude γcr is the critical strain above which systems shows non-linear response (break down of structure)

Fig. 1.17: Schematic representation of strain sweep.

γcr may be identified with the critical strain above which the structure starts to “break down”. It can also be shown that above another critical strain, G󸀠󸀠 becomes higher than G󸀠 . This is sometimes referred to as the “melting strain” at which the system becomes more viscous than elastic.

1.2.6.4 Oscillatory Sweep The strain γ0 is fixed in the linear region (taking a mid-point, i.e. not a too low strain where the results may show some “noise” and far from γcr ). G∗ , G󸀠 and G󸀠󸀠 are then measured as a function of frequency (a range of 10−3 –10−2 rad s−1 may be chosen depending on the instrument and operator patience). Fig. 1.18 shows a schematic representation of the variation of G∗ , G󸀠 and G󸀠󸀠 with frequency ω (rad s−1 ) for a viscoelastic system that can be represented by a Maxwell model. One can identify a characteristic frequency ω∗ at which G󸀠 = G󸀠󸀠 (the crossover point) which can be used to obtain the Maxwell relaxation time τm 1 τm = ∗ . (1.25) ω In the low frequency regime, i.e. ω < ω∗ , G󸀠󸀠 > G󸀠 . This corresponds to a long time experiment (time is reciprocal of frequency) and hence the system can dissipate energy as viscous flow. In the high frequency regime, i.e. ω > ω∗ , G󸀠 > G󸀠󸀠 . This corresponds to a short time experiment where energy dissipation is reduced. At sufficiently high frequency, G󸀠 ≫ G󸀠󸀠 . At such high frequency, G󸀠󸀠 → 0 and G󸀠 ≈ G∗ . The high frequency

16 | 1 Flow characteristics (rheology) of colloidal dispersions

200 G*

G/Pa

160 120

G′

80 G″ 40

G″ G* ω*

G′ 10–2

10–1

1

10

Fig. 1.18: Schematic representation of oscillatory measurements for a viscoelastic liquid.

ω/Hz

120

G′, G″/Pa

100 80

Storage modulus G′

60 40 Loss modulus G″ 20 0 0.001

0.01

0.1

1

10

100

ω/rad s–1 Fig. 1.19: Schematic representation for oscillatory measurements for a viscoelastic solid.

modulus G󸀠 (∞) is sometimes referred to as the “rigidity modulus” where the response is mainly elastic. For a viscoelastic solid, G󸀠 does not become zero at low frequency. G󸀠󸀠 still shows a maximum at intermediate frequency. This is illustrated in Fig. 1.19.

1.2.6.5 The cohesive energy density Ec The cohesive energy density, which is an important parameter for identification of the “strength” of the structure in a dispersion, can be obtained from the change of G󸀠 with γ0 (see Fig. 1.17). γcr

Ec = ∫ σ dγ, 0

(1.26)

1.3 Rheology of colloidal dispersions

|

17

where σ is the stress in the sample that is given by, σ = G󸀠 γ. γcr

Ec = ∫ G󸀠 γcr dγ =

(1.27) 1 2 󸀠 γ G. 2 cr

(1.28)

0

Note that Ec is given in J m−3 .

1.3 Rheology of colloidal dispersions Rheological measurements are useful tools for probing the microstructure of dispersions [1]. This is particularly the case if measurements are carried out at low stresses or strains as discussed above. In this case the spacial arrangement of particles is only slightly perturbed by the measurement. In other words the convective motion due to the applied deformation is less than the Brownian diffusion. The ratio of the stress applied, σ, to the “thermal stress” (that is equal to kT/6πa3 , where k is the Boltzmann constant, T is the absolute temperature and a is the particle radius) is defined in terms of a dimensionless Peclet number Pe, Pe =

6πa 3 σ . kT

(1.29)

For a colloidal particle with radius of 100 nm, σ should be less than 0.2 Pa to ensure that the microstructure is relatively undisturbed. In this case Pe < 1. In order to remain in the linear viscoelastic region, the structural relaxation by diffusion must occur on a timescale comparable to the experimental time. The ratio of the structural relaxation time to the experimental measurement time is given by the dimensionless Deborah number De, which is ≈ 1, and the dispersion appears viscoelastic. The rheology of colloidal dispersions depends on the balance between three main forces [1], namely Brownian diffusion, hydrodynamic interaction and interparticle forces. These forces are determined by three main parameters: (i) The volume fraction ϕ (total volume of the particles divided by the volume of the dispersion). (ii) The particle size and shape distribution. (iii) The net energy of interaction GT , i.e. the balance between repulsive and attractive forces. The earliest theory for prediction of the relationship between the relative viscosity ηr and ϕ was described by Einstein, which is applicable to ϕ ≤ 0.01.

18 | 1 Flow characteristics (rheology) of colloidal dispersions

1.3.1 The Einstein equation Einstein [12] assumed that particles behave as hard spheres (with no net interaction). The flow field has to dilate because the liquid has to move around the flowing particles. At ϕ ≤ 0.01 the disturbance around one particle does not interact with the disturbance around another. ηr is related to ϕ by the following expression [1] ηr = 1 + [η]ϕ = 1 + 2.5ϕ,

(1.30)

where [η] is referred to as the intrinsic viscosity and has the value 2.5 for spherical particles. For the above hard-sphere, very dilute dispersions, the flow is Newtonian, i.e. the viscosity is independent of shear rate. At higher ϕ (0.2 > ϕ > 0.1) values, one has to consider the hydrodynamic interaction suggested by Bachelor [13] that is still valid for hard spheres.

1.3.2 The Bachelor equation [13] When ϕ > 0.01, hydrodynamic interaction between the particles become important. When the particles come close to each other, the nearby stream lines and the disturbance of the fluid around one particle interacts with that around a moving particle. Using the above picture, Bachelor [13] derived the following expression for the relative viscosity ηr = 1 + 2.5ϕ + 6.2ϕ2 + Oϕ3 . (1.31) The third term in equation (1.31), i.e. 6.2ϕ2 , is the hydrodynamic term whereas the fourth term is due to higher order interactions.

1.3.3 Rheology of concentrated dispersions When ϕ > 0.2, ηr becomes a complex function of ϕ. At such high volume fractions the system mostly shows non-Newtonian flow ranging from viscous to viscoelastic to elastic response. Three responses can be considered: (i) viscous response; (ii) elastic response; (iii) viscoelastic response. These responses for any dispersion depend on the time or frequency of the applied stress or strain. Four different types of systems (with increasing complexity) can be considered as described below.

1.3 Rheology of colloidal dispersions

|

19

(1) Hard-sphere suspensions: these are systems where both repulsive and attractive forces are screened. (2) Systems with “soft” interaction: these are systems containing electrical double layers with long-range repulsion. The rheology of the dispersion is determined mainly by the double layer repulsion. (3) Sterically stabilized suspensions: the rheology is determined by the steric repulsion produced by adsorbed nonionic surfactant or polymer layers. The interaction can be “hard” or “soft” depending on the ratio of adsorbed layer thickness to particle radius (δ/R). (4) Flocculated systems: these are systems where the net interaction is attractive. One can distinguish between weak (reversible) and strong (irreversible) flocculation depending on the magnitude of the attraction.

1.3.3.1 Rheology of hard-sphere dispersions Hard-sphere dispersions (neutral stability) were developed by Krieger and co-workers [14, 15] using polystyrene latex suspensions whereby the double layer repulsion was screened by using NaCl or KCl at a concentration of 10−3 mol dm−3 or replacing water by a less polar medium such as benzyl alcohol. The relative viscosity ηr (= η/η0 ) is plotted as a function of reduced shear rate (shear rate x time for a Brownian diffusion tr ), γ̇red = γṫ r =

̇ 3 6πη0 γa , kT

(1.32)

where a is the particle radius, η0 is the viscosity of the medium, k is the Boltzmann constant and T is the absolute temperature. A plot of (η/η0 ) versus (η0 a3 /kT) is shown in Fig. 1.20 at ϕ = 0.4 for particles with different sizes. At a constant ϕ, all points fall on the same curve. The curves are shifted to higher values for larger ϕ and to lower values for smaller ϕ. The curve in Fig. 1.20 shows two limiting (Newtonian) viscosities at low and high shear rates that are separated by a shear thinning region. In the low shear rate regime, Brownian diffusion predominates over hydrodynamic interaction and the system shows a “disordered” three-dimensional structure with high relative viscosity. As the shear rate is increased, these disordered structure starts to form layers coincident with the plane of shear and this results in the shear thinning region. In the high shear rate regime, the layers can “slide” freely and hence a Newtonian region (with much lower viscosity) is obtained. In this region hydrodynamic interaction predominates over Brownian diffusion. If the relative viscosity in the first or second Newtonian region is plotted versus the volume fraction one obtains the curve shown in Fig. 1.21. The curve in Fig. 1.21 has two asymptotes: the slope of the linear portion at low ϕ values (the Einstein region) that gives the intrinsic viscosity [η] that is equal to 2.5;

20 | 1 Flow characteristics (rheology) of colloidal dispersions

the asymptote that occurs at a critical volume fraction ϕp at which the viscosity shows a sharp increase with increasing ϕ. ϕp is referred to as the maximum packing fraction for hard spheres: For hexagonal packing of equal sized spheres, ϕp = 0.74. For random packing of equal sized spheres, ϕp = 0.64. For polydisperse systems, ϕp reaches higher values as illustrated in Fig. 1.22 for one-size, two-size, three-size and four-size suspensions.

φ = 0.4 Newtonian

20

region

ηr

Shear thinning 15

Newtonian region

10

10–4

10–3

10–2

10–1

1

10

ηa3/kT

ηr

Fig. 1.20: Reduced viscosity versus reduced shear rate for hard-sphere suspensions.

10

slope = [η]

0

1 φ

φp

Fig. 1.21: Relative viscosity versus volume fraction for hard-sphere suspensions.

1.3 Rheology of colloidal dispersions

|

21

Suspension viscosities

Relative viscosity

1000

one size two size three size four size

100

10

0

0.2

0.4

0.6

0.8

1

Volume fraction of solids

Fig. 1.22: Viscosity–volume fraction curves for polydisperse dispersions.

The best analysis of the ηr –ϕ curve is due to Dougherty and Krieger [14, 15] who used a mean field approximation by calculating the increase in viscosity as small increments of the suspension are consecutively added. Each added increment corresponds to replacement of the medium by more particles. They arrived at the following simple semi-empirical equation that could fit the viscosity data over the whole volume fraction range: ηr = (1 −

ϕ −[η]ϕp . ) ϕp

(1.33)

Equation (1.33) is referred to as the Dougherty–Krieger equation [14, 15] and is commonly used for analysis of viscosity data.

1.3.3.2 Rheology of systems with “soft” or electrostatic interaction In this case the rheology is determined by the double layer repulsion particularly with small particles and extended double layers [16]. In the low shear rate regime, the viscosity is determined by Brownian diffusion and the particles approach each other to a distance of the order of ≈ 4.5 κ−1 (where κ−1 is the “double layer thickness” that is determined by electrolyte concentration and valency). This means that the effective radius of the particles Reff is much higher than the core radius R. For example, for 100 nm particles with a zeta potential ζ of 50 mV dispersed in a medium of 10−5 mol dm−3 NaCl (κ−1 = 100 nm), Reff ≈ 325 nm. The effective volume fraction ϕeff is also much higher than the core volume fraction. This results in a rapid increase in the viscosity at low core volume fraction [16]. This is illustrated in Fig. 1.23 which shows the variation of ηr with ϕ at 5 × 10−4 and 10−3 mol dm−3 NaCl (R = 85 nm and ζ = 78 mV). The low shear viscosity ηr (0) shows a rapid increase at ϕ ≈ 0.2 (the increase occurs at higher volume fraction at the higher electrolyte concentration). At ϕ > 0.2, the

22 | 1 Flow characteristics (rheology) of colloidal dispersions

107 106

ηr(0) 5 × 10–4 M

5

ηr(0)

ηr(∞)

10–3 M

ηr

10 104 103 102 101 100 10–1 0

0.2

0.4

0.6

0.8

1

ϕ Fig. 1.23: Variation of ηr with ϕ for polystyrene latex dispersions at two NaCl concentrations.

system shows “solid-like” behaviour with ηr (0) reaching very high values (> 107 ). At such high ϕ values the system shows near plastic flow. In the high shear rate regime, the increase in ηr occurs at much higher ϕ values. This is illustrated from the plot of the high shear relative viscosity ηr (∞) versus ϕ. At such high shear rates, hydrodynamic interaction predominates over Brownian diffusion and the system shows a low viscosity denoted by ηr (∞). However, when ϕ reaches a critical value, pseudoplastic flow is observed.

1.3.3.3 Rheology of sterically stabilized dispersions These are dispersions where the particle repulsion results from the interaction between adsorbed or grafted layers of nonionic surfactants or polymers [17]. The flow is determined by the balance of viscous and steric forces. Steric interaction is repulsive as long as the Flory–Huggins interaction parameter χ < 1/2 . With short chains, the interaction may be represented by a hard-sphere type with Reff = R + δ. This is particularly the case with nonaqueous dispersions with an adsorbed layer of thickness that is smaller compared to the particle radius (any electrostatic repulsion is negligible in this case). With most sterically stabilized dispersions, the adsorbed or grafted layer has an appreciable thickness (compared to particle radius) and hence the interaction is “soft” in nature as a result of the longer range of interaction. Results for aqueous sterically stabilized dispersions were produced using polystyrene (PS) latex with grafted poly(ethylene oxide) (PEO) layers [18, 19]. As an illustration, Fig. 1.24 shows the variation of η r with ϕ for latex dispersions with three particle radii (77.5, 306 and 502 nm). For comparison, the ηr –ϕ curve calculated using the Dougherty–Krieger equation is shown on the same figure. The ηr –ϕ curves are shifted to the left as a result of the presence of the grafted PEO layers. The experimental relative viscosity data may be used to obtain the grafted polymer layer thickness at various volume fractions of the dispersions. Using the Dougherty–Krieger equation, one can obtain the effective

1.3 Rheology of colloidal dispersions

700

a = 306 nm a = 502 nm

600

ηr

500

|

23

Dougherty-Krieger

a = 77.5 nm

400 200 100 0

0.4

0.45

0.5

0.55

0.6

0.65

0.7

ϕ Fig. 1.24: ηr –ϕ curves for PS latex dispersions containing grafted PEO chains.

volume fraction of the dispersion. From a knowledge of the core volume fraction, one can calculate the grafted layer thickness at each dispersion volume fraction. To apply the Dougherty–Krieger equation, one needs to know the maximum packing fraction, ϕp . This can be obtained from a plot of 1/(ηr )1/2 versus ϕ and extrapolation to 1/(ηr )1/2 , using the following empirical equation [18], K 1/2

ηr

= ϕp − ϕ.

(1.34)

The value of ϕp using equation (1.34) was found to be in the range 0.6–0.64. The intrinsic viscosity [η] was assigned a value of 2.5. Using the above calculations, the grafted PEO layer thickness δ was calculated as a function of ϕ for the three latex dispersions. For the dispersions with R = 77.5 nm, δ was found to be 8.1 nm at ϕ = 0.42, decreasing to 5.0 nm when ϕ was increased to 0.543. For the dispersions with R = 306 nm, δ = 12.0 nm at ϕ = 0.51, decreasing to 10.1 when ϕ was increased to 0.60. For the dispersions with R = 502 nm, δ = 21.0 nm at ϕ = 0.54, decreasing to 14.7 as ϕ was increased to 0.61. As mentioned above, the rheology of sterically stabilized dispersions is determined by the steric repulsion particularly for small particles with “thick” adsorbed layers. This is illustrated in Fig. 1.25 which shows the variation of G∗ , G󸀠 and G󸀠󸀠 with frequency (Hz) for polystyrene latex dispersions of 175 nm radius containing grafted poly(ethylene oxide) (PEO) with molecular weight of 2000 (giving a hydrodynamic thickness δ ≈ 20 nm) [19]. The results clearly show the transition from predominantly viscous response when ϕ ≤ 0.465 to predominantly elastic response when ϕ ≥ 0.5. This behaviour reflects the steric interaction between the PEO layers. When the surface-to-surface distance between the particles h becomes < 2δ, elastic interaction occurs and G 󸀠 > G󸀠󸀠 .

24 | 1 Flow characteristics (rheology) of colloidal dispersions

G*

30

G*, G′, G″/Pa

20 10 3 2 1 0.6 0.4 0.2 0.3 0.2 0.1

G′

G*

φ = 0.5

G′

G″

φ = 0.575

φ = 0.465

G″

G*

G″ G′ G*

φ = 0.44 10–2

G′

G″

10–1

1

ω/Hz

Fig. 1.25: Variation of G∗ , G󸀠 and G󸀠󸀠 with frequency for sterically stabilized dispersions.

The exact volume fraction at which a dispersion changes from predominantly viscous to predominantly elastic response may be obtained from plots of G∗ , G󸀠 and G󸀠󸀠 (at fixed strain in the linear viscoelastic region and fixed frequency) versus the volume fraction of the dispersion. This is illustrated in Fig. 1.26 which shows the results for the above latex dispersions. At ϕ = 0.482, G󸀠 = G󸀠󸀠 (sometimes referred to as the crossover point), which corresponds to ϕeff = 0.62 (close to maximum random packing). At ϕ > 0.482, G󸀠 becomes progressively larger than G󸀠󸀠 and ultimately the value of G󸀠 approaches G∗ and G󸀠󸀠 becomes relatively much smaller than G󸀠 . At ϕ = 0.585, G󸀠 ≈ G∗ = 4.8 × 103 and at ϕ = 0.62, G󸀠 ≈ G∗ = 1.6 × 105 Pa. Such high elastic moduli values indicate that the dispersions behave as near elastic solids (“gels”) as a result of interpenetration and/or compression of the grafted PEO chains.

G*, G′, G″/Pa

104 G*

103 102 10 G″

1 G′

10–1 0.44

0.46

0.48

0.50

0.52

0.54

0.56

0.58

ϕ Fig. 1.26: Variation of G∗ , G󸀠 and G󸀠󸀠 (at ω = 1 Hz) with ϕ for latex dispersions (a = 175 nm) containing grafted PEO chains.

1.3 Rheology of colloidal dispersions

|

25

1.3.3.4 Rheology of flocculated suspensions The rheology of unstable systems poses problems both from the experimental and the theoretical point of view. This is due to the non-equilibrium nature of the structure, resulting from weak Brownian motion [1]. For this reason, advances in the rheology of suspensions, where the net energy is attractive, have been slow and only of qualitative nature. On the practical side, control of the rheology of flocculated and coagulated suspensions is difficult, since the rheology depends not only on the magnitude of the attractive energies but also on how one arrives at the flocculated or coagulated structures in question. Various structures can be formed, e.g. compact flocs, weak and metastable structures, chain aggregates, etc. At high volume fraction of the suspension, a “three-dimensional” network of particles is formed throughout the sample. Under shear this network is broken into smaller units of flocculated spheres which can withstand the shear forces [20, 21]. The size of the units that survive is determined by the balance of shear forces which tend to break the units down, and the energy of attraction that holds the spheres together [20–22]. The appropriate dimensionless group ̇ characterizing this process (balance of viscous and van der Waals forces) is η0 a4 γ/A (where η0 is the viscosity of the medium, a is the particle radius, γ̇ is the shear rate and A is the effective Hamaker constant). Each flocculated unit is expected to rotate in the shear field, and it is likely that these units will tend to form layers as individual spheres do. As the shear stress increases, each rotating unit will ultimately behave as an individual sphere and, therefore, a flocculated suspension will show pseudoplastic flow with the relative viscosity approaching a constant value at high shear rates. The viscosity-shear rate curve will also show a pseudo-Newtonian region at low and high shear rates (similar to the case with stable systems described above). However, the values of the low and high shear rate viscosities (η0 and η∞ ) will of course depend on the extent of flocculation and the volume fraction of the suspension. It is also clear that such systems will show an apparent yield stress (Bingham yield value, σ β ) normally obtained by extrapolation of the linear portion of the σ–γ̇ curve to γ̇ = 0. Moreover, since the structural units of a weakly flocculated system change with change in shear rate, most flocculated suspensions show thixotropy as discussed above. Once shear is initiated, some finite time is required to break the network of agglomerated flocs into smaller units which persist under the shear forces applied. As smaller units are formed, some of the liquid entrapped in the flocs is liberated, thereby reducing the effective volume fraction, ϕeff , of the suspension. This reduction in ϕeff is accompanied by a reduction in ηeff and this plays a major role in generating thixotropy. It is convenient to distinguish between two types of unstable systems depending on the magnitude of the net attractive energy: (i) Weakly flocculated suspensions: the attraction in this case is weak (energy of few kT units) and reversible, e.g. in the secondary minimum of the DLVO curve or the shallow minimum obtained with sterically stabilized systems. A particular case of weak flocculation is that obtained on the addition of “free” (nonadsorbing) polymer referred to as depletion flocculation.

26 | 1 Flow characteristics (rheology) of colloidal dispersions

(ii) Strongly flocculated (coagulated) suspensions: the attraction in this case is strong (involving energies of several 100 kT units) and irreversible. This is the case of flocculation in the primary minimum or suspensions flocculated by reduction of solvency of the medium (for sterically stabilized suspensions) to worse than a θ-solvent. Study of the rheology of flocculated suspensions is difficult since the structure of the flocs is at non-equilibrium. Theories for flocculated suspensions are also qualitative and based on a number of assumptions.

Weakly flocculated dispersions As mentioned above, weak flocculation may be obtained by the addition of “free” (nonadsorbing) polymer to a sterically stabilized dispersion [1]. Several rheological investigations of such systems have been carried out by Tadros and his collaborators [23–27]. This is exemplified by a latex dispersion containing grafted PEO chains of M = 2000 to which “free” PEO is added at various concentrations. The grafted PEO chains, which were made sufficiently dense, ensure absence of adsorption of the added free polymer. Three molecular weights of PEO were used: 20 000, 35 000 and 90 000. As an illustration Fig. 1.27–1.29 show the variation of the Bingham yield value σ β with volume fraction of “free” polymer ϕp at the three PEO molecular weights studied and at various latex volume fractions ϕs . The latex radius R in this case was 73.5 nm. The results of Fig. 1.27–1.29 show a rapid and linear increase in σ β with increasing ϕp when the latter exceeds a critical value, ϕ+p . The latter is the critical free polymer volume fraction for depletion flocculation. ϕ+p decreases with increasing molecular weight M of the free polymer, as expected. There does not seem to be any dependency of ϕ+p on the volume fraction of the latex, ϕs . Similar trends were obtained using larger latex particles (with radii 217.5 and 457.5 nm). However, there was a definite trend of the effect of particle size; the larger the particle size, the smaller the value of ϕ+p . A summary of ϕ+p for the various molecular weights and particle sizes is given in Tab. 1.1.

ϕs = 0.45 ϕs = 0.40

σβ/Pa

140 120 100 80 60 40 20 0

ϕs = 0.35 ϕs = 0.30

0 0.02 0.04 0.06 0.08 0.10 ϕp

Fig. 1.27: Variation of yield value σ β with volume fraction ϕp of “free” polymer (PEO; M = 20 000) at various latex volume fractions ϕs .

1.3 Rheology of colloidal dispersions

ϕs = 0.45

80

ϕs = 0.40

60

ϕs = 0.35

40

ϕs = 0.30

σβ/Pa

100

20 0

0 0.01 0.01 0.03 0.04 0.05 ϕp

27

Fig. 1.28: Variation of yield value σ β with volume fraction ϕp of “free” polymer (PEO; M = 35 000) at various latex volume fractions ϕs .

ϕs = 0.45 ϕs = 0.40

σβ/Pa

140 120 100 80 60 40 20 0

|

ϕs = 0.35 ϕs = 0.30

0 0.005 0.01 0.015 0.02 0.025 ϕp

Fig. 1.29: Variation of yield value σ β with volume fraction ϕp of “free” polymer (PEO; M = 100 000) at various latex volume fractions ϕs .

Tab. 1.1: Volume fraction of free polymer at which flocculation starts, ϕ+p Particle radius (nm)

M (PEO)

ϕ+p

73.5 73.5 73.5 73.5 217.5 457.5

20 000 35 000 10 000 20 000 20 000 20 000

0.0150 0.0060 0.0055 0.0150 0.0055 0.0050

The results in Tab. 1.1 show a significant reduction in ϕ+p when the molecular weight of PEO is increased from 20 000 to 35 000, whereas when M is increased from 35 000 to 100 000, the reduction in ϕ+p is relatively smaller. Similarly, there is a significant reduction in ϕ+p when the particle radius is increased from 73.5 to 217.5 nm, with a relatively smaller decrease on a further increase to 457.5 nm. The straight line relationship between the extrapolated yield value and the volume fraction of free polymer can be described by the following scaling law [25, 26], σ β = Kϕsm (ϕp − ϕ+p ),

(1.35)

where K is a constant and m is the power exponent in ϕs which may be related to the flocculation process. The values of m used to fit the data of σ β versus ϕs are summarized in Tab. 1.2.

28 | 1 Flow characteristics (rheology) of colloidal dispersions

Tab. 1.2: Power law plot for σ β versus ϕs for various PEO molecular weights and latex radii. Latex R = 73.5 nm PEO 20 000

PEO 35 000

PEO 100 000

Latex R = 217.5 nm

Latex R = 457.5 nm

ϕp

m

ϕp

m

ϕp

m

ϕp

m

ϕp

m

0.040 0.060 0.080 0.100

3.0 2.7 2.8 2.8

0.022 0.030 0.040 0.050

2.9 3.0 2.8 2.9

0.015 0.020 0.025 —

2.7 2.7 2.8 —

0.020 0.040 0.060 0.080

3.0 2.9 2.8 2.8

0.020 0.030 0.040 0.050

2.7 2.7 2.8 2.7

It can be seen from Tab. 1.2 that m is nearly constant, being independent of particle size and free polymer concentration. An average value for m of 2.8 may be assigned for such a weakly flocculated system. This value is close to the exponent predicted for diffusion controlled aggregation (3.5 ± 0.2) predicted by Ball and Brown [28, 29] who developed a computer simulation method treating the flocs as fractals that are closely packed throughout the sample. The near independence of ϕ+p from ϕs can be explained on the basis of the statistical mechanical approach of Gast et al. [30], which showed such an independence when the osmotic pressure of the free polymer solution is relatively low and/or the ratio of the particle diameter to the polymer coil diameter is relatively large (> 8–9). The latter situation is certainly the case with latex suspensions with diameters of 435 and 915 nm at all PEO molecular weights. The only situation where this condition is not satisfied is with the smallest latex and the highest molecular weight. The dependency of ϕ+p on particle size can be explained from a consideration of the dependence of free energy of depletion and van der Waals attraction on particle radius as will be discussed below. Both attractions increase with increasing particle radius. Thus the larger particles would require smaller free polymer concentration at the onset of flocculation. It is possible, in principle, to relate the extrapolated Bingham yield value, σ β , to the energy required to separate the flocs into single units, Esep [23, 24], σβ =

3ϕs nEsep , 8πR3

(1.36)

where n is the average number of contacts per particle (the coordination number). The maximum value of n is 12, which corresponds to hexagonal packing of the particles in a floc. For random packing of particles in the floc, n = 8. However, it is highly unlikely that such values of 12 or 8 are reached in a weakly flocculated system and a more realistic value for n is probably 4 (a relatively open structure in the floc). In order to calculate Esep from σ β one has to assume that all particle-particle contacts are broken by shear. This is highly likely since the high shear viscosity of the weakly flocculated latex was close to that of the latex before addition of the free polymer. Values of Esep obtained using equation (1.36) with n = 4 are given in Tab. 1.3 at

1.3 Rheology of colloidal dispersions

|

29

the three PEO molecular weights for the latex with the radius of 73.5 nm. It can be seen that Esep at any given ϕp increases with increasing volume fraction ϕs of the latex. A comparison between Esep and the free energy of depletion flocculation, Gdep , can be made using the theories of Asakura and Oosawa (AO) [31, 32] and Fleer, Vincent and Scheutjens (FVS) [33]. Asakura and Oosawa [31, 32] derived the following expression for Gdep , which is valid for the case where the particle radius is much larger the polymer coil radius, Gdep 3 (1.37) = − ϕ2 βx2 , 0 < x < 1, kT 2 where k is the Boltzmann constant, T is the absolute temperature, ϕ2 is the volume concentration of free polymer that is given by, ϕ2 =

4π ∆3 N2 . 3v

(1.38)

∆ is the depletion layer thickness that is equal to the radius of gyration of free polymer, Rg , and N2 is the total number of polymer molecules in a volume v of solution. R , ∆ [∆ − (h/2)] , x= ∆

β=

(1.39) (1.40)

where h is the distance of separation between the outer surfaces of the particles. Clearly, when h = 0, i.e. at the point where the polymer coils are “squeezed out” from the region between the particles, x = 1. Fleer, Scheutjens and Vincent (FSV model) [33] developed a general approach to the interaction of hard spheres in the presence of a free polymer. This model takes into account the dependency of the range of interaction on free polymer concentration and any contribution from the non-ideal mixing of polymer solutions. This theory gives the following expression for Gdep , Gdep = 2πR(

μ1 − μ01 v01

)∆2 (1 +

2∆ ), 3R

(1.41)

where μ1 is the chemical potential at bulk polymer concentration ϕp , μ01 is the corresponding value in the absence of free polymer and v01 is the molecular volume of the solvent. The difference in chemical potential (μ1 − μ01 ) can be calculated from the volume fraction of free polymer ϕp and the polymer-solvent (Flory–Huggins) interaction parameter χ, μ1 − μ01 ϕp 1 + ( − χ)ϕ2p ], = −[ (1.42) kT n2 2 where n2 is the number of polymer segments per chain.

30 | 1 Flow characteristics (rheology) of colloidal dispersions

Tab. 1.3: Results of Esep and Gdep calculated on the basis of AO and FSV models. ϕp

ϕs

σ β (Ps)

Esep (kT )

Gdep (kT ) AO model

FSV model

(a) M(PEO) = 20 000 0.04 0.30 12.5 0.35 21.0 0.40 30.5 0.45 40.0

8.4 12.1 15.4 18.0

18.2 18.2 18.2 18.2

78.4 78.4 78.4 78.4

(b) M(PEO) = 35 000 0.03 0.30 17.5 0.35 25.7 0.40 37.3 0.45 56.8

11.8 14.8 18.9 25.5

15.7 15.7 15.7 15.7

78.6 78.6 78.6 78.6

(c) M(PEO) = 100 000 0.02 0.30 10.0 0.35 15.0 0.40 22.0 0.45 32.5

6.7 8.7 11.1 14.6

9.4 9.4 9.4 9.4

70.8 70.8 70.8 70.8

A summary of the values of Esep and Gdep calculated on the basis of AO and FSV models is given in Tab. 1.3 at three molecular weights for PEO and for a latex with a radius of 77.5 nm. It can be seen from Tab. 1.3 that Esep , at any given ϕp , increases with increasing volume fraction ϕs of the latex. In contrast, the value of Gdep does not depend on the value of ϕs . The theories on depletion flocculation only show a dependency of Gdep on ϕp and a. Thus, one cannot make a direct comparison between Esep and Gdep . The close agreement between Esep and Gdep using Asakura and Oosawa’s theory [31, 32] and assuming a value of n = 4 should only be considered fortuitous. Using equations (1.35) and (1.36), a general scaling law may be used to show the variation of Esep with the various parameters of the system, Esep =

8πR3 8πK1 3 1.8 + K1 ϕ2.8 R ϕs (ϕp − ϕ+p ). s (ϕ p − ϕ p ) = 3ϕs n 3n

(1.43)

Equation (1.43) shows the four parameters that determine Esep : the particle radius a, the volume fraction of the suspension ϕs , the concentration of free polymer ϕp and the molecular weight of the free polymer, which together with a determine ϕ+p . More insight on the structure of the flocculated latex dispersions was obtained using viscoelastic measurements [26]. As an illustration Fig. 1.30 shows the variation of the storage modulus G󸀠 with ϕp (M = 20000) at various latex (a = 77.5) volume fractions ϕs . Similar trends were obtained for the other PEO molecular weights. All results show the same trend, namely an increase in G󸀠 with increasing ϕp , reaching a plateau value at high ϕp values. These results are different from those obtained using

1.3 Rheology of colloidal dispersions

70

|

31

ϕs = 0.40

60 σβ/Pa

50 40

ϕs = 0.35

30

ϕs = 0.30

20 10 0

0

0.02

0.04

0.06

0.08

0.10

ϕp

Fig. 1.30: Variation of storage modulus G with volume fraction of polymer (PEO; M = 20 000).

steady state measurements, which show a rapid and linear increase of yield value σ β . This difference reflects the behaviour when using oscillatory (low deformation) measurements that cause little perturbation of the structure when using low amplitude and high frequency measurements. Above ϕ+p flocculation occurs and G󸀠 increases in magnitude with a further increase in ϕp until a three-dimensional network structure is reached and G󸀠 reaches a limiting value. Any further increase in free polymer concentration may cause a change in the floc structure, but this may not cause a significant increase in the number of bonds between the units formed (which determine the magnitude of G󸀠 ).

σ/Pa

Strongly flocculated (coagulated) suspensions Steady state shear stress-shear rate curves show a pseudoplastic flow curve as is illustrated in Fig. 1.31. The flow curve is characterized by three main parameters: (i) The shear rate, above which the flow curve shows linear behaviour. Above this shear rate collisions occur between the flocs and this may cause interchange between the flocculi (the smaller floc units that aggregate to form a floc). In this linear region, the ratio of the floc volume to the particle volume (ϕF /ϕp ), i.e. the floc density, remains constant. (ii) σ β the residual stress (yield stress) that arises from the residual effect of interparticle potential.

σβ

ηpl . γcrit . γ/s–1

Fig. 1.31: Pseudoplastic flow curve for a flocculated suspension.

32 | 1 Flow characteristics (rheology) of colloidal dispersions

(iii) ηpl : the slope of the linear portion of the flow curve that arises from purely hydrodynamic effects. Several theories have been proposed to analyse the flow curve of Fig. 1.31 and these are summarized below. Impulse theory: Goodeve and Gillespie [34, 35]. The interparticle interaction effects (given by σ β ) and hydrodynamic effects (given by ηpl ) are assumed to be additive, σ = σ β + ηpl γ.̇

(1.44)

To calculate σ β , Goodeve proposed that when shearing occurs, links between particles in a flocculated structure are stretched, broken and reformed. An impulse is transferred from a fast moving layer to a slow moving layer. Non-Newtonian effects are due to the effect of shear on the number of links, the lifetime of a link and any change in the size of the floc. According to Goodeve theory the yield value is given by, σβ = (

3ϕ2 )EA , 2πR3

(1.45)

where ϕ is the volume fraction of the dispersed phase, a is the particle radius and EA is the total binding energy. EA = nL εL , (1.46) where nL is the number of links with a binding energy εL per link. According to equation (1.45): σβ ∝ ϕ2 ∝ (1/a3 ) ∝ EA (the energy of attraction). Elastic floc model: Hunter and co-workers [36, 37]. The floc is assumed to consist of an open network of “girders” as schematically shown in Fig. 1.32. The floc undergoes extension and compression during rotation in a shear flow. The bonds are stretched by a small amount ∆ (that can be as small as 1 % of particle radius). To calculate σ β , Hunter considered the energy dissipation during rupture of the flocs (assumed to consist of doublets). The yield value σ β is given by the expression, ̇ σ β = α0 βλη γ(

R2floc )ϕ2s ∆CFP , R3

(1.47)

where α0 is the collision frequency, β is a constant (= (27/5), λ is a correction factor (≈ 1) and CFP is the floc density (= ϕF /ϕs ). Fractal concept for flocculation. The floc structure can be treated as fractals whereby an isolated floc with radius aF can be assumed to have uniform packing throughout that floc [38, 39].

1.3 Rheology of colloidal dispersions

|

33

No stress (a)

Apply stress

(b)

Fig. 1.32: Schematic picture of the elastic floc.

In the above case the number of particles in a floc is given by nf = ϕmf (

RF 3 ) , R

(1.48)

where ϕmf is the packing fraction of the floc. If the floc does not have constant packing throughout its structure, but is dendritic in form, the packing density of the floc begins to reduce as one goes from the centre to the edge. If this reduction is with a constant power law D, nF = (

RF D ) , R

(1.49)

where 0 < D ≤ 3. D is called the packing index and it represents the packing change with distance from the centre. Two cases may be considered: (i) Rapid aggregation (diffusion limited aggregation, DLA). When particles touch, they stick. Particle-particle aggregation gives D = 2.5; aggregate-aggregate aggregation gives D = 1.8. The lower the value of D, the more open the floc structure is. (ii) Slow aggregation (rate limited aggregation, RLA). The particles have a lower sticking probability – some are able to rearrange and densify the floc – D ≈ 2.0–2.2. The lower the value of D, the more open the floc structure is. Thus, by determining D one can obtain information on the flocculation behaviour. If flocculation of a suspension occurs by changing the conditions (e.g. increasing temperature) one can visualize sites for nucleation of flocs occurring randomly throughout the whole volume of the suspension.

34 | 1 Flow characteristics (rheology) of colloidal dispersions

The total number of primary particles does not change and the volume fraction of the floc is given by, RF 3−D (1.50) ϕF = ϕ( ) . F Since the yield stress σ β and elastic modulus G󸀠 depend on the volume fraction, one can use a power law in the form, σ β = Kϕ m , 󸀠

G = Kϕ , m

(1.51) (1.52)

where the exponent m reflects the fractal dimension. Thus by plotting log σ β or log G󸀠 versus log ϕ one can obtain m from the slope which can be used to characterize the floc nature and structure, m = 2/(3 − D).

1.4 Examples of strongly flocculated (coagulated) suspension 1.4.1 Coagulation of electrostatically stabilized suspensions by addition of electrolyte As mentioned in Chapter 7, Vol. 1, electrostatically stabilized suspensions become coagulated when the electrolyte concentration is increased above the critical coagulation concentration (CCC). This is illustrated by using a latex dispersion (prepared using surfactant-free emulsion polymerization) to which 0.2 mol dm−3 NaCl is added (well above the CCC which is 0.1 mol dm−3 NaCl). Fig. 1.33 shows the strain sweep results for latex dispersions at various volume fractions ϕ and in the presence of 0.2 mol dm−3 NaCl. It can be seen from Fig. 1.33 that G∗ and G󸀠 (which are very close to each other) remain independent of the applied strain (the linear viscoelastic region), but above a critical strain, γcr ; G∗ and G󸀠 show a rapid reduction with a further increase in strain (the nonlinear region). In contrast, G󸀠󸀠 (which is much lower than G󸀠 ) remains constant showing an ill-defined maximum at intermediate strains. Above γcr the flocculated structure becomes broken down with applied shear. Fig. 1.34 shows the variation of G󸀠 (measured at strains in the linear viscoelastic region) with frequency ν (in Hz) at various latex volume fractions. As mentioned above, G󸀠 is almost equal to G∗ since G󸀠󸀠 is very low. In all cases G󸀠 ≫ G󸀠󸀠 and it shows little dependency on frequency. This behaviour is typical of highly elastic (coagulated) structures, whereby a “continuous gel” network structure is produced at such high volume fractions. Scaling laws can be applied for the variation of G󸀠 with the volume fraction of the latex ϕ. A log-log plot of G󸀠 versus ϕ is shown in Fig. 1.35. This plot is linear and can be represented by the following scaling equation, G󸀠 = 1.98 × 107 ϕ6.0 . (1.53)

1.4 Examples of strongly flocculated (coagulated) suspension |

G/Pa

6 × 104 4 × 104 2 × 104 0

G/Pa

2 × 10

G/Pa

ϕ = 0.346

G″ 0.1

1

10

3

103 0

G/Pa

G*, G′

G*, G′

ϕ = 0.205

G″ 0.1

1

10

100 50 0 10 5 0

G*, G′ G″ 0.5

ϕ = 0.121 5

50 G*, G′

1

5

ϕ = 0.065 50

γo/10–3 Fig. 1.33: Strain sweep results for latex dispersions at various volume fractions ϕ in the presence of 0.2 mol dm−3 NaCl.

G′/Pa

4 × 104 2 × 102 0

G′/Pa

2 × 10

0.01

0.1

1

0.01

0.1

1

0.01

0.1

1

0.01

0.1

3

103 0

G′/Pa

100 50 0

G′/Pa

10 5 0

ν/Hz

1

Fig. 1.34: Variation of G󸀠 with frequency at various volume fractions.

35

36 | 1 Flow characteristics (rheology) of colloidal dispersions

G′/Pa

104 103 102 10 5 × 10–2

10–1

2 × 10–1

5 × 10–1

ϕ

Fig. 1.35: Log-log plot of G󸀠 versus ϕ for polystyrene latex dispersions.

The high power in ϕ is indicative of a relatively compact coagulated structure. This power gives a fractal dimension of 2.67 that confirms the compact structure. It is also possible to obtain the cohesive energy of the flocculated structure Ec from a knowledge of G󸀠 (in the linear viscoelastic region) and γcr . Ec is related to the stress σ in the coagulated structure by equations (1.26)–(1.28). A log-log plot of Ec versus ϕ is shown in Fig. 1.36 for such coagulated latex dispersions. The Ec versus ϕ curve can be represented by the following scaling relationship, Ec = 1.02 × 103 ϕ9.1 .

(1.54)

The high power in ϕ is indicative of the compact structure of these coagulated suspensions.

Ec/Jm–3

10–1 10–2 10–3 10–4 10–1

2 × 10–1 3 × 10–1 4 × 10–1 ϕ

Fig. 1.36: Log-log plot of Ec versus ϕ for coagulated latex dispersions.

1.4.2 Strongly flocculated sterically stabilized systems As mentioned in Chapter 14, Vol. 1, sterically stabilized dispersions show strong flocculation (sometimes referred to as incipient flocculation) when the medium for the stabilizing chain becomes worse than a θ- solvent (the Flory–Huggins interaction parameter χ > 0.5). Reduction of solvency for a PEO stabilizing chain can be achieved by addition of a nonsolvent for the chains or by addition of electrolyte such as Na2 SO4 .

1.4 Examples of strongly flocculated (coagulated) suspension | 37

Above a critical Na2 SO4 concentration (to be referred to as the critical flocculation concentration, CFC), the χ parameter exceeds 0.5 and this results in incipient flocculation. This process of flocculation can be investigated using rheological measurements without diluting the latex. This dilution may result in a change in the floc structure and hence investigations without dilution ensure absence of change of the floc structure, in particular when using low deformation (oscillatory) techniques [1]. Fig. 1.37 shows the variation of extrapolated yield value, σ β , as a function of Na2 SO4 concentration at various latex volume fractions ϕs at 25 °C. The latex had a z-average particle diameter of 435 nm and it contained grafted PEO with M = 2000. It is clear that when ϕs < 0.52, σ β is virtually equal to zero up to 0.3 mol dm−3 Na2 SO4 above which it shows a rapid increase in σ β with a further increase in Na2 SO4 concentration. When ϕs > 0.52, a small yield value is obtained below 0.3 mol dm−3 Na2 SO4 , which may be attributed to the possible elastic interaction between the grafted PEO chains when the particle–particle separation is less than 2δ (where δ is the grafted PEO layer thickness). Above 0.3 mol dm−3 Na2 SO4 , there is a rapid increase in σ β . Thus, the CFC of all concentrated latex dispersions is around 0.3 mol dm−3 Na2 SO4 . It should be mentioned that at Na2 SO4 below the CFC, σ β shows a measurable decrease with increasing Na2 SO4 concentration. This due to the reduction in the effective radius of the latex particles as a result of the reduction in the solvency of the medium for the chains. This accounts for a reduction in the effective volume fraction of the dispersion which is accompanied by a reduction in σ β .

Bingham Yield Vaue/Pa

40 φ = 0.52 φ = 0.54

30

φ = 0.50 φ = 0.45 φ = 0.40

20 φ = 0.56 10

0 0

0.1

0.2

0.3

0.4

0.5

0.6

CNa2SO4/mol dm–3 Fig. 1.37: Variation of Bingham yield value with Na2 SO4 concentration at various volume fractions of latex.

38 | 1 Flow characteristics (rheology) of colloidal dispersions Fig. 1.38 shows the results for the variation of the storage modulus G󸀠 with Na2 SO4 concentration. These results show the same trend as those shown in Fig. 1.37, i.e. an initial reduction in G󸀠 due to the reduction in the effective volume fraction, followed by a sharp increase above the CFC (which is 0.3 mol dm−3 Na2 SO4 ). Log-log plots of σ β and G󸀠 versus ϕs at various Na2 SO4 concentrations are shown in Fig. 1.39 and 1.40. All the data are described by the following scaling equations, σ β = kϕsm , 󸀠

G =k

󸀠

(1.55)

ϕsn ,

(1.56)

with 0.35 < ϕs < 0.53

60

φ = 0.50 φ = 0.52 φ = 0.45 φ = 0.40 φ = 0.54

40

G′/Pa

φ = 0.56 20

φ = 0.35

0 0

0.1

0.2

0.3

0.4

0.5

0.6

–3

CNa2SO4/mol dm

Fig. 1.38: Variation of G󸀠 with Na2 SO4 concentration at various volume fractions of latex.

The values of m and n are very high at Na2 SO4 concentrations below the CFC reaching values in the range of 30–50, which indicate the strong steric repulsion between the latex particles. When the Na2 SO4 concentration exceeds the CFC, m and n decrease very sharply reaching a value of m = 9.4 and n = 12 when the Na2 SO4 concentration increases to 0.4 mol dm−3 and a value of m = 2.8 and n = 2.2 at 0.5 mol dm−3 Na2 SO4 . These slopes can be used to calculate the fractal dimensions (see above) giving a value of D = 2.70–2.75 at 0.4 mol dm−3 and D = 1.6–1.9 at 0.5 mol dm−3 Na2 SO4 . The above results of fractal dimensions indicate a different floc structure when compared with the results obtained using electrolyte to induce coagulation. In the latter case, D = 2.67 indicating a compact structure similar to that obtained at 0.4 mol dm−3 Na2 SO4 . However, when the Na2 SO4 concentration exceeded the CFC value (0.5 mol dm−3 Na2 SO4 ) a much more open floc structure with a fractal dimension less than 2 was obtained.

1.4 Examples of strongly flocculated (coagulated) suspension |

100

39

CNa2SO4

30 0.5

σβ/Pa

10

0.1, 0.2 0.4

3 0.3

1 0.3 0.1 0.03

0.35

0.4

0.45

0.5

0.55

0.6

0.65

φs Fig. 1.39: Log-log plots of σ β versus ϕs .

50

CNa2SO4

20 0.5

G′/Pa

10

0.1, 0.2 0.4

5 0.3 2 1 0.5 0.2

0.35

0.4

0.45

0.5

0.55

0.6

0.65

φs Fig. 1.40: Log-log plots of G󸀠 versus ϕs .

Sterically stabilized dispersions with PEO chains as stabilizers also undergo flocculation on increasing the temperature. At a critical temperature (critical flocculation temperature, CFT) the Flory–Huggins interaction parameter becomes higher than 0.5 resulting in incipient flocculation. This is illustrated in Fig. 1.41 which shows the variation of the storage modulus G󸀠 and loss modulus G󸀠󸀠 with increasing temperature for a latex dispersion with a volume fraction ϕ = 0.55 and at Na2 SO4 concentration

40 | 1 Flow characteristics (rheology) of colloidal dispersions of 0.2 mol dm−3 . At this electrolyte concentration the latex is stable in the temperature range 10–40 °C. However, above this temperature (CFT) the latex is strongly flocculated.

G*

G*, G′, G″/Pa

30

G′

20 G″ 10

0 0

10

20

30

40

50

60

Temperature/°C Fig. 1.41: Variation of G∗ , G󸀠 and G󸀠󸀠 with temperature with latex dispersions (ϕs = 0.55) in 0.2 mol dm−3 Na2 SO4 .

The results of Fig. 1.41 show an initial systematic reduction in the moduli values with increasing temperature up to 40 °C. This is the result of reduction in solvency of the chains with increasing temperature. The latter increase causes a breakdown in the hydrogen bonds between the PEO chains and water molecules. This results in a reduction in the thickness of the grafted PEO chains and hence a reduction in the effective volume fraction of the dispersion. The latter causes a decrease in the moduli values. However, at 40 °C there is a rapid increase in the moduli values with a further increase of temperature. The latter indicates the onset of flocculation (the CFT). Similar results were obtained at 0.3 and 0.4 mol dm−3 Na2 SO4 , but in these cases the CFT was 35 and 15 °C respectively.

1.4.3 Models for interpretation of rheological results 1.4.3.1 Doublet floc structure model Neville and Hunter [41] introduced a doublet floc model to deal with sterically stabilized dispersions that have undergone flocculation. They assumed that the major contribution to the excess energy dissipation in such pseudoplastic systems comes from the shear field which provides energy to separate contacting particles in a floc.

1.4 Examples of strongly flocculated (coagulated) suspension |

41

The extrapolated yield value can be expressed as, σβ =

3ϕ2H Esep , 2π2 (R + δ)2

(1.57)

where ϕH is the hydrodynamic volume fraction of the particles that is equal to the effective volume fraction, δ 3 (1.58) ϕH = ϕs [1 + ] . R (R + δ) is the interaction radius of the particle and Esep is the energy needed to separate a doublet, which is the sum of van der Waals and steric attractions, Esep =

AR + Gs . 12H0

(1.59)

At a particle separation of ≈ 12 nm (twice the grafted polymer layer thickness), the van der Waals attraction is very small (1.66 kT, where k is the Boltzmann constant and T is the absolute temperature) and the contribution of Gs to the attraction is significantly larger than the van der Waals attraction. Therefore, Esep may be approximated to Gs . From equation (1.57) one can estimate Esep from σ β . The results are shown in Tab. 1.4 which shows an increase in Esep with increasing ϕs . Tab. 1.4: Results of Esep calculated from σ β for a flocculated sterically stabilized latex dispersion at various latex volume fractions. 0.4 mol dm−3 Na2 SO4

0.4 mol dm−3 Na2 SO4

ϕs

σ β (Pa)

Esep (kT )

ϕs

σ β (Pa)

Esep (kT )

0.43 0.45 0.51 0.54 0.55 0.57 0.58

1.3 2.4 3.3 5.3 7.3 9.1 17.4

97 165 179 262 336 397 736

0.25 0.29 0.33 0.37 0.41 0.44 0.47 0.49 0.52

3.5 5.4 7.4 11.4 14.1 17.0 21.1 23.1 28.3

804 910 961 1170 1190 1240 1380 1390 1510

The values of Esep given in Tab. 1.4 are unrealistically high and hence the assumptions made for calculating Esep are not fully justified and hence the data of Tab. 1.4 must be only considered as qualitative.

1.4.3.2 Elastic floc model This model [42–44] was described before and it is based on the assumption that the structural units are small flocs of particles (called flocculi) which are characterized

42 | 1 Flow characteristics (rheology) of colloidal dispersions

by the extent to which the structure is able to entrap the dispersion medium. A floc is made from an aggregate of several flocculi. The latter may range from a loose, open structure (if the attractive forces between the particles are strong) to a very closepacked structure with little entrapped liquid (if the attractive forces are weak). In the system of flocculated sterically stabilized dispersions, the structure of the flocculi depends on the volume fraction of the solid and how far the system is from the critical flocculation concentration (CFC). Just above the CFC, the flocculi are probably closepacked (with relatively small floc volume), whereas far above the CFC a more open structure is found which entraps a considerable amount of liquid. Both types of flocculi persist at high shear rates, although the flocculi with weak attraction may become more compact by maximizing the number of interactions within the flocculus. As discussed before, the Bingham yield value is given by equation (1.47) which allows one to obtain the floc radius afloc provided one can calculate the floc volume ratio CFP (ϕF /ϕs ) and assumes a value for ∆ (the distance through which bonds are stretched inside the floc by the shearing force. At high volume fractions, ϕF and hence CFP can be calculated using the Krieger equation [14, 15], m ϕF −[η]ϕs ηpl = η0 (1 − m ) , (1.60) ϕs where η0 is the viscosity of the medium, ϕsm is the maximum packing fraction which may be taken as 0.74 and [η] is the intrinsic viscosity taken as 2.5. Assuming a value of ∆ of 0.5 nm, the floc radius Rfloc was calculated using equation (1.60). Fig. 1.42 shows the variation of Rfloc with latex volume fraction at the two Na2 SO4 concentrations studied. At any given electrolyte concentration, the floc radius increases with increasing the latex volume fraction, as expected. This can be understood by assuming that the larger flocs are formed by fusion of two flocs and the smaller flocs by “splitting” of the larger ones. From simple statistical arguments, one can predict that afloc will increase with increasing ϕs because in this case larger flocs are favoured over smaller ones. In addition, at any given volume fraction of latex, the floc radius increases with increasing electrolyte concentration. This is consistent with the scaling results as discussed above.

Floc radius (μm)

90 0.5 mol dm–3 Na2SO4

70

0.4 mol dm–3 Na2SO4

50 30 0.2

0.3

0.4 ϕs

0.5

0.6

Fig. 1.42: Floc radius (afloc ) as a function of latex volume fraction at 0.4 and 0.5 mol dm−3 (above the CFC).

References | 43

The above results clearly show the correlation of viscoelasticity of flocculated dispersions with their interparticle attraction. These measurements allow one to obtain the CFC and CFT of concentrated flocculated dispersions with reasonable accuracy. In addition, the results obtained can be analysed using various models to obtain some characteristics of the flocculated structure, such as the “openness” of the network, the liquid entrapped in the floc structure and the floc radius. Clearly, several assumptions have to be made, but the trends obtained are consistent with expectations from theory.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

Tadros T. Rheology of Dispersions. Weinheim: Wiley-VCH; 2010. Mackosko CW. Rheology, Principles, Measurement and Applications. New York: Wiley-VCH; 1994. Goodwin JW. In: Tadros TF, editor. Surfactants. London: Academic Press; 1984. Goodwin JW, Hughes RW. Advances Colloid Interface Sci. 1992;42:303. Tadros TF. Advances Colloid and Interface Science. 1996;68:97. Goodwin JW, Hughes RW. Rheology for Chemists. Cambridge: Royal Society of Chemistry Publication; 2000. Whorlow RW. Rheological Techniques. Chichester: Ellis Horwood; 1980. Bingham EC. Fluidity and Plasticity. New York: McGraw Hill; 1922. Herschel WH, Bulkley R. Proc Amer Soc Test Materials. 1926;26:621; Kolloid Z. 1926;39:291. Casson N. In: Mill CC, editor. Rheology of Disperse Systems. New York; Pergamon Press; 1959. p. 84–104. Cross MM. J Colloid Interface Sci. 1965;20:417. Einstein A. Ann Physik. 1906;19:289; 1911;34:591. Bachelor GK. J Fluid Mech. 1977;83:97. Krieger IM, Dougherty TJ. Trans Soc Rheol. 1959;3:137. Krieger IM, Advances Colloid and Interface Sci. 1972;3:111. Goodwin JW, Hughes RW. Rheology for Chemists. Cambridge: Royal Society of Chemistry Publication; 2000. Napper DH. Polymeric stabilisation of colloidal dispersions. London: Academic Press; 1983. Liang W, Tadros TF, Luckham PF. J Colloid Interface Sci. 1992;153:131. Prestidge C, Tadros TF. J Colloid Interface Sci. 1988;124:660. Firth BA, Hunter RJ. J Colloid Interface Sci. 1976;57:248. van de Ven TGM, Hunter RJ. Rheol Acta. 1976;16:534. Hunter RJ, Frayane J. J Colloid Interface Sci. 1980;76:107. Heath D, Tadros TF. Faraday Disc Chem Soc. 1983;76:203. Prestidge C, Tadros TF. Colloids and Surfaces. 1988;31:325. Tadros TF, Zsednai A. Colloids and Surfaces. 1990;49:103. Liang W, Tadros TF, Luckham PF. J Colloid Interface Sci. 1993:155. Liang W, Tadros TF, Luckham PF. J Colloid Interface Sci. 1993;160:183. Ball R, Brown WD. Personal Communication. Buscall R, Mill PDA. J Chem Soc Faraday Trans I. 1988;84:4249. Gast AP, Hall CK, Russel WB. J Colloid Interface Sci. 1983;96:251. Asakura A, Oosawa F. J Chem Phys. 1954;22:1255.

44 | 1 Flow characteristics (rheology) of colloidal dispersions

[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44]

Asakura A, Oosawa F. J Polymer Sci. 1958;93:183. Fleer GJ, Scheutjens JMHM, Vincent B. ACS Symposium Series. 1984:240;245. Goodeve CV. Trans Faraday Soc. 1939;35:342. Gillespie T. J Colloid Sci. 1960;15:219. Hunter RJ, Nicol SK. J Colloid Interface Sci. 1968;28:200. Firth BA, Hunter RJ. J Colloid Interface Sci. 1976;57:248, 257, 266. Mills PDA, Goodwin JW, Grover B. Colloid Polym Sci. 1991;269:949. Goodwin JW, Hughes RW. Advances Colloid Interface Sci. 1992;42:303. Liang W, Tadros TF, Luckham PF. Langmuir. 1983;9:2077. Firth BA, Neville PC, Hunter RJ. J Colloid Interface Sci. 1974;49:214. van de Ven TGM, Hunter RJ. J Colloid Interface Sci. 1974;68:135. Hunter RJ. Advances Colloid Interface Sci. 1982;17:197. Friend JP, Hunter RJ. J Colloid Interface Sci. 1971;37:548.

2 Wetting and spreading 2.1 Introduction Wetting is important in many processes, both industrial and natural [1]. In many cases, wetting is a prerequisite for application: (i) In paint films; a paint has to wet the substrate completely in order to from a uniform paint film. (ii) In coatings such as in photographic films which are coated with a film at very high speed. In this case the dynamics of the wetting process is very important. (iii) In crop sprays applied to plants or weeds; it is essential that the spray solution wets the substrate completely and in many cases rapid spreading may be required. Again the dynamics of wetting becomes a very important factor. (iv) In personal care formulations such as creams and lotions which require good wetting of the substrate (skin). In many other applications such as hair sprays, droplet impaction and adhesion become important and this may have to be followed by wetting and spreading on the hair surface. (v) In pharmaceutical applications such as for example wetting of tablets which is essential for their disintegration and dispersion. Wetting of powders is an important prerequisite for dispersion of powders in liquids, i.e. preparation of suspensions as will be discussed in Chapter 3. It is essential to wet both the external and internal surfaces of the powder aggregates and agglomerates. Suspensions are applied in many industries such as paints, dyestuffs, printing inks, agrochemicals, pharmaceuticals, paper coatings, detergents, etc. In all the above processes one has to consider both the equilibrium and dynamic aspects of the wetting process. The equilibrium aspects of wetting can be studied at a fundamental level using interfacial thermodynamics. Wetting is a fundamental interfacial phenomenon in which one fluid phase is displaced completely or partially by another fluid phase from the surface of a solid or a liquid. When a drop of a liquid is placed on a solid, the liquid either spreads to form a thin (uniform) film or remains as a discrete drop, as illustrated in Fig. 2.1. One may identify four different situations: (i) The liquid drop does not make any contact with the surface; contact angle θ = 180°. (ii) The drop produces θ > 90°; poor wetting. (iii) The drop produces θ < 90°; partial wetting. (iv) The drop produces θ = 0°; complete wetting. Fig. 2.1 gives an illustration of complete and partial wetting [1].

https://doi.org/10.1515/9783110541953-003

46 | 2 Wetting and spreading Gas or Vapour V V Liquid L

L

Θ

S

Solid S

Complete Wetting

Liquid remains as a Discrete drop Incomplete Wetting

Fig. 2.1: Illustration of complete and partial wetting.

Under equilibrium, a drop of a liquid on a substrate produces a contact angle θ, which is the angle formed between planes tangent to the surfaces of solid and liquid at the wetting perimeter. This is illustrated in Fig. 2.2 which shows the profile of a liquid drop on a flat solid substrate. Equilibrium between vapour, liquid and solid is established with a contact angle θ (that is lower than 90°). The wetting perimeter is frequently referred to as the three-phase line (solid/liquid/vapour); the most common name is the wetting line.

γLV cos Θ

Θ γSL

γSV Wetting line

Fig. 2.2: Schematic representation of the contact angle and wetting line.

Most equilibrium wetting studies centre around measurements of the contact angle. The smaller the angle the better the liquid is said to wet the solid. Typical examples are given in Tab. 2.1 for various liquids on different substrates [1]. The above values can be roughly used as a measure of wetting of the substrate, for example water completely wets glass whereas it is a poor wetting liquid for PTFE. The dynamic process of wetting is usually described in terms of a moving wetting line which results in contact angles that change with the wetting velocity. The same name is sometimes given to contact angles that change with time.

2.2 The concept of contact angle

|

47

Tab. 2.1: Contact angles of various liquids on different substrates. Liquid, γ (mN m−1 )

Substrate

Contact angle θ (°)

Mercury, 484

PTFE*

150

Water, 72.5

PTFE Paraffin wax Polyethylene Human skin Gold Glass

112 110 103 75–90 0 0

Methylene iodide, 67

PTFE* Paraffin wax Polyethylene

85 61 46

Benzene, 28

PTFE* Graphite

46 0

n-decane, 23 n-octane, 21.6

PTFE* PTFE*

40 30

Tetradecane/water, 50.2

PTFE*

170

* Polytetrafluoroethylene

Wetting of a porous substrate may also be considered as a dynamic phenomenon. The liquid penetrates through the pores and gives different contact angles depending on the complexity of the porous structure. Studying the wetting of porous substrates is very difficult. The same applies to the wetting of agglomerates and aggregates of powders. However, even measurements of apparent contact angles can be very useful for comparing one porous substrate with another and one powder with another. Despite increasing attention being paid to the dynamics of wetting, understanding the kinetics of the process at a fundamental level has never been achieved. The spreading of liquids on substrates is also an important industrial phenomenon. A useful concept introduced by Harkins [2, 3] is the spreading coefficient. This is simply the work in destroying a unit area of solid/liquid and liquid/vapour interface to produce an area of solid/air interface. The spreading coefficient is simply determined from the contact angle θ and the liquid/vapour surface tension γLV as will be discussed below.

2.2 The concept of contact angle The contact angle θ is the angle formed between planes tangent to the surfaces of the solid and liquid at the wetting perimeter. The wetting perimeter is referred to as the three-phase line (solid/liquid/vapour) or simply the wetting line. The utility of con-

48 | 2 Wetting and spreading

tact angle measurements depends on equilibrium thermodynamic arguments (static measurements). In practical systems, one has to displace one fluid (air) with another (liquid) as quickly and as efficiently as possible. Dynamic contact angle measurements (associated with a moving wetting line) are more relevant in many practical applications. Even under static conditions, contact angle measurements are far from being simple since they are mostly accompanied by hysteresis. The value of θ depends on the history of the system and whether the liquid is tending to advance across or recedes from the solid surface. The limiting angles achieved just prior to movement of the wetting line (or just after movement ceases) are known as the advancing and receding contact angles, θA and θR , respectively. For a given system, θA > θR and θ can usually take any value between these two limits without discernible movement of the wetting line. The liquid drop takes the shape that minimizes the free energy of the system. Consider a simple system of a liquid drop (L) on a solid surface (S) in equilibrium with the vapour of the liquid (V) as was illustrated in Fig. 2.2. The sum (γSV ASV + γSL ASL + γLV ALV ) should be a minimum at equilibrium and this leads to Young’s equation [4], γSV = γSL + γLV cos θ.

(2.1)

In the above equation θ is the equilibrium contact angle. The angle that a drop assumes on a solid surface is the result of the balance between the cohesion force in the liquid and the adhesion force between the liquid and solid, i.e. γLV cos θ = γSV − γSL , or, cos θ =

γSV − γSL . γLV

(2.2)

(2.3)

If there is no interaction between solid and liquid, then γSL = γSV + γLV ,

(2.4)

i.e. θ = 180° (cos θ = −1). If there is strong interaction between solid and liquid (maximum wetting), the latter spreads until Young’s equation is satisfied (θ = 0) and, γLV = γSV − γSL .

(2.5)

The liquid spreads spontaneously on the solid surface. When the surface of the solid is in equilibrium with the liquid vapour, then one must consider the spreading pressure, πe . As a result of the adsorption of the vapour on the solid surface, its surface tension γs is reduced by πe , i.e., γSV = γs − πe ,

(2.6)

2.3 Adhesion tension

|

49

and Young’s equation can be written as, γLV cos θ = γs − γSL − πe

(2.7)

In general, Young’s equation provides a precise thermodynamic definition of the contact angle. However, it suffers from the lack of direct experimental verification since both γSV and γSL cannot be directly measured. An important criterion for applying Young’s equation is to have a common tangent at the wetting line between the two interfaces.

2.3 Adhesion tension There is no direct way by which γSV or γSL can be measured. The difference between γSV and γSL can be obtained from contact angle measurements. This difference is referred to as the “wetting tension” or “adhesion tension”, adhesion tension = γSV − γSL = γLV cos θ

(2.8)

Consider the immersion of a solid in a liquid as is illustrated in Fig. 2.3. When the plate is immersed in the liquid, an area dAγSV is lost and an area dAγSL is formed.

Θ D

dA = Area immersed

Fig. 2.3: Schematic representation of the immersion of a solid plate in a liquid.

The (Helmholtz) free energy change dF is given by, dF = dA(γSV − γSL ).

(2.9)

This is balanced by the force on the plate W dD, W dD = dA(γSV − γSL ), or W(

dD ) = (γSV − γSL ) = γLV cos θ, dA

(2.10)

(2.11)

where p = dA/dD is the plate perimeter: W = γLV cos θ. p

(2.12)

50 | 2 Wetting and spreading

Equation (2.12) forms the basis of measuring the contact angle θ using an immersed plate (Wilhelmy plate). Equation (2.12) is also the basis of measuring the surface tension of a liquid using the Wilhelmy plate technique. If the plate is made wet by the liquid completely, i.e. θ = 0 or cos θ = 1, then W/p = γLV . By measuring the weight of the plate as it touches the liquid one obtains γLV . Gibbs [5] defined the adhesion tension τ as the difference between the surface pressure at the solid/liquid interface πSL and that at the solid/vapour interface πSV , τ = πSL − πSV

(2.13)

πSL = γs − γSL

(2.14)

πSV = γs − γSV

(2.15)

Combining equations (2.13)–(2.15) with Young’s equation, Gibbs arrived at the following equation for the adhesion tension, τ = γSV − γSL = γLV cos θ,

(2.16)

which is identical to equation (2.8). Thus, the adhesion tension depends on the measurable quantities γLV and θ. As long as θ < 90°, the adhesion tension is positive.

2.4 Work of adhesion Wa Consider a liquid drop with surface tension γLV and a solid surface with surface tension γSV . When the liquid drop adheres to the solid surface it forms a surface tension γSL . This is schematically illustrated in Fig. 2.4. The work of adhesion [6, 7] is simply the difference between the surface tensions of the liquid/vapour and solid/vapour and that of the solid/liquid, Wa = γSV + γLV − γSL .

(2.17)

Wa = γLV (cos θ + 1).

(2.18)

Using Young’s equation,

γLV

γSV γSL Fig. 2.4: Representation of adhesion of a drop on a solid substrate.

2.6 Calculation of surface tension and contact angle |

51

2.5 Work of cohesion The work of cohesion Wc is the work of adhesion when the two phases are the same. Consider a liquid cylinder with unit cross-sectional area. When this liquid is subdivided into two cylinders, as is illustrated in Fig. 2.5, two new surfaces are formed. The two new areas will have a surface tension of 2γLV and the work of cohesion is simply, Wc = 2γLV . (2.19) Thus, the work of cohesion is simply equal to twice the liquid surface tension. An important conclusion may be drawn if one considers the work of adhesion given by equation (2.18) and the work of cohesion given by equation (2.19): when Wc = Wa , θ = 0°. This is the condition for complete wetting. When Wc = 2Wa , θ = 90° and the liquid forms a discrete drop on the substrate surface. Thus, the competition between the cohesion of the liquid to itself and its adhesion to a solid gives an angle of contact that is constant and specific to a given system at equilibrium. This shows the importance of Young’s equation in defining wetting.

γLV γLV Fig. 2.5: Schematic representation of subdivision of a liquid cylinder.

2.6 Calculation of surface tension and contact angle Fowler [8] was the first to calculate the surface tension of a simple liquid. The basic idea is to use the intermolecular forces that operate between atoms and molecules. For this purpose it is sufficient to consider the various van der Waals forces: Dipole-dipole (Keesom), dipole-induced dipole (Debye) and dispersion (London). Dispersion forces are the most important since they occur between all atoms and molecules and they are additive. The London expression for the dispersion interaction u between two molecules separated by a distance r is, u=−

β , r6

(2.20)

where β is the London dispersion constant (that depends on the electric polarizability of the molecules).

52 | 2 Wetting and spreading

Hamaker [9] calculated the attractive forces between macroscopic bodies using a simple additivity principle. For two semi-infinite flat plates separated by a distance d, the attractive force F is given by, A F= , (2.21) 6πd3 where A is the Hamaker constant that is given by, A = π2 n2 β,

(2.22)

where n is the number of interacting dispersion centres per unit volume. Fowler [8] used the above intermolecular theory to calculate the energy required to break a column of liquid of unit cross section and remove the two halves to infinite separation. This gave the work of cohesion and to be equal to twice the surface tension. He used statistical thermodynamic methods to integrate the intermolecular forces. These were calculated from the Lennard–Jones equation, which is like the London equation (2.21), but contains an additional term which varies as r−12 to account for very short-range intermolecular repulsion.

2.6.1 Good and Girifalco approach [10, 11] Good and Girifalco [10, 11] proposed a more empirical approach to the problem of calculating the surface and interfacial tension. The interaction constant for two different particles was assumed to be equal to the geometric mean of the interaction constants for the individual particles. This is referred to as the Berthelot principle. For two atoms i and j with London constants β i and β j , the interaction constant β ij is given by the expression, β ij = (β i β j )1/2 . (2.23) Similarly, the Hamaker constant A ij is given by the geometric mean of the individual Hamaker constants, A ij = (A i A j )1/2 . (2.24) By analogy Good and Girifalco [10] represented the work of adhesion between two different liquids Wa12 as the geometric mean of their respective works of cohesion, Wa12 = ϕ(Wc1 Wc2 )1/2 ,

(2.25)

where ϕ is a constant that depends on the relative molecular size and polar content of the interacting media. The interfacial tension γ12 is then related to the surface tension of the individual liquids γ1 and γ2 by the following expression, γ12 = γ1 + γ2 − 2ϕ(γ1 γ2 )1/2 .

(2.26)

2.6 Calculation of surface tension and contact angle |

53

For nonpolar media, equation (2.26) was found to work well with ϕ ≈ 1. For dissimilar substances such as water and alkanes, the value of ϕ ranges from 0.35 to 1.15. Good and Girifalco [10] and Good [11] extended the above treatment to the solid/ liquid interface and they obtained the following expression for the contact angle θ, cos θ = −1 + 2ϕ(

γs 1/2 πs ) − , γ1 γ1

(2.27)

where πs is the surface pressure of fluid 1 adsorbed at the solid/gas interface. Equation (2.27) gave reasonable values for γs for nonpolar substrates. Although the above analysis is semi-empirical it can be usefully applied to predict the interfacial tension between two immiscible liquids. The analysis is also useful for predicting the surface tension of a solid substrate from measurements of the contact angle of the liquid.

2.6.2 Fowkes treatment [12] Fowkes [12] proposed that the surface and interfacial tensions can be subdivided into independent additive terms arising from different types of intermolecular interactions. For water, in which both hydrogen bonding and dispersion forces operate, the surface tension can be assumed to be the sum of two contributions, γ = γh + γd .

(2.28)

For nonpolar liquids such as alkanes, γ is simply equal to γd . By applying the geometric mean relationship to γd , Fowkes [12] obtained the following expression for the work of adhesion Wa12 , (2.29) Wa12 = 2(γd1 γ22 )1/2 . Thus, the interfacial tension γ12 is given by the following expression, γ12 = γ1 + γ2 − 2(γd1 γd2 )1/2 .

(2.30)

Fowkes [12] assumed the non-dispersive contributions to γ1 and γ2 are unaltered at the 12 interface. Similar equations can be written for the solid/liquid interface. An expression for the contact angle has been derived by Fowkes [12], cos θ = −1 +

2(γds γd1 )1/2 πs − . γ1 γ1

(2.31)

Various studies showed that equations (2.30) and (2.31) are quite effective for materials that interact only through dispersion forces [13–15] and gave reasonable predictions for γd for liquids and solids in which other forces are active. The value for γd for water is

54 | 2 Wetting and spreading ≈ 21.8 mN m−1 , leaving γh ≈ 51 mN m−1 . Fowkes [16] showed that for nonpolar liquids, equation (2.31) can be obtained by summation of pairwise interactions. Paddy and Uffindell [17] used a different approach by integrating the London equation to obtain the work of cohesion. The latter is the work per unit area in breaking a slab of I and removing to two halves to infinite separation. If the average distance between neighbouring molecules or molecular groups is r0i , then this work can be calculated by integrating equation (2.21) from r0i to infinity and changing the sign. Thus, A ii Wcdi = 2γdi = , (2.32) 12π(r0i )2 and, γd1 =

A11 . 24π(r0i )2

(2.33)

Using equation (2.24), the work of adhesion between two dissimilar phases i and j, W ijd = γdi + γdj − γdij =

A ij 12π(r0ij )2

.

(2.34)

Hence, γdij = γdi + γdj − [

A ij 12π(r0ij )2

].

(2.35)

If r0ij = (r0i r0j )1/2

(2.36)

γd12 = γd1 + γd2 − 2(γd1 γd2 )1/2 .

(2.37)

then, If i and j are solid and liquid, then combining with Young’s equation gives, cos θ = −1 + [

πs,1 A s1 , ]− 0 2 γ1 12π(rs1 ) γ1

(2.38)

which is identical to equation (2.31). The method used by Paddy and Uffindell [17] gave good values for the n-hexane/ water and n-hexane/mercury interfacial tensions, and for the contact angle of water on paraffin wax. However, it is important to remember that all these calculations depend on the choice of the value of r0i ; a 0.1 nm change in r0i can change γd by 50 %. Despite its apparent success, the Hamaker microscopic summation approach suffers from a number of limitations: (i) It ignores the influence of neighbouring molecules on pairwise interactions. (ii) The Hamaker constants are evaluated with dominant interaction frequency in the ultraviolet, whereas molecular interactions cover a large part of the electromagnetic spectrum.

2.7 The spreading of liquids on surfaces |

55

(iii) With the summation procedure it is difficult to account for interactions across a third medium other than a vacuum. (iv) With polar media, temperature-dependent Keesom (dipole-dipole) and Debye (dipole-induced-dipole) forces may become significant. All the above limitations can be overcome by the use of Lifshitz macroscopic theory [18] that was discussed in Chapter 5, Vol. 1. This theory treats the interacting bodies as continuous and ascribes the interactions to fluctuating electromagnetic fields arising from spontaneous electric and magnetic polarizations within the various media. One important result is that the interactions are described completely in terms of the complex dielectric constants of the media. The significance of the macroscopic theory to the interpretation of wetting phenomena was considered by Dzyaloshinkii et al. [19, 20]. They showed how dispersion forces can give rise to liquid films that, depending upon thickness, are stable, metastable or unstable. This problem was considered in more detail by Israelachvili [21]. The treatment is based on calculating the work of cohesion and the work of adhesion, and still relies on Young’s equation for the contact angle.

2.7 The spreading of liquids on surfaces 2.7.1 The spreading coefficient S Harkins [2, 3] defined the initial spreading coefficient as the work required to destroy unit area of solid/liquid (SL) and liquid/vapour (LV) and leave unit area of bare solid (SV). This is schematically represented in Fig. 2.6. S = γSV − (γSL + γLV ).

(2.39)

S = γLV (cos θ + 1).

(2.40)

Using Young’s equation, If S is positive, the liquid will spread until it completely wets the solid so that θ = 0°. If S is negative (θ > 0°), only partial wetting occurs. Alternatively one can use the equilibrium or final spreading coefficient. V γLV V

L γSV

γSL S

S

Fig. 2.6: Schematic representation of the spreading coefficient S.

56 | 2 Wetting and spreading

2.7.2 Contact angle hysteresis For a liquid spreading on a uniform, non-deformable solid (idealized case), there is only one contact angle (the equilibrium value). With real surfaces (practical systems) a number of stable angles can be measured. Two relatively reproducible angles can be measured: Largest, advancing angle θA and smallest, receding angle θR . This is illustrated in Fig. 2.7. ΘA

ΘR

Fig. 2.7: Schematic representation of advancing and receding angles.

θA is measured by advancing the periphery of the drop over the surface (e.g. by adding more liquid to the drop). θR is measured by pulling the liquid back (e.g. by removing some liquid from the drop). The difference between θA and θ3R is termed “contact angle hysteresis”. Contact angle hysteresis can be illustrated by placing a drop on a tilted surface with an angle α from the horizontal, as illustrated in Fig. 2.8. The advancing and receding angle are clearly shown at the front and the back of the drop on the tilted surface. Due to the gravity field (mg sin α dl, where m is the mass of the drop and g is the acceleration due to gravity), the drop will slide until the difference between the work of dewetting and wetting balances the gravity force. work of dewetting = γLV (cos θR + 1)ω dl;

(2.41)

work of wetting = γLV (cos θA + 1)ω dl;

(2.42)

mg sin α dl = γLV (cos θR − cos θA )ω dl; mg sin α = γLV (cos θR − cos θA ). ω ω ΘR

ΘA

Area dewetted

dl

α Area wetted Liquid profile Fig. 2.8: Representation of drop profile on a tilted surface.

(2.43) (2.44)

2.7 The spreading of liquids on surfaces |

57

Hysteresis can be demonstrated by measuring the force on a plate that is continuously immersed in the liquid. When the plate is immersed, the force will decrease due to buoyancy. When there is no contact angle hysteresis, the relationship between depth of immersion and force will be as shown in Fig. 2.9 (a). With hysteresis, the relationship between depth of immersion and force will be as shown in Fig. 2.9 (b). Depth of immersion

Depth of immersion Immersion starts

Emersion

(a)

Force

(b)

Force

Fig. 2.9: Relationship between depth of immersion and force: (a) no hysteresis; (b) hysteresis present.

2.7.3 Reasons for hysteresis (i) Penetration of wetting liquid into pores during advancing contact angle measurements. (ii) Surface roughness: The first and rear edges meet the solid with the same intrinsic angle θ0 . The macroscopic angles θA and θR vary significantly at the front and the rear of the drop. This is illustrated in Fig. 2.10.

2.7.4 Wenzel’s equation [22] Wenzel [22] considered the true area of a rough surface A (which takes into account all the surface topography, peaks and valleys) and the projected area A󸀠 (the macroscopic or apparent area) as illustrated in Fig. 2.10. A roughness factor r can be defined as, r=

A ; A󸀠

r > 1, the higher the value of r the higher the roughness of the surface.

(2.45)

58 | 2 Wetting and spreading

ΘR ΘO

Fig. 2.10: Representation of a drop profile on a rough surface.

ΘA

The measured contact angle θ (the macroscopic angle) can be related to the intrinsic contact angle θ0 through r, (2.46) cos θ = r cos θ0 . Using Young’s equation, cos θ = r(

γSV − γSL ). γLV

(2.47)

If cos θ is negative on a smooth surface (θ > 90°), it becomes more negative on a rough surface; θ becomes larger and surface roughness reduces wetting. If cos θ is positive on a smooth surface (θ < 90°), it becomes more positive on a rough surface; θ is smaller and surface roughness enhances wetting.

2.7.5 Surface heterogeneity Most real surfaces are heterogeneous, consisting of patches (islands) that vary in their degrees of hydrophilicity/hydrophobicity. As the drop advances on such a heterogeneous surface, the edge of the drop tends to stop at the boundary of the island. The advancing angle will be associated with the intrinsic angle of the high contact angle region (the more hydrophobic patches or islands). The receding angle will be associated with the low contact angle region, i.e. the more hydrophilic patches or islands. If the heterogeneities are small compared with the dimensions of the liquid drop, one can define a composite contact angle. Cassie [23, 24] considered the maximum and minimum values of the contact angles and used the following simple expression, cos θ = Q1 cos θ1 + Q2 cos θ2 .

(2.48)

Q1 is the fraction of surface having contact angle θ1 , and Q2 is the fraction of surface having contact angle θ2 . θ1 and θ2 are the maximum and minimum contact angles respectively.

2.8 The critical surface tension of wetting |

59

2.8 The critical surface tension of wetting A systematic way of characterizing “wettability” of a surface was introduced by Fox and Zisman [25, 26]. The contact angle exhibited by a liquid on a low energy surface (with surface free energy less than 100 mJ m−2 ) is largely dependent on the surface tension of the liquid γLV . For a given substrate and a series of related liquids (such as n-alkanes, siloxanes or dialkyl ethers) cos θ is a linear function of the liquid surface tension γLV . This is illustrated in Fig. 2.11 for a number of related liquids on polytetrafluoroethylene (PTFE). The figure also shows the results for unrelated liquids with widely ranging surface tensions; the line broadens into a band which tends to be curved for high surface tension polar liquids. The dotted line in Fig. 2.11 illustrates the theoretical relationship between cos θ and γ for a nonpolar solid having γds = 20 mN m−1 calculated using Fowkes equation (2.31). 1.0

γc Related liquids

0.8 cos Θ

0.6 0.4

Theory

0.2 0.0

Unrelated liquids

–0.2 –0.4 0

10

20

30

40

50

60

70

γLV/mNm¯¹ Fig. 2.11: Variation of cos θ with γLV for related and unrelated liquids on PTFE.

The surface tension at the point where the line cuts the cos θ = 1 axis is known as the critical surface tension of wetting. γc is the surface tension of a liquid that would just spread on the substrate to give complete wetting. If γLV ≤ γc the liquid will spread, whereas for γLV > γc the liquid will form a nonzero contact angle. The above linear relationship can be represented by the following empirical equation, cos θ = 1 + b(γLV − γc ), (2.49) where the constant b usually has a value between 0.03 and 0.04. High energy solids such as glass and polyethylene terephthalate have high critical surface tension (γc > 40 mN m−1 ). Lower energy solids such as polyethylene have lower values of γc (≈ 31 mN m−1 ). The same applies to hydrocarbon surfaces such as paraffin

60 | 2 Wetting and spreading wax. Very low energy solids such as PTFE have lower γc of the order of 18 mN m−1 . The lowest known value is ≈ 6 mN m−1 , which is obtained using condensed monolayers of perfluorolauric acid. The measurement of γc requires a range of liquids with widely varying surface tensions, which is difficult to realize. Alternatively, one can use a series of aqueous surfactant solutions with varying concentrations or binary mixtures of liquids such as alcohols, glycols and water. Unfortunately, using surfactant solutions with different concentrations can give wrong values of γc due to surfactant adsorption on the solid surface, thus altering the surface characteristics.

2.9 Theoretical basis of the critical surface tension The value of γc depends to some extent on the set of liquids used to measure it. Zisman [25, 26] described γc as “a useful empirical parameter” whose relative values act as one would expect of the specific surface free energy of the solid, γ0s . Several authors were tempted to identify γc with γs (the surface tension of the solid substrate) or γd1 (the dispersion component of the surface tension). Good and Girifalco [10, 11] suggested the following expression for the contact angle, cos θ = −1 + 2ϕ(

γs 1/2 πSV , ) − γLV γLV

(2.50)

where ϕ is an empirical constant and πSV is the surface pressure of the liquid vapour adsorbed at the solid/liquid interface. With πSV = 0 and cos θ = 1, γSL = γLV = ϕ2 γs = γc .

(2.51)

For nonpolar liquids and solids, ϕ ≈ 1 and γs ≈ γc . Fowkes [12] obtained the following equation for the contact angle of a liquid on a solid substrate, 2(γds γdLV )1/2 πSV − . (2.52) cos θ = −1 + γLV γLV Again putting cos θ = 1 and πSV = 0, γSL = γLV = (γdLV γds )1/2 = γc .

(2.53)

Equations (2.50) and (2.52) predict that if πSV = 0, a plot of cos θ versus γLV should give a straight line with intercept (γc )−1/2 on the cos θ = 1 axis. The experimental results seem to support this prediction. Thus, for nonpolar solids, γc = γs , provided πSV = 0, i.e. there is no adsorption of liquid vapour on the substrate. The above condition is unlikely to be satisfied when θ = 0.

2.10 Effect of surfactant adsorption

|

61

2.10 Effect of surfactant adsorption Surfactants lower the surface tension of the liquid, γLV , and they also adsorb at the solid/liquid interface lowering γSL . The adsorption of surfactants at the liquid/air interface can be easily described by the Gibbs adsorption equation [5], dγLV = −2.303ΓRT, dC

(2.54)

where C is the surfactant concentration (mol dm−3 ) and Γ is the surface excess (amount of adsorption in mol m−2 ). Γ can be obtained from surface tension measurements using solutions with various molar concentrations (C). From a plot of γLV versus log C one can obtain Γ from the slope of the linear portion of the curve just below the critical micelle concentration (cmc). The adsorption of surfactant at the solid/liquid interface also lowers γSL . From Young’s equation, γSV − γSL . (2.55) cos θ = γLV Surfactants reduce θ if either γSL or γLV or both are reduced (when γSV remains constant). Smolders [27] obtained an equation for the change of contact angle with surfactant concentration by differentiating Young’s equation with respect to ln C at constant temperature, d(γLV cos θ) dγSV dγSL = − . (2.56) d ln C d ln C d ln C Using the Gibbs equation, sin θ(

dθ ) = RT(ΓSV − ΓSL − ΓLV cos θ). d ln C

(2.57)

Since γLV sin θ is always positive, then (dθ/d ln C) will always have the same sign as the right-hand side of equation (2.57) and three cases may be distinguished: (i) (dθ/d ln C) < 0 − ΓSV < ΓSL + ΓLV cos θ. Addition of surfactant improves wetting. (ii) (dθ/d ln C) = 0 − ΓSV = ΓSL + ΓLV cos θ (no effect). (iii) (dθ/d ln C) > 0 − ΓSV > ΓSL + ΓLV cos θ. Addition of surfactant causes dewetting. In many practical situations, the mode of behaviour may vary with surfactant concentration, e.g. from dewetting to wetting. This is particularly the case with polar surfaces such as silica. On addition of a cationic surfactant to a negatively charged silica surface (at pH > 4, i.e. above its isoelectric point of ≈ 2–3) dewetting occurs at low surfactant concentration due to the electrostatic attraction between the cationic surfactant head group with the negative charges on the surface with the alkyl chains pointing towards the solution and the surface becomes more hydrophobic resulting in dewetting. With an increase in cationic surfactant concentration a bilayer of surfactant molecules is formed by hydrophobic interaction between the alkyl chains leaving the polar positively charged head group pointing towards the solution and wetting occurs.

62 | 2 Wetting and spreading

2.11 Wetting of powders by liquids Wetting of powders by liquids is very important in their dispersion, e.g. in the preparation of concentrated suspensions. The particles in a dry powder form either aggregates or agglomerates. In the case of aggregates the particles are joined by their crystal faces. They form compact structures with relatively high bulk density. With agglomerates, the particles are joined by their edges or corners and they form loose structures with lower bulk density than those of the aggregates. It is essential in the dispersion process to wet both external and internal surfaces and displace the air entrapped between the particles [28]. Wetting is achieved by the use of surface active agents (wetting agents) of the ionic or nonionic type which are capable of diffusing quickly (i.e. lower the dynamic surface tension) to the solid/liquid interface and displace the air entrapped by rapid penetration through the channels between the particles and inside any “capillaries”. For wetting of hydrophobic powders into water, anionic surfactants, e.g. alkyl sulphates or sulphonates or nonionic surfactants of the alcohol or alkyl phenol ethoxylates are usually used (see below). The process of wetting of a solid by a liquid involves three types of wetting: adhesion wetting, Wa ; immersion wetting Wi ; spreading wetting Ws . This can be illustrated by considering a cube of solid with unit area of each side (Fig. 2.12).

γSV

S S S S

Adhesion Wa

γSL One side of cube adheres to the liquid surface

Immersion Wi

Spreading Ws

Four sides of cube are immersed in the liquid

Liquid spreads on the remaining top side

Fig. 2.12: Schematic representation of wetting of a cube of solid.

2.12 Rate of penetration of liquids: The Rideal–Washburn equation |

63

In every step one can apply Young’s equation, γSV = γSL + γLV cos θ,

(2.58)

Wa = γSL − (γSV + γLV ) = −γLV (cos θ + 1),

(2.59)

Wi = 4γSL − 4γSV = −4γLV cos θ,

(2.60)

Ws = (γSL + γLV ) − γSV = −γLV (cos θ − 1).

(2.61)

The work of dispersion, Wd , is the sum of Wa , Wi and Ws , Wd = Wa + Wi + Ws = 6γSV − γSL = −6γLV cos θ.

(2.62)

Wetting and dispersion depends on: γLV , liquid surface tension; θ, contact angle between liquid and solid. Wa , Wi and Ws are spontaneous when θ < 90°. Wd is spontaneous when θ = 0. Since surfactants are added in sufficient amounts (γdynamic is lowered sufficiently), spontaneous dispersion is the rule rather than the exception. Wetting of the internal surface requires penetration of the liquid into channels between and inside the agglomerates. The process is similar to forcing a liquid through fine capillaries. To force a liquid through a capillary with radius r, a pressure p is required that is given by, p=−

2γLV cos θ −2(γSV − γSL =[ ]. r rγLV

(2.63)

γSL has to be made as small as possible; rapid surfactant adsorption to the solid surface, low θ. When θ = 0, p ∝ γLV . Thus for penetration into pores, one requires a high γLV . Thus, wetting of the external surface requires low contact angle θ and low surface tension γLV . Wetting of the internal surface (i.e. penetration through pores) requires low θ but high γLV . These two conditions are incompatible and a compromise has to be made: γSV − γSL must be kept at a maximum. γLV should be kept as low as possible but not too low. The above conclusions illustrate the problem of choosing the best dispersing agent for a particular powder. This requires measurement of the above parameters as well as testing the efficiency of the dispersion process.

2.12 Rate of penetration of liquids: The Rideal–Washburn equation For horizontal capillaries (gravity neglected), the depth of penetration l in time t is given by the Rideal–Washburn equation [29, 30], l=[

rtγLV cos θ 1/2 ] . 2η

(2.64)

To enhance the rate of penetration, γLV has to be made as high as possible, θ as low as possible and η as low as possible. For dispersion of powders into liquids one should

64 | 2 Wetting and spreading

use surfactants that lower θ while not reducing γLV too much. The viscosity of the liquid should also be kept at a minimum. Thickening agents (such as polymers) should not be added during the dispersion process. It is also necessary to avoid foam formation during the dispersion process. For a packed bed of particles, r may be replaced by K(= r/k2 ), which contains the effective radius of the bed and a tortuosity factor K, which takes into account the complex path formed by the channels between the particles, i.e., l2 =

KtγLV cos θ . 2η

(2.65)

Thus a plot of l2 versus t gives a straight line and from the slope of the line one can obtain θ. This is illustrated in Fig. 2.13. The Rideal–Washburn equation can be applied to obtain the contact angle of liquids (and surfactant solutions) in powder beds. K should first be obtained using a liquid that produces zero contact angle. This is discussed below.

l²

rγLVcosΘ/2ƞk²

t

Fig. 2.13: Variation of l2 with t.

2.13 Measurement of contact angles of liquids and surfactant solutions on powders A packed bed of powder is prepared, say in a tube fitted with a sintered glass at the end (to retain the powder particles). It is essential to pack the powder uniformly in the tube (a plunger may be used in this case). The tube containing the bed is immersed in a liquid that gives spontaneous wetting (e.g. a lower alkane), i.e. the liquid gives a zero contact angle and cos θ = 1. By measuring the rate of penetration of the liquid (this can be carried out gravimetrically using for example a microbalance or a Kruss instrument) one can obtain K. The tube is then removed from the lower alkane liquid and left to stand for evaporation of the liquid. It is then immersed in the liquid in question and the rate of penetration is measured again as a function of time. Using equation (2.65), one can calculate cos θ and hence θ.

2.14 Assessment of wettability of powders | 65

2.14 Assessment of wettability of powders 2.14.1 Sinking time, submersion or immersion test This by far the most simple (but qualitative) method for assessment of wettability of a powder by a surfactant solution [28]. The time for which a powder floats on the surface of a liquid before sinking into the liquid is measured. 100 ml of the surfactant solution is placed in a 250 ml beaker (of internal diameter of 6.5 cm) and after 30 min standing 0.30 g of loose powder (previously screened through a 200-mesh sieve) is distributed with a spoon onto the surface of the solution. The time t for the 1 to 2 mm thin powder layer to completely disappear from the surface is measured using a stop watch. Surfactant solutions with different concentrations are used and t is plotted versus surfactant concentration as is illustrated in Fig. 2.14.

t

log surfactant Concentration

Fig. 2.14: Sinking time as a function of surfactant concentration.

2.14.2 List of wetting agents for hydrophobic solids in water The most effective wetting agent is the one that gives a zero contact angle at the lowest concentration. For θ = 0 or cos θ = 1, γSL and γLV have to be as low as possible. This requires quick reduction of γSL and γLV under dynamic conditions during powder dispersion (this reduction should normally be achieved in less than 20 seconds). This requires fast adsorption of the surfactant molecules both at the L/V and S/L interfaces [28]. It should be mentioned that reduction of γLV is not always accompanied by simultaneous reduction of γSL and hence it is necessary to have information on both interfacial tensions, which means that measurement of the contact angle is essential in the selection of wetting agents. Measurement of γSL and γLV should be carried out under dynamic conditions (i.e. at very short times). In the absence of such measurements, the sinking time described above could be applied as a guide for wetting agent selection. The most commonly used wetting agents for hydrophobic solids are listed below.

66 | 2 Wetting and spreading

To achieve rapid adsorption the wetting agent should be either a branched chain with central hydrophilic group or a short hydrophobic chain with hydrophilic end group. The most commonly used wetting agents are the following: Aerosol OT (diethylhexyl sulphosuccinate) C2H5

O

C4H9CHCH2―O―C ―CH―SO3Na C4H9CHCH2―O―C ―CH2 C2H5

O

The above molecule has a low critical micelle concentration (cmc) of 0.7 g dm−3 and at and above the cmc the water surface tension is reduced to ≈ 25 mN m−1 in less than 15 s. An alternative anionic wetting agent is sodium dodecylbenzene sulphonate with a branched alkyl chain: C6H13 ― CH3― C ―

― SO3― Na

C4H9

The above molecule has a higher cmc (1 g dm−3 ) than Aerosol OT. It is also not as effective in lowering the surface tension of water, reaching a value of 30 mN m−1 at and above the cmc. It is, therefore, not as effective as Aerosol OT for powder wetting. Several nonionic surfactants such as the alcohol ethoxylates can also be used as wetting agents. These molecules consist of a short hydrophobic chain (mostly C10 ) which is also branched. A medium chain polyethylene oxide (PEO) mostly consisting of 6 EO units or lower is used. These molecules also reduce the dynamic surface tension within a short time (< 20 s) and they have reasonably low cmc. In all cases one should use the minimum amount of wetting agent to avoid interference with the dispersant that needs to be added to maintain the colloid stability during dispersion and on storage.

2.15 Measurement of contact angles on flat surfaces [31] 2.15.1 Sessile drop or adhering gas bubble method A schematic representation of a sessile drop on a flat surface and an air bubble resting on a solid surface is given in Fig. 2.15.

2.15 Measurement of contact angles on flat surfaces |

67

Θ Θ (a)

Θ

Θ

Fig. 2.15: Schematic representation of the sessile drop (a) and air bubble (b) resting on a surface.

(b)

The contact angle can be measured using a telescope fitted with a goniometer eyepiece. Alternatively, it can be measured by taking a photograph or using image analysis. The accuracy of measurement is ±2° for θ values between 10° and 160°. For θ < 10° or > 160°, uncertainty is higher and θ can be calculated from the drop profile (applicable to drops < 10−4 ml). This is schematically shown in Fig. 2.16. 2h θ ; tan( ) = 2 d

(2.66)

24 sin3 θ d3 = . V π(2 − 3 cos θ + cos3 θ)

(2.67)

Care must be taken for kinetic effects and evaporation.

Θ

h

d

Fig. 2.16: Drop profile for calculation of contact angle.

2.15.2 Wilhelmy plate method The substrate in the form of a thin plate is attached to an electrobalance to measure the force. Two procedures may be applied: The plate is allowed to touch the surface of the liquid in question (i.e. with zero net immersion) or it is allowed to penetrate through the liquid with finite depth of immersion. This is schematically illustrated in Fig. 2.17.

68 | 2 Wetting and spreading

F

F Electrobalance

Perimeter of plate p

(a)

(b)

Fig. 2.17: Schematic representation of the Wilhelmy plate technique for measurement of contact angle; (a) zero net depth, (b) finite depth

In the first case (a), the force on the plate F is given by the equation, F = (γLV cos θ)p,

(2.68)

where p is the plate perimeter. In the second case (b), the force is given by, F = (γLV cos θ)p − ∆ρgV.

(2.69)

∆ρ is the density difference between the plate and the liquid and V is the volume of liquid displaced. The above method is convenient and allows one to measure θ as a function of time. Also θA and θR can be determined by raising and lowering the liquid in the vessel (using a lab jack).

2.15.3 Capillary rise at a vertical plate Instead of measuring the capillary pull (the Wilhelmy plate method), one can measure the capillary rise h at a vertical plate, sin θ = 1 −

∆ρgh2 . 2γLV

(2.70)

2.15 Measurement of contact angles on flat surfaces |

69

Both the Wilhelmy plate and capillary rise methods require knowledge of γLV . This may cause uncertainty with surfactant solutions (adsorption alters both γLV and θ). By combining equations (2.68) and (2.69), one can eliminate γLV to obtain θ, cos θ =

4∆m∆ρh2 p , 4(∆m)2 + p2 (∆ρ)2 h4

(2.71)

where ∆m is the weight of the measuring plate. Alternatively, θ can be eliminated to obtain γLV , γLV = (

∆mg 2 1 ∆ρgh2 + . ) p 4 ∆ρgh2

(2.72)

Thus, by combining the Wilhelmy plate with the capillary rise methods, one can obtain θ and γLV simultaneously.

2.15.4 Tilting plate method This is schematically illustrated in Fig. 2.18. The plate can be rotated around an axis normal to the plane of paper, until the liquid meniscus on one side becomes flat. The angle between the plate and the liquid meniscus is the contact angle.

Plate 2 cm wide

Fig. 2.18: Schematic representation of the tilting plate method.

Θ

2.15.5 Capillary rise or depression method The rise (or depression) h of a liquid inside a partially wetted capillary (θ < 90°) with radius r is related to the liquid surface tension and contact angle, by the following equation which gives the capillary pressure, ∆p, ∆p = ∆ρhg =

2γLV cos θ . r

(2.73)

70 | 2 Wetting and spreading

2.16 Wetting kinetics In many experimental situations, a contact angle that changes with time leads inevitably to the movement of the wetting line [1]. The result is a dynamic contact angle. This situation arises from non-equilibrium conditions and this should be distinct from contact angle hysteresis. In dynamic contact angle measurements, some movement of the wetting line is unavoidable and this must cause a temporary disturbance from equilibrium. The state of adsorption at the solid/vapour (SV) and solid/liquid (SL) interfaces will not be the same. Displacement of fluid 2 by fluid 1 (liquid on the solid by the vapour of the liquid) will result in destruction of the SL interface and creation of fresh SV interface. If the sorption-desorption processes or the accompanying transport processes to and from the various interfaces are slow compared with the rate of displacement, a non-equilibrium contact angle will result. If the external driving force is now removed, the contact angle will relax to its equilibrium value. If the relaxation processes are sufficiently slow relative to the experimental timescale, then a non-equilibrium angle may persist in an apparently stable system. It is difficult to distinguish the above situation from contact angle hysteresis. This behaviour may arise from the slow molecular orientation following the movement of the wetting line. If contact angle measurements are to be given equilibrium significance, great care must be taken to ensure that equilibrium has been reached. This is particularly the case with surfactant solutions, where there may be no ready mechanism by which a non-volatile solute (the surfactant molecules) can reach the solid/ vapour interface, except by prior contact with the solution. Kinetic factors that may cause contact angle variation can also arise from penetration of the liquid into the solid surface. The penetration of nonpolar surfaces by water has been commonly cited in the literature [32]. Timmons and Zisman [33] reported a relationship between molecular size and the extent of contact angle hysteresis. The observed behaviour simply reflects the non-attainment of a uniformly penetrated surface during the observation time. In extreme cases, penetration may cause swelling of the substrate. The wetting line will rest along a labile ridge and contact angle variation should be similar to that observed when the wetting line is pinned at an edge. Contact angle variability can be attributed to a number of factors and proper attention should be paid to attainment of equilibrium. One should be cautious in ascribing the variability to permanent features of the system such as surface roughness or intrinsic heterogeneities. With practical systems all the factors have to be considered.

