Physical Chemistry of Polymers: A Conceptual Introduction 9783110713268, 9783110713275

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Sebastian Seiffert Physical Chemistry of Polymers

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Sebastian Seiffert

Physical Chemistry of Polymers A Conceptual Introduction 2nd Edition

Author Prof. Dr. Sebastian Seiffert Johannes Gutenberg University Mainz Department of Chemistry Duesbergweg 10–14 D-55128 Mainz, Germany [email protected]

ISBN 978-3-11-071327-5 e-ISBN (PDF) 978-3-11-071326-8 e-ISBN (EPUB) 978-3-11-071339-8 Library of Congress Control Number: 2022944934 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. © 2023 Walter de Gruyter GmbH, Berlin/Boston Cover image: Sebastian Seiffert Typesetting: Integra Software Services Pvt. Ltd. Printing and binding: CPI books GmbH, Leck www.degruyter.com

Foreword to the 2nd edition The original foreword to this book’s first edition said that its intention is to converge polymer chemistry and physics, such to also converge the striking specificity and ubiquitous universality of polymers. Based on that intent, the aim of this book was sketched to be a bridge between polymeric structures & dynamics and their resulting properties, thereby arcing between macromolecular science and engineering. This bridge has been identified to be the field of physical chemistry of polymers. It was also said that a major challenge in that context is that the concepts in physical chemistry of polymers commonly appear to be rather new and unknown to students, and that it is therefore a mission of this book to converge deep conceptual understanding (with many excursions into the underlaying basic physical chemical concepts) with solid mathematical formalisms. All the foresaid still holds true just as well for this present second edition. Yet, there is a further aspect to be added: Just when the first edition appeared, the world fell into a pandemic era, which shifted minds, societies, and universities. It also shifted teaching formats, catapulting us into the digital age. Well, actually we should have been in that era way before: In the 15th century, the invention of movable-type printing by Gutenberg in Mainz ushered in the second media revolution in human history. (The first before that was the transition from spoken to written language, and the third after that was the advent of electronic mass media.) Today, the world is in the fourth media revolution: digitization and networking. However, until recently, universities (almost) exclusively taught in a way that dates back to pre-Gutenberg times: frontal lecturing. The pandemic of the SARS-CoV2 pathogen in 2020, just when the first edition of this book appeared, eventually propelled the academic world into the 21st century and established digital teaching formats. This textbook provides a basis for such a format in the subject of physical chemistry of polymers. It is a basis for a blended learning course, i.e., a teaching format that consists of a self-study phase (knowledge acquisition), a digital feedback unit (knowledge anchoring), and an interactive classroom unit (knowledge application). For this purpose, the book is divided into 23 thematically focused and modularly applicable lesson units, which can be conceived as 90-min lectures each. This number marks a plus of five new lesson units to the former eighteen ones of the book’s first edition, adding to its general polymer-physical concepts some specific material-centered contents. With that, the second book edition can be used in its entirety for a 12-week course of 2 × 90 minutes per week (this corresponds to 4 SWS in the German system). Alternatively, it can also be used for two separate, sequential courses of only 1 × 90 minutes per week each (corresponding to 2 SWS in the German system), one covering lesson units 1–9 and one adding lesson units 10–23 (perhaps cutting unit 19 if time is too tight).

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Foreword to the 2nd edition

The lesson-unit blocks each comprise about 10–20 pages, each to be worked through in about 90 minutes of self-study. Each unit concludes with a set of conceptual questions in a multiple-choice format. Lecturers can incorporate some of those into an e-learning platform so that students can solve them there right after reading the teaching unit. Many e-learning platforms even allow in-situ feedback texts to be added to the respective answer options, which then immediately indicate to the students whether and why their selected answer option is incorrect or correct. The author of this book is happy to provide lecturers such answer texts for the questions included here upon request. From the answer statistics, which can also be easily generated in many e-learning platforms, lecturers can then see which aspects of the topic are already well understood in the student group and which are not, such to then tailor the subsequent classroom session accordingly. In addition, further of these multiple-choice questions can be used in this attendance unit to further deepen the material. This may best be realized with the peer instruction method: In this approach, a conceptual multiple-choice question is projected in the classroom, and the students first answer individually with the aid of an audience response system (“clicker system”, e.g., smartphone-based). The response statistics, which are then also projected by the teacher, give the students direct and anonymous feedback on how their own selected response fits into the overall cohort. Afterwards, the students are asked to exchange themselves with their peers around them in groups of two or three, with the task to convince them of the correctness of their first chosen answer. A second round of voting after a few minutes will then almost always produce the correct result with a clear majority, simply because those who have had the correct answer in the first place also have better arguments and can understand and eliminate any gaps in their peers’ understanding — much better than any lecturer ever could. That way, the students take an active role in deepening their knowledge, are stimulated and motivated, and are interactively involved in the learning process on several levels . . . whereas the lecturer merely takes the role of a moderator. With that, the method fulfills one of the core claims of its inventor, Prof. Eric Mazur (Harvard): “good teaching is to help students learn.” As attractive as this teaching method may seem at first — it stands and falls with the quality of the questions asked; and even more with the quality of the given answer options. If it is immediately obvious which answer is the correct one, the method is at best entertaining, but not particularly instructive. If, by contrast, one of the answer options represents the “most common misconception“, i.e., the most common and typical initial misunderstanding that students often have, then this can be specifically addressed and eliminated. This is exactly where this book comes in. It aims to provide well thought-out and prepared material for all three of the above-mentioned teaching phases (self-study, e-learning feedback, and in-person consolidation unit).

Foreword to the 2nd edition

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A great acknowledgement goes to M.Ed. Julia Windhausen, who was a master student in the author’s team at Mainz University in 2021 and co-developed most of the foresaid multiple choice questions together with the author during that time. Further assistance in that task was provided by PD Dr. Wolfgang Schärtl, who is a university lecturer in the author’s team and who also co-authored other De-Gruyter textbooks on Physical Chemistry with him. Mainz, spring 2023

Foreword to the 1st edition Polymer science is a field that requires good knowledge about both chemistry and physics. This is because it is chemistry that sets the specificity of each polymer, but behind that, it is physics that sets the universality of the properties of all polymers. The convergence of both lies the ground for a huge variety of applications of polymers. In such, though, a user (=a customer on the market) usually doesn’t care about the material itself, but instead, only about the function that it provides. Thus, material designers (be it in industry or academia) are demanded to develop materials that provide functions of interest, whereby customers, however, do not actually care about the beauty of the material that does that, but instead, only about its utility. It is therefore obligatory for material designers to translate the desired functions (=the customers’ wishes) into measurable physical material parameters, and then to further translate these parameters into chemical structures. This translation can only be done on the basis of a sound understanding of structure–property relations of polymers. And that is what this book is targeted at. Physical chemistry of polymers intends to bridge the structure of polymers, as provided by chemistry, to their properties, as captured by physics. Once this bridge is erected, it can be passed in either direction, thereby allowing us to understand from what structural characteristics a certain beneficial property comes from, or vice versa, to predict what structure will provide a property (and therewith a function) of interest. A challenge in this endeavor is that at a first view, the concepts and treatments in physical chemistry of polymers appear to be rather new and unknown to students. It is therefore another goal of this book to familiarize its readers with these approaches, and to demonstrate that they are actually not at all new, but instead, quite related to concepts known from elementary physical chemistry. Hence, in this book, many relations are drawn to such classical physical chemistry contents, and the whole focus of this book is generally more on conceptual universality rather than on detailed specificity. For the purpose of the latter, all illustrations in this book are styled as lecture-like schematics. The content is ordered into six chapters that arc from fundamental polymer physical-chemical principles to actual structure–property relations, including a touch on experimental approaches to characterize both of the latter. On top of this six-chapter structuring lies a secondary structure that portions the content into 18 lesson units, which could be conceived as 90-min lectures each. The author teaches those units in two sequential classes, Physical Chemistry of Polymers 1 on bachelor level (Lessons 1–9) and Physical Chemistry of Polymers 2 on master level (Lessons 10–18). With a semester length of commonly 12 weeks, this portioning gives ample time to not only cover the respective content, but also to leave some flexibility for additional lessons to be filled with in-depthment, questions and answers, and a practicing exam. Alternatively, in a long semester of 14 weeks (as it is the case in the winter term in Germany), the full content can be covered if four lessons are left out; for example, https://doi.org/10.1515/9783110713268-203

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these may be Lessons 9, 12, 17, and 18 if there shall be no such big emphasis on analytical characterization of polymer systems, or these may be Lessons 12, 15, and 16 (plus one more) if there shall be no such a deep focus on rheology. The year of appearance of the first edition of this book is the inspiring Staudinger year 2020, the year of polymers. Based on this big anniversary of our field, this book aims to converge chemistry and physics of polymers and to make it understandable and appetizing to students at levels as early as possible. Mainz, spring 2020

Contents Foreword to the 2nd edition

V

Foreword to the 1st edition

IX

Lessons

XV

Literature basis

XVII

1 Introduction to polymer physical chemistry 1 1.1 Targets of this book 1 1.2 Definition of terms 3 1.3 Irregularity of polymers 7 1.3.1 Nonuniformity of monomer connection in a chain 7 1.3.2 Polydispersity in an ensemble of chains 9 1.3.2.1 Derivation of the Schulz–Flory distribution 13 1.3.2.2 Derivation of the Schulz–Zimm distribution 16 1.3.2.3 Characteristic averages of the Schulz–Flory and Schulz–Zimm distribution 17 1.4 Properties of polymers in contrast to low molar mass materials 19 1.5 Polymers as soft matter 20 1.6 History of polymer science 22 2 2.1 2.1.1 2.1.2 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8 2.3 2.4 2.4.1

Ideal polymer chains 31 Coil conformation 32 Micro- and macroconformations of a polymer chain 32 Measures of size of a polymer coil 35 Simple chain models 37 The random chain (phantom chain, freely jointed chain) 37 The freely rotating chain 39 The chain with hindered rotation 40 The Kuhn model 41 The rotational isomeric states model 43 Energetic versus entropic influence on the shape of a polymer 45 The persistence length 46 Summary 48 The Gaussian coil 51 The distribution of end-to-end distances 53 Random-walk statistics 53

XII

2.5 2.6 2.6.1 2.6.2 2.7 3 3.1 3.2 3.3 3.4 3.4.1 3.4.2 3.5 3.6 3.6.1 3.6.2 3.6.3 3.6.4 3.6.5 3.6.6

Contents

The free energy of ideal chains 58 Deformation of ideal chains 59 Entropy elasticity 59 A scaling argument for the deformation of ideal chains Self-similarity and fractal nature of polymers 64 Real polymer chains 71 Interaction potentials and excluded volume 72 Classification of solvents 77 Omnipresence of the Θ-state in polymer melts 79 The conformation of real chains 83 Coil expansion 83 Flory theory of a polymer in a good solvent 85 Deformation of real chains 89 Chain dynamics 96 Diffusion 96 The Rouse model 98 The Zimm model 100 Relaxation modes 102 Subdiffusion 104 Validity of the models 106