2.17 The dynamic contact angle This is usually ascribed to contact angles that change with time or those associated with moving wetting lines. Contact angles are usually dependent on the speed and di-

2.17 The dynamic contact angle

|

71

rection of the wetting line displacement. The contact angles are velocity dependent. The advancing contact angle increases and the receding contact angle decreases with increasing rate of displacement. This is illustrated in Fig. 2.19 which shows the variation of the contact angle θ with the velocity of wetting v for a system of a poorly wetting liquid with a moderate degree of contact angle hysteresis. θA and θR are the advancing and receding angles when v = 0. 180

Θ/deg

Wetting

ΘA

ΘR

Dewetting

0 Fig. 2.19: Schematic variation of the contact angle, θ, with the velocity of wetting v.

The above behaviour is clearly related to contact angle hysteresis in that both imply thermodynamic irreversibility. In the static case, this is attributed to spontaneous transitions between metastable equilibrium states. In the dynamic case, the interface may fail to attain any kind of equilibrium in the time available. Fig. 2.20 shows the velocity dependence of the contact angle [34] for aqueous glycerine solutions (0.0456 Pa s) on Mylar polyester tape. At low rates of wetting, the advancing contact angle appears to be a steep function of the wetting velocity. As v increases, the slope first decreases and then increases again as θ approaches 180° at v180 . At still higher velocities, the wetting line develops a sawtooth shape (Fig. 2.21 (a)) and air is entrained from the trailing vertices.

72 | 2 Wetting and spreading

180 160

Contact angle (Θ/deg)

140 120 100 80 60 Θ°

40 20 0 0

10

20

30

40

50

Wetting velocity (v/cm s¯ ¹)

Fig. 2.20: Velocity dependence of the contact angle [34].

0 V v0 ϕ

Surface

0 v180

ϕ 0

0 V0 0 (a)

(b)

Fig. 2.21: Formation of sawtooth wetting lines showing entrainment from trailing vertices: (a) wetting; (b) dewetting.

60

References | 73

Much of the interest in dynamic contact angles lies in maximizing the wetting velocity at the onset of entrainment. In liquid coating operations, entrainment of air leads to patchy or uneven coatings. In petroleum recovery, entrainment of crude oil by gas or water flood may reduce the efficiency of recovery. Blake and Ruschak [35] argued that the occurrence of the sawtooth wetting line shows that for a given system, v180 is the maximum velocity at which a wetting line can advance normal to itself. If an attempt is made to wet a solid at some velocity v > v180 , the wetting line must lengthen and slant at some angle ϕ relative to its orientation at velocities below v180 such that, cos ϕ =

v180 . v

(2.74)

Similarly for dewetting, if an attempt is made to dewet a solid (for which θR > 0) at a velocity greater than that at which θ becomes zero, v0 , then the wetting line again consists of at least two straight segments slanted at an angle ϕ relative to the normal orientation. In this case, v0 cos ϕ = , (2.75) v and drops of liquid or a continuous rivulet may be entrained from the trailing vertices. Blake and Ruschak [35] reported data in good agreement with these simple relationships. The very steep velocity dependence of the contact angle as v → 0 strongly suggests the kinetic origin of apparent contact angle hysteresis.

2.18 Effect of viscosity and surface tension The velocity dependence of the contact angle increases with increasing liquid viscosity η and decreasing surface tension γ. Several authors found an increase in θ with increasing capillary number Ca = (ηv/γ). The higher the viscosity, the higher the velocity dependence of θ and the lower the value of v180 . Viscous forces tend to oppose wetting. The lower the surface tension, the higher the velocity dependence of the contact angle and the lower the value of v180 . Experimental results showed that surfactants can improve the rate of wetting.

References [1] [2] [3] [4] [5] [6] [7] [8]

Blake TB. Wetting. In: Tadros TF, editor. Surfactants. London: Academic Press; 1984. Harkins WD. J Phys Chem. 1937;5:135. Harkins WD. The physical chemistry of surface films. New York: Reinhold; 1952. Young T. Phil Trans Royal Soc (London). 1805;95:65. Gibbs JW. Collected works. Vol.1. New York: Longman; 1928. Everett DH. Pure and Appl Chemistry. 1980;52:1279. Johnson RE. J Phys Chem. 1959;63:1655. Fowler RH. Proc Royal Soc Ser A. 1937;159:229.

74 | 2 Wetting and spreading

[9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

Hamaker HC. Physica. 1937;4:1058. Good RJ, Girifalco LA. J Phys Chem. 1960;64:561. Good RJ. Adv Chem Series. 1964;43:74. Fowkes FM. Adv Chem Ser. 1964;43:99. Tamai Y, Matsunaga T, Horiuchi K. J Colloid Interface Sci. 1977;60:112. J. Schultz J, Tsutsuma K, Donnet JB. J Colloid Interface Sci. 1977;59:277. Pashley RM, Israelachvili JN. J Colloid Interface Sci. 1981;80:153; 1981;83:531. Fowkes FM. SCI (Soc Chem Ind London) Monograph. 1967;25:3. Paddy JF, Uffindell ND. J Phys Chem. 1968;72:1407. Lifshitz EM. Soviet Physics JETP. 1956;2:73. Dzyaloshinskii IE, Lifshitz EM, Pitaevskii LP. Soviet Phys JETP. 1960;37:161. Dzyaloshinskii IE, Lifshitz EM, Pitaevskii LP. Soviet Phys Adv Phys. 1961;10:165. Israelachvili JN. J Chem Soc Faraday Trans. 1973;69:1729. Wenzel RN. Ind Eng Chem. 1936;28:988. Cassie ABD, Dexter S. Trans Faraday Soc. 1944;40:546. Cassie ABD. Disc Faraday Soc. 1948;3:361. Fox HW, Zisman WA. J Colloid Sci. 1950;5:514. Zisman WA. Adv Chem Ser. 1964;43:1. Smolders CA. Rec Trav Chim. 1960;80:650. Tadros T. Dispersion of powders in liquids and stabilisation of suspensions. Weinheim: WileyVCH; 2012. Davies JT, Rideal EK. Interfacial phenomena. New York: Academic Press; 1969. Rideal EK. Phil Mag. 1922;44:1152. Tadros T. Applied surfactants. Weinheim: Wiley-VCH; 2005. Kawaski K. J Colloid Sci. 1962;17:169. Timmons CO, Zisman WA. J Colloid Interface Sci. 1966;22:165. Burley R, Kennedy BS. In: Padday JF, editor. Wetting, spreading and adhesion. London: Academic Press; 1978. p. 327–360. Blake TD, Ruschak KJ. Nature. 1979;282:489.

3 Solid/liquid dispersions (suspensions) 3.1 Introduction Solid/liquid dispersions find application in almost every industrial preparation, e.g. paints, dyestuffs, paper coatings, printing inks, agrochemicals, pharmaceuticals, cosmetics, food products, detergents, ceramics, etc. The powder particles can be hydrophobic, e.g. organic pigments, agrochemicals, ceramics; or hydrophilic, e.g. silica, titania, clays [1–5]. The liquid can be aqueous or nonaqueous. The average particle size of the dispersion can be within the colloid range (1 nm–1 µm) or outside the colloid range (> 1 µm). Dispersions with dimensions within the colloid range are referred to as colloidal suspensions. In contrast, dispersions with dimensions outside the colloid range are generally referred to as coarse suspensions. They are to be distinguished from colloidal suspensions in the sense that the particles of the former settle to the bottom of the container (as a result of the gravitational field on the particles) whereas with colloidal suspensions, with particle density not significantly larger than that of the medium, the very mild mixing produced by ambient thermal fluctuations and/or Brownian motion can keep the particles uniformly dispersed in the continuous medium [1]. The formulation of suspensions and maintenance of their physical stability over long periods of time under various conditions, e.g. temperature variation, transport, etc. still remains a challenging problem to the industrial chemist or chemical engineer. This requires understanding of the various interfacial phenomena involved in their preparation and stabilization [6, 7]. The concentration of a suspension is described by its volume fraction ϕ, namely the ratio between the total volume of particles to the volume of the suspension. The value of ϕ above which a suspension may be considered “dilute”, “concentrated” or “solid” can be defined by considering the balance between the particle translational motion (Brownian diffusion) and interparticle interaction [2, 8]. If Brownian diffusion predominates over the imposed interparticle interaction, the suspension may be described as “dilute”. In this case the particle translational motion is large and only occasional contacts may occur between the particles, which are then separated by Brownian force. The particle interactions can be represented by two-body collisions. This “dilute” suspension generally has time-independent properties and if the particle size is within the colloid range and the density difference between the particles is very small, no gravitational sedimentation of particles occurs and the system maintains its properties over long periods of time. These “dilute” suspensions show Newtonian flow, i.e. their viscosity is independent of the applied shear rate [9]. In contrast, if interparticle interaction predominates over Brownian diffusion, i.e. the interparticle distance h becomes much smaller than the particle radius, the system may be described as a “solid” suspension. In this case the particles may vibrate with a distance h that is much smaller than the particle radius. The interaction produces a specific orhttps://doi.org/10.1515/9783110541953-004

76 | 3 Solid/liquid dispersions (suspensions)

der between the particles and a highly developed structure is reached. The resulting “solid” suspension becomes predominantly elastic with very little energy dissipation during flow [9]. Again, the properties of these elastic systems are time-independent. In between the two extremes one may define a volume fraction ϕ at which the suspension may be considered “concentrated”. In this case the interparticle distance h is comparable to the particle radius and the suspension shows time-dependent spatial and temporal correlations. The particle interactions occur with many body collisions and the translational motion of the particles is restricted. These “concentrated” suspensions show viscoelastic behaviour, i.e. a combination of viscous and elastic response [9]. At low stresses the suspension may show a predominantly elastic response, whereas at high stress a predominantly viscous response is obtained (see Chapter 1). There are two main processes for the preparation of suspensions [8]. The first depends on the “build-up” of particles from molecular units, i.e. the so-called “bottomup” or condensation method, which involves two main processes, namely nucleation and growth. In this case, it is necessary first to prepare a molecular (ionic, atomic or molecular) distribution of the insoluble substances; then by changing the conditions precipitation is caused, leading to the formation of nuclei that grow into the particles in question. A particular case of the condensation process is the preparation of polymer latex particles by emulsion or suspension polymerization which will be described in Chapter 8. The second procedure for preparation of suspension concentrates is usually referred to as the “top-down” or dispersion process. Dispersion is a process whereby aggregates and agglomerates of powders are dispersed into “individual” units, usually followed by a wet milling process (to subdivide the particles into smaller units) and stabilization of the resulting dispersion against aggregation and sedimentation. The larger “lumps” of the insoluble substances are subdivided by mechanical or other means into smaller units. In the whole “top-down” process, it is essential to wet both the external and internal surfaces of the aggregates and agglomerates. For hydrophobic solids dispersed in aqueous media, it is essential to use a wetting agent (surfactant) that lowers the surface tension of water (under dynamic conditions) and reduces the solid/liquid interfacial tension by adsorption on the surface of the particles. After wetting, the aggregates and agglomerates are dispersed into single particles by high speed mixing. The resulting suspension, referred to as “mill base”, is then subjected to a milling process to reduce the particle size to the desired value [10, 11]. Once a suspension is prepared, it is necessary to control its properties on storage. Three main aspects must be considered. Firstly, control of its colloid stability which requires the presence of a repulsive energy that overcomes the everlasting van der Waals attraction. Three main types of stabilization may be considered: (i) Electrostatic repulsion produced by the presence of electrical double layers surrounding the particles [2] as described in Chapter 6, Vol. 1. These double layers are extended in solution, particularly at low electrolyte concentration and low valency of ions forming the double layers. When two particles with these ex-

3.1 Introduction

|

77

tended double layers approach each other to a separation distance h that is smaller than twice the double layer extension (“thickness”), strong repulsion occurs. This repulsion Gelec increases with decreasing electrolyte concentration and valency. Combining Gelec with the van der Waals attraction GA results in an energy–distance curve that forms the basis of the theory of colloid stability due to Deryaguin–Landau–Verwey–Overbeek (DLVO theory) [12, 13]. This energy– distance curve shows a maximum (energy barrier) at intermediate separation distances (when the 1 : 1 electrolyte concentration is < 10−2 mol dm−3 ). This energy barrier prevents strong aggregation of the particles thus maintaining effective stability. (ii) Steric repulsion produced by adsorption of nonionic surfactants or polymeric surfactants as described in Chapter 13, Vol. 1. These surfactants consist of an “anchor” chain(s) that strongly adsorbs on the particle surface, and stabilizing chain(s) that remains in solution and becomes strongly solvated by the molecules of the medium [14]. One can define an adsorbed layer thickness δ which increases with increasing molar mass of the stabilizing chain(s). When two particles with adsorbed surfactant or polymer layers approach to a distance h that is lower than 2δ, the stabilizing chains may overlap or become compressed resulting in an increase in the segment concentration in the overlapped or compressed layers. Provided the latter are in good solvent conditions (highly solvated by the molecules of the medium), this effect results in an increase in the osmotic pressure in these overlapped or compressed layers. Solvent molecules will now diffuse to these layers, thus separating the particles. This repulsive effect is referred to as the mixing interaction energy Gmix (unfavourable mixing of the stabilizing chains). In the overlapped or compressed layers the configurational entropy of the chains is significantly reduced resulting in another repulsive energy, Gel (entropic, volume restriction or elastic interaction). The sum of Gmix and Gel is referred to as Gs (steric repulsive energy). Combining of Gmix and Gel with GA gives GT –h curves and this forms the basis of the theory of steric stabilization [14]. In this case, the GT –h curve shows a shallow minimum at separation distance h ≈ 2δ but when h < 2δ, GT increases sharply with a further decrease in h. (iii) Electrosteric stabilization where Gelec and Gs are combined with GA . In this case the GT –h curve shows a shallow minimum at large h values, a maximum at intermediate distances (DLVO type maximum) and a sharp increase at distances comparable to 2δ. Electrosteric stabilization is generally produced when using polyelectrolytes to stabilize the suspension or when using a mixture of nonionic or polymeric surfactant with an ionic one. The second property that must be controlled is the process of Ostwald ripening (crystal growth) that occurs with most suspensions of organic substances [1–3]. The latter have a finite solubility in the medium that may reach several hundred ppm (parts per million). Smaller particles with higher radius of curvature have a higher solubility when

78 | 3 Solid/liquid dispersions (suspensions)

compared with larger particles. This difference in solubility between small and large particles is the driving force for Ostwald ripening. With time, molecular diffusion occurs from the small to the large particles with the ultimate dissolution of these small particles and their molecules become deposited on the larger particles. Thus, on storage of the suspension, the particle size distribution shifts to larger particles. This will result in instability of the suspension, e.g. enhanced sedimentation. Another mechanism of Ostwald ripening is due to polymorphic changes. The particles (e.g. a drug) may contain two or more polymorphs with different solubility. The more stable polymorph has a lower solubility when compared with the metastable polymorph. With time, the more soluble polymorph gradually changes into the less soluble stable one. This Ostwald ripening problem may result in reduction of bioavailability of the drug and hence it must be reduced or eliminated. Several methods can be applied to reduce Ostwald ripening in suspensions, e.g. incorporation of impurities that strongly adsorb on the particle surface, thus blocking the active sites for growth. Alternatively one can use strongly adsorbed polymeric surfactants, which has the same effect as added impurities. The third instability problem with suspensions is particle sedimentation [1–4], which occurs when the particle size is outside the colloid range and the density difference between the particles and the medium is significant. In this case the gravity force (4/3)πR3 ∆ρgh (where R is the particle radius, ∆ρ is the density difference between the particle and the medium, g is the gravity force and h is the height of the container) exceeds the Brownian motion kT (where k is the Boltzmann constant and T is the absolute temperature). With most practical suspensions with a wide particle size distribution, the larger particles sediment at a higher rate than the smaller particles. A particle concentration gradient of the particles occurs across the container. Several methods may be applied to reduce sedimentation, e.g. balance of the density of the disperse phase with that of the medium, reduction of particle size (i.e. formation of nanosuspensions) and addition of thickeners. The latter can be high molecular weight polymers such as xanthan gum, or addition of “inert” fine particles such as silica or clays. In all cases these thickeners produce a “gel network” in the continuous phase, which produces a very high viscosity at low shear rates that prevents particle sedimentation. It is now important to distinguish between “colloid” and “physical” stability. Colloid stability implies absence of particle aggregation and this requires a high repulsive energy as discussed above. However, a colloidally stable suspension may undergo separation, e.g. as a result of particle sedimentation as discussed above. Physical stability, on the other hand, implies absence of any separation, ease of redispersion on gentle shaking or dilution. In many cases “physical stability” may require weak flocculation and formation of a “gel-network structure”. An important factor that controls physical stability is the bulk rheology of the suspension.

3.1 Introduction

|

79

One way to distinguish between colloid and physical stability is to consider the various states of the suspension on standing as schematically illustrated in Fig. 3.1. These states are determined by the interaction energy between the particles, the effect of gravity and addition of other components such as surfactants, polymers and thickeners [8]. States (a)–(c) correspond to a suspension that is stable in the colloid sense. The stability is obtained as a result of net repulsion due to the presence of extended double layers (i.e. at low electrolyte concentration), as the result of steric repulsion producing adsorption of nonionic surfactants or polymers, or as the result of a combination of double layer and steric repulsion (electrosteric). State (a) represents the case of a suspension with small particle size (submicron) where the Brownian diffusion overcomes the gravity force, producing uniform distribution of the particles in the suspension, i.e. kT ≫ (4/3)πR3 ∆ρgh. A good example of the above case is a latex suspension with particle size well below 1 µm that is stabilized by ionogenic groups, by an ionic surfactant or nonionic surfactant or polymer. This suspension will show no separation on storage for long periods of time. States (b) and (c) represent the case of suspensions whose particle size range is outside the colloid range (> 1 µm). In this case, the gravity force exceeds the Brownian diffusion, i.e. (4/3)πR3 ∆ρgh ≫ kT. With state (b), the particles are uniform and initially they are well dispersed, but with time and the influence of gravity they settle to form a hard sediment (technically referred to “clay” or “cake”). In the sediment, the particles are subjected to a hydrostatic pressure hρg, where h is the height of the container, ρ is the density of the particles and g is the acceleration due to gravity. Within the sediment each particle will be acting constantly with many others, and eventually an equilibrium is reached where the forces acting between the particles will be balanced by the hydrostatic pressure on the system. The forces acting between the particles will depend on the mechanism used to stabilize the particles, for example electrostatic, steric or electrosteric, the size and shape of the particles, the medium permittivity (dielectric constant), electrolyte concentration, the density of the particles, etc. Many of these factors can be incorporated to give an interaction energy in the form of a pair potential for two particles in an infinite medium. The repulsive forces between the particles allow them to move past each other until they reach small distances of separation (that are determined by the location of the repulsive barrier). Due to the small distances between the particles in the sediment, it is very difficult to redisperse the suspension by simple shaking. With case (c), consisting of a wide distribution of particle sizes, the sediment may contain larger proportions of the larger sized particles, but still a hard “clay” is produced. These “clays” are dilatant (i.e. shear thickening) and they can be easily detected by inserting a glass rod into the suspension. Penetration of the glass rod into these hard sediments is very difficult.

80 | 3 Solid/liquid dispersions (suspensions)

(a) Stable colloidal suspension

(b) Stable coarse suspension (uniform size)

(c) Stable coarse suspension (size distribution)

(d) Coagulated suspension (chain aggregates)

(e) Coagulated suspension (compact clusters)

(f) Coagulated suspension (open structure)

(g) Weakly flocculated structure

(h) Bridging flocculation

(i) Depletion flocculation

Fig. 3.1: States of the suspension.

States (d)–(f) represent the case for unstable, coagulated suspensions which either have a small repulsive energy barrier or it is completely absent. State (d) represents the case of coagulation under the condition of no stirring, in which case chain aggregates are produced that will settle under gravity forming a relatively open structure. State (e) represents the case of coagulation under stirring conditions: compact aggregates are produced that will settle faster than the chain aggregates and the sediment produced is more compact. State (f) represents the case of coagulation at high volume fraction of the particles, ϕ. In this case the whole particle will form a “one-floc” structure that is formed from chains and cross chains that extend from one wall to the other in the container. Such a coagulated structure may undergo some compression

3.1 Introduction

|

81

(consolidation) under gravity leaving a clear supernatant liquid layer at the top of the container. This phenomenon is referred to as syneresis. State (g) represents the case of weak and reversible flocculation. This occurs when the secondary minimum in the energy–distance curve is deep enough to cause flocculation [4]. This can occur at moderate electrolyte concentrations, in particular with larger particles. The same occurs with sterically and electrosterically stabilized suspensions. This occurs when the adsorbed layer thickness is not very large, particularly with large particles. The minimum depth required for causing weak flocculation depends on the volume fraction of the suspension. The higher the volume fraction, the lower the minimum depth required for weak flocculation. The above flocculation is weak and reversible, i.e. on shaking the container redispersion of the suspension occurs. On standing, the dispersed particles aggregate to form a weak “gel”. This process (referred to as sol-gel transformation) leads to reversible time dependence of viscosity (thixotropy). On shearing the suspension, the viscosity decreases and when the shear is removed, the viscosity is recovered. This phenomenon is applied in paints. On application of the paint (by a brush or roller), the gel is fluidized, allowing uniform coating of the paint. When shearing is stopped, the paint film recovers its viscosity and this avoids any dripping. State (h) represents the case in which the particles are not completely covered by the polymer chains. In this case, simultaneous adsorption of one polymer chain on more than one particle occurs, leading to bridging flocculation. If the polymer adsorption is weak (low adsorption energy per polymer segment), the flocculation could be weak and reversible. In contrast, if the adsorption of the polymer is strong, tough flocs are produced and the flocculation is irreversible. The latter phenomenon is used for solid/liquid separation, e.g. in water and effluent treatment. Case (i) represents a phenomenon, referred to as depletion flocculation, produced by addition of “free” nonadsorbing polymer [15, 16]. In this case, the polymer coils cannot approach the particles to a distance ∆ (that is determined by the radius of gyration of free polymer RG ), since the reduction of entropy on close approach of the polymer coils is not compensated for by an adsorption energy. The suspension particles will be surrounded by a depletion zone with thickness ∆. Above a critical volume fraction of the free polymer, ϕ+p , the polymer coils are “squeezed out” from between the particles and the depletion zones begin to interact. The interstices between the particles are now free from polymer coils and hence an osmotic pressure is exerted outside the particle surface (the osmotic pressure outside is higher than in between the particles) resulting in weak flocculation. The magnitude of the depletion attraction free energy, Gdep , is proportional to the osmotic pressure of the polymer solution, which in turn is determined by ϕp and molecular weight M. The range of depletion attraction is proportional to the thickness of the depletion zone, ∆, which is roughly equal to the radius of gyration, RG , of the free polymer. In this chapter I will start by describing the process of preparation of suspensions by the “bottom-up” process. The advantage of this method over the “top-down” pro-

82 | 3 Solid/liquid dispersions (suspensions)

cess is highlighted, namely the ability to control the particle size distribution. A section is devoted to the process of preparation of suspensions by precipitation. The classical theory of nucleation and growth (Gibbs–Volmer theory) is described in terms of the free energy components of the process, namely the surface free energy and the bulk energy in producing a new phase. This results in a definition of the critical nucleus size above which spontaneous particle growth occurs. The dependency of the critical nucleus size on supersaturation is described and particular reference is given to the effect of added surfactants. A section is devoted to the kinetics of precipitation and control of particle size distribution. The effect of surface modification on precipitation kinetics is described, followed by other methods that can be applied for preparation of suspension particles. A section is devoted for the process of emulsion and suspension particles for preparation of latex suspensions. The next section describes the methods of preparation of suspensions using the top-down process. The process of wetting of powder aggregates and agglomerates (both external and internal surfaces) is described with special reference to the role of surfactants (wetting agents). Wetting of the external surface requires the presence of a surfactant that lowers the surface tension of the liquid and the interfacial tension of the solid/liquid interface. This results in lowering the contact angle at the three-phase region of solid/liquid/vapour. For adequate wetting of the external surface, a contact angle approaching zero is required. Wetting of the internal surface (pores of aggregates and agglomerates) requires adequate penetration of the liquid inside the pores. Again, a low contact angle is required, but good penetration of the liquid requires a high liquid surface tension. Thus, a compromise is needed for good wetting of the external and internal surfaces, namely a low contact angle but not too low a surface tension. This shows the difficulty of choosing the right wetting agent for any particular powder. The various methods that can be applied for measuring powder wetting are described. The different classes of surfactants for enhancing wetting are described. This is followed by a section on dispersion of the aggregates and agglomerates using high speed stirrers. Finally, the methods that can be applied for size reduction are described with reference to bead milling. The main factors responsible for maintaining colloid stability are described. The next section will only give a summary of electrostatic stabilization of suspensions, since this has been described in detail in Chapter 6, Vol. 1. The origin of charge in suspensions and the structure of the electrical double layer was discussed in detail in Chapter 2, Vol. 1. The well-know theory of colloid stability due to Deryaguin–Landau– Verwey–Overbeek (DLVO theory) [12, 13] was described in detail in Chapter 6, Vol. 1. The next section gives a brief description of steric stabilization of suspensions, since this has been described in detail in Chapter 13, Vol. 1. The interaction between particles containing adsorbed surfactant or polymer layers will be briefly described. The various processes of flocculation of electrostatically and sterically stabilized suspensions are summarized since these were described in detail in Chapters 7 and 14, Vol. 1.

3.2 Preparation of suspension concentrates by the bottom-up process

| 83

A section is devoted to the process of the Ostwald ripening (crystal growth) of suspensions and its prevention. It starts with the Kelvin theory that describes the effect of curvature (particle or droplet size) on the solubility of the disperse phases. It shows the rapid increase in solubility when the particle size is reduced below 500 nm. The kinetics of Ostwald ripening is described showing the change of the cube of the mean radius with time and the effect of particle solubility on the rate. The effects of Brownian motion, phase volume fraction and surfactant micelles are described. This is followed by the methods that can be applied to reduce Ostwald ripening for suspensions. The final section deals with sedimentation of suspensions and prevention of formation of dilatant sediments. It starts with the effect of particle size and its distribution on sedimentation. The sedimentation of very dilute suspensions with a volume fraction ϕ ≤ 0.01 and application of Stokes law is described. This is followed by a description of sedimentation of moderately concentrated suspensions (with 0.2 ≥ ϕ ≥ 0.1) and the effect of hydrodynamic interaction. Sedimentation of concentrated suspensions (ϕ > 0.2) and models for its description are described. Sedimentation in nonNewtonian liquids and correlation of sedimentation rate with residual (zero shear) viscosity is illustrated. The role of thickeners (rheology modifiers) in prevention of sedimentation is discussed at a fundamental level. A section is given over to the prevention of sedimentation: balance of density, reduction of particle size and use of thickeners and finely divided inert particles. The application of depletion flocculation for reduction of sedimentation is described. The use of liquid crystalline phases for reduction of sedimentation is also described.

3.2 Preparation of suspension concentrates by the bottom-up process As mentioned in the introduction, the bottom-up process for preparation of suspensions involves the formation of particles from molecular units. A good example is the preparation of particles of inorganic materials, such as silica, titania, ZnO, etc., i.e. a process of precipitation, nucleation and growth. Another example is the preparation of polymer colloids by emulsion or dispersion polymerization, which will be described in Chapter 8. As mentioned above, one of the main advantages of the bottom-up process, over the top-down process, is the possibility to control particle size and shape distribution as well as the morphology of the resulting particles [1, 2]. By controlling the process of nucleation and growth it is possible to obtain suspensions with a narrow size distribution. This is particularly important for many practical applications such as photonic materials, and semiconductor colloids. However, for other processes such as in paints and ceramic processes, a modest polydispersity can be beneficial to enhance the random packing density of the spheres. Consequently, the viscosity of the mixtures is generally below that for monodisperse spheres at the same volume fraction [8].

84 | 3 Solid/liquid dispersions (suspensions)

A very important aspect of suspensions is the maintenance of their colloid stability, i.e. absence of any flocculation. This can be achieved by three different mechanisms as described in the introduction. The precipitation method is usually applied for preparation of inorganic particles such as metal oxides and nanoparticles of metals. As an illustration, ferric oxide and aluminium oxide or hydroxide are prepared by hydrolysis of metal salts [8], 2FeCl3 + 3H2 O → F2 O3 ↓ + 6HCl, AlCl3 + 3H2 O → AlOOOH + 3HCl.

(3.1) (3.2)

Another example of preparation of particles by precipitation is that of silica sols. These can be prepared by acidification of water glass, a strongly alkaline solution of glass (that essentially consists of amorphous silica). Acidification is necessary to achieve a highly supersaturated solution of dissolved silica. Another method for obtaining high supersaturation of silica is by changing solvent, instead of changing pH. In this method a stock solution of sodium-silica solution (Na2 O.SiO2 , 27 wt% SiO2 ) is diluted with double distilled water to 0.22 wt% SiO2 . Under vigorous stirring, 0.2 ml of this water glass solution is rapidly pipetted into 10 ml of absolute ethanol. A sudden turbidity increase manifests the formation of small, smooth silica spheres with a diameter around 30 nm and a typical polydispersity of 20–30 % [1]. A third method for preparation of silica spheres is the well-known Stober method [3]. The precursor tetraethyl silicate (TES) is dissolved in an ethanol-ammonia mixture which is gently stirred in a closed vessel. Silica spheres with a radius of about 60 nm and typical polydispersity of 10–15 % are produced. An example of nanometal particles is the reduction of metal salt [8], H2 PtCl6 + BH−4 + 3H2 O → Pt↓ + 2H2 ↑ + 6HCl + H2 BO−3 .

(3.3)

To understand the process of formation of nanoparticles by the bottom-up process, we must consider the process of homogeneous precipitation at a fundaments level. If a substance becomes less soluble by a change of some parameter, such as temperature decreases, the solution may enter a metastable state by crossing the bimodal as is illustrated in the phase diagram (Fig. 3.2) of a solution which becomes supersaturated upon cooling [8]. In the metastable region, the formation of small nuclei initially increases the Gibbs free energy. Thus, demixing by nucleation is an activation process, occurring at a rate which is extremely sensitive to the precipitation in this metastable region. In contrast, when the solution is quenched into the unstable region on crossing the spinodal (see Fig. 3.2) there is no activation barrier to form a new phase.

3.2 Preparation of suspension concentrates by the bottom-up process

stable solution

Tc

| 85

nucleation and growth

T spinodal decomposition

binodal

spinodal x

Fig. 3.2: Phase diagram of a solution which becomes supersaturated upon cooling; x is the solute mole fraction and T is the temperature.

3.2.1 Nucleation and growth Classical nucleation theory considers a precipitating particle (referred to as a nucleus or cluster) to consist of a bulk phase containing N is molecules and a shell with N iσ molecules which have a higher free energy per molecule than the bulk. The particle is embedded in a solution containing dissolved molecules i. This is schematically represented in Fig. 3.3. The Gibbs free energy of the nucleus Gs is made of a bulk part and a surface part [8], Gs = μsi N is + σA, (3.4) where μsi is the chemical potential per molecule, σ is the solid/liquid interfacial tension and A is the surface area of the nucleus.

Bulk molecules

Surface molecules With higher free energy

Fig. 3.3: Schematic representation of a nucleus.

In a supersaturated solution the activity a i is higher than that of a saturated solution a i (sat). As a result, molecules are transferred from the solution to the nucleus surface. The free energy change ∆Gs upon the transfer of a small number N i from the solution to the particle is made up of two contributions from the bulk and the surface, ∆Gs = ∆Gs (bulk) + ∆Gs (surface).

(3.5)

The first term on the right-hand side of equation (3.5) is negative (it is the driving force) whereas the second term is positive (work has to be carried out in expanding the interface). ∆Gs (bulk) is determined by the relative supersaturation, whereas ∆Gs (surface) is determined by the solid/liquid interfacial tension σ and the interfacial area A which is proportional to (N is )2/3 .

86 | 3 Solid/liquid dispersions (suspensions) ∆Gs is given by the following expression, 2/3

∆Gs = −N i kT ln S + βσN i

,

(3.6)

where k is the Boltzmann constant, T is the absolute temperature and β is a proportionality constant that depends on the shape of the nucleus. S is the relative supersaturation that is equal to a i /a i (sat). For small clusters, the surface area term dominates whereas ∆Gs only starts to decrease due to the bulk term beyond a critical value N ∗ . N ∗ can be obtained by differentiating equation (3.6) with respect to N and equating the result to 0 (dGs /dN = 0), (N ∗ )1/3 =

2σβ . 3kT ln S

(3.7)

The maximum in the Gibbs energy is given by, ∆G∗ =

1 β(N ∗ )2/3 . 3

(3.8)

Equation (3.7) shows that the critical cluster size decreases with increasing the relative supersaturation S or reducing σ by addition of surfactants. This explains why a high supersaturation and/or addition of surfactants favours the formation of small particles. A large S pushes the critical cluster size N ∗ to smaller values and simultaneously lowers the activation barrier as illustrated in Fig. 3.4, which shows the variation of ∆G with radius at increasing S. ΔG

S=0 Increasing S

ΔG*

r*

r Irreversible growth

Fig. 3.4: Schematic representation of the effect of supersaturation on particle growth.

Assuming the nuclei to be spherical, equation (3.6) can be given in terms of the nucleus radius r 4 kT (3.9) ∆G = 4πr2 σ − πr3 ( ) ln S, 3 Vm where Vm is the molecular volume.

3.2 Preparation of suspension concentrates by the bottom-up process

| 87

∆G∗ and r∗ are given by, 4 π(r∗ )2 σ, 3 2Vm σ r∗ = . kT ln S

∆G∗ =

(3.10) (3.11)

When no precautions are taken, precipitation from a supersaturated solution produces polydisperse particles. This is because nucleation of new particles and further particle growth overlap in time. This overlap is the consequence of the statistical nature of the nucleation process; near the critical size, particles may grow as well as dissolve. To narrow down the particle size distribution as much as possible, nucleation should take place over a short time, followed by equal growth of a constant number of particles. This can be achieved by rapidly creating the critical supersaturation required to initiate homogeneous nucleation after which particle growth lowers the saturation sufficiently to suppress new nucleation events. Another option is to add nuclei (seeds) to a solution with subcritical supersaturation. A fortunate consequence of particle growth is that in many cases the size distribution is self-sharpening.

3.2.2 Precipitation kinetics The kinetics of precipitation in a metastable solution can be considered to follow two regimes, according to Fig. 3.4. The initial regime is that where small particles struggle with their own solubility to pass the Gibbs energy barrier ∆G∗ . This passage is called a nucleation event, which is simply defined as the capture of one molecule by a critical cluster [8], assuming that after this capture, the cluster enters the irreversible growth regime in which a new colloid is born. In this case the number I of colloids that exist per second is proportional to cm (the concentration of single unassociated molecules) and c∗ (the concentration of critical clusters), I = kcm c∗ ,

(3.12)

where k is the rate constant. Equation (3.12) predicts second-order reaction kinetics. To quantify I, one must evaluate the frequency at which molecules encounter a spherical nucleus of radius a by diffusion. This can be evaluated using Smoluchowski’s diffusion model for coagulation kinetics [8]. The diffusion flux J of molecules through any spherical envelop with radius a is given by Fick’s first law, J = 4πr2 D

dc(a) , da

(3.13)

where D is the molecular diffusion coefficient relative to the sphere positioned at the origin a = 0. Each molecule that reaches the sphere surface irreversibly attaches to the insoluble sphere and it is assumed that the concentration cm of molecules in the

88 | 3 Solid/liquid dispersions (suspensions)

liquid far away from the sphere radius remains constant [8], c(a − r) = 0;

c(a → ∞) = cm .

(3.14)

For these boundary conditions equation (3.13) becomes, c(a − r) = 0;

c(a → ∞) = cm ,

(3.15)

if it assumed that J is independent of a, i.e. if the diffusion of the molecules towards the sphere has reached a steady state. Such a state is approached by the concentration gradient around a sphere over a time of the order of r2 /D needed by the molecules to diffuse over a sphere diameter. Assuming that the sphere growth is a consequence of stationary states, one can identify the nucleation rate I as the flux J multiplied by the concentration c∗ of spheres with critical radius r∗ , I = 4πDr∗ cm c∗ [m−3 s−1 ].

(3.16)

The concentration c∗ can be evaluated by considering the reversible work to form a cluster out of N molecules using the Boltzmann distribution principle, c(N) = cm exp(

−∆G ). kT

(3.17)

c(N) represents the equilibrium concentration of clusters composed of N molecules and ∆G is the free energy of formation of a cluster. Applying this result to clusters with a critical size r∗ , we obtain on substitution in equation (3.16), I = 4πDr∗ c2m exp( ∆G∗ = (

−∆G∗ ) kT

4π ∗ 2 )(r ) σ, 3

(3.18) (3.19)

where ∆G∗ is the height of the nucleation barrier. The exponent may be identified as the probability (per particle) that a spontaneous fluctuation will produce a critical cluster. Equation (3.18) shows that the nucleation rate is very sensitive to the value of r∗ and hence to supersaturation as given by equation (3.11). The maximum nucleation rate at very large supersaturation, i.e. the pre-exponential term in equation (3.18) can be obtained by substitution of D using the Stokes–Einstein equation, D=

kT , 6πηr∗

(3.20)

where η is the viscosity of the medium. This maximum nucleation rate is given by, I≈

kT = c2m . η

(3.21)

For an aqueous solution at room temperature with a molar concentration cm , the maximum nucleation rate is of the order of 1029 m−3 s−1 . A decrease in supersaturation to

3.2 Preparation of suspension concentrates by the bottom-up process

| 89

values around S = 5 is sufficient to reduce this very high rate to practically zero. For silica precipitation in dilute, acidified water glass solutions, S ≈ 5 and nucleation may take hours or days. As mentioned above, when no precautions are taken, precipitation from a supersaturated solution produces polydisperse particles. Fortunately in many cases, the size distribution is self-sharpening. This can be illustrated by considering the colloidal spheres with radius r, which irreversibly grow by the uptake of molecules from a solution according to the following rate law [8], dr = k0 r n , dt

(3.22)

where k0 and n are constants. This growth equation leads either to spreading or sharpening of the relative distribution, depending on the value of n. Consider at a given time t any pair of spheres with arbitrary size from the population of independently growing particles. Let 1 + ε be their size ratio such that r(1 + ε) and r are the radius of the larger and smaller spheres, respectively. The former grows according to, d r(1 + ε) = kr n (1 + ε)n , dt

(3.23)

which can be combined with the growth equation (3.22) for the smaller spheres to obtain the time evolution of the size ratio, dε = k0 r n−1 [(1 + ε)n − (1 + ε)]; dt

ε ≥ 0.

(3.24)

The relative size difference ε increases with time for n > 1, in which case particle growth broadens the distribution. For n = 1, the size ratio between the spheres remains constant, whereas for n < 1 it monotonically decreases with time. Since this decrease holds for any pair of particles in the growing population, it follows that for n < 1 the relative size distribution is self-sharpening [8]. This condition is practically realistic. For example, when the growth rate is completely determined by a slow reaction of molecules at the sphere radius, dr3 = k0 r2 , dt

(3.25)

implying that dr/dt is a constant, so n = 0. The opposite limiting case is growth governed by the rate at which molecules reach a colloid by diffusion. The diffusion flux for molecules with a diffusion coefficient D, relative to a sphere centred at the origin at a = 0, is given by equation (3.13). The saturation concentration is assumed to be maintained at the particle surface, neglecting the influence of particle size on c(sat), and keeping the bulk concentration of molecules constant [8], c(a = r) = c(sat);

c(a → ∞) = c(∞).

(3.26)

For these boundary conditions, the stationary flux towards the sphere equals, J = 4πDr[c(∞) − c(sat)],

(3.27)

90 | 3 Solid/liquid dispersions (suspensions)

showing that the rate at which the colloid intercepts the diffusing molecules is proportional to its radius and not to its surface area. If every molecule contributes a volume vm to the growing colloid, then for a homogeneous sphere, the volume increases at a rate given by, d 4 3 (3.28) πr = Jvm , dt 3 which on substitution in equation (3.27) leads to, dr = Dvm [c(∞) − c(sat)]r−1 , dt

(3.29)

with the typical scaling r2 ≈ t, as expected for a diffusion controlled process. Thus the exponent in equation (3.22) for diffusion controlled growth is n = −1, and consequently the relative width of the size distribution decreases with time. For charged species, an electrostatic interaction may be present between the growing colloids and the molecules they consume. This will either enhance or retard growth, depending on whether colloids and monomers attract or repel each other. In this case the diffusion coefficient D of the monomers in the diffusion flux J has to be replaced by an effective coefficient of the form, Deff =

D ∞

r ∫r exp(−[u(a)/kT]a−2 da

,

(3.30)

where u(a) is the interaction energy between molecule and colloid. If the molecules are ions with charge ze and the colloid sphere has a surface potential ψ0 , then for low electrolyte where the interaction is unscreened (upper estimate of the ion–colloid interaction), u(a) is given by, u(a) r = u0 , kT a zeψ0 u0 = = zy0 , kT

(3.31) (3.32)

where u0 is the colloid–ion interaction energy and y0 = (eψ0 /kT). Thus the Coulombic interaction, equation (3.30) gives, Deff = D

zy0 . exp(zy0 ) − 1

(3.33)

Thus, the growth rate is slowed down exponentially by the Coulombic interaction. For example when ψ0 = 75 mV, the effective diffusion coefficient for divalent ions is about 0.01D. Addition of electrolyte screens the colloid–ion interaction and this moderates the effect of y0 on the growth kinetics. The interaction between charged monomers and the growing colloid within the approximation given by equation (3.33) does not change the growth equation (3.29) and, therefore, does not affect the conclusion that diffusional growth sharpens the size distribution.

3.2 Preparation of suspension concentrates by the bottom-up process

| 91

The effect of Ostwald ripening on the kinetics of particle growth can be analysed using the Gibbs–Kelvin [17] equation that relates the solubility c(r) of a particle with radius r to that of an infinitely large particle c(sat), i.e. a flat surface, by the equation, ln[

c(r) 2σvm . ]= ∗ c(sat) r kT

(3.34)

The increased solubility according to equation (3.34) is referred to as the Gibbs–Kelvin effect. By considering the Gibbs energy maximum of Fig. 3.4, it clear that it represents an unstable equilibrium, which can only be maintained for particles of exactly the same size. For polydisperse particles (with the same interfacial tension), there is no single, common equilibrium solubility; particles either grow or dissolve. Clearly, the largest particles have the strongest tendency to grow owing to their low solubility. This coarsening of colloids is referred to as Ostwald ripening and it is an important ageing effect which occurs with most polydisperse systems with small particles. In a polydisperse system, the bulk concentration c(bulk) is not constant, but slowly decreases over time due to the gradual disappearance of small particles. At any moment in time there is one sphere with radius r0 which is in metastable equilibrium with the bulk concentration, c(bulk) = c(sat) exp[

2σvm ], r0 kT

(3.35)

where c(sat) is the solubility of a flat surface. If the local solute concentration near a sphere with radius r i is also fixed by the Gibbs–Kelvin equation, the steady state diffusion flux for sphere I is given by, J i = 4πDr i c(sat){exp[

2σvm 2σvm ] − exp[ ]}. r0 kT r i kT

(3.36)

It is clear that particles with radii r i < r0 dissolve because J < 0, whereas for r i > r0 the particles grow. The average particle radius and the critical radius r0 increase over time, so that the exponents in the diffusion flux can be linearized at a later stage of the ripening process. In this case, one can write for the growth or dissolution rate of sphere i the following approximate equation, σv2 1 d 3 1 r i = 6Dr i c(sat) m [ − ]. dt kT r0 r i

(3.37)

One limiting case of Ostwald ripening allows for a simple analytical solution, namely monodisperse sphere with radius r, from which dissolved matter is deposited on very large particles, or a flat substrate. If that substrate controls the bulk concentration, r0 is infinitely large and consequently, σv2 dr3 = −6Dc(sat) m . dt kT Thus, for this case the particle volume decreases at a constant rate.

(3.38)

92 | 3 Solid/liquid dispersions (suspensions)

The time evolution of a continuous size distribution was analysed by Lifshitz and Slesov [18] and Wagner [19] (referred to as the LSW theory) which predicts for large times the asymptotic result, σv2 d⟨r⟩3 8 = Dc(sat) m , dt 9 kT

(3.39)

which predicts that at a late stage of the ripening process, the average particle radius increases as t1/3 . The supersaturation falls as t−1/3 and the number of spheres as t−1 . A remarkable finding of the LSW theory is that due to Ostwald ripening the size distribution approaches a certain universal, time-independent shape, irrespective of the initial distribution.

3.2.3 Seeded nucleation and growth In the above analysis it is assumed that particle nucleation and growth occur in a solution of one solute. In practice this process of homogeneous nucleation is difficult to realize due to the presence of contaminants, dust, motes and irregularities on the vessel wall. The process of heterogeneous nucleation may have a dramatic effect on the kinetics. This process may be advantageous resulting in particle size polydispersity. This process of seed nucleation was first exploited for preparation of quite monodisperse gold colloids by using a finely divided Faraday gold sol as the seed. The seed can also differ chemically from the precipitating material, leading to the formation of core-shell colloids [8]. Good examples are the growth of silica on gold cores, and other inorganic particles for the preparation of core-shell semiconductor particles [8]. Such well-defined composite colloids are increasingly important in material science, in addition to their use in fundamental studies. The efficiency of seeds or the container wall to catalyse nucleation is due to the reduction of the interfacial Gibbs energy of a precipitating particle. Steps and kinks on the seed substrate may act as active sites because they enable more of the surface of the nucleus to be in contact with the seed, which lowers its surface excess Gibbs energy.

3.2.4 Surface modification Surface modification is the deliberate attachment of a polymeric surfactant to the surface of the colloid to change its physical properties or chemical functionality. This modification is permanent if the attached polymer is not desorbed by thermal motion. Such surface modification occurs either via a chemical bond or significant adsorption energy (lack of desorption). The polymeric surfactant provides steric repulsion for the particles as was discussed in Chapter 13, Vol. 1. Surface modification is generally

3.2 Preparation of suspension concentrates by the bottom-up process

| 93

straightforward by choosing a molecule with suitable chemical linker. For example, for metal hydroxide particles, such as silica, one can use a linkage between the –OH group on the surface of the particles and a carboxylic group or alcohol. For example, under mild conditions, the surface silane groups on silica react with silane coupling agents (SCAs) and these materials are suitable for in situ modification of the colloid in a sol. The SCAs hydrolyse to reactive silanes, which graft themselves onto silica via the formation of a silixane linkage. Once reactive oligomers or polymers attach to a colloidal core, the core-shell particle behaves as one kinetic unit with an average kinetic energy of (3/2)kT (where k is the Boltzmann constant and T is the absolute temperature). This energy has to be weighed against the replacement of a large number of solvent molecules by the adsorbed species. Even a very small Gibbs energy penalty per replacement may suffice to produce aggregates that do not break apart by thermal motion. Such aggregation can also be induced by minute changes in the nature or composition of the solvent, a subtle effect that is difficult to predict or explain. Any small change in the composition involves a large number of low-molecular species, with a net enthalpy change that easily compensates the entropy loss due to aggregation of large colloids. One obvious counterexample is any solvent adsorption on modified or unmodified colloids. Water adsorption on silica is well known, but polar organic solvents such as dimethylformamide or triethylphosphate also adsorb in significant amounts on bare silica particles, often sufficient to prevent this aggregation. It should be noted that small particles also have a disadvantage since the coagulation rate is proportional to the square of the number density. For modified, stable colloids, the small particle size becomes a benefit in view of the many functional groups per gram. One attractive option is the simultaneous synthesis and modification of inorganic colloids by nucleation and growth in the presence of the modifying agent, which also influences and controls the particle size [8].

3.2.5 Other methods for preparation of suspensions by the bottom-up process Several other methods can be applied for preparation of suspensions using the bottom-up processes of which the following are worth mentioning: (i) Precipitation of particles by addition of a nonsolvent (containing a stabilizer for the particles formed) to a solution of the compound in question. (ii) Preparation of an emulsion of the substance by using a solvent in which it is soluble following emulsification of the solvent in another immiscible solvent. This is then followed by removal of the solvent making the emulsion droplets by evaporation. (iii) Preparation of the particles by mixing two microemulsions containing two chemicals that react together when the microemulsion droplets collide with each other. (iv) Sol-gel processes, particularly used for preparation of silica particles.

94 | 3 Solid/liquid dispersions (suspensions)

(v) Production of polymer suspensions by emulsion or suspension polymerization. (vi) Preparation of polymer suspensions by polymerization of microemulsions. Below a brief description of each process is given.

3.2.5.1 Solvent–antisolvent method [8] In this method, the substance (e.g. a hydrophobic drug) is dissolved in a suitable solvent such as acetone. The resulting solution is carefully added to another miscible solvent in which the resulting compound is insoluble. This results in precipitation of the compound by nucleation and growth. The particle size distribution is controlled by using a polymeric surfactant that is strongly adsorbed on the particle surface, providing an effective repulsive barrier to prevent aggregation of the particles. The polymeric surfactant is chosen to have specific adsorption on the particle surface to prevent Ostwald ripening. This method can be adapted for preparation of low water solubility drug suspensions. In this case the drug is dissolved in acetone and the resulting solution is added to an aqueous solution of Poloxamer (an A–B–A block copolymer consisting of two A polyethylene oxide (PEO) chains and a B polypropylene oxide (PPO) chain, i.e. PEO–PPO–PEO). After precipitation of the particles, the acetone is removed by evaporation. The main problem with this method is the possibility of formation of several unstable polymorphs that will undergo crystal growth. In addition, the resulting particles may be of needle shape structure. However, by proper choice of the polymeric surfactant one can control the particle morphology and shape. Another problem may be the lack of removal of the solvent after precipitation of the particles.

3.2.5.2 Use of an emulsion In this case the compound is dissolved in a volatile organic solvent that is immiscible with water, such as methylene dichloride. The oil solution is emulsified in water using a high speed stirrer followed by high pressure homogenization [8]. A suitable emulsifier for the oil phase is used which has the same HLB number as the oil. The volatile oil in the resulting emulsion is removed by evaporation and the formed suspension particles are stabilized against aggregation by the use of an effective polymeric surfactant that could be dissolved in the aqueous phase. The main problem with this technique is the possible interaction with the emulsifier, which may result in destabilization of the resulting suspension. However, by careful selection of the emulsifier/stabilizing system one can form a colloidally stable nanosuspension.

3.2.5.3 Preparation of suspensions by mixing two microemulsions [8] Reverse microemulsions lend themselves as suitable “nonreactors” for the synthesis of particles. Inorganic salts can be dissolved in the water pools of a W/O microemul-

3.2 Preparation of suspension concentrates by the bottom-up process

| 95

sion. Another W/O microemulsion with reducing agent dissolved in the water pools is then prepared. The two microemulsions are then mixed and the reaction between the inorganic salt and the reducing agent starts at the interface and proceeds towards the centre of the droplet. The rate limiting step appears to be the droplet diffusion. Control of the exchange can be achieved by tuning the film rigidity. This procedure has been applied for the preparation of noble metal particles that could be applied in electronics, catalysis and in potential medical applications.

3.2.5.4 Sol-gel process This method is particularly applicable for preparation of silica particles [8]. This involves the development of networks through an arrangement of colloidal suspension (sol) and gelation to form a system in continuous liquid phase (gel). A sol is basically a dispersion of colloidal particles (1–100 nm) in a liquid and a gel is an interconnected rigid network with pores of submicron dimensions and polymeric chains. The sol-gel process, depending on the nature of the precursors, may be divided into two classes; namely inorganic precursors (chlorides, nitrates, sulphides, etc.) and alkoxide precursors. Extensively used precursors include tetramethyl silane and tetraethoxysilane. In this process, the reaction of metal alkoxides and water, in the presence of acid or base, forms a one phase solution that goes through a solution-to-gel transition to form a rigid, two phase system comprised of metal oxides and solvent-filled pores. The physical and electrochemical properties of the resultant materials largely depend on the type of catalyst used in the reaction. In the case of silica alkoxides, the acid catalysed reaction results in weakly crosslinked linear polymers. These polymers entangle and form additional branches leading to gelation. In base-catalysed reactions, due to rapid hydrolysis and condensation of the alkoxide silanes, the system forms highly branched clusters. The difference in cluster formation is due to the solubility of the resulting metal oxide in the reaction medium. The solubility of the silicon oxide is greater in alkaline medium than in acidic medium, which favours the interlinking of the silica clusters. A general procedure for sol-gel includes four stages; namely hydrolysis, condensation, growth and aggregation. The complete hydrolysis to form M(OH)4 is very difficult to achieve. Instead, condensation may occur between two –OH or M–OH groups and an alkoxy group to form bridging oxygen and a water or alcohol molecule. The hydrolysis and polycondensation reactions are initiated at numerous sites and the kinetics of the reaction can be very complex. When a sufficient number of interconnected M–O–M bonds are formed in a particular region, they interact cooperatively to form colloidal particles or a sol. With time, the colloidal particles link together to form three-dimensional networks. The size, shape and morphological features of the silica nanoparticles can be controlled by reaction kinetics, use of templates such as cationic, nonionic surfactants, polymers, electrolytes, etc.

96 | 3 Solid/liquid dispersions (suspensions)

3.2.5.5 Preparation of polymer particles by emulsion or suspension polymerization [8] These methods will be described in detail in Chapter 8.

3.3 Preparation of suspensions using the top-down process As mentioned in the introduction, in the top-down process one starts with the bulk material (which may consist of aggregates and agglomerates) that is dispersed into single particles (using a wetting/dispersing agent) using high speed stirrers followed by subdivision of the large particles into smaller units that fall within the required size range [1–3]. This process requires the application of intense mechanical energy that can be applied using bead milling. Finally, the resulting suspension must remain colloidally stable under all conditions (such as temperature changes, vibration, etc.) with absence of any flocculation and/or crystal growth. A schematic representation of the dispersion process is shown in Fig. 3.5. This process requires wetting of the aggregates and agglomerates (both external and internal surfaces) by molecules of the dispersion medium. This is particularly the case with hydrophobic solids dispersed in aqueous media. As will be discussed below, wetting of hydrophobic solids in aqueous media requires the presence of a wetting agent (surfactant) that lowers the surface tension of water and adsorbs very quickly at the solid/ liquid interface, thus reducing the solid/liquid interfacial tension. Once the powder is completely wetted, the aggregates and agglomerates are dispersed into single particles using high speed stirrers and addition of a dispersing agent (surfactant and/or polymer). The resulting dispersion of single particles (referred to as the “mill base”) is then subjected to a comminution (milling or particle size reduction) mostly achieved using bead mills. The final suspension must remain colloidally stable using electrostatic and/or steric stabilization as will be briefly described below. Liquid + Dispersing agent

Agglomerates (particles connected by their corners)

Aggregates (particles joined at their faces)

Wet milling

Communication Stabilisation to prevent aggregation

Fine dispersion in the range 20 – 200 nm depending on application

Fig. 3.5: Schematic representation of the dispersion process.

3.3 Preparation of suspensions using the top-down process | 97

3.3.1 Wetting of the bulk powder As discussed in Chapter 2, wetting of powders is a prerequisite for the dispersion of powders in liquids. Most chemicals are supplied as powders consisting of aggregates where the particles are joined together with their “faces” (compact structures), or agglomerates where the particles are connected at their corners (loose aggregates) as illustrated in Fig. 3.5. It is essential to wet both the external and internal surface (in the pores within the aggregate or agglomerate structures) and this requires the use of an effective wetting agent (surfactant) [1–3]. Wetting of a solid by a liquid (such as water) requires the replacement of the solid/vapour interfacial tension, γSV , by the solid/liquid interfacial tension, γSL . As discussed in Chapter 2, the equilibrium aspects of wetting can be studied at a fundamental level using interfacial thermodynamics. A useful parameter to describe wetting is the contact angle θ of a liquid drop on a solid substrate [8], which is the angle between planes tangent to the surfaces of solid and liquid at the wetting perimeter (see Chapter 2). If the liquid makes no contact with the solid, i.e. θ = 180°, the solid is referred to as non-wettable by the liquid in question. This may be the case for a perfectly hydrophobic surface with a polar liquid such as water. However, when 180° > θ > 90°, one may refer to a case of poor wetting. When 0° < θ < 90°, partial (incomplete) wetting is the case, whereas when θ = 0° complete wetting occurs and the liquid spreads on the solid substrate forming a uniform liquid film. The utility of contact angle measurements depends on equilibrium thermodynamic arguments (static measurements) using the well-known Young’s equation [8]. The value depends on: (i) the history of the system; (ii) whether the liquid is tending to advance across or recede from the solid surface (advancing angle θ A , receding angle θR ; usually θA > θR ). Under equilibrium, the liquid drop takes the shape that minimizes the free energy of the system. Three interfacial tensions can be identified: γSV , solid/vapour area ASV ; γSL , solid/liquid area ASL ; γLV , liquid/vapour area ALV . A schematic representation of the balance of tensions at the solid/liquid/vapour interface is shown in Fig. 3.6. Here, solid and liquid are simultaneously in contact with each other and the surrounding phase (air or vapour of the liquid). The wetting perimeter is referred to as the threephase line or wetting line. In this region there is an equilibrium between vapour, liquid and solid. γSV ASV + γSL ASL + γLV ALV should be a minimum at equilibrium and this leads to the well-known Young’s equation [8], γSV = γSL + γLV cos θ, γSV − γSL cos θ = γLV

(3.40) (3.41)

98 | 3 Solid/liquid dispersions (suspensions)

γLV cos Θ

Θ γSV

γSL

Wetting line Fig. 3.6: Schematic representation of the contact angle and wetting line.

The contact angle θ depends on the balance between the solid/vapour (γSV ) and solid/ liquid (γSL ) interfacial tensions. The angle which a drop assumes on a solid surface is the result of the balance between the adhesion force between solid and liquid and the cohesive force in the liquid, γLV cos θ = γSV − γSL .

(3.42)

As mentioned in Chapter 2, wetting of a powder is achieved by the use of surface active agents (wetting agents) of the ionic or nonionic type which are capable of diffusing quickly (i.e. lower the dynamic surface tension) to the solid/liquid interface and displacing the entrapped air by rapid penetration through the channels between the particles and inside any “capillaries”. For wetting of hydrophobic powders in water, anionic surfactants, e.g. alkyl sulphates or sulphonates or nonionic surfactants of the alcohol ethoxylates are usually used [1–3]. A useful concept for choosing wetting agents of the ethoxylated surfactants is the hydrophilic-lipophilic balance (HLB) concept, HLB =

% of hydrophilic groups . 5

(3.43)

Most wetting agents of this class have an HLB number in the range 7–9. As discussed in Chapter 2, the process of wetting a solid of unit surface area by a liquid involves three types of wetting [1–3]: adhesion wetting, Wa ; immersion wetting Wi ; spreading wetting Ws . In every step one can apply Young’s equation, Wa = γSL − (γSV + γLV ) = −γLV (cos θ + 1),

(3.44)

Wi = 4γSL − 4γSV = −4γLV cos θ,

(3.45)

Ws = (γSL + γLV ) − γSV = −γLV (cos θ − 1).

(3.46)

The work of dispersion of a solid with unit surface area Wd is the sum of Wa , Wi and Ws , Wd = Wa + Wi + Ws = 6γSV − γSL = −6γLV cos θ. (3.47)

3.3 Preparation of suspensions using the top-down process | 99

Wetting and dispersion depend on: γLV , liquid surface tension; θ, contact angle between liquid and solid. Wa , Wi and Ws are spontaneous when θ < 90°. Wd is spontaneous when θ = 0. Since surfactants are added in sufficient amounts (γdynamic is lowered sufficiently) spontaneous dispersion is the rule rather than the exception. The work of dispersion of a powder with surface area A, Wd , is given by [1–3], Wd = A(γSL − γSV ).

(3.48)

γSV = γSL + γLV cos θ,

(3.49)

Using Young’s equation, where γLV is the liquid/vapour interfacial tension and θ is the contact angle of the liquid drop at the wetting line. Wd = −AγLV cos θ.

(3.50)

Equation (3.50) shows that Wd depends on γLV and θ, both of which are lowered by addition of surfactants (wetting agents). If θ < 90°, Wd is negative and dispersion is spontaneous. Wetting of the internal surface requires penetration of the liquid into channels between and inside the agglomerates. The process is similar to forcing a liquid through fine capillaries. To force a liquid through a capillary with radius r, a pressure p is required that is given by, p=−

2γLV cos θ −2(γSV − γSL ) =[ ]. r rγLV

(3.51)

γSL has to be made as small as possible; rapid surfactant adsorption to the solid surface, low θ. When θ = 0, p ∝ γLV . Thus for penetration into pores one requires a high γLV . Thus, wetting of the external surface requires low contact angle θ and low surface tension γLV . Wetting of the internal surface (i.e. penetration through pores) requires low θ but high γLV . These two conditions are incompatible and a compromise has to be made: γSV − γSL must be kept at a maximum. γLV should be kept as low as possible but not too low. The above conclusions illustrate the problem of choosing the best dispersing agent for a particular powder. This requires the measurement of the above parameters as well as testing the efficiency of the dispersion process. The contact angle of liquids on solid powders can be measured by application of the Rideal–Washburn equation. For horizontal capillaries (gravity neglected), the depth of penetration l in time t is given by the Rideal–Washburn equation [8], l=[

rtγLV cos θ 1/2 ] . 2η

(3.52)

To enhance the rate of penetration, γLV has to be made as high as possible, θ as low as possible and η as low as possible. For dispersion of powders into liquids one should

100 | 3 Solid/liquid dispersions (suspensions)

use surfactants that lower θ while not reducing γLV too much. The viscosity of the liquid should also be kept at a minimum. Thickening agents (such as polymers) should not be added during the dispersion process. It is also necessary to avoid foam formation during the dispersion process. For a packed bed of particles, r may be replaced by K(= r/k2 ), which contains the effective radius of the bed r and a tortuosity factor k, which takes into account the complex path formed by the channels between the particles, i.e., l2 =

ktγLV cos θ . 2η

(3.53)

Thus a plot of l2 versus t gives a straight line and from the slope of the line one can obtain θ. The Rideal–Washburn equation can be applied to obtain the contact angle of liquids (and surfactant solutions) in powder beds. K should first be obtained using a liquid that produces zero contact angle. A packed bed of powder is prepared, say in a tube fitted with a sintered glass at the end (to retain the powder particles). It is essential to pack the powder uniformly in the tube (a plunger may be used in this case). The tube containing the bed is immersed in a liquid that gives spontaneous wetting (e.g. a lower alkane), i.e. the liquid gives a zero contact angle and cos θ = 1. By measuring the rate of penetration of the liquid (this can be carried out gravimetrically using for example a microbalance or a Kruss instrument) one can obtain K. The tube is then removed from the lower alkane liquid and left to stand for evaporation of the liquid. It is then immersed in the liquid in question and the rate of penetration is measured again as a function of time. Using equation (3.53), one can calculate cos θ and hence θ. For efficient wetting of hydrophobic solids in water, a surfactant is needed that lowers the surface tension of water very rapidly (within few ms) and quickly adsorbs at the solid/liquid interface [1–3]. To achieve rapid adsorption, the wetting agent should be either a branched chain with central hydrophilic group or a short hydrophobic chain with hydrophilic end group. As mentioned in Chapter 2, the most commonly used wetting agents are the following: Aerosol OT (diethylhexyl sulphosuccinate) C2H5

O

C4H9CHCH2―O―C ―CH―SO3Na C4H9CHCH2―O―C ―CH2 C2H5

O

The above molecule has a low critical micelle concentration (cmc) of 0.7 g dm−3 and at and above the cmc the water surface tension is reduced to ≈ 25 mN m−1 in less than 15 s. Several nonionic surfactants such as the alcohol ethoxylates can also be used as wetting agents. These molecules consist of a short hydrophobic chain (mostly C10 )

3.3 Preparation of suspensions using the top-down process | 101

which is also branched. A medium chain polyethylene oxide (PEO) mostly consisting of 6 EO units or lower is used. These molecules also reduce the dynamic surface tension within a short time (< 20 s) and they have reasonably low cmc. In all cases, to avoid interference with the dispersant one should use the minimum amount of wetting agent that needs to be added to maintain the colloid stability during dispersion and on storage.

3.3.2 Breaking of aggregates and agglomerates into individual units This usually requires the application of mechanical energy. High speed mixers (which produce turbulent flow) of the rotor-stator type [8] are efficient in breaking up the aggregates and agglomerates, e.g. Silverson mixers, Ultra-Turrax. These are the most commonly used mixers for dispersion of powders in liquids. Two main types are available. The most commonly used toothed device (schematically illustrated in Fig. 3.7) is the Ultra-Turrax (IKA works, Germany).

Fig. 3.7: Schematic representation of a toothed mixer (Ultra-Turrax).

Toothed devices are available both as in-line as well as batch mixers, and because of their open structure they have a relatively good pumping capacity. Therefore, in batch applications they frequently do not need an additional impeller to induce bulk flow even in relatively large mixing vessels. Batch radial discharge mixers such as Silverson mixers (Fig. 3.8) have a relatively simple design with a rotor equipped with four blades pumping the fluid through a stationary stator perforated with differently shaped/sized holes or slots. They are frequently supplied with a set of easily interchangeable stators enabling the same machine to be used for a range of operations e.g. blending, particle size

102 | 3 Solid/liquid dispersions (suspensions)

Fig. 3.8: Schematic representation of batch radial discharge mixer (Silverson mixer).

reduction and de-agglomeration. Changing from one screen to another is quick and simple. Different stators/screens used in batch Silverson mixers are shown in Fig. 3.9. The general purpose disintegrating stator (Fig. 3.9 (a)) is recommended for preparation of thick suspensions (“gels”) whilst the slotted disintegrating stator (Fig. 3.9 (b)) is designed for suspensions containing elastic materials such as polymers. Square hole screens (Fig. 3.9 (c)) are recommended for the preparation of suspensions whereas the standard screen (Fig. 3.9 (d)) is used for solid/liquid dispersion.

(a)

(b)

(c)

(d)

Fig. 3.9: Stators used in batch Silverson radial discharge mixers.

In all methods, there is liquid flow; unbounded and strongly confined flow. In unbounded flow any particle is surrounded by a large amount of flowing liquid (the confining walls of the apparatus are far away from most of the particles). The forces can be frictional (mostly viscous) or inertial. Viscous forces cause shear stresses to act on the interface between the particles and the continuous phase (primarily in the direction of the interface). The shear stresses can be generated by laminar flow (LV) or turbulent flow (TV); this depends on the dimensionless Reynolds numbers Re, Re =

vlρ , η

(3.54)

where v is the linear liquid velocity, ρ is the liquid density and η is its viscosity. l is a characteristic length that is given by the diameter of flow through a cylindrical tube and by twice the slit width in a narrow slit.

3.3 Preparation of suspensions using the top-down process | 103

For laminar flow Re ⪅ 1000, whereas for turbulent flow Re ⪆ 2000. Thus whether the regime is linear or turbulent depends on the scale of the apparatus, the flow rate and the liquid viscosity [1–3]. Batch toothed and radial discharge rotor-stator mixers are manufactured in different sizes ranging from the laboratory to the industrial scale. In lab applications mixing heads (assembly of rotor and stator) can be as small as 0.01 m (Ultra-Turrax, Silverson) and the volume of processed fluid can vary from several millilitres to few litres. In models used in industrial applications, mixing heads might have up to 0.5 m diameter enabling processing of several cubic metres of fluids in one batch. In practical applications the selection of the rotor-stator mixer for a specific dispersion process depends on the required morphology of the product, frequently quantified in terms of average particle size or in terms of particle size distributions, and by the scale of the process. The selection of an appropriate mixer and processing conditions for a required formulation is frequently carried out by trial and error. Initially, one can carry out lab scale dispersion of given formulations testing different type/ geometries of mixers they manufacture. Once the type of mixer and its operating parameters are determined at the lab scale the process needs to be scaled up. The majority of lab tests of dispersion are carried out in small batch vessels as this is easier and cheaper than running continuous processes. Therefore, prior to scaling up of the rotor-stator mixer it has to be decided whether industrial dispersion should be run as a batch or as a continuous process. Batch mixers are recommended for processes where formulation of a product requires long processing times typically associated with slow chemical reactions. They require simple control systems, but spatial homogeneity may be an issue in large vessels which could lead to a longer processing time. In processes where quality of the product is controlled by mechanical/hydrodynamic interactions between continuous and dispersed phases or by fast chemical reactions, but large amounts of energy are necessary to ensure adequate mixing, in-line rotorstator mixers are recommended. In-line mixers are also recommended to efficiently process large volumes of fluid. In the case of batch processing, rotor-stator devices immersed as top entry mixers are mechanically the simplest arrangement but in some processes bottom entry mixers ensure better bulk mixing; however in this case sealing is more complex. In general, the efficiency of batch rotor-stator mixers decreases as the vessel size increases and as the viscosity of the processed fluid increases because of limited bulk mixing by rotor-stator mixers. Whilst the open structure of Ultra-Turrax mixers frequently enables sufficient bulk mixing even in relatively large vessels, if the suspension has a low apparent viscosity, processing of very viscous suspensions requires an additional impeller (typically anchor type) to induce bulk flow and to circulate the dispersion through the rotor-stator mixer. On the other hand, batch Silverson rotor-stator mixers have a very limited pumping capacity and even at the lab scale they are mounted off the centre of the vessel to improve bulk mixing. At the large scale there is always a

104 | 3 Solid/liquid dispersions (suspensions)

need for at least one additional impeller and in the case of very large units more than one impeller is mounted on the same shaft. Problems associated with the application of batch rotor-stator mixers for processing large volumes of fluid discussed above can be avoided by replacing batch mixers with in-line (continuous) mixers. There are many designs offered by different suppliers (Silverson, IKA, etc.) and the main differences are related to the geometry of the rotors and stators with stators and rotors designed for different applications. The main difference between batch and in-line rotor-stator mixers is that the latter have a strong pumping capacity, therefore they are mounted directly in the pipeline. One of the main advantages of in-line over batch mixers is that for the same power duty, a much smaller mixer is required, therefore they are better suited for processing of large volumes of fluid. When the scale of the processing vessel increases, a point is reached where it is more efficient to use an in-line rotor-stator mixer rather than a batch mixer of a large diameter. Because power consumption increases sharply with rotor diameter (to the fifth power) an excessively large motor is necessary at large scales. This transition point depends on the fluid rheology, but for a fluid with a viscosity similar to water, it is recommended to change from a batch to an in-line rotor-stator process at a volume of approximately 1 to 1.5 tonnes. The majority of manufacturers supply both single and multistage mixers for the emulsification of highly viscous liquids. As mentioned above in all methods, there is liquid flow, unbounded and strongly confined flow. In unbounded flow any particle is surrounded by a large amount of flowing liquid (the confining walls of the apparatus are far away from most of the droplets); the forces can be frictional (mostly viscous) or Inertial. Viscous forces cause shear stresses to act on the interface between the particles and the continuous phase (primarily in the direction of the interface). The shear stresses can be generated by laminar flow (LV) or turbulent flow (TV); this depends on the Reynolds number Re as given by equation (3.55). For laminar flow Re ⪅ 1000, whereas for turbulent flow Re ⪆ 2000. Thus whether the regime is linear or turbulent depends on the scale of the apparatus, the flow rate and the liquid viscosity. If the turbulent eddies are much larger than the particles, they exert shear stresses on the particles. If the turbulent eddies are much smaller than the particles, inertial forces will cause disruption (TI). In bounded flow other relations hold; if the smallest dimension of the part of the apparatus in which the particles are disrupted (say a slit) is comparable to particle size, other relations hold (the flow is always laminar). Within each regime, an essential variable is the intensity of the acting forces; the viscous stress during laminar flow σviscous is given by, σviscous = ηG, where G is the velocity gradient.

(3.55)

3.3 Preparation of suspensions using the top-down process | 105

The intensity in turbulent flow is expressed by the power density ε (the amount of energy dissipated per unit volume per unit time); for turbulent flow, ε = ηG2 .

(3.56)

The most important regimes are: laminar/viscous (LV) – turbulent/viscous (TV) – turbulent/inertial (TI). For water as the continuous phase, the regime is always TI. For higher viscosity of the continuous phase (ηC = 0.1 Pa s), the regime is TV. For still higher viscosity or a small apparatus (small l), the regime is LV. For very small apparatus (as is the case with most laboratory homogenizers), the regime is nearly always LV. The mixing conditions have to be optimized: Heat generation at high stirring speeds must be avoided. This is particularly the case when the viscosity of the resulting dispersion increases during dispersion (note that the energy dissipation as heat is given by the product of the square of the shear rate and the viscosity of the suspension). One should avoid foam formation during dispersion; proper choice of the dispersing agent is essential and antifoams (silicones) may be applied during the dispersion process. Rotor-stator mixers can be characterized as energy-intensive mixing devices. The main feature of these mixers is their ability to focus high energy/shear in a small volume of fluid. They consist of a high speed rotor enclosed in a stator, with the gap between them ranging from 100 to 3000 µm. Typically, the rotor speed is between 10 and 50 m s−1 , which, in combination with a small gap, generates very high shear rates. By operating at high speed, the rotor-stator mixers can significantly reduce the processing time. In terms of energy consumption per unit mass of product, the rotor-stator mixers require high power input over a relatively short time. However, as the energy is uniformly delivered and dissipated in a relatively small volume, each element of the fluid is exposed to a similar intensity of processing. Frequently, the quality of the final product is strongly affected by its structure/morphology and it is essential that the key ingredients are uniformly distributed throughout the whole mixer volume. The most common application of rotor-stator mixers is in dispersion of powders in liquids and they are used in the manufacture of particle-based products with sizes between 1 and 20 µm, e.g. in pharmaceuticals, paints, agrochemicals and cosmetics. As mentioned above, there are a wide range of designs of rotor-stator mixers, of which the Ultra-Turrax (IKA works, Germany) and Silverson (UK) are the most commonly used. They are broadly classified according to their mode of operation such as batch or in-line (continuous) mixers. In-line radial-discharge mixers are characterized by high throughput and good pumping capacity at low energy consumption. The disperse phase can be injected directly into the high shear/turbulent zone, where mixing is much faster than by injection into the pipe or into the holding tank. They are used for manufacturing very fine solid particles of relatively narrow dispersed size distribution. They are typically supplied with a range of interchangeable screens, making them reliable and versatile in different applications. Toothed devices are available as in-line as well as batch mixers. Due to their open structure they have a relatively good

106 | 3 Solid/liquid dispersions (suspensions)

pumping capacity and they frequently do not need an additional impeller to induce bulk flow even in relatively large vessels. In rotor-stator mixers, both shear rate in laminar flow and energy dissipate flow depend on the position inside the mixer. In laminar flow in stirred vessels, the average shear rate is proportional to the rotor speed N with the proportionality constant K dependent on the type of the impeller [8], γ̇ = KN.

(3.57)

In stirred vessels the proportionality constant cannot be calculated and has to be determined experimentally. In rotor-stator mixers, the average shear rate in the gap between the rotor and stator can calculated if the rotor speed and geometry of the mixer are known, πDN (3.58) γ̇ = = K1 N, δ where D is the outer rotor diameter and δ is the rotor-stator gap width. The average energy dissipation rate ε in turbulent flow in rotor-stator mixers can be calculated from [8], P ε= , (3.59) ρc V where P is the power draw, V is the swept rotor volume and ρc is the continuous phase density. The power draw in batch rotor-stator mixers is calculated in the same way as in stirred vessels, P = P0 ρc N 3 D5 , (3.60) where P0 is the power number constant for in-line rotor-stator mixers zero flow. The power draw in in-line rotor-stator mixers in turbulent flow is given by, P = P0z ρc N 3 D5 + k1 MN 2 D2 + PL ,

(3.61)

where M is the mass flow rate and PL is the power loss term. The first term in equation (3.62) is analogous to power consumption in a batch rotor-stator mixer and the second term takes into account the effect of pumping action on total power consumption. The third term accounts for mechanical losses and is typically a few percent, and therefore can be ignored. While in turbulent flow, P0z in equation (3.61) is approximately independent of the Reynolds number Re, in laminar flow there is a strong dependency of power number on Re and in this case the power draw can be calculated from, P = k0 N 2 D3 ηc + k1 MN 2 D2 + PL

(3.62)

where ηc is the viscosity of the continuous phase and k0 is a constant that depends on the Reynolds number Re, k0 = P0z Re. (3.63) From equations (3.58)–(3.61), the average energy dissipation rate in the rotor-stator mixer can be calculated.

3.3 Preparation of suspensions using the top-down process | 107

3.3.3 Wet milling or comminution The primary dispersion (sometimes referred to as the mill base) may then be subjected to a bead milling process to produce nanoparticles. Subdivision of the primary particles into much smaller units (< 1 µm) requires application of intense energy. In some cases high pressure homogenizers (such as the Microfluidizer, USA) may be sufficient to produce small particles. This is particularly the case with many drugs. In some cases, the high pressure homogenizer is combined with application of ultrasound to produce the small particles [8]. It has been shown that high pressure homogenization is a simple technique, well established on a large scale for the production of fine suspensions and already available in the pharmaceutical industry. High pressure homogenization is also an efficient technique that has been utilized to prepare stable suspensions of several drugs such as carbazepin, bupravaquone, aphidicolin, cyclosporine, paclitaxel, prednisolone, etc. During homogenization, cavitation forces as well as collision and shear forces determine breakdown of the drug particles down to the nanometre range. Process conditions lead to an average particle size that remains constant as a result of continuous fragmentation and reaggregation processes. These high energetic forces can also induce a change of crystal structure and/or partial or total amorphization of the sample, which further enhances the solubility. For long-term storage stability of the nanosuspension formulation, the crystal structure modification must be maintained over the storage time. Microfluidization is a milling technique which results in minimal product contamination. Besides minimal contamination, this technique can be easily scaled up. In this method a sample dispersion containing large particles is made to pass through specially designed interaction chambers at high pressure. The specialized geometry of the chambers along with the high pressure causes the liquid stream to reach extremely high velocities and these streams then impinge against each other and against the walls of the chamber resulting in particle size reduction. The shear forces developed at high velocities due to attrition of particles against one another and against the chamber walls, as well as the cavitation fields generated inside the chamber are the main mechanisms of particle size reduction with this technique. In the interaction chambers the liquid feed is divided into two parts which are then made to impinge against each other and against the walls of the chambers. Particle size reduction occurs due to attrition between the particles and against the chamber walls at high velocities. Cavitation fields generated inside the chambers also contribute to particle size reduction [8]. The process of microfluidization for the preparation of suspensions varies in a complex way with the various critical processes and formulation parameters. Milling time, microfluidization pressure, stabilizer type, processing temperature and stabilizer concentration were identified as critical parameters affecting the formation of stable particles. Both ionic as well as steric stabilization were effective in stabiliz-

108 | 3 Solid/liquid dispersions (suspensions)

ing the suspensions. Microfluidization and precipitation under sonication can also be used for suspension preparation. The extreme transient conditions generated in the vicinity and within the collapsing cavitational bubbles have been used for the size reduction of the material to the nanoscale. Particles synthesis techniques include sonochemical processing and cavitation processing. In sonochemistry, an acoustic cavitation process can generate a transient localized hot zone with extremely high temperature gradient and pressure. Such sudden changes in temperature and pressure assist the destruction of the sonochemical precursor and the formation of nanoparticles [8]. A dimensionless number known as cavitation number (Cv) is used to relate the flow conditions to the cavitation intensity [8], Cv =

(P2 − Pv ) , (0.5ρV02 )

(3.64)

where P2 is the recovered downstream pressure; Pv is the vapour pressure of the liquid, ρ is the density of dispersed media and V0 is the liquid velocity at the orifice. The cavitation number at which the inception of cavitation occurs is known as the cavitation inception number Cvi . Ideally speaking, the cavitation inception should occur at 1.0. It was also reported that generally the inception of cavitation occurs from 1.0 to 2.5. This has been attributed to the presence of the dissolved gases in the flowing liquid. Cv is a function of the flow geometry and usually increases with an increase in the size of the opening in a constriction such as an orifice in a flow. Cavitation can be used, for example, for the formation of the iron oxide particles. Iron precursor, either as a neat liquid or in a decalin solution, was sonicated and this produced 10–20 nm sized amorphous iron particles. Similar experiments have been reported for the synthesis of the particles of many other inorganic materials using acoustic cavitation. To understand the mechanism of the formation of the particles during the cavitation phenomenon, the hotspot theory has been successfully applied. It explains the adiabatic collapse of a bubble, producing the hotspots. This theory claims that very high temperatures (5000–25 000 K) are obtained upon the collapse of the bubble. Since this collapse occurs in few microseconds, very high cooling rates have been obtained. These high cooling rates hinder the organization and crystallization of the products. While the explanation for the creation of amorphous products is well understood, the reason for the formation of nanostructured products under cavitation is not yet clear. The products are sometimes nanoamorphous particles, and in other cases, nanocrystalline. This depends on the temperature in the fluid ring region where the reaction takes place. The temperature in this liquid ring is lower than that inside the collapsing bubble, but higher than the temperature of the bulk liquid. In summary, in sonochemical reactions leading to inorganic products, nanomaterials have been obtained. They vary in size, shape, structure, and in their solid phase (amorphous or crystalline), but they were always of nanometre size. Cavitation being a nucleus dominated (statistical in nature) phenomenon, such variations are

3.3 Preparation of suspensions using the top-down process | 109

expected. In hydrodynamic cavitation, nanoparticles are generated through the creation and release of gas bubbles inside the sol-gel solution. By rapidly pressurizing in a supercritical drying chamber and exposing it to the cavitational disturbance and high temperature heating, the sol-gel solution is rapidly mixed. The erupting hydrodynamically generated cavitating bubbles are responsible for the nucleation, the growth of the nanoparticles, and also for their quenching to the bulk operating temperature. Particle size can be controlled by adjusting the pressure and the solution retention time in the cavitation chamber. Cavitation methods can be used to reduce the size of the rubber latex particles (styrene butadiene rubber, SBR), present in the form of aqueous suspension with micrometre particle initial size, to the nanoscale [8]. An alternative method of size reduction to produce nanoparticles, that is commonly used in many industrial applications, is wet milling, also referred to as comminution (the generic term for size reduction). Comminution is a complex process and there is little fundamental information on its mechanism. For the breakdown of single crystals or particles into smaller units, mechanical energy is required. This energy in a bead mill is supplied by impaction of the glass or ceramic beads with the particles. As a result, permanent deformation of the particles and crack initiation occur. This will eventually lead to the fracture of particles into smaller units. Since the milling conditions are random, some particles receive impacts far in excess of those required for fracture whereas others receive impacts that are insufficient for the fracture process. This makes the milling operation grossly inefficient and only a small fraction of the applied energy is used in comminution. The rest of the energy is dissipated as heat, vibration, sound, interparticulate friction, etc. The role of surfactants and dispersants in grinding efficiency is far from being understood. In most cases the choice of surfactants and dispersant is made by trial and error until a system is found that gives the maximum grinding efficiency. Rehbinder and his collaborators [10] investigated the role of surfactants in the grinding process. As a result of surfactant adsorption at the solid/liquid interface, the surface energy at the boundary is reduced and this facilitates the process of deformation or destruction. The adsorption of surfactants at the solid/liquid interface in cracks facilitates their propagation. This mechanism is referred to as the Rehbinder effect. Several factors affect the efficiency of dispersion and milling [8]: (i) the volume concentration of dispersed particles (i.e. the volume fraction); (ii) the nature of the wetting/dispersing agent; (iii) the concentration of wetter/dispersant (which determines the adsorption characteristics). For optimization of the dispersion/milling process the above parameters need to be systematically investigated. From the wetting performance of a surfactant, that can be evaluated using contact angle measurements, one can establish the nature and concentration of the wetting agent. The nature and concentration of the required dispersing agent are determined by adsorption isotherm and rheological measurements.

110 | 3 Solid/liquid dispersions (suspensions)

Once the concentration of wetting/dispersing agent is established dispersions are prepared at various volume fractions keeping the ratio of wetting/dispersing agent to the solid content constant. Each system is then subjected to the dispersion/milling process keeping all parameters constant: (i) speed of the stirrer (normally one starts at lower speed and gradually increases the speed in increments at fixed time); (ii) Volume and size of beads relative to the volume of the dispersion (an optimum value is required); (iii) speed of the mill. The change of average particle size with grinding time is established using for example the Mastersizer (Malvern, UK). Fig. 3.10 shows a schematic representation of the reduction of particle size with grinding time in minutes using a typical bead mill (see below) at various volume fractions. 5.0

Particle size/μm

4.0 3.0 Low ϕ 2.0 1.0 Optimum ϕ 0.5 High ϕ 0.0 20

40

60

80

100

120

Grinding time/minutes Fig. 3.10: Variation of particle size with grinding time in a typical bead mill.

The presentation in Fig. 3.9 is only schematic and is not based on experimental data. It shows the expected trend. When the volume fraction ϕ is below the optimum (in this case the relative viscosity of the dispersion is low) one requires a long time to achieve size reduction. In addition, the final particle size may be large and outside the nanorange. When ϕ is above the optimum value, the dispersion time is prolonged (due to the relatively high relative viscosity of the system) and the grinding time is also longer. In addition, the final particle size is larger than that obtained at the optimum ϕ. At the optimum volume fraction both the dispersion and grinding time are shorter and also the final particle size is smaller [8].

3.3 Preparation of suspensions using the top-down process | 111

For preparation of suspensions, bead mills are most commonly used. The beads are mostly made of glass or ceramics (which are preferred due to minimum contamination). The operating principle is to pump the premixed, preferably predispersed (using a high speed mixer), mill base through a cylinder containing a specified volume of say ceramic beads (normally 0.5–1 mm diameter to achieve nanosize particles). The dispersion is agitated by a single or multidisc rotor. The disc may be flat or perforated. The mill base passing through the shear zone is then separated from the beads by a suitable screen located at the opposite end of the feedport [8]. Generally speaking, bead mills may be classified to two types: (i) vertical mills with open or closed top; (ii) horizontal mills with closed chambers. The horizontal mills are more efficient and the most commonly used one are: Netzsch (Germany) and Dyno Mill (Switzerland). These bead mills are available in various sizes from 0.5 to 500 l. The factors affecting the general dispersion efficiency are known reasonably well (from the manufacturer). The selection of the correct diameter of the beads is important for maximum utilization. In general, the smaller the size of the beads and the higher their density, the more efficient the milling process [8]. To understand the operating principle of the bead mill, one must consider the centrifugal force transmitted to the grinding beads at the tip of the rotating disc which increases considerably with its weight. This applies greater shear to the mill base. This explains why the more dense beads are more efficient in grinding. The speed transmitted to the individual chambers of the beads at the tip of the disc assumes that speed and the force can be calculated [8]. The centrifugal force F is simply given by, F=

v2 , rg

(3.65)

where v is the velocity, r is the radius of the disc and g is the acceleration due to gravity.

3.3.4 Stabilization of the suspension during dispersion and milling and the resulting nanosuspension In order to maintain the particles as individual units during dispersion and milling, it is essential to use a dispersing agent that must provide an effective repulsive barrier preventing aggregation of the particles by van der Waals forces. This dispersing agent must be strongly adsorbed on the particle surface and should not be displaced by the wetting agent. As was discussed in detail in Chapter 6, Vol. 1, the repulsive barrier can be electrostatic in nature, whereby electrical double layers are formed at the solid/liquid interface [1–3]. These double layers must be extended (by maintaining low electrolyte concentration) and strong repulsion occurs on double layer

112 | 3 Solid/liquid dispersions (suspensions)

overlap. Alternatively, the repulsion can be produced by the use of nonionic surfactant or polymer layers which remain strongly hydrated (or solvated) by the molecules of the continuous medium [14] as discussed in Chapter 14, Vol. 1. On approach of the particles to a surface-to-surface separation distance that is lower than twice the adsorbed layer thickness, strong repulsion occurs as a result of two main effects: (i) unfavourable mixing of the layers when these are in good solvent conditions; (ii) loss of configurational entropy on significant overlap of the adsorbed layers. This process is referred to as steric repulsion [14]. A third repulsive mechanism is that in which both electrostatic and steric repulsion are combined, for example when using polyelectrolyte dispersants. The particles of the resulting suspension may undergo aggregation (flocculation) on standing as a result of the universal van der Waals attraction. This was discussed in detail in Chapter 5, Vol. 1, and only a summary is given in this chapter. This attractive energy becomes very large at short distances of separation between the particles. This attractive energy, GA , is given by the following expression, GA = −

AR , 12h

(3.66)

where A11(2) is the effective Hamaker constant of two identical particles with Hamaker constant A11 in a medium with Hamaker constant A22 . The Hamaker constant of any material is given by the following expression, A = πq2 β.

(3.67)

q is number of atoms or molecules per unit volume, and β is the London dispersion constant. Equation (3.66) shows that A11 has the dimension of energy. As mentioned in Chapter 6, Vol. 1, to overcome the permanent van der Waals attraction energy, it is essential to have a repulsive energy between the particles. The first mechanism is electrostatic repulsive energy produced by the presence of electrical double layers around the particles produced by charge separation at the solid/ liquid interface. The dispersant should be strongly adsorbed to the particles, produce high charge (high surface or zeta potential) and form an extended double layer (that can be achieved at low electrolyte concentration and low valency) [1–3]. When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (particle separation becomes less than twice the double layer extension), repulsion occurs. The individual double layers can no longer develop unrestrictedly, since the limited space does not allow complete potential decay [1–3]. The potential ψ H/2 half way between the plates is no longer zero (as would be the case for isolated particles at x → ∞). Combining Gelec and GA results in the well-known theory of stability of colloids (DLVO Theory) [12, 13], GT = Gelec + GA (3.68)

3.3 Preparation of suspensions using the top-down process | 113

A plot of GT versus h is shown in Fig. 3.11, which represents the case at low electrolyte concentrations, i.e. strong electrostatic repulsion between the particles. Gelec decays exponentially with h, i.e. Gelec → 0 as h becomes large. GA ∝ 1/h, i.e. GA does not decay to 0 at large h. At long distances of separation, GA > Gelec resulting in a shallow minimum (secondary minimum), which for nanosuspensions is very low (< kT). At very short distances, GA ≫ Gelec , resulting in a deep primary minimum. At intermediate distances, Gelec > GA , resulting in an energy maximum, Gmax , whose height depends on ψ0 (or ψd or zeta potential) and the electrolyte concentration and valency. At low electrolyte concentrations (< 10−2 mol dm−3 for a 1 : 1 electrolyte), Gmax is high (> 25 kT) and this prevents particle aggregation into the primary minimum. The higher the electrolyte concentration (and the higher the valency of the ions), the lower the energy maximum.

G

GT

Ge Gmax GA

Gprimary

h Gsec

Fig. 3.11: Schematic representation of the variation of GT with h according to the DLVO theory.

The second stabilization mechanism is referred to as steric repulsive energy produced by the presence of adsorbed (or grafted) layers of surfactant or polymer molecules [14] as was discussed in detail in Chapter 14, Vol. 1. In this case, the nonionic surfactant or polymer (referred to as polymeric surfactant) should be strongly adsorbed to the particle surface and the stabilizing chain should be strongly solvated (hydrated in the case of aqueous suspensions) by the molecules of the medium [14]. The most effective polymeric surfactants are those of the A–B, A–B–A block or BAn graft copolymer. The “anchor” chain B is chosen to be highly insoluble in the medium and to have strong affinity to the surface. The A stabilizing chain is chosen to be highly soluble in the medium and strongly solvated by the molecules of the medium. For nanosuspensions of hydrophobic solids in aqueous media, the B chain can be polystyrene, poly(methylmethacrylate) or poly(propylene oxide). The A chain could be poly(ethylene oxide) which is strongly hydrated by the medium. When two particles, each with a radius R and containing an adsorbed polymer layer with a hydrodynamic thickness δh , approach each other to a surface-surface separation distance h that is smaller than 2δh , the polymer layers interact with each other resulting in two main situations [14]:

114 | 3 Solid/liquid dispersions (suspensions)

(i) the polymer chains may overlap with each other; (ii) the polymer layer may undergo some compression. In both cases, there will be an increase in the local segment density of the polymer chains in the interaction region. The real situation is perhaps in between the above two cases, i.e. the polymer chains may undergo some interpenetration and some compression. Provided the dangling chains (the A chains in A–B, A–B–A block or BAn graft copolymers) are in a good solvent, this local increase in segment density in the interaction zone will result in strong repulsion as a result of two main effects [14]: (i) An increase in the osmotic pressure in the overlap region as a result of the unfavourable mixing of the polymer chains, when these are in good solvent conditions. This is referred to as osmotic repulsion or mixing interaction and it is described by a free energy of interaction Gmix . (ii) Reduction of the configurational entropy of the chains in the interaction zone; this entropy reduction results from the decrease in the volume available for the chains when these are either overlapped or compressed. This is referred to as volume restriction interaction, entropic or elastic interaction and it is described by a free energy of interaction Gel . The combination of Gmix and Gel is usually referred to as the steric interaction free energy, Gs , i.e., Gs = Gmix + Gel . (3.69) The sign of Gmix depends on the solvency of the medium for the chains. If in a good solvent, i.e. the Flory–Huggins interaction parameter χ is less than 0.5, then Gmix is positive and the mixing interaction leads to repulsion (see below). In contrast, if χ > 0.5 (i.e. the chains are in a poor solvent condition), Gmix is negative and the mixing interaction becomes attractive. Gel is always positive and hence in some cases one can produce stable nanosuspensions in a relatively poor solvent (enhanced steric stabilization). Combining Gmix and Gel with GA gives the total energy of interaction GT (assuming there is no contribution from any residual electrostatic interaction), i.e., GT = Gmix + Gel + GA

(3.70)

A schematic representation of the variation of Gmix , Gel , GA and GT with surfacesurface separation distance h is shown in Fig. 3.12. Gmix increases very sharply with decreasing h, when h < 2δ. Gel increases very sharply with decreasing h, when h < δ. GT versus h shows a minimum, Gmin , at separation distances comparable to 2δ. When h < 2δ, GT shows a rapid increase with decreasing h. The depth of the minimum depends on the Hamaker constant A, the particle radius R and adsorbed layer thickness δ. Gmin decreases with decreasing A and R. At a given A and R, Gmin decreases with increasing δ (i.e. with an increase in the molecular weight, Mw , of the

3.4 Prevention of Ostwald ripening (crystal growth) |

115

GT

G

Gel

Gmix

2δ

δ

h

Gmin

Fig. 3.12: Energy–distance curves for sterically stabilized systems.

Increasing δ/R

GT

Gmin

h Fig. 3.13: Variation of Gmin with δ/R.

stabilizer. This is illustrated in Fig. 3.13 which shows the energy–distance curves as a function of δ/R. The larger the value of δ/R, the smaller the value of Gmin . In this case the system may approach thermodynamic stability as is the case with nanosuspensions

3.4 Prevention of Ostwald ripening (crystal growth) The driving force for Ostwald ripening is the difference in solubility between the small and large particles (the smaller particles have higher solubility than the larger ones). The difference in chemical potential between different sized particles was given by Lord Kelvin [17] 2γVm (3.71) S(r) = S(∞) exp( ), rRT

Solubility Enhancement

116 | 3 Solid/liquid dispersions (suspensions)

Kelvin Equation

100

2M γ 1 r

w c(r) = e RTρ c(0)

10

1

1

10

100

Radius (nm)

1000

Fig. 3.14: Solubility enhancement with decreasing particle or droplet radius.

where S(r) is the solubility of a particle with radius r and S(∞) is the solubility of a particle with infinite radius (the bulk solubility), γ is the S/L interfacial tension, R is the gas constant and T is the absolute temperature. Equation (3.71) shows a significant increase in solubility of the particle with the reduction in particle radius, particularly when the latter becomes significantly smaller than 1 µm. A schematic representation of the enhancement of the solubility c(r)/c(0) with decreasing particle size according to the Kelvin equation is shown in Fig. 3.14. It can be seen from Fig. 3.14 that the solubility of suspension particles increases very rapidly with decreasing radius, particularly when r < 100 nm. This means that a particle with a radius of say 4 nm will have about 10 times solubility enhancement compared say with a particle with 10 nm radius which has a solubility enhancement of only 2 times. Thus, with time, molecular diffusion will occur between the smaller and larger particle or droplet, with the ultimate disappearance of most of the small particles. This results in a shift in the particle size distribution to larger values on storage of the suspension. This could lead to the formation of a suspension with average particle size > 2 µm. This instability can cause severe problems, such sedimentation, flocculation and even flocculation of the suspension. For two particles with radii r1 and r2 (r1 < r2 ), S(r1 ) 1 RT 1 ln[ ] = 2γ[ − ]. Vm S(r2 ) r1 r2

(3.72)

Equation (3.72) is sometimes referred to as the Ostwald equation and it shows that the greater the difference between r1 and r2 , the higher the rate of Ostwald ripening. That is why in preparation of suspensions, one aims at producing a narrow size distribution. A second driving force for Ostwald ripening in suspensions is due to polymorphic changes. If, for example, a drug has two polymorphs A and B, the more soluble polymorph, say A (which may be more amorphous) will have higher solubility than the less soluble (more stable) polymorph B. During storage, polymorph A will dissolve and recrystallize as polymorph B. This can have a detrimental effect on bioefficacy, since the more soluble polymorph may be more active.

3.4 Prevention of Ostwald ripening (crystal growth) |

117

The kinetics of Ostwald ripening is described in terms of the theory developed by Lifshitz and Slesov [18] and by Wagner [19] (referred to as LSW theory). The LSW theory assumes that: (i) the mass transport is due to molecular diffusion through the continuous phase; (ii) the dispersed phase particles are spherical and fixed in space; (iii) there are no interactions between neighbouring particles (the particles are separated by a distance much larger than the diameter of the particles); (iv) the concentration of the molecularly dissolved species is constant except adjacent to the particle boundaries. The rate of Ostwald ripening ω is given by: ω=

d 3 8σDS(∞)Vm 4DS(∞)α (r ) = ( )f(ϕ) = ( )f(ϕ), dr c 9RT 9

(3.73)

where rc is the radius of a particle that is neither growing nor decreasing in size, D is the diffusion coefficient of the disperse phase in the continuous phase, f(ϕ) is a factor that reflects the dependence of ω on the disperse volume fraction and α is the characteristic length scale (= 2γVm /RT). Particles with r > rc grow at the expense of smaller ones, while particles with r < rc tend to disappear. The validity of the LSW theory was tested by Kabalnov et al. [20, 21] using emulsions of 1,2 dichloroethane-in-water emulsions in which the droplets were fixed to the surface of a microscope slide to prevent their coalescence. The evolution of the droplet size distribution was followed as a function of time by microscopic investigations. The LSW theory predicts that the droplet growth over time will be proportional to r3c . LSW theory assumes that there are no interactions between the particles and it is limited to low particle volume fractions. At higher volume fractions the rate of ripening depends on the interaction between diffusion spheres of neighbouring particles. It is expected that suspensions with higher volume fractions of solid will have broader particle size distribution and faster absolute growth rates than those predicted by LSW theory. However, experimental results using high surfactant concentrations (5 %) showed the rate to be independent of the volume fraction in the range 0.01 ≤ ϕ ≤ 0.3. It has been suggested that the particles may have been screened from one another by surfactant micelles [22]. It has been suggested that micelles play a role in facilitating the mass transfer between particles by acting as carriers of solute molecules [23–26]. Three mechanisms were suggested: (i) molecules are transferred via direct particle/micelle collisions; (ii) molecules exit the particle and are trapped by micelles in the immediate vicinity of the particle; (iii) molecules exit the particles collectively with a large number of surfactant molecules to form a micelle.