4 Polymer thermodynamics 111 4.1 The Flory–Huggins mean-field theory 111 4.1.1 The entropy of mixing 115 4.1.2 The enthalpy of mixing 118 4.1.3 The Flory–Huggins parameter as a function 121 4.1.4 Microscopic demixing 124 4.1.5 Solubility parameters 125 4.2 Phase diagrams 131 4.2.1 Equilibrium and stability 131 4.2.2 Construction of the phase diagram 137 4.2.3 Mechanisms of phase separation 143 4.2.3.1 Spinodal decomposition 143 4.2.3.2 Nucleation and growth 143 4.3 Osmotic pressure 148 4.3.1 Connection of the second virial coefficient, A2, to the Flory–Huggins parameter, χ, and the excluded volume, ve 5 5.1 5.1.1

62

Mechanics and rheology of polymer systems Fundamentals of rheology 159 Elementary cases of mechanical response

159 159

151

Contents

5.1.2 5.1.3 5.2 5.2.1 5.2.1.1 5.2.1.2 5.2.1.3 5.3 5.4 5.5 5.5.1 5.5.2 5.5.2.1 5.5.2.2 5.6 5.6.1 5.6.2 5.6.3 5.7 5.7.1 5.7.2 5.7.3 5.8 5.8.1 5.8.2 5.9 5.9.1 5.9.2 5.9.3 5.10 5.10.1 5.10.2 5.10.3 5.10.4 6 6.1 6.2 6.3 6.4 6.4.1 6.4.2

Different types of deformation in rheology 161 The tensor form of Hooke’s law 163 Viscoelasticity 167 Elementary types of rheological experiments 168 The relaxation experiment 168 The creep test 168 The dynamic experiment 169 Complex moduli 170 Viscous flow 174 Methodology in rheology 180 Oscillatory shear rheology 180 Microrheology 181 Active microrheology 181 Passive microrheology 182 Principles of viscoelasticity 184 Viscoelastic fluids: the Maxwell model 184 Viscoelastic solids: the Kelvin–Voigt model 191 More complex approaches 193 Superposition principles 198 The Boltzmann superposition principle 199 The thermal activation of relaxation processes 201 Time–temperature superposition 202 Viscoelastic states of a polymer system 209 Qualitative discussion of the mechanical spectra 209 Quantitative discussion of the mechanical spectra 211 Rubber elasticity 217 Chemical thermodynamics of rubber elasticity 218 Statistical thermodynamics of rubber elasticity 220 Swelling of rubber networks 227 Terminal flow and reptation 232 The tube concept 233 Rouse relaxation and reptation 234 Reptation and diffusion 237 Constraint release 239 Scattering analysis of polymer systems Basics of scattering 243 Scattering regimes 246 Structure and form factor 247 Light scattering 255 Static light scattering 255 Dynamic light scattering 264

243

XIII

XIV

6.4.3 6.4.3.1 6.4.3.2 6.4.3.3

Contents

Light scattering on polymer gels 267 Polymer networks and gels 267 Static light scattering on gels 268 Dynamic light scattering on gels 270

7 States of polymer systems 279 7.1 Polymer solutions 279 7.1.1 Formation 279 7.1.2 Concentration regimes 280 7.1.3 Dilute solutions 282 7.1.3.1 Structure 282 7.1.3.2 Dynamics 283 7.1.4 Semi-dilute solutions 292 7.1.4.1 Peculiarity 292 7.1.4.2 Structure 292 7.1.4.3 Dynamics 296 7.2 Polymer networks and gels 304 7.2.1 Fundamentals 304 7.2.2 Gelation 305 7.2.2.1 One-dimensional bond percolation 307 7.2.2.2 Two-dimensional bond percolation 307 7.2.2.3 Mean-field model of gelation 308 7.2.2.4 The gel point 312 7.2.3 Structure of polymer networks and gels 315 7.2.4 Dynamics of polymer networks and gels 317 7.3 Glassy and crystalline polymers 321 7.3.1 The glass transition 321 7.3.1.1 Fundamentals 321 7.3.1.2 Conceptual grounds of the glass transition 324 329 7.3.1.3 Structure–property relations for Tg 329 7.3.1.4 Experimental assessment of Tg 7.3.2 Polymer crystallization 330 7.3.2.1 Fundamentals 330 7.3.2.2 Partial crystalline microstructures 330 7.3.2.3 Experimental assessment of the degree of crystallinity 335 336 7.3.2.4 Structure–property relations for Tm 7.3.3 Comparison of the glass and the crystallization transition 337 8 Index

Closing remarks 345

343

Lessons LESSON 1:

INTRODUCTION

1

LESSON 2: IDEAL CHAINS LESSON 3:

31

GAUSSIAN COILS AND BOLTZMANN SPRINGS

LESSON 4: REAL CHAINS LESSON 5:

71

FLORY EXPONENT

83

LESSON 6: CHAIN DYNAMICS

96

LESSON 7:

FLORY–HUGGINS THEORY

LESSON 8: PHASE DIAGRAMS LESSON 9: OSMOTIC PRESSURE LESSON 10: RHEOLOGY

111

131 148

159

LESSON 11: VISCOELASTICITY

167

LESSON 12: PRACTICE AND THEORY OF RHEOLOGY LESSON 13: SUPERPOSITION PRINCIPLES LESSON 14: MECHANICAL SPECTRA LESSON 15: RUBBER ELASTICITY LESSON 16: REPTATION

51

180

198 209

217

232

LESSON 17: SCATTERING METHODS IN POLYMER SCIENCE LESSON 18: LIGHT SCATTERING ON POLYMERS LESSON 19: LIGHT SCATTERING ON POLYMER GELS

255 267

243

XVI

Lessons

LESSON 20: DILUTE POLYMER SOLUTIONS

279

LESSON 21: SEMI-DILUTE POLYMER SOLUTIONS LESSON 22: POLYMER NETWORKS AND GELS LESSON 23: POLYMERS IN BULK SOLID STATES

292 304 321

Literature basis Isaac Newton once pointed out his work to be outstanding as he could “stand on the shoulders of giants” (Letter from Sir Isaac Newton to Robert Hooke; Historical Society of Pennsylvania). In the same sense, this textbook is based on the following seminal existing ones: M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, New York 2003 H. G. Elias: Makromoleküle, Wiley VCH, Weinheim 1999 and 2001 B. Tieke: Makromolekulare Chemie, Wiley VCH, Weinheim 1997 B. Vollmert: Grundriss der Makromolekularen Chemie, Springer, Berlin Heidelberg 1962 J. S. Higgins, H. C. Benoît: Polymers and Neutron Scattering, Clarendon Press, Oxford, 1994 Among those, the book by Rubinstein and Colby has given particular inspiration for the present one. In the author’s view, these colleagues’ seminal book has a physicsbased approach (as its title also says) targeted at grad students (as its backcover-text says), which would correspond to readers at a PhD-student level in the European system. As a complement to that, the present textbook provides a physical-chemistrycentered viewpoint addressing both grad and undergrad students, which corresponds to students also on bachelor and master levels in the European system. This book is based on a script to the author’s lecture series “Physical Chemistry of Polymers” at JGU Mainz. The writing of this script has been assisted by Dr. Willi Schmolke in the years 2018 and 2019, who was a PhD student in the author’s lab at Mainz then; special thanks go to Willi for this assistance with the book’s fundament, along with further thanks to PD Dr. Wolfgang Schärtl for proofreading it. An even deeper fundament has been laid by another person named “Willi”: the author’s own respected teacher in the field of polymer science at TU Clausthal in the years 2003–2008, Prof. Dr. Wilhelm Oppermann.

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1 Introduction to polymer physical chemistry LESSON 1: INTRODUCTION As a starting point of our endeavor, the first lesson of this book will introduce you to the fundamental terms and basics in the field of polymer science, along with its historical development. You will learn what a polymer actually is, why these large molecules are interesting, and what the fundamental differences are in their treatment compared to classical materials that are built up of small molecules.

1.1 Targets of this book The prime goal of polymer physical chemistry is to understand the relations between structure and properties of polymers. Once such understanding is achieved, molecular parameters can be rationally connected to macroscopic behavior of polymer-based matter. This connection bridges the fields of polymer chemistry and polymer engineering, as shown in Figure 1, thereby allowing polymer materials to be tailored by rational design. Even though these fields may first seem far apart and separated (by an impassable water body in Figure 1), the discipline of polymer physical chemistry arcs and connects them. (And in the end, by looking underneath the water, we realize that they are actually connected inherently.) To achieve this goal, along our way, we need to learn some fundamental approaches in the physical chemistry of soft condensed matter; we will also learn why polymers are a prime example of such matter. First, we need to consider the multibody nature of each polymer chain as well as the myriad of different local microscopic conformations that it can adopt. This requires us to limit our focus on averages over all these individual states, which we obtain by suitable statistical treatment as well as mean-field modeling. Second, as a direct result of such consideration of the conformational statistics of polymer chains, we will see that pretty much of all their properties are coupled to each other in the form of scaling laws. Hence, scaling discussions will be a frequent means in this book. With this treatment, we will recognize that polymers exhibit both specificity and universality. As an example, the relation between the size and the mass of a polymer chain follows a power-law proportionality, independent of what polymer exactly we look at; this is universality set by physics. The proportionality factor, however, depends on the specific polymer at hand; this is specificity set by chemistry. As a further example, note that all polymers exhibit a glass-transition temperature, Tg, at which they

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1 Introduction to polymer physical chemistry

Figure 1: Polymer physical chemistry builds a bridge between polymer chemistry, which focuses on the primary molecular-scale structure of polymer chains, and polymer engineering, which focuses on the macroscopic properties of polymer materials. This connection allows for a rational design of polymer-based matter to obtain tailored materials. To erect the bridge, three pillars are required: single chain statistics, multichain interactions, and chain dynamics. These pillars are in the focus of the second, third, and fourth chapters of this book. The final bridge on top of this basis is in the focus of the fifth chapter, which sheds a particular focus on polymers’ most important kind of properties: their mechanical ones.

change from a hard, glassy state to a soft, leather-like state upon heating (and vice versa). They do so because of universal physics shared by all polymers1. However,

 The glass transition is a phenomenon actually not only seen on polymers; basically all matter can show it. When cooling a liquid, in classical chemistry we commonly see it to solidify by crystallization. If the cooling is done rapidly, however, there is not enough time for the molecules to arrange themselves on well-defined lattice positions, but instead, we then observed freezing of the molecular dynamics only. The result is a glass: an amourphous structure (like in a fluid) with no translational dynamics (like in a solid). In polymers (and colloids as well), this kind of solidification is seen often (we might actually even say “always”), because the large size of polymers and colloids, and even more the chain connectivity of polymers, requires quite long times to undergo crystrallization. In polymers, it is not the chains as a whole that arrange themselves on a crystal lattice, but the monomeric segmental units within the chains, which requires high regularity of monomer interconnection (even proper stereo-regularity) to actually be able to do so, and long time to manage to do so. Both is often not given, and so glassy solidification is actually much more frequent upon cooling than crystallization for polymers. The point where such glassy solidification (also named “vitrification”) happens depends on the ease of monomer-segmental rotatability along the chain, because this is the primary process by which the monomer units are to be arranged on a lattice. This ease of rotatability, in turn, depends on the local primary monomer structure, such as the bulkiness and interactions of the chemical side groups along the chain backbone. This is what sets the chemical specificity of the individual glass-transition temperature of each polymer, whereas the general occurrence of a glass-transition phenomenon in polymers is physical universality.

1.2 Definition of terms

3

the exact value of each polymer’s Tg is determined by each polymer’s specific chain chemistry.

1.2 Definition of terms The term macromolecule is derived from the Greek macro, meaning “large”, and the word molecule in its usual meaning.2 Hence, this term refers to large and heavy molecules, with sizes, r, in the range of some ten to hundred nanometers, and molar masses, M, in the range of 104–107 g·mol–1, as shown in Figure 2. Delimited from that, the term polymer, which is derived from the Greek poly and meros meaning “many” and “parts”, refers to chain molecules constituted of a sequence of many repeating units. These units, named monomers, are usually covalently jointed.3 Therefore, whereas the term macromolecule generally refers to any sort of large molecules, the term polymer specifically refers to those built of a repeating sequence of monomer units. Hence, each polymer is a macromolecule, but not each macromolecule is a polymer.

Figure 2: Molar mass, M, and size range, r, of various atomic and molecular structures.

Somewhat doubtful in this context are biopolymers such as proteins and DNA. On the one hand, one may argue that they are composed of a repeating sequence of units, which are peptides in the case of proteins and nucleotides in the case of DNA, justifying them to be termed polymers. On the other hand, despite this repetition of peptides or nucleotides, it is not a monotonic sequence of just one out of the 20 natural amino acids or just one out of the four different nucleobases that are available in nature, but a diverse variety of sequences of them along the chain. This  Note: the term molecule actually derives from the Latin molecula, meaning “small mass”; as such, the word macromolecule would mean “large small mass”, which is sort of an oxymoron.  A rather new class of self-assembled matter is supramolecular polymers; it is based on chains that are noncovalently jointed.

4

1 Introduction to polymer physical chemistry

argument may then question the adequateness of terming these polymers. This is, however, a pea-picking discussion. Following this line of thought, even random copolymers, which are actually well accepted to be termed polymers, should not be named such. We therefore refrain from this argument. Instead, we point out that by virtue of their ability to store information in their primary monomer sequence, which in turn is a basis for their highly specific function and functionality, proteins and DNA are nothing less than the link between matter and life, that is, between chemistry and biology. The International Union of Pure and Applied Chemistry (IUPAC) defines a polymeric molecule as follows: A molecule of high relative molecular mass, the structure of which essentially comprises the multiple repetition of units derived, actually or conceptually, from molecules of low relative molecular mass. In many cases, [. . .], the addition or removal of one or a few of the units has a negligible effect on the molecular properties.

This definition does not only apply to molecular properties but also to macroscopic properties, as can be recognized when considering the example of simple hydrocarbons in Figure 3. Starting from methane, a family of alkanes and polyethylenes is obtained by adding –CH2– repeating units. For the first 17 additions,

Figure 3: Boiling and melting points, Tbp and Tmp, of alkanes and polyethylenes with N methylene (–CH2–) units. Whereas the boiling point increases with N and eventually becomes inexistent from N = 17 on (because beyond that, such high temperatures would be necessary for boiling that the molecules would break rather than boil), the melting point first increases with N but then eventually levels off in a plateau. In this plateau, addition or removal of a few –CH2– units does not change TM considerably. Picture redrawn from H. G. Elias: Makromoleküle, Bd. 1 – Chemische Struktur und Synthesen (6. Ed.), Wiley VCH, 1999.

1.2 Definition of terms

5

the boiling point, Tbp, is raised sharply, until then, further addition leads to solid compounds that decompose instead of boiling at high temperatures. An upward trend is also observed for the melting point, Tmp, up to even the first about hundred additions. Then, however, the picture changes. Further addition does no longer change the melting point; instead, it has reached a plateau, which denotes the polymeric regime. In this regime, addition or removal of single or few –CH2– repeating units has practically no impact on the material properties such as Tmp. The materials’ appearance in this range of high N is that of hard solids rather than that of oily or even volatile liquids at low N. From the above notion, we may conclude that polymers are macromolecules composed of a repeating sequence of units; these units are sometimes termed monomer units and sometimes termed structural units. Note the difference between these terms: the term monomer unit usually refers to the chemical species from which the polymer has been made, or more precisely, to the chemical sequence along the main chain into which the actual monomer has turned into upon polymerization, whereas the term structural unit refers to the smallest possible repetitive entity along the chain. Often this is the same, but sometimes it is not. For example, consider polyethylene again. In this polymer, the monomer is ethylene, H2C=CH2, which turns into a monomer unit of –H2C–CH2– upon polymerization, whereas the smallest possible repetitive unit, that is, the structural unit, is just the methylene group, –CH2–. Similarly, in polyamide, [–HN–(CH2)x–NH–CO–(CH2)y–CO–]n, the structural unit is –HN–(CH2)x–NH–CO–(CH2)y–CO–, whereas the monomer units are –HN–(CH2)x–NH– and –CO–(CH2)y–CO–, resulting from the respective diamines and diacids upon polymerization. With the latter examples, we have touched upon the process of polymer formation, which is generally named polymerization. This process can occur in many different ways and is a matter of own independent textbooks and lectures. For the sake of our consideration, let us just briefly and rather conceptually summarize its bottom line. In general, polymers are synthesized by the consecutive reaction of monomers. These are molecules that carry chemical groups which can form connections to each other, most typically either double bonds that can get opened to become reactive radicals that may then attack other double bonds on other monomer molecules, or pairs of mutually connectable functionalities such as nucleophilic plus electrophilic moieties, for example, alcohols or amines plus carboxylic acid groups. These different kinds of polymerizable groups can then lead to polymerization via two different pathways, as sketched illustratively in the following conceptual schematic, in which people symbolize the monomers and form chains in two different ways of growth.

6

1 Introduction to polymer physical chemistry

In the case of double-bond-containing monomers, a small fraction of them can get attacked by a chemical initiator (symbolized by the “Go” flag in our schematic) to become a radical species. This activated species then quickly attaches to a second monomer, which forms a bond with it and transfers its active radical nature to that attached partner, which in turn does the same with a third monomer, and so forth. As a result, the monomers quickly attach to each other to form a chain one by one. This mechanism is referred to as chain-growth polymerization, not because its result is a polymer chain, but because it occurs as a chain reaction, like a domino cascade. Its predominant realization is by radical polymerization, as already indicated in the process description just given. In our schematic sketch, this mechanism is illustrated as people serially joining hands, whereby it is always the terminal person in the chain that reaches out to the next. In the case of nucleophilic plus electrophilic functionalized bifunctional monomers, by contrast, these species mutually add to one another in steps, to first form dimers, then tetramers, then octamers and so forth, referred to as step-growth polymerization. Its predominant realization is by polyaddition or, if the monomer connection occurs under spin-off of a small-molecule by-product (most typically water), by polycondensation. In our schematic sketch, this mechanism is illustrated as people sequentially joining hands in larger and larger groups. Chemical modification of the resulting polymers, such as attaching further functional entities to them (like sites to bind molecular cargo or to make the polymers attach to surfaces or biological tissue) or labeling the polymers with dyes or certain isotopes (most importantly deuterium) can be achieved either at the monomer level before the polymerization, or at the polymer level after the polymerization. In our schematic illustration, such modification could mean that we equip some of the persons in the chain (either before or after their incorporation in the chain) with some kind of functional belt, vest, or backpack or with a headlight; it could also mean that one or some of our persons in the chain can be somehow distinguished from the others, for example, by wearing a different shirt. Such modification may also occur in the form of threedimensional interconnection of the chains in space, referred to as cross-linking. In our schematic, this would mean that two people on two different chains would tie together somehow, thereby joining the chains.

1.3 Irregularity of polymers

7

Because polymeric molecules consist of many monomeric units (hundreds to thousands), they can be conceived in a coarse-grained view, without specific accounting for their local molecular-scale atomic structure, but instead, simply viewed and sketched globally in the form of curved or curled lines, as also indicated in our abovementioned two schematics; this is the way how we will actually draw and conceive polymers almost all over this textbook.

1.3 Irregularity of polymers Polymers aren’t regular. This is because they are made by statistical processes during most polymerization reactions, leading to inherent nonuniformity even if the basic chain-propagation reaction step proceeds perfectly. On top of that, if there is a nonperfect course of that step, each “side product” formed during it will be incorporated into the chains and create irregular local constitutions. As a result, polymers display different types of nonuniformity and irregularity.

1.3.1 Nonuniformity of monomer connection in a chain Even along each single chain, there is structural irregularity if multiple ways of monomer–monomer interconnection are possible. One type of such irregularity is the three different possibilities of head-to-head versus head-to-tail versus tail-to-tail interconnection of asymmetric monomers such as the broad class of vinyl compounds, as shown in Figure 4.

Figure 4: Head-to-head versus head-to-tail versus tail-to-tail interconnection of asymmetric monomers such as vinyl compounds.

A second type of irregularity resulting from the connection of such asymmetric monomers is different stereochemistries along the chain, referred to as tacticity, as shown in Figure 5. This property is of high relevance when polymers are crystallized, which does only work with isotactic or syndiotactic chains. This is because crystallization requires atoms or molecules to be able to arrange themselves regularly on a lattice. In polymers, though, this is tough, because the molecules (=the

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1 Introduction to polymer physical chemistry

Figure 5: Different types of tacticities in vinyl polymers.

structural units) are connected to one another in the form of chains. Only if this connection is very regular, the building blocks can form a regular lattice. In the case of diene-monomers, even more ways of interconnectability exist, as shown in Figure 6.

Figure 6: Different possibilities of diene-monomer interconnection in the example of isoprene.

1.3 Irregularity of polymers

9

1.3.2 Polydispersity in an ensemble of chains On top of the local irregularity of the monomer connectivity along each individual chain, another relevant and very frequent kind of irregularity in polymer systems is non-uniformity of the chain lenghs and molecular weights (unit: Da) or molar massess (unit: g mol–1) in ensembles of multiple chains, referred to as polydispersity.4 As a result of the statistical nature of most polymerization processes, the distribution of these quantities can be quite marked. A typical shape of a distribution curve, in two common ways of representation, is shown in Figure 7.