118 | 3 Solid/liquid dispersions (suspensions)

In mechanism (i) the micellar contribution to the rate of mass transfer is directly proportional to the number of particle/micelle collisions, i.e. to the volume fraction of micelles in solution. In this case the molecular solubility of the particle in the LSW equation is replaced by the micellar solubility which is much higher. Large increases in the rate of mass transfer would be expected with increasing micelle concentration. Numerous studies indicate, however, that the presence of micelles affects the mass transfer to only a small extent [27]. The results showed only a two-fold increase in ω above the cmc. This result is consistent with many other studies which showed an increase in the mass transfer of only 2–5 times with increasing micelle concentration. The lack of strong dependency of mass transfer on micelle concentration for ionic surfactants may result from electrostatic repulsion between the particles and micelles, which provide a high energy barrier preventing droplet/micelle collision. In mechanism (ii), a micelle in the vicinity of a particle rapidly takes up dissolved molecules from the continuous phase. This “swollen” micelle diffuses to another particle, where the molecule is redeposited. Such a mechanism would be expected to result in an increase in the mass transfer over and above that expected from LSW theory by a factor φ given by the following equation, φ =1+

ϕs ΓDm χeq Dm = 1 + eq , D c D eq

(3.74) eq

where ϕs is the volume fraction of micelles in solution, χ = ϕs cm is the net solubility in the micelle per unit volume of micellar solution reduced by the density of the solute, eq eq Γ = cm /c ≈ 106 –1011 is the partition coefficient for the molecule between the micelle and bulk aqueous phase at the saturation point, Dm is the micellar diffusivity. When using nonionic surfactant micelles, larger increases in the Ostwald ripening rate might be expected due to the larger solubilization capacities of the nonionic surfactant micelles and absence of electrostatic repulsion between the particles and the uncharged micelles [28]. According to the above analysis, the growth of crystals can take place only under conditions of appreciable supersaturation, mostly > 1.5 %, which ensures the necessary work of formation of two-dimensional nuclei. However, experiments on the growth of various crystals have shown that crystal growth can take place at extremely low supersaturation. The existence of a critical finite supersaturation for the growth of crystals has only been established for a few materials and then for individual faces of crystals being different from case to case; at the most it is about 1 %. However, this discrepancy is not too surprising [29] since the crystals do not have a completely perfect surface needing fresh two-dimensional nucleation in order to grow. This discrepancy may be attributed to crystal dislocations and structural defects. The latter include cracks, surface kinks and surface roughness. According to Cabrera and Burton [30] and Frank [29], such defects result in the formation of steps at which crystals can grow without the need of formation of nuclei.

3.4 Prevention of Ostwald ripening (crystal growth) |

119

Screw dislocations are of special importance in the growth of real crystals. If just one dislocation of this type emerges at the centre of the face, that crystal face can grow perpetually up a “spiral staircase”. The general importance of dislocations for crystal growth accounts for many observations, such as the individual behaviour of each crystal face, particularly on the microscopic scale. The growth of a crystal in steps is due to three processes: (i) exchange of particles between adsorption layer and solution; (ii) diffusion of adsorbed particles to the steps, as well as exchange of particles between steps and adsorption layer; (iii) diffusion of particles adsorbed by steps in the direction of kinks and exchange with these kinks. Using these assumptions, Burton, Cabrera and Frank [29–31] arrived at the following expression for the growth rate R of a crystal from solution, R=

DN0 Ωaβ(x0 ) 2κ0 ρc

(3.75)

where x0 is the average distance between kinks in the steps, β(x0 ) is the supersaturation of the solution at distance x0 , D is the diffusion coefficient, N0 is the equilibrium or steady state concentration of the crystallizing substance in solution, a is the distance between two equilibrium configurations on the surface of a crystal and Ω is the volume of one molecule. The critical size of a nucleus ρc is given by, 2ρc = 2γ

a β(x0 ), kT

(3.76)

where γ is a constant (Euler constant). β(x0 ) is given by, β(x0 ) Y0 −1 2πa(δ − Y0 ) 2a + ln( )] . = [1 + β x0 Y0 x0 x0

(3.77)

Y0 is the distance between successive steps, δ is the thickness of the unsaturated layer at the surface of the crystal and β is the supersaturation elsewhere in the solution. At low supersaturation, the third term on the right-hand side of equation (3.77) is the largest and R depends on β(x0 ) in a parabolic manner. At higher supersaturations, the second term in equation (3.77) becomes more important and R = f(β) becomes linear. In this case equation (3.77) takes the form, R1 =

DN0 Ωβ . δ

(3.78)

Thus, the linear growth rate should be observed at β ≥ 10−3 . It has long been known that trace concentrations of certain additives can have pronounced effects on crystal growth and habit. These effects are of great importance

120 | 3 Solid/liquid dispersions (suspensions)

in many fields of science and technology, but the mechanism by which these additives affect crystal growth is not clear. It is generally agreed that additives must adsorb on a crystal surface in order to affect the growth on that face. Sufficient effects on growth behaviour with very small impurity additives are usually produced by large organic molecules on colloidal materials. One part in 104 or 105 of such materials may be sufficient to completely alter the growth. The effects of large molecules are usually nonspecific, presumably due to their adsorption on almost any point of the crystal. Assuming growth to be governed by creation and subsequent lateral motion of steps on the crystal surface [29, 30], it is possible to derive an expression for the effect of impurities on the flow of these steps [31]. Consider the surface of a growing crystal which contains a uniform average concentration of steps n. Suppose there is a constant flux Ji of impurity molecules deposited on the crystal surface per unit time. Assuming that the impurity molecules are immobile on the surface; as the step moves along it will be stopped by a pair of molecules that are less than 2ρc apart (where ρc is the medium radius of curvature of the step corresponding to the supersaturation) and will squeeze itself between a pair of impurities that are more than 2ρc apart. Since the steps are curved, their average velocity v will be smaller than v0 , the velocity in the absence of impurities. A rough estimate of this reduction in velocity is given by the following approximate equation, v = v0 (1 − 2ρc d1/2 )1/2 ,

(3.79)

where d is the average density of impurities just ahead of the step. Assuming, for simplicity, that that once a step has passed beyond a certain point on the crystal the impurities adsorbed then become occluded in the crystal and do not offer a significant obstacle to the advance of the flowing step, it is clear that the density d ahead of any step will be given by, d=

Ji . nv

(3.80)

This expression automatically makes the flux of impurities being adsorbed in the crystal equal Ji . Substituting equation (3.80) into (3.79) and rearranging, the following equation, which may be solved for v, is obtained, v2 = v + a = 0 v2 =

ν ; ν0

α2 =

(3.81) 4ρ2c Ji , v0

(3.82)

where ν = nv is the flow rate of a step in the presence of impurity and ν0 = nv0 is the corresponding rate in the absence of impurities. As mentioned above, when the compound used for formulation of suspensions exists in two polymorphs, crystal growth may take place as a result of reversion of the thermodynamically less stable form to the more stable form. If this is the case,

3.4 Prevention of Ostwald ripening (crystal growth)

|

121

crystal growth is virtually unaffected by temperature, i.e. it is an isothermal process, which is solvent mediated. Crystal growth involving such polymorphic changes has been carried out by various investigators [32–34]. A thermodynamic analysis based on Gibbs’ theory to account for the polymorphic changes can be made. If a crystal exists in two polymorphic forms, α and β, the Gibbs free energy is given by the expressions, N

∆G α = ∆Gvα + ∑ A i ∆Gsα ,

(3.83)

i β

N

β

∆G β = ∆Gv + ∑ A i ∆Gs ,

(3.84)

i

where V is the crystal volume and A i is the area. If ∆G α ≠ ∆G β , there exists a thermodynamic potential (driving force) to establish equilibrium by an appropriate change of phase or crystal habit. By this mechanism the less soluble phase grows at the expense of the more soluble phase. The different polymorphs can be characterized by X-ray diffraction. It is clear from the above discussion that crystal growth in suspensions where the solid particles have substantial solubility or exist in various polymorphs is the rule rather than the exception. The task of the formulation scientist is to reduce crystal growth to an acceptable level depending on the application. This is particularly the case with pharmaceutical and agrochemical suspensions, where crystal growth leads to the shift of particle size distribution to larger values. Apart from reducing the physical stability of the suspension, e.g. increased sedimentation, the increase in particle size of the active ingredient reduces its bioavailability (reduction of disease control). Unfortunately, crystal growth inhibition is still an “art”, rather than a “science”, in view of the lack of adequate fundamental understanding of the process at a molecular level. Since suspensions are prepared by using a wetting/dispersing agent, it is important to discuss how these agents can affect the growth rate. In the first place, the presence of wetting/dispersing agents influences the process of diffusion of the molecules from the surface of the crystal to the bulk solution. The wetting/dispersing agent may affect the rate of dissolution by affecting the rate of transport away from the boundary layer [1–3], although their addition is not likely to affect the rate of dissolution proper (passage from the solid to the dissolved state in the immediate adjacent layer). If the wetting/dispersing agent forms micelles which can solubilize the solute, the diffusion coefficient of the solute in the micelles is greatly reduced. However, as a result of solubilization, the concentration gradient of the solute is increased to an extent depending on the extent of solubilization. The overall effect may be an increase in crystal growth rate as a result of solubilization. In contrast, if the diffusion rate of the molecules of the wetting/dispersing agent molecules is sufficiently rapid, their presence will lower the flux of the solute molecules compared to that in

122 | 3 Solid/liquid dispersions (suspensions)

the absence of the wetting/dispersing agent. In this case, the wetting/dispersing agent will lower the rate of crystal growth. Secondly, wetting/dispersing agents are expected to influence growth when the rate is controlled by surface nucleation [1–3]. Adsorption of wetting/dispersing agents on the surface of the crystal can drastically change the specific surface energy and make it inaccessible to the solute molecules. In addition, if the wetting/dispersing agent is preferentially adsorbed at one or more of the faces of the crystal (for example by electrostatic attraction between a highly negative face of the crystal and cationic surfactant), surface nucleation is no longer possible at this particular face (or faces). Growth will then take place at the remaining faces, which are either bare or incompletely covered by the wetting/dispersing agent. This will result in a change in crystal habit. The role of surfactants in modifying the crystal habit of adipic acid has been systematically studied by Michaels and collaborators [35–37]. These authors investigated the effect of various surfactants, of the anionic and cationic type, on the growth of adipic acid crystals from aqueous solution. Microscopic measurements of the crystals permitted calculation of the individual growth rates of the (001), (010) and (110) faces. The growth rate is governed by the rate at which solute is supplied to the individual steps on the crystal faces and the spacing between them. In other words, the growth rate is proportional to the step velocity and the distance between steps. Surfactants may alter the growth rate by changing either of these. At constant step velocity, the spacing between steps may be altered, with a corresponding modification in the growth rate, by a variation in the rate of step generation. With constant step spacing, an alteration of step velocity will likewise modify the growth rate. Sodium dodecylbenzene sulphonate (NaDBS) retards the growth on the (010) and (110) faces more than on the (001) face, thus favouring the formation of prismatic or needle crystals. Cationic surfactants such as cetyltrimethyl ammonium chloride have the opposite effect, thus favouring growth of the micaceous faces. Michaels et al. [35–37] concluded that the anionic surfactants are physically adsorbed on the faces of adipic acid crystals, while the cationics appear to be chemisorbed. In all cases, the surfactants retarded crystal growth by adsorption on the crystal faces, thus reducing the area on which nucleation could occur. In fact with relatively large crystals, the influence of surfactants on crystal growth can be correlated satisfactorily with the Langmuir adsorption isotherm. Surfactants, in general, exhibit a far greater retarding influence on the crystal growth of very small crystals than on the growth of larger ones. From the above discussion, it can be seen that surfactants (wetting/dispersing agents), if properly chosen, may be used for crystal growth inhibition and control of habit formation. Inhibition of crystal growth can also be achieved by polymeric surfactants and other additives. For example, Simonelli et al. [38] found that the crystal growth of the drug sulphathiazole can be inhibited by the addition of poly(vinylpyrrolidone) (PVP). The inhibition effect depends on the concentration and molecular weight of PVP. A minimum concentration (expressed as grams PVP/100

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

123

ml) of polymer is required for inhibition, which increases with increasing molecular weight of the polymer. However, if the concentration is expressed in mol dm−3 , the reverse is true, i.e. the higher the molar mass of PVP the lower the number of moles required for inhibition. This led Simonelli et al. [38] to conclude that inhibition must involve kinetic effects, i.e. the rate of diffusion of PVP to the surfaces. If the rate of deposition of PVP is relatively slow compared to that of sulphathiazole molecules, it is buried by the “avalanche” of the precipitating sulphathiazole molecules. If, on the other hand, its rate is rapid, it in turn can bury the precipitating sulphathiazole molecules and sufficiently cover the crystal surface to cause inhibition of crystal growth. Clearly, a higher PVP concentration would be needed at higher supersaturation rates to cause inhibition. This is due to the increase in diffusion rate at higher supersaturation [38]. Carless et al. [39] reported that the crystal growth of cortisone acetate in aqueous suspensions can be inhibited by addition of cortisone alcohol. Crystal growth in this system is mainly inhibited by polymorphic transformation [39]. The authors assumed that cortisone alcohol is adsorbed onto the particles of the stable form and this prevents the arrival of new cortisone acetate molecules which would result in crystal growth. The authors also noticed that the particles change their shape, growing to long needles. This means that the cortisone alcohol fits into the most dense lattice plane of the cortisone acetate crystal, thus preventing preferential growth on that face. Many block ABA and graft BAn copolymers (with B being the “anchor” part and A the stabilizing chain) are very effective in inhibiting crystal growth. The B chain adsorbs very strongly on the surface of the crystal and sites become unavailable for deposition. This has the effect of reducing the rate of crystal growth. Apart from their influence on crystal growth, the above copolymers also provide excellent steric stabilization, providing the A chain is chosen to be strongly solvated by the molecules of the medium.

3.5 Sedimentation of suspensions and prevention of formation of hard sediments Most suspensions undergo separation on standing as a result of the density difference between the particles and the medium, unless the particles are small enough for Brownian motion to overcome gravity [1–4]. This is illustrated in Fig. 3.15 for three cases of suspensions. Case (a) represents the situation when the Brownian diffusion energy (which is in the region of kT, where k is the Boltzmann constant and T is the absolute temperature) is much larger than the gravitational potential energy (which is equal to (4/3)πR3 ∆ρgh, where R is the particle radius, ∆ρ is the density difference between the particles and medium, g the acceleration due to gravity and h is the height of

124 | 3 Solid/liquid dispersions (suspensions)

h

(a) kT > (4/3) πR³Δ ρgh

h

Ch

(b) kT < (4/3) πR³Δ ρgh Ch = Co exp (–mgh/kT) Co = conc. At the bottom Ch = conc. At time t and height h m = (4/3) πR³Δ ρ

Fig. 3.15: Schematic representation of sedimentation of suspensions.

the container). Under these conditions, the particles become randomly distributed throughout the whole system, and no separation occurs. This situation may occur with nanosuspensions with radii less than 100 nm, particularly if ∆ρ is not large, say less than 0.1. In contrast, when (4/3)πR3 ∆ρgh ≫ kT, complete sedimentation occurs as illustrated in Fig. 3.15 (b) with suspensions of uniform particles. In such a case, the repulsive force necessary to ensure colloid stability enables the particles to move past each other to form a compact layer [1–4]. As a consequence of the dense packing and small spaces between the particles, such compact sediments (which are technically referred to as “clays” or “cakes”) are difficult to redisperse. In rheological terms (see Chapter 1) the close packed sediment is shear thickening that is referred to as dilatancy, i.e. a rapid increase in the viscosity with increasing shear rate. The most practical situation is that represented by Fig. 3.15 (c), where a concentration gradient of the particles occurs across the container. The concentration of particles C can be related to that at the bottom of the container C0 by the following equation, mgh (3.85) C = C0 exp(− ), kT where m is the mass of the particles that is given by (4/3)πR3 ∆ρ (R is the particle radius and ∆ρ is the density difference between particle and medium), g is the acceleration due to gravity and h is the height of the container. For a very dilute suspension of rigid noninteracting particles (ϕ ≤ 0.01), the rate of sedimentation v0 can be calculated by application of Stokes law, where the hydrodynamic force is balanced by the gravitational force, hydrodynamic force = 6πηRv0 ,

(3.86)

gravity force = (4/3)πR ∆ρg, 3

v0 = where η is the viscosity of the medium (water).

2 9

R2 ∆ρg η

,

(3.87) (3.88)

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

125

v0 calculated for three particle sizes (0.1, 1 and 10 µm) for a suspension with density difference ∆ρ = 0.2 is 4.4 × 10−9 , 4.4 × 10−7 and 4.4 × 10−5 m s−1 respectively. The time needed for complete sedimentation in a 0.1 m container is 250 days, 60 hours and 40 minutes respectively. For moderately concentrated suspensions, 0.2 > ϕ > 0.01, sedimentation is reduced as a result of hydrodynamic interaction between the particles, which no longer sediment independently of each other [8]. Several contributions to the change in sedimentation rate as a result of increasing particle number concentration have been considered. The first of these contributions arise from the upward flux of fluid volume that accompanies the downward flux of volume of solid material in order to maintain a zero mean volume flux at each point in a homogeneous suspension. This change in fluid environment for one sphere causes the mean sedimentation rate to differ from its value at infinite dilution by an amount −ϕv0 . The second, and the largest, contribution arises from the drag down of the fluid that adheres to the spherical particles. This downward flux of fluid in the inaccessible shells surrounding the rigid spheres is accompanied by an equal upward flux of volume in the part of the fluid that is accessible to the centre of the test sphere. In other words, the reduction in sedimentation rate arises from the diffuse upward current, which compensates for the downward flux in fluid volume in the inaccessible shells surrounding the rigid sphere. This contributes −4.5ϕv0 in the mean sedimentation velocity. The third contribution to the change in sedimentation velocity arises from the motion of the spheres, which collectively generates a velocity distribution in the fluid such that the second derivative of velocity ∇2 has a nonzero mean. This property of the environment for a particular sphere changes its mean velocity by 0.5ϕv0 . The fourth contribution arises from the interaction between the spheres. When the test sphere whose velocity is being averaged is near one of the other spheres in the suspension, the interaction between these two spheres gives the test sphere a translational velocity that is significantly different from that which is estimated from the velocity distribution in the absence of the second sphere. This gives a further change in the mean sedimentation rate of −1.5ϕv0 . Therefore, the average velocity v can be related to that at infinite dilution v0 by taking into account the above four contributions, i.e., v = v0 [−ϕv0 − 4.5ϕv0 + 0.5ϕv0 − 1.55ϕv0 ], v = v0 (1 − 6.55ϕ).

(3.89)

This means that for a suspension with ϕ = 0.1, v = 0.345v0 , i.e. the rate is reduced by a factor of ≈ 3. For more concentrated suspensions (ϕ > 0.2), the sedimentation velocity becomes a complex function of ϕ. An increase in the concentration of suspension leads to a considerable increase in the complexity of the dependency of sedimentation rate on particle size. This is because there is a decrease in the distance between the particles in the disperse phase and also interactions between them occur (either directly or indirectly through the dispersion medium). In addition, an increase in the

126 | 3 Solid/liquid dispersions (suspensions)

concentration of the solid phase in the suspension brings an increase in the density and viscosity of the whole disperse system. At high values of the volume fraction of the solid phase (ϕ > 0.1), displacement of the dispersion medium occurs and of small particles sedimenting out originally by larger particles. At even higher volume fraction (ϕ > 0.4), the particles tend to sediment in what is known as “hindered sedimentation” mode, whereby all particles sediment at the same rate independent of their size. The closeness of packing prevents differential movement of any large particles through the suspension and the observed sedimentation rate becomes very much less than the Stokes sedimentation rate for any single particle. For a suspension sedimenting in this way, the solid appears to “condense” slowly to a larger volume fraction leaving a clear supernatant liquid separated from the solid sediment by a sharp interface. The above phenomenon of “hindered sedimentation” was theoretically analysed by Kynch [40] who considered the case of sedimentation of monodisperse particles. He assumed that the velocity v of any particle is a function only of the local concentration n of the particles in its immediate vicinity. The particle flux S, i.e. the number of particles passing a horizontal section, per unit area, per unit time, is given by, S = nv.

(3.90)

It is assumed everywhere that the concentration is the same across any horizontal layer. The concentration n varies from zero at the top of the sedimentation vessel to some maximum value nm at the bottom and presumably the velocity v of fall decreases from a finite value u to zero. By considering the flux of particles at various levels in a sedimenting suspension, Kynch [40] derived expressions for the decrease of the height of the suspension with time, when the initial concentration n remains constant and when the initial concentration increases towards the bottom with increasing n in the concentration n range covered during sedimentation. The x versus t diagram obtained by Kynch [40] is shown in Fig. 3.16. The curve in Fig. 3.16 is characterized by three sections: AOB, where the concentration n is the same as the initial concentration; OCD, where the concentration is a maximum nm; and OBC, where there is a continuous but extremely rapid increase in concentration from nB to the maximum concentration nm. Thus the suspension falls like a “plug” which has the initial density of the suspension (section AOB), depositing a “plug” of maximum density at the bottom (section OCD). At time t1 , there would be a suspension having the initial density above F, and a “plug” of maximum density below E. Between E and F, the density would decrease (in the upward direction) from the maximum to the initial density. Thus, for a suspension which is not flocculated, the sedimentation is constant with time in the first stage, becoming logarithmic in the third stage and the second stage is the transition between the two. Unfortunately, Kynch’s analysis [40] did not take into account the obvious change of sedimentation velocity with concentration, arising from hydrodynamic interactions [8].

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

127

A

Height x

G

B

D

C F

E 0

t1

Fig. 3.16: Fall of surface of a suspension according to Kynch [40].

Time t

Buscall et al. [41] attempted to relate the decrease in sedimentation velocity with increasing particle volume fraction ϕ to the reduction in relative viscosity with increasing ϕ. A schematic representation for the variation of v with ϕ is shown in Fig. 3.17, which also shows the variation of relative viscosity with ϕ. It can be seen that v decreases exponentially with increasing ϕ and ultimately it approaches zero when ϕ approaches a critical value ϕp (the maximum packing fraction). The relative viscosity shows a gradual increase with increasing ϕ and when ϕ = ϕp , the relative viscosity approaches infinity.

v

ƞr

ɸp

1

ɸ Fig. 3.17: Variation of v and ηr with ϕ.

[ƞ] ɸp ɸ

128 | 3 Solid/liquid dispersions (suspensions)

The maximum packing fraction ϕp can be easily calculated for monodisperse rigid spheres. For hexagonal packing ϕp = 0.74, whereas for random packing ϕp = 0.64. The maximum packing fraction increases with polydisperse suspensions. For example, for a bimodal particle size distribution (with a ratio of ≈ 10 : 1), ϕp > 0.8. It is possible to relate the relative sedimentation rate (v/v0 ) to the relative viscosity η/η0 , η0 v (3.91) ( ) = α( ). v0 η The relative viscosity is related to the volume fraction ϕ by the Dougherty–Krieger equation for hard spheres [42], ϕ −[η]ϕp η = (1 − , ) η0 ϕp

(3.92)

where [η] is the intrinsic viscosity (= 2.5 for hard spheres). Combining equations (3.91) and (3.92), v ϕ α[η]ϕp ϕ kϕp = (1 − = (1 − ) ) . v0 ϕp ϕp

(3.93)

The above empirical relationship was tested for sedimentation of polystyrene latex suspensions with R = 1.55 µm in 10−3 mol dm−3 NaCl [41]. The results are shown in Fig. 3.18. The circles are the experimental points, whereas the solid line is calculated using equation (3.93) with ϕp = 0.58 and k = 5.4. Michaels and Bolger [43] used a different model to describe the sedimentation of flocculated kaolin suspensions. Three types of sedimentation curves were considered, depending on the concentration range of the suspension. These are illustrated

1.0 Experimental data

v/vo

0.8

Line drawn with ɸp = 0.58 and k = 5.4

0.6

0.4 0.2 0.1

0.2

0.3

0.4

0.5

ɸ Fig. 3.18: Variation of sedimentation rate with volume fraction for polystyrene dispersions.

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

129

Vo

V

c

b

a

0

Fig. 3.19: Three types of sedimentation curves according to Michaels and Bolger [43].

Time

in Fig. 3.19. Curve a is the case of a dilute suspension in which the aggregates are considered to be spherical, sedimenting individually thus producing a sharp interface. The sedimentation curve starts with a linear part, with the rate v0 being a function of ϕA (the volume fraction of the aggregates). Michaels and Bolger [43] used Richardson and Zaki’s formula [44] for v0 and obtained the following equation, v0 =

g(ρ − ρ0 )d2A (1 − CA ϕ)4.56 , 18ηCA

(3.94)

where dA is the mean diameter of aggregates, ϕ = (1 − ε) is the volume fraction of the solids with ε being a measure of porosity, i.e. (1 − ε) is the volumetric density, ρ is the density of the particles and ρ0 that of the medium. CA (= (ϕA /ϕ) is a factor that characterizes the “looseness” of the aggregates (the ratio of immobilized liquid to the total volume. In the intermediate concentration range (curve b) the sedimentation curve was accounted for by Michaels and Bolger [43] by considering a network model for the aggregates. The maximum sedimentation rate is given by, v󸀠1 =

g(ρ − ρ0 )d2p 32η

(1 − CAF ϕF ),

(3.95)

where dp is the mean pore diameter in the network, CAF (= ϕA /ϕF ), where ϕF is the ratio of the volume of the flocs forming the aggregates and the volume of suspension. The characteristics of the flocs can be determined by studying the sediment volume in detail.

130 | 3 Solid/liquid dispersions (suspensions)

The sedimentation of highly concentrated flocculated suspension (curve c) shows a slow decrease in the sediment volume with time and in some cases a clear liquid layer is “squeezed out” of the liquid bound to the particle surfaces to the top of the container. This process is sometimes referred to as syneresis and the compaction of the solid aggregates is referred to as consolidation The sedimentation of particles in non-Newtonian fluids, such as aqueous solutions containing high molecular weight compounds (e.g. hydroxyethyl cellulose or xanthan gum) usually referred to as “thickeners”, is not simple since these nonNewtonian solutions are shear thinning with viscosity decreasing with increasing shear rate [9]. These solutions show a Newtonian region at low shear rates or shear stresses, usually referred to as the residual or zero shear viscosity η(0). This is illustrated in Fig. 3.20 which shows the variation of stress σ and viscosity η with shear rate γ.̇

ƞ/Pas

σ/Pa

ƞ (0)

σβ

γ/s¯¹

γ/s¯¹

Fig. 3.20: Flow behaviour of “thickeners”.

The viscosity of a polymer solution increases gradually with increasing concentration and at a critical concentration, C∗ , the polymer coils with a radius of gyration RG and a hydrodynamic radius Rh (which is higher than RG due to solvation of the polymer chains) begin to overlap and viscosity increases rapidly. This is illustrated in Fig. 3.21 which shows the variation of log η with log C. In the first part of the curve η ∝ C, whereas in the second part (above C∗ ) η ∝ C3.4 . A schematic representation of polymer coil overlap is shown in Fig. 3.22 which shows the effect of gradually increasing the polymer concentration. The polymer concentration above C∗ is referred to as the semi-dilute range [9]. C∗ is related to RG and the polymer molecular weight M by, C∗ =

3M 4πR3G Nav

.

(3.96)

Reduced Viscosity

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

18 16 14 12 10 8 6 4

131

CAC 0 0.02 0.04 0.06 0.08 0.1 0.12 C/g dm–3

(a) Dilute C < C*

(b) Onset of overlap C = C*

Fig. 3.21: Variation of reduced viscosity with HMHEC concentration.

(c) Semi-dilute C > C*

Fig. 3.22: Crossover between dilute and semi-dilute solutions.

Nav is the Avogadro number. As M increases C∗ becomes progressively lower. This shows that to produce physical gels at low concentrations by simple polymer coil overlap, one has to use high molecular weight polymers. Another method to reduce the polymer concentration at which chain overlap occurs is to use polymers that form extended chains such as xanthan gum which produces conformation in the form of a helical structure with a large axial ratio. These polymers give much higher intrinsic viscosities and they show both rotational and translational diffusion. The relaxation time for the polymer chain is much higher than a corresponding polymer with the same molecular weight but produces random coil conformation. The above polymers interact at very low concentrations and the overlap concentration can be very low (< 0.01 %). These polysaccharides are used in many formulations to produce physical gels at very low concentrations thus reducing sedimentation. The shear stress, σp , exerted by a particle (force/area) can be simply calculated [9], σp =

(4/3)πR3 ∆ρg ∆ρRg = . 3 4πR2

(3.97)

For a 10 µm radius particle with a density difference ∆ρ of 0.2 g cm−3 , the stress is equal to, 0.2 × 103 × 10 × 10−6 × 9.8 (3.98) ≈ 6 × 10−3 Pa. σp = 3 For smaller particles smaller stresses are exerted.

132 | 3 Solid/liquid dispersions (suspensions)

Thus, to predict sedimentation, one has to measure the viscosity at very low stresses (or shear rates). These measurements can be carried out using a constant stress rheometer (Carrimed, Bohlin, Rheometrics, Haake or Physica) as described in Chapter 1. Usually one obtains good correlation between the rate of sedimentation v and the residual viscosity η(0). Above a certain value of η(0), v becomes equal to 0. Clearly, to minimize sedimentation one has to increase η(0); an acceptable level for the high shear viscosity η∞ must be achieved, depending on the application. In some cases, a high η(0) may be accompanied by a high η∞ (which may not be acceptable for applications). As discussed above, the stress exerted by the particles is very small, in the region of 10−3 –10−1 Pa depending on the particle size and the density of the particles. Clearly to predict sedimentation, one needs to measure the viscosity at this low stresses [9]. This is illustrated for solutions of ethylhydroxyethylcellulose (EHEC) in Fig. 3.23.

1.48% 1.30% 1.0 1.08% EHEC η/Pas

0.86% 0.1

0.65% 0.43%

0.01

0.22%

0.01

0.1

1

10

σ/Pa Fig. 3.23: Constant stress (creep) measurements for PS latex dispersions as a function of EHEC concentration.

The results in Fig. 3.23 show that below a certain critical value of shear stress the viscous behaviour is Newtonian and this critical stress value is in the region of 0.1 Pa. Above this stress viscosity decreases with increasing shear stress, indicating shear thinning behaviour. The plateau viscosity values at low shear stress (< 0.1 Pa) give the limiting and residual viscosity η(0), i.e. the viscosity at near zero shear rate. The settling rate of a dispersion of polystyrene latex with radius 1.55 µm and at 5 % w/v was measured as a function of ethylhydroxyethylcellulose (EHEC) concentra-

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

133

v/R2 m–1s–1

tion, using the same range as in Fig. 3.23. The settling rate expressed as v/R2 , where R is the particle radius, is plotted versus η(0) in Fig. 3.24 (on a log-log scale). As is clear, a linear relationship between log(v/R2 ) and log η(0) is obtained, with a slope of −1, over three decades of viscosity, indicating that the rate of settling is proportional to [η(0)]−1 .

10–3

10–2

10–1 η(0)/Pa

100

10 Fig. 3.24: Sedimentation rate versus η(0).

The maximum shear stress developed by an isolated spherical particle as it settles through a medium of viscosity η is given by the expression [9], 3vη . (3.99) 2R For particles at the coarse end of the colloidal range the magnitude of this quantity will be in the range 10−2 –10−5 Pa. From the data obtained on EHEC solutions and given in Fig. 3.23, it can be seen that in this range of shear stresses the solutions behave as Newtonian fluids with zero shear viscosity η(0). Hence, isolated spheres should obey equation (3.89) with η0 replaced by η(0). Consequently, it can be concluded that the rate of sedimentation of a particle is determined by the zero shear rate behaviour of the medium in which it is suspended. With the present system of polystyrene latex, no sedimentation occurred when η(0) was greater than 10 Pa s. The situation with more practical dispersions is more complex due to the interaction between the thickener and the particles. Most practical suspensions show some weak flocculation and the “gel” produced between the particles and thickener may undergo some contraction as a result of the gravity force exerted on the whole network. A useful method to describe separation in these concentrated suspensions is to follow the relative sediment volume V t /V0 or relative sediment height h t /h0 (where the subscripts t and 0 refer to time t and zero time respectively) with storage time. For good physical stability, the values of V t /V0 or h t /h0 should be as close as possible to unity (i.e. minimum separation). This can be achieved by balancing the gravitational force exerted by the gel network with the bulk “elastic” modulus of the suspension. The latter is related to the high frequency modulus G󸀠 (see Chapter 1 on rheology). As mentioned before, dilatant sediments are produced with suspensions which are colloidally stable. These dilatant sediments are difficult to redisperse and hence shear stress =

134 | 3 Solid/liquid dispersions (suspensions)

they must be prevented from forming on standing. Several methods may be applied to prevent sedimentation and formation of clays or cakes in a suspension and these are summarized below. The first method is to balance of the density of the disperse phase and medium. It is clear from Stokes law that if ∆ρ = 0, v0 = 0. This method can be applied only when the density of the particles is not much larger than that of the medium (e.g. ∆ρ ≈ 0.1). For example, with many organic solids having densities in the region of 1.1–1.2 g cm−3 , by dissolving an inert substance in the continuous phase such as sugar or glycerol one may achieve density matching. However, apart from its limitation to particles with density not much larger than the medium, the method is not very practical since density matching can only occur at one temperature. Liquids usually have large thermal expansion, whereas densities of solids vary comparatively little with temperature. The second method is to reduce particle size. As mentioned above, if R is significantly reduced (to values below 0.1 µm), Brownian diffusion can overcome the gravity force and no sedimentation occurs. This is the principle of formation of nanosuspensions. In this case the Brownian diffusion kT can overcome the gravity force (4/3)πR3 ∆ρgh as shown in Fig. 3.15. The most practical method is to use high molecular weight thickeners. As discussed above, high molecular weight materials such as hydroxyethyl cellulose or xanthan gum when added above a critical concentration (at which polymer coil overlap occurs) will produce very high viscosity at low stresses or shear rates (usually in excess of several hundred Pas) and this will prevent sedimentation of the particles. In relatively concentrated suspensions, the situation becomes more complex, since the polymer molecules may lead to flocculation of the suspension, by bridging, depletion (see below), etc. Moreover, the polymer chains at high concentrations tend to interact with each other above a critical concentration C∗ (the so-called “semidilute” region discussed above). Such interaction leads to viscoelasticity (see Chapter 1 on rheology), whereby the flow behaviour shows an elastic component characterized by an elastic modulus G󸀠 (energy elastically stored during deformation) and a viscous component G󸀠󸀠 (loss modulus resulting from energy dissipation during flow). The elastic behaviour of such relatively concentrated polymer solutions plays a major role in reducing settling and prevention of formation of dilatant clays. A good example of such viscoelastic polymer solution is that of xanthan gum, a high molecular weight polymer (molecular weight in excess of 106 Daltons). This polymer shows viscoelasticity at relatively low concentration (< 0.1 %) as a result of the interaction of the polymer chains, which are very long. This polymer is very effective in reducing settling of coarse suspensions at low concentrations (in the region of 0.1–04 % depending on the volume fraction of the suspension). It should be mentioned that to arrive at the optimum concentration of polymer required to prevent settling and claying of a suspension concentrate, one needs to evaluate the rheological characteristics of the polymer solution, on the one hand, and the whole system (suspension and polymer) on the other (see Chapter 1). This

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

135

will provide the formulator with the necessary information on the interaction of polymer coils with each other and with the suspended particles. Moreover, one should be careful in applying high molecular weight materials to prevent settling, depending on the system. For example, with suspensions used as coatings, such as paints, time effects in flow are very important. In this case, the polymer used for preventing settling must show reversible time dependency of viscosity (i.e. thixotropy). In other words, the polymer used has to be shear thinning on application to ensure uniform coating, but once the shearing force is removed, the viscosity has to build up quickly in order to prevent undesirable flow. On the other hand, with suspensions that need to be diluted on application, such as agrochemical suspension concentrates, it is necessary to choose a polymer that disperses readily into water, without the need for vigorous agitation. One should also consider the temperature variation of the rheology of the polymer solution. If the rheology undergoes considerable change with temperature, the suspension may clay at high temperatures. One should also consider the ageing of the polymer, which may result from chemical or microbiological degradation. This would result in a reduction of viscosity with time, and settling or claying may occur on prolonged storage. It is worth mentioning that the use of thickeners to reduce sedimentation suffers from some serious drawbacks. Firstly, for suspensions that require dilution before application, such as agrochemical suspension concentrates, the high viscosity of the system may require vigorous agitation. Secondly, this high viscosity may prevent spontaneity of dispersion on dilution. Thirdly, with most thickened suspensions the viscosity may show a reduction with increasing temperature. This means that on storage of the suspension at high temperature (e.g. 50 °C), sedimentation may occur resulting in separation of the formulation. Such a problem may be overcome by using a mixture of high molecular weight polymer (e.g. xanthan gum) and inert finely divided solids such as clays or oxides as will be discussed below. An alternative method to the use of high molecular weight polymers is to add “inert” fine particles in the continuous phase. Several fine particulate inorganic material produce “gels” when dispersed in aqueous media, e.g. sodium montmorillonite or silica. These particulate materials produce three-dimensional structures in the continuous phase as a result of interparticle interaction. For example, sodium montmorillonite (referred to as swellable clay) forms gels at low and intermediate electrolyte concentrations. This can be understood from a knowledge of the structure of the clay particles. They consist of plate-like particles consisting of an octahedral alumina sheet sandwiched between two tetrahedral silica sheets. Several atoms in the crystal lattice undergo substitution from ones of high valency to ones of lower valency. This replacement is usually referred to as isomorphic substitution: an atom of lower positive valence replaces one of higher valence, resulting in a deficit of positive charge or excess of negative charge. This excess of negative layer charge is compensated by adsorption at the layer surfaces of cations that are too big to be accommodated in the crystal. In aqueous solution, the compensation cations on the layer surfaces may be

136 | 3 Solid/liquid dispersions (suspensions)

exchanged by other cations in solution, and hence may be referred to as exchangeable cations. With montmorillonite, the exchangeable cations are located on each side of the layer in the stack, i.e. they are present in the external surfaces as well as between the layers. This causes a slight increase of the local spacing from about 9.13 Å to about 9.6 Å; the difference depends on the nature of the counterion. When montmorillonite clays are placed in contact with water or water vapour the water molecules penetrate between the layers, causing interlayer swelling or (intra)crystalline swelling. This leads to a further increase in the basal spacing to 12.5–20 Å, depending on the type of clay and cation. This interlayer swelling leads, at most, to doubling of the volume of dry clay when four layers of water are adsorbed. The much larger degree of swelling, which is the driving force for “gel” formation (at low electrolyte concentration), is due to osmotic swelling. It has been suggested that swelling of montmorillonite clays is due to the electrostatic double layers that are produced between the charge layers and cations. This is certainly the case at low electrolyte concentration where the double layer extension (thickness) is large. However, the flat surfaces are not the only surfaces of the plate-like clay particles, they also expose an edge surface. The atomic structure of the edge surfaces is entirely different from that of the flat-layer surfaces. At the edges, the tetrahedral silica sheets and the octahedral alumina sheets are disrupted, and the primary bonds are broken. The situation is analogous to that of the surface of silica and alumina particles in aqueous solution. On such edges, therefore, an electric double layer is created by adsorption of potential determining ions (H+ and OH− ) and one may, therefore identify an isoelectric point (IEP) as the point of zero charge (pzc) for these edges. With broken octahedral sheets at the edge, the surface behaves as Al–OH with an IEP in the region of pH 7–9. Thus in most cases the edges become negatively charged above pH 9 and positively charged below pH 9. Van Olphen [45] suggested a mechanism of gel formation of montmorillonite involving interaction of the oppositely charged double layers at the faces and edges of the clay particles. This structure, which is usually referred to as a “card-house” structure, was considered to be the reason for the formation of the voluminous clay gel. However, Norrish suggested that the voluminous gel is the result of the extended double layers, particularly at low electrolyte concentrations. A schematic picture of gel formation produced by double layer expansion and “card-house” structure is shown in Fig. 3.25. Evidence for the above picture was obtained by Van Olphen [45] who measured the yield value of 3.22 % montmorillonite dispersions as a function of NaCl concentration as shown in Fig. 3.26. When C = 0, the double layers are extended and gel formation is due to double layer overlap (Fig. 3.25 (a)). First addition of NaCl causes compression of the double layers and hence the yield value decreases very rapidly. At intermediate NaCl concentrations, gel formation occurs as a result of face-to-edge association (house of cards structure) (Fig. 3.25 (b)) and the yield value increases very rapidly with increasing NaCl concentration. If the NaCl concentration is increased further, face-toface association may occur and the yield value decreases (the gel is destroyed).

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

(a) Gels produced by Double layer overlap

137

(b) Gels produced by edge-to-face association

Fig. 3.25: Schematic representation of gel formation in aqueous clay dispersions.

σß/Pa

18 16 14 12 10 8 6 4 2 0 0

10

20

30

40

50

60

70

80

CNaCl/meq dm³ Fig. 3.26: Variation of yield value with NaCl concentration for 3.22 % sodium montmorillonite dispersions.

Finely divided silica such as Aerosil 200 (produced by Degussa) produces gel structures by simple association (by van der Waals attraction) of the particles into chains and cross chains. When incorporated in the continuous phase of a suspension, these gels prevent sedimentation. In aqueous media, the gel strength depends on the pH and electrolyte concentration. By combining thickeners such as hydroxyethyl cellulose or xanthan gum with particulate solids such as sodium montmorillonite, a more robust gel structure can be produced. By using such mixtures, the concentration of the polymer can be reduced, thus overcoming the problem of dispersion on dilution (e.g. with many agrochemical suspension concentrates). This gel structure may be less temperature dependent and could be optimized by controlling the ratio of the polymer and the particles. If these combinations of say sodium montmorillonite and a polymer such as hydroxyethyl cellulose, polyvinyl alcohol (PVA) or xanthan gum, are balanced properly, they can provide a “three-dimensional structure”, which entraps all the particles and stops settling and formation of dilatant clays. The mechanism of gelation of such combined

138 | 3 Solid/liquid dispersions (suspensions)

systems depends to a large extent on the nature of the solid particles, the polymer and the conditions. If the polymer adsorbs on the particle surface (e.g. PVA on sodium montmorillonite or silica) a three-dimensional network may be formed by polymer bridging. Under conditions of incomplete coverage of the particles by the polymer, the latter becomes simultaneously adsorbed on two or more particles. In other words the polymer chains act as “bridges” or “links” between the particles. Another method for reducing sedimentation is by controlled flocculation (“selfstructured” systems). For systems where the stabilization mechanism is electrostatic in nature, for example those stabilized by ionic surfactants or polyelectrolytes, the Deryaguin–Landau–Verwey–Overbeek (DLVO) theory [12, 13] predicts the appearance of a secondary attractive minimum at large particle separations (see Chapter 6, Vol. 1). This attractive minimum can reach sufficient values, in particular for large (> 1 µm) and asymmetric particles, for weak flocculation to occur. The depth of this minimum does not only depend on particle size but also on the Hamaker constant, the surface (or zeta) potential and electrolyte concentration and valency. Thus by careful control of zeta potential and electrolyte concentration, it is possible to arrive at a secondary minimum of sufficient depth for weak flocculation. This results in the formation of a weakly structured “gel” throughout the suspension. This self-structured gel can prevent sedimentation and formation of dilatant clays is sufficiently deep for weak flocculation to occur. For systems stabilized with nonionic surfactants or macromolecules, the energy– distance curve also shows a minimum whose depth depends on particle size and shape, Hamaker constant and adsorbed layer thickness δ. Thus for a given particulate system, having a given particle size and shape and Hamaker constant, the minimum depth can be controlled by varying the adsorbed layer thickness δ. This was illustrated in Chapter 14, Vol. 1. It should be mentioned that the minimum depth needed to induce flocculation depends on the volume fraction of the suspension. This can be understood if one considers the balance between the interaction energy and entropy terms in the free energy of flocculation, i.e., ∆Gfloc = ∆Gh − T∆Sh , (3.100) where ∆Gh is the interaction energy term (which is negative), which is determined by the depth of the minimum in the energy–distance curve and ∆Sh is the entropy loss on flocculation. On flocculation configurational entropy is lost and ∆Sh is therefore negative. This means that the term T∆Sh is positive (i.e. it opposes flocculation). The condition for flocculation is ∆G ≤ 0, and therefore the ∆Gh required for flocculation depends on the magnitude of the T∆Sh term. The latter term decreases with increasing volume fraction ϕ, and hence the higher the ϕ value, the smaller the ∆Gh required for flocculation. This means that with more concentrated suspensions weak flocculation of a sterically stabilized suspension occurs at lower minimum depth.

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

139

As discussed in Chapter 14, Vol. 1, addition of free nonadsorbing polymer can produce weak flocculation above a critical volume fraction of the free polymer, ϕ+p , which depends on its molecular weight and the volume fraction of suspension. This weak flocculation produces a “gel” structure that reduces sedimentation. According to Asakura and Oosawa [15, 16], when two particles approach each other within a distance of separation that is smaller than the diameter of the free polymer coil, exclusion of the polymer molecules from the interstices between the particles takes place, leading to the formation of a polymer-free zone. This was illustrated in Chapter 14, Vol. 1, which shows the situation below and above ϕ+p . As a result of this process, an attractive energy, associated with the lower osmotic pressure in the region between the particles, is produced. The above phenomenon of flocculation can be applied to the prevention of settling and claying by forming an open structure that, under some conditions, can fill the whole volume of the suspension. Above ϕ+p , the suspension becomes weakly flocculated, and the extent of flocculation increases with a further increase in the concentration of free nonadsorbing polymer. This is illustrated for a suspension of an agrochemical (ethirimol, a fungicide) that is stabilized by a graft copolymer of poly(methyl methacrylate)/methacrylic acid with poly(ethylene oxide) side chains to which poly(ethylene oxide) (PEO) with M = 20 000, 35 000 or 90 000 is added for flocculation [8]. Rheological measurements showed that above ϕ+p a rapid increase in the yield value is produced. This is shown in Fig. 3.27. Above ϕ+p one would expect a significant reduction in the formation of dilatant sediments. This is illustrated in

18 16 14 PEO 90000 Υβ (N m–2)

12 10

PEO 35000

8 PEO 20000 6 4 2 0

0.01 0.02 0.03 0.04 0.05 φp

Fig. 3.27: Variation of yield value with PEO concentration.

140 | 3 Solid/liquid dispersions (suspensions)

Sediment height

10

T

8

Sediment height

T

6 4 2

Redispersion 0

0.05

0.1 φp

Fig. 3.28: Variation of sediment height and redispersion with ϕp for PEO 20 000.

Fig. 3.28, which shows a plot of sediment height and number of revolutions for redispersion as a function of ϕp (for PEO 20 000). It is clear that addition of PEO results in weak flocculation and redispersion becomes easier. This redispersion is maintained up to ϕp = 0.04 above which it becomes more difficult due to the increase in viscosity. Another example for the application of depletion flocculation was obtained for the same suspension but using hydroxyethyl cellulose (HEC) with various molecular weights. The weak flocculation was studied using oscillatory measurements. Fig. 3.29 shows the variation of the complex modulus G∗ with ϕp . Above a critical ϕp value (that depends on the molecular weight of HEC), G∗ increases very rapidly with a further increase in ϕp . When ϕp reaches an optimum concentration, sedimentation is prevented. This is illustrated in Fig. 3.30, which shows the sediment volume in 10 cm cylinders as a function ϕp for various volume fractions of the suspension ϕs . At sufficiently high volume fraction of the suspensions ϕs and high volume fraction of free polymer ϕp , a 100 % sediment volume is reached and this is effective in eliminating sedimentation and formation of dilatant sediments. Mw = 223,000

100

G* (Pa)

124,000 50 70,000

0.002 0.004 0.006 0.008 0.0010 φp

Fig. 3.29: Variation of G∗ with ϕp for HEC with various molecular weights.

3.5 Sedimentation of suspensions and prevention of formation of hard sediments |

Turbid

Flocculation starts

141

φs = 0.3

10

Sediment volume

φs = 0.2 8

6 φs = 0.1 4 High sediment volume with no claying

2

0.005

0.01

0.015

0.02

0.025

φp Fig. 3.30: Variation of sediment volume with ϕp for HEC (M = 70 000).

water surfactant Hexagonal

water Lamellar

½

½

¼

¼ ½

½

Cubic

Fig. 3.31: Schematic picture of liquid crystalline phases.

Another method that can be applied to reduce sedimentation is to use liquid crystalline phases. Surfactants produce liquid crystalline phases at high concentrations [6, 8]. Three main types of liquid crystals can be identified as illustrated in Fig. 3.31: Hexagonal phase (sometimes referred to as middle phase), cubic phase and lamel-

142 | 3 Solid/liquid dispersions (suspensions)

lar (neat phase). All these structures are highly viscous and they also show elastic response. If produced in the continuous phase of suspensions, they can eliminate sedimentation of the particles. These liquid crystalline phases are particularly useful for applications in liquid detergents which contain high surfactant concentrations. Their presence reduces sedimentation of the coarse builder particles (phosphates and silicates).

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

Tadros TF. Advances Colloid and Interface Science. 1980;12:141. Tadros T. Dispersion of powders in liquids and stabilisation of suspensions. Weinheim: WileyVCH; 2012. Tadros T. Formulation of disperse systems. Weinheim: Wiley-VCH; 2014. Tadros T. Suspensions. In: Tadros T, editor. Encyclopedia of colloid and interface science. Berlin: Springer; 2013. Tadros T. Nanodispersions. Berlin: De Gruyter; 2016. Tadros T. Applied surfactants. Weinheim: Wiley-VCH; 2005. Tadros T. Interfacial phenomena and colloid stability, Vol. 1. Berlin: De Gruyter; 2015. Tadros T. Suspension concentrates. Berlin: De Gruyter; 2017. Tadros T. Rheology of Dispersions. Weinheim: Wiley-VCH; 2010. Rehbinder PA. Colloid J USSR. 1958;20:493. Tadros T. Colloids in paints. Weinheim: Wiley-VCH; 2010. Deryaguin BV, Landau L. Acta Physicochem USSR. 1941;14:633. Verwey EJW, Overbeek JTG. Theory of stability of lyophobic colloids. Amsterdam: Elsevier; 1948. Napper DH. Polymeric stabilisation of colloidal dispersions. London: Academic Press; 1983. Asakura A, Oosawa F. J Chem Phys. 1954;22:1255. Asakura A, Oosawa F. J Polymer Sci. 1958;33:183. Thompson W (Lord Kelvin). Phil Mag. 1871;42:448. Lifshitz EM, Slesov VV. Soviet Physics JETP. 1959;35:331. Wagner C. Z Electrochem. 1961;35:581. Kabalnov AS, Schukin ED. Adv Colloid Interface Sci. 1992;38:69. Kabalnov AS, Makarov KN, Pertsov AV, Shchukin ED. J Colloid Interface Sci. 1990;138:98. Taylor P. Colloids and Surfaces A. 1995;99:175. Ni Y, Pelura TJ, Sklenar TA, Kinner RA, Song D. Art Cells Blood Subs Immob Biorech. 1994;22:1307. Karaboni S, van Os NM, Esselink K, Hilbers PAJ. Langmuir. 1993;9:1175. Soma J, Papadadopoulos KD. J Colloid Interface Sci. 1996;181:225. Kabalanov AS. Langmuir. 1994;10:680. Taylor P, Ottewill RH. Colloids and Surfaces A. 1994;88:303. Anainsson EAG, Wall SN, Almagren M, Hoffmann H, Kielmann I, Ulbricht W, Zana R, Lang J, Tondre C. J Phys Chem. 1976;80:905. Frank FC. Disc Faraday Soc. 1949;5:48, 67. Cabrera N, Burton W. Disc Faraday Soc. 1949;5:33, 40. Cabrera N, Vermilyea DA. Proceedings International Conference on Crystal Growth. London: John Wiley and Sons; 1958. p. 393.

References | 143

[32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]

Pearson JT, Varney G. J Pharm Pharmac Suppl. 1969;21:60. Pearson JT, Varney G. J Pharm Pharmac Suppl. 1973;25:62. Pfeiffer PR. J Pharm Pharmac. 1971;23:75. Michaels AS, Golville A Jr. J Phys Chem. 1960;64:13. Michaels AS, Tausch FW Jr. J Phys Chem. 1961;65:1730. Michaels AS, Brian PLT, Bech WF. Chem Phys Appl Surface Active Substances. Proceedings 4th Int. Congress, 2. 1967, 1053. Simonelli PA, Mehta SC, Higuchi WI. J Pharm Sci. 1970;59:633. Carless JE, Moustafa MA, Rapson HDC. J Pharm Pharmac. 1968;20:630. Kynch GJ. Trans Faraday Soc. 1952;48:166. Buscall R, Goodwin JW, Ottewill RH, Tadros TF. J Colloid Interface Sci. 1982;85:78. Krieger IM, Advances Colloid and Interface Sci. 1971;3:45. Michaels AS, Bolger JC. Ind Eng Chem. 1962;1:24. Richardson JF, Zaki WN. Trans Inst Chem Engrs. 1954;32:35. van Olphen H. Clay colloid chemistry. New York: Wiley; 1963.

4 Liquid/liquid dispersions (emulsions) 4.1 Introduction Emulsions are a class of disperse systems consisting of two immiscible liquids [1–4]. The liquid droplets (the disperse phase) are dispersed in a liquid medium (the continuous phase). Several classes may be distinguished: oil-in-water (O/W); water-in-oil (W/O); oil-in-oil (O/O). The latter class may be exemplified by an emulsion consisting of a polar oil (e.g. propylene glycol) dispersed in a nonpolar oil (paraffinic oil) and vice versa. To disperse two immiscible liquids one needs a third component, namely the emulsifier. The choice of the emulsifier is crucial in formation of the emulsion and its long-term stability [1–4]. There are many examples one could quote of naturally occurring emulsions: milk and the O/W and W/O emulsions associated with oil-bearing rocks are just two examples. Emulsion types can be classified on the basis of the nature of the emulsifier or the structure of the system as shown in Tab. 4.1. Tab. 4.1: Classification of emulsions. Nature of emulsifier – – – – – – – – –

Simple molecules and ions Nonionic surfactants Ionic surfactants Surfactant mixtures Nonionic polymers Polyelectrolytes Mixed polymers and surfactants Liquid crystalline phases Solid particles

Structure of the system – – – – – – –

Nature of internal and external phase: O/W, W/O Nanoemulsions Micellar emulsions (microemulsions) Macroemulsions Bilayer droplets Double and multiple emulsions Mixed emulsions

Several types of emulsifiers can be distinguished. The simplest type is ions such as OH− which can be specifically adsorbed on the emulsion droplet thus producing a charge. An electrical double layer can be produced which provides electrostatic repulsion. This has been demonstrated with very dilute O/W emulsions by removing any acidity. Clearly that process is not practical. The most effective emulsifiers are nonionic surfactants, such as alcohol ethoxylates with the general formula Cx H2x+1 –O–(CH2 –CH2 –O)n H, which can be used to emulsify oil in water or water in oil. In addition they can stabilize the emulsion against flocculation and coalescence. Ionic surfactants such as sodium dodecyl sulphate can also be used as emulsifiers (for O/W) but the system is sensitive to the presence of electrolytes. Surfactant mixtures, e.g. ionic and nonionic or mixtures of nonionic surfactants can be more effective https://doi.org/10.1515/9783110541953-005

146 | 4 Liquid/liquid dispersions (emulsions)

in emulsification and stabilization of the emulsion. Nonionic polymers, sometimes referred to as polymeric surfactants, e.g. Pluronics with the general formula HO– (CH2 –CH2 –O)n –(CH2 –CH(CH3 )–O)m –(CH2 –CH2 –O)n –OH or PEO–PPO–PEO are more effective in stabilizing the emulsion but they may suffer from being difficult emulsify (to produce small droplets) unless high energy is applied for the process. Polyelectrolytes such as poly(methacrylic acid) can also be applied as emulsifiers. Mixtures of polymers and surfactants are ideal in achieving ease of emulsification and stabilization of the emulsion. Lamellar liquid crystalline phases that can be produced using surfactant mixtures are very effective in emulsion stabilization. Solid particles that can accumulate at the O/W interface can also be used for emulsion stabilization. These are referred to as Pickering emulsions: particles are partially wetted by the oil phase and partially wetted by the aqueous phase. Emulsions can also be classified on the basis of the structure of the system. Emulsions (O/W or W/O) having a size range of 0.1–5 µm with an average of 1–2 µm are generally termed macroemulsions. These systems are usually opaque or milky due to the large size of the droplets and the significant difference in refractive index between the oil and water phases. Those usually with a size range 20–100 nm are classified as nanoemulsions. Like macroemulsions they are only kinetically stable. They can be transparent, translucent or opaque depending on the droplet size, the refractive index difference between the two phases and the volume fraction of the disperse phase. Another class are double and multiple emulsions that are emulsions-of-emulsions, W/O/W and O/W/O systems and they will be described separately in Chapter 5 in this volume. They are usually prepared using a two stage process. For example a W/O/W multiple emulsion is prepared by forming a W/O emulsion which is then emulsified in water to form the final multiple emulsion. Mixed emulsions are systems consisting of two different disperse droplets that do not mix in a continuous medium. A special class are micellar emulsions or microemulsions, which will be described in Chapter 9 in this volume. These usually have the size range 5–50 nm. They are thermodynamically stable and strictly speaking they should not be described as emulsions. A better description is “swollen micelles” or “micellar systems”. The present chapter will only deal with macroemulsions, their formation and stability. Several breakdown processes may occur on storage depending on droplet size distribution and density difference between the droplets and the medium, magnitude of the attractive versus repulsive forces which determines flocculation, solubility of the disperse droplets and the particle size distribution which determines Ostwald ripening, stability of the liquid film between the droplets that determines coalescence, phase inversion, where the two phases exchange, e.g. an O/W emulsion inverts to W/O and vice versa. Phase inversion can be catastrophic as is the case when the oil phase in an O/W emulsion exceeds a critical value. The inversion can be transient when for example the emulsion is subjected to temperature increase. The various breakdown processes are illustrated in Fig. 4.1. The physical phenomena involved in each breakdown process are not simple and they require analysis of

4.1 Introduction

| 147

the various surface forces involved. In addition, the above processes may take place simultaneously rather than consecutively and this complicates the analysis. Model emulsions with monodisperse droplets cannot be easily produced and hence any theoretical treatment must take into account the effect of droplet size distribution. Theories that take into account the polydispersity of the system are complex and in many cases only numerical solutions are possible. In addition, measuring surfactant and polymer adsorption in an emulsion is not easy and one has to extract such information from measurements at a planer interface.

Sedimentation Creaming

Flocculation

Phase inversion

Ostwald ripening Coalescence

Fig. 4.1: Schematic representation of the various breakdown processes in emulsions.

A summary of each of the above breakdown processes is given below and details of each process and methods for its prevention are given in separate chapters. Creaming and sedimentation, with no change in droplet size, result from external forces, usually gravitational or centrifugal. When such forces exceed the thermal motion of the droplets (Brownian motion), a concentration gradient builds up in the system with the larger droplets moving faster to the top (if their density is lower than that of the medium) or to the bottom (if their density is larger than that of the medium) of the container. In the limiting cases, the droplets may form a close-packed (random or ordered) array at the top or bottom of the system with the remainder of the volume occupied by the continuous liquid phase. Flocculation refers to aggregation of the droplets (without any change in primary droplet size) into larger units. It is the result of van der Waals attraction, which is universal with all disperse systems. The main force of attraction arises from the London dispersion force that results from charge fluctuations of the atoms or molecules in the disperse droplets. The van der Waals attraction increases with decreasing separation

148 | 4 Liquid/liquid dispersions (emulsions)

distance between the droplets and at small separation distances the attraction becomes very strong, resulting in droplet aggregation or flocculation. The latter occurs when there is not sufficient repulsion to keep the droplets apart to distances where the van der Waals attraction is weak. Flocculation may be “strong” or “weak”, depending on the magnitude of the attractive energy involved. In cases where the net attractive forces are relatively weak, an equilibrium degree of flocculation may be achieved (so-called weak flocculation), associated with the reversible nature of the aggregation process. The exact nature of the equilibrium state depends on the characteristics of the system. One can envisage the build-up of aggregate size distribution and an equilibrium may be established between single droplets and large aggregates. With a strongly flocculated system, one refers to a system in which all the droplets are present in aggregates due to the strong van der Waals attraction between the droplets. Ostwald ripening (disproportionation) results from the finite solubility of the liquid phases. Liquids which are referred to as being immiscible often have mutual solubilities which are not negligible. With emulsions which are usually polydisperse, the smaller droplets will have larger solubility when compared with the larger ones (due to curvature effects). With time, the smaller droplets disappear and their molecules diffuse to the bulk and become deposited on the larger droplets. With time the droplet size distribution shifts to larger values. Coalescence refers to the process of thinning and disruption of the liquid film between the droplets which may be present in a creamed or sedimented layer, in a floc or simply during droplet collision, with the result of fusion of two or more droplets into larger ones. This process of coalescence results in considerable change of the droplet size distribution, which shifts to larger sizes. The limiting case for coalescence is the complete separation of the emulsion into two distinct liquid phases. The thinning and disruption of the liquid film between the droplets is determined by the relative magnitudes of the attractive versus repulsive forces. To prevent coalescence, the repulsive forces must exceed the van der Waals attraction, thus preventing film rupture. Phase inversion refers to the process whereby there will be an exchange between the disperse phase and the medium. For example, with time or change of conditions an O/W emulsion may invert to a W/O emulsion. In many cases, phase inversion passes through a transition state in which multiple emulsions are produced. For example with an O/W emulsion, the aqueous continuous phase may become emulsified in the oil droplets forming a W/O/W multiple emulsion. This process may continue until all the continuous phase is emulsified into the oil phase thus producing a W/O emulsion.

4.2 Thermodynamics of emulsion formation and breakdown When two immiscible phases α and β (oil and water) come into contact, an interfacial region develops. The interfacial region is not a layer that is one molecule thick; it is a region with thickness δ with properties different from the two bulk phases α and β. In

4.2 Thermodynamics of emulsion formation and breakdown | 149

bringing phases α and β into contact, the interfacial regions of these phases undergo some changes, resulting in a concomitant change in the internal energy. If we were to move a probe from the interior of α to that of β, one would at some distance from the interface start to observe deviations in composition, in density, in structure; the closer to phase β the larger the deviations until eventually the probe arrives in the homogeneous phase β. The thickness of the transition layer will depend on the nature of the interfaces and on other factors. Gibbs [5] considered the two phases α and β to have uniform thermodynamic properties up to the interfacial region. He assumed a mathematical plane Z σ in the interfacial region in order to define the interfacial tension γ. A schematic representation of the interfacial region and the Gibbs mathematical plane is given in Fig. 4.2.

α

Uniform thermodynamic properties

β

Uniform thermodynamic properties

Mathematical diving Plane Z (Gibbs dividing line)

Fig. 4.2: The Gibbs dividing line.

Using Gibbs’ model, it is possible to obtain a definition of the interfacial tension γ. The surface free energy dG σ is made of three components: an entropy term, S σ dT; an interfacial energy term, A dγ; a composition term ∑ n i dμ i (n i is the number of moles of component i with chemical potential μ i ). The Gibbs–Deuhem equation is, dG σ = −S σ dT + A dγ + ∑ n i dμ i .

(4.1)

At constant temperature and composition, dG σ = A dγ, γ=(

∂G σ ) . ∂A T,n i

(4.2)

For a stable interface γ is positive, i.e. if the interfacial area increases, G σ increases. Note that γ is energy per unit area (mJ m−2 ) which is dimensionally equivalent to force per unit length (mN m−1 ), the unit usually used to define surface or interfacial tension. For a curved interface, one should consider the effect of the radius of curvature. Fortunately, γ for a curved interface is estimated to be very close to that of a planer surface, unless the droplets are very small (< 10 nm).

150 | 4 Liquid/liquid dispersions (emulsions)

Curved interfaces produce some other important physical phenomena which affect emulsion properties, e.g. the Laplace pressure ∆p which is determined by the radii of curvature of the droplets, 1 1 ∆p = γ ( + ), (4.3) r1 r2 where r1 and r2 are the two principal radii of curvature. For a perfectly spherical droplet r1 = r2 = r and, ∆p =

2γ . r

(4.4)

For a hydrocarbon droplet with radius 100 nm, and γO/W = 50 mN m−1 , ∆p ≈ 106 Pa (≈ 10 atm). Consider a system in which an oil is represented by a large drop 2 of area A1 immersed in a liquid 2, which is now subdivided into a large number of smaller droplets with total area A2 (A2 ≫ A1 ) as shown in Fig. 4.3. The interfacial tension γ12 is the same for the large and smaller droplets since the latter are generally in the region of 0.1 to few µm.

1

I

Formation 2 Breakdown (flocc + cool)

2 1

II

Fig. 4.3: Schematic representation of emulsion formation and breakdown.

The change in free energy in going from state I to state II is made up of two contributions: A surface energy term (that is positive) that is equal to ∆Aγ12 (where ∆A = A2 − A1 ) and an entropy of dispersions term which is also positive (since producing a large number of droplets is accompanied by an increase in configurational entropy) which is equal to T∆Sconf . From the second law of thermodynamics, ∆Gform = ∆Aγ12 − T∆Sconf .

(4.5)

In most cases ∆Aγ12 ≫ T∆Sconf , which means that ∆Gform is positive, i.e. the formation of emulsions is nonspontaneous and the system is thermodynamically unstable. In the absence of any stabilization mechanism, the emulsion will break by flocculation and coalescence as illustrated in Fig. 4.4 by the full line. In this case there are no free energy barriers either to flocculation or coalescence. The kinetics of both breakdown processes is diffusion controlled: in the case of flocculation, by the diffusion of the droplets, and in the case of coalescence, by diffusion of molecules of liquid 1 out of the thin liquid film formed between two contacting droplets of liquid 2. The dashed line in Fig. 4.4 corresponds to the case where sedimentation or creaming is superimposed upon the flocculation and coalescence. The

4.2 Thermodynamics of emulsion formation and breakdown | 151

GIV GII

GI GIII

II or IV

I or III

Fig. 4.4: Free energy path in emulsion breakdown: —, flocc. + coal., ---, flocc. + coal. + sed., ···, flocc. + coal. + sed. + Ostwald ripening.

final state of the system (state III) is now the more familiar one of two liquid phases separated by a flat interface. The dotted line in Fig. 4.4 represents the situation if, in addition to the above effects, Ostwald ripening has to be taken into account. This occurs if the initial state IV is polydisperse and the liquids have a finite mutual solubility. In the presence of a stabilizer (surfactant and/or polymer), an energy barrier is created between the droplets and therefore the reversal from state II to state I becomes noncontinuous as a result of the presence of these energy barriers. This is illustrated in Fig. 4.5. In the presence of the above energy barriers, the system becomes kinetically stable. Strictly speaking, the ∆Gflocc and ∆Gcoal are activation free energies. The intermediate state V is a metastable state and represents a flocculated emulsion that has undergone no coalescence. If ∆Gcoal is sufficiently high, it may stay in this state indefinitely. Similarly, state II is also an unstable state, and if ∆Gflocc is sufficiently high, the stable, dispersed state may persist indefinitely. However, states II and V in these cases represent states of kinetic stability, rather than true thermodynamic stability. The dashed curve in Fig. 4.5 represents the situation if there is no free energy barrier to

Gflocca

GII

Gcoala

Gflocc Gbreak

GV Gcoal GI

II

V

I

Fig. 4.5: Schematic representation of free energy path for breakdown (flocculation and coalescence) for systems containing an energy barrier.

152 | 4 Liquid/liquid dispersions (emulsions)

flocculation, but there is a large barrier to coalescence. Such a situation would arise for droplets stabilized, for example, by an adsorbed (neutral) polymer. In this case only the long-range van der Waals forces and the short-range steric repulsion forces (see Chapter 13, Vol. 1) are operating. If ∆Gflocc is not too large (say < 10 kT per droplet), then the flocculation is reversible and an equilibrium is set up as will be discussed below. From Fig. 4.5, it can be seen that, ∆Gbreak = ∆Gflocc + ∆Gcoal .

(4.6)

It is worth considering the individual contributions to ∆Gflocc and ∆Gcoal in the light of equations (4.5) and (4.6). The excess interfacial free energy G σ associated with the presence of an interface is given by, G σ = ∆Aγ12 + ∑i μ i n σi .

(4.7)

If an interface disappears due to coalescence, the change in free energy ∆Gcoal is simply given by, ∆Gcoal = −∆(γ12 ∆A). (4.8) The term ∑i μ i n σi disappears since the chemical potential of species I is the same in either bulk phase and in the interface. Considering equations (4.5), (4.6) and (4.8) leads to the conclusion that, ∆Gflocc = ∆A∆γ12 − T∆Sconf . (4.9) Since, ∆(∆Aγ12 ) = γ12 ∆∆A + ∆A∆γ12 ,

(4.10)

then ∆Gflocc is made up of two terms: the ∆A∆γ12 associated with the change in interfacial tension in the contact region of two droplets (i.e., for the two surfaces in the film separating the droplets) and the T∆Sconf term associated with the change in configurational entropy. Both terms are negative and in most cases the ∆A∆γ12 term dominates, so that ∆Gflocc is negative, i.e. flocculation is thermodynamically spontaneous. However, if (∆A∆γ12 ) is less than (T∆Sconf ), then ∆Gflocc is positive and the emulsion is then thermodynamically stable against flocculation. This means that flocculation will not occur and the emulsion has to be concentrated by creaming/sedimentation or centrifugation before coalescence can occur. The condition |∆A∆γ12 | < |T∆Sconf | may be realized if ∆γ12 is small, i.e., the secondary minimum in the energy–distance curve is small. Since |T∆Sconf | decreases as the droplet number concentration increases, one can envisage that at some initial droplet concentration ∆Gflocc = 0, i.e. below this concentration the emulsion is thermodynamically stable (∆Gflocc positive), but that beyond this concentration the emulsion becomes thermodynamically unstable (∆Gflocc negative) and reversible flocculation occurs.

4.3 Interaction forces between emulsion droplets and factors affecting their stability |

153

4.3 Interaction forces between emulsion droplets and factors affecting their stability Generally speaking there are three main interaction forces between emulsion droplets, namely van der Waals attraction, electrostatic (double layer) repulsion and steric repulsion. These interaction forces were described in detail in Chapters 5, 4 and 13, Vol. 1 respectively. As discussed in Chapter 6, Vol. 1, combining van der Waals attraction with double layer repulsion forms the basis of the theory of colloid stability due to Deryaguin–Landau–Verwey–Overbeek (DLVO theory) [6, 7]. Combining van der Waals attraction with steric repulsion leads to the theory of steric stabilization described in Chapter 13, Vol. 1. The energy–distance curves due to the DLVO theory is shown in Fig. 4.6 for emulsions at low electrolyte concentration (< 10−2 mol dm−3 ; 1 : 1 electrolyte, e.g. NaCl).

G

GT

Ge Gmax GA

Gprimary

h Gsec

Fig. 4.6: Energy–distance curves according to the DLVO theory.

Gelec decays exponentially with h, i.e. Gelec → 0 as h becomes large. GA ∝ 1/h, i.e. GA does not decay to 0 at large h. At long distances of separation, GA > Gelec resulting in a shallow minimum (secondary minimum). At very short distances, GA ≫ Gelec resulting in a deep primary minimum. At intermediate distances, Gelec > GA resulting in an energy maximum, Gmax , whose height depends on ψ0 (or ψd ) and the electrolyte concentration and valency. At low electrolyte concentrations (< 10−2 mol dm−3 for a 1 : 1 electrolyte), Gmax is high (> 25 kT) and this prevents particle aggregation into the primary minimum. The higher the electrolyte concentration (and the higher the valency of the ions), the lower the energy maximum. Under some conditions (depending on electrolyte concentration and particle size), flocculation into the secondary minimum may occur. This flocculation is weak and reversible. By increasing the electrolyte concentration, Gmax decreases until at a given concentration it vanishes and particle coagulation occurs. Emulsions stabilized with nonionic surfactants and polymeric surfactants show steric repulsion that consists of two terms, a mixing term, Gmix (that results from the unfavourable mixing of the chains when these are in good solvent conditions) and an elastic term Gel (resulting from the loss of configurational entropy on chain overlap).

154 | 4 Liquid/liquid dispersions (emulsions)

Gmix

Gel

GT

2δ

δ

Gmin GA

h

Fig. 4.7: Schematic representation of the energy–distance curve for a sterically stabilized emulsion.

Combining of Gmix and Gel with the van der Waals attraction results in the energy– distance curve shown in Fig. 4.7. Gmix increases very sharply with decreasing h, when h < 2δ. Gel increases very sharply with decreasing h, when h < δ. GT versus h shows a minimum, Gmin , at separation distances comparable to 2δ. When h < 2δ, GT shows a rapid increase with decreasing h. The depth of the minimum depends on the Hamaker constant A, the particle radius R and adsorbed layer thickness δ. Gmin increases with increasing A and R. At a given A and R, Gmin decreases with increasing δ (i.e. with increasing molecular weight, Mw , of the stabilizer). This is illustrated in Fig. 4.8 which shows the energy– distance curves as a function of δ/R. The larger the value of δ/R, the smaller the value of Gmin . In this case the system may approach thermodynamic stability, as is the case with nanoemulsions. Increasing δ/R

GT

Gmin

h

Fig. 4.8: Variation of GT with h at various δ/R values.

155

4.4 Mechanism of emulsification and the role of the emulsifier |

4.4 Mechanism of emulsification and the role of the emulsifier As mentioned above, to prepare an emulsion oil, water, surfactant and energy are needed. The composition of the system and its nature (oil-in-water, O/W, or waterin-oil, W/O) is determined by the nature of the emulsifier and the process applied [1–4]. Parameters such as the volume fraction of the disperse phase, ϕ, and the droplet size distribution are determined by the composition of the emulsifier layer around the droplets as well as the process of emulsification. In addition, the emulsifier composition and its nature determine the physical stability of the emulsion such as its flocculation behaviour, Ostwald ripening and coalescence. As mentioned above, emulsion formation is nonspontaneous and the system is thermodynamically unstable. The kinetic stability of the emulsion is determined by the balance of attractive and repulsive forces. It is important to know the process of emulsion formation and the mechanism of emulsification. The role of the emulsifier in droplet deformation and break-up must be considered at a fundamental level. The mechanism of emulsification can be considered from a consideration of the energy required to expand the interface, ∆Aγ (where ∆A is the increase in interfacial area when the bulk oil with area A1 produces a large number of droplets with area A2 ; A2 ≫ A1 and γ is the interfacial tension). Since γ is positive, the energy to expand the interface is large and positive; this energy term cannot be compensated by the small entropy of dispersion T∆Sconf (which is also positive), and the total free energy of formation of an emulsion, ∆Gform given by equation (4.5), is positive. Thus, emulsion formation is nonspontaneous and energy is required to produce the droplets. The formation of large droplets (few µm), as is the case for macroemulsions, is fairly easy and hence high speed stirrers such as the Ultra-Turrax or Silverson Mixer are sufficient to produce the emulsion. In contrast, the formation of small drops (submicron as is the case with nanoemulsions) is difficult and this requires a large amount of surfactant and/or energy. The high energy required for formation of nanoemulsions can be understood from a consideration of the Laplace pressure ∆p (the difference in pressure between inside and outside the droplet) as given by equations (4.3) and (4.4). To break up a drop into smaller ones, it must be strongly deformed and this deformation increases ∆p. Surfactants play major roles in the formation of emulsions: By lowering the interfacial tension, ∆p is reduced and hence the stress needed to break up a drop is reduced. Surfactants also prevent coalescence of newly formed drops (see below). To assess emulsion formation, one usually measures the droplet size distribution using for example laser diffraction techniques. If the number frequency of droplets as a function of droplet diameter d is given by f(d), the nth moment of the distribution is, ∞

S n = ∫ d n f(d) ∂d. 0

(4.11)

156 | 4 Liquid/liquid dispersions (emulsions)

The mean droplet size is defined as the ratio of selected moments of the size distribution, 1/(n−m) ∞ ∫ d n f(d) ∂d d nm = [ 0∞ , (4.12) ] ∫0 d m f(d) ∂d where n and m are integers and n > m and typically n does not exceed 4. Using equation (4.2) one can define several mean average diameters: – The Sauter mean diameter with n = 3 and m = 2, ∞

d32 = [ –

∫0 d3 f(d) ∂d ∞

∫0 d2 f(d) ∂d

].

(4.13)

].

(4.14)

].

(4.15)

The mass mean diameter, ∞

d43 = [ –

∫0 d4 f(d) ∂d ∞

∫0 d3 f(d) ∂d

The number mean diameter, ∞

d10 = [

∫0 d1 f(d) ∂d ∞

∫0 f(d) ∂d

In most cases d32 (the volume/surface average or Sauter mean) is used. The width of the size distribution can be given as the variation coefficient c m which is the standard deviation of the distribution weighted with d m divided by the corresponding average d. Generally, C2 will be used which corresponds to d32 . Another is the specific surface area A (surface area of all emulsion droplets per unit volume of emulsion), 6ϕ A = πS2 = . (4.16) d32 A typical droplet size distribution of an emulsion measured using the light diffraction technique (Malvern Mastersizer) is shown in Fig. 4.9. Surfactants lower the interfacial tension γ and this causes a reduction in droplet size. Droplet size decreases with decreasing γ. For laminar flow, the droplet diameter is proportional to γ; for turbulent inertial regime, the droplet diameter is proportional to γ3/5 . The surfactant can lower the interfacial tension γ0 of a clean oil-water interface to a value γ and, π = γ0 − γ (4.17) where π is the surface pressure. The dependency of π on the surfactant activity a or concentration C is given by the Gibbs equation, as discussed in Chapter 4, Vol. 1, dπ = −dγ = RT Γ d ln a = RT Γ d ln C,

(4.18)

where R is the gas constant, T is the absolute temperature and Γ is the surface excess (number of moles adsorbed per unit area of the interface).

Volume frequency (%)

12 10 8 6 4 2 0 0

5

10

(a)

15

20

25

30

35

Drop size (μm)

Cumulative volume frequency (%)

4.4 Mechanism of emulsification and the role of the emulsifier |

(b)

157

100 80 60 40 20 0

0

5

10 15 20 Drop size (μm)

25

30

Fig. 4.9: Droplet size distribution of an emulsion: (a) volume frequency in discrete classes; (b) cumulative volume distribution.

4

Γ/mgm–2

β-casein, O/W interface

β-casein, emulsion

2

SDS

0

–7

–6

–4

–2 Ceq/wt %

0

Fig. 4.10: Variation of Γ (mg m−2 ) with log Ceq (wt%). The oils are β-casein (O/W interface) toluene, β-casein (emulsions) soybean, SDS benzene.

At high a, the surface excess Γ reaches a plateau value; for many surfactants it is of the order of 3 mg m−2 . Γ increases with increasing surfactant concentration and eventually it reaches a plateau value (saturation adsorption). This is illustrated in Fig. 4.10 for various emulsifiers. It can be seen from Fig. 4.10 that the polymer (β-casein) is more surface active than the surfactant (SDS). The value of C needed to obtain the same Γ is much smaller for the polymer when compared with the surfactant. In contrast, the value of γ reached at full saturation of the interface is lower for a surfactant (mostly in the region of 1–3 mN m−1 depending on the nature of surfactant and oil) when compared with a polymer (with γ values in the region of 10–20 mN m−1 depending on the nature of polymer and oil). This is due to the much closer packing of the small surfactant molecules at the interface when compared with the much larger polymer molecule that adopts tailtrain-loop-tail conformation. The effect of reducing γ on the droplet size is illustrated in Fig. 4.11 which shows a plot of the droplet surface area A and mean drop size d32 as a function of surfactant concentration m for various systems. The amount of surfactant required to produce

158 | 4 Liquid/liquid dispersions (emulsions)

1.5

Nonionic surfactant

4 γ = 3 mNm–1 d32/μm

A/μm–1

1.0 Casinate 0.5

0 0

PVA

5

6

γ = 10 mNm–1

8 –1 10 γ = 20 mNm 15 Soy protein 30 10

m/kgm–3 Fig. 4.11: Variation of A and d32 with m for various surfactant systems.

the smallest drop size will depend on its activity a (concentration) in the bulk which determines the reduction in γ, as discussed above. Another important role of the surfactant is its effect on the interfacial dilational modulus ε, dγ ε= . (4.19) d ln A ε is the absolute value of a complex quantity, composed of an elastic and viscous term. During emulsification an increase in the interfacial area A takes place and this causes a reduction in Γ. The equilibrium is restored by adsorption of surfactant from the bulk, but this takes time (shorter times occur at higher surfactant activity). Thus ε is small at small a and also at large a. Because of the lack or slowness of equilibrium with polymeric surfactants, ε will not be the same for expansion and compression of the interface. In practice, emulsifiers are generally made of surfactant mixtures that often contain different components and these have pronounced effects on γ and ε. Some specific surfactant mixtures give lower γ values than either of the two individual components. The presence of more than one surfactant molecule at the interface tends to increase ε at high surfactant concentrations. The various components vary in surface activity. Those with the lowest γ tend to predominate at the interface, but if present at low concentrations, it may take a long time before reaching the lowest value. Polymersurfactant mixtures may show some synergetic surface activity. During emulsification, surfactant molecules are transferred from the solution to the interface and this leaves an ever lower surfactant activity [2]. Consider for example an O/W emulsion with a volume fraction ϕ = 0.4 and a Sauter diameter d32 = 1 µm. According to equation (4.16), the specific surface area is 2.4 m2 ml−1 and for a surface excess Γ of 3 mg m−2 , the amount of surfactant at the interface is 7.2 mg ml−1 emulsion, corresponding to 12 mg ml−1 aqueous phase (or 1.2 %). Assuming that the concentration of surfactant, Ceq (the concentration left after emulsification), leading to a plateau

159

4.4 Mechanism of emulsification and the role of the emulsifier |

value of Γ equals 0.3 mg ml−1 , then the surfactant concentration decreases from 12.3 to 0.3 mg ml−1 during emulsification. This implies that the effective γ value increases during the process. If insufficient surfactant is present to leave a concentration Ceq after emulsification, even the equilibrium γ value would increase. Another aspect is that the composition of surfactant mixture in solution may alter during emulsification [1, 2]. If some minor components are present that give a relatively small γ value, this will predominate at a macroscopic interface, but during emulsification, as the interfacial area increases, the solution will soon become depleted of these components. Consequently, the equilibrium value of γ will increase during the process and the final value may be markedly larger than what is expected on the basis of the macroscopic measurement. During droplet deformation, its interfacial area is increased [2]. The drop will commonly have acquired some surfactant, and it may even have a Γ value close to the equilibrium at the prevailing (local) surface activity. The surfactant molecules may distribute themselves evenly over the enlarged interface by surface diffusion or by spreading. The rate of surface diffusion is determined by the surface diffusion coefficient Ds that is inversely proportional to the molar mass of the surfactant molecule and also inversely proportional to the effective viscosity felt. Ds also decreases with increasing Γ. Sudden extension of the interface or sudden application of a surfactant to an interface can produce a large interfacial tension gradient and in such a case spreading of the surfactant can occur. Surfactants allow the existence of interfacial tension gradients which is crucial for formation of stable droplets. In the absence of surfactants (clean interface), the interface cannot withstand a tangential stress; the liquid motion will be continuous across a liquid interface (Fig. 4.12 (a)). If a liquid flows along the interface with surfactants, the latter will be swept downstream causing an interfacial tension gradient (Fig. 4.12 (b)). A balance of forces will be established, dV x dy η[ =− . (4.20) ] dy y=0 dx

Low γ Y YYY Y

High γ Y

Y

0

0

Y YYY

Oil

–y

Y

0

–y

Y

–y

Water

(a)

(b)

(c)

Fig. 4.12: Interfacial tension gradients and flow near an oil/water interface: (a) no surfactant; (b) velocity gradient causes an interfacial tension gradient; (c) interfacial tension gradient causes flow (Marangoni effect).

160 | 4 Liquid/liquid dispersions (emulsions)

If the γ-gradient can become large enough, it will arrest the interface. The largest value attainable for dγ equals about πeq , i.e. γ0 − γeq . If it acts over a small distance, a considerable stress can develop, of the order of 10 kPa. If the surfactant is applied at one site of the interface, a γ-gradient is formed that will cause the interface to move roughly at a velocity given by, v = 1.2[ηρz]−1/3 |∆γ|2/3 .

(4.21)

The interface will then drag some of the bordering liquid with it (Fig. 4.9 (c)). This is called the Marangoni effect [8–10]. Interfacial tension gradients are very important in stabilizing the thin liquid film between the droplets which is very important during the beginning of emulsification, when films of the continuous phase may be drawn through the disperse phase or when collision of the still large deformable drops causes the film to form between them. The magnitude of the γ-gradients and of the Marangoni effect depends on the surface dilational modulus ε, which for a plane interface with one surfactant-containing phase, is given by the expressions, ε=−

d − dγ/d ln Γ , d(1 + 2ξ + 2ξ 2 )1/2

dmC D 1/2 ( ) , dΓ 2ω d ln A , ω= dt ξ=

(4.22) (4.23) (4.24)

where D is the diffusion coefficient of the surfactant and ω represents a timescale (time needed for doubling the surface area) that is roughly equal to τdef . During emulsification, ε is dominated by the magnitude of the numerator in equation (4.22) because ξ remains small. The value of dmC /dΓ tends to go to very high values when Γ reaches its plateau value; ε goes to a maximum when mC is increased. However, during droplet deformation, Γ will always remain smaller. Taking reasonable values for the variables; dmC /dΓ = 102 –104 m−1 , D = 10−9 –10−11 m2 s−1 and τdef = 10−2 –10−6 s, ξ < 0.1 at all conditions. The same conclusion can be drawn for values of ε in thin films, e.g. between closely approaching drops. It may be concluded that for conditions that prevail during emulsification, ε increases with mC and follows the relation, dπ ε≈ , (4.25) d ln Γ except for very high surfactant concentration, where π is the surface pressure (π = γ0 − γ). Fig. 4.13 shows the variation of π with ln Γ; ε is given by the slope of the line. The SDS shows a much higher ε value during emulsification, when compared with the polymers β-casein and lysozome. This is because the value of Γ is higher for SDS.

4.4 Mechanism of emulsification and the role of the emulsifier |

161

30 SDS

/mNm–1

20 -casein 10

Lysozyme

0 –1

0 ln( /mg–2)

1 Fig. 4.13: π versus ln Γ for various emulsifiers.

The two proteins show a difference in their ε values which may be attributed to the conformational change that occurs upon adsorption. The presence of a surfactant means that during emulsification the interfacial tension need not to be the same everywhere (see Fig. 4.9). This has two consequences: (i) the equilibrium shape of the drop is affected; (ii) any γ-gradient formed will slow down the motion of the liquid inside the drop (this diminishes the amount of energy needed to deform and break up the drop). Another important role of the emulsifier is to prevent coalescence during emulsification. This is certainly not due to the strong repulsion between the droplets, since the pressure at which two drops are pressed together is much greater than the repulsive stresses. The counteracting stress must be due to the formation of γ-gradients. When two drops are pushed together, liquid will flow out from the thin layer between them, and the flow will induce a γ-gradient. This was shown in Fig. 4.12 (c). This produces a counteracting stress given by, 2|∆γ| τ ∆γ ≈ . (4.26) (1/2)d The factor 2 follows from the fact that two interfaces are involved. Taking a value of ∆γ = 10 mN m−1 , the stress amounts to 40 kPa (which is of the same order of magnitude as the external stress). The stress due to the γ-gradient cannot as such prevent coalescence, since it only acts for a short time, but it will greatly slow down the mutual approach of the droplets. The external stress will also act for a short time, and it may well be that the drops move apart before coalescence can occur. The effective γ-gradient will depend on the value of ε as given by equation (4.25). Closely related to the above mechanism is the Gibbs–Marangoni effect [8–10], schematically represented in Fig. 4.14. The depletion of surfactant in the thin film between approaching drops results in a γ-gradient without liquid flow being involved. This results in an inward flow of liquid that tends to drive the drops apart. Such a mechanism would only act if the drops were insufficiently covered with surfactant (Γ below the plateau value), as occurs during emulsification.

162 | 4 Liquid/liquid dispersions (emulsions)

(a)

(b)

(c)

Fig. 4.14: Schematic representation of the Gibbs– Marangoni effect for two approaching drops.

The Gibbs–Marangoni effect also explains the Bancroft rule which states that the phase in which the surfactant is most soluble forms the continuous phase. If the surfactant is in the droplets, a γ-gradient cannot develop and the drops would be prone to coalescence. Thus, surfactants with HLB > 7 tend to form O/W emulsions and those with HLB < 7 tend to form W/O emulsions. The Gibbs–Marangoni effect also explains the difference between surfactants and polymers for emulsification. Polymers give larger drops when compared with surfactants. Polymers give a smaller value of ε at small concentrations when compared to surfactants (Fig. 4.13). Various other factors should also be considered for emulsification, such as the disperse phase volume fraction ϕ. An increase in ϕ leads to an increase in droplet collision and hence coalescence during emulsification. With increasing ϕ, the viscosity of the emulsion increases and could change the flow from being turbulent to being laminar. The presence of many particles results in a local increase in velocity gradients. In turbulent flow, increasing ϕ will induce turbulence depression. This will result in larger droplets. Turbulence depression by adding polymers tends to remove the small eddies, resulting in the formation of larger droplets. If the mass ratio of surfactant to continuous phase is kept constant, increasing ϕ results in decreased surfactant concentration and hence an increase in γeq resulting in larger droplets. If the mass ratio of surfactant to disperse phase is kept constant, the above changes are reversed.

4.5 Methods of emulsification Several procedures may be applied for emulsion preparation, these range from simple pipe flow (low agitation energy, L), static mixers (toothed devices such as the UltraTurrax and batch radial discharge mixers such as the Silverson mixers) and general stirrers (low to medium energy, L–M), colloid mills and high pressure homogenizers

4.5 Methods of emulsification

|

163

(high energy, H), ultrasound generators (M–H) and membrane emulsification methods. The method of preparation can be continuous (C) or batch-wise (B): pipe flow – C; static mixers and general stirrers – B, C; colloid mill and high pressure homogenizers – C; ultrasound – B, C. In all methods, there is liquid flow; unbounded and strongly confined flow. In unbounded flow any droplet is surrounded by a large amount of flowing liquid (the confining walls of the apparatus are far away from most of the droplets). The forces can be frictional (mostly viscous) or inertial. Viscous forces cause shear stresses to act on the interface between the droplets and the continuous phase (primarily in the direction of the interface). The shear stresses can be generated by laminar flow (LV) or turbulent flow (TV); this depends on the dimensionless Reynolds numbers Re, Re =

vlρ , η

(4.27)

where v is the linear liquid velocity, ρ is the liquid density and η is its viscosity. l is a characteristic length that is given by the diameter of flow through a cylindrical tube and by twice the slit width in a narrow slit. For laminar flow Re ⪅ 1000, whereas for turbulent flow Re ⪆ 2000. Thus, whether the regime is linear or turbulent depends on the scale of the apparatus, the flow rate and the liquid viscosity [1–4]. Rotor-stator mixers are the most commonly used mixers for emulsification. Two main types are available. Toothed devices such as the Ultra-Turrax (IKA works, Germany) are the most commonly used device (schematically illustrated in Fig. 4.15). Toothed devices are available both as in-line as well as batch mixers, and because of their open structure they have a relatively good pumping capacity. Therefore in batch applications they frequently do not need an additional impeller to induce bulk flow even in relatively large mixing vessels. These mixers are used in the food

Fig. 4.15: Schematic representation of a toothed mixer (Ultra-Turrax).

164 | 4 Liquid/liquid dispersions (emulsions)

industry to manufacture ice cream, margarine and salad dressings, in cosmetic and personal care products to manufacture creams and lotions as well as in manufacturing of speciality chemicals for micro-encapsulation of waxes and paraffin. They are also popular in the paper industry to process highly viscous and non-Newtonian paper pulp and in the manufacturing of paints and coatings. Ultra-Turrax mixers have been used to manufacture emulsion-based lipid carriers with drops below 1 µm and in emulsion polymerization to produce drops of the order of 300 nm. Batch radial discharge mixers such as Silverson mixers (Fig. 4.16) have a relatively simple design with a rotor equipped with four blades pumping the fluid through a stationary stator perforated with differently shaped/sized holes or slots.

Fig. 4.16: Schematic representation of batch radial discharge mixer (Silverson mixer).

They are frequently supplied with a set of easily interchangeable stators enabling the same machine to be used for a range of operations such as emulsification, homogenization, blending, particle size reduction and de-agglomeration. Changing from one screen to another is quick and simple. Different stators/screens used in batch Silverson mixers are shown in Fig. 4.17. The general purpose disintegrating stator (Fig. 4.17 (a)) is recommended for preparing thick emulsions (gels) while the slotted disintegrating stator (Fig. 4.17 (b)) is designed for emulsions containing elastic materials such as polymers. Square holed screens (Fig. 4.17 (c)) are recommended for the preparation of emulsions whereas the standard emulsor screen (Fig. 4.17 (d)) is used for liquid/ liquid emulsification. Radial discharge high shear mixers are used in a wide range of

(a)

(b)

(c)

(d)

Fig. 4.17: Stators used in batch Silverson radial discharge mixers.

4.5 Methods of emulsification

|

165

industries ranging from foods through to chemicals, cosmetics and pharmaceutical industries. Silverson rotor-stator mixers are used in cosmetic and pharmaceutical industries to manufacture both concentrated liquid/liquid and liquid/solid emulsions such as creams, lotions, mascaras and deodorants to name the most common applications. As mentioned above, in all methods there is liquid flow; unbounded and strongly confined flow. In unbounded flow any droplet is surrounded by a large amount of flowing liquid (the confining walls of the apparatus are far away from most of the droplets); the forces can be frictional (mostly viscous) or inertial. Viscous forces cause shear stresses to act on the interface between the droplets and the continuous phase (primarily in the direction of the interface). The shear stresses can be generated by laminar flow (LV) or turbulent flow (TV); this depends on the Reynolds number Re as given by equation (4.27). For laminar flow Re ⪅ 1000, whereas for turbulent flow Re ⪆ 2000. Thus, whether the regime is linear or turbulent depends on the scale of the apparatus, the flow rate and the liquid viscosity. If the turbulent eddies are much larger than the droplets, they exert shear stresses on the droplets. If the turbulent eddies are much smaller than the droplets, inertial forces will cause disruption (TI). In bounded flow other relations hold; if the smallest dimension of the part of the apparatus in which the droplets are disrupted (say a slit) is comparable to droplet size, other relations hold (the flow is always laminar). If the turbulent eddies are much larger than the droplets, they exert shear stresses on the droplets. If the turbulent eddies are much smaller than the droplets, inertial forces will cause disruption (TI). A different regime prevails if the droplets are directly injected through a narrow capillary into the continuous phase (injection regime), i.e. membrane emulsification. Within each regime, an essential variable is the intensity of the forces acting; the viscous stress during laminar flow σviscous is given by, σviscous = ηG,

(4.28)

where G is the velocity gradient. The intensity in turbulent flow is expressed by the power density ε (the amount of energy dissipated per unit volume per unit time); for turbulent flow, ε = ηG2 .

(4.29)

The most important regimes are: laminar/viscous (LV) – turbulent/viscous (TV) – turbulent/inertial (TI). For water as the continuous phase, the regime is always TI. For higher viscosity of the continuous phase (ηC = 0.1 Pa s), the regime is TV. For still higher viscosity or a small apparatus (small l), the regime is LV. For very small apparatus (as is the case with most laboratory homogenizers), the regime is nearly always LV. For the above regimes, a semi-quantitative theory is available that can give the timescale and magnitude of the local stress σext , the droplet diameter d, timescale of droplets deformation τdef , timescale of surfactant adsorption, τads and mutual collision of droplets.

166 | 4 Liquid/liquid dispersions (emulsions)

Laminar flow can be of a variety of types, purely rotational to purely extensional. For simple shear, the flow consists of equal parts of rotation and elongation. The velocity gradient G (in reciprocal seconds) is equal to the shear rate γ. For hyperbolic flow, G is equal to the elongation rate. The strength of a flow is generally expressed by the stress it exerts on any plane in the direction of flow; it is simply equal to Gη (η is simply the shear viscosity). For elongational flow, the elongational viscosity ηel is given by, ηel = Tr η,

(4.30)

where Tr is the dimensionless Trouton number which is equal to 2 for Newtonian liquids in two-dimensional uneasily elongation flow. Tr = 3 for axisymmetric uniaxial flow and its equal to 4 and for biaxial flows. Elongational flows exert higher stresses for the same value of G than simple shear. For non-Newtonian liquids, the relationships are more complicated and the values of Tr tends to be much higher. An important parameter that describes droplet deformation is the Weber number, We (which gives the ratio of the external stress over the Laplace pressure), We =

GηC R . 2γ

(4.31)

The deformation of the drop increases with increasing We and above a critical value Wecr the drop bursts, forming smaller droplets. Wecr depends on two parameters: (i) The velocity vector α (α = 0 for simple shear and α = 1 for hyperbolic flow); (ii) the viscosity ratio λ of the oil ηD and the external continuous phase ηC , λ=

ηD . ηC

(4.32)

The variation of critical Weber number with λ at various α values is shown in Fig. 4.18. The viscosity of the oil plays an important role in the break-up of droplets; the higher the viscosity, the longer it will take to deform a drop. The deformation time τdef 100

α=0

Wecr

10 0.08 1

0.2 0.6 1 ASE

0.1 10–4

100

10–2 λ

102

Fig. 4.18: Critical Weber number for break-up of drops in various types of flow. The hatched area represents the apparent Wecr in a colloid mill.

4.5 Methods of emulsification |

167

is given by the ratio of oil viscosity to the external stress acting on the drop, τdef =

ηD . σext

(4.33)

The above ideas for simple laminar flow were tested using emulsions containing 80 % oil in water stabilized with egg yolk. A colloid mill and static mixers were used to prepare the emulsion. The results are shown in Fig. 4.19 which gives the number of droplets n in which a parent drop is broken down when it is suddenly extended into a long thread, corresponding to Web which is larger than Wecr . The number of drops increases with increasing Web /Wecr . The largest number of drops, i.e. the smallest droplet size, is obtained when λ = 1, i.e. when the viscosity of the oil phase is closer to that of the continuous phase. In practice, the resulting drop size distribution is of greater importance than the critical drop size for break-up.

λ=1

200

n

0.31 100 0.01 0 1

2

3 Web/Wecr

4

5 Fig. 4.19: Variation of n with Web /Wecr .

Turbulent flow is characterized by the presence of eddies, which means that the average local flow velocity u generally differs from the time average value u.̄ The velocity fluctuates in a chaotic way and the average difference between u and u󸀠 equals zero; however, the root mean square average u󸀠 is finite [5–8], u󸀠 = ⟨(u − u)̄ 2 ⟩1/2 .