Figure 7: Schematic of a typical shape of a molar mass (unit: g mol–1) or molecular weight (unit: Da) distribution in a polymer sample, represented as the frequency of occurrence (typically a relative mass percentage), F, plotted on the ordinate, of different molar masses in the sample, M, plotted on the abscissa. The upper-right inset shows a different representation of this distribution, which is the integral of all molar masses, plotted on the ordinate, up to a specific one considered on the abscissa.

For practical purposes, it is more convenient and often sufficient to not deal with the full distribution but just with some characteristic averages of it. These are the number-average molar mass, Mn, the weight-average molar mass, Mw, and the centrifugal-average molar mass, Mz, as represented by dashed lines in Figure 7. These averages may be calculated from the so-called moments of the distribution function, which are generally defined as follows:

 In a strict view, the word “polydispersity” is a pleonasm. The word “disperse” already refers to an ensemble with multiple different elements (here: multiple different chain lengths or chain masses, alternatively expressed as chain molar masses or molecular weights), so there is actually no need for the prefix “poly”.

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1 Introduction to polymer physical chemistry

In statistics, the moments of a distribution quantify its characteristic properties. The first moment generally corresponds to the arithmetic average, the second moment to the width of the distribution, and the third moment to its skewness. In the formula above, the argument g denotes whether number fractions, weight fractions, or z-fractions are used to weigh the contributions of different Mi in the right-hand side calculation. With this definition, we get the three characteristic average molar masses as follows: P ð1Þ X μn ðMÞ i Ni Mi = P = xi Mi – Number average: Mn = ð0Þ μn ðMÞ i Ni i where xi is the mole fraction of species i. P P ð1Þ ð2Þ Ni Mi2 μn ðMÞ X μw ð M Þ Wi Mi i – Weight average: Mw = ð0Þ = P = = Pi = ð1Þ wi Mi μw ðMÞ μn ðMÞ i Wi i Ni Mi i where wi is the weight fraction of species i. P P P ð1Þ ð2Þ Ni Mi3 μðn3Þ ðMÞ Wi Mi2 μw ðMÞ Zi Mi μz ðMÞ i i – z-average: Mz = ð0Þ = P = Pi = P = ð1Þ = ð2Þ 2 μz ðMÞ μw ðMÞ μn ðMÞ i Zi i Wi M i i Ni Mi The number average puts the same emphasis on short and long chains in the sample and simply averages over their weight or molar mass by accounting for their number fraction in the sample. An experimental means to determine this average value is by methods that probe phenomena that specifically depend on the number of molecules in a sample; a prime example of such methods is those that probe colligative properties such as the osmotic pressure of a polymer solution. In contrast to that, the weight average puts more emphasis on the long and therefore heavy chains in the sample, as it averages over them not by accounting for their number fraction in the sample but for their contribution to the total sample weight. As a result, the heavy chains in a sample receive a greater pronunciation in the averaging. The most typical experimental means to probe this average value is by static light scattering. The z-average pronounces the heavy chains in the sample even

1.3 Irregularity of polymers

11

more severely. The standard technique to probe this average value is by ultracentrifugal analysis. We may illustrate the difference of the three types of averages of the molar mass in an example. Consider a sample of 100 chains, 10 × 100,000 g·mol–1, 50 × 200,000 g·mol–1, 30 × 500,000 g·mol–1, and 10 × 1,000,000 g·mol–1. According to the above formulae, Mn = 360,000 g·mol–1, Mw = 545,000 g·mol–1, and Mz = 722,000 g·mol–1. Note that quite different averages are obtained even though they are all related to the absolute same sample. This is because Mn balances light and heavy chains equally, whereas Mw (and Mz even more) puts more emphasis on the heavy chains. Based on what we have said above,5 we may illustrate the values of Mn and Mw as follows. If the polydisperse sample with composition as listed above would be depolymerized down to its monomer units, and if these units were then reconnected with the premise of now giving the same number of chains as previously (100) but with a monodisperse, that means, an infinitesimally narrow molarmass distribution,6 the result would be a sample with 100 chains of 360,000 g·mol–1 each. This new sample would display the same osmotic properties as the original polydisperse one. We may also do the reconnection differently, with the premise of not obtaining 100 but only 66 new monodisperse chains. These chains would then all have a molar mass of 545,000 g·mol–1, and this new monodisperse sample would display the same light-scattering properties as the original polydisperse one. The preceding example shows us that we receive quite different average values from the same sample, depending on how much emphasis we put on the heavy chains in it. This phenomenon may act as a direct means to express the broadness of the chain-length and molar-mass distribution in just one quantity, which is the polydispersity index, defined as PDI = Mw/Mn. A PDI of 1 denotes a monodisperse sample, whereas a PDI greater than 1 denotes a polydisperse sample.7 In our example above, we have a PDI of 1.51. Commonly, when uncontrolled free-radical chain-growth polymerization or step-growth processes such as polycondensation or polyaddition are employed to synthesize a polymer, a PDI not smaller than 2 can be achieved. This can be calculated from the distribution function that is obtained from such polymerization processes, which is the Schulz–Zimm distribution W ð PÞ =

ð1 − αÞK + 1 K P P α K!

(1:1)

 In the latter footnote, it has been pointed out already that the word “polydisperse” is actually a doubling of terms, and strictly speaking it would be sufficient to just use the word “disperse”.  The word “monodisperse” is actually an oxymoron. If a distribution is infinitesimally narrow, then it is no distribution. So, any word of kind “disperse” is actually misleading and incorrect here in this sense.  A sample with PDI just slightly above 1, typically up to 1.1, is commonly referred to as “narrowly disperse”.

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1 Introduction to polymer physical chemistry

or its variant the Schulz–Flory distribution (with K = 1) W ðPÞ = PαP ð1 − αÞ2 ≈ PαP ln2 α Günter Victor Schulz (Figure ) was born on October , , in Łódź, which was then part of the Russian Empire, and moved to Berlin in . He did his undergraduate studies in Freiburg and Munich, working together with Heinrich Wieland and Gustaf Mie, before returning to Berlin for his graduate studies on the thermodynamics of solvation equilibria in colloidal solutions of proteins at the Kaiser Wilhelm Institute of Physical Chemistry and Electrochemistry. He obtained his PhD in , being examined by no other than Fritz Haber on the subject of physical chemistry. He acquired his habilitation in Freiburg under Hermann Staudinger in  and then held an associate professorship at Rostock from , before he was appointed as full professor at the JGU Mainz in , where he established a new Institute of Physical Chemistry out of World War II ruins, which he then headed until his retirement. He died in Mainz on February , .

(1:2)

Figure 8: Portrait of Günter V. Schulz. Image reprinted with permission from Macromol. Chem. Phys. 2005, 206(19), 1913–1914. Copyright 2005 Wiley-VCH.

In eqs. (1.1) and (1.2), W(P) denotes the frequency of occurrence of a certain degree of polymerization P in the sample, in the form of its weight fraction in the sample, W(P) = mP/m. The parameter α denotes the conversion of functional groups (not monomers!) in the case of step-growth processes or the probability for chain propagation relative to the sum of all probabilities (propagation + termination + transfer) for chain-growth processes. The parameter K denotes the degree of coupling, that is, how many individually growing chains form one macromolecule in the end, which is K = 1 in the case of termination by disproportionation or K = 2 in the case of termination by recombination. In the Schulz–Flory case, with K = 1, we get a number-average degree of polymerization of Pn = 1/(1 – α) and a weight-average degree of polymerization of Pw = (1 + α) / (1 – α), and hence, PDI = Pw/Pn = 1 + α, which approaches 2 if the reaction approaches full conversion (α ! 1) in the case of step-growth processes. In chain-growth reactions, by contrast, we will get a PDI greater than 2, as in this case, we get a Schulz–Flory distribution for each momentary degree of conversion during the reaction. This is because the parameter α in the Schulz–Flory equation denotes the probability for chain propagation relative to the sum of all probabilities (propagation + termination + transfer) in the case of a chain-growth reaction, and therefore it depends on the monomer concentration,

1.3 Irregularity of polymers

13

which changes continuously during the process. As a result, the final overall distribution will be an overlay of all these momentary Schulz–Flory distributions, yielding a broad sum distribution with PDI greater than 2. Typically, uncontrolled freeradical polymerization gives PDI = 3. . .10, whereas with the aid of controlling chain-transfer agents, the PDI can be narrowed down to about 2.5 or so. Even narrower distributions with PDI smaller than 2, however, can only be obtained by truly living or quasi-living polymerization processes. In the best controlled case of a living anionic polymerization, the distribution function obtained will be the Poisson distribution W ð PÞ =

νP − 1 P expð − νÞ ðP − 1Þ! ðν + 1Þ

(1:3)

In this equation, ν is the “kinetic chain length”, which is the degree of polymerization built from one active, growing molecule before termination; often ν = Pn. With this function, we obtain a PDI of ν Pw =1+ Pn ðν + 1Þ2

(1:4)

that approaches 1 with increasing ν = Pn. This is because in a living anionic polymerization, a constant number of chains grow steadily. Any initial imbalance of the lengths of these chains will then get less and less significant if they grow longer. As an analogue, consider two kids that are 4 and 2 years old. At that stage, their “age-PDI” is 1.11, but 70 year later, at an age of 74 and 72, their age-PDI is just 1.0002. Nevertheless, though, note that even such a narrow distribution in a polymer sample still has a finite width. For example, consider a sample with number-average degree of polymerization Pn = 500 and Pw/Pn = 1.04. In this sample, still about 32% of the chains have degrees of polymerization of less than 400 or more than 600. The PDI is related to the general measure of the width of a distribution, which is its variance, σ2, by the relation σ2 = Mn2 (PDI – 1). As a result, the PDI yields information on σ/Mn, but not on σ itself. This means that if two samples have different Mn, the sample with higher PDI may not have the larger σ! 1.3.2.1 Derivation of the Schulz–Flory distribution Due to its fundamental importance and universal applicability to both chain-growth (i.e., free-radical) and step-growth (i.e., polycondensation and polyaddition) processes of polymer synthesis, the Schulz–Flory and Schulz–Zimm distribution functions, eqs. (1.1) and (1.2), shall be derived in the following. This analytical derivation

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was originally reported by Schulz for the case of (free-radical) chain-growth polymerization.8 On top of that, Schulz and Flory were both working on the sister problem of step-growth polycondensation,9 and a further contribution was made by Zimm even later,10 but Schulz’ work on that matter was first held back by his supervisor Staudinger, so that Flory’s report appeared earlier. Both approaches are based on appraising the probability for chain extension steps to occur. The probability for chain growth in a free-radical polymerization, wgrowth, which is given by the ratio of the rate of chain growth relative to the sum of the rates of chain growth and termination, is equal to the probability for a certain functional group to have reacted during a step-growth polymerization, α: wgrowth =

νgrowth =α νgrowth + νtermination

(1:5)

In turn, the probability for chain termination in free-radical polymerization, wtermination, and the probability for a certain functional group not to have reacted in step-growth polymerization are wtermination =

νtermination =1−α νgrowth + νtermination

(1:6)

From these two equations, we can derive an expression for the probability of formation of a chain with degree of polymerization P, which we name as wP . To form such a chain, we need P growth events, the likelihood of which is wgrowthP, and we need one termination event, the likelihood of which is wtermination: wP = wgrowth P · wtermination = αP ð1 − αÞ =

NP N

(1:7)

This probability directly translates to the number fraction of chains with degree of polymerization P in the sample, NP =N, just like the frequency of occurrence of a certain number of points in a dice game with many rolls directly translates to the probability of occurrence of that number in each single dice-rolling event. Note that in the polymer literature, αP – 1 is sometimes used instead of αP in eq. (1.7). This is because the start of a polymerization reaction can be seen as R* + M or as R–M* + M, where R is the residue of an initiator, M is the monomer, and * denotes some kind of active species, typically a radical or an ion. However, as P is usually much greater than 1, we may say that P ≈ P – 1.

 G. V. Schulz, Z. Phys. Chem. 1935, 30B(1), 379–398.  P. J. Flory, J. Am. Chem. Soc. 1936, 58(10), 1877–1885; G. V. Schulz, Z. Phys. Chem. 1938, 182A(1), 127–144.  B. H. Zimm, J. Chem. Phys. 1948, 16(12), 1099–1116.

1.3 Irregularity of polymers

15

The total number of macromolecules in the sample, N, can also be expressed as the total number of chain terminations during the reaction, n(1 – α), because each such event forms one chain: N = nð1 − αÞ

(1:8)

where n denotes the total number of all reaction steps. We may limit our focus to cases of α close to 1, as only in that limit, long chains will be present. In the limit of α ≈ 1, n is equal to the number of addition reactions, nα: n ≈ nα

(1:9)

From this, it follows that n is practically identical with the number of monomer units in the N macromolecules. The total weight of these N macromolecules, m, in turn, can be expressed as the number of these monomer units times the mass of each unit: m = n · mmonomer

(1:10)

Note that m is not the total weight of the sample, but only that of the N macromolecules in it, without including residual unreacted monomer or other components. Analogously, the total weight of all macromolecules with the degree of polymerization P is given by mP = NP · mmonomer · P

(1:11)

With these two equations, along with eqs. (1.7) and (1.8), we can formulate an expression for the weight fraction of chains with degree of polymerization P mP NP · mmonomer · P NP · P = = = P · αP ð1 − αÞ2 m n · mmonomer n

(1:12a)

This is the Schulz–Flory distribution. In the polymer literature, a variant form is often found as well: mP = P · αP ln2 α m

(1:12b)

P This is because Schulz’ original derivation leads to ðmP =mÞ = P · αP ð1= PαP Þ. The P P sum in the denominator is a series of type Pα = α + 2α2 + 3α3 + 4α4 + · · · . As 2 α < 1, this series converges to 1=ð1 − αÞ , so that eq. (1.12a) is recovered. Alternatively, if the derivation is conducted in a continuous fashion with the integral  Ð P P P Pα , we obtain 1 ln2 α instead of 1=ð1 − αÞ2 ,11 thereby Pα dP instead of the sum leading to eq. (1.12b).

 Also note that ln α = ln (1 – (1 – α)), which may be expanded into a Taylor series with a first term of –(1 – α), and so we again get ln2α ≈ (1 – α)2.

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1.3.2.2 Derivation of the Schulz–Zimm distribution Let us now examine how such a distribution looks like for the case of free-radical polymerization with chain termination by recombination. There are P/2 ways for two macroradicals to recombine and form a chain with degree of polymerization P: 1 + (P – 1); 2 + (P – 2); 3 + (P – 3);. . . . . . . . .; P/2 + P/2.12 The probability of two macroradicals with degrees of polymerization X and Y to be present at the same time, which is a prerequisite to meet and combine with one another, is wX + Y = 2 · wX wY

(1:13)

The factor of 2 in eq. (1.13) accounts for the two possible combinations X + Y and Y + X. As a result of each such combination, we can obtain a macromolecule with degree of polymerization P = X + Y. Rearrangement of that term yields Y = P – X, so that with eq. (1.7), we can formulate the likelihood of two macroradicals with degrees of polymerization X and Y = P – X to be present as follows: wX = αX ð1 − αÞ

(1:14a)

wP − X = αP − X ð1 − αÞ

(1:14b)

This can be inserted into eq. (1.13) to yield wX + ðP − XÞ = wP = 2 · αX ð1 − αÞ · αP − X ð1 − αÞ = 2 · αP ð1 − αÞ2

(1:15)

When we multiply this equation with P/2, we generate an expression that approximates the number fraction of macroradicals that can yield a chain with degree of polymerization P upon combination: NP = P · αP ð1 − αÞ2 N

(1:16)

Note that in the literature, αP – 2 is sometimes used instead of αP. From that, we directly get an expression for the number of chains with degree of polymerization P that results from recombination of these macroradicals: NP =

1 · N · P · αP ð1 − αÞ2 2

(1:17)

The factor ½ in the latter equation is necessary because recombination of two macroradicals yields only one polymer chain. By inserting eq. (1.8) into the latter one, we generate

12 This is basically Gauss’ math class puzzle, in which he was asked to calculate the sum of all numbers from 1 to 100. Gauss did so by a clever approach: (100 + 1) + (99 + 2) + (98 + 3) +. . .+ P (51 + 50) + (50 + 51), which is 50 × 101. In general: nk = 1 k = nðn2+ 1Þ.

1.3 Irregularity of polymers

NP =

1 · n · P · αP ð1 − αÞ3 2

17

(1:18)

Analogous to eq. (1.12), we can derive an expression for the weight fraction of chains with degree of polymerization P: mP 1 2 P = · P · α ð1 − αÞ3 m 2

(1:19)

In general: W ð PÞ =

ð1 − αÞK + 1 K P P ·α K!

(1:20)

where K is the degree of coupling. Two boundary cases can be distinguished: If K = 1, the reaction is only terminated by disproportionation, whereas K = 2 accounts for termination mostly by recombination. 1.3.2.3 Characteristic averages of the Schulz–Flory and Schulz–Zimm distribution From eq. (1.7), in its variant with αP – 1 rather than αP, we can derive NP = N · αP − 1 ð1 − αÞ, from which we may further derive the number-average degree of polymerization, Pn : P∞ P∞ P∞ P−1 P−1 ð1 − αÞ ð1 − α Þ P = 1 P · N ðPÞ P=1 P · N · α P=1 P · α P P = = (1:21a) Pn = P ∞ ∞ ∞ P − 1 P − 1 ð1 − αÞ N ð P Þ N · α ð 1 − α Þ α P=1 P=1 P=1 where the numerator is the first moment of N(P) and the denominator is the zeroth moment of N(P). By factoring out (1 – α), we can write Pn =

1 − α 1α0 + 2α1 + 3α2 + 4α3 +    = · 1−α α0 + α1 + α2 + α3 +   

1 ð1 − αÞ2 1 1−α

=

1 1−α

(1:21b)

Analogously, from the distribution of weight fractions, mP =m = P · αP − 1 ð1 − αÞ2 , eq. (1.12a), we can derive the weight-average degree of polymerization, Pw : P∞ 2 P∞ 2 P − 1 P · N ð PÞ P · α ð1 − α Þ = PP∞= 1 (1:22a) Pw = PP∞= 1 P − 1 ð1 − αÞ P · N ð P Þ P=1 P=1 P · α where the numerator is the second moment of N(P) and the denominator is the first moment of N(P). Again we can factor out (1 – α) to generate Pw =

1 − α 1α0 + 4α1 + 9α2 + 16α3 +    1+α = · 1 − α 1α0 + 2α1 + 3α2 + 4α3 +    1−α

(1:22b)

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The PDI, which is the ratio of weight and number-average degree of polymerization, then turns out to be α!1 Pw =1+α ) 2 Pn

The derivation for the case of chain termination via recombination, that is, K = 2, is analogous to the latter one. It yields Pn =

2 1−α

(1:23)

Pw =

2+α 1−α

(1:24)

and

From this, a PDI of 1.5 follows in the limit α ! 1 Pw 2 + α α!1 = ) 1.5 Pn 2

(1:25)

We can also calculate the maximum of the Schulz–Flory distribution. We do so for the case of chain termination via disproportionation (K = 1): h i ∂ ð1 − αÞ2 · P · αP ! = ð1 − αÞ2 · P · αP ln α + αP ð1 − αÞ2 = 0 ∂P , ð1 − αÞ2 · αPmax · ðPmax ln α + 1Þ = 0 , Pmax ln α + 1 = 0 , Pmax =

−1 1 ≈ = Pn ln α 1 − α

(1:26)

Figure 9: Graphical representation of the number fraction, N(P), and the weight fraction, W(P), of each degree of polymerization P in a polymer sample according to the Schulz–Flory distribution.

1.4 Properties of polymers in contrast to low molar mass materials

19

We see that the maximum of the distribution is equal to the number average Pn . The same is true for the case of chain termination via recombination (K = 2) (Figure 9): Pmax =

2 = Pn 1−α

(1:27)

1.4 Properties of polymers in contrast to low molar mass materials Due to their macromolecular, usually ill-defined structure, polymers exhibit properties that are fundamentally different from their low molar mass counterparts. A comparison of a few such different properties is compiled in Table 1. Table 1: Comparison of some properties of low molar mass versus polymeric materials.

Composition Solubility Melt behavior Purification

Low molar mass materials

Polymer materials

Uniform Finite solubility, no swelling Low viscous Distillation, crystallization

Polydisperse and irregular “All or nothing”, often with swelling Highly viscous or even viscoelastic Precipitation

First, in contrast to low molar mass materials that have a uniform and well-defined molecular constitution (often even in view of their stereochemistry), polymers have a polydisperse and irregular composition. As detailed in Section 1.3, this is because they are commonly synthesized in statistical chain-growth or step-growth processes, leading to nonuniform chain lengths in a polymer sample, and also leading to irregularities along each individual chain, such as head-to-head versus head-totail versus tail-to-tail monomer–monomer connections. Second, low molar mass compounds can usually be dissolved up to a certain solubility threshold, which is high or low depending on the chemical similarity of the solvent and the compound. This is fundamentally different for polymers: their solubility often practically exhibits an “all or nothing behavior”, meaning that when a solvent is good it can dissolve virtually any amount of a polymer (at least in the practically relevant concentration range), whereas if a solvent is not good it can hardly dissolve any of the polymer. This is because usually, mixing a compound A (the solute) with a compound B (the solvent) is energetically unfavorable (as A–A and B–B contacts are trivially more similar and therefore more favorable than A–B contacts)13 but entropically favorable (as entropy generally likes mixing, as that  “Birds of a feather flock together.”