(4.34)

The value of u󸀠 generally depends on direction, but for very high Re (> 50 000) and at small length scales the turbulence flow can be isotropic, and u󸀠 does not depend on direction. Turbulent flow shows a spectrum of eddy sizes (l); the largest eddies have the highest u󸀠 , they transfer their kinetic energy to smaller eddies, which have a smaller u󸀠 but larger velocity gradient u󸀠 /l. The smallest eddy size is given by, le = η3/4 ρ−1/2 ε−1/4 .

(4.35)

ε is called the power density, i.e. the amount of energy dissipated per unit volume of liquid and per unit time (given in W m−3 ).

168 | 4 Liquid/liquid dispersions (emulsions)

The local flow velocities depend on the distance scale x considered, and for a scale comparable to the size of an energy-bearing eddy (x ≈ le ). The velocity near that eddy is given by u󸀠 (x) = ε1/3 x1/3 ρ−1/3 . (4.36) The velocity gradient in an eddy is given by u󸀠 (x)/(x) and it increases strongly with decreasing size. The eddies have a short lifetime given by, τ(le ) = le /u󸀠 (le ) = le ε−1/3 ρ1/3 . 2/3

(4.37)

The local flow velocity for scales much smaller than the size of an energy-bearing eddy (x ≪ l e ) is given by, u󸀠 (x) = ε1/2 xη−1/2 . (4.38) The break-up of droplets in turbulent flow due to inertial forces may be represented by local pressure fluctuations near energy-bearing eddies, ∆p(x) = ρ[u󸀠 (x)]2 = ε2/3 x2/3 ρ1/3 ,

(4.39)

where ε is the power density, i.e. the amount of energy dissipated per unit volume, ρ is the density and x is the distance scale. If ∆p is larger than the Laplace pressure (p = 2γ/R) near the eddy, the drop would be broken up. Break-up would be most effective if d = le . Putting x = dmax , the following expression gives the largest drops that are not broken up in the turbulent field, dmax = ε−2/5 γ3/5 ρ−1/5 .

(4.40)

The validity of equation (4.40) is subject to two conditions: (i) The droplet size obtained cannot be much smaller than l0 . This equation is fulfilled for small ηC . (ii) The flow near the droplet should be turbulent. This depends on the droplet Reynolds number given by, Redr = du󸀠 (d)ρC /ηC .

(4.41)

The condition Redr > 1 and combination with equation (4.36) leads to, d > η2C /γρ.

(4.42)

Provided that (i) ϕ is small, (ii) ηC is not much larger than 1 mPa s, (iii) ηD is fairly small, (iv) γ is constant and (v) the machine is fairly small, equation (4.42) seems to hold well even for nonisotropic turbulence with Reynolds number much smaller than 50 000. The smallest drops are produced at the highest power density. Since the power density varies from place to place (particularly if Re is not very high) the droplet size distribution can be very wide. For break-up of drops in TI regime, the flow near the

4.5 Methods of emulsification |

169

drop is turbulent. For laminar flow, break-up by viscous forces is possible. If the flow rate near the drop (u) varies greatly with distance d, the local velocity gradient is G. A pressure difference is produced over the drop of (1/2)∆ρ(u2 ) = ρGd. At the same time a shear stress ηC G acts on the drop. The viscous forces will predominate for ηC G > ρGd, leading to the condition, ̄ udρ/η (4.43) C = Redr < 1. 1/2

The local velocity gradient is ηC G = ε1/2 ηC . This results in the following expression for dmax −1/2 dmax = Wecr γε−1/2 ηC . (4.44) The value of Wecr is rarely > 1, since the flow has an elongational component. For not very small ηC , dmax is smaller for TV than TI. The viscosity of the oil plays an important role in the break-up of droplets; the higher the viscosity, the longer it will take to deform a drop. The deformation time τdef is given by the ratio of oil viscosity to the external stress acting on the drop, τdef =

ηD . ηC

(4.45)

The viscosity of the continuous phase ηC plays an important role in some regimes: For turbulent inertial regimes, ηC has no effect on droplets size. For turbulent viscous regimes, larger ηC leads to smaller droplets. For laminar viscous regimes, the effect is even stronger. The value of ηC and the size of the apparatus determines which regime prevails, via the effect on Re. In a large machine and low ηC , Re is always very large and the resulting average droplet diameter d is proportional to PH − 0.6 (where PH is the homogenization pressure). If ηC is higher and Redr < 1, the regime is TV and d ∝ PH − 0.75. For a smaller machine, as used in the lab, where the slit width of the valve may be of the order of µm, Re is small and the regime is LV; d ∝ PH − 1.0. If the slit is made very small (of the order of droplet diameter), the regime can become TV. Fig. 4.20 shows the variation of average droplet diameter d43 with PH at low and high Re for 20 % soybean oil/water emulsion stabilized with sodium caseinate (30 mg ml−1 ). Fig. 4.21 shows the variation of width of distribution with number of passages at low and high R. Addition of high molecular weight polymers in the continuous phase increases ηC resulting in turbulence depression increasing d32 while decreasing c2 . In membrane emulsification, the disperse phase is passed through a membrane and droplets leaving the pores are immediately taken up by the continuous phase. The membrane is commonly made of porous glass or of ceramic materials. The general configuration is a membrane in the shape of a hollow cylinder; the disperse phase is pressed through it from outside, and the continuous phase pumped through the cylinder (cross flow). The flow also causes detachment of the protruding droplets from the membrane [1].

170 | 4 Liquid/liquid dispersions (emulsions) 0.4 a

Log d34/μm

Low Re, slope –0.94

0.0 High Re, slope –0.61

–o.4 0.4

0.8

1.2

Log(PH/MPa) Fig. 4.20: Comparison of droplet size distribution obtained with two high pressure homogenizers; a very small one (low Re) and a large one (high Re).

1.5 b

1.0 c2

Low Re 0.5

High Re 0

20

40

Number of passages Fig. 4.21: Relative distribution width c2 as a function of number of passes through the homogenizers.

Several requirements are necessary for the process: (i) For a hydrophobic disperse phase (O/W emulsion) the membrane should be hydrophilic, whereas for a hydrophilic disperse phase (W/O emulsion) the membrane should be hydrophobic, since otherwise the droplets cannot be detached. (ii) The pores must be sufficiently far apart to prevent the droplets coming out from touching each other and coalescing.

4.5 Methods of emulsification |

171

(iii) The pressure over the membrane should be sufficiently high to achieve drop formation. This pressure should be at least of the order of Laplace pressure of a drop of diameter equal to the pore diameter. For example, for pores of 0.4 µm and γ = 5 mN m−1 , the pressure should be of the order of 105 Pa, but larger pressures are needed in practice, this would amount to 3 × 105 Pa, also to obtain a significant flow rate of the disperse phase through the membrane. The smallest drop size obtained by membrane emulsification is about three times the pore diameter. The main disadvantage is its slow process, which can be of the order of 10−3 m3 per m2 per second. This implies that very long circulation times are needed to produce even small volume fractions. The most important variables that affect the emulsification process are the nature of the oil and emulsifier, the volume fraction of the disperse phase ϕ and the emulsification process. The effect of the nature of the oil and emulsifier were discussed before. The method of emulsification and the regime (laminar or turbulent) have a pronounced effect on the process and the final droplet size distribution. The effect of the volume fraction of the disperse phase requires special attention. It affects the rate of collision between droplets during emulsification, and thereby the rate of coalescence. As a first approximation, this would depend on the relation between τads and τcoal (where τads is the average time it takes for surfactant adsorption and τcoal is the average time it takes until a droplet collides with another one). In the various regimes, the hydrodynamic constraints are the same for τads . For example, in regime LV, τcoal = π/8ϕG. Thus for all regimes, the ratio of τads /τcoal is given by [1, 2], κ≡

τads ϕΓ ∝ , τcoal mC d

(4.46)

where the proportionality factor would be at least of order 10. For example, for ϕ = 0.1, Γ/mC = 10−6 m and d = 10−6 m (total surfactant concentration of the emulsion should then be about 05 %), κ would be of the order of 1. For κ ≫ 1, considerable coalescence is likely to occur, particularly at high ϕ. The coalescence rate would then markedly increase during emulsification, since both mC and d become smaller during the process. If emulsification proceeds long enough, the droplet size distribution may then be the result of a steady state of simultaneous break-up and coalescence. The effect of an increase in ϕ can be summarized as follows [1, 2]: (i) τcoal is shorter and coalescence will be faster unless κ remains small. (ii) Emulsion viscosity ηem increases, hence Re decreases. This implies a change of flow from turbulent to laminar (LV). (iii) In laminar flow, the effective ηC becomes higher. The presence of many droplets means that the local velocity gradients near a droplet will generally be higher than the overall value of G. Consequently, the local shear stress ηG does increase with increasing ϕ, which is as if ηC increases.

172 | 4 Liquid/liquid dispersions (emulsions)

(iv) In turbulent flow, an increase in ϕ will induce turbulence depression leading to larger d. (v) If the mass ratio of surfactant to continuous phase is constant, an increase in ϕ gives a decrease in surfactant concentration; hence an increase in γeq , an increase in κ, an increase in d produced by an increase in coalescence rate. If the mass ratio of surfactant to disperse phase is kept constant, the above mentioned changes are reversed, unless κ ≫ 1. It is clear from the above discussion that general conclusions cannot be drawn, since several of the above mentioned mechanisms may come into play. Using a high pressure homogenizer, Walstra [11] compared the values of d with various ϕ values up to 0.4 at constant initial mC , regime TI probably changing to TV at higher ϕ. With increasing ϕ (> 0.1), the resulting d increased and the dependency on homogenizer pressure pH (Fig. 4.22). This points to increased coalescence (effects (i) and (v)). Fig. 4.22 shows a comparison of the average droplet diameter versus power consumption using different emulsifying machines. It can be seen the smallest droplet diameters were obtained when using the high pressure homogenizers.

Colloid mill 0.025

log (d32/ m)

1 Ultra turrax

1.5

Homogenizer

0

30 –0.8 4.5

5

6

us

7

log (p/Jm3) Fig. 4.22: Average droplet diameters obtained in various emulsifying machines as a function of energy consumption p. The number near the curves denote the viscosity ratio λ. The results for the homogenizer are for ϕ = 0.04 (solid line) and ϕ = 0.3 (broken line); us means ultrasonic generator.

4.6 Selection of emulsifiers Several surfactants and their mixtures are used for the preparation and stabilization of oil-in-water (O/W) and water-in-oil (W/O) emulsions. A summary of the most commonly used surfactants was given in Chapter 8, Vol. 1. Two general methods can be applied for selecting emulsifiers, namely the hydrophilic-lipophilic-balance (HLB) and the phase inversion temperature (PIT) concepts. The hydrophilic-lipophilic balance (HLB number) is a semi-empirical scale for selecting surfactants developed by Griffin

4.6 Selection of emulsifiers

| 173

[12]. This scale is based on the relative percentage of hydrophilic to lipophilic (hydrophobic) groups in the surfactant molecule(s). For an O/W emulsion droplet, the hydrophobic chain resides in the oil phase whereas the hydrophilic head group resides in the aqueous phase. For a W/O emulsion droplet, the hydrophilic group(s) reside in the water droplet, whereas the lipophilic groups reside in the hydrocarbon phase. Tab. 4.2 gives a guide to the selection of surfactants for a particular application. The HLB number depends on the nature of the oil. As an illustration, Tab. 4.3 gives the required HLB numbers to emulsify various oils. Examples of HLB numbers of a list of surfactants are given in Tab. 4.4. Tab. 4.2: Summary of HLB ranges and their applications. HLB range

Application

3–6 7–9 8–18 13–15 15–18

W/O emulsifier Wetting agent O/W emulsifier Detergent Solubilizer

Tab. 4.3: Required HLB numbers to emulsify various oils. Oil

W/O emulsion

O/W emulsion

Paraffin oil Beeswax Linolin, anhydrous Cyclohexane Toluene Silicone oil (volatile) Isopropyl myristate Isohexadecyl alcohol Castor oil

4 5 8 — — — — 11–12

10 9 12 15 15 7–8 11–12 14

Tab. 4.4: HLB numbers of some surfactants. Surfactant

Chemical name

HLB

Span 85 Span 80 Brij 72 Triton X-35 Tween 85 Tween 80

Sorbitan trioleate Sorbitan monooleate Ethoxylated (2 mol ethylene oxide) stearyl alcohol Ethoxylated octylphenol Ethoxylated (20 mol ethylene oxide) sorbitan trioleate Ethoxylated (20 mol ethylene oxide) sorbitan monooleate

1.8 4.3 4.9 7.8 11.0 15.0

174 | 4 Liquid/liquid dispersions (emulsions)

The relative importance of the hydrophilic and lipophilic groups was first recognized when using mixtures of surfactants containing varying proportions of a low and high HLB number. The efficiency of any combination (as judged by phase separation) was found to pass a maximum when the blend contained a particular proportion of the surfactant with the higher HLB number. This is illustrated in Fig. 4.23 which shows the variation of emulsion stability, droplet size and interfacial tension with % surfactant with high HLB number. Emulsion stability

Droplet size interfacial Tension

0

Fig. 4.23: Variation of emulsion stability, droplet size and interfacial tension with % surfactant with high HLB number.

100

% Surfactant with high HLB

The average HLB number may be calculated from additivity, HLB = x1 HLB1 + x2 HLB2 .

(4.47)

x1 and x2 are the weight fractions of the two surfactants with HLB1 and HLB2 . Griffin [12] developed simple equations for calculating the HLB number of relatively simple nonionic surfactants. For a polyhydroxy fatty acid ester, HLB = 20(1 −

S ). A

(4.48)

S is the saponification number of the ester and A is the acid number. For a glyceryl monostearate, S = 161 and A = 198; the HLB is 3.8 (suitable for a W/O emulsion). For a simple alcohol ethoxylate, the HLB number can be calculated from the weight percent of ethylene oxide (E) and polyhydric alcohol (P), HLB =

E+P 5

(4.49)

If the surfactant contains PEO as the only hydrophilic group, the contribution from one OH group is neglected, E HLB = . (4.50) 5 For a nonionic surfactant C12 H25 –O–(CH2 –CH2 –O)6 , the HLB is 12 (suitable for an O/W emulsion). The above simple equations cannot be used for surfactants containing propylene oxide or butylene oxide. They also cannot be applied for ionic surfactants. Davies [13, 14] devised a method for calculating the HLB number for surfactants from their

4.6 Selection of emulsifiers

| 175

chemical formulae, using empirically determined group numbers. A group number is assigned to various component groups. A summary of the group numbers for some surfactants is given in Tab. 4.5. The HLB is given by the following empirical equation, HLB = 7 + ∑(hydrophilic group numbers) − ∑(lipohilic group numbers).

(4.51)

Davies has shown that the agreement between HLB numbers calculated from the above equation and those determined experimentally is quite satisfactory. Various other procedures have been developed to obtain a rough estimate of the HLB number. Griffin found good correlation between the cloud point of 5 % solution of various ethoxylated surfactants and their HLB number. Davies [13, 14] attempted to relate the HLB values to the selective coalescence rates of emulsions. Such correlations were not realized since it was found that the emulsion stability and even its type depend to a large extent on the method of dispersing the oil into the water and vice versa. At best the HLB number can only be used as a guide for selecting optimum compositions of emulsifying agents. Tab. 4.5: HLB group numbers. Group number Hydrophilic –SO4 Na+ –COOK –COONa N(tertiary amine) Ester (sorbitan ring) –O– CH–(sorbitan ring)

38.7 21.2 19.1 9.4 6.8 1.3 0.5

Lipophilic (–CH–), (–CH2 –), CH3

0.475

Derived –CH2 –CH2 –O –CH2 –CHCH3 –O–

0.33 0.11

One may take any pair of emulsifying agents, which fall at opposite ends of the HLB scale, e.g. Tween 80 (sorbitan monooleate with 20 mol EO, HLB = 15) and Span 80 (sorbitan monooleate, HLB = 5) using them in various proportions to cover a wide range of HLB numbers. The emulsions should be prepared in the same way, with a few percent of the emulsifying blend. For example, a 20 % O/W emulsion is prepared by using 4 % emulsifier blend (20 % with respect to oil) and 76 % water. The stability of the emulsions is then assessed at each HLB number from the rate of coalescence or qualitatively by measuring the rate of oil separation. In this way one may be able to

176 | 4 Liquid/liquid dispersions (emulsions)

find the optimum HLB number for a given oil. For example with a given oil, the optimum HLB number is found to be 10.3. The latter can be determined more exactly by using mixtures of surfactants with narrower HLB range, say between 9.5 and 11. Having found the most effective HLB value, various other surfactant pairs are compared at this HLB value, to find the most effective pair. This is illustrated in Fig. 4.24 which schematically shows the difference between three chemical classes of surfactants. Although the different classes give a stable emulsion at HLB = 12, mixture A gives the best emulsion stability. The HLB value of a given magnitude can be obtained by mixing emulsifiers of different chemical types. The “correct” chemical type is as important as the “correct” HLB number. This is illustrated in Fig. 4.25, which shows that an emulsifier with unsaturated alkyl chain such as oleate (ethoxylated sorbitan mono-oleate, Tween 80) is more suitable for emulsifying an unsaturated oil [15]. An emulsifier with saturated alkyl chain (stearate in Tween 60) is better for emulsifying a saturated oil. Chemical class A

Stability

B C

12

Fig. 4.24: Stabilization of emulsion by different classes of surfactants as a function of HLB.

HLB value of emulsifier mixtures

(a)

Oleate

(b)

Unsaturated oil

Stearate

Saturated oil

Oil Water

POE

POE POE POE

POE

Fig. 4.25: Selection of Tween type to correspond to the type of the oil to be emulsified.

POE

4.6 Selection of emulsifiers | 177

Various procedures have been developed to determine the HLB of different surfactants. Griffin [12] found a correlation between the HLB and the cloud points of 5 % aqueous solution of ethoxylated surfactants as illustrated in Fig. 4.26. A titration procedure was developed [16] for estimating the HLB number. In this method, a 1 % solution of surfactant in benzene plus dioxane is titrated with distilled water at constant temperature until a permanent turbidity appears. Griffin found a good linear relationship between the HLB number and the water titration value for polyhydric alcohol esters as shown in Fig. 4.27. However, the slope of the line depends on the class of material used.

Temperature (°C) “Cloud point”

100

80

60

Cloud point vs HLB 5 % aqueous solutions heated to clouding temperature

40

20 12

13

14

15

16

17

18

Estimated HLB

Fig. 4.26: Relationship between cloud point and HLB.

18 16

With ethylene oxide

14 (1)

HLB

12 10 8

(2)

6

Whithout ethylene oxide

4 2 2

4

6

8

10

Water number

12

14 Fig. 4.27: Correlation of HLB with water number.

178 | 4 Liquid/liquid dispersions (emulsions)

Gas liquid chromatography (GLC) could also be used to determine the HLB number [16]. Since in GLC the efficiency of separation depends on the polarity of the substrate with respect to the components of the mixture, it should be possible to determine the HLB directly by using the surfactant as the substrate and passing an oil phase down the column. Thus, when a 50 : 50 mixture of ethanol and hexane is passed down a column of a simple nonionic surfactant, such as sorbitan fatty acid esters and polyoxyethylated sorbitan fatty acid esters, two well defined peaks, corresponding to hexane (which appears first) and ethanol appear on the chromatograms. A good correlation was found between the retention time ratio Rt (ethanol/hexane) and the HLB value. This is illustrated in Fig. 4.28. Statistical analysis of the data gave the following empirical relationship between Rt and HLB, HLB = 8.55Rt − 6.36,

(4.52)

where, Rt =

RETOH t Rhexane t

.

(4.53)

3.5 3.0

Rt

2.5 2.0 1.5 1.0 0.5 0

2

4

6

8 10 12 14 16 18 HLB

Fig. 4.28: Correlation between retention time and HLB of sorbitan fatty acid esters and polyoxyethylated fatty acid esters.

The phase inversion temperature (PIT) concept was developed by Shinoda and coworkers [17, 18] who found that many O/W emulsions stabilized with nonionic surfactants undergo a process of inversion at a critical temperature (PIT). The PIT can be determined by following the emulsion conductivity (small amount of electrolyte is added to increase the sensitivity) as a function of temperature as illustrated in Fig. 4.29. The conductivity of the O/W emulsion increases with increasing temperature until the PIT is reached, above which there will be a rapid reduction in conductivity (W/O emulsion is formed). Shinoda and co-workers [17, 18] found that the PIT is influenced by the HLB number of the surfactant [19], as shown in Fig. 4.30. For any given oil, the PIT increases with increasing HLB number. The size of the emulsion droplets was found to depend on the temperature and HLB number of the emulsifiers. The droplets are less stable towards coalescence close to the PIT. However, by rapid cooling of the emulsion a stable system may be produced. Relatively stable O/W emulsions were obtained when

Conductivity

4.6 Selection of emulsifiers | 179

O/ W

W/O

PIT Increase of temperature

Fig. 4.29: Variation of conductivity with temperature for an O/W emulsion.

HLB number (at 25°C)

16

14

C6H6 m–(CH3)2C6H4 c–C6H12

12 n–C7H16

n–C16H34

10 20

60

100

140

Phase inversion Temperature (°C) Fig. 4.30: Correlation between HLB number and PIT for various O/W (1 : 1) emulsions stabilized with nonionic surfactants (1.5 wt%).

the PIT of the system was 20–65 °C higher than the storage temperature. Emulsions prepared at a temperature just below the PIT followed by rapid cooling generally have smaller droplet sizes. This can be understood if one considers the change of interfacial tension with temperature as illustrated in Fig. 4.31. The interfacial tension decreases with increasing temperature reaching a minimum close to the PIT, after which it increases. Thus, the droplets prepared close to the PIT are smaller than those prepared at lower temperatures. These droplets are relatively unstable towards coalescence near the PIT, but by rapid cooling of the emulsion one can retain the smaller size. This procedure may be applied to prepare mini- (nano-)emulsions.

180 | 4 Liquid/liquid dispersions (emulsions) 20

γ/mN m–1

10 1 0.1 0.01 PIT Temperature

Increase

Fig. 4.31: Variation of interfacial tension with temperature increase for an O/W emulsion.

The optimum stability of the emulsion was found to be relatively insensitive to changes in the HLB value or the PIT of the emulsifier, but instability was very sensitive to the PIT of the system. It is essential, therefore to measure the PIT of the emulsion as a whole (with all other ingredients). At a given HLB value, stability of the emulsions against coalescence increases markedly as the molar mass of both the hydrophilic and lipophilic components increases. The enhanced stability using high molecular weight surfactants (polymeric surfactants) can be understood from a consideration of the steric repulsion which produces more stable films. Films produced using macromolecular surfactants resist thinning and disruption, thus reducing the possibility of coalescence. The emulsions showed maximum stability when the distribution of the PEO chains was broad. The cloud point is lower but the PIT is higher than in the corresponding case for narrow size distributions. The PIT and HLB number are directly related parameters. Addition of electrolytes reduces the PIT and hence an emulsifier with a higher PIT value is required when preparing emulsions in the presence of electrolytes. Electrolytes cause dehydration of the PEO chains and in effect this reduces the cloud point of the nonionic surfactant. One needs to compensate for this effect by using a surfactant with higher HLB. The optimum PIT of the emulsifier is fixed if the storage temperature is fixed. In view of the above correlation between PIT and HLB and the possible dependency of the kinetics of droplet coalescence on the HLB number, Sherman and coworkers suggested the use of PIT measurements as a rapid method for assessing emulsion stability [16]. However, one should be careful in using such methods for assessing the long-term stability since the correlations were based on a very limited number of surfactants and oils. Measurement of the PIT can at best be used as a guide for the preparation of stable emulsions. An assessment of the stability should be evaluated by following the droplet size distribution as a function of time using a Coulter Counter or light diffrac-

4.7 Creaming/sedimentation of emulsions and its prevention | 181

tion techniques. Following the rheology of the emulsion as a function of time and temperature may also be used for assessing the stability against coalescence [20]. Care should be taken in analysing the rheological results. Coalescence results in an increase in the droplet size and this is usually followed by a reduction in the viscosity of the emulsion. This trend is only observed if the coalescence is not accompanied by flocculation of the emulsion droplets (which results in an increase in the viscosity). Ostwald ripening can also complicate the analysis of the rheological data.

4.7 Creaming/sedimentation of emulsions and its prevention When the density of the droplets and the medium are not equal, the process of creaming or sedimentation of emulsions is the result of gravity. When the density of the disperse phase is lower than that of the medium (as with most oil-in-water O/W emulsions), creaming occurs, whereas if the density of the disperse phase is higher than that of the medium (as with most W/O emulsions), sedimentation occurs. Fig. 4.32 gives a schematic picture for creaming of emulsions for three cases [1–4].

h

(a) kT > (4/3)πR³Δρgh

h

(b) kT < (4/3)πR³Δρgh

Ch

Ch = Co exp(–mgh/kT) Co = conc. At the bottom Ch = conc. At time t and height h m = (4/3)πR³Δρ

Fig. 4.32: Representation of creaming of emulsions.

Case (a) represents the situation for small droplets (< 0.1 µ, i.e. nanoemulsions) in which the gravitational force is opposed by a diffusional force (i.e. associated with the translational kinetic energy of the droplets). A Boltzmann distribution is set up, in which the droplet concentration C h at height h is related to that at the top by, C(h) = C0 exp(−

mgh ), kT

(4.54)

where mgh is the potential energy of a droplet at height h, with m being the effective mass of a droplet, given by 4 m = πR3 ∆ρ, (4.55) 3

182 | 4 Liquid/liquid dispersions (emulsions)

where R is the hydrodynamic radius of the droplets and ∆ρ is the density difference between the two liquid phases, k is the Boltzmann constant and T is the absolute temperature. For no separation to occur, i.e. C h = C0 , kT ≫

4 3 πR ∆ρgL. 3

(4.56)

Case (b) represents emulsions consisting of “monodisperse” droplets with radius > 1 µm. In this case, the concentration forces are much greater than the opposing diffusional force, so that the emulsion separates into two distinct layers with the droplets forming a cream or sediment leaving the clear supernatant liquid, kT ≪

4 3 πR ∆ρgL. 3

(4.57)

Case (c) is that for a polydisperse (practical) emulsion, in which case the droplets will cream or sediment at various rates. In the last case, a concentration gradient builds up, as predicted by equation (4.54), with the larger droplets staying at the top of the cream layer, with some smaller droplets remaining in the bottom layer. The rate of creaming or sedimentation can be calculated following the same principles discussed for solid/liquid dispersions (Chapter 3). For very dilute noninteracting emulsion droplets, the rate could be calculated using Stokes’ law which balances the hydrodynamic force with gravity force [1], v0 =

2 ∆ρgR2 . 9 η0

(4.58)

v0 is the Stokes velocity and η0 is the viscosity of the medium. For an O/W emulsion with ∆ρ = 0.2 in water (η0 ≈ 10−3 Pa s), the rate of creaming or sedimentation is ≈ 4.4 × 10−5 ms−1 for 10 µm droplets and ≈ 4.4 × 10−7 ms−1 for 1 µm droplets. This means that in a 0.1 m container creaming or sedimentation of the 10 µm droplets is complete in ≈ 0.6 h and for the 1 µm droplets this takes ≈ 60 h. If the droplets are deformable, a liquid drop moving within a second liquid phase has an internal circulation imparted to it. As a result of this, the motion through the continuous phase has a “rolling” as well as a “sliding” component. This situation has been treated theoretically resulting in the following equation [1], v=

2 ∆ρgR2 η0 + η , 3 η0 3η0 + 2η

(4.59)

where η is the viscosity of the internal phase. For the case η ≫ η0 , equation (4.59) predicts v to be 50 % higher than v0 , whereas for two liquids having similar viscosities (η ≈ η0 ), v is only 20 % greater than v0 . However, these theoretical predictions do not agree well with the experimental data for v. This could be due to neglecting the interfacial viscosity contribution.

4.7 Creaming/sedimentation of emulsions and its prevention

| 183

With large droplets, shape distortion may occur due to changes in pressure with “vertical height” of the droplet. The difference in pressure between “top” and “bottom” of the droplet is ∆ρgd, where d is the distorted vertical diameter. Any deviation from spherical geometry leads to an increase in surface area ∆A. Thus the distorting force due to gravity is opposed by the work necessary to increase the surface area (γ∆A). For small deformities, d ≈ 2R and the fractional change in the radius is ∆ρgR2 /γ. For ∆ρ = 0.1 g cm−3 and γ = 2 mN m−1 , a 2 µm diameter droplet would undergo a distortion in radius of ≈ 5 × 10−5 %, whereas for a 200 µm diameter droplet the distortion would be ≈ 0.5 %. Thus, this effect is really only significant for large droplets. For moderately concentrated emulsions (0.2 > ϕ > 0.1), one has to take into account the hydrodynamic interaction between the droplets and the Stokes rate is reduced to v, v = v0 (1 − 6.55ϕ). (4.60) Equation (4.60) is referred to as Bachelor’s equation [21], which shows that for ϕ = 0.1, the rate of creaming or sedimentation is reduced by about 65 %. For more concentrated emulsions (ϕ > 0.2), the rate of creaming or sedimentation becomes a complex function of ϕ; as illustrated in Fig. 4.33 which also shows the change of relative viscosity ηr with ϕ. As can be seen from this figure, v decreases with increasing ϕ and ultimately it approaches zero when ϕ exceeds a critical value, ϕp , which is the so called “maximum packing fraction”. The value of ϕp for monodisperse “hard spheres” ranges from 0.64 (for random packing) to 0.74 for hexagonal packing. The value of ϕp exceeds 0.74 for polydisperse systems. Also for emulsions which are deformable, ϕp can be much larger than 0.74. Fig. 4.33 also shows that when ϕ approaches ϕp , ηr approaches ∞. In practice, most emulsions are prepared at ϕ values well below ϕp , usually in the range 0.2–0.5, and under these conditions creaming or sedimentation is the rule rather than the exception. Several procedures may be applied to reduce or eliminate creaming or sedimentation and these are discussed below.

v

ƞr

ɸp

1

ɸ Fig. 4.33: Variation of v and ηr with ϕ.

[ƞ] ɸp ɸ

184 | 4 Liquid/liquid dispersions (emulsions)

The structure of the creamed layer shown in Fig. 4.32 (c) represents the equilibrium volume where the mutual distances between the droplets are determined by the balance between the external gravitational field and the mutual interdroplet forces (electrostatic and/or steric forces associated with an interfacial polymer layer). Due to the polydispersity of the emulsion and deformation from spherical geometry, the maximum packing in the creamed layer can exceed that for monodisperse spheres (ϕ = 0.74 for hexagonal packing and 0.64 for random packing). The effects of polydispersity are important in that the smaller droplets may fit into the voids between the larger droplets in a packed cream. Packings of greater than 0.90 can be achieved in this way. If the droplets are deformable, the volume of packed cream layer is reduced and ϕ values in the range 0.95–0.99 can be reached. The greater the size of the droplets and density difference between the two liquid phases, the greater the tendency for deformation to occur. In many cases, the droplets distort to polyhedral cells resembling a foam in structure, with a corresponding network of more-or-less planer thin liquid films of one liquid separating cells of the other liquid. The stability of the system to coalescence then depends on the stability to rupture of these films as will be discussed below. If flocculation occurs in the emulsion, the overall rate of creaming will be faster since the flocs are larger in size. However, the final cream volume will be greater due to the more open structure. The methods that can be applied for preventing creaming or sedimentation are roughly the same as those discussed in Chapter 3, for preventing sedimentation of suspensions. The first method is to match the density of oil and aqueous phases. Clearly if ∆ρ = 0, v = 0 as shown by equation (4.58). Consider for example an O/W emulsion where the oil density is 0.9 g cm−3 . To match the density of the aqueous phase (≈ 1 g cm−3 ), the density of the oil must be increased; this can be achieved by replacing 20 % of the oil with another oil with density 1.4 g cm−3 . However, this method is seldom practical, since if density matching is possible, it only occurs at one temperature. The second method for reducing creaming or sedimentation is to reduce the droplet size to values well below 1 µm. Since the gravity force is proportional to R3 , then if R is reduced by a factor of 10, the gravity force is reduced by 1000. Below a certain droplet size (which also depends on the density difference between oil and water), Brownian diffusion may exceed gravity, as shown by equation (4.56), and creaming or sedimentation is prevented. This is the principle of formulation of nanoemulsions (with size range 20–200 nm) which may show very little or no creaming or sedimentation. The most practical method for reducing creaming or sedimentation of macroemulsions (with size > 1 µm) is to use “thickeners” These are high molecular weight polymers, natural or synthetic such as xanthan gum, hydroxyethyl cellulose, alginates, carrageenans, etc. To understand the role of these “thickeners”, let us consider the physical gels obtained by chain overlap. Flexible polymers that produce random coils

4.7 Creaming/sedimentation of emulsions and its prevention

| 185

in solution can produce “gels” at a critical concentration C∗ , referred to as the polymer coil “overlap” concentration. This picture can be realized if one considers the coil dimensions in solution. Considering the polymer chain to be represented by a random walk in three dimensions, one may define two main parameters, namely the root mean square end-to-end length ⟨r2 ⟩1/2 and the root mean square radius of gyration ⟨s2 ⟩1/2 (sometimes denoted by RG ). The two are related by, ⟨r2 ⟩1/2 = 61/2 ⟨s2 ⟩1/2 .

(4.61)

log ƞ

The viscosity of a polymer solution increases gradually with increasing concentration and at a critical concentration, C∗ , the polymer coils with a radius of gyration RG and a hydrodynamic radius Rh (which is higher than RG due to solvation of the polymer chains) begin to overlap and this shows a rapid increase in viscosity. This is illustrated in Fig. 4.34 which shows the variation of log η with log C.

C*

log C

Fig. 4.34: Variation of log η with log C.

In the first part of the curve, η ∝ C, whereas in the second part (above C∗ ), η ∝ C3.4 . The polymer concentration above C∗ is referred to as the semi-dilute range. C∗ is related to RG and the polymer molecular weight M by, C∗ =

3M 4πR3G Nav

.

(4.62)

Nav is the Avogadro number. As M increases, C∗ becomes progressively lower. This shows that to produce physical gels at low concentrations by simple polymer coil overlap, one has to use high molecular weight polymers. Another method to reduce the polymer concentration at which chain overlap occurs is to use polymers that form extended chains such as xanthan gum, which produces conformation in the form of a helical structure with a large axial ratio. These polymers give much higher intrinsic viscosities and they show both rotational and translational diffusion. The relaxation time for the polymer chain is much higher than a corresponding polymer with the same molecular weight but produces random coil conformation. These polymers interact at very low concentrations and the overlap concentration can be very low

186 | 4 Liquid/liquid dispersions (emulsions)

(< 0.01 %). These polysaccharides are used in many formulations to produce physical gels at very low concentrations. At sufficiently high polymer concentration, the elastic properties of the physical gels can arrest the oil droplets, thus preventing creaming or sedimentation. Let us consider the gravitational stresses exerted during creaming or sedimentation, 4 stress = mass of drop × acceleration of gravity = πR3 ∆ρg. (4.63) 3 To overcome such stress one needs a restoring force, restoring force = area of drop × stress of drop = 4πR2 σp .

(4.64)

Thus, the stress exerted by the droplet σp is given by, σp =

∆ρRg . 3

(4.65)

The maximum shear stress developed by an isolated spherical droplet through a medium of viscosity η is given by the expression [18], σp =

3vη . 2R

(4.66)

For droplets at the coarse end of the colloidal range, σp is in the range 10−5 –10−2 Pa. As an illustration, Fig. 4.32 shows the variation of viscosity with shear stress for a typical thickener, namely ethylhydroxyethyl cellulose (EHEC). The results in Fig. 4.35 show that below a critical value of the shear stress (≈ 0.2 Pa) the viscous behaviour is Newtonian. Above this stress value the viscosity decreases, indicating the shear thinning behaviour. The plateau value at σ < 0.2 give the limiting and residual viscosity η(0), sometimes referred to as the zero shear viscosity. Thus for systems containing thickeners, the viscosity of the medium in equation (4.58) may be replaced by the residual viscosity η(0). This results in considerable reduction of the creaming or sedimentation rate. For example, for a very dilute emulsion containing 1.48 % EHEC with η(0) ≈ 2 Pa s, the rate of creaming for 10 µm droplets with ∆ρ = 0.2 g cm−3 is ≈ 2.2 × 10−8 m s−1 which is about three orders of magnitude lower than the value in water (≈ 4.4 × 10−5 m s−1 ). This means that in a 10 cm tube, complete creaming will take more than a month. For 1 µm droplets, complete creaming time will take about two years. In more concentrated emulsions this creaming time will be much longer. The above analysis implies that the residual viscosity η(0) can be used to predict creaming or sedimentation of emulsions. This is illustrated in Fig. 4.36 which shows a plot of v/R2 (m−1 s−1 ) versus η(0) for a latex with volume fraction 0.05 in solutions of EHEC. A linear correlation between v/R2 and η(0) is clearly shown. An effective method for reducing creaming/sedimentation of emulsions is to use associative thickeners. These are hydrophobically modified polymer molecules in which alkyl chains (C12 –C16 ) are either randomly grafted on a hydrophilic polymer

4.7 Creaming/sedimentation of emulsions and its prevention | 187

1.48 % 1.30 %

1.0 η/Pas

1.08 % EHEC 0.86 % 0.1

0.65 % 0.43 % 0.22 %

0.01

0.1

0.01

1

σ/Pa

10

v/R2 m–1s–1

Fig. 4.35: Viscosity as a function of shear stress for various EHEC concentrations.

10–3

10–2

10–1 η(0)/Pa

100

10 Fig. 4.36: Sedimentation rate versus η(0).

molecule such as hydroxyethyl cellulose (HEC), or simply grafted at both ends of the hydrophilic chain. An example of hydrophobically modified HEC is Natrosol plus (Hercules) which contains 3–4 C16 randomly grafted onto hydroxyethyl cellulose. Another example of a polymer that contains two alkyl chains at both ends of the molecule is HEUR (Rohm and Haas) that is made of polyethylene oxide (PEO) that is capped at both ends with linear C18 hydrocarbon chains. The above hydrophobically modified polymers form gels when dissolved in water. Gel formation can occur at relatively lower polymer concentrations when compared with the unmodified molecule. The most likely explanation of gel formation is due to hydrophobic bonding (association) between the alkyl chains in the molecule. This effectively causes an apparent increase in the molecular weight. These associative structures are similar to micelles, except the aggregation numbers are much smaller.

188 | 4 Liquid/liquid dispersions (emulsions)

Fig. 4.37 shows the variation of viscosity (measured using a Brookfield at 30 rpm) as a function of the alkyl content (C8 , C12 and C16 ) for hydrophobically modified HEC (i.e. HMHEC). The viscosity reaches a maximum at a given alkyl group content that decreases with increasing alkyl chain length. The viscosity maximum increases with increasing alkyl chain length.

C16 HMEC

η/mPas

104

C12 HMEC C8 HMEC

103

102 0

1

2

3

4

5

Alkyl Content

6

Fig. 4.37: Variation of viscosity of 1 % HMHEC versus alkyl group content of the polymer.

η

Associative thickeners also show interaction with surfactant micelles that are present in the formulation. The viscosity of the associative thickeners shows a maximum at a given surfactant concentration that depends on the nature of surfactant. This is shown schematically in Fig. 4.38. The increase in viscosity is attributed to the hydrophobic interaction between the alkyl chains on the backbone of the polymer with the surfactant micelles. A schematic picture showing the interaction between HM polymers and surfactant micelles is shown in Fig. 4.39. At higher surfactant concentration, the “bridges” between the HM polymer molecules and the micelles are broken (free micelles) and η decreases. The viscosity of hydrophobically modified polymers shows a rapid increase a critical concentration, which may be defined as the critical aggregation concentration (CAC) as illustrated in Fig. 4.40 for HMHEC (WSP-D45 from Hercules). The assumption is made that the CAC is equal to the coil overlap concentration C∗ . From a knowledge of C∗ and the intrinsic viscosity [η] one can obtain the number of chains in each aggregate. For the above example [η] = 4.7 and C∗ [η] = 1 giving an aggregation number of ≈ 4. At C∗ the polymer solution shows non-Newtonian flow (shear thinning behaviour) and its shows a high viscosity at low shear rates. This is

Surfactant concentration

Fig. 4.38: Schematic plot of viscosity of HM polymer with surfactant concentration.

4.7 Creaming/sedimentation of emulsions and its prevention | 189

Mixed micelles

Surfactant

Free micelles

Surfactant

Reduced Viscosity

Fig. 4.39: Schematic representation of the interaction of polymers with surfactants.

18 16 14 12 10 8 6 4

CAC 0 0.02 0.04 0.06 0.08 0.1 0.12 C/g dm–3

Fig. 4.40: Variation of reduced viscosity with HMHEC concentration.

illustrated in Fig. 4.41 which shows the variation of apparent viscosity with shear rate (using a constant stress rheometer). Below ≈ 0.1 s−1 , a plateau viscosity value η(0) (referred to as residual or zero shear viscosity) is reached (≈ 200 Pa s). With increasing polymer concentrations above C∗ , the zero shear viscosity increases with increasing polymer concentration. This is illustrated in Fig. 4.42. 1000

η/Pas

100 10 1 0.1 0.01 0.001 0.01 0.1

1

10 100 1000

Shear rate/s–1

Fig. 4.41: Variation of viscosity with shear rate for HMEC WSP-47 at 0.75 g/100 cm−3 .

190 | 4 Liquid/liquid dispersions (emulsions)

105 104 η(0)

103 102 10 1 0

0.2

0.4

0.6

0.8

1.0

1.2

Polymer concentration (%)

Fig. 4.42: Variation of η(0) with polymer concentration.

3000 2500

G'

G' /Pa G''/Pa

2000 1500 G''

1000

G'

500

G'' ω*

0 –3

10

10

–2

10–1

1

10

ω/rad s–1 Fig. 4.43: Variation of G󸀠 and G󸀠󸀠 with frequency for 5.24 HM PEO.

The above hydrophobically modified polymers are viscoelastic; this is illustrated in Fig. 4.43 for a solution 5.25 % of C18 end-capped PEO with M = 35 000, which shows the variation of the storage modulus G󸀠 and loss modulus G󸀠󸀠 with frequency ω (rad s−1 ). G󸀠 increases with increasing frequency and ultimately it reaches a plateau value at high frequency. G󸀠󸀠 (which is higher than G󸀠 in the low frequency regime) increases with increasing frequency, reaches a maximum at a characteristic frequency ω∗ (at which G󸀠 = G󸀠󸀠 ) and then decreases to near zero value in the high frequency regime. The above variation of G󸀠 and G󸀠󸀠 with ω is typical for a system that shows Maxwell behaviour. From the crossover point ω∗ (at which G󸀠 = G󸀠󸀠 ) one can obtain the relaxation time τ of the polymer in solution, 1 τ = ∗. (4.67) ω For the above polymer, τ = 8 s.

4.7 Creaming/sedimentation of emulsions and its prevention | 191

The above gels (sometimes referred to as rheology modifiers) are used for reducing creaming or sedimentation of emulsions. These hydrophobically modified polymers can also interact with hydrophobic oil droplets in an emulsion forming several other associative structures. Another method for reducing creaming or sedimentation of emulsions is to apply the principle of controlled flocculation. As discussed before, the total energy– distance of separation curve for an electrostatically stabilized emulsion shows a shallow minimum (secondary minimum) at relatively long distance of separation between the droplets. By addition of small amounts of electrolyte such a minimum can be made sufficiently deep for weak flocculation to occur. The same applied for sterically stabilized emulsions, which show only one minimum whose depth can be controlled by reducing the thickness of the adsorbed layer. This can be achieved by reducing the molecular weight of the stabilizer and/or addition of a nonsolvent for the chains (e.g. electrolyte). The above phenomenon of weak flocculation may be applied to reduce creaming or sedimentation, in particular for concentrated emulsions. In the latter case, the attractive energy required for inducing weak flocculation can be small (of the order of few kT units). This can be understood if one considers the free energy of flocculation that consists of two terms, an energy term determined by the depth of the minimum (Gmin ) and an entropy term that is determined by a reduction in configurational entropy on aggregation of droplets, ∆Gflocc = ∆Hflocc − T∆Sflocc .

(4.68)

With concentrated emulsions, the entropy loss on flocculation is small when compared with that for a dilute emulsion. Hence for flocculation of a concentrated emulsion, a small energy minimum is sufficient when compared with the case with a dilute emulsion. A useful method for reducing creaming or sedimentation is to apply the phenomenon of depletion flocculation that was described in detail in Chapter 14, Vol. 1. Many thickeners such as HEC or xanthan gum may not adsorb on the droplets. This is described as “free” (nonadsorbing) polymer in the continuous phase [22]. At a critical concentration, or volume fraction of free polymer, ϕ+p , weak flocculation occurs, since the free polymer coils become “squeezed-out” from between the droplets. This is illustrated in Fig. 4.44 which shows the situation when the polymer volume fraction exceeds the critical concentration. Since the polymer coils do not adsorb on the droplet surface, a “polymer-free” zone with thickness ∆ (that is proportional to the radius of gyration RG of the free polymer) is produced. When two droplets approach each other such that the interdroplet distance h ≤ 2∆, the polymer coils become “squeezed out” from between the droplets as illustrated in Fig. 4.44. The osmotic pressure outside the droplets is higher than that in between the droplets and this results in attraction whose magnitude depends on the concentration of the free polymer and its molecular weight, as well as the droplet size and ϕ.

192 | 4 Liquid/liquid dispersions (emulsions) ∝ ∝ ∝ ∝

∝ ∝ ∝ ∝ ∝ ∝

Polymer coil

∝ ∝

∝ ∝

∝ ∝ ∝

∝ ∝ ∝ ∝

∝ Depletion zone ∝ ∝ ∝ ∝

ɸp+ ∝

∝

∝ ∝ ∝ ∝

∝ ∝ ∝ ∝ ∝

∝

∝ ∝

Osmotic pressure

∝ ∝ ∝ ∝

Fig. 4.44: Schematic representation of depletion flocculation.

The value of ϕ+p decreases with increasing molecular weight of the free polymer. It also decreases as the volume fraction of the emulsion increases. Initial addition of “free polymer” produces weak flocs that show an increase in the rate of creaming. With a further increase in the “free polymer” concentration, the rate of creaming increases and the emulsion separates into two layers: a cream layer at the top and a clear liquid layer at the bottom of the container. However, above a critical concentration of “free polymer”, the emulsion shows no creaming. This behaviour was recently demonstrated by Abend et al. [23] who investigated the effect of addition of xanthan gum to an O/W emulsion (the oil being Isopar V, a hydrocarbon oil) with a volume fraction ϕ = 0.5, that was stabilized using an A–B–A block copolymer (with A being polyethylene oxide, PEO, and B is polypropylene oxide, PPO, containing 26.5 EO units and 29.7 PO units). The xanthan gum concentration was varied between 0.01 and 0.67 %. Fig. 4.45 shows the visual observation of the emulsions after eight months of storage.

Fig. 4.45: Visual observation (after eight months storage) of O/W emulsions (ϕ = 0.5) stabilized with an A–B–A block copolymer (PEO–PPO–PEO) at various xanthan gum concentrations; from left to right: 0.0, 0.01, 0.05, 0.1, 0.25, 0.50, 0.67 % xanthan gum.

4.7 Creaming/sedimentation of emulsions and its prevention

| 193

It can be seen from Fig. 4.45 that initial addition of xanthan gum up to 0.1 % enhances the creaming of the emulsion as a result of depletion flocculation. The emulsion containing 0.25 % xanthan gum shows less creaming, but phase separation did occur. However at 0.5 and 0.67 % xanthan gum, no creaming occurred after storage for eight months. At a concentration of 0.5 and 0.67 % xanthan gum, the arrested network formed by the droplets appears to be strong enough to withstand the gravitational stress (creaming) and no clear water phase is visible at the bottom of the vessel, even after eight months of storage. Thus, the above weak flocculation can be applied to reduce creaming or sedimentation although it suffers from the following drawbacks: (i) Temperature dependency; as the temperature increases, the hydrodynamic radius of the free polymer decreases (due to dehydration) and hence more polymer will be required to achieve the same effect at lower temperatures. (ii) If the free polymer concentration is below a certain limit, phase separation may occur and the flocculated emulsion droplets may cream or sediment faster than in the absence of the free polymer. As discussed in Chapter 3 in this volume, sedimentation of suspensions can be reduced by addition of “inert” fine particles in the continuous phase. The same principle can be applied to reduce creaming or sedimentation of emulsions. Several fine particulate inorganic material produce “gels” when dispersed in aqueous media, e.g. sodium montmorillonite or silica. These particulate materials produce three-dimensional structures in the continuous phase as a result of interparticle interaction. For example, sodium montmorillonite (referred to as swellable clay) forms gels at low and intermediate electrolyte concentrations. This can be understood from a knowledge of the structure of the clay particles. They consist of plate-like particles composed of an octahedral alumina sheet sandwiched between two tetrahedral silica sheets. In the tetrahedral sheet, tetravalent Si is sometimes replaced by trivalent Al. In the octahedral sheet there may be replacement of trivalent Al by divalent Mg, Fe, Cr or Zn. The small size of these atoms allows them to take the place of small Si and Al. This replacement is usually referred to as isomorphic substitution in which an atom of lower positive valence replaces one of higher valence, resulting in a deficit of positive charge or excess of negative charge. This excess of negative layer charge is compensated by adsorption at the layer surfaces of cations that are too big to be accommodated in the crystal. In aqueous solution, the compensation cations on the layer surfaces may be exchanged by other cations in solution, and hence may be referred to as exchangeable cations. With montmorillonite, the exchangeable cations are located on each side of the layer in the stack, i.e. they are present in the external surfaces as well as between the layers. When montmorillonite clays are placed in contact with water or water vapor the water molecules penetrate between the layers, causing interlayer swelling or (intra)crystalline swelling. This interlayer swelling leads, at most, to doubling of the volume of the dry clay when four layers of water are adsorbed. The much larger

194 | 4 Liquid/liquid dispersions (emulsions)

degree of swelling, which is the driving force for “gel” formation (at low electrolyte concentration), is due to osmotic swelling. It has been suggested that swelling of montmorillonite clays is due to the electrostatic double layers that are produced between the charge layers and cations. This is certainly the case at low electrolyte concentration where the double layer extension (thickness) is large. As discussed above, the clay particles carry a negative charge as a result of isomorphic substitution of certain electropositive elements by elements of lower valency. The negative charge is compensated by cations, which in aqueous solution form a diffuse layer, i.e. an electric double layer is formed at the clay plate/solution interface. This double layer has a constant charge, which is determined by the type and degree of isomorphic substitution. However, the flat surfaces are not the only surfaces of the plate-like clay particles, they also expose an edge surface. The atomic structure of the edge surfaces is entirely different from that of the flat-layer surfaces. At the edges, the tetrahedral silica sheets and the octahedral alumina sheets are disrupted, and the primary bonds are broken. The situation is analogous to that of the surface of silica and alumina particles in aqueous solution. On such edges, therefore, an electric double layer is created by adsorption of potential determining ions (H+ and OH− ) and one may, therefore identify an isoelectric point (IEP) as the point of zero charge (pzc) for these edges. With broken octahedral sheets at the edge, the surface behaves as Al–OH with an IEP in the region of pH 7–9. Thus in most cases the edges become negatively charged above pH 9 and positively charged below pH 9. Van Olphen [24] suggested a mechanism of gel formation of montmorillonite involving interaction of the oppositely charged double layers at the faces and edges of the clay particles. This structure, which is usually referred to as a “card-house” structure, was considered to be the reason for the formation of the voluminous clay gel. However, Norrish [25] suggested that the voluminous gel is the result of the extended double layers, particularly at low electrolyte concentrations. A schematic picture of gel formation produced by double layer expansion and “card-house” structure is shown in Fig. 4.46.

(a) Gels produced by Double layer overlap

(b) Gels produced by edge-to-face association

Fig. 4.46: Schematic representation of gel formation in aqueous clay dispersions.

4.7 Creaming/sedimentation of emulsions and its prevention | 195

Finely divided silica such as Aerosil 200 (produced by Degussa) produces gel structures by simple association (by van der Waals attraction) of the particles into chains and cross chains. When incorporated in the continuous phase of an emulsion, these gels prevent creaming or sedimentation. By combining thickeners such as hydroxyethyl cellulose or xanthan gum with particulate solids such as sodium montmorillonite, a more robust gel structure can be produced. By using such mixtures, the concentration of the polymer can be reduced, thus overcoming the problem of dispersion on dilution. This gel structure may be less temperature dependent and could be optimized by controlling the ratio of the polymer and the particles. If these combinations of say sodium montmorillonite and a polymer such as hydroxyethyl cellulose, polyvinyl alcohol (PVA) or xanthan gum are balanced properly, they can provide a “three-dimensional structure”, which entraps all the droplets and stops creaming or sedimentation of the emulsion. The mechanism of gelation of such combined systems depends to a large extent on the nature of the droplets, the polymer and the conditions. If the polymer adsorbs on the particle surface (e.g. PVA on sodium montmorillonite or silica) a three-dimensional network may be formed by polymer bridging. Under conditions of incomplete coverage of the particles by the polymer, the latter becomes simultaneously adsorbed on two or more particles. In other words the polymer chains act as “bridges” or “links” between the particles. Another method for reducing creaming or sedimentation of emulsions is to use liquid crystalline phases. As discussed in Chapter 9, Vol. 1, surfactants produce liquid crystalline phases at high concentrations [3]. Three main types of liquid crystals can be identified as illustrated in Fig. 4.47: hexagonal phase (sometimes referred to as middle phase), cubic phase and lamellar (neat phase). All these structures are highly viscous and they also show elastic response. If produced in the continuous phase of emulsions, they can eliminate creaming or sedimentation of the droplets. These liquid crystalline phases are particularly useful for applications in hand creams which contain high surfactant concentrations. Surfacant

Water

1/2 1/4

Surfactant Hexagonal

Water Lamellar

1/2

Cubic

Fig. 4.47: Schematic picture of liquid crystalline phases.

1/2 1/4 1/2

196 | 4 Liquid/liquid dispersions (emulsions)

4.8 Flocculation of emulsions Flocculation is the process in which the emulsion drops aggregate, without rupture of the stabilizing layer at the interface, if the pair interaction free energy becomes appreciably negative at a certain separation. This negative interaction is the result of van der Waals attraction GA that is universal for all disperse systems. As shown in Chapter 5, Vol. 1, GA is inversely proportional to the droplet-droplet distance of separation h and it depends on the effective Hamaker constant A of the emulsion system. Flocculation may be weak (reversible) or strong (not easily reversible) depending on the strength of the interdroplet forces. Flocculation usually leads to enhanced creaming because the flocs rise faster than individual drops due to their larger effective radius. Exceptions occur in concentrated emulsions where the formation of gel-like network structures can have a stabilizing influence. Flocculation is enhanced by polydispersity since the differential creaming speeds of small and large drops cause them to come into close proximity more often than would occur in a monodisperse system [1]. The cream layer formed towards the end of the creaming process is actually a concentrated floc. The rate of flocculation can be estimated from the product of a frequency factor (how often drops encounter each other) and a probability factor (how long they stay together). The former can be calculated for the case of Brownian motion (perikinetic flocculation) or under shear flow (orthokinetic flocculation) as will be discussed below. Orthokinetic flocculation depends on the interaction energy, i.e. the free energy required to bring drops from infinity to a specified distance apart. In calculating the interaction energy as a function of interdroplet distance, three terms are normally considered: van der Waals attraction (which depends on droplet radius and the effective Hamaker constant), electrostatic repulsion, produced for example by adsorption of ionic surfactants (which depends on the surface or zeta potential, drop radius and ionic strength of the medium) and steric repulsion, produced by adsorption of nonionic surfactants or polymers (which depends on the adsorption density, conformation of the chain at the O/W interface and solvent quality). Flocculation can occur if the energy barrier is small or absent (for electrostatically stabilized emulsions) or when the stabilizing chains reach poor solvency (for sterically stabilized emulsions, i.e. the Flory–Huggins interaction parameter χ > 0.5). As mentioned above, the condition for kinetic stability described by the Deryaguin–Landau–Verwey–Overbeek (DLVO) theory [6, 7] is the magnitude of the energy maximum, G max , at intermediate separation of droplets. For an emulsion to remain kinetically stable (with no flocculation), Gmax > 25kT. When Gmax < 5kT, or is completely absent, flocculation occurs. Two types of flocculation kinetics may be distinguished: Fast flocculation with no energy barrier, and slow flocculation when an energy barrier exists. Fast flocculation kinetics was treated by Smoluchowki [26], who considered the case where there is no interaction between the two colliding droplets until they come into contact, whereupon they adhere irreversibly. The process is simply represented by second-order kinetics and it is simply diffusion controlled. The

4.8 Flocculation of emulsions | 197

number of particles n at any time t may be related to the final number (at t = 0) n0 by the following expression, n0 , (4.69) n= 1 + k0 n0 t where k0 is the rate constant for fast flocculation that is related to the diffusion coefficient of the droplets D, i.e., k0 = 8πDR. (4.70) D is given by the Stokes–Einstein equation, D=

kT . 6πηR

(4.71)

Combining equations (4.70) and (4.71), k0 =

4 kT = 5.5 × 10−18 m3 s−1 3 η

for water at 25 °C.

(4.72)

The half-life t1/2 (n = (1/2)n0 ) can be calculated at various n0 or volume fraction ϕ as given in Tab. 4.6. Tab. 4.6: Half-life of emulsion flocculation. R (µm)

0.1 1.0 10.0

ϕ 10−5

10−2

10−1

5 × 10−1

765 s 21 h 4 month

76 ms 76 s 21 h

7.6 ms 7.6 s 2h

1.5 ms 1.5 s 25 min

It can be seen from Tab. 4.6 that the timescales involved at various droplet sizes and volume fractions range over some 10 orders of magnitude. A dilute emulsion of large droplets may show no visible sign of flocculation over a day or so, whereas a high concentration of small droplets would appear to be instantly flocculated. Smoluchowski’s analysis also leads to an expression for the number of aggregates at time t, n0 (t/t1/2 )i−1 ni = . (4.73) (1 + t/t1/2 )i+1 Most emulsions are polydisperse and this causes an increase in flocculation rate compared to that of monodisperse emulsions. Another factor that affects the rate of fast flocculation arises from hydrodynamic effects that show a dependency on droplet size and number concentration. The maximum value of k0 found for these systems is 3 × 10−18 m3 s−1 compared to the Smoluchowski value of 5.5 × 10−18 m3 s−1 . This is due to the fact that the hydrodynamic interactions for diffusing droplets obey Stokes law

198 | 4 Liquid/liquid dispersions (emulsions)

only when the droplets are isolated from each other. In a real emulsion, extra hydrodynamic interactions have to be taken into account. Thus, for approaching droplets, the assumption made in the Smoluchowski analysis that the relative diffusion coefficient D12 , given by the sum of their two individual diffusion coefficients D1 and D2 , is no longer valid. Smoluchowski [26] originally accounted for the effect of an energy barrier Gmax , arising from interparticle interaction, on the kinetics of flocculation by introducing a correction parameter α, where α is the fraction of collisions which are “effective”, i.e. leading to irreversible flocculation. This idea was based on the Arrhenius equation for chemical reactions, i.e., Gmax k = k0 exp(− (4.74) ) kT The analogy with chemical reactions is not exact. In chemical kinetics the rate of disappearance of a reactant is given by the absolute concentration of the transition-state species times the frequency of the bond in that species which has to break to form the products. In flocculation kinetics the rate depends on the flux of particles to any chosen particle. Fuchs [27] showed that when particle interactions are present, the flux is made up of two contributions, one due to the Brownian diffusion of the particles, the other due to the interactions. In this way, the rate constant k of slow flocculation is related to the Smoluchowski rate k0 by the stability constant W, W=

k0 . k

(4.75)

W is related to Gmax by the following expression [28], ∞

W = 2R ∫ exp(

Gmax −2 )r dr, kT

(4.76)

2R

where r equals the centre-centre separation of two interacting droplets. W depends on the extent of flocculation and the morphology of the flocs in a given system. For this reason it is usual to only consider the early stages of flocculation (t → 0), where doublets are the only aggregated structures. For charge-stabilized emulsions, W is given by the following expression [28], W=

1 Gmax exp( ), 2κR kT

(4.77)

where κ is the Debye–Huckel parameter that is given by, 1/2

κ=(

2Z 2 e2 C ) εr ε0 kT

,

(4.78)

where Z is the valency of counterions, e is the electronic charge, C is the electrolyte concentration in bulk solution, εr is the relative permittivity of the medium, ε0 is the

4.8 Flocculation of emulsions | 199

permittivity of free space, k is the Boltzmann constant and T is the absolute temperature. Since Gmax is determined by the electrolyte concentration C and valency, one can derive an expression relating W to C and Z, log W = const − 2.06 × 109 (

Rγ2 ) log C, Z2

(4.79)

where γ is a function that is determined by the surface potential ψ0 , γ=[

exp(Zeψ0 /kT) − 1 .] exp(ZEψ0 /kT) + 1

(4.80)

Plots of log W versus log C are shown in Fig. 4.48, which shows a linear relationship in the slow flocculation regime. In the fast flocculation regime, Gmax = 0 and d(log W)/d(log C) = 0. The condition log W = 0 (W = 1) is the onset of fast flocculation. The electrolyte concentration at this point defines the critical flocculation concentration CFC. Above the CFC, W < 1 (due to the contribution of van der Waals attraction which accelerates the rate above the Smoluchowski value). Below the CFC, W > 1 and it increases with decreasing electrolyte concentration. Fig. 4.48 also shows that the CFC decreases with increasing valency in accordance with the Schultz–Hardy rule. 1:1 Electrolyte

W=1

log W

2:2 Electrolyte

0

10¯³

10¯²

10¯¹ log C

Fig. 4.48: log W–log C curves for electrostatically stabilized emulsions.

As described in Chapter 6, Vol. 1, the energy–distance curve described by DLVO theory [6, 7] shows the presence of a shallow energy well, namely the secondary minimum (Gmin ) which is few kT units. In this case flocculation is weak and reversible and hence one must consider both the rate of flocculation (forward rate kf ) and deflocculation (backward rate kb ). The rate of decrease of particle number with time is given by the expression, dn − (4.81) = −kf n2 + kb n. dt The backward reaction (break-up of weak flocs) reduces the overall rate of flocculation. kb may depend on the floc size and the exact way in which the flocs break down. Another complication in the analysis of weak (reversible) flocculation is the effect

200 | 4 Liquid/liquid dispersions (emulsions)

of droplet number concentration. Flocculation of this type is a critical phenomenon rather than a chain (or sequential) process. Thus, a critical droplet number concentration, ncrit has to be exceeded before flocculation occurs. The process of flocculation occurs under shearing conditions and is referred to as orthokinetic (to distinguish it from the diffusion controlled perikinetic process). The simplest analysis is for laminar flow, since for turbulent flow with chaotic vortices (as is the case in a high speed mixer) the particles are subjected to a wide and unpredictable range of hydrodynamic forces [29]. For laminar flow, the particle will move at the velocity of the liquid at the plane coincident with the centre of the particle, vp . In this case the total collision frequency due to flow, cf , is given by the following expression, 16 2 3 dv cf = n R ( ). (4.82) 3 p dx As the particles approach in the shear field, the hydrodynamic interactions cause the colliding pair to rotate and with the combination of the slowing approach due to liquid drainage (lubrication stress) and Brownian motion, not all collisions will lead to aggregation. Equation (4.82) must be reduced by a factor α (the collision frequency) to account for this, 16 dv cf = α n2p R3 ( ). (4.83) 3 dx The collision frequency α is of the order 1 and a typical value would be α ≈ 0.8. (dv/dx) is the shear rate so that equation (4.83) can be written as, cf = α

16 2 3 n R γ.̇ 3 p

(4.84)

And the rate of orthokinetic flocculation is given by, −

dn 16 = α n2p R3 γ.̇ dt 3

(4.85)

A comparison can be made between the collision frequency or rate of orthokinetic and perikinetic flocculation, cf and cB respectively, 2αη0 R3 γ̇ cf = . cB kT

(4.86)

If the particles are dispersed in water at a temperature of 25 °C, the ratio in equation (4.86) becomes, cf ≈ 4 × 1017 R3 γ.̇ (4.87) cB When a liquid is stirred in a beaker using a rod the velocity gradient r shear rate is in the range 1–10 s−1 , with a mechanical stirrer it is about 100 s−1 and at the tip of a turbine in a large reactor it can reach values as high as 1000–10 000 s−1 . This means that the particle radius R must be less than 1 µm if even slow mixing can be disregarded. This shows how the effect of shear can increase the rate of aggregation.

4.8 Flocculation of emulsions | 201

It should be mentioned that the above analysis is for the case where there is no energy barrier, i.e. the Smoluchowski case [26]. In the presence of an energy barrier, i.e. potential limited aggregation, one must consider the contribution due to the hydrodynamic forces acting on the colliding pair [29]. The flocculation of sterically stabilized emulsions occurs when the solvency of the medium for the stabilizing chain becomes worse than a θ-solvent or the Flory–Huggins interaction parameter χ > 0 (see Chapter 14, Vol. 1). The Flory–Huggins interaction parameter χ may be conveniently changed by varying the temperature, adding a nonsolvent or increasing the electrolyte concentration in the external phase. As an illustration, Fig. 4.49 shows the variation of Gmix , Gel , GA and GT with h as χ is increased from < 0.5 to > 0.5. GT Gel

Gmix

GT

Gel

G

Reduce solvency δ

h

2δ

δ

2δ

h

Gmix GA Χ < 0.5

Gmin

Χ > 0.5

Fig. 4.49: Schematic representation of the interaction free energy-separation curves for two droplets stabilized by high molar mass polymer.

It can be seen that when χ > 0.5 (i.e. the medium for the A chains becomes worse than a θ-solvent), a significant value of Gmin is attained resulting in catastrophic flocculation (sometimes referred to as incipient flocculation). With many systems good correlation between the flocculation point and the θ point is obtained. For example, the emulsion will flocculate at a temperature (referred to as the critical flocculation temperature, CFT) that is equal to the θ-temperature of the stabilizing chain. The emulsion may flocculate at a critical volume fraction of a nonsolvent (CFV) which is equal to the volume of nonsolvent that brings it to a θ-solvent. It should be mentioned, however, that some emulsions flocculate on cooling and others on heating. Generally speaking (but not always), the former occurs when a nonaqueous solvent is the external phase (e.g. W/O emulsions), while the latter occurs when water is the external phase (O/W emulsions).

202 | 4 Liquid/liquid dispersions (emulsions)

As discussed in Chapter 13, Vol. 1, the energy distance curve of sterically stabilized emulsions shows a shallow minimum, Gmin , at separation distances h ≈ 2δ, whose depth can be of the order of few kT units. The minimum depth depends on droplet radius R, Hamaker constant A and adsorbed layer thickness δ (i.e. with decreasing molecular weight of the stabilizer). For a given R and A, G min increases with decreasing δ. This is illustrated in Fig. 4.50 which shows the energy–distance curves as a function of δ/R. The smaller the value of δ/R, the larger the value of Gmin . Increasing δ/R

GT

Gmin

h Fig. 4.50: Variation of Gmin with δ/R.

The minimum depth required for causing weak flocculation depends on the volume fraction of the emulsion. The higher the volume fraction, the lower the minimum depth required for weak flocculation. This can be understood if one considers the free energy of flocculation that consists of two terms, an energy term determined by the depth of the minimum (Gmin ) and an entropy term that is determined by a reduction in configurational entropy on aggregation of droplets, ∆Gflocc = ∆Hflocc − T∆Sflocc .

(4.88)

With dilute emulsions, the entropy loss on flocculation is larger than with concentrated emulsions. Hence for flocculation of a dilute emulsion, a higher energy minimum is required when compared with the case of concentrated emulsions. This flocculation is weak and reversible, i.e. on shaking the container redispersion of the emulsion occurs. On standing, the dispersed droplets aggregate to form a weak “gel”. This process (referred to as sol ↔ gel transformation) leads to reversible time dependency of viscosity (thixotropy). On shearing the emulsion, the viscosity decreases and when the shear is removed the viscosity is recovered. As mentioned in Chapter 14, Vol. 1, depletion flocculation is produced by addition of “free” nonadsorbing polymer [21]. In this case, the polymer coils cannot approach the droplets to a distance ∆ (that is determined by the radius of gyration of free polymer RG ), since the reduction of entropy on close approach of the polymer coils is not

4.8 Flocculation of emulsions | 203

compensated by an adsorption energy. The emulsion droplets will be surrounded by a depletion zone with thickness ∆. Above a critical volume fraction of the free polymer, ϕ+p , the polymer coils are “squeezed out” from between the droplets and the depletion zones begin to interact. The interstices between the droplets are now free from polymer coils and hence an osmotic pressure is exerted outside the droplet surface (the osmotic pressure outside is higher than in between the particles) resulting in weak flocculation [21]. A schematic representation of depletion flocculation was shown in Fig. 4.44. The magnitude of the depletion attraction free energy, Gdep , is proportional to the osmotic pressure of the polymer solution, which in turn is determined by ϕp and molecular weight M. The range of depletion attraction is proportional to the thickness of the depletion zone, ∆, which is roughly equal to the radius of gyration, RG , of the free polymer. A simple expression for Gdep is [21], Gdep =

2πR∆2 2∆ (μ1 − μ01 )(1 + ), V1 R

(4.89)

where V1 is the molar volume of the solvent, μ1 is the chemical potential of the solvent in the presence of free polymer with volume fraction ϕp and μ01 is the chemical potential of the solvent in the absence of free polymer. (μ1 − μ01 ) is proportional to the osmotic pressure of the polymer solution. Certain long-chain polymers may adsorb in such a way that different segments of the same polymer chain are adsorbed on different droplets, thus binding or “bridging” the droplets together, despite the electrical repulsion [30]. With polyelectrolytes of opposite charge to the droplets, another possibility exists; the droplet charge may be partly or completely neutralized by the adsorbed polyelectrolyte, thus reducing or eliminating the electrical repulsion and destabilizing the droplets. Effective flocculants are usually linear polymers, often of high molecular weight, which may be nonionic, anionic or cationic in character. Ionic polymers should be strictly referred to as polyelectrolytes. The most important properties are molecular weight and charge density. There are several polymeric flocculants that are based on natural products, e.g. starch and alginates, but the most commonly used flocculants are synthetic polymers and polyelectrolytes, e.g. polyacrylamide and copolymers of acrylamide and a suitable cationic monomer such as dimethylaminoethyl acrylate or methacrylate. Other synthetic polymeric flocculants are poly(vinyl alcohol), poly(ethylene oxide) (nonionic), sodium polystyrene sulphonate (anionic) and polyethyleneimine (cationic). As mentioned above, bridging flocculation occurs because segments of a polymer chain adsorb simultaneously on different droplets thus linking them together. Adsorption is an essential step and this requires favourable interaction between the polymer segments and the droplets. Several types of interactions are responsible for adsorption that is irreversible in nature:

204 | 4 Liquid/liquid dispersions (emulsions)

(i) Electrostatic interaction when a polyelectrolyte adsorbs on a surface bearing oppositely charged ionic groups, e.g. adsorption of a cationic polyelectrolyte on a negative emulsion surface. (ii) Hydrophobic bonding that is responsible for adsorption of nonpolar segments on a hydrophobic surface, e.g. partially hydrolyzed poly(vinyl acetate) (PVA) on a hydrophobic surface such as hydrocarbon oil. (iii) Hydrogen bonding as for example the interaction of the amide group of polyacrylamide with hydroxyl groups on an emulsion surface. (iv) Ion binding as is the case with adsorption of anionic polyacrylamide on a negatively charged surface in the presence of Ca+2 . Effective bridging flocculation requires the adsorbed polymer extends far enough from the droplet surface to attach to other droplets and that there is sufficient free surface available for adsorption of these segments of extended chains. When excess polymer is adsorbed, the droplets can be restabilized, either because of surface saturation or by steric stabilization as discussed before. This is one explanation for the fact that an “optimum dosage” of flocculant is often found; at low concentration there is insufficient polymer to provide adequate links and with larger amounts restabilization may occur. A schematic picture of bridging flocculation and restabilization by adsorbed polymer is given in Fig. 4.51.

Fig. 4.51: Schematic illustration of bridging flocculation (left) and restabilization (right) by adsorbed polymer.

If the fraction of droplet surface covered by polymer is θ then the fraction of uncovered surface is (1 − θ) and the successful bridging encounters between the droplets should be proportional to θ(1 − θ), which has its maximum when θ = 0.5. This is the well-known condition of “half-surface coverage” that has been suggested as giving the optimum flocculation.

4.8 Flocculation of emulsions | 205

An important condition for bridging flocculation with charged droplets is the role of electrolyte concentration. This determines the extension (“thickness”) of the double layer which can reach values as high as 100 nm (in 10−5 mol dm−5 1 : 1 electrolyte such as NaCl). For bridging flocculation to occur, the adsorbed polymer must extend far enough from the surface to a distance over which electrostatic repulsion occurs (> 100 nm in the above example). This means that at low electrolyte concentrations quite high molecular weight polymers are needed for bridging to occur. As the ionic strength is increased, the range of electrical repulsion is reduced and lower molecular weight polymers should be effective. In many practical applications, it has been found that the most effective flocculants are polyelectrolytes with a charge opposite to that of the droplets. In aqueous media most droplets are negatively charged, and cationic polyelectrolytes such as polyethyleneimine are often necessary. With oppositely charged polyelectrolytes it is likely that adsorption occurs to give a rather flat configuration of the adsorbed chain, due to the strong electrostatic attraction between the positive ionic groups on the polymer and the negative charged sites on the droplet surface. This would probably reduce the probability of bridging contacts with other particles, especially with fairly low molecular weight polyelectrolytes with high charge density. However, the adsorption of a cationic polyelectrolyte on a negatively charged droplet will reduce the surface charge of the latter, and this charge neutralization could be an important factor in destabilizing the particles. Another mechanism for destabilization has been suggested by Gregory [10] who proposed an “electrostatic patch” model. This applies to cases where the droplets have a fairly low density of immobile charges and the polyelectrolyte has a fairly high charge density. Under these conditions, it is not physically possible for each surface site to be neutralized by a charged segment on the polymer chain, even though the droplet may have sufficient adsorbed polyelectrolyte to achieve overall neutrality. There are then “patches” of excess positive charge, corresponding to the adsorbed polyelectrolyte chains (probably in a rather flat configuration), surrounded by areas of negative charge, representing the original particle surface. Droplets which have this “patchy” or “mosaic” type of surface charge distribution may interact in such a way that the positive and negative “patches” come into contact, giving quite strong attraction (although not as strong as in the case of bridging flocculation). A schematic illustration of this type of interaction is given in Fig. 4.52. The electrostatic patch concept (which can be regarded as another form of “bridging”) can explain a number of features of flocculation of negatively charged droplets with positive polyelectrolytes. These include the rather small effect of increasing the molecular weight and the effect of ionic strength on the breadth of the flocculation dosage range and the rate of flocculation at optimum dosage. Flocculation of emulsions can be reduced by controlling the parameters that affect their stabilization. For charge stabilized emulsions, e.g. using ionic surfactants, the most important criterion is to make Gmax as high as possible; this is achieved by three main conditions, namely high surface or zeta potential, low electrolyte concen-

206 | 4 Liquid/liquid dispersions (emulsions)

+ ++ + + ++++ + + +

++ + + + + + + + + + + + +

+

++ + + + + ++ + ++ +++ + + + +

+

+ + + + + + + + + Fig. 4.52: “Electrostatic patch” model for the interaction of negatively charged droplets with adsorbed cationic polyelectrolytes.

tration and low valency of ions. For sterically stabilized emulsions, four main criteria are necessary: (i) Complete coverage of the droplets by the stabilizing chains. (ii) Firm attachment (strong anchoring) of the chains to the droplets. This requires the chains to be insoluble in the medium and soluble in the oil. However, this is incompatible with stabilization which requires a chain that is soluble in the medium and strongly solvated by its molecules. These conflicting requirements are solved by the use of A–B, A–B–A block or BAn graft copolymers (B is the “anchor” chain and A is the stabilizing chain(s)). Examples for the B chains for O/W emulsions are polystyrene, polymethylmethacrylate, polypropylene oxide and alkyl polypropylene oxide. For the A chain(s), polyethylene oxide (PEO) or polyvinyl alcohol are good examples. For W/O emulsions, PEO can form the B chain, whereas the A chain(s) could be polyhydroxy stearic acid (PHS) which is strongly solvated by most oils. (iii) Thick adsorbed layers; the adsorbed layer thickness should be in the region of 5–10 nm. This means that the molecular weight of the stabilizing chains could be in the region of 1000–5000. (iv) The stabilizing chain should be maintained in good solvent conditions (χ < 0.5) under all conditions of temperature change on storage.

4.9 Ostwald ripening in emulsions and its prevention The driving force of Ostwald ripening is the difference in solubility between the smaller and larger droplets [1]. The small droplets with radius r1 will have higher solubility than the larger droplet with radius r2 . This can be easily recognized from the Kelvin equation [31] which relates the solubility of a particle or droplet S(r) to that of a particle or droplet with infinite radius S(∞),

207

4.9 Ostwald ripening in emulsions and its prevention |

S(r) = S(∞) exp(

2γVm ), rRT

(4.90)

Solubility Enhancement

where γ is the solid/liquid or liquid/liquid interfacial tension, Vm is the molar volume of the disperse phase, R is the gas constant and T is the absolute temperature. The quantity (2γVm /RT) has the dimension of length and it termed the characteristic length with an order of ≈ 1 nm. A schematic representation of the enhancement of the solubility c(r)/c(0) with decreasing droplet size according to the Kelvin equation is shown in Fig. 4.53. Kelvin Equation

100

2M γ 1 r

w c(r) = e RTρ c(0)

10

1

1

10

100

Radius (nm)

1000

Fig. 4.53: Solubility enhancement with decreasing particle or droplet radius.

It can be seen from Fig. 4.53 that the solubility of droplets increases very rapidly with decreasing radius, particularly when r < 100 nm. This means that a droplet with a radius of say 4 nm will have about 10 times solubility enhancement compared say with a droplet with 10 nm radius, which has a solubility enhancement of only 2 times. Thus with time, molecular diffusion will occur between the smaller and larger droplets, with the ultimate disappearance of most of the small droplets. This results in a shift in the droplet size distribution to larger values on storage of the emulsion. This could lead to the formation of a dispersion droplet size > µm. This instability can cause severe problems, such as creaming or sedimentation, flocculation and even coalescence of the emulsion. For two droplets with radii r1 and r2 (r1 < r2 ), RT S(r1 ) 1 1 ln[ ] = 2γ[ − ]. Vm S(r2 ) r1 r2

(4.91)

Equation (4.91) is sometimes referred to as the Ostwald equation and it shows that the larger the difference between r1 and r2 , the higher the rate of Ostwald ripening. That is why in preparation of an emulsion, one aims at producing a narrow size distribution. The kinetics of Ostwald ripening is described in terms of the theory developed by Lifshitz and Slesov [32] and by Wagner [33] (referred to as LSW theory). LSW theory assumes that: (i) the mass transport is due to molecular diffusion through the continuous phase; (ii) the dispersed phase droplets are spherical and fixed in space;

208 | 4 Liquid/liquid dispersions (emulsions)

(iii) there is no interaction between neighbouring droplets (the droplets are separated by a distance much larger than the diameter of the droplets); (iv) the concentration of the molecularly dissolved species is constant, except adjacent to the droplet boundaries. The rate of Ostwald ripening ω is given by: ω=

d 3 8γDS(∞)Vm 4DS(∞)α (r ) = ( )f(ϕ) = ( )f(ϕ), dr c 9RT 9

(4.92)

where rc is the radius of a particle or droplet that is neither growing nor decreasing in size, D is the diffusion coefficient of the disperse phase in the continuous phase, f(ϕ) is a factor that reflects the dependency of ω on the disperse volume fraction and α is the characteristic length scale (= 2γVm /RT). Droplets with r > rc grow at the expense of smaller ones, while droplets with r < rc tend to disappear. The validity of the LSW theory was tested by Kabalnov et al. [5] who used 1,2 dichloroethane-in-water emulsions in which the droplets were fixed to the surface of a microscope slide to prevent their coalescence. The evolution of the droplet size distribution was followed as a function of time by microscopic investigations. LSW theory predicts that the droplet growth over time will be proportional to r3c . This is illustrated in Fig. 4.54 for dichloroethane-in-water emulsions.

1

0

2 3 5

4

Fig. 4.54: Variation of average cube radius with time during Ostwald ripening in emulsions of: (1) 1,2 dichloroethane; (2) benzene; (3) nitrobenzene; (4) toluene; (5) p-xylene.

Another consequence of the LSW theory is the prediction that the size distribution function g(u) for the normalized droplet radius u = r/r c adopts a time-independent form given by: g(u) =

81eu2 exp[1/(2u/3 − 1)] 321/3 (u + 3)7/3 (1.5 − u)11/3

for 0 < u ≤ 1.5

(4.93)

and g(u) = 0 for u > 1.5.

(4.94)

4.9 Ostwald ripening in emulsions and its prevention |

209

A characteristic feature of the size distribution is the cut-off at u > 1.5. A comparison of the experimentally determined size distribution (dichloroethane-in-water emulsions) with the theoretical calculations based on the LSW theory is shown in Fig. 4.55. The influence of the alkyl chain length of the hydrocarbon on the Ostwald ripening rate of nanoemulsions was systematically investigated by Kabalanov et al. [34, 35]. Increasing the alkyl chain length of the hydrocarbon used for the emulsion results in a decrease in the oil solubility. According to LSW theory, this reduction in solubility should result in a decrease in the Ostwald ripening rate. This was confirmed by the results of Kabalnov et al. [34, 35] who showed that the Ostwald ripening rate decreases with increasing alkyl chain length from C9 –C16 .

Fig. 4.55: Comparison between theoretical function g(u) (full line) and experimentally determined functions obtained for 1,2 dichloroethane droplets at time 0 (open triangles) and 300 s (inverted solid triangles).

Although the results showed the linear dependency of the cube of the droplet radius with time in accordance with LSW theory, the experimental rates were ≈ 2–3 times higher than the theoretical values. The deviation between theory and experiment has been ascribed to the effect of Brownian motion [34, 35]. LSW theory assumes that the droplets are fixed in space and that molecular diffusion is the only mechanism of mass transfer. For droplets undergoing Brownian motion, one must take into account the contributions of molecular and convective diffusion. LSW theory assumes that there are no interactions between the droplets and it is limited to low oil volume fractions. At higher volume fractions the rate of ripening depends on the interaction between diffusion spheres of neighbouring droplets. It is expected that emulsions with higher volume fractions of oil will have broader droplet size distribution and faster absolute growth rates than those predicted by LSW theory. However, experimental results using high surfactant concentrations (5 %) showed the rate to be independent of the volume fraction in the range 0.01 ≤ ϕ ≤ 0.3. It has been suggested that the emulsion droplets may have been screened from one another by surfactant micelles [34, 35]. A strong dependency on volume fraction has been observed for fluorocarbon-in-water emulsions [34, 35]. A threefold increase in ω was found when ϕ was increased from 0.08 to 0.52.

210 | 4 Liquid/liquid dispersions (emulsions)

It has been suggested that micelles play a role in facilitating the mass transfer between emulsion droplets by acting as carriers of oil molecules [34, 35]. Three mechanisms were suggested: (i) Oil molecules are transferred via direct droplet/micelle collisions. (ii) Oil molecules exit the oil droplet and are trapped by micelles in the immediate vicinity of the droplet. (iii) Oil molecules exit the oil droplet collectively with a large number of surfactant molecules to form a micelle. In mechanism (i) the micellar contribution to the rate of mass transfer is directly proportional to the number of droplet/micelle collisions, i.e. to the volume fraction of micelles in solution. In this case the molecular solubility of the oil in the LSW equation is replaced by the micellar solubility which is much higher. Large increases in the rate of mass transfer would be expected with increasing micelle concentration. Numerous studies indicate, however, that the presence of micelles affects the mass transfer to only a small extent [36]. Results were obtained for decane-in-water emulsions using sodium dodecyl sulphate (SDS) as emulsifier at concentrations above the critical micelle concentration (cmc). The results showed only a two-fold increase in ω above the cmc. This result is consistent with many other studies which showed an increase in the mass transfer of only 2–5 times with increasing micelle concentration. The lack of strong dependency of mass transfer on micelle concentration for ionic surfactants may result from electrostatic repulsion between the emulsion droplets and micelles, which provide a high energy barrier preventing droplet/micelle collisions. In mechanism (ii), a micelle in the vicinity of an emulsion droplet rapidly takes up dissolved oil from the continuous phase. This “swollen” micelle diffuses to another droplet, where the oil is redeposited. Such a mechanism would be expected to result in an increase in the mass transfer over and above that expected from LSW theory. To account for the discrepancy between theory and experiment in the presence of surfactant micelles, Kabalanov [37] considered the kinetics of micellar solubilization and he proposed that the rate of oil monomer exchange between the oil droplets and the micelles is slow, and rate determining. Thus at low micellar concentration, only a small proportion of the micelles are able to rapidly solubilize the oil. This leads to a small, but measurable increase in the Ostwald ripening rate with micellar concentration. Taylor and Ottewill [38] proposed that micellar dynamics may also be important. According to Aniansson et al. [39], micellar growth occurs in a step-wise fashion and is characterized by two relaxation times τ1 and τ2 . The short relaxation time τ1 is related to the transfer of monomers in and out of the micelles, while the long relaxation time τ2 is the time required for break-up and reformation of the micelle. At low SDS (0.05 mol dm−3 ) concentration τ2 ≈ 0.01 s, whereas at higher SDS concentration (0.2 mol dm−3 ) τ2 ≈ 6 s. Taylor and Ottewill [38] suggested that, at low SDS concentration, τ2 may be fast enough to have an effect on the Ostwald ripening rate, but at

4.9 Ostwald ripening in emulsions and its prevention

|

211

5 % SDS τ2 may be as long as 1000 s (taking into account the effect of solubilization on τ2 ), which is too long to have a significant effect on the Ostwald ripening rate. When using nonionic surfactant micelles, larger increases in the Ostwald ripening rate might be expected due to the larger solubilization capacities of the nonionic surfactant micelles and absence of electrostatic repulsion between the nanoemulsion droplets and the uncharged micelles. This was confirmed by Weiss et al. [40] who found a large increase in the Ostwald ripening rate in tetradecane-in-water emulsions in the presence of Tween 20 micelles. Several methods have been suggested for reducing Ostwald ripening. Huguchi and Misra [41] suggested that the addition of a second disperse phase which is virtually insoluble in the continuous phase, such as squalane, can significantly reduce the Ostwald ripening rate. In this case, significant partitioning between different droplets is predicted, with the component having the low solubility in the continuous phase (e.g. squalane) being expected to be concentrated in the smaller droplets. During Ostwald ripening in a two-component disperse system, equilibrium is established when the difference in chemical potential between different sized droplets, which results from curvature effects, is balanced by the difference in chemical potential resulting from partitioning of the two components. Huguchi and Misra [41] derived the following expression for the equilibrium condition, wherein the excess chemical potential of the medium soluble component, ∆μ1 , is equal for all of the droplets in a polydisperse medium, a1 a1 r0 3 ∆μ i = ( ) + ln(1 − Xeq2 ) = ( ) − X02 ( ) = const, RT req req req

(4.95)

where ∆μ1 = μ1 − μ∗1 is the excess chemical potential of the first component with respect to the state μ∗1 when the radius r = ∞ and X02 = 0, r0 and req are the radii of an arbitrary drop under initial and equilibrium conditions respectively, X02 and Xeq2 are the initial and equilibrium mole fractions of the medium insoluble component 2, a1 is the characteristic length scale of the medium soluble component 1. The equilibrium determined by equation (4.95) is stable if the derivative ∂∆μ1 / ∂req is greater than zero for all the droplets in a polydisperse system. Based on this analysis, Kabalanov et al. [42] derived the following criterion, X02 >

2a1 , 3d0

(4.96)

where d0 is the initial droplet diameter. If the stability criterion is met for all droplets, two patterns of growth will result, depending on the solubility characteristic of the secondary component. If the secondary component has zero solubility in the continuous phase, then the size distribution will not deviate significantly from the initial one, and the growth rate will be equal to zero. In the case of limited solubility of the secondary component, the distribution is governed by rules similar to those of LSW theory, i.e. the distribution function is time variant. In this case, the Ostwald ripening

212 | 4 Liquid/liquid dispersions (emulsions)

rate ωmix will be a mixture growth rate that is approximately given by the following equation [42], ϕ1 ϕ2 −1 ωmix = ( + (4.97) ) , ω1 ω2 where ϕ1 is the volume fraction of the medium soluble component and ϕ2 is the volume fraction of the medium insoluble component respectively. If the stability criterion is not met, a bimodal size distribution is predicted to emerge from the initially monomodal one. Since the chemical potential of the soluble component is predicted to be constant for all the droplets, it is also possible to derive the following equation for the quasi-equilibrium component 1, 2a1 X02 + = const, (4.98) d where d is the diameter at time t. Kabalanov et al. [43] studied the effect of addition of hexadecane to a hexanein-water nanoemulsion. Hexadecane, which is less soluble than hexane, was studied at three levels X02 = 0.001, 0.01 and 0.1. For the higher mole fraction of hexadecane, namely 0.01 and 0.1, the emulsion had a physical appearance similar to that of an emulsion containing only hexadecane and the Ostwald ripening rate was reliably predicted by equation (4.95). However, the emulsion with X02 = 0.001 quickly separated into two layers, a sedimented layer with a droplet size of ca. 5 µm and a dispersed population of submicron droplets (i.e. a bimodal distribution). Since the stability criterion was not met for this low volume fraction of hexadecane, the observed bimodal distribution of droplets is predictable. The second method that can be applied to reduce Ostwald ripening is to modify the interfacial layer. According to LSW theory, the Ostwald ripening rate ω is directly proportional to the interfacial tension γ. Thus by reducing γ, ω is reduced. This could be confirmed by measuring ω as a function of SDS concentration for decane-in-water emulsion below the critical micelle concentration (cmc). Below the cmc, γ shows a linear decrease with increasing log[SDS] concentration. Several other mechanisms have been suggested to account for the reduction in Ostwald ripening rate achieved by modification of the interfacial layer. For example, Walstra [44] suggested that emulsions could be effectively stabilized against Ostwald ripening by the use of surfactants that are strongly adsorbed at the interface and which do not desorb during the Ostwald ripening process. In this case the increase in interfacial dilational modulus ε and decreases in interfacial tension γ would be observed for the shrinking droplets. Eventually the difference in ε and γ between droplets would balance the difference in capillary pressure (i.e. curvature effects) leading to a quasi-equilibrium state. In this case, emulsifiers with low solubilities in the continuous phase such as proteins would be preferred. Long-chain phospholipids with a very low solubility (cmc ≈ 10−10 mol dm−3 are also effective in reducing Ostwald ripening of some emulsions. The phospholipid would have to have a solubility in water about three orders of magnitude lower than that of the oil [19].

4.10 Emulsion coalescence and its prevention | 213

4.10 Emulsion coalescence and its prevention When two emulsion droplets come into close contact in a floc or creamed layer or during Brownian diffusion, a thin liquid film or lamella forms between them [1]. This is illustrated in Fig. 4.56.

1 +2 1

2

1

2

1

thin liquid film (a)

(b)

2

1 2 (c)

Fig. 4.56: Droplet coalescence, adhesion and engulfment.

Coalescence results from the rupture of this film as illustrated in Fig. 4.56 (c) at the top. If the film cannot be ruptured, adhesion (Fig. 4.56 (c) middle) or engulfment (Fig. 4.56 (c) bottom) may occur. Film rupture usually commences at a specific “spot” in the lamella, arising from thinning in that region. This is illustrated in Fig. 4.57 where the liquid surfaces undergo some fluctuations forming surface waves. The surface waves may grow in amplitude and the apices may join as a result of the strong van der Waals attraction (at the apex, the film thickness is the smallest). The same applies if the film thins to a small value (critical thickness for coalescence). In order to understand the behaviour of these films, one has to consider two aspects of their physics: (i) the nature of the forces acting across the film, these determine whether the film is thermodynamically stable, metastable or unstable; (ii) the kinetic aspects associated with local (thermal or mechanical) fluctuations in film thickness.

Fig. 4.57: Schematic representation of surface fluctuations.

Several approaches have been considered to analyse the stability of thin films between emulsion droplets in terms of the relevant interactions. Deryaguin [45] introduced the concept of disjoining pressure to account for the stability of the liquid film. Deryaguin [45] suggested that a “disjoining pressure” π(h) is produced in the film

214 | 4 Liquid/liquid dispersions (emulsions)

which balances the excess normal pressure, π(h) = P(b) − P0 ,

(4.99)

where P(h) is the pressure of a film with thickness b and P0 is the pressure of a sufficiently thick film such that the net interaction free energy is zero. π(h) may be equated to the net force (or energy) per unit area acting across the film, dGT π(h) = − , (4.100) db where GT is the total interaction energy in the film. π(h) is made up of three contributions due to electrostatic repulsion (πE ), steric repulsion (πs ) and van der Waals attraction (πA ), π(h) = πE + πs + πA .

(4.101)

To produce a stable film πE + πs > πA and this is the driving force for prevention of coalescence which can be achieved by two mechanisms and their combination: (i) increased repulsion, both electrostatic and steric; (ii) dampening the fluctuation by enhancing the Gibbs elasticity. In general, smaller droplets are less susceptible to surface fluctuations and hence coalescence is reduced. Van den Tempel [46] derived an expression for the coalescence rate of emulsion droplets by assuming the rate to be proportional to the number of contact points between the droplets in an aggregate. Both flocculation and coalescence are taken into account simultaneously. The average number of primary droplets na in an aggregate at time t is given by Smoluchowski theory as described above. The number of droplets n which have not yet combined into aggregates at time t is given by, n=

n0 , (1 + kn0 t)2

(4.102)

where n0 is the initial number of droplets. The number of aggregates nv is given by, nv =

kn20 t . (1 + kn0 t)2

(4.103)

The total number of primary droplets in all aggregates is given by, n0 − nt = n0 [1 − Hence, na =

1 ]. (1 + kn0 t)2

(n0 − nt ) = 2 + an0 t, n0

where a now denotes the rate of flocculation.

(4.104)

(4.105)

4.10 Emulsion coalescence and its prevention | 215

If m is the number of separate droplets existing in an aggregate, then m < na as some coalescence will have occurred; m will be only slightly lower than na if coalescence is slow, whereas m → 1 if coalescence is very rapid. The rate of coalescence is then proportional to m − 1, i.e., the number of contacts between droplets in an aggregate. In a small aggregate, van den Tempel [46] observed that, in sufficiently dilute emulsions, small aggregates generally contain one large droplet together with one or two small ones and are built up linearly. Thus nv decreases in direct proportion to m − 1, whereas m increases at the same time by adhesion to other droplets. The rate of increase caused by flocculation is given by, dm = an0 − K(m − 1), dt

(4.106)

where K is the rate of coalescence. Integrating equation (4.106), for the boundary conditions m = 2 for t = 0, m−1=

an0 an0 + (1 − ) exp(−Kt). K K

(4.107)

The total number of droplets, whether flocculated or not, in a coagulating emulsion at time t is obtained by adding the number of unreacted primary droplets to the number of droplets in aggregates, n = nt + nv m =

kn20 t nv kn0 kn0 + + (1 − ) exp(−Kt)]. [ 1 + kn0 t (1 + kn0 t)2 K K

(4.108)

The first term on the right-hand side of equation (4.108) represents the number of droplets which would have been present if each droplet has been counted as a single droplet. The second term gives the number of droplets which arise when the composition of the aggregates is taken into account. In the limiting case K → ∞, the second term on the right-hand side of equation (4.108) is equal to zero, and the equation reduces to the Smoluchowski equation. On the other hand, if K = 0, i.e. no coalescence occurs, n = n0 for all values of t. For the case 0 < K < ∞, the effect of a change in the droplet number concentration on the rate of flocculation is given by equation (4.108). This clearly shows that the change in droplet number concentration with time depends on the initial droplet number concentration n0 . This illustrates the difference between emulsions and suspensions. In the latter case, the rate of increase of 1/n with time is independent of the particle number concentration. Some calculations, using reasonable values for the rate of flocculation (denoted by a that is equivalent to k in equation (4.108)) and rate of coalescence K are shown in Fig. 4.58 for various values of n0 . It is clear that the rate of increase of 1/n (or decrease in the droplet number concentration) with t rises more rapidly as n0 decreases. Van den Tempel plotted ∆(1/n), i.e. the decrease in droplet number concentration after 5 minutes, versus the initial droplet number concentration n0 for two values of K and k (or a). The results are shown in Fig. 4.59, which clearly shows the rate of flocculation, as measured by

216 | 4 Liquid/liquid dispersions (emulsions)

a = 2 × 10–11 cm3 sec–1 K = 10–3 sec–1

2.0

no = 107

(1/n) × 108

1.5

no = 108 1.0

0.5

no = 109 0

1000

2000

3000

Sec

1.0 K = 4 × 10–3

Fig. 4.58: Variation of 1/n with t at various n0 values.

a = 5 × 10–11 cm3 sec–1 a = 5 × 10–11 cm3 sec–1

Δ (1/n) × 108

0.8

0.6 K = 4 × 10–3 0.4 K = 10–3 0.2

K = 10–3

0

7

8 log no

9

Fig. 4.59: Plot of ∆(1/n) versus log n0 for two values of K and a (or k).

the value of 1/n, does not change significantly with n0 , either for dilute or for concentrated emulsions. However, in the region where kn0 K is of the order of unity, the rate of flocculation decreases sharply with increasing n0 . To simplify equation (4.108) van den Tempel made three approximations: (i) In a flocculating concentrated emulsion kn0 ≫ K. In most real systems K is generally much smaller than unity and kn0 ≥ 1 is sufficient to satisfy this condition. Thus kn0 rapidly becomes larger than unity and the contribution from unreacted primary droplets may be neglected. In this case equation (4.108) reduces to,

4.10 Emulsion coalescence and its prevention

| 217

kn20 t kn0 [1 − exp(−kt)]. (4.109) (1 + kn0 t)2 K Since kn0 t ≫ 1, then 1 + kn0 t ≈ kn0 t, so that, n0 n= [1 − exp(−kt)]. (4.110) Kt This means that the rate of coalescence no longer depends on the rate of flocculation for concentrated emulsions. Van den Tempel calculated the change in droplet number concentration with time for concentrated (n0 > 1010 cm−3 ) and dilute emulsion (n0 = 109 cm−3 ) and for values of k = 5 × 10−11 cm3 s−1 and K = 103 s−1 ; the results are shown in Fig. 4.60. n=

1.0

0.8 no > 1010 0.6 n/no

no = 109

0.4 a = 5 × 10–11 cm3 sec–1 K = 10–3 sec–1

0.2

0

1000

2000 sec

3000

Fig. 4.60: Change in droplet number concentration with time for dilute and concentrated emulsions.