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creates disorder). If the compound A has a low molar mass, there’s many molecules of A in a given mass of this substance, and hence, the mixing entropy is considerably high, thereby compensating the commonly unfavorable mixing energy and leading to mixing up to a certain threshold concentration at which the solution is saturated. By contrast, if the compound A is a polymer, a given mass amount of it contains just few molecules; hence, the entropy of mixing is so little that it cannot compensate the usually unfavorable energy of mixing, thereby prohibiting dissolution more or less completely. Only if the polymer A and the solvent B are chemically very similar, which is a situation referred to as “good solvent state” (or, even better, “athermal solvent state”), the just little entropy of mixing is still powerful enough to compensate the then also just minorly unfavorable energy of mixing. In this case, the polymer will dissolve in almost any quantities, only limited by practical impairment such as overly high viscosities at high concentrations. Third, tying in to the preceding sentence, low molar mass molecules usually show low viscosities in the melt and flow like Newtonian fluids (with exemptions such as glycerol or related compounds that have strong mutual intermolecular associative forces causing high viscosities). This is because they are small and can relax (i.e., change their positions relative to each other) comparably fast, without need for input of extraordinarily large additional energies of activation. Polymers, by contrast, are large molecules that need great energies of activation for relaxation and flow, corresponding to high viscosities. Especially long-chain molecules can also undergo mechanical entanglement with one another, which imparts further hindrance on their motion. The result is a rich mechanical behavior that is very different from that of small molecules, ranging from elastic snap over viscoelastic creep and relaxation to viscous flow. Whereas the macromolecular chain-like shape of polymers causes this to hold true for all polymers in a universal fashion, the chemical specificity of a given polymer determines what type of mechanical response will be observed at what temperature and on which timescale exactly. Fourth, polymers do not exist in the vapor phase, because they decompose before boiling (see Figure 3). Therefore, standard purification procedures such as distillation cannot be applied for polymers. Fortunately, however, tying in to our second point above, their “all or nothing” solubility allows polymers to be precipitated from a solution by adding an excess of a nonsolvent, thereby keeping low molar mass impurities dissolved while the polymer separates from the solution and can be harvested just by filtration.

1.5 Polymers as soft matter Polymers are a prime example of a material class named soft matter. This term refers to materials that (i) consist of building blocks with sizes in the colloidal domain, 10–1000 nm, and that (ii) have interaction energies in the range of

1.5 Polymers as soft matter

21

10–100 RT (as a benchmark to that, note that RT at room temperature is about 8 J mol–1 K–1 · 300 K ≈ 2.5 kJ mol–1).14 To put this into context, let us consider that a covalent C–C bond has a dissociation energy of 350 kJ mol–1, which is more than 100 RT at room temperature; this means that it requires a huge amount of external energy to be broken. A transient OH···H interaction, by contrast, has a dissociation energy of just 18 kJ mol–1, and can therefore be broken with just little additional energy. Together, the large size and weak interactions of the building blocks in soft matter make it soft. We can appraise that by estimating the cohesive energy density, e = E/r³, where E is the interaction energy and r is the distance between the building blocks. For hard matter, we have E in the range of 10–18 J and r in the range of 0.1 nm, which results in values of e around 1012 Nm–2; this is a typical elastic modulus for materials such as diamond. For soft matter, in contrast, we have E in the range of 10–20 J and r in the range of 10–100 nm, which results in values of e around 104–107 Nm–2. This is a typical elastic modulus for a polymeric or colloidal melt or gel. The weak interaction energy also gives rise to another characteristic property of soft matter: its ability to be assembled, disassembled, and reassembled in a dynamic manner, thereby displaying a rich dynamic and stimuli-sensitive phase behavior. Due to that and due to their colloidalscale size, these materials (iii) have relaxation times, which are the times needed for their building blocks to move a distance of their own size, in the practically relevant range of milliseconds to seconds. As a result, soft matter materials often show rich viscoelasticity, depending on the timescale of experimentation and temperature. Following this definition, polymers can be classified as a prime example of soft matter. First, the shape of a polymer chain is, in most cases, that of a random coil with a couple of 10 nm in diameter. This puts polymers right in the colloidal domain. Second, due to their colloidal-scale size, the polymeric coils exhibit relaxation times in the range of a couple milliseconds to seconds.15 Third, multiple polymer chains can interact with each other usually via (random-fluctuating, induced, or permanent) dipolar interactions to form higher-order assemblies. These interactions have energies in the range of a couple 10 RT. Consequently, polymeric assemblies are stable at

 A little footnote in this context: the value of RT at room temperature should be in the mind of each physical chemist. This is because it is the fundamental benchmark of the energy landscape, which is a land that physical chemists often have to pass. As a comparison, consider the following analogy: an information like “the building over there is 10 m tall” is only valuable for you if you know what 1 m is. If you do not know that, then you have no clue whether 10 m is tall or short. The same applies to energies, with which you need to argue over and over again in physical chemistry. You can only appraise whether a 350 kJ mol–1 covalent C–C bond or a 20 kJ mol–1 hydrogen bonding interaction is strong or weak if you know the baseline of RT at room temperature.  The exact time depends on the architecture and the environment of the polymer chain. A short chain will relax faster than a long one. It will also relax faster when it is surrounded only by solvent molecules, as it would be the case in a dilute polymer solution, than when it is in direct contact with other polymer chains that constrain its movement, as it would be the case in a polymer melt.

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room temperature, but can be restructured with relative ease, which is one out of two reasons why nature often uses polymeric structures as building blocks for dynamic complex architectures. (The second reason will be discussed in the context of the Flory–Huggins theory in Chapter 4.) Aside from this fundamental role in nature, the soft-matter characteristics of polymers have made tremendous impact on science. The basic physics of ordering phenomena is the same for colloidal-scale soft matter and for classical atomic- or small-molecule-scale hard matter, but soft matter, due to the much larger size and slower motion of its building blocks compared to that of atoms and small molecules, can be powerfully observed and studied. This notion led Pierre-Gilles de Gennes to use polymers as fundamental studying objects in condensed-matter physics, which is much more elementary than “only” studying them for their vast practical applicability. This groundshifting inspiration was honored with the Physics Nobel Prize in 1991. Pierre-Gilles de Gennes (Figure ) was born on October , , in Paris. He was homeschooled to the age of . By the age of , he had adopted adult reading habits and was visiting museums. Later on, he studied at the École Normale Supérieure, from which he majored in . From  to , he was a research engineer at the Atomic Energy Center (Saclay), the heart of French nuclear research at that time, and obtained his PhD in  from the University of Paris. After postdoctoral research in Berkeley in  and a -month stint in the French Navy, he became an assistant professor in Orsay in . Later, in , he became full professor at the Collège de France. From  to , he was the director of the Ecole de Physique et Chimie. After receiving the Physics Nobel Prize in , de Gennes decided to give talks on science, innovation, and common sense to high school students. He died in Orsay on May , .

Figure 10: Portrait of Pierre-Gilles de Gennes. Image reproduced with permission from Nature 2007, 448(7150), 149. Copyright 2007 Springer Nature.

1.6 History of polymer science For centuries, natural polymers have been employed without any knowledge about their macromolecular nature. Table 2 shows a brief history of mankind’s use of them.

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Table 2: Use of natural polymers in human history. Time

Material

Discovery

 BC  BC  BC  AD  AD  AD

Cotton Silk Bitumen Rubber Guttapercha (trans-,-polyisoprene) Vulcanized rubber (cis-,-polyisoprene)

Mexico China Orient Mexico (Mayas) Goodyear

Gutta “rubber” and percha “tree”.

1

In this list, Goodyear’s vulcanization of natural rubber, which denotes the turnover of viscous natural caoutchouc into an elastic rubber by crosslinkage of its polyisoprene chains through sulfur bridges, is a special milestone, because it is the first industrial-scale modification of a natural polymeric material, and therefore signifies the advent of a chemical industry in this field.16 From thereon, chemists modified many biopolymers, but still had no knowledge about their structure or their macromolecular nature. It was believed at that time that covalently jointed structures with molar masses of several thousand grams per mole simply could not exist. Instead, a common belief was that polymers were colloidal structures, built up by small molecular entities aggregated by noncovalent forces. The reason for this notion might have been because organic chemistry was successful in assessing small molecule structures to that time, whereas physical chemistry was successful in assessing their intermolecular forces, so that scientists were prone to apply both polymer-type matters as well. The high molar masses of these materials that were measured in laboratories were therefore viewed to be that of the colloidal aggregates, and hence, sort of artifacts. Table 3 displays the hypothetic and actual molecular structures of caoutchouc and cellulose as they were perceived before 1920 and as we know them today. In 1920, Hermann Staudinger was able to prove that polymers, in fact, have actual rather than artifactual high molar masses. He did this by chemically modifying  The turnover of natural caoutchouc to rubber has in fact been discovered much earlier, by Amazon natives, who extrachted the sap of the rubber tree and smeared it onto their feet, which then gave rubbery boots due to crosslinking of the polyisoprene chains by oxygen from air. A lot later, though, Charles Goodyear achieved the same effect in a better and more powerful way through the use of sulfur. This was in fact achieved by accident: in 1839, Goodyear (who was a self-trained chemist) dropped a caoutchouc–sulfur mixture onto a hotplate, thereby obtaining a dry, durable, elastic substance. Despite that luck and the tremendous impact it had on various industries (such as the automotive business), Goodyear was never successful in commerce. He was even sentenced to prison several times, as he could not pay his debt. There is a saying after which a newspaper once wrote about him: “If you see a man in shoes, coat, and a hat of rubber, but without a cent in his pocket, then you are looking at Charles Goodyear.”

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Table 3: “Colloidal” and actual structure of caotchouc and cellulose.

known polymers, as shown in Figure 11, with the premise that such chemical change in their “colloidal” structures should severely alter their aggregation, just in the same way that a colloidal fatty acid micelle breaks up upon esterification. What Staudinger found instead was staggering: despite drastic changes in many other properties, the high molar masses were retained in all the compounds upon such modification (which has therefore been referred to as “polymer analogous modification” in the following). This finding strongly supports Staudinger’s hypothesis that polymers are not associated by physical interactions only, but instead, covalently jointed chain molecules with an inherently high molar mass. This mass doesn’t change upon polymer analogous modification, whereas other properties do, as the modification alters the side groups of the polymer, but not the length of its chain backbone.

Figure 11: Hermann Staudinger’s polymer analogous reactions that proved a high molar mass to be an actual rather than just artificial polymeric property.