For concentrated emulsions equations (4.109) and (4.110) yield similar results, whereas for dilute emulsions, equation (4.110) gives rise to a serious deviation at values of t less than 1000 s. Moreover, the droplet number concentration is found to decrease approximately exponentially with time, until Kt becomes large compared to unity. Another limitation for the application of equations (4.109) and (4.110) is the assumption made in their derivation, where the number of contact points between m droplets was taken to be m − 1. This is certainly not the case for concentrated emulsions, where the aggregates contain a large number of contacts. In a closely packed aggregate of spheres with the same size, each droplet touches 12 other droplets. The number of contact points will be proportional to m rather than m − 1. In a heterodisperse system, one droplet may even touch 12 other droplets. This can be taken into account by rewriting equation (4.107) as, dm = kn0 − pKm, dt

(4.111)

218 | 4 Liquid/liquid dispersions (emulsions)

where p has a value between 1 and 6. On integrating equation (4.111), one obtains, m=

kn0 kn0 + (2 − ) exp(−pKt), pK pK

(4.112)

which replaces equation (4.107) for concentrated emulsions. This means that for concentrated emulsions the rate of coalescence increases with droplet concentration in a manner dependent on the droplet size distribution, the degree of packing, and on the size of aggregates. (ii) In a very dilute emulsion, kn0 /K can be much smaller than unity if coalescence occurs very rapidly. After flocculation has proceeded for sufficient time, such that Kt ≫ 1, the second term on the right-hand side of equation (4.108) may be neglected relative to the first term. This equation reduces to Smoluchowski’s equation, i.e. the rate of flocculation is independent of any coalescence. (iii) If the degree of coalescence is very small, the exponential term in equation (4.108) may be expanded in a power series, retaining the first two terms only when K ≪ 1, such that the equation reduces to, n = n0 [1 −

Kt Kt + ]. (1 + kn0 t) (1 + kn0 t)2

(4.113)

Equation (4.113) predicts only a very small decrease in droplet number concentration with time, as expected. When flocculation has proceeded for a sufficiently long time, Kt may be much greater than unity. In this case the exponential term may be neglected and then kn0 /t ≫ 1 (in the denominator), then, n=

1 n0 + . Kt kt

(4.114)

With large n0 , the first term on the right-hand side of equation (4.114) predominates, and hence, 1/kt can be neglected. Davies and Rideal [14] discussed the problem of coalescence, incorporating an energy barrier term into the Smoluchowski equation, in order to account for the slow coalescence for emulsions stabilized by sodium oleate. The Smoluchowski equation may be written in terms of the mean volume V of emulsion droplets, V=

ϕ + 4πDRϕt, n0

(4.115)

where ϕ is the volume fraction of the dispersed phase, D is the diffusion coefficient of the droplets and R is the collision radius. D can be calculated using the Stokes– Einstein equation, kT D= . (4.116) 6πηR k is the Boltzmann constant, T is the absolute temperature, η is the viscosity of the medium and R is the droplet radius.

4.10 Emulsion coalescence and its prevention

| 219

Equation (4.115) predicts that the mean volume of the droplets should be doubled in about 43 seconds, whereas experiments show that in the presence of sodium oleate, this takes about 50 days. To account for this an energy barrier (∆Gcoal ) was introduced in equation (4.115), ∆Gcoal V = V0 + 4πDRϕt exp(− (4.117) ). kT Substituting for D from equation (4.116) and differentiating with respect to t, the rate of coalescence for an O/W emulsion is given by, dV 4ϕkT ∆Gcoal ∆Gcoal exp(− = ) = C1 exp(− ), dt 3ηw kT kT

(4.118)

where ηw is the viscosity of the continuous phase (water for O/W emulsion) and C1 is the collision factor defined by equation (4.118). For a W/O emulsion, the corresponding relation would be, ∆Gcoal ∆Gcoal dV 4(1 − ϕ)kT exp(− = ) = C2 exp(− ), dt 3η0 kT kT

(4.119)

where η0 is the viscosity of the oil continuous phase and C2 is the corresponding collision factor. Davies and Rideal considered the energy barrier in terms of the electrical potential ψ0 at the surface of the oil droplets, arising from drops stabilized by ionic surfactants. The energy barrier preventing coalescence is proportional to ψ20 according to DLVO theory [6, 7] as described above, ∆Gcoal = Bψ20 ,

(4.120)

where B is a constant that depends on the radius of curvature of the droplets. When two approaching droplets tend to flatten in the region of contact in a lamella, the radius of curvature to be used for emulsion droplets may be considerably different from the actual droplet radius. However, the degree of flattening is negligible for small emulsion droplets (< 1 µm in diameter). If there is specific adsorption of counterions, the electric potential to be used in evaluating the electrical double repulsion will be less than ψ0 . In this case one has to use the Stern potential ψd at the plane of specifically adsorbed ions, i.e., ∆Gcoal = Bψ2d . (4.121) Several methods can be applied to reduce coalescence. It has long been known that mixed surfactants can have a synergistic effect on emulsion stability, with respect to coalescence rates. For example, Schulman and Cockbain [47] found that the stability of Nujol/water emulsions increases markedly on addition of cetyl alcohol or cholesterol to an emulsion prepared using sodium cetyl sulphate. The enhanced stability was assumed to be associated with the formation of a densely packed interfacial layer. The maximum effect is obtained when using a water soluble surfactant (cetyl sulphate)

220 | 4 Liquid/liquid dispersions (emulsions)

and an oil soluble surfactant (cetyl alcohol), sometimes referred to as cosurfactant, in combination. Suitable combinations lead to enhanced stability as compared to the individual components. These mixed surfactant films also produce a low interfacial tension, in the region of 0.1 mN m−1 or lower. This reduction in interfacial tension may be due to the cooperative adsorption of the two surfactant molecules, as predicted by the Gibbs adsorption equation for multicomponent systems. For a multicomponent system i, each with an adsorption Γ i (mol m−2 , referred to as the surface excess), the reduction in γ, i.e. dγ, is given by the following expression, dγ = − ∑ Γ i dμ i = − ∑ Γ i RT d ln C i ,

(4.122)

where μ i is the chemical potential of component i, R is the gas constant, T is the absolute temperature and C i is the concentration (mol dm−3 ) of each surfactant component. The reason for the lowering of γ when using two surfactant molecules can be understood by considering the Gibbs adsorption equation for multicomponent systems [9]. For two components ‘sa’ (surfactant) and ‘co’ (cosurfactant), equation (4.122) becomes, dγ = −Γsa RT d ln Csa − Γco RT d ln Cco . (4.123) Integration of equation (4.123) gives, Csa

Cco

γ = γ0 − ∫ Γsa RTdlnCsa − ∫ Γco RT d ln Cco , 0

(4.124)

0

which clearly shows that γ0 is lowered by two terms, from both the surfactant and the cosurfactant. The two surfactant molecules should adsorb simultaneously and they should not interact with each other, otherwise they lower their respective activities. Thus, the surfactant and cosurfactant molecules should vary in nature, one predominantly water soluble (such as an anionic surfactant) and one predominantly oil soluble (such as a long chain alcohol). The synergistic effect of surfactant mixtures can be accounted for by the enhanced lowering of interfacial tension of the mixture when compared with individual components. For example, addition of cetyl alcohol to an O/W emulsion stabilized by cetyl trimethyl ammonium bromide results in lowering of the interfacial tension, and a shift of the critical micelle concentration (cmc) to lower values, probably due to the increased packing of the molecules at the O/W interface [1]. Another effect of using surfactant mixtures is due to the enhanced Gibbs dilational elasticity, ε, given by equation (4.19) Prins and van den Tempel [48] showed that the surfactant mixture sodium laurate plus lauric acid gives a very high Gibbs elasticity (of the order of 103 mN m−1 ) when compared with that of sodium laurate alone. In the presence of laurate ions,

4.10 Emulsion coalescence and its prevention | 221

lauric acid has an extremely high surface activity. At half coverage, the interface contains 1.3 mol dm−3 laurate ions and 4.8 × 10−7 mol dm−3 lauric acid. Thus, under these conditions, the minor constituent can contribute more to the Gibbs elasticity than the major constituent. Similar results were obtained by Prins et al. [49] who showed that ε increases markedly in the presence of lauryl alcohol for O/W emulsions stabilized by sodium lauryl sulphate. A correlation between film elasticity and coalescence rate has been observed for O/W emulsions stabilized with proteins [50]. It has long been assumed that a high interfacial viscosity could account for the stability of liquid films. This must play a role under dynamic conditions, i.e. when two droplets approach each other. Under static conditions, the interfacial viscosity does not play a direct role. However, a high interfacial viscosity is often accompanied by a high interfacial elasticity and this may be an indirect contribution to the increased stability of the emulsion. Prins and van den Tempel [48] argued against there being any role played by the interfacial viscosity due to two main observations, namely the small changes in film stability with change in temperature (which should have a significant effect on the interfacial viscosity) and the sudden decrease in interfacial viscosity with a slight increase in the concentration of the major component. Thus, Prins and van den Tempel [48] attributed the enhanced emulsion stability resulting from the presence of a minor component to be solely due to an increase in interfacial elasticity. Another possible explanation of the enhanced stability in the presence of mixed surfactants could be connected to the hindered diffusion of the surfactant molecules in the condensed film. This would imply that the desorption of surfactant molecules is hindered on the approach of two emulsion droplets, and hence thinning of the film is prevented. Another effect of using surfactant mixtures is to produce liquid crystalline phase formation. Friberg and co-workers [51] attributed the enhanced stability of emulsions formed with mixtures of surfactants to the formation of three-dimensional structures, namely, liquid crystals. These structures can form, for example, in a three-component system of surfactant, alcohol and water, as illustrated in Fig. 4.61. The lamellar liquid crystalline phase, denoted by N (neat phase), in the phase diagram is particularly important for stabilizing the emulsion against coalescence. In this case the liquid crystals “wrap” around the droplets in several layers as will be illustrated below. These multilayers form a barrier against coalescence as will be discussed below. Friberg et al. [51] gave an explanation in terms of the reduced attractive potential energy between two emulsion droplets, each surrounded by a layer of liquid crystalline phase. They also considered changes in the hydrodynamic interactions in the interdroplet region; this affects the aggregation kinetics. Friberg et al. [51] calculated the effect on the van der Waals attraction of the presence of a liquid crystalline phase surrounding the droplets. A schematic representation of the flocculation and coalescence of droplets with and without a liquid crystalline layer is shown in Fig. 4.62.

Concentration, alcohol

222 | 4 Liquid/liquid dispersions (emulsions)

N C A

Liquid crystal D

Micellar solution L

B

CMC

Concentration, soap

Fig. 4.61: Phase diagram of surfactant-alcohol-water system.

d OIL

WATER

WATER

A

m WATER

WATER

O

F

WATER

O W O W O W O W O

WATER

M

m l m l m l m l m d W O W O W

OIL

W O W O W

B

Fig. 4.62: Schematic representation of flocculation and coalescence in the presence and absence of liquid crystalline phases.

The upper part of Fig. 4.62 (A to F) represents the flocculation process when the emulsifier is adsorbed as a monomolecular layer. The distance d between the water droplets decreases to a distance m at which the film ruptures and the droplets coalesce; m is chosen to correspond to the thickness of the hydrophilic layers in the liquid crystalline phase. This simplifies the calculations and facilitates comparison with the case in which the liquid crystalline layer is adsorbed around the droplets. The flocculation process for the case of droplets covered with liquid crystalline layers is illustrated in the lower part of Fig. 4.62 (B to M). The oil layer between the droplets thins to thickness m. The coalescence process which follows involves the

4.11 Phase inversion and its prevention

|

223

removal of successive layers between the droplets until a thickness of one layer is reached (F); the final coalescence step occurs in a similar manner to the case for a monomolecular layer of adsorbed surfactant. An effective method to reduce coalescence is to use polymeric surfactants. The most convenient polymeric surfactants are those of the block and graft copolymer type. A block copolymer is a linear arrangement of blocks of variable monomer composition. The nomenclature for a diblock is poly-A-block–poly-B and for a triblock is poly-A-block–poly-B–poly-A. One of the most widely used triblock polymeric surfactants are the “Pluronics” (BASF, Germany) which consists of two poly-A blocks of poly(ethylene oxide) (PEO) and one block of poly (propylene oxide) (PPO). Several chain lengths of PEO and PPO are available. The above polymeric triblocks can be applied as emulsifiers, whereby the assumption is made that the hydrophobic PPO chain resides at the hydrophobic surface, leaving the two PEO chains dangling in aqueous solution and hence providing steric repulsion, and this reduces or eliminates the coalescence of emulsions. A graft copolymers based on polysaccharides on inulin, a linear polyfructose chain with a glucose end, has been developed for stabilization of emulsions [1]. This molecule is used to prepare a series of graft copolymers by random grafting of alkyl chains (using alky isocyanate) onto the inulin backbone. The first molecule of this series is INUTEC® SP1 that is obtained by random grafting of C12 alkyl chains. It has an average molecular weight of ≈ 5000 Daltons. The main advantages of INUTEC® SP1 as a stabilizer for emulsions are: (i) Strong adsorption to the droplet by multipoint attachment with several alkyl chains. This ensures lack of desorption and displacement of the molecule from the interface. (ii) Strong hydration of the linear polyfructose chains both in water and in the presence of high electrolyte concentrations and high temperatures. This ensures effective steric stabilization. Evidence for the high stability of the liquid film between emulsion droplets when using INUTEC® SP1 was obtained by Exerowa et al. [52] using disjoining pressure measurements.

4.11 Phase inversion and its prevention Phase inversion is the process whereby the internal and external phases of an emulsion suddenly invert, e.g., O/W to W/O or vice versa [1–4]. Catastrophic inversion is induced by increasing the volume fraction of the disperse phase. This type of inversion is not reversible [1]; the value of the water : oil ratio at the transition when oil is added to water is not the same as that when water is added to oil. The inversion point depends on the intensity of agitation and the rate of liquid addition to the emulsion.

224 | 4 Liquid/liquid dispersions (emulsions)

Phase inversion can also be transitional, induced by changing factors that affect the HLB of the system, e.g. temperature and/or electrolyte concentration. The average droplet size decreases and the emulsification rate (defined as the time required to achieve a stable droplet size) increases as inversion is approached. Both trends are consistent with O/W interfacial tension reaching a minimum near the inversion point. Catastrophic inversion is illustrated in Fig. 4.63 which shows the variation of viscosity and conductivity with the oil volume fraction ϕ. As can be seen, inversion occurs at a critical ϕ, which may be identified with the maximum packing fraction [1]. At ϕcr , η suddenly decreases; the inverted W/O emulsion has a much lower volume fraction. κ also decreases sharply at the inversion point since the continuous phase is now oil. Similar trends are observed if water is added to a W/O emulsion, but in this case the conductivity of the emulsion increases sharply at the inversion point. For example, if one starts with a W/O emulsion, then, on increasing the volume fraction of the water phase (the disperse phase), the viscosity of the emulsion increases gradually until a maximum value is obtained, generally ≈ 0.74. When inversion takes place to an O/W emulsion, the volume fraction of the disperse phase (the oil) will now be ≈ 0.26; hence the dramatic decrease in viscosity. κ η κ

η

O/W

κ

η

W/O ϕcr ϕ

Fig. 4.63: Variation of conductivity and viscosity with volume fraction of oil.

In the early theories of phase inversion, it was postulated that inversion took place as a result of the difficulty in packing the emulsion droplets above a certain volume fraction. For example, according to Ostwald [53] an assembly of spheres of equal radii should occupy 74 % of the total volume. Thus, at phase volume ϕ > 0.74, the emulsion droplets have to be packed more densely than is possible. This means that any attempt to increase the phase volume beyond that point should result in distortion, breaking or inversion. However, several investigations showed the invalidity of this argument, inversion being found to take place at phase volumes much greater or smaller than this critical value. For example, Shinoda and Saito [54] showed that inversion of olive oil/water emulsions takes place at ϕ = 0.25. Moreover, Sherman [55] showed that the volume fraction at which inversion takes place depends to a large extent on the nature of the emulsifier. It should be mentioned that Ostwald’s theory [53] applies only to the packing of rigid, nondeformable spheres of equal size. Emulsion droplets are neither

4.11 Phase inversion and its prevention |

225

resistant to deformation, nor are they, in general, of equal size. The wide distribution of droplet size makes it possible to achieve a higher internal phase volume fraction by virtue of the fact that the smaller droplets can be fitted into the interstices between the larger ones. If one adds to this the possibility that the droplets may be deformed into polyhedra, even denser packing is possible. This is the principle of preparing high internal phase emulsions (HIPE) reaching ϕ > 0.95. A useful index to characterize phase inversion is to measure the emulsion inversion point (EIP). The EIP is related to the inversion of W/O emulsions to O/W emulsions at constant temperature [56]. An aqueous phase is added (incrementally) to a finite amount of oil which contains a known amount of surfactant. The mixture is agitated by a turbine blender for 15 s on each addition and the emulsion type is determined. The EIP is simply the ratio of the volume of the aqueous phase at the inversion point to the volume of the oil phase. A plot of EIP versus HLB is made and the results show that the EIP decreases with increasing HLB number until a minimum is observed. The value of the HLB at the minimum is the required HLB of the oil to produce a stable emulsion. However, the exact position of the EIP minimum can be affected by the agitation conditions. EIP experiments have yielded several findings: (i) At the EIP minimum, the inversion from W/O to O/W occurs and produces emulsions with very small drops. (ii) The EIP increases with increasing lipophilic surfactant, whereas the EIP decreases with increasing concentration of hydrophilic surfactant. (iii) In a series of alkanes, the higher the EIP the lower the required HLB. (iv) Highest viscosity and lowest interfacial tension occur at the EIP. (v) For aromatic hydrocarbons, with increasing methyl group substitutions the EIP and the required HLB decrease. (vi) The EIP shows changes in the required HLB of an oil brought about by addition of additives, e.g. alcohols and poly(ethylene glycol). When catastrophic inversion is brought about by addition of the aqueous phase to the oil phase (high HLB), two drop types can be present before phase inversion: Unstable water drops containing surfactant micelles in a continuous oil phase (i.e. Wm /O), and stable oil drops within water drops in a continuous oil phase (i.e. O/Wm /O). When catastrophic inversion is brought about by adding of the oil phase to the water phase (low HLB), two drop types can be present before phase inversion: Unstable oil drops containing surfactant micelles in a continuous aqueous phase (i.e. Om /W), and stable water drops within oil drops in a continuous aqueous phase (i.e. W/Om /W). After catastrophic inversion has taken place the resulting emulsion consists of stable oil drops in a continuous water phase containing surfactant micelles (i.e. O/Wm ) when the initial continuous phase is oil. When the initial continuous phase is aqueous, the resulting emulsion consists of stable water drops in a continuous oil phase containing surfactant micelles (i.e. W/Om ).

226 | 4 Liquid/liquid dispersions (emulsions)

15

0.5

3-Phase region

0.0 Oil drops

10

–0.5 Water drops 5

–1.0 –1,5 5

7

9

11 HLB

13

Log (rate constant/min–1)

Suater mean diameter

Ostwald [53] first modelled catastrophic inversions as being caused by the complete coalescence of the dispersed phase at the close packed condition, whereas Marzall [57] has shown that catastrophic inversion can occur over a wide range of water : oil ratios. This may be due to the formation of double emulsion drops (O/Wm /O) boosting the actual volume of the dispersed phase. The dynamic factors affecting catastrophic inversion have been connected with the movement of inversion boundaries with either changes in the system composition, or changes in the system’s dynamics such as the effect of agitation conditions. For systems that do not contain stabilizing surfactant, inversion hysteresis has been shown to occur. As the viscosity of the oil phase increases, the more likely it is that the oil becomes the dispersed phase. Inversion is shifted to a higher dispersed phase fraction as the stirrer speed increases. Transitional inversion is caused by a change of the system HLB at constant temperature using surfactant mixtures. This is illustrated in Fig. 4.64 which shows the change in the droplet Sauter diameter, d32 (volume/area mean diameter) and rate constant (min−1 ) as a function of the HLB of (nonionic) surfactant mixtures [2].

15

Fig. 4.64: Emulsion drop diameters (circles) and rate constant for attending steady size (squares) as a function of surfactant HLB in cyclohexane/0.067 mol dm−3 KCl containing nonylphenol ethoxylates at 25 °C.

It can be seen from Fig. 4.64 that the average droplet diameter decreases and the emulsification rate (defined as the time required to achieve a stable droplet size) increases as inversion is approached. The results are consistent with the oil/water interfacial tension passing through a minimum within the HLB range where three-phase region forms. It was also noted that emulsions formed by transitional inversion are finer and they require less energy than those made by direct emulsification. Several other conditions can cause transitional phase inversion, such as addition of electrolyte and/or increasing temperature, in particular for emulsions based on nonionic surfactants of the ethoxylate. In order to understand the process of phase inversion, one must consider the surfactant’s affinity to the oil and water phases, as described by the Winsor R0 ratio [58], which is the ratio of the intermolecular attraction of oil molecules (O) and lipophilic portion of surfactant (L), CLO , to that of water (W) and hydrophilic portion (H), CHW , R0 =

CLO . CHW

(4.125)

4.11 Phase inversion and its prevention |

227

CLL, COO, CLO (at oil side) CHH, CWW, CHW (at water side) CLW, CHO, CLH (at the interface) L H Oil O Water W

Fig. 4.65: Various interaction parameters at the oil and water phases.

Several interaction parameters may be identified at the oil and water sides of the interface. One can identify at least nine interaction parameters as schematically represented in Fig. 4.65. In Fig. 4.65, CLL , COO and CLO refer to the interaction energies between the two lipophilic parts of the surfactant molecule, the interaction energy between two oil molecules and the interaction energy between the lipophilic chain and oil respectively. CHH , CWW and CHW refer to the interaction energies between the two hydrophilic parts of the surfactant molecule, the interaction energy between two water molecules and the interaction energy between the hydrophilic chain and water respectively. CLW , CHO and CLH refer to the interaction energies at the interface between the lipophilic part of the surfactant molecule and water, the interaction energy between the hydrophilic part and oil and the interaction energy between the lipophilic and hydrophilic parts respectively. The three cases, R0 < 1, R0 > 1 and R0 = 1, correspond to type I (O/W), type II (W/O) and type III (flat interface) phase behaviour respectively. For example, for R0 < 1, increasing temperature results in a reduction of the hydration of the hydrophilic part of the surfactant molecule and the emulsion changes from Winsor I to Winsor III to Winsor II and this causes phase inversion from an O/W to a W/O emulsion. This inversion occurs at a particular temperature, referred to as the phase inversion temperature (PIT) as will be discussed below. In Winsor’s type I systems (R0 < 1), the affinity of the surfactant for the water phase exceeds its affinity to the oil phase. Thus, the interface will be convex towards water and the nonionic surfactant-oil-water (n-SOW) system can have one or two phases. A system in the two-phase system will split into an oil phase containing dissolved surfactant monomers at the cmco (critical micelle concentration in the oil phase) and an aqueous microemulsion-water phase containing solubilized oil in normal surfactant micelles. In Winsor’s type II systems (R0 > 1), the affinity of the surfactant for the oil phase exceeds its affinity to the water phase. Thus, the interface will be convex towards oil and the nonionic surfactant-oil-water (n-SOW) system can have one or two phases. A system in the two-phase system will split into a water phase containing dissolved

228 | 4 Liquid/liquid dispersions (emulsions)

surfactant monomers at the cmcw (critical micelle concentration in the water phase) and an oleic microemulsion phase containing solubilized water in inverse surfactant micelles. In Winsor’s type III system (R0 = 1), the surfactants affinity for the oil and water phases is balanced. The interface will be flat and the n-SOW system can have one, two or three phases depending on its composition. In the multiphase region, the system can be: (i) two-phase, a water phase and an oleic microemulsion; (ii) two-phase, an oil phase and an aqueous microemulsion; (iii) three-phase, a water phase containing surfactant monomers at cmcw , an oil phase containing surfactant monomers at cmco and a “surfactant phase”. The latter phase may have a bicontinuous structure, being composed of cosolubilized oil and water separated from each other by a layer of surfactant. The “surfactant phase” is sometimes called the middle phase because its intermediate density causes it to appear between the oil and water phases in a phase-separated type III n-SOW system. One way to alter the affinity in an n-SOW system is by changing the temperature which changes the surfactant’s affinity to the two phases. At high temperature the nonionic surfactant becomes soluble in the oil phase whereas at low temperature it becomes more soluble in the water phase. Thus, at a constant surfactant concentration, the phase behaviour will change with temperature [9]. Thus with increasing temperature the surfactant’s affinity to the oil phase increases and the system changes from Winsor I to Winsor III and finally to Winsor II, i.e. the emulsion will invert from an O/W to a W/O system at a particular temperature, referred to as the phase inversion temperature (PIT). A schematic representation of the change in phase behaviour in an n-SOW system is shown in Fig. 4.66. At low temperature, over the Winsor I region, O/W emulsions can be easily formed and are quite stable as shown in Fig. 4.66 (a). On raising the temperature (as shown by the arrow in Fig. 4.66) the O/W emulsion stability decreases, and the macroemulsion finally resolves when the system reaches Winsor III state (represented in Fig. 4.66 (c)). Within this region, both O/W and W/O emulsions are unstable, with the minimum stability in the balanced region. At higher temperature, over the Winsor II region, W/O emulsions become stable as represented in Fig. 4.66 (e). This behaviour is always observed in nonionic systems if the surfactant concentration is above the cmc and the volume fractions of the components are not extreme. The macroemulsion’s stability is essentially symmetrical with respect to the balanced point, just as the phase behaviour is. At positive spontaneous curvature, O/W emulsions are stable, while at negative spontaneous curvature, W/O emulsions are stable. Within the Winsor III region, the stability of the macroemulsion is very temperature sensitive. Although exactly in the balanced state, macroemulsions are very unstable and break within minutes; the system becomes stable only several tens of degrees away from the balanced point, while still being in the Winsor III region.

4.11 Phase inversion and its prevention |

Increasing temperature

w

o

s

o

s

s w

o

Increasing temperature

s

s

w

229

o

w w

o

Oil Middle phase Water

Increasing temperature Fig. 4.66: Effect of increasing temperature on the phase behaviour of an n-SOW system. The PIT concept.

The macroemulsion’s stability pattern is not completely symmetric. W/O emulsions reach maximum stability at ≈ 20 °C above the balanced point, after which the stability starts to decrease. On the other hand, there is no similar maximum stability for the O/W emulsion’s stability at very low temperatures. The macroemulsion phase behaviour can be “tuned”, not only by changes in temperature, but also by addition of “cosolvents”, cosurfactants or electrolytes [59]. For example, the balanced point of an n-C8 H18 –C10 E5 –water system is at ≈ 45 °C, while that of the n-C8 H18 –C10 E5 –10 % NaCl is at ≈ 28 °C. The changes in the macroemulsion phase behaviour induced by additives leads to a similar shift in the macroemulsion stability profile. Thus, when 10 % of NaCl is added to the system, the new balanced point is now established at 28 °C and now macroemulsions prepared below 28 °C will have an O/W type. The same effect is found when the balanced location point is controlled by adding cosolvent to oil and water, changing the chain length of the oil, and adding cosurfactants. The overall pattern of macroemulsion stability as a function of salinity or temperature is illustrated by considering an O/W emulsion of n-octane/water stabilized by a nonionic surfactant such as C12 E5 . Fig. 4.67 shows a visual inspection of the emulsion as a function of NaCl concentration at room temperature. At low salinities, the macroemulsion has an O/W type. As the salinity increases, the system changes from very stable O/W to very stable W/O type with the inversion at the three-phase equilibrium range. O/W emulsions can be distinguished from W/O by the fact that the former forms a cream layer at the top of the container, while the latter forms a milky sediment at the bottom of the container. A schematic representation of emulsion inversion on increasing temperature or increasing the NaCl concentration (at room temperature) is shown in Fig. 4.68, which shows the variation of log lifetime (log τ1/2 ) with increas-

230 | 4 Liquid/liquid dispersions (emulsions)

ing temperature or NaCl concentration. Both O/W and W/O emulsions are very stable far away from the balanced point. The behaviour is not completely identical when the spontaneous curvature is controlled by temperature. Although at low temperature the O/W emulsion is very stable, the W/O emulsion’s stability passes through the maximum and then decreases.

C12E5–n–C8H18–H2O–NaCl System, 20 ± 3 °C CNaCl, wt% 0

1

2

3

7

11 12 13

14

Fig. 4.67: Visual observation of the emulsion type as a function of NaCl concentration.

7

6

log (τ1/2), S

5

Winsor I

Winsor II

O/W

W/O

4

3

2

Winsor III

1

0

5

20

25

30

35

40

45

T, °C

1

5

9

13

17

CNaCl, wt% Fig. 4.68: Logarithmic macroemulsion lifetime (log τ1/2 ) versus temperature or NaCl concentration.

References | 231

In the Winsor III region, the macroemulsion is extremely temperature and salinity sensitive. This is illustrated in Fig. 4.69 which shows that changes by only several tenths of a degree or several tenths of NaCl increase the macroemulsion stability from minutes to days. 7

6

O/W

W/O

log (τ1/2), S

5

4

3

2

1 32.2 32.3 32.4

32.5 32.6 32.7 32.8 32.9 33.0 33.1 T, °C

8.9

9.0

9.1

9.2

9.3

9.4

9.5

9.6

CNaCl, wt% Fig. 4.69: log τ1/2 versus temperature (upper curve) or NaCl concentration (lower curve).

References [1] [2] [3] [4] [5] [6]

Tadros T. Emulsions. Berlin: De Gruyter; 2016. Binks BP, editor. Modern aspects of emulsion science. Cambridge: The Royal Society of Chemistry Publication; 1998. Tadros T. Applied surfactants. Weinheim: Wiley-VCH; 2005. Tadros T. Emulsion Formation Stability and Rheology. In: Tadros T, editor. Emulsion formation and stability. Weinheim: Wiley-VCH; 2013. Chapter 1. Gibbs JW. Collected papers. Vol. 1. Thermodynamics. New York: Dover; 1961. Deryaguin BV, Landau L. Acta Physicochem USSR. 1941;14:633.

232 | 4 Liquid/liquid dispersions (emulsions)

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

Verwey EJW, Overbeek JTG. Theory of stability of lyophobic colloids. Amsterdam: Elsevier; 1948. Lucassen-Reynders EH. Colloids and Surfaces. 1994;A91:79. Lucassen J. In: Lucassen-Reynders EH, editor. Anionic surfactants. New York: Marcel Dekker; 1981. van den Tempel M. Proc Int Congr Surf Act. 1960;2:573. Walstra P, Smolders PEA. In: Binks BP, editor. Modern aspects of emulsions. Cambridge: The Royal Society of Chemistry; 1998. Griffin WC. J Cosmet Chemists. 1949;1:311; 1954;5:249. Davies JT. Proc Int Congr Surface Activity, Vol. 1. 1959. p. 426. Davies JT, Rideal EK. Interfacial phenomena. New York: Academic Press; 1969. Mollet H, Grubenmann A. Formulation Technology. Weinheim: Wiley-VCH; 2001. Tadros TF, Vincent B. In: Becher P, editor. Encyclopedia of Emulsion Technology. New York: Marcel Dekker; 1983. Shinoda K. J Colloid Interface Sci. 1967;25:396. Shinoda K, Saito H. J Colloid Interface Sci. 1969;30:258. Shinoda K. J Chem Soc Japan. 1968;89:435. Tadros T. Rheology of Dispersions. Weinheim: Wiley-VCH; 2010. Bachelor GK. J Fluid Mech. 1972;52:245. Asakura A, Oosawa F. J Chem Phys. 1954;22:1235; J Polymer Sci. 1958;33:183. Abend S, Holtze C, Tadros T, Schutenberger P. Langmuir. 2012;28:7967. van Olphen H. Clay colloid chemistry. New York: Wiley; 1963. Norrish K. Discussion Faraday Soc. 1954;18:120. Smoluchowski MV. Z Phys Chem. 1927;92:129. Fuchs N. Z Physik. 1936;89:736. Reerink H, Overbeek JTG. Discussion Faraday Soc. 1954;18:74. Tadros T. Interfacial phenomena and colloid stability, Vol. 1. Berlin: De Gruyter; 2015. Gregory J. In: Tadros T, editor. Solid/liquid dispersions. London: Academic Press; 1987. Thompson W (Lord Kelvin). Phil Mag. 1871;42:448. Lifshitz EM, Slesov VV. Soviet Physics JETP. 1959;35:331. Wagner C. Z Electrochem. 1961;35:581. Kabalnov AS, Schukin ED. Adv Colloid Interface Sci. 1992;38:69. Kabalnov AS, Makarov KN, Pertsov AV, Shchukin ED. J Colloid Interface Sci. 1990;138:98. Taylor P. Colloids and Surfaces A. 1995;99:175. Kabalanov AS. Langmuir. 1994;10:680. Taylor P, Ottewill RH. Colloids and Surfaces A. 1994;88:303. Anainsson EAG, Wall SN, Almagren M, Hoffmann H, Kielmann I, Ulbricht W, Zana R, Lang J, Tondre C. J Phys Chem. 1976;80:905. Weiss J, Coupland JN, Brathwaite D, McClemments DJ. Colloids and Surfaces A. 1997;121:53. Higuchi WI, Misra J. J Pharm Sci. 1962;51:459. Kabalnov AS, Pertsov AV, Shchukin ED. Colloids and Surfaces. 1987;24:19. Kabalnov AS, Pertsov AV, Aprosin YD, Shchukin ED. Kolloid Zh. 1995;47:1048. Walstra P. In: Binks BP, editor. Encyclopedia of emulsion technology, Vol.4. New York: Marcel Dekker; 1996. Deryaguin BV, Scherbaker RL. Kolloid Zh. 1961;23:33. van den Tempel M. REc Trav Chim. 1953;72:433, 442. Schulman JH, Cockbain EG. Transaction Faraday Soc. 1940;36:661. Prins A, van den Tempel M. Proc Int Congr Surface Activity (4th), Vol. II. London: Gordon and Breach; 1967. p. 1119.

References | 233

[49] [50] [51] [52] [53] [54] [55] [56]

Prins A, Arcuri C, van den Tempel M. J Colloid and Interface Sci. 1967;24:811. Biswas B, Haydon DA. Proc Roy Soc. 1963;A271:296; 1963;A2:317. Friberg S, Jansson PO, Cederberg E. J Colloid Interface Sci. 1976;55:614. Exerowa D, Gotchev G, Kolarev T, Khristov K, Levecke B, Tadros T. Langmuir. 2007;23:1711. Ostwald WO. Kolloid Z. 1910;6:103; 1910;7:64. Shinoda K, Saito H. J Colloid Interface Sci. 1969;30:258. Sherman P. J Oc Chem Inc (London). 1950;69(Suppl. No. 2):570. Brooks BW, Richmond HN, Zefra M. Phase inversion and drop formation in agitated liquidliquid dispersions. In: Binks BP, editor. Modern aspects of emulsion science. Cambridge: The Royal Society of Chemistry Publication; 1998. [57] Marzall L. In: Schick MJ, editor. Nonionic surfactants: Physical chemistry. Surfactant Science Series, Vol. 23. New York: Marcel Dekker; 1967. [58] Winsor PA. Trans Faraday Soc. 1948;44:376. [59] Kabalnov AS. Coalescence in emulsions. In: Binks BP, editor. Modern aspects of emulsion science. Cambridge: The Royal Society of Chemistry Publication; 1998.

5 Multiple emulsions 5.1 Introduction Multiple emulsions are complex systems of “emulsions of emulsions”. Both the waterin-oil-in-water (W/O/W) and oil-in-water-in-oil (O/W/O) multiple emulsions have potential applications in various fields. The W/O/W multiple emulsion may be considered a water/water emulsion in which the internal water droplets are separated by an “oily layer” (membrane). The internal droplets could also consist of a polar solvent such as glycol or glycerol which may contain a dissolved or dispersed active ingredient (AI). The O/W/O multiple emulsion can be considered an oil/oil emulsion separated by an “aqueous layer” (membrane). Multiple emulsions are ideal systems for applications in several industrial systems, e.g. pharmaceuticals, cosmetics, the food industry and agrochemicals. Due to the oily liquid or aqueous membrane formed, multiple emulsions ensure complete protection of the entrapped drug, controlled release of the drug from the internal to the external phase and possible drug targeting due to the vesicular character of these systems. In the cosmetics field, multiple emulsions offer several advantages such as protection of fragile ingredients, separation of incompatible ingredients, prolonged hydration of the skin and in some cases formation of a firm gelled structure. In addition, a pleasant skin feel like that of an O/W emulsion combined with the well-known moisturizing properties of W/O emulsions are obtained with W/O/W multiple emulsions. In the food industry, the existence of an encapsulated water (or oil) phase in a W/O/W (or O/W/O) multiple emulsion facilitates the protection of reactive food ingredients or volatile flavours as well as control of their release. Since less oil phase is required to make a W/O/W multiple emulsion compared to an O/W emulsion with the same disperse volume fraction, multiple emulsions can be used in developing low calorie, reduced-fat food products, e.g. dressings, mayonnaise or spreads. In the agrochemical industry, multiple emulsions allow one to have three active ingredients (AIs) in one formulation and one can incorporate adjuvants in three compartments. Multiple emulsions can be usefully applied for controlled release by controlling the rate of the breakdown process of the multiple emulsion on application. Initially, one prepares a stable multiple emulsion (with a shelf life of two years) which on dilution breaks down in a controlled manner thus releasing the AI also in a controlled manner (slow or sustained release). As will be discussed later, the formulated W/O/W multiple emulsion is osmotically balanced but on dilution the system breaks down as a result of the lack of this balance.

https://doi.org/10.1515/9783110541953-006

236 | 5 Multiple emulsions

5.2 Preparation of multiple emulsions Multiple emulsions are usually prepared in a two-stage process. For example a W/O/W multiple emulsion is formulated by first preparing a W/O emulsion using a surfactant with a low hydrophilic-lipophilic balance (HLB number 5–6) using a high speed mixer (e.g. an Ultra-Turrax or Silverson). The resulting W/O emulsion is further emulsified in aqueous solution containing a surfactant with a high HLB number (9–12) using a low speed stirrer (e.g. a paddle stirrer). A schematic representation of the preparation of multiple emulsions is given in Fig. 5.1. Electrolyte solution

Aqueous electrolyte

Emulsifier 1 (low HLB) plus oil High shear mixer small drops (~1 μm)

Emulsifier 2

Emulsifier 1

Oil

Emulsifier 2 (high HLB) plus electrolyte solution Low shear mixer large drops (10–100 μm)

Oil Electrolyte solution

Emulsifier 1

Fig. 5.1: Scheme for the preparation of a W/O/W multiple emulsion.

The yield of the multiple emulsion can be determined using dialysis for W/O/W multiple emulsions. A water soluble marker is used and its concentration in the outside phase is determined. % multiple emulsion =

Ci × 100, Ci + Ce

(5.1)

where Ci is the amount of marker in the internal phase and Ce is the amount of marker in the external phase. It has been suggested that if a yield of more than 90 % is required, the lipophilic (low HLB) surfactant used to prepare the primary emulsion must be ≈ 10 times higher in concentration than the hydrophilic (high HLB) surfactant.

5.3 Types of multiple emulsions and their breakdown processes | 237

5.3 Types of multiple emulsions and their breakdown processes Florence and Whitehill [1] classified multiple emulsions into three main categories A, B and C as illustrated in Fig. 5.2.

W O W W OW

(a)

W OW W OW

(b)

(c)

Fig. 5.2: Schematic representation of three structures of W/O/W multiple emulsions: (a) one large internal water droplet (Brij 30); (b) several small internal water droplets (Triton X-165); (c) large number of very small internal water droplets (3 : 1 Span 80 : Tween 80).

Type (a) contains one large internal water droplet similar to that described by Matsumoto et al. [2]. This type was produced when polyoxyethylene (4) lauryl ether (Brij 30) was used as emulsifier at 2 %. Type (b) contains several small internal water droplets. These were prepared using 2 % polyoxyethylene (16.5) nonyl phenyl ether (Triton X165). Type (c) drops entrapped a very large number of small internal droplets. These were prepared using a 3 : 1 Span 80:Tween 80 mixture. It should be mentioned that type (a) multiple emulsions are not encountered often in practice. Type (c) is difficult to prepare since a large number of small water internal droplets (which are produced in the primary emulsification process) results in a large increase in viscosity. Thus, the most common multiple emulsions used in practice are those represented by type (b). Florence and Whitehill [1] identified several types of breakdown processes. The external oil drops may coalesce with other oil drops (which may or may not contain internal aqueous droplets). Alternatively, the internal aqueous droplets may be expelled individually or more than one at a time, or they may be less frequently expelled in one step. The internal droplets may coalesce before being expelled or water may pass by diffusion through the oil phase resulting in shrinkage of the internal droplets. A schematic picture of the breakdown processes in multiple emulsions [1] is given in Fig. 5.3. All the above processes are influenced by the nature of the two emulsifiers used for preparation of the multiple emulsion. Most papers published in the literature on multiple emulsions are based on conventional nonionic surfactants. Unfortunately, most of these surfactant systems produce multiple emulsions with limited shelf life, particularly if the system is subjected to large temperature variations. During the past few years, we have formulated multiple emulsions using polymeric surfactants for both the primary and multiple emulsion preparation. These polymeric surfactants proved to be superior over the conventional nonionic surfactants in maintaining the physical stability of the multiple emulsion and they now could be successfully applied

238 | 5 Multiple emulsions

Coalescence (a) (l)

(m)

(j)

(k)

(b)

(g)

(f)

(i) (n)

(e)

(c)

(h) (d)

Fig. 5.3: Schematic representation of the possible breakdown pathways in W/O/W multiple emulsions: (a) coalescence; (b)–(e) expulsion of one or more internal aqueous droplets; (g) less frequent expulsion; (h), (i) coalescence of water droplets before expulsion; (j), (k) diffusion of water through the oil phase; (l)–(n) shrinking of internal droplets.

for the formulation of agrochemical multiple emulsions. The results obtained using these polymeric surfactants offer several potential applications in formulations. The key in the latter cases is to use polymeric surfactants that are approved by the FDA for pharmacy and food, CTA for cosmetics and EPA for agrochemicals. One of the main instabilities of multiple emulsions is the osmotic flow of water from the internal to the external phase or vice versa. This leads to shrinkage or swelling of the internal water droplets respectively. This process assumes the oil layer to act as a semi-permeable membrane (permeable to water but not to solute). The volume flow of water, JW , may be equated with the change of droplet volume with time dv/dt, dv JW = (5.2) = −Lp ART(g2 c2 − g1 c1 ) dt Lp is the hydrodynamic coefficient of the oil “membrane”, A is the cross-sectional area, R is the gas constant and T is the absolute temperature. The flux of water ϕW is, JW , (5.3) ϕW = Vm where Vm is the partial molar volume of water. An osmotic permeability coefficient Po can be defined, Po =

Lp RT . Vm

(5.4)

Combining equations (5.3) and (5.4), ϕW = −Po A(g2 c2 − g1 c1 ).

(5.5)

The diffusion coefficient of water DW can be obtained from Po and the thickness of the diffusion layer ∆x, DW − Po = . (5.6) ∆x For isopropyl myristate W/O/W emulsions, ∆x ≈ 8.2 µm and DW ≈ 5.15 × 10−8 cm2 s−1 , the value expected for diffusion of water in reverse micelles.

5.4 Factors affecting stability of multiple emulsions and criteria for their stabilization |

239

5.4 Factors affecting stability of multiple emulsions and criteria for their stabilization The stability of the resulting multiple emulsion depends on a number of factors: (a) the nature of the emulsifiers used for the preparation of the primary and multiple emulsion; (b) the osmotic balance between the aqueous droplets in the multiple emulsion drops and that in the external aqueous phase; (c) the volume fractions of the disperse water droplets in the multiple emulsion drops and the final volume fraction of the multiple emulsions; (d) the temperature range to which the multiple emulsion is subjected; (e) the process used to prepare the system; (f) the rheology of the whole system which can be modified by the addition of thickeners in the external aqueous phase. The main criteria for the preparation of a stable multiple emulsion are: (i) Two emulsifiers – one with low (emulsifier I) and one with high (emulsifier II) HLB number. (ii) Emulsifier I should provide a very effective barrier against coalescence of the water droplets in the multiple emulsion drop. Emulsifier II should also provide an effective barrier against flocculation and/or coalescence of the multiple emulsion drops. (iii) The amount of emulsifiers used in the preparation of the primary and the multiple emulsion is critical. Excess emulsifier I in the oil phase may result in further emulsification of the aqueous phase into the multiple emulsion with the ultimate production of a W/O emulsion. Excess emulsifier II in the aqueous phase may result in solubilization of the low HLB number surfactant with the ultimate formation of an O/W emulsion. (iv) Optimum osmotic balance of the internal and external aqueous phases. If the osmotic pressure of the internal aqueous droplets is higher than the external aqueous phase, water will flow to the internal droplets, resulting in “swelling” of the multiple emulsion drops with the ultimate production of a W/O emulsion. In contrast, if the osmotic pressure in the outside external phase is higher, water will diffuse in the opposite direction and the multiple emulsion will revert to an O/W emulsion. Various formulation variables must be considered: (i) primary W/O emulsifier; various low HLB number surfactants are available of which the following may be mentioned: decaglycerol decaoleate, mixed triglycerol trioleate and sorbitan trioleate, A–B–A block copolymers of PEO and polyhydroxystearic acid;

240 | 5 Multiple emulsions

(ii) (iii) (iv)

(v) (vi) (vii)

(viii)

primary volume fraction of the W/O or O/W emulsion; usually volume fractions between 0.4 and 0.6 are produced, depending on the requirements; nature of the oil phase; various paraffinic oils (e.g. heptamethyl nonane), silicone oil, soybean and other vegetable oils may be used; secondary O/W emulsifier; high HLB number surfactants or polymers may be used, e.g. Tween 20, polyethylene oxide-polypropylene oxide block copolymers (Pluronics) may be used; secondary volume fraction; this may be varied between 0.4 and 0.8 depending on the consistency required; electrolyte nature and concentration; e.g. NaCl, CaCl2 , MgCl2 or MgSO4 ; thickeners and other additives; in some cases a gel coating for the multiple emulsion drops may be beneficial, e.g. polymethacrylic acid or carboxymethyl cellulose. Gels in the outside continuous phase for a W/O/W multiple emulsion may be produced using xanthan gum (Keltrol or Rhodopol), Carbopol or alginates; process; for the preparation of the primary emulsion, high speed mixers such as Ultra-Turrax or Silverson may be used. For the secondary emulsion preparation, a low shear mixing regime is required, in which case paddle stirrers are probably the most convenient. The mixing times, speed and order of addition need to be optimized.

The present chapter will deal with the formulation of multiple emulsions with particular reference to the use of polymeric surfactants for the preparation of W/O/W multiple emulsions for the above applications. The various classes of polymeric surfactants have been described in Chapter 11, Vol. 1 and in this chapter the most commonly used polymeric surfactants for the preparation of multiple emulsions are described. The fundamental principles of stabilization of emulsions using polymeric surfactants were described in Chapter 13, Vol. 1 and a brief summary of these principles is given. The final section will show some examples of formulations using these polymeric surfactants.

5.5 Polymeric surfactants used for preparation of multiple emulsions As discussed in Chapter 11, Vol. 1, the most convenient polymeric surfactants are those of the block (A–B or A–B–A) and graft (BAn ) types which exhibit considerable surface activity. For an O/W emulsifier, the B chain is chosen to have high affinity to the oil phase (or be soluble in it), whereas the A chain is chosen to be highly soluble in the aqueous medium and strongly solvated by the water molecules. This configuration is the most suitable for effective stabilization as will be discussed below. The high affinity of the B chain to the oil phase ensures strong “anchoring” of the chain to the surface and this prevents any displacement of the molecule on the approach

5.5 Polymeric surfactants used for preparation of multiple emulsions | 241

of two oil droplets, thus preventing any flocculation and/or coalescence. Examples of the A–B–A block copolymers for stabilizing O/W emulsions are the “Pluronics” (trade name of BASF), schematically represented in Fig. 5.4 which also shows the adsorption and conformation of the polymer. These triblock copolymers consist of two poly-A blocks of PEO and one block poly-B of polypropylene oxide (PPO). Several chain lengths of PEO and PPO are available. The PPO chain, which is hydrophobic, adsorbs at the oil surface either due to its high affinity to the oil molecules or as result of “rejection anchoring” (since the PPO chain is insoluble in water). The two PEO chains reside in the aqueous medium and they become strongly solvated by the water molecules (as a result of hydrogen bonding with water molecules). These PEO chains provide the strong repulsion on close approach of two emulsion droplets as will be discussed later. PEO = H(OCH2-CH 2)n

PEO

PPO

PEO

PPO = (O-CH2-CH)m CH 3

Oil Drop

Fig. 5.4: Schematic representation of the structure of “Pluronics” and their adsorption at the O/W interface.

For water-in-oil emulsions, the B chain should have high affinity to the water droplets (or be soluble in water), whereas the A chains should be soluble in the oil phase and strongly solvated by the oil molecules. A good example of such a triblock is polyhydroxystearic acid-polyethylene oxide-polyhydroxystearic acid (PHS–PEO–PHS). A schematic representation of the structure of this block copolymer is shown in Fig. 5.5. Its adsorption and conformation at the W/O interface are shown in Fig. 5.6. The PEO chain is soluble in the water droplets and it provides a strong “anchor” to the interface, whereas the PHS chains are highly soluble in most hydrocarbon solvents as well as some of the polar ones. These PHS chains provide the strong repulsion on approach of the water droplets. The PHS–PEO–PHS molecules also lower the interfacial tension of the W/O interface to very low values (approaching zero) and hence emulsification of water in oil is very efficient allowing one to prepare highly concentrated W/O emulsions that have low viscosity.

242 | 5 Multiple emulsions

O

PHS

O C O O

PHS chain

C HO O

O

O

PEO O

PEO chain

Fig. 5.5: Structure of PHS–PEO–PHS block copolymer.

PHS

PEO

PHS

PEO

Fig. 5.6: Conformation of PHS–PEO–PHS block copolymer at the W/O interface.

5.6 Interaction between oil or water droplets containing an adsorbed polymeric surfactant – steric stabilization Let us consider the case for two water droplets containing an adsorbed polymeric surfactant such as PHS–PEO–PHS (which is the molecule that is used for the preparation of the primary W/O emulsion). As discussed above, this molecule has a very high surface activity at the W/O interface. This is due to the fact that the PEO chain (which has a molecular weight of ≈ 1500) resides in the water droplets, leaving the two PHS chains (each with a molecular weight of ≈ 1000) in the oil phase. Surface pressure (π)–area per molecule (A) isotherms at the water/air (W/A) and water/oil (W/A) interfaces obtained using a semi-automatic Langmuir trough [4] are shown in Fig. 5.7. The W/O isotherm is more expanded than the W/O one, although the two curves tend to approach each other as the surface pressure is increased. The W/O monolayer would

5.6 Interaction between oil or water droplets | 243

Surface Pressure π mNm–1

be expected to be more expanded since the oil will reduce the van der Waals attraction between the hydrocarbon chains. Collapse at the W/A interface was well defined, occurring at a surface pressure of 38.5 mN m−1 . This possibly involves formation of bilayers of PHS–PEO–PHS at the surface, particularly in view of the lamellar liquid crystalline structure of the bulk material. For the W/O case, the maximum surface pressure approached 51 mN m−1 , implying an interfacial pressure approaching zero. These interfacial tension results show that the PHS–PEO–PHS block copolymer is an excellent W/O emulsifier. Indeed, W/O emulsions with a water volume fraction ϕ of 0.9 can be prepared using this block copolymer. 60 50 40 30 W/A

20

W/O

10 0

0

10

20 A/nm2

30

Fig. 5.7: π–A curves (25 °C) for PHS–PEO– PHS block copolymer at the air/water and oil/water interfaces.

The PHS chains are highly solvated by the oil molecules and they extend at the W/O interface giving a layer thickness of the order of 7–8 nm. This was confirmed using film thickness measurements [4]. A thin film consisting of oil plus surfactant was formed between two aqueous droplets, when they were brought into contact and the thickness of the film was measured using light reflectance. A film thickness of 14.2 nm was measured, indicating a layer thickness of the PHS chains of the order of 7 nm. The layer thickness could also be measured using rheological measurements. Shear stress (τ)–shear rate (γ) curves were obtained for W/O emulsions at various volume fractions of water. The results showed a change from Newtonian to non-Newtonian flow behaviour as the volume fraction of the water in the emulsion is increased. The non-Newtonian flow reflects the droplet-droplet interaction as these approach to closer distances. The data were analysed using the Bingham model [5], τ = τ β + ηpl γ.

(5.7)

Fig. 5.8 shows plots of ηpl versus ϕ. The results show that ηpl increases gradually with increasing ϕ, but above ϕ = 0.6, there is a rapid increase in both parameters with a further increase in the volume fraction. Such behaviour is typical for concentrated dispersions [6, 7] which show a rapid increase in the rheological parameters when the distance of separation between the particles become comparable to the range of repulsive interaction.

244 | 5 Multiple emulsions 250

Viscosity (m Pas)

200 Φexp 150

Φeff

100 50 0 0.4

0.5

0.6 0.7 Volume fraction

0.8

0.9

Fig. 5.8: Viscosity–volume fraction curves for W/O emulsions stabilized with PHS–PEO–PHS block copolymer.

Assuming the W/O emulsion behaves as a near “hard-sphere” dispersion, it is possible to apply the Dougherty–Krieger equation [8, 9] to obtain the effective volume fraction, ϕeff . The assumption that the W/O emulsion behaves as a near hard-sphere dispersion is reasonable since the water droplets are stabilized with a block copolymer with relatively short PHS chains (of the order of 10 nm and less). Any lateral displacement of the polymer will be opposed by the high Gibbs elasticity of the adsorbed polymer layer and the droplets will maintain their spherical shape up to high volume fractions. For hard-sphere dispersions, the relative viscosity, ηr , is related to the effective volume fraction by the following expression, ηr = [1 − (

ϕeff −[η]ϕp )] ϕp

(5.8)

In equation (5.8) ϕ is replaced by ϕeff which includes the contribution from the adsorbed layer. [η] is the intrinsic viscosity, which for hard spheres is equal to 2.5, 1/2 whereas ϕp is the maximum packing fraction. It was shown that a plot of 1/ηr versus ϕ is linear with an intercept that is equal to ϕp . For the present W/O emulsion, such a plot gave a ϕp value of 0.84. This value is higher than the theoretical maximum packing fraction for monodisperse spheres (0.74 for hexagonal packing). However, this high value is not unreasonable considering the polydispersity of the W/O emulsion. The high ϕp value shows without doubt that the PHS–PEO–PHS block copolymer is very suitable for the preparation of high volume fraction W/O emulsions. Using ϕp and the measured ηr , ϕeff was calculated at each ϕ value using equation (5.8) and the results are plotted in Fig. 5.8. From ϕeff , the adsorbed layer thickness, δ, was calculated using the following equation, δ 3 ϕeff = ϕ[1 + ( )] , R

(5.9)

where R is the droplet size (which could be determined using dynamic light scattering).

5.6 Interaction between oil or water droplets | 245

A plot of δ versus ϕ showed a linear decrease of δ with increasing ϕ. The value of δ at ϕ = 0.4 is ≈ 10 nm, which is a measure of the fully extended PHS chains. At such low ϕ value, there will be no interpenetration of the PHS chains since the distance between the droplets is relatively large. The reduction in δ with increasing ϕ is due to the possible interpenetration and/or compression of the PHS chains on close approach of the droplets. This is also possible in the thin liquid film studies, which showed a layer thickness in the region of 7–8 nm. When two water droplets each containing adsorbed copolymers of PHS–PEO– PHS approach to a distance of separation h that is smaller than twice the fully extended PHS chains, i.e. h becomes smaller than about 20 nm, repulsion occurs as a result of two main effects [10]. The first repulsive force arises from the unfavourable mixing of the PHS chains when these are in a good solvent. The PHS chains are soluble in most hydrocarbon solvents and are strongly solvated by their molecules over a wide range of temperatures. The unfavourable mixing of polymer chains in good solvent conditions was considered by Flory and Krigbaum [11] whose theory could be applied to the present case of mixing two PHS chains in a hydrocarbon solvent. Before overlap, the chains have a volume fraction ϕ2 and the solvent has a chemical potential μ1α . In the overlap region, the volume fraction of the chains is ϕ󸀠2 , which β

is higher than ϕ2 , and the solvent has a chemical potential, μ1 , which is lower than μ1α . This is equivalent to an increase in the osmotic pressure in the overlap region. As a result, solvent diffuses from the bulk to the overlap region and the two water droplets are separated apart, i.e. this results in strong repulsion. The latter is referred to mixing or osmotic repulsion. Using the Flory–Krigbaum theory [11], one can calculate the free energy of mixing, Gmix , due to this unfavourable overlap, i.e., Gmix 4π 2 h 2 1 h ϕ2 Nav ( − χ)(δ − ) (3R + 2δ + ), = kT 3V1 2 2 2

(5.10)

where k is the Boltzmann constant, T is the absolute temperature, V1 is the molar volume of the solvent and Nav is the Avogadro constant. It is clear from equation (5.10) that when the Flory–Huggins interaction parameter, χ, is less than 0.5, i.e. the chains are in good solvent conditions, Gmix is positive and the interaction is repulsive and it increases very rapidly with decreasing h, when the latter is lower than 2δ. This explains the strong repulsion obtained between water droplets surrounded by PHS chains. The latter are highly soluble in the hydrocarbon medium and any attempt to overlap the chains results in very strong repulsion as a result of the above mentioned unfavourable mixing. Equation (5.10) also shows that when χ > 0.5, i.e. when the solvency of the medium for the chains becomes poor, Gmix is negative and the interaction becomes attractive. Thus, one has to be sure that the solvent used for preparation of the W/O emulsion is a good solvent for the PHS chains, otherwise flocculation of the water droplets (which could be followed by their coalescence) may occur. Fortunately, the

246 | 5 Multiple emulsions

PHS chains are soluble in most hydrocarbon solvents used in most formulations. The condition χ = 0.5 is referred to as θ-solvent and this denotes the onset of change of repulsion to attraction. Thus to ensure steric stabilization by the above mechanism, one has to ensure that the chains are kept in better than θ-solvent. The second repulsive force arises from the loss of configuration entropy when the chains overlap to some extent. When the two surfaces of the water droplets are separated at infinite distance, each chain will have a number of configurations, Ω∞ , that are determined by the volume of the hemisphere swept by the PHS chain. When the two surfaces approach to a distance h that is significantly smaller than the radius of the hemisphere swept by the PHS chain, the volume available to the chain becomes smaller and this results in a reduction in the configurational entropy to a value Ω (that is smaller than Ω∞ ). This results in strong repulsion and the effect is referred to as entropic, volume restriction or elastic repulsion and is given by the following expression, Gel = 2ν ln

Ω , Ω∞

(5.11)

where ν is the number of polymer chains per unit area of the surface. It should be mentioned that Gel is always repulsive in any solvent and it becomes very high on considerable overlap of the PHS chains. This can be illustrated from rheological measurements as will be discussed later. Plots of Gmix , Gel and GA (the van der Waals energy) versus h are illustrated in Fig. 5.9. This figure shows that Gmix is positive (when χ < 0.5) and it increases very rapidly with decreasing h as soon as h becomes less than 2δ (≈ 20 nm for the PHS chains). Gel (which is always positive) increases very rapidly with decreasing h, becoming very large at short distances (when h becomes smaller than δ; about 10 nm for PHS).

GT

G

Gel

Gmix

δ

2δ

h

Gmin

Fig. 5.9: Schematic representation of the variation of Gmix , Gel , GA and GT with h.

5.6 Interaction between oil or water droplets | 247

Combining Gmix , Gel and GA results in the total GT –h curve shown in Fig. 5.9. This curve shows a shallow minimum, Gmin (weak attraction) at h ≈ 2δ, i.e. at h ≈ 20 nm for the present W/O emulsion based on PHS–PEO–PHS block copolymer. When h < 2δ, GT increases very rapidly with any further decrease in h. The depth of the minimum, Gmin , depends on the adsorbed layer thickness. In the present W/O emulsion based on a PHS layer thickness of about 10 nm, Gmin is very small (fraction of kT). This shows that with the present sterically stabilized W/O emulsion, there is only very weak attraction at relatively long distance of separation between the water droplets, which is overcome by Brownian diffusion (which is of the order of 1 kT). Thus, one can say that the net interaction is repulsive and this ensures the long-term physical stability of the W/O emulsion (which approaches thermodynamic stability). Another important use of the PHS–PEO–PHS block copolymer is the formation of a viscoelastic film around the water droplets [12]. This results from the dense packing of the molecule at the W/O interface which results in an appreciable interfacial viscosity. This viscoelastic film prevents transport of water from the internal water droplets in the multiple emulsion drops to the external aqueous medium and this ensures the long-term physical stability of the multiple emulsion when using polymeric surfactants. The viscoelastic film can also reduce the transport of any active ingredient in the internal water droplets to the external phase. This is desirable in many cases when protection of the ingredient in the internal aqueous droplets is required and release is provided on application of the multiple emulsion. When the W/O emulsion is emulsified in an aqueous solution containing another polymeric surfactant with high HLB number, the multiple emulsion drops become surrounded with another polymer surfactant layer. This is illustrated, for example, when using Pluronic PEF127 as the secondary polymer emulsifier. This triblock copolymer consists of two PEO chains of ≈ 100 EO units each and a PPO chain of ≈ 55 units. The PPO chain adsorbs relatively strongly at the O/W interface leaving the two PEO chains in the aqueous continuous phase. The PEO layer thickness is probably larger than 10 nm and hence at separation distance h between the multiple emulsion drops smaller than ≈ 20 nm strong repulsion between the multiple emulsion drops becomes very strong, thus preventing any flocculation and/or coalescence. It should be emphasized that polymeric surfactants prevent coalescence of the water droplets in the multiple emulsion drops as well as coalescence of the latter drops themselves. This is due to the interfacial rheology of the polymeric surfactant films. As a result of the strong lateral repulsion between the stabilizing chains at the interface (PHS chains at the W/O interface and PEO chains at the O/W interface), these films resist deformation under shear and hence they produce a viscoelastic film. On approach of two droplets, this film prevents deformation of the interface and hence coalescence is prevented. From the above discussion one can summarize the role of polymeric surfactants in the stabilization of W/O and W/O/W multiple emulsions. Firstly, the polymeric surfactant ensures complete coverage of the droplets and by virtue of its strong adsorption,

248 | 5 Multiple emulsions

displacement of the film on close approach is prevented. This is essential for eliminating coalescence of the emulsion droplets. Secondly, since the stabilizing chains (PHS for the W/O emulsion and PEO for the W/O/W multiple emulsion) are in good solvent conditions, the mixing interaction is positive, leading to strong repulsion between the drops on close approach. This together with the elastic interaction provides a system that is repulsive at short separation distances. This prevents any flocculation and/or coalescence between the drops. The polymeric surfactant chains should be sufficiently long to prevent any weak flocculation, which may result if the depth of the minimum becomes large, i.e. when δ becomes small (say less than 5 nm). As mentioned above, the strong repulsive force between sterically stabilized emulsion droplets can be investigated using rheological measurements, in particular dynamic (oscillatory) measurements. In these measurements, the emulsion is placed in the gap between two concentric cylinders or a cone and plate geometry. A sinusoidal strain with small amplitude γ0 is applied on one of the plates (say the cup of the concentric cylinder or the plate of the cone-plate geometry). The stress on the other plate is measured simultaneously during the oscillation. The response in stress of a viscoelastic material subjected to a sinusoidally varying strain is monitored as a function of strain amplitude and frequency (see Chapter 1). The stress amplitude, τ0 , is also a sinusoidally varying function in time, but in a viscoelastic material it is shifted out of phase with the strain. The phase angle shift between stress and strain, δ, is given by, δ = ∆tω,

(5.12)

where ω is the frequency in rad s−1 (ω = 2πv, where v is the frequency in Hertz). From the amplitudes of stress and strain and the phase angle shift, one can obtain the following viscoelastic parameters, τ0 , γ0

(5.13)

G󸀠 = |G∗ | cos δ,

(5.14)

|G∗ | =

󸀠󸀠

∗

G = |G | sin δ.

(5.15)

G∗ is the complex modulus, G󸀠 is the elastic component of the complex modulus (which is a measure of the energy stored by the system in a cycle) and G󸀠󸀠 is the viscous component of the complex modulus (which is a measure of the energy dissipated as viscous flow in a cycle). In viscoelastic measurements, one measures the viscoelastic parameters as a function of strain amplitude (at a fixed frequency) in order to obtain the linear viscoelastic region. The strain amplitude is gradually increased from the smallest possible value at which a measurement can be made and the rheological parameters are monitored as a function of strain amplitude γ0 . Initially, the rheological parameters remain virtually constant and independent of the strain amplitude. However, above a critical value of strain amplitude (γcr ), the rheological parameters show a

5.7 Examples of multiple emulsions using polymeric surfactants | 249

change with a further increase in the amplitude above γcr . The linear viscoelastic region is the range of strain amplitudes below γcr . Once this region is established, measurements are made as a function of frequency keeping γ0 below γcr . By fixing the frequency region, while changing the volume fraction of the emulsion one can obtain information on the interdroplet interaction. As an illustration, Fig. 5.10 shows the variation of G󸀠 and G󸀠󸀠 (measured in the linear viscoelastic region and at a frequency of 1 Hz) versus the water volume fraction ϕ.

80 G′

G/Pa

60

40

20

0

G″

0.6

0.65

0.7

0.75

γ Fig. 5.10: Variation of G󸀠 and G󸀠󸀠 with ϕ for W/O emulsions stabilized with a block copolymer of PHS–PEO–PHS.

The results show a transition from predominantly viscous to predominantly elastic response as ϕ exceeds 0.67. This is a direct manifestation of the strong elastic interaction that occurs at and above this critical ϕ. At this volume fraction, the interdroplet distance is comparable to twice the thickness of PHS chains, resulting in their interpenetration and/or compression. As ϕ exceeds 0.7, the storage modulus increases very sharply with a further increase in ϕ and this is a reflection of the very strong repulsion between the water droplets.

5.7 Examples of multiple emulsions using polymeric surfactants Several examples of W/O emulsions and W/O/W multiple emulsions based on the block copolymer of PHS–PEO–PHS have been produced. As an illustration a typical formulation of a W/O/W multiple emulsion is described below, using two different thickeners, namely Keltrol (xanthan gum from Kelco) and Carbopol 980 (a crosslinked polyacrylate gel produced by BF Goodrich). These thickeners were added to reduce creaming of the multiple emulsion. A two-step process was used in both cases.

250 | 5 Multiple emulsions

The primary W/O emulsion was prepared using PHS–PEO–PHS. Four g of PHS– PEO–PHS were dissolved in 30 g of a hydrocarbon oil. For quick dissolution, the mixture was heated to 75 °C. The aqueous phase consisted of 65.3 g water, 0.7 g MgSO4 .7H2 O and a preservative. This aqueous solution was also heated to 75o C. The aqueous phase was added to the oil phase slowly while stirring intensively using a high speed mixer. The W/O emulsion was homogenized for 1 minute and allowed to cool to 40–45 °C followed by further homogenization for another minute and stirring was continued UNTIL the temperature reached ambient. The primary W/O emulsion was emulsified in an aqueous solution containing the polymeric surfactant Pluronic PEF127. Two g of the polymeric surfactant were dissolved in 16.2 g water containing a preservative by stirring at 5 °C. 0.4 g MgSO4 .7H2 O were then added to the aqueous polymeric surfactant solution. 60 g of the primary W/O emulsion were slowly added to the aqueous PFE127 solution while stirring slowly at 700 rpm (using a paddle stirrer). An aqueous Keltrol solution was prepared by slowly adding 0.7 g Keltrol powder to 20.7 g water, while stirring. The resulting thickener solution was further stirred for 30–40 minutes until a homogeneous gel was produced. The thickener solution was slowly added to the multiple emulsion while stirring at low speed (400 rpm) and the whole system was homogenized for 1 minute followed by gentle stirring at 300 rpm until the thickener completely dispersed in the multiple emulsion (about 30 minutes stirring was sufficient). The final system was investigated using optical microscopy to ensure that the multiple emulsion was produced. The formulation was left standing for several months and the droplets of the multiple emulsion were investigated using optical microscopy. The rheology of the multiple emulsion was also measured at various intervals to ensure that the consistency of the product remained the same on long storage. The above multiple emulsion was made under the same conditions except using Carbopol 980 as a thickener (gel). In this case, no MgSO4 was added, since the Carbopol gel is affected by electrolytes. The aqueous PEF127 polymeric surfactant solution was made by dissolving 2 g of the polymer in 23 g water. 15 g of 2 % master gel of Carbopol were added to the PEF127 solution while stirring until the Carbopol was completely dispersed. 60 g of the primary W/O emulsion were slowly added to the aqueous solution of PEF127/Carbopol solution, while stirring thoroughly at 700 rpm. Triethanolamine was added slowly, while gently stirring until the pH of the system reached 6.0–6.5.

5.8 Characterization of multiple emulsions 5.8.1 Droplet size measurements For measuring the droplet size distribution of the primary emulsion (W/O in W/O/W or O/W in O/W/O) that has a submicron range (with an average radius of 0.5–1.0 µm) one

5.8 Characterization of multiple emulsions | 251

can apply the dynamic light scattering technique, referred to as photon correlation spectroscopy (PCS). Details of this method will be described in Chapter 12. Basically, one measures the intensity fluctuation of scattered light by the droplets as they undergo Brownian diffusion. From this intensity fluctuation one can obtain the diffusion coefficient of the droplets from which the radius can be obtained using the Stokes– Einstein equation. For measuring the droplet size distribution of the resulting multiple emulsion (with diameters greater than 5 µm), one can use optical microscopy combined with image analysis. An alternative method for measuring the droplet size distribution is to use light diffraction and to apply Fraunhofer’s diffraction theory. Details of the method will be described in Chapter 12. Basically a laser beam, enlarged and monochromatic by passage through a spatial filter, traverses the sample in which all the droplets to be measured must be well separated. These droplets diffract the light and the intensity of the scattered light is a function of the droplets’ dimensions. Detailed information on the size and structure of the multiple emulsion drops can be obtained using electron microscopy and freeze fracture techniques. The technique consists of four essential steps, namely cryofixation of the sample by rapid cooling to avoid formation of ice crystals, fracture and etching of the cryofixed sample, replication of the exposed surface by coating with platinum-carbon and cleaning of the sample by washing using convenient chemicals. Initially measurements are carried out using the primary W/O or O/W emulsion to obtain information on the size and structure of the primary droplets. This is followed by measurements using the W/O/W or O/W/O multiple emulsion.

5.8.2 Rheological measurements Both steady state shear stress-shear rate, and dynamic (oscillatory) techniques can be applied to study the stability of the multiple emulsions. These techniques were described in detail in Chapter 1. In the steady state method the sample is placed in the gap between two concentric cylinders (or a cone-plate geometry) and the inner or outer cylinder (or cone or plate) is subjected to a constant shear rate that can be gradually increased from the lowest value (usually 0.1 s−1 ) to a maximum value of 100–500 s−1 . The stress is simultaneously measured at each shear rate. In this way a plot of shear stress and viscosity as a function of shear rate is obtained. The rheological results are analysed using models for non-Newtonian flow to obtain the yield value and viscosity. By following the rheology over time of storage of the multiple emulsion (both at room temperature, lower and higher temperature in the range 10–50 °C) one can obtain information on the stability of the multiple emulsion. For example, if the viscosity and yield value of the system do not show any change with storage time this indicates a stable multiple emulsion. If for example a W/O/W multiple emulsion shows a gradual diffusion of water from the external to the internal water droplets, this re-

252 | 5 Multiple emulsions

sults in swelling of the multiple emulsion droplets that is accompanied by a gradual increase in the viscosity and yield value with time until a maximum of the values is reached when maximum swelling occurs. However, when the multiple emulsion droplets break down to form an O/W emulsion, a sudden reduction in viscosity and yield value occurs after a certain storage time. A more sensitive rheological technique for following the stability of multiple emulsions is to use oscillatory techniques. In this case a sinusoidal strain or stress is applied on the sample that is placed in the gap of the concentric cylinder or cone and plate geometry. The resulting stress or strain sine wave is followed at the same time. For a viscoelastic system, as is the case with multiple emulsions, the stress and strain sine waves oscillate with the same frequency but out of phase. From the time shift of the stress and strain amplitudes ∆t and the frequency ω (rad s−1 ) the phase angle shift δ is calculated (δ = ∆tω). From the amplitudes of stress (σ0 ) and strain (γ0 ) and δ one can calculate the various viscoelastic parameters as described above: complex modulus G∗ = (σ0 )/(γ0 ); storage (elastic) modulus G󸀠 = G∗ cos δ; loss (viscous) modulus is G∗ sin δ. Two main experiments are carried out. In the first experiment (strain sweep), the frequency is kept constant at 1 Hz (or 6.28 rad s−1 ) and G∗ , G󸀠 and G󸀠󸀠 are measured as a function of strain amplitude. This allows one to obtain the linear viscoelastic region, where G∗ , G󸀠 and G󸀠󸀠 remain constant and independent of the applied strain. After a critical strain value, that depends on the system, G∗ and G󸀠 start to decrease and G󸀠󸀠 starts to increase with increasing applied strain (nonlinear response). In the second experiment, the strain is kept constant in the linear region and G∗ , G󸀠 and G󸀠󸀠 are measured as a function of frequency. For a viscoelastic liquid, as is the case with a multiple emulsion, both G∗ and G󸀠 are low at low frequency (long timescale) and they gradually increase with increasing frequency, reaching a plateau value at high frequency. In contrast, G󸀠󸀠 is high at low frequency but it increases with increasing frequency, reaching a maximum at which G󸀠 = G󸀠󸀠 and then decreases with any further increase in frequency. The characteristic frequency ω∗ at which G󸀠 = G󸀠󸀠 (tan δ = 1), is equal to the reciprocal of the relaxation time of the system. By following the above measurements as a function of storage time one can assess the stability of the multiple emulsion. For example for a W/O/W multiple emulsion in which diffusion of water occurs from outside to inside the water droplets (as a result of osmotic imbalance), swelling of the multiple emulsion with time results in an increase in the storage time obtained in the linear viscoelastic region, a shift of the characteristic frequency ω∗ to higher values (increase in the relaxation time of the system). However, after sudden breakdown of the multiple emulsion droplets, G󸀠 and ω∗ show a sudden decrease with increasing storage time.

References | 253

References [1] Florence AT, Whitehill D. J Colloid Interface Sci. 1981;79:243. [2] Matsumoto S, Kita Y, Yonezawa D. J Colloid Interface Sci. 1976;57:353. [3] Tadros TF. Int J Cosmet. Sci. 1992;14:93. [4] Aston M. PhD Thesis, Reading University; 1987. [5] Whorlow RW. Rheological Techniques. Chichester: Ellis Horwood; 1980. [6] Tadros TF. Advances Colloid and Interface Science. 1980;12:141. [7] Mewis J, Spaul AJB. Advances Colloid Interface Sci. 1976;6:173. [8] Krieger IM, Dougherty TJ. Trans Soc Rheol. 1959;3:137. [9] Krieger IM, Advances Colloid and Interface Sci. 1972;3:111. [10] Napper DH. Polymeric stabilisation of colloidal dispersions. London: Academic Press; 1983. [11] Flory PJ. Principles of polymer chemistry. New York: Cornell University Press; 1953. [12] Tadros TF. Colloids and Surfaces. 1994;91:39.

6 Gas (air)/liquid dispersions (foams) 6.1 Introduction A gas/liquid dispersion or foam consists of gas bubbles separated by liquid layers [1]. Because of the significant density difference between the gas bubbles and the medium, the system quickly separates into two layers with the gas bubbles rising to the top, which may undergo deformation to form polyhedral structures as will be discussed below. Pure liquids cannot foam unless a surface active material, mostly a surfactant, is present. When a gas bubble is introduced below the surface of a liquid, it burst almost immediately as a soon as the liquid has drained away. With dilute surfactant solutions, as the liquid/air interface expands and the equilibrium at the surface is disturbed, a resorting force is set up which tries to establish the equilibrium. The restoring force arises from the Gibbs–Marangoni effect which was discussed in Chapter 4. As a result of the presence of surface tension gradients dγ (due to incomplete coverage of the film by surfactant), a dilational elasticity ε is produced (Gibbs elasticity). This surface tension gradient induces flow of surfactant molecules from the bulk to the interface and these molecules carry liquid with them (the Marangoni effect). The Gibbs–Marangoni effect prevents thinning and disruption of the liquid film between the air bubbles and this stabilizes the foam. Several surface active foaming materials may be distinguished, e.g. ionic, nonionic and zwitterionic surfactants, polymers (polymeric surfactants). Particles that accumulate at the air/solution interface can also stabilize the foam. In some cases, specifically adsorbed cations or anions from inorganic salts may also stabilize the foam bubbles. Many of the surfactants can cause foaming at extremely low concentrations (as low as 10−9 mol dm−3 ). In kinetic terms, foams may be classified into two main types, namely unstable, transient foams (lifetime of seconds) and metastable or permanent foams (lifetimes of hours or days).

6.2 Foam preparation Like most disperse systems, foams can be obtained by condensation (top-down) and dispersion (bottom-up) methods. The condensation methods for generating foam involve the creation of gas bubbles in the solution by decreasing the external pressure, by increasing temperature or as a result of chemical reaction. Thus, bubble formation may occur through homogeneous nucleation that occurs at high supersaturation or heterogeneous nucleation (e.g. from catalytic sites) that occurs at low supersaturation. The most frequently applied technique for generating foam is by a simple https://doi.org/10.1515/9783110541953-007

256 | 6 Gas (air)/liquid dispersions (foams)

dispersion technique (mechanical shaking or whipping). This method is not satisfactory since accurate control of the amount of air incorporated is difficult to achieve. The most convenient method is to pass a flow of gas (sparging) through an orifice with well-defined radius r0 . The size of the bubbles (produced at an orifice) r may be roughly estimated from the balance of the buoyancy force Fb with the surface tension force Fs [1], Fb = (4/3)πr3 ρg,

(6.1)

Fs = 2πr0 γ,

(6.2)

r=(

3γr0 1/3 ) . 2ρg

(6.3)

r and r0 are the radii of the bubble and orifice, ρ is the specific gravity of liquid, g is the acceleration due to gravity and γ is the gas/liquid surface tension. Since the dynamic surface tension of the growing bubble is higher than the equilibrium tension, the contact base may spread, depending on the wetting conditions. Thus, the main problem is the value of γ to be used in equation (6.3). Another important factor that controls bubble size is the adhesion tension, γ cos θ, where θ is the dynamic contact angle of the liquid on the solid of the orifice. With a hydrophobic surface, a bubble develops with a greater size than the hole. One should always distinguish between the equilibrium contact angle θ and the dynamic contact angle, θdyn , during bubble growth. As the bubble detaches from the orifice, the dimensions of the bubble will determine the velocity of its rise. The rise of the bubble through the liquid causes a redistribution of surfactant on the bubble surface, with the top having a reduced concentration and the polar base having a higher concentration than the equilibrium value. This unequal distribution of surfactant on the bubble surface has an important role in foam stabilization (due to the surface tension gradients). When the bubble reaches the interface, a thin liquid film is produced on its top. The life time of this thin film depends on many factors, e.g. surfactant concentration, rate of drainage, surface tension gradient, surface diffusion and external disturbances.

6.3 Foam structure Two main types of foams may be distinguished [1]: (i) spherical foam (“Kugelschaum”) consisting of gas bubbles separated by thick films of viscous liquid produced in freshly prepared systems. This may be considered as a temporary dilute dispersion of bubbles in the liquid. (ii) Polyhedral gas cells produced on aging; thin flat “walls” are produced with junction points of the interconnecting channels (plateau borders). Due to the interfacial curvature, the pressure is lower and the film is thicker in the plateau border. A capillary suction effect of the liquid occurs from the centre of the film to its periphery.

6.3 Foam structure

|

257

The pressure difference between neighbouring cells, ∆p, is related to the radius of curvature (r) of the plateau border by, 2γ . (6.4) r In a foam column, several transitional structures may be distinguishes as illustrated in Fig. 6.1. Near the surface, a high gas content (polyhedral foam) is formed, with a much lower gas content structure near the base of the column (bubble zone). A transition state may be distinguished between the upper and bottom layers. The drainage of excess liquid from the foam column to the underlying solution is initially driven by hydrostatic pressure, which causes the bubble to become distorted. The foam collapse usually occurs from top to bottom of the column. The films in the polyhedral foam are more susceptible to rupture by shock, temperature gradient or vibration.

Drainage of liquid

∆p =

Polyhedralschaum concentrated gas system with high gas volume and thin films

Upflow of bubbles

Kugelschaum (dilute system) with lower gas volume and thick films

Bubble zone

Fig. 6.1: Schematic representation of a foam structure in a column.

Another mechanism of foam instability is due to Ostwald ripening (disproportionation). The driving force for this process is the difference in Laplace pressure between the small and the larger foam bubbles. The smaller bubbles have higher Laplace pressure than the larger ones. The gas solubility increases with pressure and hence gas molecules will diffuse from the smaller to the larger bubbles. This process only occurs with spherical foam bubbles. This process may be opposed by the Gibbs elasticity effect. Alternatively, rigid films produced using polymers may resist Ostwald ripening as a result of the high surface viscosity. With polyhedral foam with planer liquid lamella, the pressure difference between the bubbles is not large and hence Ostwald ripening is not the mechanism for foam instability in this case. With polyhedral foam, the main driving force for foam collapse is the surface forces that act across the liquid lamella.

258 | 6 Gas (air)/liquid dispersions (foams)

To keep the foam stable (i.e. to prevent complete rupture of the film), this capillary suction effect must be prevented by an opposing “disjoining pressure” that acts between the parallel layers of the central flat film (see below). The generalized model for drainage involves the plateau borders forming a “network” through which the liquid flows due to gravity.

6.4 Classification of foam stability As with most disperse systems, all foams are thermodynamically unstable [1]. This is due to the high interfacial free energy ∆Aγ (where ∆A is the increase in interfacial area when producing the foam bubbles and γ is the surface tension) that exceeds the entropy of dispersion T∆S (where T is the absolute temperature and ∆S is the increase in entropy in forming a large number of air bubbles. Thus, the free energy of formation of a foam ∆G, ∆G = ∆Aγ − T∆S (6.5) is positive and foam formation is non-spontaneous and the resulting foam is thermodynamically unstable. For convenience, foams are classified according to the kinetics of their breakdown: (i) Unstable (transient) foams, with a lifetime of seconds. These are generally produced using “mild” surfactants, e.g. short chain alcohols, aniline, phenol, pine oil, short chain undissociated fatty acid. Most of these compounds are sparingly soluble and may produce a low degree of elasticity. (ii) Metastable (“permanent”) foams, with a lifetime of hours or days. These metastable foams are capable of withstanding ordinary disturbances (thermal or Brownian fluctuations). They can collapse from abnormal disturbances (evaporation, temperature gradients, etc.). The above metastable foams are produced from surfactant solutions near or above the critical micelle concentration (cmc). The stability is governed by the balance of surface forces (see below). The film thickness is comparable to the range of intermolecular forces. In the absence of external disturbances, these foams may stay stable indefinitely. They are produced using proteins, long chain fatty acids or solid particles. Gravity is the main driving force for foam collapse, directly or indirectly through the plateau border. Thinning and disruption may be opposed by surface tension gradients at the air/water interface. Alternatively, the drainage rate may be decreased by increasing the bulk viscosity of the liquid (e.g. addition of glycerol or polymers). Stability may be increased in some cases by the addition of electrolytes that produce a “gel network” in the surfactant film. Foam stability may also be enhanced by increasing the surface viscosity and/or surface elasticity. High packing of surfactant films (high cohe-

6.5 Drainage and thinning of foam films |

259

sive forces) may also be produced using mixed surfactant films or surfactant/polymer mixtures. For investigating foam stability one must consider the role of the plateau border under dynamic and static conditions. One should also consider foam films with intermediate life times, i.e. between unstable and metastable foams.

6.5 Drainage and thinning of foam films As mentioned above, gravity is the main driving force for film drainage. Gravity can act directly on the film or through capillary suction in the plateau borders. As a general rule, the rate of drainage of foam films may be decreased by increasing the bulk viscosity of the liquid from which the foam is prepared. This can be achieved by adding glycerol or high molecular weight poly(ethylene oxide). Alternatively, the viscosity of the aqueous surfactant phase can be increased by addition of electrolytes that form a “gel” network (liquid crystalline phases may be produced). Film drainage can also be decreased by increasing the surface viscosity and surface elasticity. This can be achieved, for example, by addition of proteins, polysaccharides and even particles. These systems are applied in many food foams. Most quantitative studies on film drainage have been carried out by Scheludko and co-workers [2–5] who studied the drainage of small horizontal films using a specially designed measuring system, illustrated in Fig. 6.2. The foam film c is formed in the middle of a biconcave drop b, situated in a glass tube of radius R, by withdrawing liquid from it (A and B) and in the hole of a porous plate g (C). A suitable tube diameter in A and B is 0.2–0.6 mm and the film radius ranges from 100 to 500 nm. In C, the hole radius can be considerably smaller, in the range of 120 µm and the film radius is 10 µm. The film can be observed under the microscope and when it thins to form the so-called “black” film, black spots are observed under the microscope. The film thickness is determined by interferometry, which is based on comparison between the intensities of the light falling on the film and that reflected from it [4]. The drainage time T is determined and compared to the theoretical value for a flat film calculated from the Reynolds equation [6], h0

T=∫

dh . V

(6.6)

ht

h0 is the initial film thickness and h t is the value after time t. V is the velocity of thinning V = −dh/dt. For a horizontal, fairly thick film (> 100 nm), Scheludko [3] derived an expression for the thinning between two disc surfaces under the influence of a uniform external pressure. The change in film thickness with drainage time, Vre , is given by the

260 | 6 Gas (air)/liquid dispersions (foams)

a 2r b

2R c

d dʹ

e

2r g

f

c

2R

Fig. 6.2: Cell used for studying macroscopic foam films: (A) in a glass tube; (B) with a reservoir of surfactant solution d′; (C) in a porous plate. a. glass tube film holder; b. biconcave drop; c. macroscopic foam film; d. glass capillary; e. surfactant solution; f. optically flat glass; g. porous plate.

expression, Vre = −

dh 2h3 ∆P , = dt 3ηR2

(6.7)

where R is the radius of the disc, η is the viscosity of the liquid and ∆P is the difference in pressure between the film and the bulk solution. ∆P was taken to be equal to the capillary pressure in the plateau border. For very thin films, the pressure gradient also includes the disjoining pressure (see below). Equation (6.7) applies for the following conditions: (i) the liquid flows between parallel plane surfaces; (ii) the film surfaces are tangentially immobile; (iii) the rate of thinning due to evaporation is negligible compared to the thinning due to drainage.

6.5 Drainage and thinning of foam films

| 261

Experimental results obtained by Scheludko and co-workers [2–4] with comparatively thick rigid films produced from dilute solutions of sodium oleate and isoamyl alcohol, gave reasonably good agreement with the drainage equation. However, deviation from the Reynolds equation was observed in many cases due to tangential surface mobility. Surface viscosity can also slow down the drainage. A schematic representation of film drainage is given in Fig. 6.3 [5]. (a) Drainage

Drainage

Zero velocity

Maximum velocity

Surfaces are immobile

(b) Drainage

Maximum velocity

Drainage Surfaces are immobile Force

Force (d)

Tension

(c)

Bulk flux Surface diffusion flux Force

Surfactant concentration

Absorption Diffusion

Convective flux Bulk flux

Fig. 6.3: Schematic representation of film drainage: (a) thick rigid films; (b) mobile surfactant films; (c) interfacial tension gradients; (d) diffusion along the surface and from bulk solution [5].

In Fig. 6.3, (a) represents the case for thick film (> 100 nm), where the drainage velocity can be determined from Reynold’s equation; (b) represents the case for most surfactant films, where the surfaces are not rigid and the tangential velocity at the surface is not zero; (c) represents the case where surface mobility during drainage causes interfacial tension gradients; (d) represents the case where surface diffusion along the surface and from the bulk solution occurs with both adsorption and convective flow. For thinner films, large electrostatic repulsive interactions can reduce the driving force for film drainage and may lead to stable films. For thick films that contain high surfactant concentrations (> cmc), the micelles present in the film can produce a repulsive structural mechanism.

262 | 6 Gas (air)/liquid dispersions (foams)

The drainage of vertical films was investigated by pulling a frame out of a reservoir containing a surfactant solution. Three stages could be identified: (i) initial formation of the film that is determined by the withdrawal velocity; (ii) drainage of the film within the lamella which causes thinning with time; (iii) aging of the film, which may result in the formation of a metastable film. Assuming that the monolayer of the surfactant film at the boundaries of the film is rigid, film drainage may be described by the viscous flow of the liquid under gravity between two parallel plates. As the process proceeds, thinning can also occur by a horizontal mechanism, known as marginal regeneration [7–9], in which the liquid is drained from the film near the border region and exchanged within the low pressure plateau border. Marginal regeneration is probably the most important cause of film drainage of films with mobile surfaces, i.e. at surfactant concentrations above the cmc.

6.6 Theories of foam stability 6.6.1 Surface viscosity and elasticity theory The adsorbed surfactant film is assumed to control the mechanical-dynamical properties of the surface layers by virtue of its surface viscosity and elasticity. This concept may be true for thick films (> 100 nm) in which intermolecular forces are less dominant (i.e. foam stability under dynamic conditions). Surface viscosity reflects the speed of the relaxation process which restores the equilibrium in the system after imposing a stress on it. Surface elasticity is a measure of the energy stored in the surface layer as a result of an external stress. The viscoelastic properties of the surface layer is an important parameter. The most useful technique to study the viscoelastic properties of surfactant monolayers is the surface scattering method. When transversal ripples occur, periodic dilation and compression of the monolayer occurs and this can be accurately measured. This enables one to obtain the viscoelastic behaviour of monolayers under equilibrium and non-equilibrium conditions, without disturbing the original sate of the adsorbed layer. Some correlations have been found between surface viscosity and elasticity and foam stability, e.g. when adding lauryl alcohol to sodium lauryl sulphate which tends to increase the surface viscosity and elasticity [10].

6.6.2 The Gibbs–Marangoni effect theory The Gibbs coefficient of elasticity, ε, was introduced as a variable resistance to surface deformation during thinning: ε = 2(

dγ dγ ) = −2( ). d ln A d ln h

(6.8)

6.6 Theories of foam stability | 263

d ln h is the relative change in lamella thickness. ε is the “film elasticity of compression modulus” or “surface dilational modulus”. ε is a measure of the ability of the film to adjust its surface tension to an instant stress. In general, the higher the value of ε the more stable the film is. ε depends on surface concentration and film thickness. For a freshly produced film to survive, a minimum ε is required [8]. The main deficiency of early studies on Gibbs elasticity was that they were applied to thin films and the diffusion from the bulk solution was neglected. In other words, the Gibbs theory applies to the case where there are insufficient surfactant molecules in the film to diffuse to the surface and lower the surface tension. This is clearly not the case with most surfactant films. For thick lamella under dynamic conditions, one should consider diffusion from the bulk solution, i.e. the Marangoni effect. The Marangoni effect tends to oppose any rapid displacement of the surface (Gibbs effect) and may provide a temporary restoring force to “dangerous” thin films. In fact, the Marangoni effect is superimposed on the Gibbs elasticity, so that the effective restoring force is a function of the rate of extension, as well as the thickness. When the surface layers behave as insoluble monolayers, then the surface elasticity has its greatest value and is referred to as the Marangoni dilational modulus, εm . The Gibbs–Marangoni effect explains the maximum foaming behaviour at intermediate surfactant concentration [5]. At low surfactant concentrations (well below the cmc), the greatest possible differential surface tension will only be relatively small (Fig. 6.4 (a)) and little foaming will occur. At very high surfactant concentration (well above the cmc), the differential tension relaxes too rapidly because of the supply of surfactant which diffuses to the surface (Fig. 6.4 (c)). This causes the restoring force to have time to counteract the disturbing forces, producing a dangerously thinner film and foaming is poor. It is the intermediate surfactant concentration range that produces maximum foaming (Fig. 6.4 (b)).

6.6.3 Surface forces theory (disjoining pressure π) This theory operates under static (equilibrium) conditions in relatively dilute surfactant solutions (h < 100 nm). In the early stages of formation, foam films drain under the action of gravitation or capillary forces. Provided the films remain stable during this drainage stage, they may approach a thickness in the range of 100 nm. At this stage, surface forces come into play, i.e. the range of the surface forces now becomes comparable to the film thickness. Deryaguin and co-workers [11, 12] introduced the concept of disjoining pressure which should remain positive to slow down further drainage and film collapse. This is the principle of formation of thin metastable (equilibrium) films. In addition to the Laplace capillary pressure, three additional forces can operate at surfactant concentration below the cmc: electrostatic double layer repulsion πel ,

264 | 6 Gas (air)/liquid dispersions (foams)

(a)

(b)

(c)

Original state of thin film

Local mechanism caused a thin section

High local surface tension developed

Action of elasticity pulls back surfactant molecules into thinned section dγ dγ

Film thinned and easily ruptured

Surfactant molecules diffuse from bulk solution to surface

Returning surfactant molecules drag back underlaying layers of liquid with them

Surfactant concentration gradient in surface removed

Surface film repaired by surface transport mechanism

Film thinned and easily ruptured

Fig. 6.4: Schematic representation of the Gibbs–Marangoni effect: (a) low surfactant concentration (< cmc); (b) intermediate surfactant concentration; (c) high surfactant concentration (> cmc).

van der Waals attraction πvdW and steric (short range) forces πst , π = πel + πvdW + πst .

(6.9)

In the original definition of disjoining pressure by Deryaguin [11, 12], he only considered the first two terms on the right-hand side of equation (6.9). At low electrolyte concentrations, double layer repulsion predominates and πel can compensate the capillary pressure, i.e. πel = Pc . This results in the formation of an equilibrium-free film

6.6 Theories of foam stability | 265

which is usually referred to as the thick common film CF (≈ 50 nm thickness). This equilibrium metastable film persists until thermal or mechanical fluctuations cause rupture. The stability of the CF can be described in terms of the theory of colloid stability due to Deryaguin, Landau [13] and Verwey and Overbeek [14] (DLVO theory). The critical thickness value at which the CF ruptures (due to thickness perturbations) fluctuates and an average value hcr may be defined. However, an alternative situation may occur as hcr is reached and instead of rupturing a metastable film (high stability) may be formed with a thickness h < hcr . The formation of this metastable film can be experimentally observed through the formation of “islands of spots” which appear black in light reflected from the surface. This film is often referred to a “first black” or “common black” film. The surfactant concentration at which this “first black” film is produced can be 1–2 orders of magnitude lower than the cmc. Further thinning can cause an additional transformation into a thinner stable region (a stepwise transformation). This usually occurs at high electrolyte concentrations which leads to a second, very stable, thin black film usually referred to as Newton secondary black film, with a thickness in the region of 4 nm. Under these conditions, the short range steric or hydration forces control the stability and this provided the third contribution to the disjoining press, πst described in equation (6.9). Fig. 6.5 shows a schematic representation of the variation of disjoining pressure π with film thickness h, which shows the transition from the common film to the common black film and to the Newton black film. The common black film has a thickness in the region of 30 nm, whereas the Newton black film has a thickness in the region of 4–5 nm, depending on electrolyte concentration.

Disjoining Pressure

NBF CBF,CF πmax p h2

h1

h

Film thickness Fig. 6.5: Variation of disjoining pressure with film thickness.

Several investigations have been carried out to study the above transitions from common film to common black film and finally to Newton black film. For sodium dodecyl sulphate, the common black films have thicknesses ranging from 200 nm in a very dilute system to about 5.4 nm. The thickness depends strongly on electrolyte concentration and the stability may be considered to be caused by the secondary minimum in the energy–distance curve. In cases where the film thins further and overcomes the primary energy maximum, it will fall into the primary minimum potential energy

266 | 6 Gas (air)/liquid dispersions (foams)

sink where very thin Newton black films are produced. The transition from common black films to Newton black films occurs at a critical electrolyte concentration which depends on the type of surfactant. The rupture mechanisms of thin liquid films were considered by de Vries [15] and by Vrij and Overbeek [16]. It was assumed that thermal and mechanical disturbances (having a wave-like nature) cause film thickness fluctuations (in thin films) leading to rupture or coalescence of bubbles at a critical thickness. Vrij and Overbeek [16] carried out a theoretical analysis of the hydrodynamic interfacial force balance, and expressed the critical thickness of rupture in terms of the attractive van der Waals interaction (characterized by the Hamaker constant A), the surface or interfacial tension γ and disjoining pressure. The critical wavelength, λcrit , for the perturbation to grow (assuming the disjoining pressure just exceeds the capillary pressure) was determined. Film collapse occurs when the amplitude of the fast growing perturbation was equal to the thickness of the film. The critical thickness of rupture, hcr , was defined by the following equation, 1/7

hcr = 0.267(

af A2 ) 6πγ∆p

,

(6.10)

where af is the area of the film. Many poorly foaming liquids with thick film lamella are easily ruptured, e.g. pure water and ethanol films (with thickness between 110 and 453 nm). Under these conditions, rupture occurs by growth of disturbances which may lead to thinner sections [17]. Rupture can also be caused by spontaneous nucleation of vapour bubbles (forming gas cavities) in the structured liquid lamella [18]. An alternative explanation for rupture of relatively thick aqueous films containing low level of surfactants is the hydrophobic attractive interaction between the surfaces that may be caused by bubble cavities [19, 20].

6.6.4 Stabilization by micelles (high surfactant concentrations > cmc) At high surfactant concentrations (above the cmc), micelles of ionic or nonionic surfactants can produce organized molecular structures within the liquid film [21, 22]. This will provide an additional contribution to the disjoining pressure. Thinning of the film occurs through a stepwise drainage mechanism, referred to as stratification [23]. The ordering of surfactant micelles (or colloidal particles) in the liquid film due to the repulsive interaction provides an additional contribution to the disjoining pressure and this prevents the thinning of the liquid film.

6.7 Foam inhibitors

| 267

6.6.5 Stabilization by lamellar liquid crystalline phases This is particularly the case with nonionic surfactants that produce lamellar liquid crystalline structures in the film between the bubbles [24, 25]. These liquid crystals reduce film drainage as a result of the increase in viscosity of the film. In addition, the liquid crystals act as a reservoir of surfactant of the optimal composition to stabilize the foam.

6.6.6 Stabilization of foam films by mixed surfactants It has been found that a combination of surfactants gives slower drainage and improved foam stability. For example, mixtures of anionic and nonionic surfactants or anionic surfactant and long chain alcohol produce much more stable films than the single components. This could be attributed to several factors. For example, addition of a nonionic surfactant to an anionic surfactant causes a reduction in the cmc of the anionic. The mixture can also produce lower surface tension compared to the individual components. The combined surfactant system also has a high surface elasticity and viscosity when compared with the single components.

6.7 Foam inhibitors Two main types of inhibition may be distinguished, namely antifoamers that are added to prevent foam formation, and defoamers that are added to eliminate an existing foam. For example alcohols such as octanol are effective as defoamers but ineffective as antifoamers. Since the drainage and stability of liquid films is far from being fully understood, it is very difficult at present to explain the antifoaming and foam breaking action obtained by addition of substances. This is also complicated by the fact that in many industrial processes foams are produced by unknown impurities. For these reasons, the mechanism of action of antifoamers and defoamers is far from being understood [26]. A summary of the various methods that can be applied for foam inhibition and foam breaking is given below.

6.7.1 Chemical inhibitors that lower viscosity and increase drainage Chemicals that reduce the bulk viscosity and increase drainage can cause a decrease in foam stability. The same applies for materials that reduce surface viscosity and elasticity (swamping the surface layer with excess compound of lower viscosity). It has been suggested that a spreading film of antifoam may simply displace the stabilizing surfactant monolayer. As the oil lens spreads and expands on the surface,

268 | 6 Gas (air)/liquid dispersions (foams)

the tension will be gradually reduced to a lower uniform value. This will eliminate the stabilizing effect of the interfacial tension gradients, i.e. elimination of surface elasticity. Reduction of surface viscosity and elasticity may be achieved by low molecular weight surfactants. This will reduce the coherence of the layer, e.g. by addition of small amounts of nonionic surfactants. These effects depend on the molecular structure of the added surfactant. Other materials that are not surface active can also destabilize the film by acting as cosolvents which reduce the surfactant concentration in the liquid layer. Unfortunately, these nonsurface-active materials, such as methanol or ethanol, need to be added in large quantities (> 10 %).

6.7.2 Solubilized chemicals which cause antifoaming It has been demonstrated that solubilized antifoamers such as tributyl phosphate and methyl isobutyl carbinol, when added to surfactant solutions such as sodium dodecyl sulphate and sodium oleate, may reduce foam formation [27]. In cases where the oils exceed the solubility limit, the emulsifier droplets of oil can have a great influence on the antifoam action. It has been claimed [27] that the oil solubilized in the micelle causes a weak defoaming action. Mixed micelle formation with extremely low concentrations of surfactant may explain the actions of insoluble fatty acid esters, alkyl phosphate esters and alkyl amines.

6.7.3 Droplets and oil lenses which cause antifoaming and defoaming Undissolved oil droplets form in the surface of the film and this can lead to film rupture. Several examples of oils may be used: alkyl phosphates, diols, fatty acid esters and silicone oils (polydimethyl siloxane). A widely accepted mechanism for the antifoaming action of oils considers two steps: The oil drops enter the air/water interface and the oil spreads over the film causing rupture. The antifoaming action can be rationalized [28] in terms of the balance between the entering coefficient E and the Harkins [29] spreading coefficientS which are given by the following equations, E = γW/A + γW/O − γO/A ,

(6.11)

S = γW/A − γW/O − γO/A ,

(6.12)

where γW/A , γO/A and γW/O are the macroscopic interfacial tensions of the aqueous phase, oil phase and interfacial tension of the oil/water interface respectively. Ross and McBain [30] suggested that for efficient defoaming, the oil drop must enter the air/water interface and spread to form a duplex film at both sides of the original

6.7 Foam inhibitors

| 269

film. This leads to displacement of the original film, leaving an oil film which is unstable and can easily break. Ross [28] used the spreading coefficient (equation (6.12)) as a defoaming criterion. For antifoaming, both E and S should be > 0 for entry and spreading. A schematic representation of oil entry and the balance of the relevant tensions is given in Fig. 6.6 [5]. Aqueous solution

(a) Oil drop

Drop moves towards surface and enters E = γW/A + γW/O – γO/A

(b)

γO/A Air α1

γW/A

α2

Oil

γW/O For equilibrium γW/A = γO/Acos α1 + γW/Ocos α2

Fig. 6.6: Schematic representation of entry of oil droplet into the air/water interface (a) and its further spreading (b).

A typical example of this type of spreading/breaking is illustrated for a hydrocarbon surfactant stabilized film. For most surfactant systems, γAW = 35–45 mN m−1 and γOW = 5–10 mN m−1 and hence for an oil to act as an antifoaming agent γOA should be less than 25 mN m−1 . This shows why low surface tension silicone oils with surface tensions as low as 10 mN m−1 are effective.

6.7.4 Surface tension gradients (induced by antifoamers) It has been suggested that some antifoamers act by eliminating the structure tension gradient effect in foam films by reducing the Marangoni effect. Since spreading is driven by a surface tension gradient between the spreading front and the leading edge of the spreading front, thinning and foam rupture can occur by this surface tension gradient acting as a shear force (dragging the underlying liquid away from

270 | 6 Gas (air)/liquid dispersions (foams)

the source). This could be achieved by solids or liquids containing surfactant other than that stabilizing the foam. Alternatively, liquids which contain foam stabilizers at higher concentrations than that present in the foam may also act by this mechanism. A third possibility is the use of adsorbed vapours of surface active liquids.

6.7.5 Hydrophobic particles as antifoamers Many solid particles with some degree of hydrophobicity have been shown to cause destabilization of foams, e.g. hydrophobic silica, PTFE particles. These particles exhibit a finite contact angle when adhering to the aqueous interface. It has been suggested that many of these hydrophobic particles can deplete the stabilizing surfactant film by rapid adsorption and can cause weak spots in the film. A further mechanism was suggested based on the degree of wetting of the hydrophobic particles [31] and this led to the idea of particle bridging. For large smooth particles (large enough to touch both surfaces and with a contact angle θ > 90°), dewetting can occur. Initially, the Laplace pressure in the film adjacent to the particle becomes positive and causes liquid to flow away from the particle leading to enhanced drainage and formation of a “hole”. In the case of θ < 90°, initially the situation is the same as for θ > 90°, but as the film drains it attains a critical thickness where the film is planar and the capillary pressure becomes zero. At this point, further drainage reverses the sign of the radii of curvature causing unbalanced capillary forces which prevent drainage occurring. This can cause a stabilizing effect for certain types of particles. This means that a critical receding contact angle is required for efficient foam breaking. With particles containing rough edges, the situation is more complex, as demonstrated by Johansson and Pugh [32], using finely ground quartz particles of different size fractions. The particle surfaces were hydrophobized by methylation. These studies and others reported in the literature confirmed the importance of size, shape and hydrophobicity of the particles on foam stability.

6.7.6 Mixtures of hydrophobic particles and oils as antifoamers The synergetic antifoaming effect of mixtures of insoluble hydrophobic particles and hydrophobic oils when dispersed in aqueous medium has been well established in the patent literature. These mixed antifoamers are very effective at very low concentrations (10–100 ppm). The hydrophobic particles could be hydrophobized silica and the oil is polydimethyl siloxane (PDMS) One possible explanation of the synergetic effect is that the spreading coefficient of PDMS oil is modified by the addition of hydrophobic particles. It has been suggested that the oil-particle mixtures form composite entities where the particles can adhere

6.8 Physical properties of foams

|

271

to the oil-water interface. The presence of particles adhering to the oil-water interface may facilitate the emergence of oil droplets into the air-water interface to form lenses leading to rupture of the oil-water-air film.

6.8 Physical properties of foams 6.8.1 Mechanical properties The compressibility of a foam is determined by the ability of the gas to compress, its wetting power is determined by the properties of the foaming solution [4]. As any disperse system, a foam may acquire the properties of a solid body, i.e. it can maintain its shape and it possesses a shear modulus (see below). One of the basic mechanical properties of foams is its compressibility [4] (elasticity) and a bulk modulus Ev may be defined by the following expression, Ev = −

dp0 , d ln V

(6.13)

where p0 is the external pressure, causing deformation and V is the volume of the deforming system. By taking into account the liquid volume VL , the modulus of bulk elasticity of the “wet” foam Ev ’ is given by the expression, Ev =

dp0 VF dp0 VL = = Ev (1 + ). d ln VF d(VL + VG ) VG

(6.14)

Thus, the real modulus of bulk elasticity (“wet” foam) is higher than Ev (“dry” foam).

6.8.2 Rheological properties Like any disperse system, foams produce non-Newtonian systems and to characterize their rheological properties one needs to obtain information on the elasticity modulus (modulus of compressibility and expansion), the shear modulus, yield stress and effective viscosity, elastic recovery, etc. It is difficult to study the rheological properties of a foam since on deformation its properties change. The most convenient geometry to measure foam rheology is to use a parallel plate. The rheological properties could be characterized by a variable viscosity [4], τβ η = η∗ (γ)̇ + , (6.15) γ̇ where γ is the shear rate.

272 | 6 Gas (air)/liquid dispersions (foams)

The shear modulus of a foam is given by, τ β ∆l , (6.16) H where ∆l is the shear deformation and H is the distance between parallel plates in the rheometer. Deryaguin [33] obtained the following expression for the shear modulus, G=

G=

2 2 4γ 2 , p γ = ( γε) ≈ 5 5 3 3Rv

(6.17)

where Rv is the average volume of the bubble and ε is the specific surface area. Bikerman [34] obtained the following equation for the yield stress of a foam, τ β = 0.5

Nf γ p γ cos θ ≈ cos θ, NF − 1 R

(6.18)

where Nf is the number of films contacting the plate per unit area and θ is the average angle between the plate and the film. Princen [35] used a two-dimensional hexagonal package model to derive an expression for the shear modulus and yield stress of a foam, taking into account the foam expansion ratio and the contact angles, γ cos θ 1/2 (6.19) ϕ , R γ cos θ 1/2 (6.20) ϕ Fmax , τ β = 1.05 R where Fmax is a coefficient that is equal to 0.1–0.5, depending on the gas volume fraction ϕ. For a “dry” foam (ϕ → 1), the yield stress can be calculated from the expression, G = 0.525

θ . R For real foams, the value of τ β can be expressed by the general expression, τ β = 0.525γ cos

γ cos θ 1/3 ϕ Fmax , R where C is a coefficient that is approximately equal to 1. τβ = C

(6.21)

(6.22)

6.8.3 Electrical properties Only the liquid phase in a foam possesses electrical conductivity. The specific conductivity of a foam, κF , depends on the liquid content and its specific conductivity κL , κL , (6.23) κF = nB where n is the foam expansion ratio and B is a structural coefficient that depends on the foam expansion ratio and the liquid phase distribution between the plateau borders. B changes monotonically from 1.5 to 3 with increasing foam expansion factor.

6.9 Experimental techniques for studying foams |

273

6.8.4 Electrokinetic properties In foams with a charged gas/liquid interface, one can obtain various electrokinetic parameters, such as streaming potential and zeta potential. For example, the relation between the volumetric flow of a liquid flowing through a capillary or membrane and zeta potential can be given by the Smoluchowski equation, Q=

εε0 I εε0 r2 ∆V = , ηκ η L

(6.24)

where ε is the permittivity of the liquid and ε0 is the permittivity of free space, I is the value of the electric current, η is the viscosity of the liquid, r is the capillary radius, L is its length and ∆V is the potential distance between the electrodes placed at the capillary ends. The interpretation of electrokinetic results is complicated because of surface mobility and borders, and film elasticity, which causes large non-homogeneities in density and border radii at hydrostatic equilibrium and liquid motion.

6.8.5 Optical properties The extinction of the luminous flux passing through a foam layer occurs as a result of light scattering (reflection, refraction, interference and diffraction from the foam elements) and light absorption by the solution [4]. In a polyhedral foam, there are three structural elements that are clearly distinct by optical properties: films, plateau border and vertexes. The optical properties of single foam films have been extensively studied, but those of the foam as a disperse system are poorly considered. It has been concluded that the extinction of luminous flux (I/I0 , where I is the intensity of the light passing through the foam and I0 is the intensity of the incident light) is a linear function of the specific foam area. This could be used to determine the specific surface area of a foam.

6.9 Experimental techniques for studying foams 6.9.1 Techniques for studying foam films Most quantitative studies on foams have been carried out using foam films. As discussed above, microscopic horizontal films were studied by Scheludko and co-workers [2–4]. The foam thickness was determined by interferometry. Studies on vertical films were carried out by Mysels and collaborators [5, 7]. One of the most important characteristics of foam films is the contact angle θ appearing at the contact of the film with the bulk phase (solution) from which it is

274 | 6 Gas (air)/liquid dispersions (foams)

formed. This could be obtained by a topographic technique (that is suitable for small contact angles) that is based on determination of the radii of the interference Newton rings when the film is observed in a reflected monochromatic light. Another technique for studying foam films is to use α-particle irradiation, which can destroy the film. Depending on the intensity of the α-source, the film either ruptures instantaneously or lives for a much shorter time than required for its spontaneous rupture. The lifetime τa of a black film subjected to irradiation is considered as a parameter characterizing the destructive effect of α-particles. A third technique for studying foam films is fluorescence recovery after photobleaching (FRAP). This technique was applied by Clark et al. [36] for lateral diffusion in foam films. The technique involves irreversible photobleaching by intense laser light of fluorophore molecules in the sample. The time of redistribution of probe molecules (assumed to be randomly distributed within the constitutive membrane lipids in the film) is monitored. The lateral diffusion coefficient, D, is calculated from the rate of recovery of fluorescence in the bleaching region due to the entry of unbleaching fluoroprobes from adjacent parts of the membranes. Deryaguin and Titijevskaya [37] measured the isotherms of disjoining pressure of microscopic foam films (common thin films) in a narrow range of pressures. At equilibrium, the capillary p σ pressure in the flat horizontal foam film is equal to the disjoining pressure π in it, p σ = π = pg − pL , (6.25) where pg is the pressure in the gas phase and pL is the pressure in the liquid phase. Several other techniques have been applied for measuring foam films, e.g. ellipsometry, FT-IR spectroscopy, X-ray reflection and measurement of gas permeability through the film. These techniques are described in detail in the text by Exerowa and Kruglyakov [4] to which the reader should refer.

6.9.2 Techniques for studying structural parameters of foams Polyhedral foam consists of gas bubbles with polyhedral shape whose faces are flat or slightly bent liquid films, the edges are the plateau borders and the edge cross points are the vortexes. Several techniques can be applied to obtain the analytical dependency of these characteristics and the structural parameters of the foam [4]. The foam expansion ratio can be characterized by the liquid volume fraction in the foam, which is the sum of the volume fractions of the films, plateau borders and vertexes. Alternatively, one can use the foam density as a measure of the foam expansion ratio. The reduced pressure in the foam’s plateau border can be measured using a capillary manometer [4]. The bubble size and shape distribution in a foam can be determined by microphotography of the foam. Information about the liquid distribution between films and plateau borders is obtained from the data on the border radius

References | 275

of curvature, the film thickness and the film to plateau border number ratio obtained in an elementary foam cell.

6.9.3 Measuring foam drainage After foam formation the liquid starts to drain out of the foam. The “excess” liquid in the foam film drains into the plateau borders, then through them flows down from the upper to the lower foam layers following the direction of gravity until the gradient of the capillary pressure equalizes the gravity force, dp σ = ρg, dl

(6.26)

where l is a coordinate in the opposite direction to gravity. Simultaneously with drainage from films into borders, the liquid begins to flow out from the foam when the pressure in the lower foam films outweighs the external pressure. This process is similar to gel syneresis and it is sometimes referred to as “foam syneresis” and “foam drainage”. The rate of foam drainage is determined by the hydrodynamic characteristics of the foam as well as the rate of internal foam collapse and breakdown of the foam column. The foam drainage is determined by measuring the quantity of liquid that drains from the foam per unit time. Various types of vessels and graduated tubes can be used for measuring the liquid quantity draining from a foam. Alternatively, one can measure the change in electrical conductivity of the layer at the vessel mouth compared to the electrical conductivity of the foaming solution [4].

6.9.4 Measuring foam collapse This can be followed by measuring the bubble size distribution as a function of time, for example by microphotography or by counting the number of bubbles. Alternatively, one can measure the specific surface area or average bubble size as a function of time. Other techniques such as light scattering or ultrasound can also be applied.

References [1] [2] [3] [4] [5] [6]

Tadros T. Formulation of disperse systems. Weinheim: Wiley-VCH; 2014. Scheludko A. Colloid Science. Amsterdam: Elsevier; 1966. Scheludko A. Adavances Colloid Interface Sci. 1971;1:391. Exerowa D, Kruglyakov PM. Foam and Foam Films. Amsterdam: Elsevier; 1997. Pugh RJ. Advances Colloid and Interface Sci. 1995. Reynolds O. Phil Trans Royal Soc London Ser. A. 1886;177:157.

276 | 6 Gas (air)/liquid dispersions (foams)

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

Mysels KJ. J Phys Chem. 1964;68:3441. Lucassen J. In: Lucassen-Reynders EH, editor. Anionic surfactants. New York: Marcel Dekker; 1981. p.217 Stein HN. Advances Colloid Interface Sci. 1991;34:175. Davies JT. In: Schulman JH, editor. Proceedings of the second International Congress of Surface Activity, Vol. 1. London: Butterworth; 1957. Deryaguin BV, Scherbaker RL. Kolloid Zh. 1976;38:438. Deryaguin BV. Theory of stability of colloids and thin films. New York: Consultant Bureau; 1989. Deryaguin BV, Landau L. Acta Physicochem USSR. 1941;14:633. Verwey EJW, Overbeek JTG. Theory of stability of lyophobic colloids. Amsterdam: Elsevier; 1948. de Vries AJ. Disc Faraday Soc. 1966;42:23. Vrij A, Overbeek JTG. J Amer Chem Soc. 1968;90:3074. Radoev B, Scheludko A, Manev E. J Colloid Interface Sci. 1983;95:254. Gleim VG, Shelomov IV, Shidlovskii BR. J Appl Chem USSR. 1959;32:1069. Pugh RJ, Yoon RH. J Colloid Interface Sci. 1994;163:169. Claesson PM, Christensen HK. J Phys Chem. 1988;92:1650. Johnott ES. Philos Mag. 1906;11:746. Perrin J. Ann Phys. 1918;10:160. Loeb L, Wasan DT. Langmuir. 1993;9:1668. Frieberg S. Mol Cryst Liq Cryst. 1977;40:49. Perez JE, Proust JE Saraga TM. In: Ivanov IB, editor. Thin liquid films. New York: Marcel Dekker; 1988. p. 70. Garrett PR, editor. Defoaming. Surfactant Science Series Vol. 45. New York: Marcel Dekker; 1993. Ross S, Haak RM. J Phys Chem. 1958;62:1260. Robinson JV, Woods WW. J Soc Chem Ind. 1948;67:361. Harkins WD. J Phys Chem. 1941;9:552. Ross S, Mc Bain JW. Ind Chem Eng. 1944;36:570. Garett PR. J Colloid Interface Sci. 1979;69:107. Johansson G, Pugh RG. Int J Mineral Process. 1992;34:1. Deryaguin BV. Kolloid Z. 1933;64:1. Bickermann JJ. Foams. New York: Springer-Verlag; 1973. Princen H. J Colloid Interface Sci. 19833;91:160. Clark D, Dann R, Mackie A, Mingins J, Pinder A, Purdy P, Russel E, Smith L, Wilson D. J Colloid Interface Sci. 1990;138:195. Deryaguin BV, Titijevskaya AS. Kolloid Z. 1953;15:416.

7 Liquid/solid dispersions (gels) 7.1 Introduction A gel is a “semi-solid” consisting of a “network” in which the solvent is “entrapped”. It may be classified as a “liquid-in-solid” dispersion [1]. A gel shows some solidlike properties as well as liquid-like properties, i.e. it is a viscoelastic system (see Chapter 1). Depending on the gel strength, the system may behave as a viscoelastic solid or a viscoelastic liquid depending on the stress applied on the gel. For “strong” gels (such as those produced by chemical crosslinking), the system may behave as a viscoelastic solid up to high stresses and the gel could also show a significant yield value. For “weaker” gels, e.g. those produced by associative thickeners, the system may show viscoelastic liquid-like behaviour at lower applied stresses when compared with chemical gels. Gels are applied in many industries of which pharmaceuticals and cosmetics are probably the most useful examples. In the pharmaceutical industry gels find use as delivery systems for oral administration, as gels proper or as capsule shells made from gelatin. For topical drugs, gels are applied directly to the skin, mucous membrane or eye and for long acting forms of drugs injected intramuscularly. Gelling agents are also useful as binders in tablet granulations, protective colloids in suspensions, thickeners in oral liquids and suppository bases. Cosmetically, gels are used in shampoos, fragrant products, dentifrices and skin and hair preparations.

7.2 Classification of gels Several classes of gels can be identified [1]: (i) Gels produced as a result of repulsive interaction, e.g. expanded double layers. (ii) Self-structured systems in which one induces weak flocculation to produce a “gel” by the particles or droplets. This requires control of the particle size and shape, volume fraction of the dispersion and depth of the secondary minimum. (iii) Thickeners consisting of high molecular weight polymers or finely divided particulate systems that interact in the continuous phase forming a “three-dimensional” structure. (iv) Self-assembled structures such as associative thickeners. (v) Crosslinked polymers (chemical gels). (vi) Liquid crystalline structures of the hexagonal, cubic or lamellar phases.

https://doi.org/10.1515/9783110541953-008

278 | 7 Liquid/solid dispersions (gels)

7.3 Gel-forming materials The most commonly used materials to produce a gel network are polymers both natural and synthetic. However, many colloidal particulate solids can form gels by some specific interactions between the particles. High concentrations of nonionic surfactants can also produce clear gels in systems containing up to 15 % mineral oil. Natural gums have been used for a long time as gelling agents. These gums are typically branched-chain polysaccharides, mostly anionic although some such as guar gum are uncharged. Unfortunately, these gums are subject to microbial degradation and they require the addition of a preservative. An important gum that is used to produce gels is xanthan gum that is produced from sugar by microbiological preparation. Several other gums have been used to produce gels, of which alginates, carrageenan, tragacanth, pectin, gelatine and agar are probably the most important. Alginates are derived from brown seaweeds in the form of monovalent and divalent salts. Sodium alginate is the most widely used gum. Gelation occurs by reduction of pH or reaction with divalent cations. Carrageenan is produced by extraction from red seaweeds and it is a mixture of sodium, potassium, ammonium, calcium and magnesium sulphate esters of polymerized galactose and 3,6=anhydrogalactose. The main copolymer types are labelled kappa-, iota and lamda-carrageenan. They are all anionic in nature. Alpha and iota fractions form thermally reversible gels in water. At high temperatures, the copolymers exist as random coils and on cooling they result in the formation of double helices which act as crosslinks. Tragacanth is produced by extraction from special plants grown in the Middle East. It is a complex material composed chiefly of an acidic polysaccharide (tragacanthic acid) containing calcium, magnesium and potassium, as well as a smaller amount of neutral polysaccharide (tragacanthin). The gum swells in water and a concentration of 2 % or above of a “high quality” gum produces a gel. Hydration takes place over a period of time so that development of maximum gel strength requires several hours. Pectin is a polysaccharide extracted from the inner rind of citrus fruits or apple pomace. The gel is formed at an acid pH in aqueous solution containing calcium and possibly other agents that act to dehydrate the gum. Gelatin is used as a bodying agent and gel former in the food industry and occasionally in pharmaceutical products. Agar can be used to make firm gels; it is most frequently used in culture media. Gellan gum has been more recently used as a substitute for agar. Several synthetic gel forming materials have been developed for various applications. Cellulose derivatives (such as hydroxyethyl cellulose and carboxymethyl cellulose) are frequently used as gelling agents in several disperse systems (such as suspensions and emulsions). Cellulose is a natural structural polymer found in plants. Treatment in the presence of various active substances results in breakdown of the cellulose backbone as well as substitution of a portion of its hydroxyl moieties. The major factors affecting the gel characteristics (and rheological properties) of the resulting

7.4 Rheological behaviour of a “gel” |

279

material are the nature of the substituent, degree of substitution and average molecular weight of the resultant polymer. Cellulose derivatives are subject to enzymatic degradation and sterilization of the aqueous system and/or addition of preservatives are used to prevent viscosity reduction resulting from depolymerization due to enzyme production by microorganisms. One synthetic gelling agent that is frequently used in the cosmetic industry is carbomer (sold under trade name “Carbopol”). It is an acrylic polymer crosslinked with a polyalkenyl ether. The polymer can be easily dispersed in aqueous media and on neutralization with a suitable base (such as NaOH or ethanolamine) a gel is produced. Carbomer can produce gels at concentrations as low as 0.5 %. Various forms of polyethylene and its copolymers are used to gel hydrophobic liquids. Polyethylene is a suitable gelling agent for simple aliphatic hydrocarbon liquids but it may lack compatibility with many other oils found in personal care products. In this case copolymers with vinyl acetate and acrylic acid may be used, perhaps with the aid of a cosolvent. To produce the gel, it is necessary to disperse the polymer at high temperature (above 80 °C) and then shock cool to precipitate fine crystals that make up the matrix. Several finely divided solids, such as sodium montmorillonite and silica, can also be used as gelling agents. The mechanisms by which these particulate solids form gels will be discussed in detail below.

7.4 Rheological behaviour of a “gel” One of the most effective techniques to characterize a gel is to investigate its rheological (viscoelastic) behaviour, in particular under conditions of low deformation [1, 2]. The basic principles of these rheological techniques have been discussed in detail in Chapter 1 of this volume and here a brief summary is given. As discussed in Chapter 1, three methods can be applied to investigate the viscoelastic properties of a gel.

7.4.1 Stress relaxation (after sudden application of strain) One of the most useful ways to describe a gel is to consider the relaxation time of the system. Consider a “gel” with the components in some sort of a “three-dimensional” structure. To deform it instantly, a stress is required and energy is stored in the system (high energy structure). To maintain the new shape (constant deformation) the stress required becomes smaller since the components of the “gel” undergo some diffusion resulting in a lower energy structure to be approached (structural or stress relaxation). At long times, deformation becomes permanent with complete relaxation of the structure (new low energy structure) and viscous flow will occur.

280 | 7 Liquid/solid dispersions (gels)

The above exponential decay of the stress can be represented by the following equation, t σ(t) = σ0 exp(− ), (7.1) τ where τ is the stress relaxation time. If the stress is divided by the strain, one obtains the modulus G t G(t) = G0 exp( ). τ

(7.2)

G0 is the instantaneous modulus (the spring constant). The above behaviour is schematically represented in Fig. 7.1 if the modulus (after sudden application of strain) is plotted as a function of time. This representation is for a viscoelastic liquid (Maxwell element represented by a spring and dashpot in series) with complete relaxation of the springs at infinite time (see Chapter 1). In other words, the modulus approaches zero at infinite time. 1

Go

0.8

G/Pa

0.6 τ = σo /e

0.4 0.2 0 20

40

60 t/s

80

100

Fig. 7.1: Variation of modulus with time for a viscoelastic liquid.

Many crosslinked gels behave like viscoelastic solids (Kelvin model) with another spring in parallel having an elasticity Ge . The modulus does not decay to zero. The relaxation modulus is given by, t G(t) = G0 exp( ) + Ge . τ

(7.3)

Fig. 7.2 shows the variation of G(t) with time for a viscoelastic solid. A useful way to distinguish between the various gels is to consider the Deborah number De, τ De = . (7.4) te

7.4 Rheological behaviour of a “gel” |

281

120 G = 80 Pa

100

G(t)/Pa

80 60

Ge

G η

40 η/G = τ = 1 s

20

Ge = 30 Pa

0 0.001

0.1

1

10

100

1000

t/s Fig. 7.2: Variation of G(t) with t for a viscoelastic solid.

For a gel that shows “solid-like” behaviour (“three-dimensional structure”) De is large when compared with a gel that behaves as a viscoelastic liquid.

7.4.2 Constant stress (creep) measurements In this case a constant stress σ is applied and the strain (deformation) γ or compliance J (= γ/σ, Pa−1 ) is followed as a function of time [1]. A gel that consists of a strong “three-dimensional” structure (e.g. crosslinked) behaves as a viscoelastic solid as is illustrated in Fig. 7.3. This behaviour may occur up to high applied stresses. In other words, the critical stress above which significant deformation occurs can be quite high. A weaker gel (produced for example by high molecular weight polymers that are

σ removed σ applied

J/Pa–1

Viscoelastic liquid

σ

Viscoelastic solid

t=0

t

Fig. 7.3: Viscoelastic solid and viscoelastic liquid response for gels.

282 | 7 Liquid/solid dispersions (gels)

physically attached) behaves as a viscoelastic liquid as shown in Fig. 7.3. In this case viscoelastic solid behaviour only occurs at much lower stresses than that observed with the crosslinked gels.

7.4.3 Dynamic (oscillatory) measurements A sinusoidal strain (or stress) with amplitude γ0 and frequency ω (rad s−1 ) is applied on the system and the resulting stress (or strain) with amplitude σ0 is simultaneously measured [1]. This is illustrated in Fig. 7.4.

γ0

τ0

Δ0

Fig. 7.4: Strain and stress sine waves for a viscoelastic system.

For any gel δ < 90° and the smaller the value of δ, the stronger the gel. From the amplitudes of stress and strain (σ0 and γ0 ) and the phase angle shift δ, one can obtain the various viscoelastic parameters. σ0 |G∗ | = (7.5) γ0 storage (elastic) modulus G󸀠 = |G∗ | cos δ 󸀠󸀠

∗

loss (viscous) modulus G = |G | sin δ tan δ =

G󸀠󸀠

. G󸀠 For gels, tan δ < 1 and the smaller the value the stronger the gel.

(7.6) (7.7) (7.8)

7.5 Polymer gels 7.5.1 Physical gels obtained by chain overlap Flexible polymers that produce random coils in solution can produce “gels” at a critical concentration C∗ , referred to as the polymer coil “overlap” concentration. This picture can be realized if one considers the coil dimensions in solution: Considering the polymer chain to be represented by a random walk in three-dimensions, one may define two main parameters, namely the root mean square end-to-end length ⟨r2 ⟩1/2 and the root mean square radius of gyration ⟨s2 ⟩1/2 (sometimes denoted by RG ). The two are related by, ⟨r2 ⟩1/2 = 61/2 ⟨s2 ⟩1/2 . (7.9)

7.5 Polymer gels

| 283

log ƞ

The viscosity of a polymer solution increases gradually as its concentration increases and at a critical concentration, C∗ , the polymer coils with a radius of gyration RG and a hydrodynamic radius Rh (which is higher than RG due to solvation of the polymer chains) begin to overlap and this yields a rapid increase in viscosity. This is illustrated in Fig. 7.5, which shows the variation of log η with log C.

C*

log C

Fig. 7.5: Variation of log η with log C.

In the first part of the curve η ∝ C, whereas in the second part (above C∗ ) η ∝ C3.4 . A schematic representation of polymer coil overlap is shown in Fig. 7.6 which shows the effect of gradually increasing the polymer concentration. The polymer concentration above C∗ is referred to as the semi-dilute range [3].

(a) Dilute

(b) Onset of overlap

(c) Semi-dilute

C < C*

C = C*

C > C*

Fig. 7.6: Crossover between dilute and semi-dilute solutions.

C∗ is related to RG and the polymer molecular weight M by, 4 Nav C∗ = ( )πR3G ( ) ≈ 1. 3 M

(7.10)

Nav is the Avogadro number. As M increases C∗ becomes progressively lower. This shows that to produce physical gels at low concentrations by simple polymer coil overlap, one has to use high molecular weight polymers. Another method to reduce the polymer concentration at which chain overlap occurs is to use polymers that form extended chains, such as xanthan gum which produces conformation in the form of a helical structure with a large axial ratio. These

284 | 7 Liquid/solid dispersions (gels)

polymers give much higher intrinsic viscosities and they show both rotational and translational diffusion. The relaxation time for the polymer chain is much higher than a corresponding polymer with the same molecular weight but produces random coil conformation. The above polymers interact at very low concentrations and the overlap concentration can be very low (< 0.01 %). These polysaccharides are used in many formulations to produce physical gels at very low concentrations.

7.5.2 Gels produced by associative thickeners Associative thickeners are hydrophobically modified polymer molecules in which alkyl chains (C12 –C16 ) are either randomly grafted onto a hydrophilic polymer molecule such as hydroxyethyl cellulose (HEC) or simply grafted at both ends of the hydrophilic chain [4]. An example of hydrophobically modified HEC is Natrosol plus (Hercules) which contains 3–4 C16 randomly grafted onto hydroxyethyl cellulose. Another example of a polymer that contains two alkyl chains at both ends of the molecule is HEUR (Rohm and Haas) that is made of polyethylene oxide (PEO) that is capped at both ends with linear C18 hydrocarbon chains. The above hydrophobically modified polymers form gels when dissolved in water. Gel formation can occur at relatively lower polymer concentrations when compared with the unmodified molecule. The most likely explanation of gel formation is due to hydrophobic bonding (association) between the alkyl chains in the molecule [4]. This effectively causes an apparent increase in the molecular weight. These associative structures are similar to micelles, except the aggregation numbers are much smaller. Fig. 7.7 shows the variation of viscosity (measured using a Brookfield at 30 rpm) as a function of the alkyl content (C8 , C12 and C16 ) for hydrophobically modified HEC (i.e. HMHEC). The viscosity reaches a maximum at a given alkyl group content that decreases with increasing alkyl chain length. The viscosity maximum increases with increasing alkyl chain length [4]. Associative thickeners also show interaction with surfactant micelles that are present in the formulation. The viscosity of the associative thickeners shows a maximum at a given surfactant concentration that depends on the nature of surfactant. This is shown schematically in Fig. 7.8. The increase in viscosity is attributed to the hydrophobic interaction between the alkyl chains on the backbone of the polymer with the surfactant micelles. A schematic picture showing the interaction between HM polymers and surfactant micelles is shown in Fig. 7.9. At higher surfactant concentration, the “bridges” between the HM polymer molecules and the micelles are broken (free micelles) and η decreases. The viscosity of hydrophobically modified polymers shows a rapid increase at a critical concentration, which may be defined as the critical aggregation concentration

Conductivity

7.5 Polymer gels |

O/ W

285

W/O

PIT Fig. 7.7: Variation of viscosity of 1 % HMHEC versus alkyl group content of the polymer.

η

Increase of temperature

Surfactant concentration

Fig. 7.8: Schematic plot of viscosity of HM polymer with surfactant concentration.

Mixed micelles

Surfactant

Free micelles

Surfactant

Fig. 7.9: Schematic representation of the interaction of polymers with surfactants.

(CAC) as illustrated in Fig. 7.10 for HMHEC (WSP-D45 from Hercules). The assumption is made that the CAC is equal to the coil overlap concentration C∗ . From a knowledge of C∗ and the intrinsic viscosity [η] one can obtain the number of chains in each aggregate. For the above example [η] = 4.7 and C∗ [η] = 1 giving an aggregation number of ≈ 4 At C∗ the polymer solution shows non-Newtonian flow (shear thinning behaviour) and its shows a high viscosity at low shear rates. This is illustrated in Fig. 7.11 which shows the variation of apparent viscosity with shear rate (using a constant stress rheometer). Below ≈ 0.1 s−1 , a plateau viscosity value η(o) (referred to as residual or zero shear viscosity) is reached (≈ 200 Pa s).

Reduced Viscosity

286 | 7 Liquid/solid dispersions (gels) 18 16 14 12 10 8 6 4

CAC 0 0.02 0.04 0.06 0.08 0.1 0.12 C/g dm–3

Fig. 7.10: Variation of reduced viscosity with HMHEC concentration.

105 104 η(0)

103 102 10 1 0

0.2

0.4

0.6

0.8

1.0

Polymer concentration (%)

1.2

Fig. 7.11: Variation of η(0) with polymer concentration.

With increasing polymer concentrations above C∗ the zero shear viscosity increases with increasing polymer concentration. This is illustrated in Fig. 7.11. The above hydrophobically modified polymers are viscoelastic. This is illustrated in Fig. 7.12 for a solution 5.25 % of C18 end-capped PEO with M = 35 000, which shows the variation of the storage modulus G󸀠 and loss modulus G󸀠󸀠 with frequency ω (rad s−1 ). G󸀠 increases with increasing frequency and ultimately it reaches a plateau value at high frequency. G󸀠󸀠 (which is higher than G󸀠 in the low frequency regime) increases with increasing frequency, reaches a maximum at a characteristic frequency ω∗ (at which G󸀠 = G󸀠󸀠 ) and then decreases to near zero value in the high frequency regime. The above variation of G󸀠 and G󸀠󸀠 with ω is typical for a system that shows Maxwell behaviour. From the crossover point ω∗ (at which G󸀠 = G󸀠󸀠 ) one can obtain the relaxation time τ of the polymer in solution, 1 τ = ∗. (7.11) ω For the above polymer, τ = 8 s. The above gels (sometimes referred to as rheology modifiers) are used in many formulations to produce the right consistency and also for reducing sedimentation or creaming of suspensions and emulsions. These hydrophobically modified polymers

7.5 Polymer gels

| 287

3000 2500

G'

G' /Pa G''/Pa

2000 1500 G''

1000

G'

500

G'' ω*

0 –3

10

10

–2

10–1

1

10

ω/rad s–1 Fig. 7.12: Variation of G󸀠 and G󸀠󸀠 with frequency for 5.24 HM PEO.

can also interact with hydrophobic particles in a suspension forming several other associative structures. The high frequency modulus, sometimes referred to as the network modulus, can be used to obtain the number of “links” in the gel network structure. Using the theory of rubber elasticity, the network modulus GN is related to the number of elastically effective links N and a factor A that depends on the junction functionality, GN = ANkT,

(7.12)

where k is the Boltzmann constant and T is the absolute temperature. For an end-capped PEO (i.e. HEUR) the junctions should be multifunctional (A = 1); for tetra-functional junctions, A = 1/2.

7.5.3 Crosslinked gels (chemical gels) Many commercially available gels are made using crosslink agents to produce what is sometimes referred to as “microgels”. These microgel particles are dispersed in the liquid and they undergo solvent swelling which may also be enhanced by some chemical modification, e.g. pH adjustment in aqueous systems [1]. As mentioned before, an acrylic polymer crosslinked with a polyalkenyl ether, namely carbomer, that is commercially sold under the trade name “Carbopol” (B.F. Goodrich) forms gels at concentrations as low as 0.5 %. The polymer can be easily dispersed in aqueous media and on neutralization with a suitable base (such as NaOH or ethanolamine) a gel is produced. The polymer swells as a result of the ionization of the polyacrylic acid chains. The ionization of polyacrylic acid occurs when the pH

288 | 7 Liquid/solid dispersions (gels) is increased above ≈ 5 and these ionized chains form extended double layers forming gels at low microgel concentration (mostly less than 0.5 % of the microgel particles). Another example of microgels is that based on N-isopropyl acrylamide crosslinked with NN′′-methylene bisacrylamide (poly-NIPAM). These microgel particles are swollen by temperature changes. At temperatures > 35 °C, the crosslinked polymer is in a collapsed state. When the temperature is reduced the crosslinked polymer swells, absorbing water that reaches several order of magnitudes its volume. These polymer gels are sometimes referred to as “smart” colloids and they have several applications in controlled release.

7.6 Particulate gels Two main interactions can cause gel formation with particulate materials: (i) Long-range repulsions between the particles, e.g. using extended electrical double layers or steric repulsion resulting from the presence of adsorbed or grafted surfactant or polymer chains. (ii) Van der Waals attraction between the particles (flocculation) which can produce three-dimensional gel networks in the continuous phase. All the above systems produce non-Newtonian systems that shows a “yield value” and high viscosity at low shear stresses or shear rates. Several examples may be quoted to illustrate the above particulate gels: (i) Swellable clays, e.g. sodium montmorillonite (sometimes referred to as Bentonite) at low electrolyte concentration. These produce gels as a result of the formation of extended double layers. At moderate electrolyte concentrations the clay particles may form association structures as a result of face-to-edge flocculation (see below). These clays can be modified by interaction with alkyl ammonium salts (cationic surfactants) to produce hydrophobically modified clays, sometimes referred to as organo-clays or Bentones. These can be dispersed in nonaqueous media and swollen by addition of polar solvents. (ii) Finely divided oxide, e.g. silica which can produce gels by aggregation of the particles to form three-dimensional gel structures. In many cases, particulate solids are combined with high molecular weight polymers to enhance gel formation, e.g. as a result of “bridging” or “depletion flocculation”.

7.6.1 Aqueous clay gels Sodium montmorillonite (referred to as swellable clay) forms gels at low and intermediate electrolyte concentrations [5]. This can be understood from a knowledge of the structure of the clay particles. They consist of plate-like particles consisting of an

7.6 Particulate gels |

289

octahedral alumina sheet sandwiched between two tetrahedral silica sheets. This is shown schematically in Fig. 7.13, which also shows the change in the spacing of these sheets. In the tetrahedral sheet, tetravalent Si is sometimes replaced by trivalent Al. In the octahedral sheet, there may be replacement of trivalent Al by divalent Mg, Fe, Cr or Zn. The small size of these atoms allows them to take the place of small Si and Al. Distance between atom centers Tetrahedral sheel

Octahedral sheel

Tetrahedral sheel

Charge 6O 4 Si

0.60 Å 1.60 Å

–12 +16

4O –10 2 OH 4 AI

2.20 Å

+12

2 OH –10 4O

1.60 Å

4 Si 6O

0.60 Å

–12 –44

+16 +14

Surface of unit cell: 5.15 × 8.9 Ų O OH Si AI

Formula of unit cell: [AI₂(OH)₂(Si₂O5)₂]₂

C-spacing 9.2 Å

Unit-cell weight: 720 Hydroxyl water: 5%

Fig. 7.13: Atom arrangement in one unit cell of 2 : 1 layer mineral.

This replacement is usually referred to as isomorphic substitution in which an atom of lower positive valence replaces one of higher valence, resulting in a deficit of positive charge or excess of negative charge. This excess of negative layer charge is compensated by adsorption at the layer surfaces of cations that are too big to be accommodated in the crystal. In aqueous solution, the compensation cations on the layer surfaces may be exchanged by other cations in solution, and hence may be referred to as exchangeable cations. With montmorillonite, the exchangeable cations are located on each side of the layer in the stack, i.e. they are present in the external surfaces as well as between the layers. This causes a slight increase in the local spacing from about 9.13 Å to about 9.6 Å; the difference depends on the nature of the counterion. When montmorillonite clays are placed in contact with water or water vapour the water molecules penetrate between the layers, causing interlayer swelling or (intra)crystalline swelling. This leads to a further increase in the basal spacing to

290 | 7 Liquid/solid dispersions (gels)

12.5–20 Å, depending on the type of clay and cation. This interlayer swelling leads, at most, to doubling of the volume of dry clay when four layers of water are adsorbed. The much larger degree of swelling, which is the driving force for “gel” formation (at low electrolyte concentration), is due to osmotic swelling. It has been suggested that swelling of montmorillonite clays is due to the electrostatic double layers that are produced between the charge layers and cations. This is certainly the case at low electrolyte concentration where the double layer extension (thickness) is large (see Fig. 7.14). As discussed above, the clay particles carry a negative charge as a result of isomorphic substitution of certain electropositive elements by elements of lower valency. The negative charge is compensated by cations, which in aqueous solution form a diffuse layer, i.e. an electric double layer is formed at the clay plate/solution interface. This double layer has a constant charge, which is determined by the type and degree of isomorphic substitution. However, the flat surfaces are not the only surfaces of the plate-like clay particles, they also expose an edge surface. The atomic structure of the edge surfaces is entirely different from that of the flat-layer surfaces. At the edges, the tetrahedral silica sheets and the octahedral alumina sheets are disrupted, and the primary bonds are broken. The situation is analogous to that of the surface of silica and alumina particles in aqueous solution. On such edges, therefore, an electric double layer is created by adsorption of potential determining ions (H+ and OH− ) and one may, therefore identify an isoelectric point (IEP) as the point of zero charge (pzc) for these edges. With broken octahedral sheets at the edge, the surface behaves as Al–OH with an IEP in the region of pH 7–9. Thus in most cases the edges become negatively charged above pH 9 and positively charged below pH 9. Van Olphen [5] suggested a mechanism of gel formation of montmorillonite involving interaction of the oppositely charged double layers at the faces and edges of the clay particles. This structure, which is usually referred to as a “card-house” structure, was considered to be the reason for the formation of the voluminous clay gel. However, Norrish suggested that the voluminous gel is the result of the extended double layers, particularly at low electrolyte concentrations. A schematic picture of gel formation produced by double layer expansion and “card-house” structure is shown in Fig. 7.14. Evidence for the above picture was obtained by Van Olphen [5] who measured the yield value of 3.22 % montmorillonite dispersions as a function of NaCl concentration

(a) Gels produced by Double layer overlap

(b) Gels produced by edge-to-face association

Fig. 7.14: Schematic representation of gel formation in aqueous clay dispersions.

7.6 Particulate gels |

291

as shown in Fig. 7.15. When C = 0, the double layers are extended and gel formation is due to double layer overlap (Fig. 7.14 (a)). First addition of NaCl causes compression of the double layers and hence the yield value decreases very rapidly. At intermediate NaCl concentrations, gel formation occurs as a result of face-to-edge association (house of cards structure) (Fig. 7.14 (b)) and the yield value increases very rapidly with increasing NaCl concentration. If the NaCl concentration is increased further, face-toface association may occur and the yield value decreases (the gel is destroyed).

σß/Pa

18 16 14 12 10 8 6 4 2 0 0

10

20

30

40

50

60

70

80

CNaCl/meq dm³ Fig. 7.15: Variation of yield value with NaCl concentration for 3.22 % sodium montmorillonite dispersions.

7.6.2 Organo-clays (bentones) These are produced by exchanging the Na + ions with alkyl ammonium ions, e.g. dodecyl or cetyl trimethyl ammonium ions. In some cases dialkyl ammonium ions are also used. In this case the clay particle surface will be covered with hydrophobic alkyl groups and hence it can be dispersed in organic solvents, e.g. hydrocarbon or silicone oils. The exchange is not carried out completely, leaving a few hydrophilic groups on the surface. The dispersed organo-clays are then activated by addition of a polar solvent such as propylene carbonate, alcohols, glycols, etc. The gel is produced by hydrogen bonding between the polar groups on the surface of the clay and the polar solvent added. Several types of organo-clays are commercially available depending on the application and the type of solvent in which a gel is required. In some cases, organo-clays can be supplied already activated. Organo-clays are applied to “thicken” many personal care products, e.g. foundations, nonaqueous creams, nail polish, lipsticks, etc. The procedure for dispersion of the organo-clay particles and their subsequent activation is crucial and it requires good process control.

292 | 7 Liquid/solid dispersions (gels)

7.6.3 Oxide gels The most commonly used oxide gels are based on silica. Various forms of silicas can be produced, the most common are referred to as fumed and precipitated silicas. Fumed silica such as Aerosil 200 is produced by reaction of silicon tetrachloride with steam. The surface contains siloxane bonds and isolated silanol groups (referred to as vicinal). Precipitated silicas are produced from sodium silicate by acidification. The surface is more populated with silanol groups than fumed silica. It contains germinal OH groups (two attached to the same Si atom). Both fumed and precipitated silicas can produce gels both in aqueous and nonaqueous systems. Gelation results from aggregation of silica particles thus producing three-dimensional gel networks with a yield value. In aqueous media, the gel strength depends on the pH and electrolyte concentration [6]. As an illustration, Fig. 7.16 shows the variation of viscosity and yield value with Aerosil silica (which has been dispersed by sonication) concentration at three different pH values. In all cases, the viscosity and yield value shows a rapid increase above a certain silica concentration that depends on the pH of the system.

pH = 7

η /Pas

80

pH = 9

pH = 3

η

η

η

10 8

σß

60

6

40

4

20

2 5

6

7

8

9

10

11

σß/Pa

100

12

% v/v Fig. 7.16: Variation of viscosity η and yield value σ β with Aerosil 200 concentration at three pH values.

At pH = 3 (near the isoelectric point of silica), the particles are aggregated (forming flocs) and the increase in viscosity occurs at relatively high silica concentration (> 11 % v/v). At pH = 7, the silica becomes negatively charged and the double layers stabilize the silica particles against aggregation. In this case the particles remain as small units and the viscosity and yield value increases sharply above 7 % v/v. At pH = 9, some aggregation occurs as a result of the electrolyte released on adjusting

7.7 Gels produced by mixtures of polymers and finely divided particulate solids

|

293

the pH; in this case the viscosity increases at higher concentration (> 9 % v/v) when compared with the results at pH = 7. These results clearly indicate the importance of pH and electrolyte concentration in gelation of silica. It seems that the optimum gel formation occurs at neutral pH.

7.7 Gels produced by mixtures of polymers and finely divided particulate solids By combining the thickeners such as hydroxyethyl cellulose or xanthan gum with particulate solids such as sodium montmorillonite, a more robust gel structure can be produced. By using such mixtures, the concentration of the polymer can be reduced, thus overcoming the problem of dispersion on dilution (e.g. with many agrochemical suspension concentrates). This gel structure may be less temperature dependent and could be optimized by controlling the ratio of the polymer and the particles. If these combinations of say sodium montmorillonite and a polymer such as hydroxyethyl cellulose, polyvinyl alcohol (PVA) or xanthan gum, are balanced properly, they can provide a “three-dimensional structure”, which entraps all the particles and stops settling and formation of dilatant clays. The mechanism of gelation of such combined systems depends to a large extent on the nature of the solid particles, the polymer and the conditions. If the polymer adsorbs on the particle surface (e.g. PVA on sodium montmorillonite or silica) a three-dimensional network may be formed by polymer bridging. Under conditions of incomplete coverage of the particles by the polymer, the latter becomes simultaneously adsorbed on two or more particles. In other words, the polymer chains act as “bridges” or “links” between the particles. The optimum composition of these particulate-polymer mixtures can be obtained using rheological measurements. By measuring the yield value as a function of polymer concentration at a fixed particulate concentration, one can obtain the optimum polymer concentration required. In most cases, the yield value reaches a maximum at a given ratio of particulate solid to polymer. This trend may be due to bridging flocculation, which reaches an optimum at a given surface coverage of the particles (usually at 0.25–0.5 surface coverage). All the above mentioned gels produce thixotropy, i.e., reversible decrease of viscosity on application of shear (at constant shear rate) and recovery of the viscosity on standing. This thixotropic behaviour finds application in many systems in personal care, e.g. in creams, toothpastes, foundations, etc. One of the most useful techniques to study thixotropy is to follow the change of modulus with time after application of shear, i.e. after subjecting the dispersion to a constant shear rate, oscillatory measurements are carried out at low strains and high frequency and the increase in modulus with time (which is exponential) can be used to characterize the recovery of the gel.

294 | 7 Liquid/solid dispersions (gels)

7.8 Gels based on surfactant systems In dilute solutions surfactants tend to form spherical micelles with aggregation numbers in the range 50–100 units. These micellar solutions are isotropic with low viscosity. At much higher surfactant concentration (> 30 % depending on the surfactant nature) they produce liquid crystalline phases of the Hexagonal (H1 ) and Lamellar (Lα ) phases which are anisotropic with much higher viscosities [7]. A schematic representation of the hexagonal and lamellar phases is shown in Fig. 7.17 and 7.18. These liquid crystalline phases, which are viscoelastic, can be used as rheology modifiers. However, for practical applications, such as in shampoos, such very high surfactant concentrations are undesirable [8]. One way to increase the viscosity of a surfactant solution at lower concentrations is to add an electrolyte that causes a change from spherical to cylindrical micelles which can grow in length and at above

Water

Surfactant

Fig. 7.17: Schematic representation of hexagonal phase.

Surfacant

Water

Fig. 7.18: Schematic representation of lamellar phase.

7.8 Gels based on surfactant systems |

ϕ < ϕ*

ϕ*

ϕ > ϕ*

295

Fig. 7.19: Overlap of thread-like micelles.

a critical surfactant volume fraction ϕ∗ these worm-like micelles begin to overlap, forming a “gel” as illustrated in Fig. 7.19. An alternative method to produce gels in emulsions is to use mixtures of surfactants. By proper choice of the surfactant types (e.g. their hydrophilic-lipophilic balance, HLB) one can produce lamellar liquid crystalline structures that can “wrap” around the oil droplets and extend in solution to form gel networks [7, 8]. These structures (sometimes referred to as oleosomes) are schematically shown in Fig. 7.20. Alternatively, the liquid crystalline structures may produce a “three-dimensional” gel network and the oil droplets become entrapped in the “holes” of the network. These structures are sometimes referred to as hydrosomes and are illustrated in Fig. 7.21.

Oil

Oil w w Oil

Oil w Fig. 7.20: Schematic representation of “oleosomes”.

d

a

d

c

d e

c

b b b d

a

c a

a: Nyaropnobic part b: Trapped water c: Hydrophilic part d: Bilk water e: Oil

Fig. 7.21: Schematic representation of “hydrosomes”.

296 | 7 Liquid/solid dispersions (gels)

The above mentioned surfactant systems are used in most personal care and cosmetic formulations [8]. Apart from giving the right consistency for application (e.g. good skin feel) they are also effective in stabilizing emulsions against creaming or sedimentation, flocculation and coalescence. Liquid crystalline structures can also influence the delivery of active ingredients both of the lipophilic and hydrophilic types. Since lamellar liquid crystals mimic the skin structure (in particular the stratum corneum) they can offer prolonged hydration potential.

References [1] [2] [3] [4] [5] [6] [7] [8]

Tadros T. Rheology of Dispersions. Weinheim: Wiley-VCH; 2010. Ferry JD. Viscoelastic properties of polymers. New York: John Wiley & Sons; 1980. de Gennes PG. Scaling concepts of polymer physics. Ithaca: Cornell University Press; 1979. Goddard ED. In: Goddard ED, Gruber JV, editors. Polymer/surfactant interaction. New York: Marcel Dekker; 1999, Chapters 4 and 5. van Olphen H. Clay colloid chemistry. New York: Wiley; 1963. Heath D, Tadros TF. J Colloid Interface Sci. 1983;90:207, 320 Tadros T. Applied surfactants. Weinheim: Wiley-VCH; 2005. Tadros T. Formulation of Cosmetics and Personal Care., Berlin: De Gruyter; 2016.

8 Polymer colloids (latexes) 8.1 Introduction Polymers (latexes) are widely used in many industrial applications, e.g. in paints and coatings (film formers), as adhesives, as diagnostic markers for certain diseases, etc. In paints and coatings, latexes are used in aqueous emulsion paints that are used for home decoration. These aqueous emulsion paints are applied at room temperature and the latexes coalesce on the substrate forming a thermoplastic film. Sometimes functional polymers are used for crosslinking in the coating system. The polymer particles are typically submicron (0.1–0.5 µm). Generally speaking, there are three methods for preparing polymer dispersions, namely emulsion, dispersion and suspension polymerization. In emulsion polymerization, monomer is emulsified in a nonsolvent, commonly water, usually in the presence of a surfactant [1]. A water soluble initiator is added, and particles of polymer form and grow in the aqueous medium as the reservoir of the monomer in the emulsified droplets is gradually used up. In dispersion polymerization (which is usually applied for the preparation of nonaqueous polymer dispersion, commonly referred to as nonaqueous dispersion polymerization, NAD) monomer, initiator, stabilizer (referred to as protective agent) and solvent initially form a homogeneous solution. The polymer particles precipitate when the solubility limit of the polymer is exceeded. The particles continue to grow until the monomer is consumed. In suspension polymerization, the monomer is emulsified in the continuous phase using a surfactant or polymeric suspending agent. The initiator (which is oil soluble) is dissolved in the monomer droplets and the droplets are converted into insoluble particles, but no new particles are formed. A description of both emulsion and dispersion polymerization is given below, with particular reference to the control of their particle size and colloid stability which is greatly influenced by the emulsifier or dispersant used. Particular emphasis will be given to the effect of polymeric surfactants that have been recently used for the preparation of emulsion polymers.

8.2 Emulsion polymerization As mentioned above, in emulsion polymerization, the monomer, e.g. styrene or methyl methacrylate that is insoluble in the continuous phase, is emulsified using a surfactant that adsorbs at the monomer/water interface [1]. The surfactant micelles in bulk solution solubilize some of the monomer. A water soluble initiator such as potassium persulphate K2 S2 O8 is added and this decomposes in the aqueous phase forming free radicals that interact with the monomers forming oligomeric chains. It has long been https://doi.org/10.1515/9783110541953-009

298 | 8 Polymer colloids (latexes)

assumed that nucleation occurs in the “monomer swollen micelles”. The reasoning behind this mechanism was the sharp increase in the rate of reaction above the critical micelle concentration and that the number of particles formed and their size depend to a large extent on the nature of the surfactant and its concentration (which determines the number of micelles formed). However, later this mechanism was disputed and it was suggested that the presence of micelles means that excess surfactant is available and molecules will readily diffuse to any interface. The most accepted theory of emulsion polymerization is referred to as the coagulative nucleation theory [2, 3]. A two-step coagulative nucleation model has been proposed by Napper and co-workers [2, 3]. In this process the oligomers grow by propagation and this is followed by a termination process in the continuous phase. A random coil is produced which is insoluble in the medium and this produces a precursor oligomer at the θ-point. The precursor particles subsequently grow primarily by coagulation to form true latex particles. Some growth may also occur by further polymerization. The colloidal instability of the precursor particles may arise from their small size, and the slow rate of polymerization can be due to reduced swelling of the particles by the hydrophilic monomer [2, 3]. The role of surfactants in these processes is crucial since they determine the stabilizing efficiency and the effectiveness of the surface active agent, ultimately determining the number of particles formed. This was confirmed by using surface active agents of different nature. The effectiveness of any surface active agent in stabilizing the particles was the dominant factor and the number of micelles formed was relatively unimportant. A typical emulsion polymerization formulation contains water, 50 % monomer blended for the required glass transition temperature, Tg , surfactant (and often colloid), initiator, pH buffer and fungicide. Hard monomers with a high Tg used in emulsion polymerization may be vinyl acetate, methyl methacrylate and styrene. Soft monomers with a low Tg include butyl acrylate, 2-ethylhexyl acrylate, vinyl versatate and maleate esters. Most suitable monomers are those with low, but not too low, water solubility. Other monomers such as acrylic acid, methacrylic acid, adhesion promoting monomers may be included in the formulation. It is important that the latex particles coalesce as the diluent evaporates. The minimum film forming temperature (MFFT) of the paint is a characteristic of the paint system. It is closely related to the Tg of the polymer but the latter can be affected by materials present, such as surfactant, and the inhomogeneity of the polymer composition at the surface. High Tg polymers will not coalesce at room temperature and in this case a plasticizer (“coalescing agent”) such as benzyl alcohol is incorporated in the formulation to reduce the Tg of the polymer, thus reducing the MFFT of the paint. Clearly, for any paint system one must determine the MFFT since, as mentioned above, the Tg of the polymer is greatly affected by the ingredients in the paint formulation. Several types of surfactants (anionic, cationic, zwitterionic and nonionic) can be used in emulsion polymerization; the various classes have been described in detail in Chapter 8, Vol. 1.

8.2 Emulsion polymerization

|

299

The role of surfactants is two-fold, firstly to provide a locus for the monomer to polymerize and secondly to stabilize the polymer particles as they form. In addition, surfactants aggregate to form micelles (above the critical micelle concentration) and these can solubilize the monomers. In most cases a mixture of anionic and nonionic surfactants is used for optimum preparation of polymer latexes. Cationic surfactants are seldom used, except for some specific applications where a positive charge is required on the surface of the polymer particles. In addition to surfactants most latex preparations require the addition of a polymer (sometimes referred to as “protective colloid”) such as partially hydrolysed polyvinyl acetate (commercially referred to as polyvinyl alcohol, PVA), hydroxyethyl cellulose or a block copolymer of polyethylene oxide (PEO) and polypropylene oxide (PPO). These polymers can be supplied with various molecular weights or proportions of PEO and PPO. When used in emulsion polymerization, they can be grafted by the growing chain of the polymer being formed. They assist in controlling the particle size of the latex, enhancing the stability of the polymer dispersion and controlling the rheology of the final paint. A typical emulsion polymerization process involves two stages known as the seed stage and the feed stage. In the feed stage, an aqueous charge of water, surfactant, and colloid is raised to the reaction temperature (85–90 °C) and 5–10 % of the monomer mixture is added along with a proportion of the initiator (a water soluble persulphate). In this seed stage, the formulation contains monomer droplets stabilized by surfactant, a small amount of monomer in solution as well as surfactant monomers and micelles. Radicals are formed in solution from the breakdown of the initiator and these radicals polymerize the small amount of monomer in solution. These oligomeric chains will grow to some critical size, the length of which depends on the solubility of the monomer in water. The oligomers build up to a limiting concentration and this is followed by a precipitous formation of aggregates (seeds), a process similar to micelle formation, except in this case the aggregation process is irreversible (unlike surfactant micelles which are in dynamic equilibrium with monomers). In the feed stage, the remaining monomer and initiator are fed together and the monomer droplets become emulsified by the surfactant remaining in solution (or by extra addition of surfactant). Polymerization proceeds as the monomer diffuses from the droplets, through the water phase, into the already forming growing particles. At the same time radicals enter the monomer-swollen particles causing both termination and re-initiation of polymerization. As the particles grow, the remaining surfactant from the water phase is adsorbed onto the surface of the particles to stabilize the polymer particles. The stabilization mechanism involves both electrostatic and steric repulsion. The final stage of polymerization may include a further shot of initiator to complete the conversion. According to the theory of Smith and Ewart [4] of the kinetics of emulsion polymerization, the rate of propagation Rp is related to the number of particles N formed

300 | 8 Polymer colloids (latexes)

in a reaction by the equation, −

d[M] = Rp kp Nnav [M], dt

(8.1)

where [M] is the monomer concentration in the particles, kp is the propagation rate constant and nav is the average number of radicals per particle. According to equation (8.1), the rate of polymerization and the number of particles are directly related to each other, i.e. an increase in the number of particles will increase the rate. This has been found for many polymerizations, although there are some exceptions. The number of particles is related to the surfactant concentration [S] by the equation [4] N ≈ [S]3/5 . (8.2) Using the coagulative nucleation model, Napper et al. [2, 3] found that the final particle number increases with increasing surfactant concentration with a monotonically diminishing exponent. The slope of d(log Nc )/d(log t) varies from 0.4 to 1.2. At high surfactant concentration, the nucleation time will be long in duration since the new precursor particles will be readily stabilized. As a result, more latex particles are formed and eventually will outnumber the very small precursor particles at long times. The precursor/particle collisions will become more frequent and fewer latex particles are produced. The dNc /dt will approach zero and at long times the number of latex particles remain constant. This shows the inadequacy of the Smith–Ewart theory which predicts a constant exponent (3/5) at all surfactant concentrations. For this reason, the coagulative nucleation mechanism has now been accepted as the most probable theory for emulsion polymerization. In all cases, the nature and concentration of surfactant used is very crucial and this is very important in the industrial preparation of latex systems. Most reports on emulsion polymerization have been limited to commercially available surfactants, which in many cases are relatively simple molecules such as sodium dodecyl sulphate and simple nonionic surfactants. However, studies on the effect of surfactant structure on latex formation have revealed the importance of the structure of the molecule. Block and graft copolymers (polymeric surfactants) are expected to be better stabilizers when compared to simple surfactants. The use of these polymeric surfactants in emulsion polymerization and the stabilization of the resulting polymer particles are discussed below. Most aqueous emulsion and dispersion polymerization reported in the literature are based on a few commercial block and graft copolymers, with a broad molecular weight distribution and varying block composition. The results obtained from these studies could not establish what effect the structural features of the block copolymer has on their stabilizing ability and effectiveness in polymerization. Fortunately, model block copolymers with well-defined structures could be synthesized and their role in emulsion polymerization has been investigated using model polymers and model latexes.

8.2 Emulsion polymerization

|

301

A series of well-defined A–B block copolymers of polystyrene-block–polyethylene oxide (PS–PEO) were synthesized [5] and used for emulsion polymerization of styrene. These molecules are “ideal” since the polystyrene block is compatible with the polystyrene formed and thus it forms the best anchor chain. The PEO chain (the stabilizing chain) is strongly hydrated with water molecules and it extends into the aqueous phase forming the steric layer necessary for stabilization. However, the PEO chain can become dehydrated at high temperature (due to the breakage of hydrogen bonds), thus reducing the effective steric stabilization. Thus the emulsion polymerization should be carried out at temperatures well below the theta (θ)-temperature of PEO. Five block copolymers were synthesized [5] with various molecular weights of the PS and PEO blocks. The molecular weight of the polystyrene block and the resulting PS–PEO polymer was determined using gel permeation chromatography. The mole percent of ethylene oxide and the percent of PEO in the block was determined using H1 NMR spectroscopy. The molecular weight of the blocks varied from Mn = 1000–7000 for PS and Mw = 3000–9000 for PEO. These five block copolymers were used for emulsion polymerization of styrene at 50 °C (well below the θ-temperature of PEO). The results indicated that for efficient anchoring, the PS block need not be more than 10 monomer units. The PEO blocks should have a Mw ≥ 3000. However, the ratio of the two blocks is very important; for example if the wt% of PEO is ≤ 3000 the molecule becomes insoluble in water (not sufficiently hydrophilic) and no polymerization could occur when using this block copolymer. In addition, the 50 % PEO block could produce a latex but it was unstable and coagulated at 35 % conversion. It became clear from these studies that the % PEO in the block copolymer plays an important role and this should exceed 75 %. However, the overall molecular weight of the block copolymer is also very important. For example if one uses a PS block with Mn = 7000, the PEO molecular weight has to be 21 000, which is too high and may result in bridging flocculation unless one prepares a very dilute latex. Another systematic study of the effect of block copolymer on emulsion polymerization was carried out using blocks of poly(methylmethacrylate)-block–polyethylene oxide (PMMA–PEO) for the preparation of PMMA latexes [5]. The ratio and molecular weight of PMMA to PEO in the block copolymer was varied. Ten different PMMA–PEO blocks were synthesized [5] with Mn for PMMA varying between 400 and 2500. The Mw of PEO was varied between 750 and 5000. The recipe for MMA polymerization consisted of 100 monomer, 800 g water, 20 g PMMA–PEO block copolymer and 0.5 g potassium persulphate. The polymerization was carried out at 45 °C which is well below the θ-temperature of PEO. The rate of polymerization Rp was calculating by using latex samples drawn from the reaction mixture at various intervals of time (the amount of latex was determined gravimetrically). The particle size of each latex was determined by dynamic light scattering (photon correlation spectroscopy, PCS). The number of particles, N, in each case was calculated from the weight of the latex and the z-average diameter. The results obtained were used to study the effect of the an-

302 | 8 Polymer colloids (latexes)

Tab. 8.1: Effect of PMMA and PEO molecular weight in the diblock. Mn PMMA

Mw PEO

wt% PEO

Rp × 104 (mol/l s)

D (nm)

N × 10−13 (cm−3 )

400 400 400 900 800 800 1300 1200 1900 2500

750 2000 5000 750 2000 5000 2000 5000 5000 5000

65 83 93 46 71 86 61 81 72 67

1.3 1.5 2.4 Unstable latex 3.4 3.2 2.4 4.6 3.4 2.2

213 103 116 — 92 106 116 99 110 322

1.7 14.7 10.3 — 20.6 13.5 10.3 16.6 11.4 0.4

Tab. 8.2: Effect of total molecular weight of the PMMA–PEO diblock. Mw

wt% PEO

Rp × 104 (mol/l s)

D (nm)

N × 10−13 (cm−3 )

1150 2400 2800 3300 6200 6900 7500

65 83 71 61 81 72 67

1.3 1.5 3.4 2.4 4.6 3.4 2.2

213 103 92 99 99 110 322

1.7 14.7 20.6 16.6 16.6 11.4 0.4

choring group PMMA, molecular weight, the effect of PEO molecular weight and the effect of the total molecular weight of the block copolymer. The results are summarized in Tab. 8.1 and 8.2. These results of the systematic study (Tab. 8.1 and 8.2) of varying the PMMA and PEO block molecular weight, the % PEO in the chain as well as the overall molecular weight clearly show the effect of these factors on the resulting latex. For example, when using a block copolymer with 400 molecular weight of PMMA and 750 molecular weight of PEO (i.e. containing 65 wt% PEO) the resulting latex has fewer particles when compared with the other surfactants. The most dramatic effect was obtained when the PMMA molecular weight was increased to 900 while keeping the PEO molecular weight (750) the same. This block copolymer contains only 46 wt% PEO and it became insoluble in water due to the lack of hydrophilicity. The latex produced was unstable and it collapsed at the early stage of polymerization. The PEO molecular weight of 750 is insufficient to provide effective steric stabilization. By increasing the molecular weight of PEO to 2000 or 5000 while keeping the PMMA molecular weight at 400 or 800, a stable latex was produced with a small particle diameter and large number of particles. The best results were obtained by keeping the molecular weight of PMMA at 800 and that of PEO at 2,000. This block copolymer gave the highest conversion rate,

8.2 Emulsion polymerization |

303

the smallest particle diameter and the largest number of particles (see Tab. 8.2). It is interesting to note that by increasing the PEO molecular weight to 5000 while keeping the PMMA molecular weight at 800, the rate of conversion decreased, the average diameter increased and the number of particles decreased when compared with the results obtained using 2000 molecular weight for PEO. It seems that when the PEO molecular weight is increased the hydrophilicity of the molecule increased (86 wt% PEO) and this reduced the efficiency of the copolymer. It seems that by increasing hydrophilicity of the block copolymer and its overall molecular weight the rate of adsorption of the polymer to the latex particles and its overall adsorption strength may have decreased. The effect of the overall molecular weight of the block copolymer and its overall hydrophilicity have a big effect on the latex production (see Tab. 8.3). Increasing the overall molecular weight of the block copolymer above 6200 resulted in a reduction in the rate of conversion, an increase in the particle diameter and a reduction in the number of latex particles. The worst results were obtained with an overall molecular weight of 7500 while reducing the PEO wt%. In this case particles with 322 nm diameter were obtained and the number of latex particles is significantly reduced. The importance of the affinity of the anchor chain (PMMA) to the latex particles was investigated by using different monomers [5]. For example, when using styrene as the monomer the resulting latex was unstable and it showed the presence of coagulum. This can be attributed to the lack of chemical compatibility of the anchor chain (PMMA) and the polymer to be stabilized, namely polystyrene. This clearly indicates that block copolymers of PMMA–PEO are not suitable for emulsion polymerization of styrene. However, when using vinyl acetate monomer, where the resulting poly(vinyl acetate) latex should have strong affinity to the PMMA anchor, no latex was produced when the reaction was carried out at 45 °C. It was speculated that the water solubility of the vinyl acetate monomer resulted in the formation of oligomeric chain radicals which could exist in solution without nucleation. Polymerization at 60 °C, which did nucleate particles, was found to be controlled by chain transfer of the vinyl acetate radical with the surfactant, resulting in broad molecular weight distributions Emulsion polymerization of MMA using triblock copolymers was carried using PMMA-block–PEO–PMMA with the same PMMA molecular weight (800 or 900) while varying the PEO molecular weight from 3400 to 14 000 in order to vary the loop size. Although the rate of polymerization was not affected by the loop size, the particles with the smallest diameter were obtained with the 10 000 molecular weight PEO. Comparison of the results obtained using the triblock copolymer with those obtained using diblock copolymer (while keeping the PMMA block molecular weight the same) showed the same rate of polymerization. However, the average particle diameter was smaller and the total number of particles larger when using the diblock copolymer. This clearly shows the higher efficacy of the diblock copolymer when compared with the triblock copolymer.

304 | 8 Polymer colloids (latexes)

The first systematic study of the effect of graft copolymers were carried out by Piirma and Lenzotti [6] who synthesized well characterized graft copolymers with different backbone and side chain lengths. Several grafts of poly(p-methylstyrene)graft–polyethylene oxide, (PMSt)–(PEO)n ,were synthesized and used in styrene emulsion polymerization. Three different PMSt chain length (with molecular weight of 750, 2000 and 5000) and three different PEO chain lengths were prepared. In this way the structure of the amphipathic graft copolymer could be changed in three different ways: (i) three different PEO graft chain lengths; (ii) three different backbone chain lengths with the same wt% PEO; (iii) four different wt% PEO grafts. Piirma and Lenzotti [6] first investigated the graft copolymer concentration required to produce the highest conversion rate, the smallest particle size and the largest number of latex particles. The monomer-to-water ratio was kept at 0.15 to avoid overcrowding of the resulting particles. They found that a concentration of 18 g/100 g monomer (2.7 % aqueous phase) was necessary to obtain the above results, after which a further increase in graft copolymer concentration did not have any significant effect on increasing the rate of polymerization or increasing the number of particles used. Using the graft copolymer concentration of 2.7 % aqueous phase, the results showed an increase in the number of particles with increasing conversion reaching a steady value at about 35 % conversion. Obviously, before that conversion, new particles are still being stabilized from the oligomeric precursor particles, after which all precursor particles are assimilated by the existing particles. The small size of the latex produced, namely 30–40 nm, clearly indicates the efficiency with which this graft copolymer stabilizes the dispersion. Three different backbone chain lengths Mn of 1140, 4270 and 24 000 were used while the weight percent of PEO (82 %) was kept the same, that is equivalent to 3, 10 and 55 PEO chains per backbone respectively. The results showed that the rate of polymerization, particle diameter and number of particles was similar for the three cases. Since the graft copolymer concentration was the same in each case, it can be concluded that one molecule of the highest molecular graft is just as effective as 18 molecules of the lowest molecular weight graft in stabilizing the particles. Four graft copolymers were synthesized with a PMSt backbone with Mw = 4540 while increasing the wt% of PEO: 68, 73, 82 and 92 wt% (corresponding to 4.8, 6, 10 and 36 grafts per chain). The results showed a sharp decrease (by more than one order of magnitude) in the number of particles as the wt% of PEO is increased from 82 to 94 %. The reason for this reduction in the number of particles is the increased hydrophilicity of the graft copolymer which could result in desorption of the molecule from the surface of the particle. In addition, a graft with 36 side chains does not leave enough space for anchoring by the backbone. The effect of PEO side chain length on emulsion polymerization using graft copolymers was systematically studied by keeping the backbone molecular weight the

8.2 Emulsion polymerization

|

305

same (1380) while gradually increasing the PEO molecular weight of the side chains from 750 to 5000. For example, by increasing Mw of PEO from 750 to 2000 while keeping the wt% of PEO roughly the same (84 and 82 wt% respectively) the number of side chains in the graft decreases from 10 to 3. The results showed a decrease in the rate of polymerization as the number of side chains in the graft increases. This is followed by a sharp reduction in the number of particles produced. This clearly shows the importance of spacing of the side chains to ensure anchoring of the graft copolymer to the particle surface, which is stronger with the graft containing a smaller number of side chains. If the number of side chains for the PEO with Mw of 2000 is increased from 3 to 9 (93 wt% of PEO) the rate of polymerization and number of particles decrease. Using a PEO chain with Mw of 5000 (92 wt% PEO) and 3 chains per graft gives the same result as the PEO 2000 with 3 side chains. Any increase in the number of side chains in the graft results in a reduction in the rate of polymerization and the number of latex particles produced. This clearly shows the importance of spacing of the side chains of the graft copolymer. Similar results were obtained using a graft copolymer of poly(methyl methacrylate-co-2-hydroxypropyl methacrylate)-graft–polyethylene oxide, PMMA(PEO)n , for emulsion polymerization of methyl methacrylate. As with PMSt(PEO)n graft, the backbone molecular weight had little effect on the rate of polymerization or the number of particles used. The molecular weight of the PEO side chains was varied at constant Mw of the backbone (10 000). Three PEO grafts with Mw of 750, 2000 and 5000 were used. Although the rate of polymerization was similar for the three graft copolymers, the number of particles was significantly lower with the graft containing PEO 750. This shows that this short PEO chain is not sufficient for stabilizing the particles. The overall content of PEO in the graft also has a big effect. Using the same backbone chain length while changing the wt% of PEO 200, it was found that the molecule containing 67 wt% PEO is not sufficient for stabilizing the particles when compared with a graft containing 82 wt% PEO. This shows that a high concentration of PEO in the adsorbed layer is required for effective steric stabilization. The chemical nature of the monomer also plays an important role. For example stable latexes could be produced using PMSt(PEO)n graft but not with PMMA(PEO)n graft. A novel graft copolymer of hydrophobically modified inulin (INUTEC® SP1) has been used in emulsion polymerization of styrene, methyl methacrylate, butyl acrylate and several other monomers [7]. All lattices were prepared by emulsion polymerization using potassium persulphate as initiator. The z-average particle size was determined by photon correlation spectroscopy (PCS) and electron micrographs were also taken. Emulsion polymerization of styrene or methylmethacrylate showed an optimum weight ratio of (INUTEC)/Monomer of 0.0033 for PS and 0.001 for PMMA particles. The (initiator)/(Monomer) ratio was kept constant at 0.001 25. The monomer conversion was higher than 85 % in all cases. Latex dispersions of PS reaching 50 %

306 | 8 Polymer colloids (latexes)

0.2 μm

0.2 μm

0.2 μm

0.2 μm

(5 wt%)

(10 wt%)

(30 wt%)

(40 wt%)

(a) PS latexes

0.2 μm

0.2 μm

0.2 μm

(10 wt%)

(20 wt%)

(30 wt%)

(b) PMMA Latexes Fig. 8.1: Electron micrographs of the latexes.

and of PMMA reaching 40 % could be obtained using such low concentration of INUTEC® SP1. Fig. 8.1 shows the variation of particle diameter with monomer concentration. The stability of the latexes was determined by determining the critical coagulation concentration (CCC) using CaCl2 . The CCC was low (0.0175–0.05 mol dm−3 ) but this was higher than that for the latex prepared without surfactant. Post addition of INUTEC® SP1 resulted in a large increase in the CCC, as illustrated in Fig. 8.2 that shows log W–log C curves (where W is the ratio between the fast flocculation rate constant to the slow flocculation rate constant, referred to as the stability ratio) at various additions of INUTEC® SP1. As with the emulsions, the high stability of the latex when using INUTEC® SP1 is due to the strong adsorption of the polymeric surfactant on the latex particles and formation of strongly hydrated loops and tails of polyfructose that provide effective steric stabilization. Evidence for the strong repulsion produced when using INUTEC® SP1 was obtained from atomic force microscopy investigations [8] in which the force between hydrophobic glass spheres and a hydrophobic glass plate, both containing an adsorbed layer of INUTEC® SP1, was measured as a function of distance of separation both in water and in the presence of various Na2 SO4 concentrations. The results are shown in Fig. 8.3 and 8.4.

8.3 Polymeric surfactants for stabilizing preformed latex dispersions

|

307

1.4 1.2 PS 5 wt% 1

PS 5 wt% synthesized with 0.01 wt% INUTEC

log W

0.8 0.6

PS 5 wt% + post-added 0.01 wt% INUTEC

0.4

PS 5 wt% + post-added

0.2 0 –3

–2

–1

0

log ([CaCl2]/M)

Fig. 8.2: Influence of post addition of INUTEC® SP1 on the latex’s stability.

F/2πr (μN/m)

0.12

Approach

0.1 Withdrawal

0.08 0.06 0.04 0.02 0 –0.02 –0.04 0

20

40

60

h/nm Fig. 8.3: Force–distance curves between hydrophobized glass surfaces containing adsorbed INUTEC® SP1 in water.

8.3 Polymeric surfactants for stabilizing preformed latex dispersions For this purpose polystyrene (PS) latexes were prepared using the surfactant-free emulsion polymerization [9]. Two latexes with z-average diameter of 427 and 867 (as measured using photon correlation spectroscopy, PCS) that are reasonably monodisperse were prepared. Two polymeric surfactants, namely Hypermer CG-6 and Atlox 4913 (CRODA, UK) were used. Both are graft (“comb”) type, consisting of polymethylmethacrylate/polymethacrylic acid (PMMA/PMA) backbone with methoxy-capped

308 | 8 Polymer colloids (latexes) 0.1

F/2πr (μN/m)

0.08 0.3 mol dm¯³

0.06

0.04

0.8 mol dm¯³ 1 mol dm¯³

0.02

1.5 mol dm¯³

0

–0.02 0

40

20

60

Separation (nm) Fig. 8.4: Force–distance curves for hydrophobized glass surfaces containing adsorbed INUTEC® SP1 at various Na2 SO4 concentrations.

polyethylene oxide (PEO) side chains (M = 750 Daltons). Hypermer CG-6 is the same graft copolymer as Atlox 4913 but it contains a higher proportion of methacrylic acid in the backbone. The average molecular weight of the polymer is ≈ 5000 Daltons. Fig. 8.5 shows a typical adsorption isotherm of Atlox 4913 on the two latexes. Similar results were obtained for Hypermer CG-6 but the plateau adsorption was lower (1.2 mg m−2 compared with 1.5 mg m−2 for Atlox 4913). It is likely that the backbone of Hypermer

Adsorption Γ/mgm²

2

1.5

1.0 Latex D = 427 nm 0.5

Latex D = 867 nm

0.0 0.05

0.1

0.15

0.2

Atlox 4913 concentration/mgml¯ ¹ Fig. 8.5: Adsorption isotherms of Atlox 4913 on the two latexes at 25 °C.

8.3 Polymeric surfactants for stabilizing preformed latex dispersions

|

309

Adsorption Γ/mgm²

2

1.5

1.0 40°C 0.5

20°C

0.0 0.05

0.1

0.15

0.2

Atlox 4913 concentration/mgml¯ ¹ Fig. 8.6: Effect of temperature on adsorption of Atlox 4913 on PS.

CG-6 that contains more PMA is more polar and hence less strongly adsorbed. The amount of adsorption was independent of particle size. The influence of temperature on adsorption is shown in Fig. 8.6. The amount of adsorption increases with increasing temperature. This is due to the poorer solvency of the medium for the PEO chains. The PEO chains become less hydrated at higher temperature and the reduction of solubility of the polymer enhances adsorption. The adsorbed layer thickness of the graft copolymer on the latexes was determined using rheological measurements (see Chapter 1). Steady state (shear stress σ– shear rate γ)̇ measurements were carried out and the results were fitted to the Bingham equation to obtain the yield value σ β and the high shear viscosity η of the suspension, σ = σ β + η γ.̇

(8.3)

As an illustration, Fig. 8.7 shows a plot of σ β versus volume fraction ϕ of the latex for Atlox 4913. Similar results were obtained for latexes stabilized using Hypermer CG-6. At any given volume fraction, the smaller latex has higher σ β when compared to the larger latex. This is due to the higher ratio of adsorbed layer thickness to particle radius, ∆/R, for the smaller latex. The effective volume fraction of the latex ϕeff is related to the core volume fraction ϕ by the equation, ϕeff = ϕ[1 +

∆ 3 ] . R

(8.4)

ϕeff can be calculated from the relative viscosity ηr using the Dougherty–Krieger equation [10, 11], ϕeff −[η]ϕp ηr = [1 − ( , (8.5) )] ϕp where ϕp is the maximum packing fraction.

310 | 8 Polymer colloids (latexes)

25

D = 427 nm

σβ/Pa

20

D = 867 nm

15

10

5

0 0.2

0.3

0.4

0.5

0.6

ϕ Fig. 8.7: Variation of yield stress with latex volume fraction for Atlox 4913.

The maximum packing fraction ϕp can be calculated using the following empirical equation [9], 1/2 (ηr − 1) 1 (8.6) = ( )(η1/2 − 1) + 1.25. ϕ ϕp The results showed a gradual decrease of adsorbed layer thickness ∆ with increasing volume fraction ϕ. For the latex with diameter D of 867 nm and Atlox 4913, ∆ decreased from 17.5 nm at ϕ = 0.36 to 6.5 at ϕ = 0.57. For Hypermer CG-6 with the same latex, ∆ decreased from 11.8 nm at ϕ = 0.49 to 6.5 at ϕ = 0.57. The reduction of ∆ with increasing ϕ may be due to overlap and/or compression of the adsorbed layers as the particles come close to each other at higher volume fraction of the latex. The stability of the latexes was determined using viscoelastic measurements, as discussed in Chapter 1. For this purpose, dynamic (oscillatory) measurements were used to obtain the storage modulus G∗ , the elastic modulus G󸀠 and the viscous modulus G󸀠󸀠 as a function of strain amplitude γ0 and frequency ω (rad s−1 ). The method relies on application of a sinusoidal strain or stress and the resulting stress or strain is measured simultaneously. For a viscoelastic system, the strain and stress sine waves oscillate with the same frequency but out of phase. From the time shift ∆t and ω one can obtain the phase angle shift δ. The ratio of the maximum stress σ0 to the maximum strain γ0 gives the complex modulus |G∗ | σ0 |G∗ | = . (8.7) γ0 |G∗ | can be resolved into two components: storage (elastic) modulus G󸀠 , the real component of the complex modulus; and loss (viscous) modulus G󸀠󸀠 , the imaginary component of the complex modulus. The complex modulus can be resolved into G󸀠 and

8.4 Dispersion polymerization

| 311

G󸀠󸀠 using vector analysis and the phase angle shift δ, G󸀠 = |G∗ | cos δ, 󸀠󸀠

(8.8)

∗

G = |G | sin δ.

(8.9)

G󸀠 is measured as a function of electrolyte concentration and/or temperature to assess the latex stability. As an illustration, Fig. 8.8 shows the variation of G󸀠 with temperature for latex stabilized with Atlox 4913 in the absence of any added electrolyte and in the presence of 0.1, 0.2 and 0.3 mol dm−3 Na2 SO4 . In the absence of electrolyte, G󸀠 showed no change with temperature up to 65 °C.

3000 1000

0.3 mol dm–3 Na2SO4

G′/Pa

300

0.2 mol dm–3 Na2SO4 0.1 mol dm–3 Na2SO4

100 30 10

Water

3 1

10

20

30

40

50

60

70

t/°C Fig. 8.8: Variation of G󸀠 with temperature in water and at various Na2 SO4 concentrations.

In the presence of 0.1 mol dm−3 Na2 SO4 , G󸀠 remained constant up to 40 °C above which G󸀠 increased with a further increase in temperature. This temperature is denoted as the critical flocculation temperature (CFT). The CFT decreases with increasing electrolyte concentration reaching ≈ 30 °C in 0.2 and 0.3 mol dm−3 Na2 SO4 . This reduction in CFT with increasing electrolyte concentration is due to the reduction in solvency of the PEO chains with increasing electrolyte concentrations. The latex stabilized with Hypermer CG-6 gave relatively higher CFT values when compared with that stabilized using Atlox 4913.

8.4 Dispersion polymerization This method is usually applied for the preparation of nonaqueous latex dispersions and hence it is referred to as NAD [12]. The method has also been adapted to prepare aqueous latex dispersions by using an alcohol-water mixture.

312 | 8 Polymer colloids (latexes)

In the NAD process the monomer, normally an acrylic, is dissolved in a nonaqueous solvent, normally an aliphatic hydrocarbon and an oil soluble initiator and a stabilizer (to protect the resulting particles from flocculation, sometimes referred to as “protective colloid”) is added to the reaction mixture. The most successful stabilizers used in NAD are block and graft copolymers. These block and graft copolymers are assembled in a variety of ways to provide the molecule with an “anchor chain” and a stabilizing chain. The anchor chain should be sufficiently insoluble in the medium and have a strong affinity to the polymer particles produced. In contrast, the stabilizing chain should be soluble in the medium and strongly solvated by its molecules to provide effective steric stabilization. The length of the anchor and stabilizing chains has to be carefully adjusted to ensure strong adsorption (by multipoint attachment of the anchor chain to the particle surface) and sufficiently “thick” layer of the stabilizing chain that prevents close approach of the particles to a distance where the van der Waals attraction becomes strong. Several configurations of block and graft copolymers are possible as illustrated in Fig. 8.9.

A-B block

A-B graft with one B chain

A-B-A block

B-A-B block

Anchor chain A ABn graft with several B chains

Stabilizing chain B

Fig. 8.9: Configurations of block and graft copolymers.

Typical preformed graft stabilizers based on poly(12-hydroxy stearic acid) (PHS) are simple to prepare and effective in NAD polymerization. Commercial 12-hydroxystearic acid contains 8–15 % palmitic and stearic acids which limits the molecular weight during polymerization to an average of 1500–2000. This oligomer may be converted to a “macromonomer” by reacting the carboxylic group with glycidyl methacrylate. The macromonomer is then copolymerized with an equal weight of methyl methacrylate (MMA) or similar monomer to give a “comb” graft copolymer with an average molecular weight of 10 000–20 000. The graft copolymer contains on average 5–10 PHS chains pendent from a polymeric anchor backbone of PMMA. This graft copolymer can sta-

8.4 Dispersion polymerization | 313

bilize latex particles of various monomers. The major limitation of the monomer composition is that the polymer produced should be insoluble in the medium used. Several other examples of block and graft copolymers that are used in dispersion polymerization are given in Tab. 8.3 which also shows the continuous phase and disperse polymer that can be used with these polymers. Tab. 8.3: Block and graft copolymers used in emulsion polymerization. Polymeric surfactant

Continuous phase

Disperse polymer

Polystyrene-block–poly(dimethyl siloxane) Polystyrene-block–poly(methacrylic acid) Polybutadiene-graft–poly(methacrylic acid) Poly(2-ethylhexyl acrylate)-graft–poly(vinyl acetate)

Hexane Ethanol Ethanol Aliphatic hydrocarbon

Polystyrene-block–poly(t-butylstyrene)

Aliphatic hydrocarbon

Polystyrene Polystyrene Polystyrene Poly(methyl methacrylate) Polystyrene

Two main criteria must be considered in the process of dispersion polymerization: (i) the insolubility of the formed polymer in the continuous phase; (ii) the solubility of the monomer and initiator in the continuous phase. Initially, dispersion polymerization starts as a homogeneous system but after sufficient polymerization, the insolubility of the resulting polymer in the medium forces them to precipitate. Initially, polymer nuclei are produced which then grow to polymer particles. The latter are stabilized against aggregation by the block or graft copolymer that is added to the continuous phase before the process of polymerization starts. It is essential to choose the right block or graft copolymer that should have a strong anchor chain A and good stabilizing chain B as schematically represented in Fig. 8.9. Dispersion polymerization may be considered as a heterogeneous process which may include emulsion, suspension, precipitation and dispersion polymerization. In dispersion and precipitation polymerization, the initiator must be soluble in the continuous phase, whereas in emulsion and suspension polymerization the initiator is chosen to be soluble in the disperse phase of the monomer. A comparison of the rate of polymerization of methylmethacrylate at 80 °C for the three systems was given by Barrett and Thomas [13] as illustrated in Fig. 8.10. The rate of dispersion polymerization is much faster than that of precipitation or solution polymerization. The enhancement of the rate in precipitation polymerization over solution polymerization has been attributed to the hindered termination of the growing polymer radicals. Several mechanisms have been proposed to explain the mechanism of emulsion polymerization; however, no single mechanism can explain all happenings in emulsion polymerization. Barrett and Thomas [13] suggested that particles are formed in emulsion polymerization by two main steps:

Fractional conversion

314 | 8 Polymer colloids (latexes)

Dispersion polymerization 0.8

Precipitation polymerization

0.6 0.4

Solution polymerization

0.2 0

50

100 Time/minutes

150

Fig. 8.10: Comparison of rates of polymerization.

(i) initiation of monomer in the continuous phase and subsequent growth of the polymer chains until the latter become insoluble. This process clearly depends on the nature of the polymer and medium. (ii) The growing oligomeric chains associate with each other forming aggregates that, below a certain size, are unstable and they become stabilized by the block or graft copolymer added. As mentioned before, this aggregative nucleation theory cannot explain all happenings in dispersion polymerization. An alternative mechanism based on Napper’s theory [2, 3] for aqueous emulsion polymerization can be adapted to the process of dispersion polymerization. This theory includes coagulation of the nuclei formed and not just association of the oligomeric species. The precursor particles (nuclei) being unstable can undergo one of the following events to become colloidally stable: (i) homocoagulation, i.e. collision with other precursor particles; (ii) growth by propagation, adsorption of stabilizer; (iii) swelling with monomer. The nucleation terminating events are diffusional capture of oligomers and heterocoagulation. The number of particles formed in the final latex does not depend on particle nucleation alone, since other steps are involved which determine how many precursor particles created are involved in the formation of a colloidally stable particle. This clearly depends on the effectiveness of the block or graft copolymer used in stabilizing the particles (see below). In most cases, increasing polymeric surfactant concentration (at any given monomer amount) results in the production of a larger number of particles with smaller size. This is to be expected since the larger number of particles with smaller size (i.e. larger total surface area of the disperse particles) require more polymeric surfactant for their formation. The molecular weight of the polymeric surfactant can also influence the number of particles formed. For example, Dawkins and Taylor [14] found that in dis-

8.4 Dispersion polymerization | 315

persion polymerization of styrene in hexane, increasing the molecular weight of the block copolymer of polydimethyl siloxane-block-polystyrene resulted in the formation of smaller particles, which was attributed to the more effective steric stabilization by the higher molecular weight block. A systematic study of the effect of monomer solubility and concentration in the continuous phase was carried out by Antl and co-workers [15]. Dispersion polymerization of methyl methacrylate in hexane mixed with a high boiling point aliphatic hydrocarbon was investigated using poly(12-hydoxystyearic acid)-glycidyl methacrylate block copolymer. They found that the methyl methacrylate concentration had a drastic effect on the size of the particles produced. When the monomer concentration was kept below 8.5 %, very small particles (80 nm) were produced and these remained very stable. However, between 8.5 and 35 % monomer the latex produced was initially stable but flocculated during polymerization. An increase in monomer concentration from 35 to 50 % resulted in the formation of a stable latex but the particle size increased sharply from 180 nm to 2.6 µm as the monomer concentration increased. The authors suggested that the final particle size and stability of the latex are strongly affected by increased monomer concentration in the continuous phase. The presence of monomer in the continuous phase increases the solvency of the medium for the polymer formed. In a good solvent for the polymer, the growing chain is capable of reaching higher molecular weight before it is forced to phase separate and precipitate. NAD polymerization is carried out in two steps: (i) seed stage: the diluent, portion of the monomer, portion of dispersant and initiator (azo or peroxy type) are heated to form an initial low-concentration fine dispersion; (ii) growth stage: the remaining monomer together with more dispersant and initiator are then fed over the course of several hours to complete the growth of the particles. A small amount of transfer agent is usually added to control the molecular weight. Excellent control of particle size is achieved by proper choice of the designed dispersant and correct distribution of dispersant between the seed and growth stages. NAD acrylic polymers are applied in automotive thermosetting polymers and hydroxy monomers may be included in the monomer blend used. Two main factors must be considered when considering the long-term stability of a nonaqueous polymer dispersion. The first and very important factor is the nature of the “anchor chain” A. As mentioned above, this should have a strong affinity to the produced latex and in most cases it can be designed to be “chemically” attached to the polymer surface. Once this criterion is satisfied, the second and important factor in determining the stability is the solvency of the medium for the stabilizing chain B. As will be discussed in detail, the solvency of the medium is characterized by the Flory–Huggins interaction parameter χ. Three main conditions can be identified: χ < 0.5 (good solvent for the stabilizing chain); χ > 0.5 (poor solvent for the stabilizing

316 | 8 Polymer colloids (latexes) chain); and χ = 0.5 (referred to as the θ-solvent). Clearly, to maintain stability of the latex dispersion, the solvent must be better than a θ-solvent. The solvency of the medium for the B chain is affected by addition of a nonsolvent and/or temperature changes. It is, therefore, essential to determine the critical volume fraction (CFV) of a nonsolvent above which flocculation (sometimes referred to as incipient flocculation) occurs. One should also determine the critical flocculation temperature at any given solvent composition, below which flocculation occurs. The correlation between CFV or CFT and the flocculation of the nonaqueous polymer dispersion has been demonstrated by Napper [16] who investigated the flocculation of poly(methyl methacrylate) dispersions stabilized by poly(12-hydroxy stearic acid) or poly(n-lauryl methacrylate-co-glycidyl methacrylate) in hexane by adding a nonsolvent such as ethanol or propanol and cooling the dispersion. The dispersions remained stable until the addition of ethanol transformed the medium to a θ-solvent for the stabilizing chains in solution. However, flocculation did occur under conditions of slightly better than θ-solvent for the chains. The same was found for the CFT, which was 5–15 K above the θ-temperature. This difference was accounted for by the polydispersity of the polymer chains. The θ-condition is usually determined by cloud point measurements and the least soluble component will precipitate first, giving values that are lower than the CFV or higher than the CFT.

8.5 Particle formation in polar media The process of dispersion polymerization has been applied in many cases using completely polar solvents such as alcohol or alcohol-water mixtures [17]. The results obtained showed completely different behaviour when compared with dispersion polymerization in nonpolar media. For example, results obtained by Lok and Ober [17] using styrene as monomer and hydroxypropyl cellulose as stabilizer showed a linear increase of particle diameter with increasing weight percent of the monomer. There was no region in monomer concentration where instability occurred (as has been observed for the dispersion polymerization of methyl methacrylate in aliphatic hydrocarbons). Replacing water in the continuous phase with 2-methoxyethanol, Lok and Ober were able to grow large, monodisperse particles up to 15 µm in diameter. They concluded from these results that the polarity of the medium is the controlling factor in the formation of particles and their final size. The authors suggested a mechanism in which the polymeric surfactant molecule grafts to the polystyrene chain, forming a physically anchored stabilizer (nuclei). These nuclei grow to form the polymer particles. Paine [18] carried out dispersion polymerization of styrene by systematically increasing the alcohol chain length from methanol to octadecanol and using hydroxypropyl cellulose as stabilizer. The results showed an increase in particle diameter with increasing number of carbon atoms in the alcohol, reaching a maximum when

References | 317

hexanol was used as the medium, after which there was a sharp decrease in the particle diameter with any further increase in the number of carbon atoms in the alcohol. Paine explained his results in terms of the solubility parameter of the dispersion medium. The largest particles are produced when the solubility parameter of the medium is closest to those of styrene and hydroxypropyl cellulose.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

Blakely DC. Emulsion polymerization. London; Elsevier Applied Science; 1975. Litchi G, Gilbert RG, Napper DH. J Polym Sci. 1983;21:269. Feeney PJ, Napper DH, Gilbert RG. Macromolecules. 1984;17:2520; 1987;20:2922. Smith WV, Ewart RH. J Chem Phys. 1948;16:592. Piirma I. Polymeric surfactants. Surfactant Science Series, No.42. New York: Marcel Dekker; 1992. Piirma I, Lenzotti JR. Br Polymer J. 1959;21:45. Nestor J, Esquena J, Solans C, Levecke B, Booten K, Tadros TF. Langmuir. 2005;21:4837. Nestor J, Esquena J, Solans C, Luckham PF, Levecke B, Tadros TF. J Colloid Interface Sci. 2007;311:430. Liang W, Bognolo G, Tadros TF. Langmuir. 1995;11:2899. Krieger IM, Dougherty TJ. Trans Soc Rheol. 1959;3:137. Krieger IM, Advances Colloid and Interface Sci. 1972;3:111. Barrett KEJ, editor. Dispersion polymerization in organic media. Chichester: John Wiley & Sons, Ltd; 1975. Barrett KEJ and Thomas HR. J Polym Sci Part A1. 1969;7:2627. Dawkins JV, Taylor G. Polymer. 1987;20:171. Antl I, Goodwin JW, Hill RD, Ottewill RH, Owen SM, Papworth S, Waters JA. Colloids Surf. 1986;1:67. Napper DH. Polymeric stabilisation of colloidal dispersions. London: Academic Press; 1983. Lok KP, Ober CK. Can J Chem. 1985;63:209. Paine AJ. J Polymer Sci Part A. 1990;28:2485.

9 Microemulsions 9.1 Introduction Microemulsions, which are better described as swollen micelles, are a special class of nanodispersions (transparent or translucent) which actually have little in common with emulsions. The term microemulsion was first introduced by Hoar and Schulman [1, 2] who discovered that by titration of a milky emulsion (stabilized by soap such as potassium oleate) with a medium chain alcohol such as pentanol or hexanol, a transparent or translucent system was produced. A schematic representation of the titration method adopted by Schulman and co-workers is given below, O/W emulsion → stabilized by soap

Add cosurfactant, → e.g. C5 H11 OH, C6 H13 OH

Transparent or translucent

The final transparent or translucent system is a W/O microemulsion. A convenient way to describe microemulsions is to compare them with micelles. The latter, which are thermodynamically stable, may consist of spherical units with a radius that is usually less than 5 nm. Two types of micelles may be considered: normal micelles with the hydrocarbon tails forming the core and the polar head groups in contact with the aqueous medium; and reverse micelles (formed in nonpolar media) with a water core containing the polar head groups and the hydrocarbon tails now in contact with the oil. The normal micelles can solubilize oil in the hydrocarbon core forming O/W microemulsions, whereas the reverse micelles can solubilize water forming a W/O microemulsion. A schematic representation of these systems is shown in Fig. 9.1. A rough guide to the dimensions of micelles, micellar solutions and macroemulsions is as follows: micelles, R < 5 nm (they scatter little light and are transparent); macroemulsions, R > 50 nm (opaque and milky); micellar solutions or microemulsions, 5–50 nm (transparent, 5–10 nm, translucent 10–50 nm).

Normal micelle

Inverse micelle

O/W microemulsion

W/O microemulsion

https://doi.org/10.1515/9783110541953-010

Fig. 9.1: Schematic representation of microemulsions.

320 | 9 Microemulsions

The classification of microemulsions based on size is not adequate. Whether a system is transparent or translucent depends not only on the size but also on the difference in refractive index between the oil and the water phases. A microemulsion with small size (in the region of 10 nm) may appear translucent if the difference in refractive index between the oil and the water is large. Relatively large sized microemulsion droplets (in the region of 50 nm) may appear transparent if the refractive index difference is very small. The best definition of microemulsions is based on the application of thermodynamics as is discussed below.