Staudinger expressed his findings rather carefully first: If one is willing to imagine the formation and constitution of such high-molecular compounds, one may assume that primarily, a combination of unsaturated molecules has taken place, similar to the formation of four- or six-membered rings, but that for some reason, potentially a

1.6 History of polymer science

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steric one, the four- or six-ring closure didn’t happen, and now many, perhaps hundreds of molecules are jointed.17

Staudinger’s notion received immediate and harsh criticism by the scientific community of the early twentieth century, such as the following, which was expressed by Heinrich Wieland, his predecessor at the University of Freiburg and Nobel Prize laureate for the structural characterization of bile acids: Dear colleague, refrain from the imagination of large molecules; organic molecules with a molecular weight above 5000 do not exist. Purify your products, such as the caoutchouc, and then they will crystallize and turn out to be low-molecular compounds.18

But Staudinger proved persistent, and, as his hypothesis held true, grew increasingly self-confident later, when he said: Even though the formation of millions of compounds whose molecules are built of hundred or several hundreds of atoms is possible, this realm of low-molecular organic chemistry is just a pre-stage to actual organic chemistry, namely chemistry of compounds that are of fundamental importance for processes in life. This is because the latter compounds contain molecules that are not built from some hundreds, but from thousands, ten-thousands, and perhaps even millions of atoms.19

Staudinger eventually received the Nobel Prize for his work in 1953. By then, the field of macromolecular chemistry that he founded had advanced impressively, and also his above statement about polymers as the “true organic molecules” and backbones of life held true, as in the same year, Watson and Crick published the structure of DNA.

 The original German wording is: “Will man sich eine Vorstellung über die Bildung und Konstitution solcher hochmolekularen Stoffe machen, so kann man annehmen, daß primär eine Vereinigung von ungesättigten Molekülen eingetreten ist, ähnlich einer Bildung von Vier- und Sechsringen, daß aber aus irgendeinem, evtl. sterischen, Grunde der Vier- oder Sechsringschluß nicht stattfand, und nun zahlreiche, evtl. hunderte von Molekülen sich zusammenlagern.”  The original German wording is: “Lieber Herr Kollege, lassen Sie doch die Vorstellung mit den großen Molekülen, organische Moleküle mit einem Molekulargewicht über 5000 gibt es nicht. Reinigen Sie Ihre Produkte, wie z.B. den Kautschuk, dann werden diese kristallisieren und sich als niedermolekulare Stoffe erweisen.”  The original German wording is: “Wenn auch die Darstellung von Millionen von Stoffen möglich ist, deren Moleküle aus Hundert oder einigen Hundert Atomen aufgebaut sind, so bliebe dieses Gebiet der niedermolekularen organischen Chemie trotzdem nur die Vorstufe der eigentlichen organischen Chemie, nämlich der Chemie der Stoffe, die für die Lebensprozesse von grundlegender Bedeutung sind. Denn in letzteren Verbindungen liegen Moleküle vor, die nicht aus einigen Hundert, sondern aus Tausenden, Zehntausenden und vielleicht sogar Millionen von Atomen bestehen.”

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Hermann Staudinger (Figure ) was born on March , , in Worms. His father, a leading figure in the German union movement, wanted his son to enter a hands-on profession, so Hermann became an apprentice in carpentry first. After his training, he went to Halle to study chemistry and obtained his PhD there. He then held professorships in Karlsruhe, Zurich, and finally at Freiburg, where he conducted his Nobel Prize research. He died in Freiburg on September , .

Figure 12: Portrait of Hermann Staudinger. Image reproduced with permission from ETH-Bibliothek Zürich, Bildarchiv.

In the 40 years following Staudinger’s groundbreaking experiments (1920–1960), the core concepts of polymer physics were developed: – Kuhn model for polymer chain renormalization (Chapter 2) – Flory theory of coil conformation in a good solvent (Chapter 3) – Rouse and Zimm models of polymer dynamics (Chapter 3) – Flory–Huggins mean-field theory of polymer thermodynamics (Chapter 4) – Kuhn, Grün, Guth, Mark: statistical theory of rubber elasticity (Chapter 5) Between 1960 and 1980, these concepts were further developed and refined to move from the description of single chains to chain assemblies, culminating in the Nobel Prize for Pierre-Gilles de Gennes in 1991 for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular, to liquid crystals and polymers. We just name two bullet points to summarize these milestones: – Doi, Edwards, De Gennes: tube concept and reptation theory (Chapter 5) – De Gennes: “soft matter physics” Current topics of polymer research (with a special focus on the physical chemistry branch of it) have introduced dynamics and adaptivity into polymer systems by actively incorporating or making use of transient bonding. This leads to selfassembly processes that create complex architectures potentially soon rivaling those in nature; it also leads to sensitivity, responsiveness, and adaptivity of these complex systems and materials. To freely quote Bruno Vollmert’s book Grundriss der Makromolekularen Chemie: “Polymers are the highest level of complexity in

Questions to Lesson Unit 1

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chemistry, and the lowest level of complexity in biology.” Current polymer research focuses right on the bridge between these realms. Other modern topics include: – Supramolecular polymers (J. M. Lehn, E. W. Meijer, T. Aida, M. A. Cohen-Stuart, and many others) – Conductive polymers and polymers for energy conversion and storage – Biopolymer science and biomimetics – Large-scale theoretical modeling and simulation Aside and along with this academic development, industrial polymers have turned mankind’s twentieth century into the “polymer age”.20 This vast and versatile use of polymers throughout decades, however, has posed environmental challenges that modern polymer research must address, in the form of reversing the tremendous amounts of polymer-based waste that has been and still is being produced by mankind. Modern research, both industrial and academic, therefore must have a special focus on degradable polymeric materials. On top of that, it will be a sociological challenge for us to learn about the great value of polymer-based everyday life products, given their remarkable properties (mostly their strong mechanics compared to their light weight), such as to not only use them once but multiple times before discarding them.

Questions to Lesson Unit 1 (1) Which statement is correct? a. Each polymer is a macromolecule. b. Each macromolecule is a polymer. c. Polymer and macromolecule are synonym terms. d. Polymer and macromolecule are different terms that do not overlap in meaning. (2) Which of the following biological macromolecules is a polymer? a. DNA b. Proteins c. Lignin d. Polysaccharides

 Just as much longer before, mankind’s main materials of use led us to entitle former ages as the stone age, the bronze age, and the iron age, and just as our world today is certainly justified to be termed to be in the silicon age.

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(3) Given are two samples A and B with ten chains each. The molar mass distribution is listed in this table: Molar mass (g mol–) Number of chains in sample A Number of chains in sample B

,  –

,  

,  

,  

,  –

Which statement regarding the number average Mn and weight average Mw is correct? a. Mw is identical for both samples, since both samples have identical Mn. b. Mn is identical but Mw is different, because chains with higher molar mass are weighted more. c. Mn is identical and also Mw because sample A has a chain with higher molar mass of 500,000 g mol−1 but also a chain with lowest molar mass 100,000 g mol−1, which practically cancels out. d. Mn is identical and also Mw since both samples are symmetrically distributed. (4) Complete the following sentence: Soft matter________ a. consists of building blocks smaller than a few nanometers, which can be easily displaced against each other due to their small size. b. has a cohesive energy density that is five to eight orders of magnitude lower than that of conventional hard matter. c. consists of building blocks of a few nanometers in size that are linked by strong covalent bonds but can be displaced against each other due to the small number of bonds. d. consists of building blocks that are linked together via long-range interactions, that is, based on a soft repulsive potential (as opposed to a hardsphere potential). (5) On which experimental observation was Staudinger’s description of polymers based? a. Polymers are aggregates of small molecules in the colloidal domain; Staudinger made that visible with the (at his time newly invented) method of electron microscopy. b. Polymers are aggregates of small molecules in the colloidal domain; upon chemical modification, these aggregates change their size due to a change in interactions. c. Polymers consist of covalently bonded monomers; chemical modification usually affects their side groups only, but not their connectivity. So, the chain length is retained.

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d. Polymers consist of covalently bonded monomers; chemical modification usually changes their connectivity, either by chain breakage or by chain extension/linking/branching. Hence, the chain length is changed. (6) Consider four samples with identical number-average molar mass, Mn, but different distributions:

From the following possibilities, select the polymerization-type assignment that matches the molar mass distributions from left to right: a. Biopolymer – living anionic polymerization – polyaddition – free radical polymerization. b. Living anionic polymerization – radical polymerization – polyaddition – biopolymer. c. Polyaddition – radical polymerization – living anionic polymerization – biopolymer. d. Radical polymerization – polyaddition – biopolymer – living anionic polymerization. (7) Which method is suitable for the purification of polymers and why? a. Crystallization, because polymers based on covalent linkages can be purified by adjusting the polymer concentration in solution as high as possible. b. Crystallization, because the long polymer chains can easily change into a glassy arrangement upon melting and crystallize upon cooling. c. Precipitation, because the mixing process of a polymer and a solvent is entropy-driven, and the entropy in a polymer system is particularly high due to the long chain length. d. Precipitation, since the mixing process of a polymer and a solvent is energy-driven, whereas entropy hardly plays a role in polymer–solvent mixtures.

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(8) Polymers are a prime representative of soft matter, because ______ a. they consist of small monomer units bound by weak interactions such as van der Waals interactions. b. they exhibit fast relaxation times of a few microseconds that are accessible only to very sophisticated methods. c. they consist of individual chains that have the shape of a random coil in the colloidal size range and interact with each other via van der Waals or similar interactions. d. they consist of covalently linked monomer units forming long chains that interact with themselves.

2 Ideal polymer chains LESSON 2: IDEAL CHAINS In your elementary physical chemistry classes, the first and most simple type of matter that you dealt with was the ideal gas. In polymer science, this has an analog. The first and most simple type of polymeric matter that we will deal with is the ideal chain. In this lesson, you will get to know the basics of the ideal chain model, see how this is conceptually very related to the ideal gas analog, and you will see how this model allows us to simply capture and quantify the structure of a polymer.