9.2 Thermodynamic definition of microemulsions A thermodynamic definition of microemulsions can be obtained from a consideration of the energy and entropy terms for formation of microemulsions, schematically represented in Fig. 9.2. The figure shows schematically the process of formation of a microemulsion from a bulk oil phase (for O/W microemulsion) or bulk water phase (for a W/O microemulsion).

A1

12

12

I

Formation

A2 II

Fig. 9.2: Schematic representation of microemulsion formation.

A1 is the surface area of the bulk oil phase and A2 is the total surface area of all the microemulsion droplets. γ12 is the O/W interfacial tension. The increase in surface area when going from state I to state II is ∆A(= A2 − A1 ) and the surface energy increase is equal to ∆Aγ12 . The increase in entropy when going from state I to state II is T∆Sconf (note that state II has higher entropy since a large number of droplets can arrange themselves in several ways, whereas state I with one oil drop has much lower entropy). According to the second law of thermodynamics, the free energy of formation of microemulsions ∆Gm is given by the following expression, ∆Gm = ∆Aγ12 − T∆Sconf .

(9.1)

With macroemulsions, ∆Aγ12 ≫ T∆Sconf and ∆Gm > 0. The system is non-spontaneous (it requires energy for the formation of emulsion drops) and it is thermodynamically unstable. With microemulsions, ∆Aγ12 ≤ T∆Sconf (this is due to the ultra-low interfacial tension accompanying microemulsion formation) and ∆Gm ≤ 0. The system is produced spontaneously and it is thermodynamically stable.

9.3 Mixed film and solubilization theories of microemulsions | 321

The above analysis shows the contrast between emulsions and microemulsion: With emulsions, an increase in mechanical energy and an increase in surfactant concentration usually results in the formation of smaller droplets which become kinetically more stable. With microemulsions, neither mechanical energy nor increasing surfactant concentration can result in their formation. The latter is based on a specific combination of surfactants and specific interaction with the oil and the water phases and the system is produced at optimum composition. Thus, microemulsions have nothing in common with macroemulsions and in many cases it is better to describe the system as “swollen micelles”. The best definition of microemulsions is as follows [3]: “system of water + oil + amphiphile that is a single optically isotropic and thermodynamically stable liquid solution”. Amphiphile refers to any molecule that consist of hydrophobic and hydrophilic portions, e.g. surfactants, alcohols, etc. The driving force for microemulsion formation is the low interfacial energy which is overcompensated by the negative entropy of dispersion term. The low (ultra-low) interfacial tension is produced in most cases by combining two molecules, referred to as the surfactant and cosurfactant (e.g. medium chain alcohol).

9.3 Mixed film and solubilization theories of microemulsions 9.3.1 Mixed film theories [4] The film (which may consist of surfactant and cosurfactant molecules) is considered as a liquid “two-dimensional” third phase in equilibrium with both oil and water. Such a monolayer could be a duplex film, i.e. giving different properties on the water side and oil side. The initial “flat” duplex film (see Fig. 9.3) has different tensions at the oil and water sides. This is due to the different packing of the hydrophobic and hydrophilic groups (these groups have different sizes and cross-sectional areas). It is convenient to define a two dimensional surface pressure π, π = γ0 − γ.

(9.2)

γ0 is the interfacial tension of the clean interface, whereas γ is the interfacial tension with adsorbed surfactant. One can define two values for π at the oil and water phases, πo and πw , which for a 󸀠 . As a result of the difference in tensions, the film will flat film are not equal, i.e. πo󸀠 ≠ πw 󸀠 󸀠 bend until πo = πw . If πo > πw , the area at the oil side has to expand (resulting in re󸀠 > π󸀠 , duction of πo󸀠 ) until πo = πw . In this case a W/O microemulsion is produced. If πw o the area at the water side expands until πw = πo . In this case an O/W microemulsion is produced. A schematic representation of film bending for the production of W/O or W/O microemulsions is illustrated in Fig. 9.3.

322 | 9 Microemulsions Oil

πo’ > πW’

Water W/O microemulsion

Area at oil side expands πW’ > πo’

Oil

Area at water side expands Water O/W microemulsion Fig. 9.3: Schematic representation of film bending.

According to the duplex film theory, the interfacial tension γT is given by the following expression [5], γT = γ(O/W) − π, (9.3) where (γo/w )a is the interfacial tension that is reduced by the presence of the alcohol. The value of (γo/w )a is significantly lower than γo/w in the absence of the alcohol. For example, for hydrocarbon/water, γo/w is reduced from 50 to 15–20 mN m−1 on the additional of a significant amount of a medium chain alcohol like pentanol or hexanol. Contributions to π are considered to be due to crowding of the surfactant and cosurfactant molecules and penetration of the oil phase into the hydrocarbon chains of the interface. According to equation (9.3), if π > (γo/w )a , γT becomes negative and this leads to expansion of the interface until γT reaches a small positive value. Since (γo/w )a is of the order of 15–20 mN m−1 , surface pressures of this order are required for γT to approach a value of zero. The above duplex film theory can explain the nature of the microemulsion: The surface pressures at the oil and water sides of the interface depend on the interactions of the hydrophobic and hydrophilic potions of the surfactant molecule at both sides respectively. If the hydrophobic groups are bulky in nature relative to the hydrophilic groups, then for a flat film such hydrophobic groups tend to crowd, forming a higher surface pressure at the oil side of the interface; this results in bending and expansion at the oil side forming a W/O microemulsion. An example for a surfactant with bulky hydrophobic groups is Aerosol OT (dioctyl sulphosuccinate). If the hydrophilic groups are bulky, such as is the case with ethoxylated surfactants containing more than 5 ethylene oxide units, crowding occurs at the water side of the interface. This produces an O/W microemulsion.

9.3 Mixed film and solubilization theories of microemulsions

| 323

9.3.2 Solubilization theories These concepts were introduced by Shinoda and co-workers [6] who considered microemulsions to be swollen micelles that are directly related to the phase diagram of their components. Consider the phase diagram of a three-component system of water, ionic surfactant and medium chain alcohol as described in Fig. 9.4. At the water corner and at low alcohol concentration, normal micelles (L1 ) are formed since in this case there are more surfactant than alcohol molecules. At the alcohol (cosurfactant) corner, inverse micelles (L2 ) are formed, since in this region there are more alcohol than surfactant molecules. Alcohol L2 W/O

Liquid crystal

L2 W/O

Water

Surfactant

Fig. 9.4: Schematic representation of a threecomponent phase diagram.

These L1 and L2 are not in equilibrium but are separated by a liquid crystalline region (lamellar structure with equal number of surfactant and alcohol molecules). The L1 region may be considered as an O/W microemulsion, whereas the L2 may be considered as a W/O microemulsion. Addition of a small amount of oil miscible with the cosurfactant, but not with the surfactant and water, changes the phase diagram only slightly. The oil may be simply solubilized in the hydrocarbon core of the micelles. Addition of more oil leads to fundamental changes of the phase diagram as illustrated in Fig. 9.5 where 50 : 50 of W : O are used. To simplify the phase diagram, the 50 W/50 O are presented on one corner of the phase diagram. Near the cosurfactant (co) corner the changes are small compared to the three phase diagram (Fig. 9.5). The O/W microemulsion near the water-surfactant (sa) axis is not in equilibrium with the lamellar phase, but with a non-colloidal oil + cosurfactant phase. If co is added to such a two-phase equilibrium at fairly high surfactant concentration, all oil is taken up and a one-phase microemulsion appears. Addition of co at low sa concentration may lead to separation of an excess aqueous phase before all oil is taken up in the microemulsion. A three phase system is formed, containing a microemulsion that cannot be clearly identified as W/O or W/O and that presumably is similar to the lamellar phase swollen with oil or to a more irregular intertwining

324 | 9 Microemulsions Co W/O W

O M W

2 W/O 3

M 1 O/W 2

oil 2 O/W

500 : 50 W

SA

Fig. 9.5: Schematic representation of the pseudoternary phase diagram of oil/water/surfactant/ cosurfactant.

of aqueous and oily regions (bicontinuous or middle phase microemulsion). The interfacial tensions between the three phases are very low (0.1–10−4 mN m−1 ). Further addition of co to the three phase system makes the oil phase disappear and leaves a W/O microemulsion in equilibrium with a dilute aqueous sa solution. In the large one phase region continuous transitions from O/W to middle phase to W/O microemulsions are found. Solubilization can also be illustrated by considering the phase diagrams of nonionic surfactants containing poly(ethylene oxide) (PEO) head groups. Such surfactants do not generally need a cosurfactant for microemulsion formation. A schematic representation of oil and water solubilization by nonionic surfactants is given in Fig. 9.6. At low temperatures, the ethoxylated surfactant is soluble in water and at a given concentration is capable of solubilizing a given amount of oil. The oil solubilization increases rapidly with increasing temperature near the cloud point of the surfactant. This is illustrated in Fig. 9.6 which shows the solubilization and cloud point curves of the surfactant. Between these two curves, an isotropic region of O/W solubilized system exists. At any given temperature, any increase in the oil weight fraction above

Oil 쎵 water 쎵 surfactant u r ve point c O/W r ve opic r t o n cu o Is i t liza ubi Sol

W/O solobilized 쎵 water Solub

T

T

Cloud

W/O solobilized 쎵 water (a)

wt fraction of oil

ilizat ion c u r ve Isotr opic W/O Haz e po int c u r ve

Oil 쎵 water 쎵 surfactant (b)

wt fraction of water

Fig. 9.6: Schematic representation of solubilization: (a) oil solubilized in a nonionic surfactant solution; (b) water solubilized in an oil solution of a nonionic surfactant.

9.4 Thermodynamic theory of microemulsion formation

| 325

the solubilization limit results in oil separation (oil solubilized + oil). At any given surfactant concentration, any increase in temperature above the cloud point results in separation into oil, water and surfactant. If one starts from the oil phase with dissolved surfactant and adds water, solubilization of the latter takes place and solubilization increases with reducing the temperature to near the haze point. Between the solubilization and haze point curves, an isotropic region of W/O solubilized system exists. At any given temperature, any increase in water weight fraction above the solubilization limit results in water separation (W/O solubilized + water). At any given surfactant concentration, any decrease in temperature below the haze point results in separation into water, oil and surfactant. With nonionic surfactants, both types of microemulsions can be formed depending on the conditions. With such systems temperature is the most crucial factor since the solubility of surfactant in water or oil depends on temperature. Microemulsions prepared using nonionic surfactants have a limited temperature range.

9.4 Thermodynamic theory of microemulsion formation The spontaneous formation of a microemulsion with decreasing free energy can only be expected if the interfacial tension is so low that the remaining free energy of the interface is over compensated for by the entropy of dispersion of the droplets in the medium [7, 8]. This concept forms the basis of the thermodynamic theory proposed by Ruckenstein and Chi, and Overbeek [7, 8].

9.4.1 Reason for combining two surfactants Single surfactants do lower the interfacial tension γ, but in most cases the critical micelle concentration (cmc) is reached before γ is close to zero. Addition of a second surfactant of a completely different nature (i.e. predominantly oil soluble such as an alcohol) then lowers γ further and very small, even transiently negative values may be reached [9]. This is illustrated in Fig. 9.7 which shows the effect of addition of the cosurfactant on the γ–log Csa curve. It can be seen that addition of cosurfactant shifts the whole curve to low γ values and the cmc is shifted to lower values. For a multicomponent system i, each with an adsorption Γ i (mol m−2 , referred to as the surface excess), the reduction in γ, i.e. dγ, is given by the following expression, dγ = − ∑ Γ i dμ i = − ∑ Γ i RT d ln C i ,

(9.4)

where μ i is the chemical potential of component i, R is the gas constant, T is the absolute temperature and C i is the concentration (mol dm−3 ) of each surfactant component.

326 | 9 Microemulsions

One surfactant

g

Add co-surfactant

c.m.c 0 Log Csa

Fig. 9.7: γ–log Csa curves for surfactant + cosurfactant.

The reason for the lowering of γ when using two surfactant molecules can be understood from considering the Gibbs adsorption equation for multicomponent systems [9]. For two components sa (surfactant) and co (cosurfactant), equation (9.4) becomes, dγ = −Γsa RT d ln Csa − Γco RTd ln Cco . (9.5) Integrating equation (9.5) gives, Csa

Cco

γ = γ0 − ∫ Γsa RT d ln Csa − ∫ Γco RT d ln Cco , 0

(9.6)

0

which clearly shows that γ0 is lowered by two terms, from both surfactant and cosurfactant. The two surfactant molecules should adsorb simultaneously and they should not interact with each other, otherwise they lower their respective activities. Thus, the surfactant and cosurfactant molecules should vary in nature, one predominantly water soluble (such as an anionic surfactant) and the other predominantly oil soluble (such as a medium chain alcohol). In some cases a single surfactant may be sufficient for lowering γ far enough for microemulsion formation to become possible, e.g. Aerosol OT (sodium diethyl hexyl sulphosuccinate) and many nonionic surfactants.

9.4.2 Free energy of formation of a microemulsion A simple model was used by Overbeek [9] to calculate the free energy of formation of a model W/O microemulsion: The droplets were assumed to be of equal size. The droplets are large enough to consider the adsorbed surfactant layer to have constant composition. The microemulsion is prepared in a number of steps and for each step one calculates the Helmholtz free energy F (this was chosen since the pressure inside the drop is higher by the Laplace pressure 2γ/a (where a is the droplet radius) than the pressure in the medium.

9.4 Thermodynamic theory of microemulsion formation

| 327

A summary of the four steps involved in the preparation of a model W/O microemulsion is given below: (i) Prepare the oil phase in its final concentration, F1 = ∑ n󸀠i μ󸀠i − p1 V1 ,

(9.7)

where n󸀠i and μ󸀠i are the amount and chemical potential of oil and cosurfactant in the continuous phase, without droplets being mixed in. p1 is the atmospheric pressure and V1 is the volume of the oil phase. (ii) Prepare the aqueous phase in its final concentration, F2 = ∑ n󸀠i μ󸀠i − p1 V2 ,

(9.8)

where i are now water, surfactant and salt and V2 is the volume of the water phase. (iii) Form the water phase into droplets closely packed in the oil phase (i.e. with a packing fraction ϕ = 0.74) and add all the adsorbed material, γ F3 = γA + Γsa A[μ󸀠sa + (2 )V̄ sa ] + Γ i Aμ i , a

(9.9)

where i refers to cosurfactant and oil. The oil must be negatively adsorbed in order to keep the volume of the adsorption layer at zero (in accordance with the Gibbs dividing surface). It is assumed in equation (9.9) that the Gibbs plane (the surface of tension in this case) lies close to the surface (where Γwater = 0). (iv) Allow the close packed emulsion to expand to its final concentration (volume fraction ϕ), F4 = ndr RTf(ϕ). (9.10) ndr is the amount of drops (in mol) and f(ϕ) is a function of ϕ. f(ϕ) may be simply written as f(ϕ) = ln ϕ − ln 0.74. (9.11) More accurately, f(ϕ) may be calculated using a hard-sphere model [10], f(ϕ) = ln ϕ + ϕ[

4 − 3ϕ ] − 19.25. (1 − ϕ)2

(9.12)

Combining equations (9.7)–(9.12) gives the Helmholtz free energy of the complete emulsion. The free energy is minimized with respect to a change in the interfacial area A. This involves transfer of adsorbed components to or from the interface, thereby changing the bulk concentration and thus γ; the result is, γ = −const ×

1 × g(ϕ), a2

where g(ϕ) is similar but not identical to f(ϕ).

(9.13)

328 | 9 Microemulsions

The droplet radius a can be calculated from a knowledge of the total interfacial area A, nsa A= − nsa Nav (area/molecule). (9.14) Γsa The area per molecule of an anionic surfactant such as sodium dodecyl sulphate (SDS) varies between 0.7 to 1.1 nm2 , depending on the concentration of cosurfactant (pentanol) and salt concentration. The area per pentanol molecule is about 0.3 nm2 . This means that the average area per surfactant molecule is about 0.9 nm2 . The radius of the droplet can be calculated from the ratio of the volume of the drop to its area, 3 × (4/3)πa3 3V a= = , (9.15) A 4πa2 where V is the total volume of the droplets and A is the total interfacial area. The radius a of the microemulsion droplet has to fit both equations (9.13) and (9.15); γ is the most easily varied quantity in these equations. The correct value of γ is obtained by adaptation of Csa . According to equation (9.13), any value of a is allowed in the accessible range of γ. If γ is close to zero very large radii can be obtained, i.e. very large water/sa ratios are allowed. However, the phase diagram shows that at such high ratios demixing occurs. This analysis shows the inadequacy of the above simple model and it is necessary to add an explicit influence of the radius of curvature on the interfacial tension. The curvature effect is manifested in the packing of the tails and head groups at the O/W interface. With W/O microemulsions the packing of the short chains and the packing of the head groups will favour W/O curvature with a ratio of 3 or more for co/sa. With O/W microemulsions a ratio of sa/co of 2 or less is required. Thus O/W microemulsions need less cosurfactant than W/O microemulsions.

9.4.3 Factors determining W/O versus O/W microemulsions The duplex film theory predicts that the nature of the microemulsion formed depends on the relative packing of the hydrophobic and hydrophilic portions of the surfactant molecule, which determine the bending of the interface. For example, a surfactant molecule such as Aerosol OT, O O C2H5 \\ \ \ | C CH2–CH–CH2 –CH2–CH2–CH3 \ CH2 \ Na+–O3 –CH \ C CH2–CH–CH2 –CH2–CH2–CH3 \\ \ \ | O O C2H5

9.4 Thermodynamic theory of microemulsion formation

| 329

favours the formation of a W/O microemulsion, without the need of a cosurfactant. As a result of the presence of a stumpy head group and large volume to length (V/l) ratio of the nonpolar group, the interface tends to bend with the head groups facing onwards, thus forming a W/O microemulsion. The molecule has V/l > 0.7 which is considered necessary for formation of a W/O microemulsion. For ionic surfactants such as SDS for which V/l < 0.7, microemulsion formation needs the presence of a cosurfactant (the latter has the effect of increasing V without changing l). The importance of geometric packing was considered in detail by Mitchell and Ninham [11] who introduced the concept of the packing ratio P, P=

V , lc a0

(9.16)

where a0 is the head group area and lc is the maximum chain length. P gives a measure of the hydrophilic-lipophilic balance. For values of P < 1 (usually P ≈ 1/3), normal or convex aggregates are produced (normal micelles). For values of P > 1, inverse micelles are produced. P is influenced by many factors: hydrophilicity of the head group, ionic strength and pH of the medium and temperature. P also explains the nature of the microemulsion produced using nonionic surfactants of the ethoxylate type: P increases with increasing temperature (as a result of the dehydration of the PEO chain). A critical temperature (PIT) is reached at which P reaches 1 and above this temperature inversion occurs to a W/O system. The influence of the surfactant structure on the nature of the microemulsion can also be predicted from thermodynamic theory. The most stable microemulsion would be that in which the phase with the smaller volume fraction forms the droplets (the osmotic pressure increases with increasing ϕ). For a W/O microemulsion prepared using an ionic surfactant such as Aerosol OT, the effective volume (hard-sphere volume) is only slightly larger than the water core volume, since the hydrocarbon tails may penetrate to a certain extent when two droplets come together. For an O/W microemulsion, the double layers may expand to a considerable extent, depending on the electrolyte concentration (the double layer thickness is of the order of 100 nm in 10−5 mol dm−3 1 : 1 electrolyte and 10 nm in 10−3 mol dm−3 electrolyte). Thus the effective volume of O/W microemulsion droplets can be significantly higher than the core oil droplet volume and this explains the difficulty of preparation of O/W microemulsions at high ϕ values when using ionic surfactants. A schematic representation of the effective volume for W/O and O/W microemulsions is shown in Fig. 9.8.

330 | 9 Microemulsions

Reff = R+lc Reff ~ R High φ can be reached with W/O

–+ – +– –+–+ – –+ +– –+ + –+ ––+–+ +– –+ + – +– ––+ +– –+ –+ –+ +– +– +– +– +– + –+ + –+ Extended + +– + ––+ double layer ++ – + –+ Reff >> R Limited φ reached with O/W

Fig. 9.8: Schematic representation of W/O and O/W microemulsion droplets.

9.5 Characterization of microemulsions using scattering techniques Scattering techniques provide the most obvious methods for obtaining information on the size, shape and structure of microemulsions. The scattering of radiation, e.g. light, neutrons, X-rays, etc. by particles has been successfully applied for the investigation of many systems such as polymer solutions, micelles and colloidal particles. In all these methods, measurements can be made at sufficiently low concentration to avoid complications arising from particle-particle interactions. The results obtained are extrapolated to infinite dilution to obtain the desired property such as the molecular weight and radius of gyration of a polymer coil, the size and shape of micelles, etc. Unfortunately, this dilution method cannot be applied for microemulsions, which depend on a specific composition of oil, water and surfactants. The microemulsions cannot be diluted by the continuous phase since this dilution results in breakdown of the microemulsion. Thus, when applying the scattering techniques to microemulsions measurements have to be made at finite concentrations and the results obtained have to be analysed using theoretical treatments to take into account the droplet-droplet interactions. Three scattering methods will be discussed below: Time-average (static) light scattering, dynamic (quasi-elastic) light scattering referred to as photon correlation spectroscopy and neutron scattering.

9.5.1 Time-average (static) light scattering The intensity of scattered light I(Q) is measured as a function of scattering vector Q, Q=(

4πn θ ) sin( ), λ 2

(9.17)

9.5 Characterization of microemulsions using scattering techniques |

331

where n is the refractive index of the medium, λ is the wavelength of light and θ is the angle at which the scattered light is measured. For a fairly dilute system, I(Q) is proportional to the number of particles N, the square of the individual scattering units V p and some property of the system (material constant) such as its refractive index, I(Q) = [(material constant)(instrument constant)]NVp2 .

(9.18)

The instrument constant depends on the geometry of the apparatus (the light path length and the scattering cell constant). For more concentrated systems, I(Q) also depends on the interference effects arising from particle-particle interaction, I(Q) = [(instrument constant)(material constant)]NVp2 P(Q)S(Q),

(9.19)

where P(Q) is the particle form factor which allows the scattering from a single particle of known size and shape to be predicted as a function of Q. For a spherical particle of radius R, (3 sin QR − QR cos QR) 2 P(Q) = [ (9.20) ] . (QR)3 S(Q) is the so called “structure factor” which takes into account the particle–particle interaction. S(Q) is related to the radial distribution function g(r) (which gives the number of particles in shells surrounding a central particle) [12], ∞

S(Q) = 1 −

4πN ∫ [g(r) − 1]r sin QR dr. Q

(9.21)

0

For a hard-sphere dispersion with radius RHS (which is equal to R + t, where t is the thickness of the adsorbed layer), S(Q) =

1 , [1 − NC(2QRHS )]

(9.22)

where C is a constant. One usually measures I(Q) at various scattering angles θ and then plots the intensity at some chosen angle (usually 90°), i90 as a function of the volume fraction ϕ of the dispersion. Alternatively, the results may be expressed in terms of the Rayleigh ratio R90 , i90 (9.23) R90 = ( )r2s . I0 I0 is the intensity of the incident beam and rs is the distance from the detector. R90 = K0 MCP(90)S(90).

(9.24)

K0 is an optical constant (related to the refractive index difference between the particles and the medium). M is the molecular mass of scattering units with weight fraction C.

332 | 9 Microemulsions For small particles (as is the case with microemulsions), P(90) ≈ 1 and, M=

4 3 πR NA , 3 c

(9.25)

where NA is the Avogadro constant. C = ϕc ρc ,

(9.26)

where ϕc is the volume fraction of the particle core and ρc is their density. Equation (9.24) can be written in the simple form, R90 = K1 ϕc R3c S(90),

(9.27)

where K1 = K0 (4/3)NA ρ2c . Equation (9.27) shows that to calculate Rc from R90 one needs to know S(90). The latter can be calculated using equations (9.21) and (9.22). The above calculations were obtained using a W/O microemulsion of water/ xylene/sodium dodecyl benzene sulphonate (NaDBS)/hexanol [11]. The microemulsion region was established using the quaternary phase diagram. W/O microemulsions were produced at various water volume fractions using increasing amounts of NaDBS: 5, 10.9, 15 and 20 %. The results for the variation of R90 with the volume fraction of the water core droplets at various NaDBS concentrations are shown in Fig. 9.9. With the exception of the 5 % NaDBS results, all the others showed an initial increase in R90 with increasing ϕ, reaching a maximum at a given ϕ, after which R90 decreases with any further increase in ϕ. These results were used to calculate R as a function of ϕ using the hard-sphere model discussed above (equation (9.27)). This is also shown in Fig. 9.9. It can be seen that with increasing ϕ, at constant surfactant concentration, R increases (the ratio of surfactant to water decreases with increasing ϕ). At any volume fraction of water, increasing surfactant concentration results in decreasing microemulsion droplet size (the ratio of surfactant to water increases). 10.9 % NaDBS 5% NaDBS 15% NaDBS

60 10.9 % NaDBS

5% NaDBS

40

40

20% NaDBS

R/Å

R90 쎹 104 cm–1

60

15% NaDBS 20

20

S (Q) calculated using hard sphere model

20% NaDBS 0

0 0.2

0.4 H 2O

0.6

0.2

0.4

0.6

H 2O

Fig. 9.9: Variation of R90 and R with the volume fraction of water for a W/O microemulsion based on xylene–water–NaDBS–hexanol.

9.5 Characterization of microemulsions using scattering techniques |

333

9.5.2 Calculating droplet size from interfacial area If one assumes that all surfactant and cosurfactant molecules are adsorbed at the interface, it is possible to calculate the total interfacial area of the microemulsion from a knowledge of the area occupied by the surfactant and cosurfactant molecules. total interfacial area = total number of surfactant molecules × area per surfactant molecule As + total number of cosurfactant molecules × area per cosurfactant molecule Aco . The total interfacial area A per kg of microemulsion is given by the expression, A=

(ns NA As + nco NA Aco ) . ϕ

(9.28)

ns and nco are the number of moles of surfactant and cosurfactant. A is related to the droplet radius R (assuming all the droplets are of the same size) by, 3 A= . (9.29) Rρ Using reasonable values for As and Aco (30A2 for NaDBS and 20A2 for hexanol) R was calculated and the results were compared with those obtained using light scattering results. Two conditions were considered: (a) all hexanol molecules were adsorbed 1A1; (b) part of the hexanol was adsorbed to give a molar ratio of hexanol to NaDBS of 2 : 1 (1A2). Good agreement is obtained between the light scattering data and R calculated from interfacial area, particularly for 1A2.

9.5.3 Dynamic light scattering (photon correlation spectroscopy, PCS) In this technique one measures the intensity fluctuation of scattered light by the droplets as they undergo Brownian motion [13]. When a light beam passes through a colloidal dispersion, an oscillating dipole movement is induced in the particles, thereby radiating the light. Due to the random position of the particles, the intensity of scattered light, at any instant, appears as random diffraction (“speckle” pattern). As the particles undergo Brownian motion, the random configuration of the pattern will fluctuate, such that the time taken for an intensity maximum to become a minimum (the coherence time), corresponds approximately to the time required for a particle to move one wavelength λ. Using a photomultiplier of active area about the diffraction

334 | 9 Microemulsions

maximum (i.e. one coherent area) this intensity fluctuation can be measured. The analogue output is digitized (using a digital correlator) that measures the photocount (or intensity) correlation function of scattered light. The photocount correlation function G(2) (τ) is given by, g (2) = B[1 + γ2 g(1) (τ)]2 ,

(9.30)

where τ is the correlation delay time. The correlator compares g(2) (τ) for many values of τ. B is the background value to which g (2) (τ) decays at long delay times. g (1) (τ) is the normalized correlation function of the scattered electric field and γ is a constant (≈ 1). For monodispersed non-interacting particles, g (1) (τ) = exp(−Γγ).

(9.31)

Γ is the decay rate or inverse coherence time that is related to the translational diffusion coefficient D, (9.32) Γ = DK 2 , where K is the scattering vector, K=(

θ 4πn ) sin( ). λ0 2

(9.33)

The particle radius R can be calculated from D using the Stokes–Einstein equation, D=

kT , 6πη0 R

(9.34)

where η0 is the viscosity of the medium. The above analysis only applies for very dilute dispersions. With microemulsions, which are concentrated dispersions, corrections are needed to take into account the interdroplet interaction. This is reflected in plots of ln g (1) (τ) versus τ which become nonlinear, implying that the observed correlation functions are not single exponentials. As with time-average light scattering, one needs to introduce a structure factor in calculating the average diffusion coefficient. For comparative purposes, one calculates the collective diffusion coefficient D, which can be related to its value at infinite dilution D0 by [14], D = D0 (1 + αϕ), (9.35) where α is a constant that is equal to 1.5 for hard spheres with repulsive interaction.

9.5 Characterization of microemulsions using scattering techniques |

335

9.5.4 Neutron scattering Neutron scattering offers a valuable technique for determining the dimensions and structure of microemulsion droplets. The scattering intensity I(Q) is given by, I(Q) = (instrument constant)(ρ − ρ0 )NVp2 P(Q)S(Q),

(9.36)

where ρ is the mean scattering length density of the particles and ρ0 is the corresponding value for the solvent. One of the main advantages of neutron scattering over light scattering is the Q range at which one operates: With light scattering, the range of Q is small (≈ 0.0005– 0.0015 A−1 ) while for small angle neutron scattering the Q range is large (0.02– 0.18 A−1 ). In addition, neutron scattering can give information on the structure of the droplets. As an illustration, Fig. 9.10 shows plots of I(Q) versus Q for W/O microemulsions (xylene/water/NaDBS/hexanol) [15]. The Q values at the maximum can be used to calculate the lattice spacing using Bragg’s equation. Alternatively, one can use a hard-sphere model to calculate S(Q) and then fit the data of I(Q) versus Q to obtain the droplet radius R. φc = 0.147

φc = 0.198

φc = 0.301

φc = 0.380

φc = 0.436

φc = 0.533

4

I (Q)

12 8 4 12

φc

8 4 0.02

0.08

0.02 Q/Å

–1

0.08

Fig. 9.10: I(Q) versus Q for W/O microemulsions at various water volume fractions.

9.5.5 Contrast matching for determining the structure of microemulsions By changing the isotopic composition of the components (e.g. using deuterated oil and H2 O–D2 O) one can match the scattering length density of the various components: By matching the scattering length density of the water core with that of the oil, one can investigate the scattering from the surfactant “shell”. By matching the scattering length density of the surfactant “shell” and the oil, one can investigate the scattering from the water core.

336 | 9 Microemulsions

9.6 Characterization of microemulsions using conductivity Conductivity measurements may provide valuable information on the structural behaviour of microemulsions. In the early applications of conductivity measurements, the technique was used to determine the nature of the continuous phase. O/W microemulsions should give fairly high conductivity (that is determined by that of the continuous aqueous phase) whereas W/O microemulsions should give fairly low conductivity (that is determined by that of the continuous oil phase). As an illustration, Fig. 9.11 shows the change in electrical resistance (reciprocal of conductivity) with the ratio of water to oil (Vw /Vo ) for a microemulsion system prepared using the inversion method [2]. Fig. 9.11 indicates the change in optical clarity and birefringence with the ratio of water to oil.

106

Clear

Turbid

Clear

Birefringent K-oleate = 0.20g/ml oil 105

Hexanol

= 0.40

Ω

Oil

104

103

102 0

0.2

0.4

0.6

Fig. 9.11: Electrical resistance versus Vw /Vo .

0.8 VW/VO

1.0

1.2

1.4

1.6

9.6 Characterization of microemulsions using conductivity

Water/toluene/k-oleate/butanol

5

337

|

Water/hexadecane/k-oleate/hexanol

Rapid increase above critical φW

κ/Ω–1m–1

κ ohm–1m–1 × 104

10

φW

κ remains low and show maxima and minima

5

φW’’

φ W’ 0.5

0.1

Percolating

0.2 0.3 Non-percolating

0.4

Fig. 9.12: Conductivity versus water volume fraction for two W/O microemulsion systems.

At low Vw /Vo , a clear W/O microemulsion is produced with a high resistance (oil continuous). As Vw /Vo increases, the resistance decreases, and in the turbid region, hexanol and lamellar micelles are produced. Above a critical ratio, inversion occurs and the resistance decreases producing O/W microemulsion. Conductivity measurements were also used to study the structure of the microemulsion, which is influenced by the nature of the cosurfactant. This is illustrated in Fig. 9.12 for two systems based on water/toluene/potassium oleate/butanol and water/hexadecane/potassium oleate/hexanol [17]. The difference between the two systems is in the nature of the cosurfactant, namely butanol (C4 alcohol) and hexanol (C6 alcohol). The first system based on butanol shows a rapid increase in κ above a critical water volume fraction value, whereas the second system based on hexanol shows much lower conductivity values with a maximum and minimum at two water volume fractions values ϕ󸀠w and ϕ󸀠󸀠 w. In the first case (when using butanol), the κ–ϕw curve can be analysed using the percolation theory of conductivity [18]. In this model, the effective conductivity is practically zero as long as the volume fraction of the conductor (water) is below a critical p value ϕw (the percolation threshold). Beyond this value, κ suddenly takes a nonzero value and it increases rapidly with a further increase in ϕw . In the above case (percolating microemulsions), the following equations were theoretically derived. p

κ = (ϕw − ϕw )8/5 κ=

p (ϕw

− ϕw )

−0.7

when ϕw > ϕw ,

p

(9.37)

p ϕw .

(9.38)

when ϕw
400 nm), large unilamellar vesicles (LUVs > 100 nm) and small unilamellar vesicles (SUVs < 100 nm). Other types reported are the Giant Vesicles (GV), which are unilamellar vesicles of diameter between 1–5 µm and large oligolamellar vesicles (LOV) where a few vesicles are entrapped in the LUV or GV.

10.3 Driving force for formation of vesicles

| 345

10.3 Driving force for formation of vesicles The driving force for formation of vesicles has been described in detail by Israelachvili et al. [5–7]. From equilibrium thermodynamics, small aggregates, or even monomers, are entropically favoured over larger ones. This entropic force explains the aggregation of single-chain amphiphiles into small spherical micelles instead of into bilayers or cylinders, as the aggregation number of the latter aggregates is much higher. Israelachvili et al. [5–7] attempted to describe the thermodynamic drive for vesicle formation by biological lipids. From equilibrium thermodynamics of self-assembly, the chemical potential of all molecules in a system of aggregated structures such as micelles or bilayers will be the same, μ0N +

kT XN ln( ) = const; N N

N = 1, 2, 3, . . . ,

(10.1)

where μ0N is the free energy per molecule in the aggregate, X N is the mole fraction of molecules incorporated into the aggregate, with an aggregation number N, k is the Boltzmann constant and T is the absolute temperature. For monomers in solution with N = 1, μ0N +

kT XN ln( ) = μ01 + kT ln X1 . N N

(10.2)

Equation (10.1) can be written as, X N = N(

N(μ0M − μ0N X M N/M exp( ) ), M kT

(10.3)

where M is any arbitrary state of reference of aggregation number M. The following assumptions are made to obtain the free energy per molecule: (i) the hydrocarbon interior of the aggregate is considered to be in a fluid-like state; (ii) geometric consideration and packing constrains in term of aggregate formation are excluded; (iii) strong long-range forces (van der Waals and electrostatic) are neglected. By considering the “opposing forces” approach of Tanford [8], the contributions to the chemical potential, μ0N , can be estimated. A balance exists between the attractive forces mainly of hydrophobic (and interfacial tension) nature and the repulsive forces due to steric repulsion (between the hydrated head group and alkyl chains), electrostatic and other forces [9]. The free energy per molecule is thus, μ0N = γa +

C . a

(10.4)

The attractive contribution (the hydrophobic free energy contribution) to μ0N is γa, where γ is the interfacial free energy per unit area and a is the molecular area measured at the hydrocarbon/water interface. C/a is the repulsive contribution, where C is

346 | 10 Liposomes and vesicles

a constant term used to incorporate the charge per head group, e, and includes terms such as the dielectric constant at the head group region, ε, and curvature corrections. This fine balance yields the optimum surface area, a0 , for the polar head groups of the amphiphile molecules at the water interface, at which the total interaction free energy per molecule is a minimum, C =0 a ∂μ0N C =γ− 2 =0 ∂a a C 1/2 a = a0 = ( ) γ

μ0N (min) = γa +

(10.5) (10.6) (10.7)

Using the above equations, the general form relating the free energy per molecule μ0N with a0 can be expressed as, μ0N = γ(a +

a20 γ ) = 2a0 γ + (a − a0 )2 . a a

(10.8)

Equation (10.8) shows that: (i) μ0N has a parabolic (elastic) variation about the minimum energy; (ii) amphiphilic molecules, including phospholipids, can pack in a variety of structures in which their surface areas will remain equal or close to a0 . Both single-chain and double-chain amphiphiles have very much the same optimum surface area per head group (a0 ≈ 0.5–0.7 nm2 ), i.e. a0 is not dependent on the nature of the hydrophobe. Thus, by considering the balance between entropic and energetic contributions to the double-chain phospholipid molecule one arrives at the conclusion that the aggregation number must be as low as possible and a0 for each polar group is of the order of 0.5–0.7 nm2 (almost the same as that for a single-chain amphiphile). For phospholipid molecules containing two hydrocarbon chains of 16–18 carbon atoms per chain, the volume of the hydrocarbon part of the molecule is double the volume of a single-chain molecule, while the optimum surface area for its head group is of the same order as that of a single-chain surfactant (a0 ≈ 0.5–0.7 nm2 ). Thus, the only way for this double-chain surfactant is to form aggregates of the bilayer sheet or the close bilayer vesicle type. This will be further explained using the critical packing parameter concept (CPP) described by Israelachvili et al. [5–7]. The CPP is a geometric expression given by the ratio of the cross-sectional area of the hydrocarbon tail(s) a to that of the head group a0 . a is equal to the volume of the hydrocarbon chain(s) v divided by the critical chain length lc of the hydrocarbon tail. Thus the CPP is given by [10], v . (10.9) CPP = a0 lc Regardless of shape, any aggregated structure should satisfy the following criterion: no point within the structure can be farther from the hydrocarbon-water surface

10.3 Driving force for formation of vesicles | 347

than lc , which is roughly equal to, but less than the fully extended length l of the alkyl chain. For a spherical micelle, the radius r = lc and from simple geometry CPP = v/a0 lc ≤ 1/3. Once v/a0 lc > 1/3, spherical micelles cannot be formed and when 1/2 ≥ CPP > 1/3 cylindrical micelles are produced. When the CPP > 1/2 but < 1, vesicles are produced. These vesicles will grow until CPP ≈ 1 when planer bilayers will start forming. A schematic representation of the CPP concept is given in Tab. 10.1. According to Israelachvili et al. [5–7], the bilayer sheet lipid structure is energetically unfavourable to the spherical vesicle, because of the lower aggregation number of the spherical structure. Without the introduction of packing constrains (described above), the vesicles should shrink to such a small size that they would actually form micelles. For double-chain amphiphiles three considerations must be considered: (i) an optimum a0 (almost the same as that for single-chain surfactants) must be achieved by considering the various opposing forces; (ii) structures with minimum aggregation number N must be formed; (iii) aggregates into bilayers must be the favourite structure. A schematic picture of the formation of bilayer vesicle and tubule structures was introduced by Israelachvili and Mitchell [10] and is shown in Fig. 10.3. Bilayer Tubule

Water Lipid influx Vesicle Growth

Fusion

Fig. 10.3: Bilayer vesicle and tubule formation [10].

Israelachvili et al. [5–7] believe that steps A and B are energetically favourable. They considered step C to be governed by packing constraints and thermodynamics in terms of the least aggregation number. They concluded that the spherical vesicle is an equilibrium state of the aggregate in water and it is certainly more favoured over extended bilayers. The main drawback of applications of liposomes in cosmetic formulations is their metastability. On storage, the liposomes tend to aggregate and fuse to form larger polydisperse systems and finally the system reverses into a phospholipid lamellar phase in water. This process takes place relatively slowly because of the slow exchange between the lipids in the vesicle and the monomers in the surrounding medium. Therefore, it is essential to investigate both the chemical and physical stability of the liposomes. Examining the process of aggregation can be achieved by measuring their size as a function of time. Maintenance of the vesicle structure can be assessed using freeze fracture and electron microscopy.

348 | 10 Liposomes and vesicles

Tab. 10.1: CPP concept and various shapes of aggregates. Lipid

Critical packing parameter v/a0 lc

Critical packing shape

Structures formed

Single-chained lipids (surfactants) with large head-group areas: – SDS in low salt

< 1/3

Cone

Spherical micelles do

v

lc

Single-chained lipids with small head-group areas: – SDS and CTAB in high salt – nonionic lipids

1/3–1/2

Truncated cone

Cylindrical micelles

Double-chained lipids with large head-group areas, fluid chains: – phosphatidyl choline (lecithin) – phosphatidyl serine – phosphatidyl glycerol – phosphatidyl inositol – phosphatidic acid – sphingomyelin, DGDGa – dihexadecyl phosphate – dialkyl dimethyl ammonium – salts

1/2–1

Truncated cone

Flexible bilayers, vesicles

Double-chained lipids with small head-group areas, anionic lipids in high salt, saturated frozen chains: – phosphatidyl ethanaiamine – phosphatidyl serine + Ca2+

≈1

Cylinder

Planar bilayers

Double-chained lipids with small head-group areas, nonionic lipids, poly(cis) unsaturated chains, high T : – unsat. phosphatidyl ethanolamine – cardiolipin + Ca2+ – phosphatidic acid + Ca2+ – cholesterol, MGDGb

>1

Inverted truncated cone or wedge

Inverted micelles

a DGDG: digalactosyl diglyceride, diglucosyldiglyceride b MGDG: monogalactosyl diglyceride, monoglucosyl diglyceride

10.3 Driving force for formation of vesicles

| 349

Several methods have been applied to increase the rigidity and physicochemical stability of the liposome bilayer of which the following methods are the most commonly used; hydrogenation of the double bonds within the liposomes, polymerization of the bilayer using synthesized polymerizable amphiphiles and inclusion of cholesterol to rigidify the bilayer [3]. Other methods to increase the stability of the liposomes include modification of the liposome surface, for example by physical adsorption of polymeric surfactants onto the liposome surface (e.g. proteins and block copolymers). Another approach is to covalently bond the macromolecules to the lipids and subsequent formation of vesicles. A third method is to incorporate the hydrophobic segments of the polymeric surfactant within the lipid bilayer. This latter approach have been successfully applied by Kostarelos et al. [4] who used A–B–A block copolymers of polyethylene oxide (A) and polypropylene oxide (PPO), namely Poloxamers (Pluronics). Two different techniques of adding the copolymer were attempted [4]. In the first method (A), the block copolymer was added after formation of the vesicles. In the second method, the phospholipid and copolymer are first mixed together and this is followed by hydration and formation of SUV vesicles. These two methods are briefly described below. The formation of small unilamellar vesicles (SUVs) was carried out by sonication of 2 % w/w of the hydrated lipid (for about 4 hours). This produced SUV vesicles with a mean vesicle diameter of 45 nm (polydispersity index of 1.7–2.4). This is followed by the addition of the block copolymer solution and dilution of 100 times to obtain a lipid concentration of 0.02 % (method A). In the second method (I) SUV vesicles were prepared in the presence of the copolymer at the required molar ratio. In method (A), the hydrodynamic diameter increases with increasing block copolymer concentration, particularly those with high PEO content, reaching a plateau at a certain concentration of the block copolymer. The largest increase in hydrodynamic diameter (from ≈ 43 nm to ≈ 48 nm) was obtained using Pluronic F127 (that contains a molar mass of 8330 PPO and molar mass of 3570 PEO). In method I the mean vesicle diameter showed a sharp increase with increasing % w/w copolymer reaching a maximum at a certain block copolymer concentration, after which a further increase in polymer concentration showed a sharp reduction in average diameter. For example with Pluronic F127, the average diameter increased from ≈ 43 nm to ≈ 78 nm at 0.02 % w/w block copolymer and then it decreased sharply with a further increase in polymer concentration, reaching ≈ 45 nm at 0.06 % w/w block copolymer. This reduction in average diameter at high polymer concentration is due to the presence of excess micelles of the block copolymer. A schematic representation of the structure of the vesicles obtained on addition of the block copolymer using methods (A) and (I) is shown in Fig. 10.4. With method (A), the triblock copolymer is adsorbed on the vesicle surface by both PPO and PEO blocks. These “flat” polymer layers are prone to desorption due to the weak binding onto the phospholipid surface. In contrast, with the vesicles prepared using method (I), the polymer molecules are more strongly attached to the lipid

350 | 10 Liposomes and vesicles

Method (A)

Method (I)

Fig. 10.4: Schematic representation of vesicle structure in the presence of triblock copolymer prepared using methods (A) and (I) [4].

bilayer with PPO segments “buried” in the bilayer environment surrounded by the lipid fatty acids. The PEO chains remain at the vesicle surfaces free to dangle in solution and attain the preferred conformation. The resulting sterically stabilized vesicles [(I) system] have several advantages over the (A) system with the copolymer simply coating their outer surface. The anchoring of the triblock copolymer using method (I) results in irreversible adsorption and lack of desorption. This is confirmed by dilution of both systems. With method (A), dilution of the vesicles results in reduction of the diameter to its original bare liposome system, indicating polymer desorption. In contrast, dilution of the vesicles prepared by method (I) showed no significant reduction in diameter size indicating strong anchoring of the polymer to the vesicle. A further advantage of constructing the vesicles with bilayer-associated copolymer molecules is the possibility of increased rigidity of the lipid-polymer bilayer [3, 4].

References [1] [2] [3] [4] [5] [6] [7]

Tadros T. Formulation of Cosmetics and Personal Care. Berlin: De Gruyter; 2016. Tadros T. Nanodispersions. Berlin: De Gruyter; 2016. Kostarelos K. PhD Thesis. Imperial College, London; 1995. Kostarelos K, Tadros TF, Luckham PF. Langmuir. 1999;15:369. Israelachvili JN, Mitchell DJ, Ninham BW. J Chem Soc Faraday Trans II. 1976;72:1525. Israelachvili JN, Marcelja S, Horn RG. Q Rev Biophys. 1980;13(2):121. Israelachvili JN. Intermolecular and surface forces, with special applications to colloidal and biological systems. San Diego: Academic Press; 1991. [8] Tanford C. The hydrophobic effect. New York: Wiley; 1980. [9] Tanford C. In: Biomembranes. Proc Int Sch Phys Enrico Ferm. 1985;90:547. [10] Israelachvili JN, Mitchell DJ. Biochim Biophys Acta. 1975;389:13.

11 Deposition of particles at interfaces and their adhesion 11.1 Introduction The deposition of particles to surfaces, illustrated in Fig. 11.1 [1, 2], is an important process in many industrial applications, since it governs many practical processes such as waste water filtration, flotation, separation of toner and ink particles, coatings, paper making, etc. Particle suspesion

Deposition

Interface (collector)

Particle monolayer

Fig. 11.1: Schematic representation of particle deposition [1, 2].

Particle monolayers formed by controlled deposition processes have potential applications in the production of nano- and microstructured materials. In other processes such as membrane filtration, flotation and production of microelectronic or optical devices, particle deposition processes are undesirable. Controlled deposition of bioparticles, e.g. proteins, on various surfaces is a prerequisite for their different separation and purification by chromatograph, filtration, for biosensing, bioreactors and immunological assays. However, deposition of bioparticles is undesirable in such processes as thrombosis, artificial organ failure, dental plaque formation and membrane filtration units. Besides the above mentioned practical applications, fundamental studies of particle deposition provide one with valuable information on the interactions between particles and interfaces and between attached and moving particles. By measuring particle deposition in model systems, information can also be gained on the mechanism and kinetics of molecular adsorption, which is inaccessible to direct experimenhttps://doi.org/10.1515/9783110541953-012

352 | 11 Deposition of particles at interfaces and their adhesion

tal studies. In this way the link between irreversible (colloid) and reversible (molecular) systems can be established.

11.2 Particle deposition Particle deposition can be conveniently split into three major steps: (i) transfer of particles from the bulk dispersion over macroscopic distances to the surface; (ii) transfer of the particles through the boundary layer adjacent to the interface; (iii) formation of a permanent adhesive contact with the surface or previously deposited particles leading to particle immobilization (attachment). Particle transfer involved in step (i) is mainly governed by flow (forced or natural convection) and external forces such as electrostatic or gravity force arising from the density difference between the particle and the medium. This leads to particle migration or sedimentation in the case of gravitational force. At distances comparable to particle dimensions, diffusion starts to play a significant role. At much smaller distances of the order of 100 nm (step (ii)), particles enter the specific fields generated by the interfaces, which change rapidly with distance. For such small distances, electrostatic interactions play the dominant role. Depending on the sign of the charge on the particles and boundary surface, the electrostatic interactions can be either positive (repulsion) or negative (attractive). The range and magnitude of the electrostatic interactions can be varied within broad limits by the addition of electrolytes (see Chapter 6, Vol. 1), change of pH and adsorption of surfactants (see Chapter 9, Vol. 1) or charged polymers (see Chapter 12, Vol. 1). At separation distances significantly smaller than particle dimensions, van der Waals interactions occur (see Chapter 5, Vol. 1), stemming from the induced spontaneous polarization of the molecules, which are attractive for all practical systems. Contrary to the electrostatic interactions, the range and magnitude of the van der Waals interactions cannot be regulated in any systematic way. However, the van der Waals interactions are largely affected by the geometry of the interacting objects, becoming very weak for rough surfaces and strong for smooth surfaces. Additional complications of the deposition processes arise because of the presence of previously accumulated particles, which exert forces on the depositing particles coming into their vicinity. Since these interactions are usually repulsive, the presence of pre-adsorbed particles diminishes the rate of deposition by blocking the available surface area of the boundary surfaces [1, 2]. In most practical applications one is interested in describing the kinetic of deposition, i.e. predicting the number of particles attached to the surface N as a function of time for various physicochemical parameters characterizing the dispersion, namely

11.2 Particle deposition

t1

| 353

t2

N

Number of particles on surface

Nmx Linear range

No 0 Deposition time, t Fig. 11.2: Schematic representation of particle deposition kinetics.

particle and transport mechanism. A schematic representation of the variation of particle number N with time t is shown in Fig. 11.2 [1, 2]. The first relaxation time t1 can be estimated from the simple expression, t1 =

δ2d , D

(11.1)

where δd is the diffusion boundary layer thickness and D is the particle diffusion coefficient. For colloid particles in the size range 10–1000 nm, D varies between 10−11 – 10−12 m2 s−1 and δd is of the order of 10−6 m (1 µm) under typical transport conditions, t1 varies between 0.1–10 seconds. This is a negligible value in comparison with the time of particle deposition experiments usually lasting 102 to 105 seconds [3]. The second relaxation time t2 under convection transport conditions can be estimated from the formula, N∞ , (11.2) t2 = kc nb where kc is the mass transfer rate under steady state conditions [1, 2] and nb is the particle concentration of particles in the bulk. Equation (11.2) predicts that the relaxation time t2 decreases proportionally to the bulk concentration of the particles. It can be estimated from equations (11.1) and (11.2) that for typical parameters describing colloidal dispersions N∞ = 1013 m−2 , kc = 10−6 m s−1 , nb = 1016 m−3 , t2 = 103 seconds.

354 | 11 Deposition of particles at interfaces and their adhesion

In the case of diffusion-controlled transport, t2 can be calculated from, t2 =

2 N∞ . Dn2b

(11.3)

In this case the relaxation time decreases proportionally to the square of bulk concentration. From equation (11.3) typical parameters characterizing colloidal dispersions are: N∞ = 1013 m−2 , kc = 10−6 m s−1 , nb = 1016 m−3 , D = 10−12 m2 s−1 , t2 = 106 seconds which is much larger than the convection relaxation time. This shows that diffusiontransport conditions are orders of magnitude less effective than the convection transport deposition regimes. From the above analysis it is clear that two main parameters quantitatively describe deposition kinetics at interfaces: (i) the initial particle deposition rate (characterized by the slope of the linear part of the kinetic run shown in Fig. 11.2), when the blocking effects stemming from the presence of adsorbed particles are negligible; (ii) the maximum or jamming coverage N∞ governed by the particle shape and interface properties (shape, size, roughness, etc.). Usually the initial particle deposition rates are determined using the convective diffusion theory [4–6]. On the other hand, the more complicated problem of predicting the maximum, mono- and multilayer coverage of particles, as well as the structure of particle monolayers, can be solved by applying various simulation procedures, based on a random sequential adsorption model [1]. Generally speaking, the convective diffusion theory is based on the continuity (conservation) equation. Using this approach, expressions can be derived enabling one to calculate the particle deposition kinetics for various transfer mechanisms such as external force (sedimentation), diffusion, convection, barrier-limited transport, etc. [1]. These expressions shows that the deposition rate for smaller particles (< 500 nm) increases proportionally to the flow velocity to the power of one third, and inversely proportional to particle size (hydrodynamic radius) to the power of two-thirds. It was also revealed that for such small particles, the role of attractive electrostatic attraction is negligible. For larger particles (> 500 nm) and typical collector shapes (spherical, cylindrical), the deposition rate is proportional to the square of the particle size due to the interception effect. The deposition is enhanced by specific surface interactions resulting from the presence of double layers (see Chapter 5, Vol. 1). It was shown that the barrier controls the deposition rates, especially for particles > 500 nm, if its height exceeds 5kT (where k is the Boltzmann constant and T is the absolute temperature). The theoretical results [1] enable one to predict the blocking effect and the maximum coverage (jamming limit) for particles of various shapes. It was shown [1] that the maximum random monolayer coverage of spherical particles is 0.547, which is much smaller than their hexagonal packing in two dimensions that is equal to 0.907. For elongated (needle-like) particles, the maximum coverage is even smaller and is equal to 0.445 for a cylinder with an axial ratio of 15.

11.2 Particle deposition

| 355

The maximum coverage is also reduced by the repulsive electrostatic interactions for smaller particles (< 100 nm) and low electrolyte concentrations. It was demonstrated [1] that electrostatic interactions play an essential role in particle deposition. They may enhance deposition for oppositely charged particles and interfaces or diminish deposition by creating a barrier when the surface charges of the particle and interface are of the same sign. Particle deposition is determined by long-range forces: van der Waals Attraction, electrostatic repulsion or attraction and the presence of adsorbed or grafted surfactants, polymers or polyelectrolytes (referred to as steric interaction). These forces were discussed in Chapters 4–6, Vol. 1. The interaction forces involved in particle deposition consisting of van der Waals attraction, GA , and double layer repulsion, Gelec have been described in detail in Chapters 4 and 5, Vol. 1. The combination of GA and Gelec gives the total energy of interaction GT , as was illustrated in Chapter 6, Vol. 1. A schematic representation of the variation of GA and Gelec and GT with separation distance h between the particle and substrate is shown in Fig. 11.3. This forms the basis of the theory of colloid stability due to Deryaguin–Landau–Verwey–Overbeek (DLVO theory) [10, 11]. The energy–distance curve is characterized by two minima, a shallow secondary minimum (weak and reversible attraction) and a primary deep minimum (strong and irreversible attraction).

G

GT

Ge Gmax GA

Gprimary

h Gsec

Fig. 11.3: Energy–distance curve (DLVO theory).

Particles deposited under conditions of secondary minimum will be weakly attached, whereas particles deposited under conditions of primary minimum will be strongly attached. At intermediate distances of separation, an energy maximum is obtained whose height depends on the surface or zeta potential, electrolyte concentration and valency of the ions. This maximum prevents particle deposition. The magnitude of the energy minima and the energy maximum depends on electrolyte concentration and valency. This is illustrated in Fig. 11.4 for a 1 : 1 electrolyte (e.g. NaCl) at various concentrations. It can be seen that Gmax decreases with increasing NaCl concentration and eventually it disappears at 10−1 mol dm−3 . Thus, particle deposition for particles with the same sign as the surface will increase with increasing electrolyte concentrations. This

356 | 11 Deposition of particles at interfaces and their adhesion (1/κ) = 1000 nm 10¯⁷ mol dm¯³ (1/κ) = 100 nm 10¯⁵ mol dm¯³ (1/κ) = 10 nm 10¯³ mol dm¯³

G

h (1/κ) = 1 nm 10¯¹ mol dm¯³

Fig. 11.4: GT –h curves at various NaCl concentrations.

above trend was confirmed by Hull and Kitchener [12] using a rotating disc coated with a negative film and negative polystyrene latex particles. The number of polystyrene particles deposited was found to increase with increasing NaCl concentration, reaching a maximum at CNaCl > 10−1 mol dm−3 . The ratio of maximum number of particles deposited Nmax to the number deposited at any other NaCl concentration Nd (the so called stability ratio W) was calculated and plotted versus NaCl concentration, Nmax . (11.4) W= Nd Fig. 11.5 shows such plots which clearly shows that W decreases with increasing NaCl concentration reaching a minimum above 10−3 mol dm−3 , where maximum deposition occurs. Similar results were obtained by Tadros [13] using a rotating cylinder apparatus and the results are shown in Fig. 11.6. It can be concluded from the above results that deposition of particles on substrates of the same sign will increase with increasing electrolyte concentration. However, the situation with a surface that is has the opposite charge to the particles being deposited is very different. In this case, attraction between oppositely charged surfaces will occur, a phenomenon referred to as heteroflocculation. This is schematically illustrated in Fig. 11.7 for positively charged polystyrene latex particles on a negative surface; both surfaces were covered by a nonionic polymer layer. 200 150 W 100 50 –2

–1 log CNaCl

0

Fig. 11.5: Variation of W with log CNaCl using rotating disc.

11.3 Effect of polymers and polyelectrolytes on particle deposition |

357

100

25 Nd 20

W

Nd 15

10 W

10

5

–4

–3

–2

–1

0

log CNaCl Fig. 11.6: Nd and W versus CNaCl using a rotating cylinder.

++

+

Fig. 11.7: Deposition of positively charged particles on negatively charged surface.

The effect of adding electrolyte in this case will be opposite to that observed with surfaces of the same charge. Attraction between oppositely charged double layers will be higher at lower electrolyte concentrations. In other words, adding electrolyte in this case will decrease deposition.

11.3 Effect of polymers and polyelectrolytes on particle deposition Polymers and polyelectrolytes, both of the natural and synthetic types, are commonly used in most paint formulations. These materials are used as thickening agents, film formers, resinous powder and humectants. For example, thickening agents, sometimes referred to as rheology modifiers, are used in many formulations to maintain the product stability.

358 | 11 Deposition of particles at interfaces and their adhesion

Adsorbed Amount

No of Particles Deposited

Polymers and surfactants are present in many formulations and interaction between them can produce remarkable effects. Several structures can be identified and the aggregates produced can have profound effects on particle deposition. With many paint formulations polymers are added and these are mostly polyelectrolytes with cationic charges which are essential for strong attachment to the negatively charged surface. For convenience, I will consider the effect of three classes of polymers on particle deposition namely, nonionic polymers, anionic polyelectrolytes and cationic polyelectrolytes [14]. Nonionic polymers can be of the synthetic type such as polyvinylpyrrolidone or natural such as many polysaccharides. The role of nonionic polymers in particle deposition depends on the manner in which they interact with the surface and the particle to be deposited. With many high molecular weight polymers, the chains adopt a conformation forming loops and tails that may extends several nm from the surface. If there is not sufficient polymer to fully cover the surfaces, bridging may occur, resulting in enhancement of deposition. In contrast, if there is sufficient polymer to cover both surfaces, the loops and tails provide steric repulsion resulting in a reduction of deposition. One may be able to correlate particle deposition to the adsorption isotherm. Under conditions of incomplete coverage, i.e. well before the plateau value is reached, particle deposition is enhanced. Under conditions of complete coverage, one observes reduction in deposition and at sufficient coverage deposition may be prevented altogether. This is schematically shown in Fig. 11.8 which shows the correlation of the adsorption isotherm to particle surface deposition.

Polymer concentration Fig. 11.8: Correlation of particle deposition to polymer adsorption.

11.3 Effect of polymers and polyelectrolytes on particle deposition |

359

The most commonly used nonionic polymers in most aqueous formulations are polysaccharide based. Polysaccharides perform a number of functions in these systems: rheology modifiers, suspending agents and emulsifying agents. Polysaccharides are sometimes referred to as “polyglycans” or “hydrocolloids”. The majority of polysaccharides are comprised primarily of six-membered cyclic structures known as a pyranose ring (five carbon atoms and one oxygen atom). Many polysaccharides form helices which is a tertiary spatial configuration, arranged to minimize the total energy of the polysaccharide (e.g. xanthan gum). The behaviour of polysaccharides is critically influenced by the nature of the substituent groups bound to the individual monosaccharides (natural or synthetic). Anionic charges may also occur in natural polysaccharides and this will have a big influence on the adsorption and conformation of the polymer chain. The effect of polysaccharides on particle deposition is rather complex and it depends on the structure of the molecule and interaction with other ingredients in the formulation. Many formulations contain anionic polymers mostly of the polyacrylate and polysaccharide type. The role of the anionic polymers in particle deposition is complex since these polyelectrolytes interact with ions in the formulation, e.g. Ca2+ as well as with the surfactants used. Two of the most commonly used anionic polysaccharides are carboxymethyl cellulose and carboxymethyl chitin obtained by carboxymethylation of cellulose and chitin respectively. Several naturally occurring anionic polysaccharides exist: alginic acid, pectin, carrageenans, xanthan gum, hyaluronic acidic, gum excudates (gum arabic, karaya, traganth, etc.). Crosslinking sites that occur when a polyvalent cation (e.g. Ca+ ) causes interpolysaccharide binding are called “junction zones”. The above complexes, which may produce colloidal particles, will have a big influence on the deposition of other particles in the formulation. They may enhance binding, simply by a cooperative effect in which the polysaccharide complex interacts with the particles and increases the attraction to the surface. The pH of the whole system plays a major role since it affects the dissociation of the carboxylic groups. Many of the anionic polysaccharides and their complexes affect the rheology of the system and this has a pronounced effect on particle deposition. Any increase in the viscosity of the system will reduce the flux of the particles to the surface and this may reduce particle deposition. This reduction may be offset by specific interaction between the particles and the polyanion or its complex. Polycationic polyelectrolytes have a pronounced effect on particle deposition due to their interaction with the substrate and the particles. One of the earliest polycationic polymers is polyethyleneimine (PEI), which was used in some formulations. Later, an important class of cellulosic polycationic polymers were introduced with the trade name “Polymer JR” (Amerchol Corporation). Other synthetic polycationic polymers from Calgon Corporation are Merquat 100 (based on dimethyl diallyl amine chloride) and Merquat 550 (based on acrylamide/dimethyl diallyl amine chloride).

360 | 11 Deposition of particles at interfaces and their adhesion

Several naturally occurring polycationic polymers exist: chitosan (polyglycan with cationic charges) that is positively charged at pH < 7, cationic hydroxyethyl cellulose, cationic guar gum. These polycationic polymers interact with anionic surfactants present in the formulation and at a specific surfactant concentration a rapid increase in the viscosity of the solution is observed. At higher surfactant concentration precipitation of the polymer-surfactant complex occurs and at even higher surfactant concentration, repeptization may occur. Clearly the above interactions will have a pronounced effect on particle deposition. In the absence of any other effects, addition of cationic polyelectrolytes can enhance particle deposition either by simple charge neutralization or “bridging” between the particle and the surface. At high polyelectrolyte concentrations, when there is sufficient molecules to coat both particle and surface, repulsion may occur resulting in reduction in deposition. However, the above effects are complicated by the interaction of the polycationic polymer with surfactants in the formulation and this complicates the prediction of particle deposition. Investigations of the interactions that take place between the polycation, surfactants and other ingredients in the formulation are essential before a complete picture of particle deposition is possible.

11.4 Experimental methods for studying kinetics of particle deposition The most universal and convenient methods are the indirect methods, where changes in particle concentration are measured before and after contact with the adsorbent (collector). The depletion of the solution concentration is determined by measuring optical density changes (turbidimetry), by interferometry or nephelometry or by applying high pressure liquid chromatography (HPLC). For larger colloidal particles one can use on-line particle concentration detection using light scattering. For protein deposition, one can directly detect their concentration in the supernatant (for example collected by membrane filtration) using UV light absorption, circular dichroism, photoluminescence spectroscopy, etc. [15]. A major advantage of the depletion methods is that they can be applied for any solid adsorbent, e.g. larger particles, packed bed columns, etc. Unfortunately, the methods can be subject to errors due to the possibility of adsorption of the particles on the container walls or particles that may be trapped in pores of the adsorbed. In addition, the depletion methods cannot be applied to study the kinetics of deposition. Analogously, average information about the mass of the adsorbed layer can be acquired using the quartz microbalance technique [15]. However, the method has a disadvantage in the case of liquid phase measurements, because the vibration frequency depends in a complicated way on the hydrodynamic resistance of the adsorbed layer rather than its mass. Ellipsometry can be applied for studying biopolymers [15]. The method is based on the principle that the state of po-

11.4 Experimental methods for studying kinetics of particle deposition

|

361

larization of an oblique light beam changes upon reflection from the interface. One measures the changes in the complex refractive index of the monolayer rather than its mass. Unfortunately, the relationship between the refractive index change and mass of deposited substance is not straightforward. Reflectometry is another optical method used in studying deposition of proteins, polymers and nanoparticles [16]. The method depends on detection of the reflectivity changes caused by an adsorbed layer having a refractive index different from the suspending medium. Both ellipsometry and reflectivity measurements are not sensitive enough at low coverage. A more sensitive method in this case is the total internal fluorescence method [15], but the method is tedious to apply. Another method for studying deposition is based on measuring the electrokinetic or ζ - potential of the solid/liquid interface in an electrolyte solution. The zeta potential is sensitive to the amount of adsorbed substance due to the shift of shear plane (see Chapter 10, Vol. 1). Usually, the streaming potential in parallel plate channel, formed by two substrate surfaces (for example mica sheets) is measured using a pair of reversible electrodes (see Chapter 3, Vol. 1). In this way, precise calibration measurements for colloid particles and polyelectrolytes can be performed. A quantitative interpretation of these measurements is achieved in terms of the hydrodynamic theoretical model [17, 18]. In this way the streaming potential method provides a precision of particle or protein coverage of about 1 %. The electrokinetic measurements can also be performed for capillaries covered by particles or proteins and for larger particle suspensions used as carriers of proteins. In the latter case the zeta potential is measured using microelecrophoresis (see Chapter 3, Vol. 1). Unfortunately, electrokinetic methods become less accurate at high electrolyte concentrations and for high coverage of particles. Direct methods for determining particle concentration as a function of various physicochemical parameters are based on optical and electron microscopy as well as atomic force microscopy (AFM). For suspensions with large particle size (> µm), optical microscopy coupled with image analysis is most easy to use. The number of particles can be determined in situ in a continuous manner using well defined transport conditions, e.g. using the impinging-jet cells [19] or the parallel-plate channel [20]. AFM was used for direct in situ imaging of latex particles on mica [3]. This method allows one to measure not only the number of adsorbed particles but also the size of individual particles and their distribution over the surface. However, AFM is inconvenient for measuring particle deposition in liquid cells because of various artefacts stemming from tip-induced aggregation of the suspended particles, convolution of the tip and particle signal, adhesion of particles to the tip, etc. A better resolution can be achieved by imaging the particles in air upon drying the sample. However, drying can cause errors due to the possibility of particle removal or changes in their position and monolayer structure due to strong capillary forces. In all the above methods, it is necessary to characterize the substrate and particle surfaces. The most convenient method for characterizing the surfaces is based on elec-

362 | 11 Deposition of particles at interfaces and their adhesion

trokinetic or zeta potential measurements (see Chapter 3, Vol. 1). For flat substrates such as mica, a parallel-plate cell is made by clamping together the two surfaces that are separated by a polymeric spacer with thickness 100–400 µm. The cell is completed with two compartments containing streaming potential electrodes (Ag/AgCl) connected with a high resistance electrometer. The other electrode pair (platinum) is used for measuring the electric conductance of the cell. The electrolyte or suspension flow through the channel is driven by gas (nitrogen) pressure, pumping or hydrostatic pressure difference regulated by the level of the two electrolyte reservoirs. The major advantages of the streaming potential in cells are the possibility of direct in situ microscope observation of the surfaces without drying, and the simple theoretical interpretation of the results. However, the technique is tedious and time consuming, due to the heterogeneity of the channel side walls, sealing problems and the necessity of determining correction for surface conductance occurring in the cell. Streaming potential measurements can be effectively done for mica, which can be produced to be molecularly smooth by cleavage of the layered mineral. Mica sheets are homogeneous, elastic and mechanically stable and the surface acquires a negative charge as a result of isomorphic substitution of positive ions with higher valency to ones with lower valency (see Chapter 2, Vol. 1). Using streaming potential measurements, Scales et al. [21, 22] measured the zeta potential ζ of mica as a function of ionic strength I in the presence of various monovalent electrolytes (LiCl, NaCl, KCl, CsCl). At low ionic strength (I < 10−4 mol dm−3 ) the limiting value of ζ of mica was −120 mV, independent of the cation. The negative ζ -potential decreased to −80 mV for 10−2 mol dm−3 LiCl and NaCl and −40 mV for 10−2 mol dm−3 KCl. This dependency on the type of cation is due to the specific adsorption of the counterion on the mica surface. The dependency of ζ -potential of mica on pH, at a constant ionic strength of 10−3 mol dm−3 KCl showed a change from −30 mV at pH of 4 to −80 mV for pH > 6. For divalent cations such as Ca2+ , the negative ζ -potential decreased to −10 mV for 10−2 mol dm−3 CaCl2 at pH 5.8. For LaCl3 the ζ -potential inversed sign to positive values when the LaCl3 concentration was increased above 3 × 10−5 mol dm−3 reaching a maximum value of +35 mV for 10−2 mol dm−3 solution of LaCl3 . From a knowledge of ζ -potential, one can calculate the electrokinetic charge of mica σe at any given pH and ionic strength, which is a parameter of primary significance for particle deposition kinetics. Using the Gouy–Chapman relationship, Adamczyk et al. [3] calculated σe and obtained the following results: σe = −0.042 e nm−2 at pH = 3.5 and 10−3 mol dm−3 NaCl; σe = −0.090 e nm−2 at pH = 3.5 and 10−2 mol dm−3 NaCl; σe = −0.107 e nm−2 at pH = 7.4 and 10−3 mol dm−3 NaCl; and σe = −0.17 e nm−2 at pH = 3.5 and 10−2 mol dm−3 NaCl. This shows that the electrokinetic charge density decreases with increasing ionic strength and pH. Even the lowest value of −0.17 e nm−2 is still much higher than the lattice charge of mica that is equal to −2.1 e nm−2 and this confirms the significant compensation of the lattice charge of mica in electrolyte solutions. These results also demonstrate that the zeta potential of mica and poly-

11.5 Linear deposition regime

| 363

meric substrates can be adjusted within broad limits by varying the pH, electrolyte concentration and valency of the counterion. This can be applied in model studies for understanding the mechanisms of particle and protein deposition on solid substrates. The most commonly used colloidal particles for fundamental investigation of deposition kinetics are polystyrene latexes with ≈ 1 µm diameter which can be prepared fairly monodisperse (with relative standard deviation well below 10 %) using surfactant-free emulsion polymerization. These particles are negatively charged when using potassium persulphate initiator in the emulsion polymerization process (the negative charge is due to the presence of –SO4− groups). Positively charged latex particles can be produced by using other initiators having amino groups. These positively charged latex particles are convenient for studying deposition on negatively charged surfaces such as mica. The size of the particles can be accurately determined using transmission or scanning electron microscopy, by laser diffraction or dynamic light scattering. The surface properties of the latex particles are determined by electrophoresis measurements (see Chapter 3, Vol. 1), which allows one to obtain the particle mobility (at unit field strength) that can be converted to zeta potential using the Smoluchowski equation (see Chapter 3, Vol. 1). Results [24] obtained for negative latex particles with 800 nm diameter at an ionic strength I = 10−2 mol dm−3 NaCl gave a ζ -potential of −110 mV at pH = 3.5 and −120 mV at pH = 11.5, indicating very small change of charge with change of the pH. For the positively charged amidine latex particles at the same ionic strength of 10−2 mol dm−3 NaCl, the ζ -potential is +80 mV at pH = 3.5, decreasing monotonically with increasing pH reaching zero at pH ≈ 10 (the isoelectric point of the latex) and became negative above this pH reaching −50 mV at pH = 11. This reflects the amphoteric character of this latex.

11.5 Linear deposition regime Many experimental results are devoted to studying the kinetic aspects of particle deposition, with the aim of determining the role of particle size, bulk concentration, flow rate, ionic strength and other relevant physicochemical parameters. Reliable results were obtained for latex suspensions and mica substrates using the impinging-jet cells of various geometry [2]. Such cells produce laminar flow of particle suspension due to the hydrostatic pressure difference. The jet produced in the capillary impinges against the substrate surface perpendicularly oriented to its direction and then leaves the cell. The deposited particles are observed using an optical microscope through transparent substrate surfaces. For non-transparent substrates, direct microscopic observation becomes feasible if the impinging jet is oriented obliquely to the interface. Using the above procedure one can measure the number of particles over various surface areas of the substrate and this allows one to obtain the average number of particles per unit area ⟨Np ⟩. The deposition kinetics can be quantitatively evaluated as the dependency of ⟨Np ⟩ on the deposition time. Usually instead of surface concentration,

364 | 11 Deposition of particles at interfaces and their adhesion

deposition kinetics is expressed in terms of part defined as, θ = Sg ⟨Np ⟩,

(11.5)

where Sg is the characteristic cross-sectional area for a particle, which for a spherical particle is πa2 . θ represents the fraction of the entire surface area of the interface occupied by particles, i.e. it is a two-dimensional density of particles. As an illustration, Fig. 11.9 shows the deposition kinetics of 1 µm diameter negative latex particles with various bulk concentrations, nb on positive mica surface (obtained by chemisorption of silane compounds).

Θ

0.02 1

0.01 2

3 0 0

5

10

15

20

25

t [min]

Fig. 11.9: Deposition kinetics of negative polystyrene latex particles of diameter 1 µm on silanized mica determined using the impinging-jet cell at Re = 8.4, I = 10−4 mol dm−3 , pH = 5.5. Curve 1, particle number concentration nb = 7.1 × 1013 m−3 ; curve 2, nb = 3.5 × 1013 m−3 ; curve 3, nb = 7.1 × 1012 m−3 ; the continuous lines denote the linear regression fit.

The kinetics was determined using the optical microscope method in the impingingjet cell under laminar conditions for a Reynolds number Re equal to 8.4. The Reynolds number is defined as, RV∞ , (11.6) Re = ν where R is the radius of the impinging jet, V∞ is the average fluid velocity in the capillary and ν is the kinematic viscosity.

11.5 Linear deposition regime

| 365

1

Θ

0.015

0.01

2

0.005 3

0 0

5

10

15

20 t [min]

Fig. 11.10: Deposition kinetics of 1.5 µm latex particles on silanized mica determined using the impinging-jet cell at I = 10−4 mol dm−3 , pH = 5.5, nb = 3.5 × 1013 m−3 . Curve 1, Re = 16; curve 2, Re = 8; curve 3, Re = 2. The continuous lines denote the linear regression fit.

The effect of Reynolds number (2, 8, 16) on particle deposition was investigated by using 1.5 µm latex particles at a bulk concentration nb = 3.5 × 1013 m−3 . The results are shown in Fig. 11.10. The above results and many others reported in the literature [3] confirm the existence of the linear deposition regime shown schematically in Fig. 11.2. The particle flux j can be calculated from the slope of the θ versus time dependence, −j=

∆θ . ∆t πa2p

(11.7)

The minus sign appears because the flux is directed to the interface, i.e. opposite to the direction of a coordinate system fixed at the interface. The flux defined by equation (11.7) is referred to as the limiting, maximum or initial flux. The flux depends on the bulk particle concentration nb and, therefore, instead of the flux one often introduces the mass transfer constant kc which is the normalized flux that is independent of nb and is defined as, kc =

∆θ . ∆tnb πa2p

(11.8)

366 | 11 Deposition of particles at interfaces and their adhesion

kc contains interesting information on the mechanisms of particle deposition, allowing one to evaluate the role of transport conditions (flow intensity), particle size, density and other parameters describing specific interactions between particles and interfaces. As an illustration, Fig. 11.11 shows the dependency of kc on particle size. As can be seen, for the low flow intensity characterized by Re = 30 (curve 2 in Fig. 11.11), the rate decreases sharply with increasing particle radius, reaching a constant plateau for particles with radii > 0.5 µm. This is in accordance with the exact theoretical results derived by the numerical solution of the governing transport equation [25, 26]. However, the experimental results and the exact theoretical results are well above the analytical results (showed by the dashed line in Fig. 11.11) derived from the Smoluchowski– Levich model for point-like particles, where the particle surface interactions are neglected [1]. The difference becomes much more pronounced at high Re of 150 (upper curve in Fig. 11.11). In this case kc reaches a deep minimum for particle radii of about 0.5 µm in accordance with the exact numerical predictions. The minimum is net of two different transport mechanisms:

2.0

k c [μm/s]

1.5

1.0 1

0.5 2

0

0.5

1.0

1.5 a [μm]

Fig. 11.11: Dependency of kc on particle radius. I = 10−3 mol dm−3 , pH = 5.5. Curve 1, Re = 150; curve 2, Re = 30. The continuous lines denote the theoretical results calculated numerically and the dashed line represents the analytical results calculated using the Smoluchowski–Levich approximation [26].

11.5 Linear deposition regime

| 367

(i) diffusion, whose rate decreases with increasing particle radius; (ii) convection (the interception effect), whose significance increases with increasing square of particle radius [1]. The role of electrostatic interactions in particle deposition has been systematically investigated. This interaction becomes attractive for oppositely charged particles and interfaces. As discussed above, the electrostatic attraction increases with decreasing electrolyte concentration. This is illustrated in Fig. 11.12 which shows the dependency of kc on Re at various ionic strengths I for latex particles with 0.87 µm diameter. It can be seen from Fig. 11.12 that for the higher ionic strength of 10−3 mol dm−3 (curve 3), the experimental results approach the theoretical predictions derived from the Smoluchowski–Levich approximation (dashed line in Fig. 11.12) which neglects particle–surface interactions [27]. This is because at such high ionic strength the range of interaction is of the order of 10 nm, i.e. much smaller than particle size. At lower ionic strength (I = 10−4 mol dm−3 and 2 × 10−5 mol dm−3 ), kc increases significantly

1.5

1

1.0 k c [μm/s]

2

3 0.5

0.0 0

10

20

20

40

50

Re Fig. 11.12: Dependency of kc on Re: curve 1, 2 × 10−5 mol dm−3 ; curve 2, 10−4 mol dm−3 ; curve 3, 10−3 mol dm−3 ; the continuous lines denote exact theoretical results; the dashed line represents the analytical results calculated by the Smoluchowski–Levich approximation [27].

368 | 11 Deposition of particles at interfaces and their adhesion

with increasing Re, in accordance with the theoretical predictions derived by numerical solution of the exact transport equation (solid lines in Fig. 11.12). This is due to the increased range of electrostatic interaction, which promotes deposition due to attraction of the negative latex particles near the positive mica surface. This illustrates the coupling between hydrodynamic (flow) and electrostatic interactions. The enhancement of particle deposition is proportional to the square of particle size, particularly when the particle size is greater than 100 nm and high intensity flows. For smaller sized particles and diffusion-controlled transport conditions, the role of attractive electrostatic interactions becomes negligible. Typical results that were obtained for 16 nm silver nanoparticles [28] are shown in Fig. 11.13 where Ns /cb (Ns is the number of particles per square micrometre and cb is the bulk concentration of silver particles expressed in parts per million, ppm) is plotted versus the square root of time t1/2 . Ns was determined by atomic force microscopy (hollow points) and scanning tunnelling microscopy (full points) The bulk concentration of nanoparticles in ppm can be determined by a high precision densitometer.

Ns/cb [μm¯¹]

30

20

10

0 0

5

10

15 t½ [min ½]

Fig. 11.13: Kinetics of 16 nm silver particle deposition on mica modified by a polyelectrolyte layer determined by atomic force microscopy (hollow point) and scanning tunnelling microscopy (full points). I = 10−2 mol dm−3 , T = 293 K bulk suspension concentration 19 and 35 ppm; diffusioncontrolled transport. The points denote experimental results and the solid line denotes the linear fit of experimental data [28].

11.5 Linear deposition regime

| 369

The bulk number concentration nb is connected to the weight concentration cb by, nb =

10−6 cb , mp

(11.9)

where mp is the mass of a single particle. It can be seen from Fig. 11.13 that the dependency of Ns /cb on t1/2 is well reflected by a linear relationship with a slope sD equal to 2.1 ppm−1 min−1/2 . This is in accordance with the theoretical predictions [1] that postulate the following relationship, Ns 2 D 1/2 1/2 = ( ) t . cb mp π

(11.10)

Knowing the slope (Ns /cb )/t1/2 , one can calculate the mass of a single silver particle, 2 D 1/2 ( ) . sD π

mp =

(11.11)

Assuming that,

π 3 (11.12) d ρp , 6 m where dm is the equivalent particle diameter, i.e. the diameter of a sphere having the same mass as the silver particle and ρp is the particle specific density. From equation (11.12) one can obtain the diameter of a single particle, mp =

dm = (

12D1/2 ) ρp π3/2 sD

1/3

.

(11.13)

The diffusion coefficient D can be determined from dynamic light scattering or it can be calculated from the Stokes–Einstein equation, D=

kT , 3πηdm

(11.14)

where k is the Boltzmann constant, T is the absolute temperature and η is the viscosity of the medium. In this way, dm = (

2/7 12 kT 1/7 ) . ) ( 3η ρp π2 sD

(11.15)

Equation (11.15) can be used for precise determination of the equivalent particle diameter in the suspension. However, all the results discussed above are strictly valid for the initial deposition regimes, where the particle coverage remains low and the surface blocking effects can be neglected. The significance of blocking effects at high surface coverage is discussed in the next section.

370 | 11 Deposition of particles at interfaces and their adhesion

11.6 Nonlinear particle deposition (high coverage) The kinetics of particle deposition at high coverage was determined for latex, gold and dendrimer suspensions using direct experimental methods, such as optical and atomic force microscopy [3, 27]. Most of these studies were carried out under diffusioncontrolled transport with the aim of collecting reliable reference data for protein deposition kinetics. As an illustration, Fig. 11.14 shows the results for deposition of amidine (positive) latex particles on bare (negative) mica surface [3]. The measurement of deposition kinetics can provide information on the particle distribution on the substrate. It can also be used to evaluate the lateral interactions between the particles. For a particle coverage θ = 0.4 and I = 10−3 mol dm−3 NaCl, a shortrange, liquid-like ordering of particles was observed in the optical micrographs. The structure of such particle monolayers can be quantitatively characterized in terms of the pair correlation function g(r/a) [3]. As the ionic strength increases, the range of repulsive lateral interaction between the particles decreases (due to compression of the double layers) and the particles can form a more compact monolayer, characterized by higher surface coverage. It is, therefore, important to determine the maximum surface coverage θmax as a function of the double layer parameter a/Le (where Le is the double layer thickness) that is given by, 1/2 a2 kT a =( 2 ) . (11.16) Le 2e I As an illustration, Fig. 11.15 shows the kinetics of 16 nm silver particles on mica modified by adsorption of a cationic polyelectrolyte (polyallylamine chloride) at pH = 5.5 and I = 10−2 mol dm−3 NaCl. The surface concentration of the silver particles was determined by atomic force microscopy and scanning tunnelling microscopy [28]. The kinetics of silver particle deposition on mica controlled by diffusion transport was expressed as the dependency of the surface concentration Ns [µm−2 ] on the square root of adsorption time t1/2 . The inset of Fig. 11.15 shows the silver particle monolayer determined by scanning tunnelling microscopy. The points represent the experimental results and the solid line denotes the theoretical results calculated from the random sequential adsorption model. The main features of the results shown in Fig. 11.15 are: (i) the linear increase of surface concentration with square root of deposition time for t1/2 < 20 min1/2 ; (ii) the abrupt decrease in deposition rate for longer time due to the surface blocking effect. For t1/2 > 20 min1/2 , the limiting surface concentration of silver particles is reached and is equal to 1400 µm−1 . From a knowledge of the maximum surface concentration, one can calculate the coverage of particles from the constitutive dependence of θmx

11.6 Nonlinear particle deposition (high coverage)

| 371

Θ 0.3

0.2

0.1

0.0 0

20

40

60

80

0

20

40

60

80

t½ [min]

0.5 Θ 0.4

0.3

0.2

0.1

0.0 t½ [min]

Fig. 11.14: Deposition kinetics of positively charged latex particles (0.8 µm diameter) on mica determined by optical microscopy (hollow circles) and atomic force microscopy (full circles). Diffusion transport conditions, pH = 5.5, part “a” I = 2 × 10−5 mol dm−3 NaCl; part “b” I = 10−3 mol dm−3 NaCl. The solid lines denote the exact theoretical results obtained using the random sequential adsorbed model and the dotted line represents the theoretical results obtained using the Langmuir adsorption model [3].

372 | 11 Deposition of particles at interfaces and their adhesion Ns [μm¯²] 1400 1200 1000 800 600 400 200 0 0

10

20

30

t½ [min½]

Fig. 11.15: Kinetics of 16 nm silver particle deposition on mica modified by polyallylamine polyelectrolyte. Hollow points (atomic force microscopy); full points (scanning tunnelling microscopy). pH = 5.5, I = 10−2 mol dm−3 NaCl, T = 293 K, suspension concentration 35 ppm, diffusioncontrolled transport. The points denote the experimental points and the solid line denotes the theoretical results calculated from the random sequential model. The inset shows the scanning tunnelling microscopy micrographs of particle monolayers (Ns = 615[µm−2 ]) [28].

on Nmax , θmx =

πd2m Nmax . 4

(11.17)

For dm = 16 nm, one obtains θmx = 0.28 for the ionic strength 10−2 mol dm−3 NaCl. This value is in accordance with the theoretical predictions stemming from the effective hard particle model [1], which predicts the following expression for maximum coverage, 1 θmx = θ∞ , (11.18) (1 + 2h∗ /dp )2 where θ∞ is the maximum coverage for hard (noninteracting) particles that is equal to 0.547 for spherical particles and homogeneous surfaces, and h∗ is the effective interaction range characterizing the repulsive double layer interactions, 2h∗ 1 1 ϕ0 ϕ0 = − ln[1 + ln {ln ]}, dp κdp 2ϕch κdp 2ϕch

(11.19)

11.6 Nonlinear particle deposition (high coverage)

| 373

where κ is the reciprocal Debye length, ϕ0 is the repulsive interaction energy between deposited particles depending on their zeta potentials and size, and ϕch is the scaling interaction energy that is close to kT. Using equations (11.18) and (11.19) the theoretical value of θmx = 0.29, which is in close agreement with the experimentally determined value of 0.28. From a knowledge of the maximum coverage one can theoretically predict silver particle deposition for the entire range of times, by numerically solving the governing mass transport equation. The surface blocking effect at high coverage can be considered in an exact way as the nonlinear boundary condition for the bulk transport equation analogously as shown in Fig. 11.14 for the latex particles. As can be seen in Fig. 11.15, the theoretical results calculated using this approach (depicted by the solid lines) are in accordance with the experimental data for the entire range of time. The role of electrostatic interaction was investigated by carrying out deposition experiments as a function of ionic strength [28]. The results are shown in Fig. 11.16, which shows that the maximum coverage of silver particles increases systematically with increasing ionic strength from the value of 0.12 at I = 10−4 mol dm−3 NaCl to 0.39 for I = 3 × 10−2 mol dm−3 NaCl. This effect is due to the decrease in the range of lateral electrostatic among adsorbed particles as reflected by equations (11.18) and (11.19). The results of Fig. 11.14 and 11.16 show that one can regulate the maximum coverage and structure of deposited particle monolayers in a controlled manner by controlling the ionic strength of the particle suspension, which is the parameter of primary importance for all particle deposition problems. Analogous kinetic runs were observed in the case of convection (flow) controlled particle deposition. Examples of such kinetic measurements were carried out for gold particles (average diameter 13.4 nm) at a silicon substrate covered by a thick silica layer [16]. The impinging-jet cell was used in this study, combined with the reflectometric method. This is shown in Fig. 11.17 where the particle coverage is plotted versus time at various ionic strengths. The results show the significant role of ionic strength, whose increase considerably increases the maximum coverage of gold particles. For example at I = 0.0136 mol dm−3 the maximum coverage is 0.22 and for I = 0.0036 mol dm−3 it is equal to 0.15 .This effect is due to the increased repulsion between the deposited gold particles at low ionic strength. The experimental results shown in Fig. 11.17 are well reflected by the theoretical random sequential adsorption model, which postulates that the overall transport resistance is the sum of the bulk resistance governed by the convective diffusion and the surface resistance stemming from the surface blocking effect [1]. As can be seen from Fig. 11.17, the surface blocking effects play a significant role for coverage close to the maximum coverage. For lower particle coverage, the deposition kinetics remains a linear function of time with a rate independent of ionic strength.

374 | 11 Deposition of particles at interfaces and their adhesion Θ 0.5

0.4

0.3

0.2

0.1

0.0 10¯⁴

10¯³

10¯²

I [M]

Fig. 11.16: Maximum coverage versus ionic strength for silver particle deposition on modified mica. The points denote the experimental results and the solid line denotes the theoretical results calculated from the random sequential adsorption model, equations (11.23) and (11.24). The dashed line shows the limiting coverage pertinent to monodisperse, hard particles, θ = 0.547 [28].

Θ

0.20

0.15

0.10

0.05

0 0

5

10

15

t [min]

Fig. 11.17: Kinetics of 13.4 nm gold particles on silicon with a 25 nm silica layer determined by reflectometry using the impinging-jet cell. The ionic strength of the solutions are: 0.0036 mol dm−3 ( ⃝ ); 0.0061 mol dm−3 (◻); 0.0086 mol dm−3 (⬦); 0.0136 mol dm−3 (△). The coverage is calculated from the reflectometer signal using the thin island film theory. The solid lines represent the theoretical results calculated using the random sequential model for K = 45 [16].

11.7 Particle deposition on heterogeneous surfaces |

375

11.7 Particle deposition on heterogeneous surfaces Most real surfaces are heterogeneous with respect to surface charge (zeta potential) distribution and topology of the interfaces such as roughness. Study of particle deposition on such heterogeneous surfaces is of vital importance for many practical phenomena such as protein, antibody and antigen deposition. To study the effect of surface heterogeneity, model deposition experiments were performed involving an initial step of site deposition [19]. The coverage and distribution sites were controlled by the deposition time and the ionic strength of the suspension [19]. A useful parameter characterizing the deposition is the ratio of site dimension to particle size λ. Depending on this ratio, surface clusters (aggregates) of a controlled architecture are produced. This is illustrated in Fig. 11.18 which shows the micrographs of such aggregates at λ = 0.95. These clusters were produced by a two-phase deposition procedure: (i) deposition of positively charged latex particles (sites) under diffusion transport on bare mica; (ii) deposition of negatively charged latex particles on the site monolayer. Because the size ratio of particles to sites is close to one, the sites are able to coordinate a few particles. Theoretically, the average coordination number for this value of λ is 5.5. Indeed, the micrographs (top of Fig. 11.18) show that most of the clusters contain either 5 or 6 particles coordinated to the central site [19]. When λ increases to 2 (particle size two times larger than the site size) the coordination number theoretically predicted is 2.47. This can be seen in the micrograph of Fig. 11.18 (bottom) where there are two or three particles coordinated by one site. The kinetics of particle deposition on heterogeneous surfaces has been studied using the optical microscope technique [27]. These measurements allow one to obtain the functional relationship between the coverage of sites and the initial deposition rate of particles. Typical results are shown in Fig. 11.19 as the normalized deposition rate of particles k󸀠c (θs )/kc (where k󸀠c (θs ) is the mass transfer for a given coverage of sites and kc is the limiting mass transfer rate for homogeneous surfaces as discussed above). In these experiments, positively charged polystyrene latex particles 0.45 µm diameter were first deposited on a bare mica surface under convectivediffusion transport conditions using the impinging-jet cell [27]. Afterward, negatively charged polystyrene latex particles 0.9 µm diameter were deposited on the heterogeneous surface created in the first step (λ = 2). Fig. 11.19 shows that the normalized deposition rate increases abruptly with site coverage and reaches a maximum value pertinent to continuous surfaces for site coverage θs as low as 5 %. This behaviour is in contrast to the commonly used intuitive model which postulates that the deposition rate should increase proportionally to the favourable surface area of the substrate, i.e. it should be proportional to θs . This discrepancy can be explained by the fact that because the sites are spherical in shape, the effective fraction of the substrate surface area available for depositing particles is 4λθ

376 | 11 Deposition of particles at interfaces and their adhesion

Fig. 11.18: Top: micrographs of negative latex particles (0.9 µm diameter) deposited on positive sites of latex particle (0.95 µm diameter); I = 10−3 mol dm−3 , λ = 0.95, θp = 0.06. Bottom: micrographs of negative latex particles (0.9 µm) deposited at sites of positive latex particles (0.45 µm); I = 10−3 mol dm−3 , λ = 2, θs = 0.014, θp = 0.075.

k′c/kc

1.0

0.8

0.6

0.4

0.2

0.0 0.00

0.05

0.10

0.15

0.20

Θs

Fig. 11.19: Normalized deposition rate of particles at heterogeneous surface kc󸀠 /kc versus spherical site coverage θs ; the points denote the experimental results for λ = 2. The solid line shows the theoretical results calculated from the generalized random sequential model by considering the coupling (K = 8) and the dashed line shows the theoretical results calculated by neglecting the coupling (K = 0) [27].

11.7 Particle deposition on heterogeneous surfaces |

377

(8θ), for λ = 2. However, even by considering this fact, the theoretical results shown by the dashed line in Fig. 11.19 significantly underestimate the experimental results. Good agreement with experiments can only be achieved using the expended random sequential model [1, 6]. In this model, coupling between the bulk transport and the surface transport is explicitly considered. Coupling reflects the fact that after a failed deposition attempt at a bare interface area a particle does not immediately return to the bulk, but has a finite chance to adsorb in the vicinity. Thus, the effective number of deposition attempts increases proportionally to the coupling constant K that is equal to k a /kc (where ka is the adsorption constant describing particle transport through the adsorption boundary layer having the thickness comparable to the size of sites). Because ka is inversely proportional to the thickness of the adsorption layer, the coupling constant attains large values for smaller colloid particles, especially at moderate flow rates [6]. As can be seen in Fig. 11.19, the theoretical results based on random sequential adsorption model are in good agreement with the experimental data. Another parameter that characterizes deposition on heterogeneous surfaces is the maximum coverage of particles, θmx , which depends on the site coverage θs (heterogeneity degree) and λ. Extensive theoretical calculations performed using the random sequential adsorption model enable one to calculate θmx for an arbitrary site coverage and λ varying between 10 and 1 [6]. It was predicted that for low θs , the maximum particle coverage can be calculated from, θmx = λ2 ns θs ,

(11.20)

where ns is the site coordination number that is given by, ns =

5.97 . λ − 0.517

(11.21)

Thus, ns = 1 for λ > 4. For higher site coverage, it was predicted that θmx should attain the value of 0.547 pertinent to deposition on homogeneous surfaces. Experimental verification of the predictions was performed using the optical microscope technique for the above system of positive sites and negative polystyrene latex particles (λ = 2, mica substrate). The results are shown in Fig. 11.20 for ionic strength of 10−2 and 10−3 mol dm−3 NaCl [19]. The experimental data are adequately reflected by the theoretical predictions (solid line of Fig. 11.20) derived from the random sequential adsorption model for noninteracting hard-sphere particles. This agreement is due to the fact that the effective interaction range at such ionic strength is only few percent of the adsorbing particle dimension. For low site coverage range, the experimental results are adequately described by equation (11.21), with ns = 2.47, calculated numerically. It is interesting to note that for I = mol dm−2 , a maximum of θmx appears at site coverage θs of 0.22. The height of the maximum is 0.57, which slightly exceeds the jamming limit predicted theoretically for homogeneous surfaces (that is equal to 0.547).

378 | 11 Deposition of particles at interfaces and their adhesion

Θmx

Θ∞

0.5

0.4

0.3

0.2

0.1

0.0 10¯³

10¯²

10¯¹

Θs

Fig. 11.20: Dependency of maximum (jamming) coverage θmx of negative latex particles (diameter 0.9 µm) on the positive sites of smaller latex particles (diameter 0.45 µm), coverage θs . The points are the experimental results: I = 10−2 (•), I = 10−3 (⬦) mol dm−3 . The solid line denotes the theoretical Monte Carlo simulations; the dashed line shows the analytical results derived from the dependence θmx = λ2 ns θs .

This confirms that particle deposition at spherically shaped sites is indeed a threedimensional process with the surface area available for particles larger than the geometrical area of the substrate surface. The above results obtained for model systems of colloidal particles can be exploited as reference data for interpreting more complicated deposition processes involving molecular species, such as proteins or polyelectrolytes. Most kinetic studies with such systems were performed using streaming potential measurements [3, 23], which is the only method for in situ measurements under wet conditions.

11.8 Particle–surface adhesion Adhesion is the force necessary to separate adherents; it is governed by short-range forces [29, 30]. Adhesion is more complex than deposition and more difficult to measure. No quantitative theory is available that can describe all adhesion phenomena:

11.8 Particle–surface adhesion

|

379

chemical and non-chemical bonds operate. Adequate experimental techniques for measuring adhesion strength are still lacking. When considering adhesion one must consider elastic and non-elastic deformation that may take place at the point of attachment. The short-range forces could be strong, e.g. primary bonds or intermediate, e.g. hydrogen and charge transfer bonds. In 1934, Deryaguin [31, 32] considered the force of adhesion F in terms of the free energy of separation of two surfaces [G(h∞ ) − G(h0 )] from a distance h0 to infinite separation distance h∞ . For the simple case of parallel plates, ∞

− ∫ F dh = G(h∞ ) − G(h0 ).

(11.22)

h0

F is made up of three contributions, F = Fm + Fc + Fe .

(11.23)

Fm is the molecular component and it is made up of two components, an elastic deformation component FS and a surface energy component FH , Fm = FS + FH .

(11.24)

Fc is the component that depends on prior electrification. Fe is the electrical double layer contribution. When a sphere adheres to a plane surface, elastic deformation occurs and one can distinguish the radius of the adhesive area r0 . A schematic representation of elastic deformation is shown in Fig. 11.21. Usually r0 /R ≪ 1, where R is the particle radius. The adhesive area can be calculated from a knowledge of the time dependency of the modulus of the sphere and the time dependency of the hardness of the plate. Two approaches may be applied to consider the process of adhesion.

R R Fo

Fo

2ro (a)

P– (b)

Fig. 11.21: Elastic deformation on adhesion of a sphere to a plane surface.

380 | 11 Deposition of particles at interfaces and their adhesion

11.8.1 Fox and Zisman critical surface tension approach This approach was initially used to obtain the critical surface tension of wetting of liquid on solid substrates. Fox and Zisman [33] found that a plot of cos θ (where θ is the contact angle of a liquid drop on the substrate) versus γLV (the liquid surface tension) for a number of related liquids gives a straight line which when extrapolated to cos θ = 1 gives the critical surface tension of wetting γc . This is shown in Fig. 11.22 for a number of solids. 1.0

Glass Cellulose acetate

cos Θ

0.5

Silicone

0 Vaseline –0.5

Teflon

–1.0 0

20

40

60

80

γLV/mNm¯¹ Fig. 11.22: cos θ versus γLV for a number of substrates.

A liquid with γLV < γc will give complete wetting of the substrate. Surfaces with high γc > 40 mN m−1 and small slopes are high energy surfaces (e.g. glass and cellulose). Surfaces with very low γ c (< 22 mN m−1 ) and high slope are low energy surfaces, e.g. Teflon. Hydrocarbon surfaces such as Vaseline produce intermediate values (γc ≈ 30 mN m−1 ). The above approach could also be applied for adhesion of “soft” particles to solid substrates. One can define three surface tensions γPL (particle/liquid), γPS (particle/ surface) and γSL (solid/liquid). The Helmholtz free energy of adhesion ∆F is given by the following expression, ∆F = (γPS − γPL − γSL )πr20 .

(11.25)

For adhesion to occur, ∆F should be negative. If ∆F is positive, no adhesion occurs.

11.8.2 Neumann’s equation of state approach Neumann [34] simplified the analysis by using a simple equation of state approach.

11.8 Particle–surface adhesion

|

381

40

γLV cosΘ/mNm¯¹

30 20 10 0 –10 30

50

40

60

70

80

γLV/mNm¯¹ Fig. 11.23: Plot of γLV cos θ versus γLV .

He showed that a plot of γLV cos θ versus γLV gives a smooth curve as illustrated in Fig. 11.23. The above simple analysis allows one to obtain γS from a single contact angle measurement. Several methods can be applied to measure particle surface adhesion. Basically one has to measure the force required to remove the particle from the substrate and this can be obtained by measuring the centrifugal force to remove the particle from the substrate [35]. This is given by the expression [35], Fc =

4 3 πa (ρs − ρw )(ω2 x + g), 3

(11.26)

where a is the particle radius, ρs is the particle density, ρw is the density of water, ω is the centrifuge speed, x is the distance from the rotor and g is the acceleration due to gravity. To remove small particles from substrates one needs to apply very high g values (as high as 107 ). Thus this method is of little practical application. The second method that can be applied to obtain an estimate of particle surface adhesion depends on measuring the hydrodynamic force required to remove the particles [36]. For this purpose a rotating cylinder apparatus is used and after the particles are deposited on the inner cylinder, the speed is increased [36]. The percentage detachment is measured (microscopically) as a function of the speed of rotation. The hydrodynamic force required to remove 50 % of the particles is taken as a measure of the force of adhesion. The above method was applied by Tadros [37] in 1980 to measure the force of adhesion of polystyrene latex on polyethylene. The results are shown in Fig. 11.24 which shows the percentage of particles removed versus the speed of rotation of the cylinder.

382 | 11 Deposition of particles at interfaces and their adhesion

% of Particles romoved

60 50 40 30 20 10 0 0

100

200

300

400

500

600

Speed of rotation, n (r.p.m.) Fig. 11.24: Percentage of particles removed versus speed of rotation.

From knowledge of the hydrodynamic force required for particle removal one could calculate the force of adhesion. The force of adhesion could be compared with the attractive force calculated from the Hamaker equation. The results showed that the force of adhesion was about two orders of magnitude lower than the theoretical value calculated from the van der Waals attraction. It was concluded from the above results that the latex particles are not perfectly smooth (“hairy” surface) and hence not in intimate contact with the surface.

References [1]

Adamczyk Z. Deposition. In: Tadros T, editor. Encyclopedia of colloid and interface science. Berlin: Springer; 2013. [2] Adamczyk Z. Particle Deposition. In: Tadros T, editor. Encyclopedia of colloid and interface science. Berlin: Springer; 2013. [3] Adamczyk Z, Nattich M, Barbaszk J. Adsorption and deposition of particles, polyelectrolytes and biopolymers. In: Starov V, editor. Nanoscience: Colloid and Interface Aspects. London, New York: CRC Press; 2010. p. 503–648. [4] Levich VG. Physicochemical hydrodynamics. Englewood Cliffs: Prentice-Hall; 1962. [5] Elimelech M, Gregory J, Jia X, Williams R. Particle deposition and aggregation, measurement, modeling and simulation. Bodmin, Cornwell: Butterworth Heinman; 1995. [6] Adamczyk Z. Particles at interfaces, interactions, deposition, structure. Amsterdam: Elsevier; 2006. [7] Hamaker HC. Physica. 1937;4:1058. [8] Gouy G. J Phys. 1910;9:457; Ann Phys. 1917;7:129. Chapman DL. Phil Mag. 1913;25:475. [9] Stern O. Z Electrochem. 1924;30:508. [10] Deryaguin BV, Landau L. Acta Physicochem USSR. 1941;14:633. [11] Verwey EJW, Overbeek JTG. Theory of stability of lyophobic colloids. Amsterdam: Elsevier; 1948.

References | 383

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

Hull M, Kitchener SA. Trans Faraday Soc. 1969;65:3039. Tadros TF. Unpublished results. Tadros T. Colloids in paints. Weinheim: Wiley-VCH; 2010. Ramsden JJ. Experimental methods for investigating protein adsorption kinetics at surfaces. Quarterly Revs Biophysics. 1993;27:41. Brouwer A, Kooij ES, Hakbijl M, Wormeester H, Poelsema B. Colloids and Surfaces A. 205;267:133. Sadlej K, Wajnryb E, Blawzdziewicz J, Ekiel-Jezewska ML, Adamczyk Z. J Chem Phys. 2009;130:144706. Adamczyk Z, Nattich M, Barbaszk J. Advances Colloid and Interface Sci. 2009;1:147–148. Adamczyk Z, Jaszczolt K, Minha A, Siwek B, Szyk-Warszynska J, Zembale M. Advances Colloid and Interface Sci. 2005;25. Toscano A, Santore MM. Langmuir. 2006;22:2588. Scales PJ, Grieser F, Healy TW. Langmuir. 1990;6:582. Scales PJ, Grieser F, Healy TW. Langmuir. 1992;8:965. Adamczyk Z, Nattich M, Wasilewska M, Zaucha M. Advances Colloid and Interface Sci. 2011;3:168. Adamczyk Z, Zaucha M, Zembala M. Langmuir. 2010;26:9368. Adamczyk Z. J Colloid Interface Sci. 2000;229:477. Adamczyk Z. Irreversible adsorption of particles. In: Toth J, editor. Adsorption: Theory, modeling and analysis. New York: Marcel Dekker; 2001. Adamczyk Z, Saslej K, Wajnryb E, Nattich M, Blawzdziewicz J, Ekiel-Jezewska ML. Advances Colloid and Interface Sci. 2010;1:153. Ocwieja M, Adamczyk Z, Morga M, Michna A. J Colloid Interface Sci. 2011;39:364. Huntsberger JR. The mechanism of adhesion. In: Kilpatrik RL, editor. Treatise on adhesion and adhesives. New York: Marcel Dekker; 1967. Chapter 4. Tadros TF. Particle-surface adhesion. In: Berkeley RCW, et al., editors. Microbial adhesion to surfaces. Elis Horwood Ltd.; 1980. Chapter 5. Deryaguin BV, Smilga VP. Proc Intern Congr Surface Activity 3rd, Cologne, II, Dec. B. 1960, p. 349. Deryaguin BV. Kolloid Z. 1934;69:155. Fox HW, Zisman WA. J Colloid Sci. 1952;7:109, 428. Neumann AW. Advances Colloid and Interface Sci. 1974;4:105. Krupp H. Advances Colloid Interface Sci. 1967;1:111. Visser J. J Colloid Interface Sci. 1970;34:26. Tadros TF. Unpublished results.

12 Characterization, assessment and prediction of stability of colloidal dispersions 12.1 Introduction For full characterization of the properties of colloidal dispersions, three main types of investigations are needed: (i) Fundamental investigation of the system at a molecular level. This requires investigation of the structure of the solid/liquid interface, namely the structure of the electrical double layer (for charge stabilized suspensions), adsorption of surfactants, polymers and polyelectrolytes and conformation of the adsorbed layers (e.g. the adsorbed layer thickness). It is important to know how each of these parameters changes with the conditions, such as temperature, solvency of the medium for the adsorbed layers and effect of addition of electrolytes. (ii) Investigation of the state of dispersion on standing, namely flocculation rates, flocculation points with sterically stabilized systems, spontaneity of dispersion on dilution and Ostwald ripening or crystal growth. All these phenomena require accurate determination of the particle size distribution as a function of storage time. (iii) Bulk properties of the suspension, which are particularly important for concentrated systems. This requires measurement of the rate of sedimentation and equilibrium sediment height. More quantitative techniques are based on assessment of the rheological properties of the suspension (without disturbing the system, i.e. without its dilution and measurement under conditions of low deformation) and how these are affected by long-term storage. In this chapter, I will start with a summary of the methods that can be applied to assess the structure of the solid/liquid interface, measurement of surfactant and polymeric surfactant adsorption and conformation at the interface. This is followed by more detailed sections on assessment of creaming/sedimentation, flocculation and Ostwald ripening. For the latter (flocculation and Ostwald ripening) one needs to obtain information on the particle size distribution. Several techniques are available for obtaining this information on diluted systems. It is essential to dilute the concentrated dispersion with its own dispersion medium in order not to affect the state of the dispersion during examination. The dispersion medium can be obtained by centrifugation of the dispersion whereby the supernatant liquid is produced at the top of the centrifuge tube for suspensions and at the bottom of the tube for emulsions. Care should be taken on dilution of the concentrated system with its supernatant liquid in order not to break up any flocs that are formed. This dilution process should be carried out with minimum shear. For assessing the structure of the dispersion without dilution (which may affect

https://doi.org/10.1515/9783110541953-013

386 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

its structure), rheological techniques (described in Chapter 1) are the most suitable methods. These techniques can also be applied for predicting the long-term physical stability of the suspension as will be discussed in this chapter.

12.2 Assessment of the structure of the solid/liquid interface The most common method for the investigation of double layers in disperse systems is to use titration techniques that can be applied to obtain the surface charge as a function of surface potential at different electrolyte concentrations and types [1]. The first established example of such a procedure was obtained using silver iodide sols since the charge-determining mechanism is well established. Both stable positive and negative sols can be made and the material is rather inert and insoluble and not particularly sensitive to light. In addition, Ag/AgI electrodes are very stable and the sol can be titrated with Ag+ and I− ions while measuring the pAg and pI. Using a material balance one can determine the surface concentration of Ag+ and I− ions and this allows one to calculate the surface charge σ0 . The titration method has been successfully applied for double layer investigations on oxides. σ0 can be directly determined by titration of an oxide suspension in an aqueous solution of indifferent electrolyte (e.g. KCl) using a cell of the type, E1 | oxide suspension | E2 , where E1 is an electrode reversible to H+ and OH− ions, such as a glass electrode, and E2 is a reference electrode such as Ag–AgCl. A known mass m of solid oxide with a specific surface area A (m2 g−1 ) is added to a known volume V of electrolyte solution such as KCl or KNO3 (assuming there is no specific adsorption of ions) of known concentration (e.g. 10−2 mol dm−3 KCl or KNO3 ) [1–7]. The initial pH (called pH0 ) is noted and the sample is titrated with, say, 10−2 mol dm−3 NaOH and the volume required to achieve each solution is recorded as described in detail in Chapter 2, Vol. 1. A suitable time must elapse after each addition to establish equilibrium with the surface. The surface charge density σ0 is given by, σ0 = F(ΓH+ − ΓOH− ).

(12.1)

Whereas the surface potential ψ0 is given by the Nernst equation, ψ0 =

RT a H+ , ln F (aH+ )pzc

(12.2)

where R is the gas constant, T is the absolute temperature, F is the Faraday constant; aH+ is the activity of H+ ions in bulk solution and (aH+ )pzc is the value at the point of zero charge. Determining the surface charge density σ0 (µC cm−2 ) requires accurate determination of the specific surface area A. The particles are usually irregular in shape and they

12.2 Assessment of the structure of the solid/liquid interface | 387

may undergo Ostwald ripening on standing. This causes problems for determining A from average particle size obtained say by electron microscopy. In addition, most oxide particles are not smooth and this may underestimate the area obtained using gas adsorption and application of the BET equation. Measurement of the surface area using dye adsorption requires knowledge of the effective cross-sectional area of the dye molecule and this may vary from one substrate to another. An appropriate method for measuring the surface area is to measure the expulsion of the co-ions from the increase in its concentration in solution, referred to as negative adsorption [4]. Several examples of σ0 –pH curves for oxide sols were given in Chapter 2, Vol. 1. Another method that can be applied for double layer investigations is electrokinetic or zeta potential measurements. The principles of electrokinetic phenomena and measurement of the zeta potential were discussed in detail in Chapter 3, Vol. 1. There are essentially three techniques for measuring the electrophoretic mobility and zeta potential, namely the ultramicroscopic method, laser velocimetry and electroacoustic methods. The ultramicroscopic technique (microelectrophoresis) is the most commonly used method since it allows direct observation of the particles using an ultramicroscope (suitable for particles that are larger than 100 nm). Microelectrophoresis has many advantages since the particles can be measured in their normal environment [8]. It is preferable to dilute the suspension with the supernatant liquid which can be produced by centrifugation. Basically, a dilute suspension is placed in a cell (that can be circular or rectangular) consisting of a thin walled (≈ 100 µm) glass tube that is attached to two larger bore tubes with sockets for placing the electrodes. The cell is immersed in a thermostat bath (accurate to ±0.1 °C) that contains an attachment for illumination and a microscope objective for observing the particles. It is also possible to use a video camera for directly observing the particles. The laser velocimetry technique is suitable for small particles that undergo Brownian motion [2]. The light scattered by small particles will show intensity fluctuations as a result of Brownian diffusion (Doppler shift). When a light beam passes through a colloidal dispersion, an oscillating dipole movement is induced in the particles, thereby radiating the light. Due to the random position of the particles, the intensity of scattered light, at any instant, will appear as random diffraction (“speckle” pattern). As the particles undergo Brownian motion, the random configuration of the pattern will fluctuate, such that the time taken for an intensity maximum to become a minimum (the coherence time), corresponds approximately to the time required for a particle to move one wavelength λ. Using a photomultiplier of active area about the diffraction maximum (i.e. one coherent area), this intensity fluctuation can be measured. The analogue output is digitized (using a digital correlator) to measures the photocount (or intensity) correlation function of scattered light. If an electric field is placed at right angles to the incident light and in the plane defined by the incident and observation beam, the line broadening is unaffected, but the centre frequency of the scattered light is shifted to an extent determined by the

388 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

electrophoretic mobility. The shift is very small compared to the incident frequency (≈ 100 Hz for and incident frequency of ≈ 6 × 1014 Hz) but with a laser source it can be detected by heterodyning (i.e. mixing) the scattered light with the incident beam and detecting the output of the difference frequency. A homodyne method may be applied, in which case a modulator is used to generate an apparent Doppler shift at the modulated frequency. To increase the sensitivity of the laser Doppler method, the electric fields are much higher than those used in conventional electrophoresis. Joule heating is minimized by pulsing of the electric field in opposite directions. The Brownian motion of the particles also contributes to the Doppler shift and an approximate correction can be made by subtracting the peak width obtained in the absence of an electric field from the electrophoretic spectrum. An He–Ne Laser is used as the light source and the output of the laser is split into two coherent beams which are cross-focused in the cell to illuminate the sample. The light scattered by the particle, together with the reference beam, is detected by a photomultiplier. The output is amplified and analysed to transform the signals to a frequency distribution spectrum. At the intersection of the beams, interferences of known spacing are formed. The magnitude of the Doppler shift ∆v is used to calculate the electrophoretic mobility. In the electroacoustic methods, the mobility of a particle in an alternating field is termed dynamic mobility, to distinguish it from the electrophoretic mobility in a static electric field described above [8]. The principle of the technique is based on the creation of an electric potential by a sound wave transmitted through an electrolyte solution, as described by Debye [9, 10]. The potential, termed the ionic vibration potential (IVP), arises from the difference in the frictional forces and the inertia of hydrated ions subjected to ultrasound waves. The effect of the ultrasonic compression is different for ions of different masses and the displacement amplitudes are different for anions and cations. Hence the sound waves create periodically changing electric charge densities. This original theory of Debye was extended to include electrophoretic, relaxation and pressure gradient forces [11, 12]. A much stronger effect can be observed in colloidal dispersions. The sound waves transmitted by the dispersion of charged particles generate an electric field because the relative motion of the two phases is different. The displacement of a charged particle from its environment by the ultrasound waves generates an alternating potential, termed colloidal vibration potential (CVP). The IVP and CVP are both called ultrasound vibration potential (UVP). The converse effect, namely the generation of sound waves by an alternating electric field [13–17] in a colloidal dispersion can be measured and is termed the electrokinetic sonic amplitude (ESA). The theory for the ESA effect has been developed by O’Brian and co-workers [14–18]. The dynamic mobility can be determined by measuring either UVP or ESA, although in general the ESA is the preferred method. Several commercial instruments are available for measuring dynamic mobility:

12.3 Measurement of surfactant and polymer adsorption

|

389

(i) the ESA-8000 system from Matec Applied Sciences that can measure both CVP and ESA signals; (ii) the Pen Kem System 7000 Acoustophoretic titrator that measures the CVP, conductivity, pH, temperature, pressure amplitude and sound velocity. To convert the ESA signal to dynamic mobility one needs to know the density of the disperse phase and the dispersion medium, the volume fraction of the particles and the velocity of sound in the solvent. To convert mobility to zeta potential one needs to know the viscosity of the dispersion medium and its relative permittivity. Because of the inertia effects in dynamic mobility measurements, the weight average particle size has to be known. The ESA measurements can also be applied for determining the particle size in a dispersion from particle mobilities. The electric force acting upon a particle is opposed by the hydrodynamic friction and inertia of the particles. At low frequencies of alternating electric field, the inertial force is insignificant and the particle moves in the alternating electric field with the same velocity as it would have moved in a constant field. The particle mobility at low frequencies can be measured to calculate the zeta potential. At high frequencies the inertia of the particle increases, causing the velocity of the particle to decrease and the movement of the particle to lag behind the field.

12.3 Measurement of surfactant and polymer adsorption Surfactant and polymer adsorption are key to understanding how these molecules affect the stability/flocculation of the suspension. The various techniques that may be applied for obtaining information on surfactant and polymer adsorption have been described in Chapters 10 and 12, Vol. 1. Surfactant (both ionic and nonionic) adsorption is reversible and the process of adsorption can be described using the Langmuir isotherm [19]. Basically, representative samples of the solid with mass m and surface area A (m2 g−1 ) are equilibrated with surfactant solutions covering various concentrations C1 (a wide concentration range from values below and above the critical micelle concentration, cmc). The particles are dispersed in the solution by stirring and left to equilibrate (preferably overnight while stirred over rollers) after which the particles are removed by centrifugation and/or filtration (using millipore filters). The concentration in the supernatant solution C2 is determined using a suitable analytical method. The latter must be sensitive enough to determine very low surfactant concentrations. The surface area of the solid can be determined using gas adsorption and application of the BET equation. Alternatively, the surface area of the “wet” solid (which may be different from that of the dry solid) can be determined using dye adsorption [19]. From a knowledge of C1 and C2 , m and A one can calculate the amount of adsorption Γ (mg m−2 or mol m−2 ) as a function of equilibrium concentration C2 (ppm or

390 | 12 Characterization, assessment and prediction of stability of colloidal dispersions mol dm−3 ),

C1 − C2 . (12.3) mA With most surfactants, a Langmuir-type isotherm is obtained; Γ increases gradually with increasing C2 and eventually reaches a plateau value Γ∞ which corresponds to saturation adsorption. The results can be fitted to the Langmuir equation, Γ=

Γ=

Γ∞ bC2 , 1 + bC2

(12.4)

where b is a constant that is related to the free energy of adsorption ∆Gads , b = exp(−∆Gads /RT).

(12.5)

A linearized form of the Langmuir equation may be used to obtain Γ∞ and b, 1 1 1 + . = Γ Γ∞ Γ∞ bC2

(12.6)

A plot of 1/Γ versus 1/C2 gives a straight line with intercept 1/Γ∞ and slope 1/Γ∞ b from which both Γ∞ and b can be calculated. From Γ∞ the area per surfactant ion or molecule can be calculated, area/molecule =

1018 1 [m2 ] = [nm2 ]. Γ∞ Nav Γ∞ Nav

(12.7)

As discussed before in Chapter 10, Vol. 1, the area per surfactant ion or molecule gives information on the orientation of surfactant ions or molecules at the interface. This information is relevant for the stability of the dispersion. For example, for vertical orientation of surfactant ions, e.g. dodecyl sulphate anions, which is essential to produce a high surface charge (and hence enhanced electrostatic stability), the area per molecule is determined by the cross-sectional area of the sulphate group which is in the region of 0.4 nm2 . With nonionic surfactants consisting of an alkyl chain and poly(ethylene oxide) (PEO) head group, adsorption on a hydrophobic surface is determined by the hydrophobic interaction between the alkyl chain and the hydrophobic surface. For vertical orientation of a monolayer of surfactant molecules, the area per molecule depends on the size of the PEO chain. The latter is directly related to the number of EO units in the chain. If the area per molecule is smaller than that predicted from the size of the PEO chain, the surfactant molecules may associate on the surface forming bilayers, hemimicelles, etc. as discussed in detail in Chapter 10, Vol. 1. This information can be directly related to the stability of the suspension. The adsorption of polymers is more complex than surfactant adsorption, since one must consider the various interactions (chain-surface, chain-solvent and surfacesolvent) as well as the conformation of the polymer chain on the surface [19]. As discussed in Chapter 12, Vol. 1, complete information on polymer adsorption may be obtained if one is able to determine the segment density distribution, i.e. the segment

12.4 Assessment of creaming/sedimentation of dispersions

|

391

concentration in all layers parallel to the surface. However, such information is generally unavailable, and therefore one determines three main parameters: the amount of adsorption Γ per unit area, the fraction p of segments in direct contact with the surface (i.e. in trains) and the adsorbed layer thickness δ. The amount of adsorption Γ can be determined in the same way as for surfactants, although in this case the adsorption process may take a long equilibrium time. Most polymers show a high affinity isotherm. This implies that the first added molecules are completely adsorbed and the isotherm cuts the y-axis at C2 = 0. For desorption to occur, the polymer concentration in the supernatant liquid must approach zero and this implies irreversible adsorption. As discussed in Chapter 6, Vol. 1, the magnitude of saturation adsorption depends on the molecular weight of the polymer, the temperature and the solvency of the medium for the chains. The fraction of segments p in trains can be determined using spectroscopic techniques such as IR, ESR and NMR. p depends on surface coverage, polymer molecular weight and solvency of the medium for the chains. Several techniques may be applied for determining the adsorbed layer thickness δ and these were described in detail in Chapter 12, Vol. 1.

12.4 Assessment of creaming/sedimentation of dispersions Most dispersions undergo creaming or sedimentation on standing due to gravity and the density difference ∆ρ between the particles and the dispersion medium. This is particularly the case when the particle radius exceeds 50 nm and when ∆ρ > 0.1. In this case Brownian diffusion cannot overcome the gravity force and creaming or sedimentation occurs, resulting in increasing droplet or particle concentration from the bottom to the top (for creaming) or from the top to the bottom (for sedimentation) of the container. As discussed in Chapters 3 and 4, to prevent droplet creaming or particle sedimentation, “thickeners” (rheology modifiers) are added in the continuous phase. The creaming of emulsions or sedimentation of suspensions is characterized by the creaming/sedimentation rate, cream/sediment volume, the change of droplet or particle size distribution during creaming/settling and the stability of the dispersion to creaming/sedimentation. Assessment of creaming/sedimentation of a dispersion depends on the force applied to the droplets or particles in the dispersion, namely gravitational, centrifugal and electrophoretic. The creaming/sedimentation processes are complex and subject to various errors in creaming/sedimentation measurements [19]. A dispersion is usually agitated before measuring creaming/sedimentation, to ensure an initially homogeneous system of droplets/particles in random motion. Vigorous agitation or the use of ultrasonic cavitation must be avoided to prevent any breakdown of aggregates and change of the droplet/particle size distribution. The practical measurement of creaming or sedimentation is hindered by the opacity of the dispersion [20]. If there is any variation in the speed of the droplets or par-

392 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

ticles due to polydispersity or density variation, the slower moving fraction obscures the movement of the faster particles or droplets. Analysis of creaming or sedimentation rates requires a knowledge of the droplet or particle concentration with height and time. Two methods can be applied to obtain such information, namely the use of backscattering of near-infrared (NIR). A schematic representation of an instrument that can be used for such measurement, namely the Turbiscan is shown in Fig. 12.1. This technique consists in sending photons (light) into the sample. After being scattered by the emulsion droplets these photons emerge from the sample and are detected by the measurement device of the Turbiscan. A mobile reading head, composed of an NIR diode and two detectors (transition T) and backscattering BS, scans a cell containing the emulsion. The Turbiscan software then facilitates the easy interpretation of the obtained data.

Transmission detector

Light source (near IR) Backscatter detector Fig. 12.1: Schematic representation of the Turbiscan.

The measurement enables the quantification of several parameters, as BS and T values are linked to droplet average diameter (d) and volume fraction (ϕ), d BS = f ( ). ϕ

(12.8)

A schematic representation of an ultrasonic method is shown in Fig. 12.2. The velocity of ultrasound through a dispersion is sensitive to composition. This is the principle of the ultrasound monitor, shown in Fig. 12.2, which measures the ultrasonic velocity as a function of height. The time-of-flight of a pulse of ultrasound is measured across

12.4 Assessment of creaming/sedimentation of dispersions |

% v/v

393

Height controller

height

Thermostat

Timer

Pulse generator

Transmit amplifier

Receiver amplifier

Oscilloscope

Fig. 12.2: Schematic representation of the ultrasonic creaming meter.

a rectangular sample cell immersed in a thermostated water bath. The time-of-flight data are converted to ultrasonic velocity by reference to measurements made in two calibration fluids. The ultrasonic velocity data may be used to calculate the volume fraction of the disperse phase using simple mixing theory. The speed at which ultrasound propagates through an emulsion is a complex function of the droplet volume fraction, size and properties of the particles or droplets and continuous phase. However, when the particles or droplets are much smaller than the wavelength of ultrasound and there is a significant difference between the speed of sound in the bulk dispersed and continuous phases, the effect of volume fraction greatly outweighs all other effects so that the speed of ultrasound V may be calculated by assuming the system behaves like a simple mixture using the equation, V=

Vc2 (

) ρc Vc2 ρd (1 − ϕ(1 − ))(1 − ϕ(1 − )) 2 ρc ρd Vd

1/2

,

(12.9)

394 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

where ρd , ρc , Vd and Vc are the densities and speeds of ultrasound through the dispersed and continuous phases respectively and ϕ is the volume fraction of the dispersed phase. Several other techniques have been designed to monitor creaming or sedimentation of dispersions of which photosedimentation, X-ray sedimentation and laser anemometry are perhaps worth mentioning [20]. The simplest sedimentation test is based on visual observation of settling. The turbidity of the dispersion is estimated visually, or the height of the cream/sediment and cream/sediment volume are recorded as a function of time. This visual estimation of creaming or sedimentation is only qualitative but is adequate in many practical situations. However, the characterization of dispersions and the determination of the particle or droplet size distribution require quantitative creaming/sedimentation methods. Instrumental techniques have been developed for measuring the turbidity of the dispersion as a function of time, either by measuring the turbidity of the bulk dispersion or by withdrawing a sample at a given height of the creaming/settling dispersion. The earlier instruments used for measuring the turbidity of dispersions, called nephelometers, have evolved into instruments with a more sophisticated optical system. Photosedimentometers monitor gravitational particle or droplet sedimentation by photoelectric measurement of incident light under steady-state conditions. A horizontal beam of parallel light is projected through a dispersion in a creaming/sedimentation column to a photocell. Double-beam photosedimentometers using matched photocells, one for the sample and the other for the reference beam, were developed later. A more sophisticated method was introduced later, using a linear charge-coupled photodiode array as the image sensor to convert the light intensity attenuated by the particles or droplets into an electric signal. The output of each of the photodetectors is handled independently by a computer. Hence the creaming/settling distance between any point in the liquid and the surface of the liquid can be measured accurately without using a mechanical device. As a consequence, particle measurement is rapid, requiring only about 5 minutes to determine a particle or droplet size distribution. The use of fiber optics has made it possible to scan the sedimentation column without moving parts or with a fiber optic probe that is moved inside the sedimentation column. Laser anemometry, also described as laser Doppler velocity measurement (LVD) is a sensitive technique that can extend the range of photosedimentation methods. It has been applied in a sedimentometer to measure particle sizes as low as 0.5 µm. X-ray sedimentometers measure X-ray absorption to determine concentration gradients in sedimenting dispersions. The use of X-ray and γ-rays has been proposed as transmittance probes that correlate transmitted radiation with the density of suspension. X-ray transmittance T is directly related to the weight of particles by an exponential relationship, analogous to the Lambert–Beer law governing transmittance of visible radiation, ln T = −Aϕs , (12.10)

12.5 Assessment of flocculation |

395

where A is a particle-, medium- and equipment constant and ϕs is the volume fraction of particles in the suspension. The concentration of particles remaining in the liquid at various sedimentation depths is determined by using a finely collimated beam of X-rays. The time required for the sedimentation measurement is shortened by continuously changing the effective sedimentation depth. The concentration of particles remaining at various depths is measured as a function of time. The X-ray sedimentometers can be used for particles containing elements with atomic numbers above 15 and, therefore, the method cannot be applied to measure sedimentation of organic pigments. It should be mentioned that gravitational sedimentation is often too slow, particularly if the particles are small and having a density that is not appreciably higher than that of the medium. Application of a centrifugal force accelerates sedimentation, allowing one to obtain results within a reasonable time. However, the data obtained by centrifugation do not always correlate with those resulting from settling under gravity. This is particularly the case with suspensions that are weakly flocculated, where the loose structure may break up on application of a centrifugal force. The interaction between the particles may also change on application of a high gravitational force. This casts doubt on the use of centrifugation as an accelerated test for predicting sedimentation.

12.5 Assessment of flocculation Assessing the flocculation of a dispersion requires measuring the particle size and shape distribution as a function of time. In most practical applications, one usually measures an average particle or droplet diameter that can be defined by the n-th moment of the distribution, ∞

S n = ∫ d n f(d) ∂d,

(12.11)

0

where f(d) is the number frequency of particles or droplets with diameter d. The mean droplet size is defined as the ratio of selected moments of the size distribution, 1/(n−m) ∞ ∫0 d n f(d) ∂d d nm = [ ∞ , (12.12) ] ∫0 d m f(d) ∂d where n and m are integers and n > m and typically n does not exceed 4. Using equation (12.12) one can define several mean average diameters: – The Sauter mean diameter with n = 3 and m = 2, ∞

d32 = [

∫0 d3 f(d) ∂d ∞

∫0 d2 f(d) ∂d

].

(12.13)

396 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

–

The mass mean diameter, ∞

d43 = [ –

∫0 d4 f(d) ∂d ∞

∫0 d3 f(d) ∂d

].

(12.14)

].

(12.15)

The number mean diameter, ∞

d10 = [

∫0 d1 f(d) ∂d ∞

∫0 f(d) ∂d

In most cases d32 (the volume/surface average or Sauter mean) is used. The width of the size distribution can be given as the variation coefficient c m which is the standard deviation of the distribution weighted with d m divided by the corresponding average d. Generally, C2 will be used, which corresponds to d32 . Several techniques may be applied for obtaining the particle or droplet size distribution and the mean particle or droplet diameter and these are summarized below [19]. Optical microscopy is by far the most valuable tool for a qualitative or quantitative examination of the dispersion. Information on the size, shape, morphology and aggregation of particles or droplets can be conveniently obtained with minimum time required for sample preparation. Since individual particles or droplets can be directly observed and their shape examined, optical microscopy is considered as the only absolute method for particle or droplet characterization. However, optical microscopy has some limitations: (i) The minimum size that can be detected. The practical lower limit for accurate measurement of particle size is 1.0 µm, although some detection may be obtained down to 0.3 µm. (ii) Image contrast may not be good enough for observation, particularly when using a video camera which is mostly used for convenience. The contrast can be improved by decreasing the aperture of the iris diaphragm but this reduces the resolution. The contrast of the image depends on the refractive index of the particles or droplets relative to that of the medium. Hence the contrast can be improved by increasing the difference between the refractive index of the particles or droplets and the immersion medium. Unfortunately, changing the medium for the dispersion is not practical since this may affect the state of the dispersion. Fortunately, water with a refractive index of 1.33 is a suitable medium for most organic particles or droplets with a refractive index usually > 1.4. The ultramicroscope by virtue of dark field illumination extends the useful range of optical microscopy to small particles not visible in a bright light illumination. Dark field illumination utilizes a hollow cone of light at a large angle of incidence. The image is formed by light scattered from the particles against a dark background. Particles about 10 times smaller than those visible by bright light illumination can be detected.

12.5 Assessment of flocculation |

397

However, the image obtained is abnormal and the particle size cannot be accurately measured. For that reason, the electron microscope (see below) has displaced the ultramicroscope, except for dynamic studies by flow ultramicroscopy. Three main attachments to the optical microscope are possible, namely phase contrast, differential interference contrast and polarizing microscopy. Phase contrast utilizes the difference between the diffracted waves from the main image and the direct light from the light source. The specimen is illuminated with a light cone and this illumination is within the objective aperture. The light illuminates the specimen and generates zero order and higher orders of diffracted light. The zero order light beam passes through the objective and a phase plate which is located at the objective back focal plane. The difference between the optical path of the direct light beam and that of the beam diffracted by a particle causes a phase difference. The constructive and destructive interferences result in brightness changes which enhance the contrast. This produces sharp images allowing one to obtain particle size measurements more accurately. The phase contrast microscope has a plate in the focal plane of the objective back focus. Instead of a conventional iris diaphragm, the condenser is equipped with a ring matched in its dimension to the phase plate. Differential interference contrast (DIC) gives better contrast than the phase contrast method. It utilizes a phase difference to improve contrast, but the separation and recombination of a light beam into two beams is accomplished by prisms. DIC generates interference colours and the contrast effects indicate the refractive index difference between the particle and the medium. In polarized light microscopy, the sample is illuminated with linearly or circularly polarized light, either in a reflection or transmission mode. One polarizing element, located below the stage of the microscope, converts the illumination to polarized light. The second polarizer is located between the objective and the ocular and is used to detect polarized light. Linearly polarized light cannot pass the second polarizer in a crossed position, unless the plane of polarization has been rotated by the specimen. Various characteristics of the specimen can be determined, including anisotropy, polarization colours, birefringence, polymorphism, etc. For sample preparation in optical microscopy, a drop of the dispersion is placed on a glass slide and covered with a cover glass. If the dispersion has to be diluted, the dispersion medium (that can be obtained by centrifugation and/or filtration of the dispersion) should be used as the diluent in order to avoid aggregation. At low magnifications the distance between the objective and the sample is usually adequate for manipulating the sample, but at high magnification the objective may be too close to the sample. An adequate working distance can be obtained, while maintaining high magnification, by using a more powerful eyepiece with a low power objective. For dispersions encountering Brownian motion (when the particle or droplet size is relatively small), microscopic examination of moving particles or droplets can become difficult. In this case one can record the image on a photographic film or video tape or disc (using computer software).

398 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

The optical microscope can be used to observe dispersed particles or drops and flocs. Particle or droplet sizing can be carried out using manual, semiautomatic or automatic image analysis techniques. In the manual method (which is tedious) the microscope is fitted with a minimum of 10× and 43× achromatic or apochromatic objectives equipped with high numerical apertures (10×, 15× and 20×), a mechanical XY stage, a stage micrometer and a light source. The direct measurement of particle or droplet size is aided by a linear scale or globe-and-circle graticules in the ocular. The linear scale is mainly useful for spherical particles or droplets with a relatively narrow particle or droplet size distribution. The globe-and-circle graticules are used to compare the projected particle area with a series of circles in the ocular graticule. The size of spherical particles or droplets can be expressed by the diameter, but for irregularly shape particles various statistical diameters are used. One of the difficulties with the evaluation of dispersions by optical microscopy is the quantification of data. The number of particles or droplets in at least six different size ranges must be counted to obtain a distribution. This problem can be alleviated by the use of automatic image analysis which can also give an indication of the floc size and its morphology. Electron microscopy utilizes an electron beam to illuminate the sample. The electrons behave as charged particles which can be focused by annular electrostatic or electromagnetic fields surrounding the electron beam. Due to the very short wavelength of electrons, the resolving power of an electron microscope exceeds that of an optical microscope by ≈ 200 times. The resolution depends on the accelerating voltage that determines the wavelength of the electron beam and magnifications as high as 200 000 can be reached with intense beams, but this could damage the sample. Mostly the accelerating voltage is kept below 100–200 kV and the maximum magnification obtained is below 100 000. The main advantage of electron microscopy is the high resolution, sufficient for resolving details separated by only a fraction of a nanometre. The increased depth of field, usually by about 10 µm or about 10 times that of an optical microscope, is another important advantage of electron microscopy. Nevertheless, electron microscopy also has some disadvantages such as sample preparation, selection of the area viewed and interpretation of the data. The main drawback of electron microscopy is the potential risk of altering or damaging the sample that may introduce artefacts and possible aggregation of the particles or droplets during sample preparation. The dispersion has to be dried or frozen and the removal of the dispersion medium may alter the distribution of the particles. If the particles or droplets do not conduct electricity, the sample has to be coated with a conducting layer, such as gold, carbon or platinum to avoid negative charging by the electron beam. Two main types of electron microscopes are used, namely transmission (TEM) and scanning (SEM). TEM displays an image of the specimen on a fluorescent screen and the image can be recorded on a photographic plate or film. TEM can be used to examine particles or droplets in the range 0.001–5 µm. The sample is deposited on a Formvar (polyvinyl formal) film resting on a grid to prevent charging of the simple. The sample is usually observed as a replica by coating with an electron transparent material (such as gold or

12.5 Assessment of flocculation |

399

graphite). The preparation of the sample for the TEM may alter the state of dispersion and cause aggregation. Freeze fracturing techniques have been developed to avoid some of the alterations of the sample during sample preparation. Freeze fracturing allows the dispersions to be examined without dilution and replicas can be made of dispersions containing water. It is necessary to have a high cooling rate to avoid the formation of ice crystals. SEM can show particle topography by scanning a very narrowly focused beam across the particle or droplet surface. The electron beam is directed normally or obliquely at the surface. The backscattered or secondary electrons are detected in a raster pattern and displayed on a monitor screen. The image provided by secondary electrons exhibits good three-dimensional detail. The backscattered electrons, reflected from the incoming electron beam, indicate regions of high electron density. Most SEMs are equipped with both types of detectors. The resolution of the SEM depends on the energy of the electron beam which does not exceed 30 kV and hence the resolution is lower than that obtained by the TEM. A very important advantage of SEM is elemental analysis by energy dispersive X-ray analysis (EDX). If the electron beam impinging on the specimen has sufficient energy to excite atoms on the surface, the sample will emit X-rays. The energy required for X-ray emission is characteristic of a given element and since the emission is related to the number of atoms present, quantitative determination is possible. Scanning transmission electron microscopy (STEM) coupled with EDX has been used for the determination of metal particle sizes. Specimens for STEM were prepared by ultrasonically dispersing the sample in methanol and one drop of the suspension was placed onto a Formvar film supported on a copper grid. Confocal laser scanning microscopy (CLSM) is a very useful technique for the identification of dispersions. It uses a variable pinhole aperture or variable width slit to illuminate only the focal plane by the apex of a cone of laser light. Out-of-focus items are dark and do not distract from the contrast of the image. As a result of extreme depth discrimination (optical sectioning) the resolution is considerably improved (up to 40 % when compared with optical microscopy). The CLSM technique acquires images by laser scanning or uses computer software to subtract out-of-focus details from the in-focus image. Images are stored as the sample is advanced through the focal plane in elements as small as 50 nm. Three-dimensional images can be constructed to show the shape of the particles. Scanning probe microscopy (SPM) can measure physical, chemical and electrical properties of the sample by scanning the particle surface with a tiny sensor of high resolution. Scanning probe microscopes do not measure a force directly; they measure the deflection of a cantilever which is equipped with a tiny stylus (the tip) functioning as the probe. The deflection of the cantilever is monitored by (i) a tunnelling current, (ii) laser deflection beam from the back side of the cantilever, (iii) optical interferometry,

400 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

(iv) laser output controlled by the cantilever used as a mirror in the laser cavity, (v) change in capacitance. SPM generates a three-dimensional image and allows calibrated measurements in three (x, y, z) coordinates. SPM not only produces a highly magnified image, but provides valuable information on sample characteristics. Unlike EM which requires vacuum for its operation, SPM can be operated under ambient conditions and, with some limitation, in liquid media. Scanning tunnelling microscopy (STM) measures an electric current that flows through a thin insulating layer (vacuum or air) separating two conductive surfaces. The electrons are visualized to “tunnel” through the dielectric and generate a current, I, that depends exponentially on the distance, s, between the tiny tip of the sensor and the electrically conductive surface of the sample. The STM tips are usually prepared by etching a tungsten wire in an NaOH solution until the wire forms a conical tip. Pt/Ir wire has also been used. In the contrast current imaging mode, the probe tip is raster-scanned across the surface and a feedback loop adjusts the height of the tip in order to maintain a constant tunnel current. When the energy of the tunnelling current is sufficient to excite luminescence, the tip-surface region emits light and functions as an excitation source of subnanometre dimensions. In situ STM has revealed a two-dimensional molecular lamellar arrangement of long chain alkanes adsorbed on the basal plane of graphite. Thermally induced disordering of adsorbed alkanes was studied by variable temperature STM and atomic scale resolution of the disordered phase was claimed by studying the quenched high-temperature phase Atomic force microscopy (AFM) allows one to scan the topography of a sample using a very small tip made of silicon nitride. The tip is attached to a cantilever that is characterized by its spring constant, resonance frequency and a quality factor. The sample rests on a piezoceramic tube which can move the sample horizontally (x, y motion) and vertically (z motion). The displacement of the cantilever is measured by the position of a laser beam reflected from the mirrored surface on the top side of the cantilever. The reflected laser beam is detected by a photodetector. AFM can be operated in either a contact or a non-contact mode. In the contact mode the tip travels in close contact with the surface, whereas in the non-contact mode the tip hovers 5–10 nm above the surface. Scattering techniques are by far the most useful methods for the characterization of dispersions and in principle they can give quantitative information on the particle or droplet size distribution, floc size and shape. The only limitation of the methods is the need to use sufficiently dilute samples to avoid interference such as multiple scattering which makes interpretation of the results difficult. However, recently backscattering methods have been designed to allow one to measure the sample without dilution. In principle one can use any electromagnetic radiation such as light, X-ray or neutrons but in most industrial labs only light scattering is applied (using lasers).

12.5 Assessment of flocculation | 401

Scattering methods can be conveniently divided into the following classes: (i) Time-average light scattering; static or elastic scattering. (ii) Turbidity measurements which can be carried out using a simple spectrophotometer. (iii) Light diffraction techniques. (iv) Dynamic (quasi-elastic) light scattering that is usually referred to as photon correlation spectroscopy (PCS). This is a rapid technique that is very suitable for measuring submicron particles or droplets (nano-size range). (v) Backscattering techniques that are suitable for measuring concentrated samples. The application of any of these methods depends on the information required and availability of the instrument. In time-average light scattering methods the dispersion that is sufficiently diluted to avoid multiple scattering and the dispersion is illuminated by a collimated light (usually laser) beam and the time-average intensity of scattered light is measured as a function of scattering angle θ. Static light scattering is termed elastic scattering. Three regimes can be identified: (i) Rayleigh regime, where the particle radius R is smaller than λ/20 (where λ is the wavelength of incident light). The scattering intensity is given by the equation, I(Q) = [instrument constant][material constant]NVp2 .

(12.16)

Q is the scattering vector that depends on the wavelength of light λ used and is given by, θ 4πn (12.17) Q=( ) sin( ), λ 2 where n is the refractive index of the medium. The material constant depends on the difference between the refractive index of the particle and that of the medium. N is the number of particles and Vp is the volume of each particle. Assuming that the particles are spherical one can obtain the average size using equation (12.16). The Rayleigh equation reveals two important relationships: (a) The intensity of scattered light increases with the square of the particle volume and consequently with the sixth power of the radius R. Hence, scattering from larger particles may dominate scattering from smaller particles. (b) The intensity of scattering is inversely proportional to λ4 . Hence a decrease in the wavelength will substantially increase the scattering intensity. (ii) Rayleigh–Gans–Debye regime (RGD) λ/20 < R < λ that is more complicated than the Rayleigh regime and the scattering pattern is no longer symmetrical about the line corresponding to the 90° angle but favours forward scattering (θ < 90°) or backscattering (180° > θ > 90°). Since the preference for forward scattering increases with increasing particle size, the ratio I45° /I135° can indicate the particle size.

402 | 12 Characterization, assessment and prediction of stability of colloidal dispersions (iii) Mie regime R > λ; the scattering behaviour is more complex than the RGD regime and the intensity exhibits maxima and minima at various scattering angles depending on particle size and refractive index. The Mie theory for light scattering can be used to obtain the particle size distribution using numerical solutions. One can also obtain information on particle shape. Turbidity (total light scattering technique) can be used to measure particle size, flocculation and particle sedimentation. This technique is simple and easy to use; a single or double beam spectrophotometer or a nephelometer can be used. For nonabsorbing particles the turbidity τ is given by, τ = (1/L)ln(I0 /I),

(12.18)

where L is the path length, I0 is the intensity of incident beam and I is the intensity of transmitted beam. The particle size measurement assumes that the light scattered by a particle is singular and independent of other particles Any multiple scattering complicates the analysis. According to Mie theory the turbidity is related to the particle number N and their cross section πr2 (where r is the particle radius) by τ = Qπr2 N,

(12.19)

where Q is the total Mie scattering coefficient. Q depends on the particle size parameter α (which depends on particle diameter and wavelength of incident light λ) and the ratio of refractive index of the particles and medium m. Q depends on α in an oscillatory mode that exhibits a series of maxima and minima whose position depends on m. For particles with R < (1/20)λ, α < 1 and it can be calculated using the Rayleigh theory. For R > λ, Q approaches 2 and between these two extremes, the Mie theory is used. If the particles are not monodisperse (as is the case with most practical systems), the particle size distribution must be taken into account. Using this analysis one can establish the particle size distribution using numerical solutions. Light diffraction techniques are a rapid and nonintrusive technique for determining particle size distributions in the range 2–300 µm with good accuracy for most practical purposes. Light diffraction gives an average diameter over all particle orientations as randomly oriented particles pass the light beam. A collimated and vertically polarized laser beam illuminates a particle dispersion and generates a diffraction pattern with the undiffracted beam in the centre. The energy distribution of diffracted light is measured by a detector consisting of light sensitive circles separated by isolating circles of equal width. The angle formed by the diffracted light increases with decreasing particle size. The angle-dependent intensity distribution is converted by Fourier optics into a spatial intensity distribution I(r). The spatial intensity distribution is converted into a set of photocurrents and the particle size distribution is calculated using

12.5 Assessment of flocculation |

r Optics

S F

Particle presentation area Fourier transform F = Focal length of lens lens S = Radial distance in detectoer plane f r = Partical radius 0.6328 He/Ne laser

403

Expanded laser beam

Detector

Fig. 12.3: Schematic illustration of light diffraction particle sizing system.

a computer. Several commercial instruments are available, e.g. Malvern Mastersizer (Malvern, UK), Horiba (Japan) and Coulter LS Sizer (USA). A schematic illustration of the set-up is shown in Fig. 12.3. In accordance with the Fraunhofer theory (which was introduced by Fraunhofer over 100 years ago), the special intensity distribution is given by, Xmax

I(r) = ∫ Ntot q0 (x)I(r, x) dx,

(12.20)

Xmin

where I(r, x) is the radial intensity distribution at radius r for particles of size x, Ntot is the total number of particles and q0 (x) describes the particle size distribution. The radial intensity distribution I(r, x) is given by, 2

I(r, x) = I0 (

πx2 Ji (k) 2 ) ( ) . 2f k

(12.21)

with k = (πxr)/(λf). Where r is the distance to the centre of the disc, λ is the wavelength, f is the focal length, and Ji is the first-order Bessel function. The Fraunhofer diffraction theory applies to particles whose diameter is considerably larger than the wavelength of illumination. As shown in Fig. 12.3, an He/Ne laser is used with λ = 632.8 nm for particle sizes mainly in the 2–120 µm range. In general, the diameter of a sphere-shaped particle should be at least four times the wavelength of the illumination light. The accuracy of the particle size distribution determined by light diffraction is not very good if a large fraction of particles with diameter < 10 µm is present in the suspension. For small particles (diameter < 10 µm), the Mie theory is more accurate if the necessary optical parameters such as refractive index of particles and medium, and the light absorptivity of the dispersed particles are known. Most commercial instruments combine light diffraction with forward light scattering to obtain a full particle size distribution covering a wide range of sizes. As an illustration, Fig. 12.4 shows the result of particle sizing using a six component mixture of standard polystyrene lattices (using a Mastersizer).

Latex mixture (volume %)

404 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

10

0

Mixture of 6 lattices measured on mastersizer S

1

10 100 Particle diameter (μm)

1000

Fig. 12.4: Single measurement of a mixture of six standard lattices using the Mastersizer.

Most practical suspensions are polydisperse and generate a very complex diffraction pattern. The diffraction pattern of each particle size overlaps with the diffraction patterns of other sizes. The particles of different sizes diffract light at different angles and the energy distribution becomes a very complex pattern. However, manufacturers of light diffraction instruments (such as Malvern, Coulters and Horiba) have developed numerical algorithms relating diffraction patterns to particle size distribution. Several factors can affect the accuracy of Fraunhofer diffraction: (i) particles smaller than the lower limit of Fraunhofer theory; (ii) non-existent “ghost” particles in a particle size distribution obtained by Fraunhofer diffraction that was applied to systems containing particles with edges, or a large fraction of small particles (below 10 µm); (iii) computer algorithms that are unknown to the user and vary with the manufacturer’s software version; (iv) the composition-dependent optical properties of the particles and the dispersion medium; (v) if the density of all particles is not the same, the result may be inaccurate. Dynamic light scattering (DLS) is a method that measures the time-dependent fluctuation of scattered intensity. It is also referred to as quasi-elastic light scattering (QELS) or photon correlation spectroscopy (PCS). The latter is the most commonly used term for describing the process since most dynamic scattering techniques employ autocorrelation. PCS is a technique that utilizes Brownian motion to measure the particle size. As a result of Brownian motion of dispersed particles the intensity of scattered light undergoes fluctuations that are related to the velocity of the particles. Since larger particles or droplets move less rapidly than the smaller ones, the intensity fluctuation (intensity versus time) pattern depends on particle or droplet size as illustrated in Fig. 12.5. The velocity of the scatterer is measured in order to obtain the diffusion coefficient. In a system where Brownian motion is not interrupted by creaming, sedimentation or particle-particle interaction, the movement of particles is random. Hence, the intensity fluctuations observed after a long time interval do not resemble those fluctuations observed initially, but represent a random distribution of particles. Conse-

12.5 Assessment of flocculation |

405

Intensity

Large particles

Time

Intensity

Small particles

Time

Fig. 12.5: Schematic representation of the intensity fluctuation for large and small particles or droplets.

quently, the fluctuations observed at large time delay are not correlated with the initial fluctuation pattern. However, when the time differential between the observations is very small (a nanosecond or a microsecond) both positions of a particle are similar and the scattered intensities are correlated. When the time interval is increased, the correlation decreases. The decay of correlation is particle or droplet size-dependent. The smaller the particles or droplets are, the faster is the decay. The fluctuations in scattered light are detected by a photomultiplier and are recorded. The data containing information on the particle’s motion are processed by a digital correlator. The latter compares the intensity of scattered light at time t, I(t), to the intensity at a very small time interval τ later, I(t + τ), and it constructs the second-order autocorrelation function G2 (τ) of the scattered intensity, G2 (τ) = ⟨I(t)I(t + τ)⟩.

(12.22)

The experimentally measured intensity autocorrelation function G 2 (τ) depends only on the time interval τ, and is independent of t, the time when the measurement started. PCS can be measured in a homodyne where only scattered light is directed towards the detector. It can also be measured in heterodyne mode where a reference beam split from the incident beam is superimposed on scattered light. The diverted light beam functions as a reference for the scattered light from each particle. In the homodyne mode, G2 (τ) can be related to the normalized field autocorrelation function g1 (τ) by, G2 (τ) = A + Bg12 (τ), (12.23)

406 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

where A is the background term designated as the baseline value and B is an instrument-dependent factor. The ratio B/A is regarded as a quality factor of the measurement or the signal-to-noise ratio and expressed sometimes as the % merit. The field autocorrelation function g1 (τ) for a monodisperse suspension decays exponentially with τ, g1 (τ) = exp(−Γτ), (12.24) where Γ is the decay constant (s−1 ). Substituting equation (12.24) into equation (12.23) yields the measured autocorrelation function, G2 (τ) = A + B exp(−2Γτ). (12.25) The decay constant Γ is linearly related to the translational diffusion coefficient DT of the particle, Γ = DT q2 . (12.26) The modulus q of the scattering vector is given by, q=

θ 4πn sin( ), λ0 2

(12.27)

where n is the refractive index of the dispersion medium, θ is the scattering angle and λ0 is the wavelength of the incident light in vacuum. PCS determines the diffusion coefficient and the particle or droplet radius R is obtained using the Stokes–Einstein equation, D=

kT , 6πηR

(12.28)

where k is the Boltzmann constant, T is the absolute temperature and η is the viscosity of the medium. The Stokes–Einstein equation is limited to noninteracting, spherical and rigid spheres. The effect of particle interaction at relatively low particle concentration c can be taken into account by expanding the diffusion coefficient into a power series of concentration, D = D0 (1 + kD c), (12.29) where D0 is the diffusion coefficient at infinite dilution and kD is the virial coefficient that is related to particle or droplet interaction. D0 can be obtained by measuring D at several particle or droplet number concentrations and extrapolating to zero concentration. For polydisperse dispersions, the first-order autocorrelation function is an intensity-weighted sum of autocorrelation functions of particles contributing to the scattering, ∞

g1 (τ) = ∫ C(Γ) exp(−Γτ) dΓ. 0

C(Γ) represents the distribution of decay rates.

(12.30)

12.5 Assessment of flocculation |

407

For narrow particle or droplet size distribution the cumulant analysis is usually satisfactory The cumulant method is based on the assumption that for monodisperse suspensions g1 (τ) is monoexponential. Hence log g1 (τ) versus τ yields a straight line with a slope equal to Γ, ln g1 (τ) = 0.5 ln(B) − Γτ, (12.31) where B is the signal-to-noise ratio. The cumulant method expands the Laplace transform about an average decay rate, ∞

⟨Γ⟩ = ∫ ΓC(Γ) dΓ.

(12.32)

0

The exponential in equation (12.31) is expanded about an average and integrated term, ln g1 (τ) = ⟨Γ⟩τ + (μ2 τ2 )/2! − (μ3 τ3 )/3! + ⋅ ⋅ ⋅ .

(12.33)

An average diffusion coefficient is calculated from ⟨Γ⟩ and the polydispersity (termed the polydispersity index) is indicated by the relative second moment, μ2 /⟨Γ⟩2 . A constrained regulation method (CONTIN) yields several numerical solutions to the particle size distribution and this is normally included in the software of the PCS machine. PCS is a rapid, absolute, and non-destructive method for particle size measurements. It has some limitations. The main disadvantage is the poor resolution of particle size distribution. Also it suffers from the limited size range (absence of any sedimentation) that can be accurately measured. Several instruments are commercially available, e.g. by Malvern, Brookhaven, Coulters, etc. Backscattering techniques are based on the use of fiber optics, sometimes referred to as fiber optic dynamic light scattering (FODLS) and they allow one to measure at high particle or droplet number concentrations. The FODLS employ either one or two optical fibers. Alternatively, fiber bundles may be used. The exit port of the optical fiber (optode) is immersed in the sample and the scattered light in the same fiber is detected at a scattering angle of 180° (i.e. backscattering). The above technique is suitable for on-line measurements during manufacture of a dispersion. Several commercial instruments are available, e.g. Lesentech (USA). Two general techniques may be applied for measuring the rate of flocculation of dispersions, both of which can only be applied for dilute systems. The first method is based on measuring the scattering of light by the particles or droplets. For monodisperse particles or droplets with a radius that is less than λ/20 (where λ is the wavelength of light) one can apply the Rayleigh equation, where the turbidity τ0 is given by, τ0 = A󸀠 n0 V12 , (12.34) where A󸀠 is an optical constant (which is related to the refractive index of the particle and medium and the wavelength of light) and n0 is the number of particles, each with a volume V1 .

408 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

By combining the Rayleigh theory with the Smoluchowski–Fuchs theory of flocculation kinetics [19], one can obtain the following expression for the variation of turbidity with time, τ = A󸀠 n0 V12 (1 + 2n0 kt), (12.35) where k is the rate constant of flocculation The second method for obtaining the rate constant of flocculation is by direct particle or droplet counting as a function of time. For this purpose optical microscopy or image analysis may be used, provided the particle or droplet size is within the resolution limit of the microscope. Alternatively, the particle or droplet number may be determined using electronic devices such as the Coulter Counter or the flow ultramicroscope. The rate constant of flocculation is determined by plotting 1/n versus t, where n is the number of particles after time t, i.e., 1 1 ( ) = ( ) + kt. n n0

(12.36)

The rate constant k of slow flocculation is usually related to the rapid rate constant k0 (the Smoluchowski rate) by the stability ratio W, W=(

k ). k0

(12.37)

One usually plots log W versus log C (where C is the electrolyte concentration) to obtain the critical coagulation concentration (CCC), which is the point at which log W = 0. A very useful method for measuring flocculation is to use the single-particle or droplet optical method. The particles, or droplets of the dispersion, which are dispersed in a liquid, flow through a narrow uniformly illuminated cell. The dispersion is made sufficiently dilute (using the continuous medium) so that particles or droplets pass through the cell individually. A particle or droplet passing through the light beam illuminating the cell generates an optical pulse detected by a sensor. If the particle or droplet size is greater than the wavelength of light (> 0.5 µm), the peak height depends on the projected area of the particle or droplet. If the particle or droplet size is smaller than 0.5 µm, scattering dominates the response. For particles or droplet > 1 µm, a light obscuration (also called blockage or extinction) sensor is used. For particles or droplets smaller than 1 µm, a light scattering sensor is more sensitive. The above method can be used to determine the size distribution of aggregating dispersions. The aggregated particles or droplets pass individually through the illuminated zone and generate a pulse which is collected at small angle (< 3°). At sufficiently small angles, the pulse height is proportional to the square of the number of monomeric units in an aggregate and independent of the aggregate shape or its orientation.

12.6 Measurement of Ostwald ripening |

409

Measurement of incipient flocculation can be carried out for sterically stabilized dispersions when the medium for the chains becomes a θ-solvent. This occurs, for example, on heating an aqueous dispersion stabilized with poly(ethylene oxide) (PEO) or poly(vinyl alcohol) chains. Above a certain temperature (the θ-temperature) that depends on electrolyte concentration, flocculation of the dispersion occurs. The temperature at which this occurs is defined as the critical flocculation temperature (CFT). This process of incipient flocculation can be followed by measuring the turbidity of the suspension as a function of temperature. Above the CFT, the turbidity of the dispersion rises very sharply. For this purpose, the cell in the spectrophotometer that is used to measure the turbidity is placed in a metal block that is connected to a temperature programming unit (which allows one to increase the temperature rise at a controlled rate).

12.6 Measurement of Ostwald ripening Ostwald ripening is the result of the difference in solubility S between small and large particles or droplets. The smaller particles or droplets have larger solubility than the larger particles. The effect of particle size on solubility is described by the Kelvin equation [19], 2σVm (12.38) S(r) = S(∞) exp( ), rRT where S(r) is the solubility of a particle with radius r, S(∞) is the solubility of a particle with infinite radius, σ is the interfacial tension, Vm is the molar volume of the disperse phase, R is the gas constant and T is the absolute temperature. For two particles or droplets with radii r1 and r2 , S1 1 1 RT ln( ) = 2σ( − ). Vm S2 r1 r2

(12.39)

R is the gas constant, T is the absolute temperature and Vm is the molecular weight. To obtain a measure of the rate of Ostwald ripening, the particle or droplet size distribution of the dispersion is followed as a function of time, using either a Coulter Counter, a Mastersizer or an optical disc centrifuge. One usually plots the cube of the average radius versus time which gives a straight line from which the rate of Ostwald can be determined (the slope of the linear curve), r3 =

8 S(∞)γVm D [ ]t. 9 ρRT

(12.40)

D is the diffusion coefficient of the disperse phase in the continuous phase and ρ is the density of the particles.

410 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

12.7 Assessment of coalescence of emulsions As discussed in Chapter 4, coalescence is the combination of two or more droplets to form a larger drop. The rate of coalescence can be expressed in terms of a first-order rate equation if the droplets will have flocculated in a time which is much shorter than the time scale of coalescence. If K (s−1 ) is the coalescence rate constant, then −

dn = Kn, dt

(12.41)

where n is the number of droplets at time t, or n = n0 exp(−Kt),

(12.42)

n0 is the number of droplets at t = 0. Equation (12.42) shows that a plot of log n versus t gives a straight line and the slope is equal to K. The number of droplets at any time in an emulsion can be measured using a Coulter counter. Alternatively, one can measure the average diameter d of the droplets as a function of time using the Mastersizer described above, d = d0 exp(Kt).

(12.43)

Again a plot of log d versus time gives a straight line with a positive slope equal to K.

12.8 Bulk properties of dispersions: Equilibrium sediment or cream volume (or height) and redispersion For a “structured” dispersion, obtained by “controlled" flocculation” or addition of “thickeners” (such as polysaccharides, clays or oxides), the “flocs” sediment or cream at a rate depending on their size and the porosity of the aggregated mass. After this initial sedimentation or creaming, compaction and rearrangement of the floc structure occurs, a phenomenon referred to as consolidation. Normally in sediment or cream volume measurements, one compares the initial volume V0 (or height H0 ) with the ultimately reached value V (or H). A colloidally stable dispersion gives a “close-packed” structure with relatively small sediment or cream volume (dilatant sediment or cream). A weakly “flocculated” or “structured” dispersion gives a more open sediment or cream and hence a higher sediment or cream volume. Thus by comparing the relative sediment or cream volume V/V0 or height H/H0 , one can distinguish between a stable and a flocculated dispersion.

12.9 Application of rheological techniques |

411

12.9 Application of rheological techniques for the assessment and prediction of the physical stability of dispersions [21] 12.9.1 Rheological techniques for predicting sedimentation or creaming and syneresis As mentioned in Chapters 4 and 5, creaming or sedimentation is prevented by addition of “thickeners” that form a “three-dimensional elastic” network in the continuous phase. If the viscosity of the elastic network, at shear stresses (or shear rates) comparable to those exerted by the particles or droplets, exceeds a certain value, then sedimentation or creaming is completely eliminated. The shear stress, σp , exerted by a particle or droplet (force/area) can be simply calculated, (4/3)πR3 ∆ρg ∆ρRg = . (12.44) σp = 3 4πR2 For a 10 µm radius particle or droplet with a density difference ∆ρ of 0.2 g cm−3 , the stress is equal to, σp =

0.2 × 103 × 10 × 10−6 × 9.8 ≈ 6 × 10−3 Pa. 3

(12.45)

V

For smaller particles or droplets, smaller stresses are exerted. Thus, to predict sedimentation or creaming, one has to measure the viscosity at very low stresses (or shear rates). These measurements can be carried out using a constant stress rheometer (Carrimed, Bohlin, Rheometrics, Haake or Physica). Usually one obtains good correlation between the rate of creaming or sedimentation v and the residual viscosity η(0). This is illustrated in Fig. 12.6. Above a certain value of η(0), v becomes equal to 0. Clearly, to minimize sedimentation or creaming one has to increase η(0); an acceptable level for the high shear viscosity η∞ must be achieved, depending on the application. In some cases, a high η(0) may be accompanied by a high η∞ (which may not be acceptable for the application, for example if spontaneous dispersion on dilution is required). If this is the case, the formulation chemist should look for an alternative thickener.

η(o)

Fig. 12.6: Variation of creaming or sedimentation rate with residual viscosity.

412 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

Another problem encountered with many dispersions is that of “syneresis”, i.e. the appearance of a clear liquid film at the top of the container. “Syneresis” occurs with most “flocculated” and/or “structured” (i.e. those containing a thickener in the continuous phase) dispersions. “Syneresis” may be predicted from measurement of the yield value (using steady state measurements of shear stress as a function of shear rate) as a function of time or using oscillatory techniques (in which the storage and loss modulus are measured as a function of strain amplitude and frequency of oscillation). The oscillatory measurements are perhaps more useful, since to prevent separation the bulk modulus of the system should balance the gravity forces that is given by h∆ρg (where h is the height of the disperse phase, ∆ρ is the density difference and g is the acceleration due to gravity). The bulk modulus is related to the storage modulus G󸀠 . A more useful predictive test is to calculate the cohesive energy density of the structure Ec that is given by equation (12.46), γcr

Ec = ∫ G󸀠 γ dγ =

1 󸀠 2 G γcr . 2

(12.46)

0

% Separation

The separation of a formulation decreases with increasing Ec . This is illustrated in Fig. 12.7, which schematically shows the reduction in percentage separation with increasing Ec . The value of Ec that is required to stop complete separation depends on the particle size distribution, the density difference between the particle and the medium as well as on the volume fraction ϕ of the dispersion.

Ec

Fig. 12.7: Schematic representation of the variation of percentage separation with Ec .

The correlation of sedimentation with residual (zero shear) viscosity is illustrated for model dispersions of aqueous polystyrene latex in the presence of ethylhydroxyethyl cellulose as a thickener. As mentioned above, thickeners reduce creaming or sedimentation by increasing the residual viscosity η(0) which must be measured at stresses compared to those exerted by the particles (mostly less than 0.1 Pa). At such low stresses, η(0) increases very rapidly with increasing “thickener” concentration. This rapid increase is not observed at high stresses and this illustrates the need for measurement at low stresses (using constant stress or creep measurements).The variation of η with applied stress σ for

12.9 Application of rheological techniques | 413

ethylhydroxyethyl cellulose (EHEC), a thickener that is applied in some formulations, is shown in Fig. 12.8. The limiting residual viscosity increases rapidly with increasing EHEC concentration. A plot of sedimentation rate for 1.55 µm PS latex particles versus η(0) is shown in Fig. 12.9, which shows an excellent correlation. In this case a value of η(0) ≥ 10 Pa s is sufficient for reducing the rate of sedimentation to 0. 1.48 % 1.30 %

1.0 η/Pas

1.08 % EHEC 0.86 % 0.1

0.65 % 0.43 % 0.22 %

0.01

0.1

0.01

1

σ/Pa

10

v/R2 m–1s–1

Fig. 12.8: Constant stress (creep) measurements for PS latex dispersions as a function of EHEC concentration.

10–3

10–2

10–1 η(0)/Pa

100

10 Fig. 12.9: Sedimentation rate versus η(0).

414 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

12.9.2 Prediction of emulsion creaming For predicting emulsion creaming, some model emulsions were prepared using mixtures of oils and commercial surfactants. The oil phase of the emulsion consisted of 10 parts Arlamol HD (Isohexadecane), 2 parts of Estol 3603 (Caprylic/capric triglyceride), 1 part of sunflower oil (Florasen 90, helanthus Annus) and 1 part of avocado oil (Persea Gratissima). Two emulsifier systems were used for the preparation of oilin-water (O/W) emulsions. The first emulsifier was Pluronic PEF 127, an A–B–A block copolymer of polyethylene oxide, PEO (the A chains, about 100 EO units each) and propylene oxide PPO (the B chain, about 55 PO units). The second emulsifier system was Arlatone V-100, which is a nonionic emulsifier system made of a blend of Steareth100 (stearyl alcohol with 100 EO units), Steareth-2 (Stearyl alcohol with 2 EO units), glyceryl stearate citrate, sucrose and a mixture of two polysaccharides, namely mannan and xanthan gum. In some emulsions, xanthan gum was used as a thickener. All emulsions contained a preservative (Nipaguard BPX). The rate of creaming and cream volume were measured using graduated cylinders. The creaming rate was assessed by comparing the cream volume Vc with that of the maximum value V∞ obtained when the emulsion was stored at 55 °C. The time t0.3 taken to reach a value of Vc /V∞ = 0.3 (i.e. 30 % of the maximum rate) was calculated [21]. All rheological measurements were carried out using a Physica UDS 200 (Universal Dynamic Spectrometer). A cone and plate geometry was used with a cone angle of 2°. The emulsions were also investigated using optical microscopy and image analysis. Fig. 12.10 shows the results for creaming rates obtained at various temperatures, using a 20/80 O/W emulsion stabilized with Synperonic PEF 127. As is clear, t0.3 decreases with increasing temperature. The most useful method to predict creaming is to use constant stress (creep) measurements. From these measurements one can obtain the residual (zero shear) viscosity η(0). Results were obtained for 20/80 % v/v emulsions as a function of Arlatone V-100 concentration. The results are shown in Fig. 12.11 after several periods of time of storage (1 day, 1 week, 2 weeks and 1 month). η(0) showed a large decrease after 1 day which could be due to equilibration of the structure. The results after one week, two weeks and one month are close to each other. There is a significant increase in η(0) when the Arlatone V-100 concentration increased above 0.8 %. The creaming rate of the emulsion also showed a sharp decrease above 0.8 % Arlatone V-100, indicating the correlation between η(0) and creaming rate. A very useful method for predicting creaming is to measure the cohesive energy density as given by equation (12.46). As an illustration, Fig. 12.12 shows the variation of cohesive energy density Ec with Arlatone V-100 concentration. The results clearly show a rapid increase in Ec above 0.8 % Arlatone V-100. Ec seems to show a decrease

12.9 Application of rheological techniques |

415

100 50°C

80 % Creaming

40°C

RT

60

4°C

40 20 0

0

5

10

15

20

25

30

Time/days

η(0)/Pas

Fig. 12.10: Percentage creaming versus time at various temperatures.

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 0

1 day 1 week 2 weeks 1 month

0.4

0.5

0.6

0.7

0.8

0.9

1

% Arlatone V 100

Ec/mJm–3

Fig. 12.11: Variation of residual viscosity with Arlatone V-100 concentration at various storage times.

1 day 1 week

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0

2 weeks 1 month

0.4

0.5

0.6

0.7

0.8

0.9

% Arlatone V 100

Fig. 12.12: Variation of Ec with % Arlatone V-100 in the emulsion.

1

416 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

in the values after storage for two weeks. This may be due a small increase in droplet size (as a result of some coalescence) which results in reduction in the cohesive energy density. This small increase in droplet size could not be detected by microscopy since the change was very small.

12.9.3 Assessment and prediction of flocculation using rheological techniques Steady state rheological investigations may be used to investigate the state of flocculation of a dispersion. Weakly flocculated dispersions usually show thixotropy and the change of thixotropy with applied time may be used as an indication of the strength of this weak flocculation. Steady state shear stress-shear rate measurements are by far the most commonly used method in many industrial laboratories. Basically, the dispersion is stored at various temperatures and the yield value σ β and plastic viscosity ηpl are measured at various intervals of time. Any flocculation in the formulation should be accompanied by an increase in σ β and ηpl . A rapid technique to study the effect of temperature changes on the flocculation of a formulation is to carry out temperature sweep experiments, running the samples from say 5–50 °C. The trend in the variation of σ β and ηpl with temperature can quickly give an indication of the temperature range at which a dispersion remains stable (during that temperature range σ β and ηpl remain constant). If Ostwald ripening occurs simultaneously, σ β and ηpl may change in a complex manner with storage time. Ostwald ripening results in a shift of the particle size distribution to higher diameters. This has the effect of reducing σ β and ηpl . If flocculation occurs simultaneously (having the effect of increasing these rheological parameters), the net effect may be an increase or decrease of the rheological parameters. This trend depends on the extent of flocculation relative to Ostwald ripening and/or coalescence. Therefore, following σ β and ηpl with storage time requires knowledge of Ostwald ripening. Only in the absence of these latter breakdown processes can one use rheological measurements as a guide for the assessment of flocculation [21]. The above methods are only qualitative and one cannot use the results in a quantitative manner. This is due to the possible breakdown of the structure on transferring the formulation to the rheometer and also during the uncontrolled shear experiment. Better techniques to study flocculation of a formulation are constant stress (creep) or oscillatory measurements. By careful transfer of the sample to the rheometer (with minimum) shear, the structure of the flocculated system may be maintained. A very important point that must be considered in any rheological measurement is the possibility of “slip” during the measurements. This is particularly the case with highly concentrated dispersions in which the flocculated system may form a “plug” in the gap of the plates, leaving a thin liquid film at the walls of the concentric cylinder or cone-and-plate geometry. This behaviour is caused by some “syneresis” of the for-

12.9 Application of rheological techniques |

417

mulation in the gap of the concentric cylinder or cone and plate. To reduce “slip”, one should use roughened walls for the plates. A vane rheometer may also be used. Constant stress (creep) experiments are more sensitive for following flocculation. As mentioned in Chapter 1, a constant stress σ is applied on the system and the compliance J (Pa−1 ) is plotted as a function of time. These experiments are repeated several times, increasing the stress from the smallest possible value (that can be applied by the instrument) in small increments. A set of creep curves are produced at various applied stresses. From the slope of the linear portion of the creep curve (after the system reaches a steady state), the viscosity at each applied stress, η σ , is calculated. A plot of η σ versus σ allows one to obtain the limiting (or zero shear) viscosity η(0) and the critical stress σcr (which may be identified with the “true” yield stress of the system. The values of η(0) and σcr may be used to assess the flocculation of the dispersion on storage. If flocculation occurs on storage (without any Ostwald ripening), the values of η(0) and σcr may show a gradual increase with increasing storage time. As discussed in the previous section (on steady state measurements), the trend becomes complicated if Ostwald ripening occurs simultaneously (both have the effect of reducing η(0) and σcr ). The above measurements should be supplemented by particle size distribution measurements of the diluted dispersion (making sure that no flocs are present after dilution) to assess the extent of Ostwald ripening. Another complication may arise from the nature of the flocculation. If the latter occurs in an irregular way (producing strong and tight flocs), η(0) may increase, while σcr may show some decrease and this complicates the analysis of the results. In spite of these complications, constant stress measurements may provide valuable information on the state of the dispersion on storage. Carrying out creep experiments and ensuring that a steady state is reached can be time consuming. One usually carries out a stress sweep experiment, whereby the stress is gradually increased (within a predetermined time period to ensure that one is not too far from reaching the steady state) and plots of η σ versus σ are established. These experiments are carried out at various storage times (say every two weeks) and temperatures. From the change of η(0) and σcr with storage time and temperature, one may obtain information on the degree and the rate of flocculation of the system. Clearly, interpretation of the rheological results requires expert knowledge of rheology and measurement of the particle size distribution as a function of time. One main problem in carrying out the above experiments is sample preparation. When a flocculated dispersion is removed from the container, care should be taken not to cause much disturbance to that structure (minimum shear should be applied on transferring the formulation to the rheometer). It is also advisable to use separate containers for assessing flocculation; a relatively large sample is prepared and this is then transferred to a number of separate containers. Each sample is used separately at a given storage time and temperature. One should be careful in transferring the sample to the rheometer. If any separation occurs in the formulation, the sample is

418 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

gently mixed by placing it on a roller. It is advisable to use as minimum shear as possible when transferring the sample from the container to the rheometer (the sample is preferably transferred using a “spoon” or by simple pouring from the container). The experiment should be carried out without an initial pre-shear. Another rheological technique for assessing flocculation is oscillatory measurement. As mentioned in Chapter 1, one carries out strain sweep measurements, where the oscillation frequency is fixed (say at 1 Hz) and the viscoelastic parameters are measured as a function of strain amplitude. G∗ , G󸀠 and G󸀠󸀠 remain virtually constant up to a critical strain value, γcr . This region is the linear viscoelastic region. Above γcr , G∗ and G󸀠 start to fall, whereas G󸀠󸀠 starts to increase; this is the nonlinear region. The value of γcr may be identified with the minimum strain above which the “structure” of the dispersion starts to break down (for example breakdown of flocs into smaller units and/or breakdown of a “structuring” agent). From γcr and G󸀠 , one can obtain the cohesive energy Ec (J m−3 ) of the flocculated structure using equation (12.46). Ec may be used in a quantitative manner as a measure of the extent and strength of the flocculated structure in a dispersion. The higher the value of Ec the more flocculated the structure is. Clearly, Ec depends on the volume fraction of the dispersion as well as the particle size distribution (which determines the number of contact points in a floc). Therefore, for quantitative comparison between various systems, one has to make sure that the volume fraction of the disperse particles is the same and the dispersions have very similar particle size distributions. Ec also depends on the strength of the flocculated structure, i.e. the energy of attraction between the droplets. This depends on whether the flocculation is in the primary or secondary minimum. Flocculation in the primary minimum is associated with a large attractive energy and this leads to higher values of Ec when compared with the values obtained for secondary minimum flocculation (weak flocculation). For a weakly flocculated dispersion, such as is the case with secondary minimum flocculation of an electrostatically stabilized system, the deeper the secondary minimum, the higher the value of Ec (at any given volume fraction and particle size distribution of the dispersion). With a sterically stabilized dispersion, weak flocculation can also occur when the thickness of the adsorbed layer decreases. Again, the value of Ec can be used as a measure of the flocculation; the higher the value of Ec , the stronger the flocculation. If incipient flocculation occurs (on reducing the solvency of the medium for the change to worse than θ-condition) a much deeper minimum is observed and this is accompanied by a much larger increase in Ec . To apply the above analysis, one must have an independent method for assessing the nature of the flocculation. Rheology is a bulk property that can give information on the interparticle interaction (whether repulsive or attractive) and to apply it in a quantitative manner one must know the nature of these interaction forces. However, rheology can be used in a qualitative manner to follow the change of the formulation on storage. Providing the system does not undergo any Ostwald ripening, the change of the moduli with time and in particular the change of the linear viscoelastic region

12.9 Application of rheological techniques | 419

may be used as an indication of flocculation. Strong flocculation is usually accompanied by a rapid increase in G󸀠 and this may be accompanied by a decrease in the critical strain above which the “structure” breaks down. This may be used as an indication of formation of “irregular” and tight flocs which become sensitive to the applied strain. The floc structure will entrap a large amount of the continuous phase and this leads to an apparent increase in the volume fraction of the dispersion and hence an increase in G󸀠 . In the case of oscillatory measurements, the strain amplitude is kept constant in the linear viscoelastic region (one usually takes a point far from γcr but not too low, i.e. in the midpoint of the linear viscoelastic region) and measurements are carried out as a function of frequency. Both G∗ and G󸀠 increase with increasing frequency and ultimately, above a certain frequency, they reach a limiting value and show little dependency on frequency. G󸀠󸀠 is higher than G󸀠 in the low frequency regime; it also increases with increasing frequency and at a certain characteristic frequency ω∗ (that depends on the system) it becomes equal to G󸀠 (usually referred to as the crossover point), after which it reaches a maximum and then shows a reduction with any further increase in frequency. From ω∗ one can calculate the relaxation time τ of the system, τ=

1 ω∗

(12.47)

The relaxation time may be used as a guide for the state of the dispersion. For a colloidally stable dispersion (at a given particle size distribution), τ increases with increasing volume fraction of the disperse phase, ϕ. In other words, the crossover point shifts to lower frequency with increasing ϕ. For a given dispersion, τ increases with increasing flocculation, provided that the particle size distribution remains the same (i.e. no Ostwald ripening). The value of G󸀠 also increases with increasing flocculation, since aggregation of particles usually results in liquid entrapment and the effective volume fraction of the dispersion shows an apparent increase. With flocculation, the net attraction between the particles also increases and this results in an increase in G󸀠 . The latter is determined by the number of contacts between the particles and the strength of each contact (which is determined by the attractive energy). It should be mentioned that in practice one may not obtain the full curve, due to the frequency limit of the instrument and, furthermore, measurements at low frequency are time consuming. Usually one obtains part of the frequency dependency of G󸀠 and G󸀠󸀠 . In most cases, one has a more elastic than viscous system. Most disperse systems used in practice are weakly flocculated and they also contain “thickeners” or “structuring” agents to reduce sedimentation and to acquire the right rheological characteristics for the application, e.g. in hand creams and lotions. The exact values of G󸀠 and G󸀠󸀠 required depend on the system and its application. In

420 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

most cases a compromise has to be made between acquiring the right rheological characteristics for the application and the optimum rheological parameters for long-term physical stability. Applying rheological measurements to achieve these conditions requires a great deal of skill and understanding of the factors that affect rheology. Several examples have been published to illustrate the use of rheological measurements for studying flocculation of a dispersion. For example, Hunter and Nicol [22] studied the flocculation and restabilization of kaolinite suspensions using rheology and zeta potential measurements. Fig. 12.13 shows plots of the yield value σ β and electrophoretic mobility as a function of cetyl trimethyl ammonium bromide (CTAB) concentration at pH = 9. σ β increases with increasing CTAB concentration, reaching a maximum at the point where the mobility reaches zero (the isoelectric point, IEP, of the clay) and then decreases with a further increase in CTAB concentration. This trend can be explained on the basis of flocculation and restabilization of the clay suspension. u × 10–4 2.6

σβ

u

0.6

2.4 0.4

σβ/Pa

2.2 2

0.2

1.8

0

1.6

–0.2

1.4 –0.4

1.2 1

–0.6 –5

–4

–3

log [C16TAB] Fig. 12.13: Variation of yield value σ β and electrophoretic mobility u with C16 TAB concentration.

Initial addition of CTAB causes a reduction in the negative surface charge of the clay (by adsorption of CTA+ on the negative sites of the clay). This is accompanied by a reduction in the negative mobility of the clay. When complete neutralization of the clay particles occurs (at the IEP), maximum flocculation of the clay suspension occurs and this is accompanied by a maximum in σ β . On a further increase in CTAB concentration, further adsorption of CTA+ occurs, resulting in charge reversal and restabilization of the clay suspension. This is accompanied by a reduction in σ β . Neville and Hunter [23] studied the flocculation of sterically stabilized dispersions using polymethylmethacrylate (PMMA) latex stabilized with poly(ethylene oxide) (PEO). Flocculation was induced by addition of electrolyte and/or increase of temperature. Fig. 12.14 shows the variation of σ β with increasing temperature at constant electrolyte concentration.

421

2.0 1.0

1.0 0.5

σβ/Pa

Relative hydrodynamic volume

12.9 Application of rheological techniques |

Fig. 12.14: Variation of σ β and hydrodynamic volume v with temperature.

0 30

40

50

It can be seen that σ β increases with increasing temperature, reaching a maximum at the critical flocculation temperature (CFT) and then decreases with a further increase in temperature. The initial increase is due to the flocculation of the latex with increasing temperature, as a result of reduction of solvency of the PEO chains with increasing temperature. The reduction in σ β after the CFT is due to the reduction in the hydrodynamic volume of the dispersion. The flocculation of sterically stabilized emulsions was investigated using a system stabilized with an A–B–A block copolymer of PEO-PPO-PEO (Pluronic F127). Flocculation was induced by addition of NaCl. Fig. 12.15 shows the variation of the yield value, calculated using the Herschel–Bulkley model as a function of NaCl concentration at various storage times. In the absence of NaCl, the yield value did not change with storage time over a period of 1 month, indicating absence of flocculation. In the presence of NaCl, the yield value increased with increasing storage time and this increase was very significant when the NaCl concentration was increased above 0.8 mol dm−3 NaCl. 12

Yield value/Pa

10

2 weeks

8

1 week

6 4 1 day

2 0

0

0.2

0.4

0.6

0.8

1.0

CNaCl/mol dm–3 Fig. 12.15: Variation of yield value with NaCl concentration.

1.2

422 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

The above increase in yield value indicated flocculation of the emulsion and this was confirmed by optical microscopy. The smaller increase in yield value below 0.8 mol dm−3 NaCl is indicative of weak flocculation and this could be confirmed by redispersion of the emulsion by gently shaking. Above 0.8 mol dm−3 NaCl, the flocculation was strong and irreversible. In this case, the solvency of the medium for the PEO chains becomes poor, resulting in incipient flocculation. Further evidence of flocculation was also obtained from dynamic (oscillatory) measurements. Fig. 12.16 shows the variation of G󸀠 with NaCl concentration at various storage times. Below 0.8 mol dm−3 NaCl, G󸀠 shows a modest increase in G󸀠 with storage time over a period of two weeks, indicating weak flocculation. Above 0.8 mol dm−3 NaCl, G󸀠 shows a rapid increase in G󸀠 with increasing storage time, indicating strong flocculation. This strong (incipient) flocculation is due to the reduction of solvency of PEO chains (worse than θ-solvent) resulting in strong attraction between the droplets which are difficult to redisperse. 600 G’/Pa 400

2 weeks 1 week 500 400 300 200 1 day 100 0

0.2

0.4

0.6

0.8

1.0

1.2

CNaCl/mol dm–3 Fig. 12.16: Variation of G󸀠 with NaCl concentration.

12.9.4 Assessment and prediction of emulsion coalescence using rheological techniques As mentioned in Chapter 4, the driving force of emulsion coalescence is the thinning and disruption of the liquid film between the droplets. When two emulsion droplets come into contact, say in a cream layer or a floc, or even during Brownian collision, the liquid film between them undergoes some fluctuation in thickness; the thinnest part of the film will have the highest van der Waals attraction and this is the region where coalescence starts. Alternatively, the surfaces of the emulsion droplets may undergo fluctuation producing waves, which may grow in amplitude; the strongest van der Waals attraction is at the apices of these fluctuations and coalescence occurs by

12.9 Application of rheological techniques | 423

further growth of the fluctuation. One may define a critical film thickness below which coalescence occurs. The rate of coalescence is determined by the rate at which the film thins and this usually follows a first-order kinetics as given by equations (12.42) and (12.43). Providing the emulsion does not undergo any flocculation, the coalescence rate can be simply measured by following the number of droplets or average diameter as a function of time. A given volume of the emulsion is carefully diluted into the isotone solution of the Coulter counter and the number of droplets is measured. The average diameter can be obtained using laser diffraction methods (e.g. using the Mastersizer). By following this procedure at various time periods, one can obtain the coalescence rate constant K. Usually one plots log n or log d versus t and the slope of the line in the initial period gives the rate of coalescence K. Clearly, the higher the value of K, the higher the coalescence of the emulsion. An accelerated test may be used by subjecting the system to higher temperatures; usually the rate of coalescence increases with increasing temperature (although this is not always the case). One should be careful in the dilution procedure, particularly if the oil is significantly soluble (say greater than 10 ppm) in the isotone solution or in the tank of the Mastersizer. In this case, one should saturate the solution with the oil before diluting the concentrated emulsion for droplet counting or sizing. Emulsion coalescence can be assessed using rheological techniques as described in Chapter 1. In the absence of any flocculation, coalescence of an emulsion results in the reduction of its viscosity. At any given volume fraction of oil, an increase in droplet size results in a reduction in viscosity; this is particularly the case with concentrated emulsions. Thus, by following the decrease in emulsion viscosity with time one may obtain information on its coalescence. However, one should be careful in applying simple viscosity measurements, particularly if flocculation occurs simultaneously (which results in an increase in the viscosity). It is possible in principle to predict the extent of viscosity reduction on storage, if one combines the results of droplet size analysis (or droplet number) as a function of time with the reduction in viscosity in the first few weeks. Freshly prepared emulsions with various droplet sizes are prepared (by controlling the speed of the stirrer used for emulsification). The emulsifier concentration in these experiments should be kept constant and care should be taken that excess emulsifier is not present in the continuous phase. The viscosity of these freshly prepared emulsions is plotted versus the average droplet diameter. A master curve is produced that relates the emulsion viscosity to the average droplet size; the viscosity decreases monotonically with increasing average droplet size. Using the Coulter counter or Mastersizer one can determine the rate of coalescence by plotting log of the average droplet diameter versus time in the first few weeks. This allows one to predict the average droplet diameter over a longer period (say 6–12 months). The predicted average droplet diameter is used to obtain the vis-

424 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

cosity that is reached on storage using the master curve of viscosity versus average droplet size. The above procedure is quite useful for setting the limit of viscosity that may be reached on storage as a result of coalescence. With many creams, the viscosity of the system is not allowed to drop below an acceptable limit (which is important for the application). The limit that may be reached after one year storage may be predicted from the viscosity and rate constant measurements over the first few weeks. Another rheological method is to measure the yield value as a function of time. Since the yield value σ β of an emulsion depends on the number of contacts between the droplets, any coalescence should be accompanied by a reduction in the yield value. This trend is only observed if no flocculation occurs (this causes an increase in σ β ). The above change was measured using O/W emulsions that were stabilized with an A–B–A block copolymer of polyethylene oxide (PEO, A) and polypropylene oxide (PPO, B); Pluronic PEF127. 60 : 40 O/W emulsions were prepared using 0.5, 1.0, 1.5, 2.0, 3, 4 and 5 % emulsifier. Fig. 12.17 shows the variation of droplet size with time at various Pluronic PEF 127 concentrations. At emulsifier concentration > 2 % there is no change of droplet size with time, indicating absence of coalescence. Below 2 % the droplet size increased with time, indicating coalescence. Measuring the storage modulus G󸀠 as a function of time is perhaps the most sensitive method for predicting coalescence. G󸀠 is a measure of the contact points of the emulsion droplets as well as their strength. Providing no flocculation occurs (which results in an increase in G󸀠 ), any reduction in G󸀠 on storage indicates coalescence. The above trend was confirmed using the emulsions described above. The emulsions containing less than 3 % Pluronic PEF 127 showed a rapid reduction in G󸀠 when 8

0.5% 1.0% 2.0% 3.0% 4.0% 5.0%

Droplet size (μm)

7 6 5 4 3 2 1 2

4

6

8

10

12

14

Time/days Fig. 12.17: Variation of droplet size with time at various Pluronic PEF 127 concentrations.

12.9 Application of rheological techniques | 425

1000

G′/Pa

100

1 day

10

1 week 2 weeks 1 1

2

3

4

5

% Synperonic PEF 127 Fig. 12.18: Variation of G󸀠 with Pluronic PEF 127 concentration at various storage times.

compared with those containing > 3 % which showed virtually no change in G󸀠 over a two-week period. This is illustrated in Fig. 12.18. The correlation between the emulsion elastic modulus and coalescence rate can be easily represented if one calculates the relative decrease in G󸀠 after 2 weeks, relative decrease of G󸀠 = (

Ginitial − Gafter 2 weeks ) × 100. Ginitial

(12.48)

Fig. 12.19 shows the variation in the relative decrease of G󸀠 and relative increase in droplet size with Synperonic PEF127 concentration. The correlation between the relative decrease in G󸀠 and relative increase in droplet size as a result of coalescence is now very clear.

Relative increase in droplet size

Relative decrease in G′

100 80 60 40 20 0 1

2

3

4

5

% Synperonic PEF 127

Fig. 12.19: Correlation of relative decrease in G󸀠 with relative increase in droplet size.

426 | 12 Characterization, assessment and prediction of stability of colloidal dispersions

The cohesive energy Ec (given by equation 12.46) is the most sensitive parameter for assessing coalescence. Any coalescence results in a decrease in the number of contact points and causes a reduction in Ec . Using the above mentioned emulsions, Ec was found to decrease with increasing droplet size (as a result of coalescence). At and above 3 % Pluronic PEF 127, Ec remained virtually unchanged, indicating absence of coalescence. Fig. 12.20 shows the variation of the relative decrease of Ec with the relative increase in droplet size; the correlation is clear.

Relative increase in droplet size

Relative decrease in Ec

100 80 60 40 20 0 1

2

3

4

5

% Synperonic PEF 127

Fig. 12.20: Correlation of relative decrease in Ec with relative increase in droplet size.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

Lyklema J. Structure of the solid/liquid interface and the electrical double layer. In: Tadros TF, editor. Solid/Liquid dispersions. London: Academic Press; 1987. Tadros T. Interfacial phenomena and colloid stability, Vol. 1. Berlin: De Gruyter; 2015. Lyklema J. In: Bockris JO’M, Rand DAJ, Welch BJ, editors. Trends in Electrochemistry. New York: Plenum Publishing Corp.; 1977. Ven den Hul HJ, Lyklema J. J Colloid Sci. 1967;23:500. Penners NHG. The preparation and stability of monodisperse colloidal haematite (α-Fe2 O3 ). PhD thesis. Agricultural University, Wageningen, Netherlands. Tadros TF, Lyklema J. J Electroanal Chem. 1968;17:267. Lyklema J. J Electroanal Chem. 1968;18:341. Hunter RJ. Zeta potential in colloid science. London: Academic Press; 1981. Debye P. J Chem Phys. 1933;1:13. Bugosh J, Yeager E, Hovarka F. J Chem Phys. 1947;15:542. Yeager E, Bugosh J, Hovarka F, McCarthy J. J Chem Phys. 1949;17:411. Dukhin AS, Goetz PJ. Colloids and Surfaces. 1998;144:49. Oja T, Petersen GL, Cannon DC. US Patent 4,497,208 (1985). O’Brian RW. J Fluid Mech. 1988;190:71. O’Brian RW. J Fluid Mech. 1990;212:81.

References | 427

[16] [17] [18] [19] [20] [21] [22] [23]

O’Brian RW, Garaside P, Hunter RJ. Langmuir. 1994;10:931. O’Brian RW, Cannon DW, Rowlands WN. J Colloid Interface Sci. 1995;173:406. Rowlands WN, O’Brian RW. J Colloid Interface Sci. 1995;175:190. Tadros T. Dispersions of powders in liquids and stabilisation of suspensions. Weinheim: WileyVCH; 2012. Kissa E. Dispersions: Characterization, testing and measurement. New York: Marcel Dekker; 1999. Tadros T. Rheology of Dispersions. Weinheim: Wiley-VCH; 2010. Hunter RJ, Nicol SK. J Colloid Interface Sci. 1968;28:250. Neville PC, Hunter RJ. J Colloid Interface Sci. 1974;49:204.

Index adhesion tension 49 adsorbed layer thickness 244, 245 aggregation – breaking of 101 – rapid 33, 34 – slow 33, 34 antifoamers 268, 269 associative thickeners 186–189, 284–286 Bachelor equation 18 Bingham plastic 3 bottom up 84 bridging flocculation 204–206 Casson model 4 clay gels 288–290 coalescence of emulsions 410 cohesive energy density 16, 17 colloid stability 78, 79 comminution 107 complex modulus 14 concentric cylinder 2 cone and plate 2 condensation methods 75, 83, 84 contact angle 45, 46, 51 – advancing 56, 57 – concept of 47 – dynamic 70, 71 – measurement of 64, 66 – of various liquids 47 – receding 56, 57 contact angle hysteresis 56–58 controlled flocculation 191 creaming/sedimentation – assessment using Turboscan 392 – assessment using ultrasound velocity 393 creep curves 7–9 critical packing parameter 346, 348 Cross equation 5 cross-linked gels 207 defoaming 268, 269 depletion flocculation 26–29, 139, 191, 192, 202, 203 deposition of particles 351 dispersion polymerization 311–316

dilatant system 4 disjoining pressure 213, 214, 263 DLVO theory 153 Dougherty–Krieger equation 21, 244 Einstein equation 18 elastic floc model 32, 33, 41, 42 elastic response 246 electrostatic repulsion 76, 77 elongational viscosity 166 emulsification 145, 151 emulsifier – selection of 172 emulsion polymerization 297, 300, 303, 306, 307 emulsions – breakdown processes of 147 – coalescence of 213 – coalescence rate of 214 – creaming and sedimentation of 181–183 – prevention of 184–186 – flocculation of 196 – Ostwald ripening in 207, 211, 212 – phase inversion of 223 flocculation – assessment of 395, 396, 400, 402, 403 Flory–Huggins interaction parameter 245 Flory–Krigbaum theory 245 foam drainage – measurement of 275 foam films – critical thickness of 266 – drainage of 259–261 – rupture of 265 – stability of 256, 257 – stabilized by lamellar phases 267 – stabilized by micelles 266 – stabilized by mixed surfactants 267 – theories of stability of 266 – thickness of 259 foam inhibition – by hydrophobic particles 267 – by hydrophobic particles and oil 270, 271 foams 255 – electrical properties of 272

430 | Index

– electrokinetic properties of 273 – experimental techniques of study of 273 – mechanical properties of 271 – optical properties of 273, 274 – preparation of 255 – rheological properties of 271 – structure of 254 fractal dimension 32, 33 free energy of mixing 245 gel-forming materials 238, 239 gels 277 – based on surfactant systems 294, 295 – classification of 277 Gibbs–Deuhem equation 149 Gibbs–Marangoni effect 161–163 Good and Grifalco approach 52 hard-sphere dispersion 19, 20 Herschel–Bulkley model 4 hydrophilic-lipophilic balance (HLB) 172, 173, 175–179 interaction forces between droplets 153, 242 interfacial dilational modulus 158, 160 interfacial region 148 interfacial tension gradient 159 Laplace pressure 150 latexes 297 – in polar media 316 – polymeric surfactants for 307–311 liquid crystalline phases 141, 195 lipids 344 liposomes 343 loss modulus 14 membrane emulsification 169, 170 microemulsions 318 – characterization of 380 – by conductivity 336 – by dynamic light scattering 333 – by light scattering 330 – by NMR 339 – formulation of 339, 340 – free energy of formation of 326 – mixing film theory of 321 – solubilisation theory of 323, 324 – thermodynamic definition of 320 – thermodynamic theory of 325

microfluidization 107–109 mixing interaction 245, 246 multiple emulsions 234 – breakdown processes of 235, 238 – factors affecting stability of 239 – formulation of 240 – formulation variables of 239, 240 – main criteria of stability of 239 – rheological measurements of 248–252 – stability of 252, 253 – yield of 236 nucleation and growth 85 – effect of supersaturation on 86 Newton films 265 Newtonian systems 3 orthokinetic flocculation 200 oscillatory measurements 12, 13, 282 oscillatory sweep 15, 16 Ostwald ripening 91, 409 – effect of wetting agents on 122 – rate of 117 – prevention of 115, 116, 123 oxide gels 292–294 particle deposition 352 – experimental methods of studying of 360 – in linear regime 363–366 – in non-linear regime 370, 372–374 – on heterogeneous surfaces 275, 376, 377 particulate gels 288–290 particle/surface adhesion 378, 379 – Fox–Zisman critical surface tension approach 380 – Neuman equation of state approach 380, 381 Peclet number 17 phase angle shift 13 phase inversion temperature (PIT) 178, 179 polymer colloids 297 polymer gels 282–284 polymeric surfactants 242 power density 105, 165 precipitation kinetics 87–90 pseudoplastic system 3 reduced shear rate 19 residual viscosity 10 Reynold number 109, 163

Index |

rheological techniques 1 – for prediction of creaming/sedimentation 411–413, 416 – for prediction of coalescence 422, 423, 425, 426 – for prediction of flocculation 416, 418, 420, 421 rheology – of colloidal dispersions 17, 20 – of concentrated dispersions 18, 19 – of electrostatically stabilized dispersions 21–23 – of flocculated dispersions 25–28 – of sterically stabilized dispersions 22, 24 Rideal–Washburn equation 63, 99 rotor-stator mixer 100–103, 162–164 shear rate 1 shear stress 1 shear thickening 4 shear thinning 3 spreading coefficient 55 stability ratio 198, 199, 408 steady state 1 storage modulus 10, 11, 279, 280 strain relaxation 7, 281 strain sweep 15 stress relaxation 10, 11, 279, 280

431

surface elasticity 262 surface heterogeneity 58 surface roughness 57 surface tension 51 surface viscosity 262 surfactant and polymer adsorption 389–391 thixotropy 5, 7 velocity gradient 104 vesicles 343 – driving force for formation of 345 – formation of 347 – stabilization of 349, 350 viscoelastic liquid 12 viscoelastic solid 12 viscosity–shear rate relationship 2 Weber number 166 Wenzel equation 57, 58 wet milling 107, 109–111, 114, 115 wetting agents 65, 66, 97, 100 Wilhelmy plate method 67–69 work of adhesion 50, 98 work of cohesion 51 work of dispersion 62, 98 Young’s equation 48, 49, 97