The goal of this book is to establish relations between the structure and properties of polymers. To do so, we have to find a way to comprehensively describe the structure of the elementary building block of each polymeric material: the single polymer chain. Polymers are made from a large number of repeating units; as such, a polymer chain is a complex, multibodied entity that is hard to describe analytically. We therefore have to rely on models that simplify the complexity while simultaneously retaining the physical realities that we observe in experiments. In the following, we will introduce a few of these models, starting with the simplest one: the ideal polymer chain. Let us recall a concept that you have encountered in the beginning of your basic physical chemistry lectures, where you have considered the most simple state of matter imaginable: the ideal gas. The description of an ideal gas is based on the assumptions that (i) the gas molecules are point-like particles with no volume and (ii) no interactions (other than simple elastic collisions). These assumptions are, of course, not true: real gas molecules do have a finite volume, and they do interact with each other in many ways, especially at short mutual distances, which is given at high pressure. In many cases, however (for instance, at usual temperature21 and pressure), gases in fact widely behave ideal, allowing us to use the above very simplified model to treat them, which gives us (iii) a very simple equation of state: pV = nRT, with p the pressure, V the volume, n the molar number of gas particles (atoms or molecules) in our system, R the gas constant, and T the absolute temperature. The concept of the ideal polymer chain is based on the same simplistic principle: it imagines the chain to consist of (i) monomer segments with no volume and (ii) no interactions with one another or the environment (other than the trivial interaction that they have due to their mutual connectivity, which, as we will see in Chapter 3, impairs their independent statistical motion). The model of the ideal polymer chain is the basis for any further development of refined polymer chain

 Strictly speaking: at a temperature well above the critical one. https://doi.org/10.1515/9783110713268-002

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models and can therefore rightfully be called a starting point of polymer physics. We will see later in this chapter that this model will give us (iii) an equation of state that is strikingly similar to the ideal gas law, both in view of its mathematical appearance and in view of its simplicity but yet powerful utility.

2.1 Coil conformation 2.1.1 Micro- and macroconformations of a polymer chain Let us begin by considering how a polymer chain might look like. The shape of a polymer is determined by three factors, as illustrated in Figure 13(A). The first factor is the monomer segmental length, that is, the length of the elementary structural unit plus the bond that connects it to the next. In many simple vinyl polymers, such as polyethylene (PE), polypropylene (PP), polyvinylchloride, or polystyrene (PS), this is a covalent carbon–carbon bond with a length of 1.54 Å. The second factor is the monomer bonding angle, φ. It has a value of 109.6° for an sp3-hybridized carbon–carbon bond. Both of these parameters are determined by the specific chemistry of the monomer, which means that they are fixed for a given polymer. The third factor is the monomer bond torsion angle, θ. This factor is not fixed, because the bonds can rotate quite easily. As a consequence, the macroconformation of a chain will result from the sequence of its monomer bond torsional microconformations. A polymer chain may therefore adopt many different shapes, two of which are depicted in Figure 13(B) and (C). In Figure 13(B), you see a structure in which all the bonds are trans-conformed. This all-trans conformation leads to an ordered, rod-like macroconformation. Entropy, however, does not favor such ordered states. More favorable, by contrast, is the structure depicted in Figure 13(C). Here, the microconformation is a random cis–trans mix that leads to a macroconformation of a random coil, which does not show a high degree of order. The actual reason for the entropic favor of this random-coil macroconformation is because it is realized by many different microconformations of kind as the one shown in Figure 13(C), because there are many different possibilities for arrangement of a random cis–trans mix along the chain contour, whereas the rod macroconformation is realized by just one microconformation, namely the all-trans chain. With that, the core principle of statistical thermodynamics applies: a macrostate that is realized by a high number of microstates has a higher entropy and is more likely to be observed (both over time and ensemble) than a macrostate that is realized by just a few microstates. So far, we have discussed the monomer bond torsional angle θ and the resulting structure of a polymer chain primarily from an entropic perspective. There is, however, also an energy side to the argument. This is because some monomer bond torsional angles are energetically favorable, whereas others are not. The trans or gauche conformations have lower free energies, ΔE, than the cis or the anti ones, as shown in

2.1 Coil conformation

33

Figure 13: (A) Characteristic quantities of the elementary chemical bonds in the backbone of a polyethylene chain. In this simplest case of a polymer, the bond length between two methylene units, here denoted as dark-shaded dots, corresponds to the segmental length, l. The monomer bonding angle, φ, is the angle between two segment–segment (here: methylene–methylene) bonds, and the monomer bond torsion angle, θ, defines the conformational positions of each unit. Picture taken from H. G. Elias: Makromoleküle, Bd. 1 – Chemische Struktur und Synthesen (6. Ed.), Wiley VCH, 1999. Depending on the angle θ, many different macroconformations of the polymer chain can be realized, two of which are shown here. (B) If all microconformations of the chain are trans, this leads to an ordered, rod-like macroconformation that is entropically unfavorable. (C) In contrast, if a more favorable random microconformational mix of cis and trans is present, this leads to a macroconformation of a random coil. This unordered structure is favored by entropy, because it is realized by more microconformations than the all-trans structure in (B).

Figure 14 for the very typical case of a carbon main chain. The energy difference between those conformations, however, is just small: the energy of the trans or gauche conformations differs by only 3 kJ mol–1, which corresponds to just about 1.2 RT at room temperature. Furthermore, a conformational change between these states has an energy of activation of only 13 kJ mol–1, that is, just about 5 RT. Even the biggest energy gap, which is between the cis and the trans conformation, is only about 17 kJ mol–1, that is, 7 RT. This means that all the conformations available to the

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2 Ideal polymer chains

Figure 14: Dependence of the molar energy, ΔE, on the bond torsion angle, θ, for basic carbohydrates. Conformations are abbreviated as C = cis, G = gauche, A = anti, T = trans. The energy has an absolute minimum at the trans conformation and two further local minima for the gauche conformations G+ and G– that are 3 kJ mol–1 higher (ΔETG). The energy of activation needed to change between T and G+ or G– is 13 kJ mol–1 (ΔE‡TG), and the energy gap between the most favorable and the least favorable conformation, T and C, is 17 kJ mol–1 (ΔE‡). Picture redrawn from H. G. Elias: Makromoleküle, Bd. 1 – Chemische Struktur und Synthesen (6. Ed.), Wiley VCH, 1999.

building blocks of a polymer chain are not very different in energy. Moreover, it is easy to pass from one conformation to the other, because a conformational change needs an activation energy in the range of only a couple RT at room temperature. As a result, these changes happen fast, which is the prime reason why polymers are flexible above their glass transition temperature. This flexibility is lost if there is a more severe energy barrier between the monomer segmental conformations, which is the case for monomer units that carry bulky substituents. This is the reason why PS, which has a bulky benzene side group, has a higher glass transition temperature than PE, where no side groups are present, because it takes more thermal energy to activate conformational change dynamics in the more obstructed PS backbone than in the flexible PE backbone. (See: here we have our very first qualitative structure–property relation!) As another direct consequence of these low energy values, the entropy aspect is mainly responsible for the macroconformation of a polymer chain and outweighs the energy aspect. Therefore, the most probable structure of a polymer chain is that of a flexible random coil. There are some cases where energy outweighs entropy, but these only occur when an extremely high energy gain is associated with the adoption of a certain specific chain structure. Proteins are a prime example: they have specific, energetically favorable folded shapes, such as the α-helix or the β-sheet. The energy they gain through secondary interactions when adopting these shapes outweighs the entropy penalty that such an ordered structure imposes. This ordered structure, in turn, is the basis for their biological function, often referred to as key-and-lock principle. This notion has led Vollmert to formulate that biopolymers such as proteins are the highest level of complexity in chemistry and the

2.1 Coil conformation

35

lowest level of complexity in biology. Most synthetic polymers, by contrast, do not have such preferred structures, but instead shape up as random coils. Now that we know that a polymer chain looks like a coil, how do we describe it mathematically? Even if we restrict ourselves to just the energetically favored trans and gauch microconformations, T, G+, and G–, there are 3N – 2 possible macroconformations that result in a chain with N segments. If we consider N to be 100, which is still a comparably short polymer chain, that would mean that there are 6 × 1046 possible different macroconformations! On top of that, conformational change is very fast due to the low energies involved. An appraisal based on an Eyring-type equation with an energy barrier of 13 kJ mol–1, or 5 RT at room temperature, for the T ! G transition shows that they happen on a timescale of just nanoseconds at room temperature. All this means that the shape of a polymer chain cannot be described analytically. It can, however, be described mathematically using suitable average values.

2.1.2 Measures of size of a polymer coil Two common averages are used to describe the size of a polymer coil. These are the end-to-end distance, ~ r, and the radius of gyration, Rg, both depicted in Figure 15. The end-to-end distance, ~ r, is calculated by simply summing up all individual bond vectors, which creates a vector that points from one end of the polymer chain to the other: X ~ ~ r2 +~ r3 +    = ri (2:1) r =~ r1 +~ It can be easily visualized and imagined, but it is hard to be determined experimentally, which is why it is often preferred by polymer theorists. Note that the end-toend distance is only meaningful for linear chains, which have a defined start and end of their polymer chain. Branched macromolecules, by contrast, have many end-to-end distances, because they have multiple chain ends. The radius of gyration, Rg, is calculated by the sum over all mass segments, m, weighted by their distances to the center of mass: ð  1X 1 2 1 2 2 2 2 2 ~ ~ ~ ~ (2:2) r 1 m1 + r 2 m2 + r 3 m3 +    = ri mi = r dm Rg = m m m Rg corresponds to about the 1.3-fold of the radius of a sphere with the same moment of inertia and the same density as our actual object of interest. It is not limited to linear polymers, but can also be determined for branched or cross-linked polymers. In fact, Rg can be calculated for any geometrical object. It is also much easier to assess experimentally than the end-to-end distance, which is the reason why it is often preferred by polymer experimentalists.

36

2 Ideal polymer chains

Figure 15: Schematic representation of two common average measures of the size of a polymer coil: (A) The end-to-end distance ~ r, simply calculated as the sum of all bond vectors, and (B) the radius of gyration, Rg, calculated by a normalized sum over all mass segments weighted by their distances to the center of mass of the polymer coil; Rg corresponds to the radius of a solid sphere (illustrated by a dashed green circle) with the same moment of inertia and the same density as the fuzzy polymer coil.

Both quantities are connected to each other by simple relations that depend on the geometry of the compound investigated. Table 4 compiles a selection of some of these relations. Table 4: Relation between the end-to-end distance and the radius of gyration for various types of objects.

Coil in good solvent

– characteristic length  2  2 ~ r Rg = 6  2 ~  2 r Rg = ð2ν + 1Þð2ν + 2Þ

Rod with length L



Disc with radius r

 2  r2 Rg = 2

r ⁓ m1=2

Sphere with radius r

 2  3r 2 Rg = 5

r ⁓ m1=3

Type of molecule/ object Random coil

 L2 Rg 2 = 12

Characteristic length–mass  2  2 ~ r ⁓ Rg ⁓ m  2 ~ r ⁓ Rg 2 ⁓ m2ν

L⁓m

ν = Flory exponent (Chapter ) → ideal coils: . → with rep. interact: .

2.2 Simple chain models

37

2.2 Simple chain models 2.2.1 The random chain (phantom chain, freely jointed chain) The simplest model to describe a polymer chain is that of the random chain, also referred to as freely jointed chain or phantom chain. It is based on the assumption that all monomer bond conformations, assessed by the torsional angle, θ, and also even all monomer–monomer bonding angles, φ, are possible. This is somewhat unintuitive, as the bonding angle is in fact set by the monomer’s specific chemistry. Releasing this constraint and allowing this angle to be free would allow two monomer segments to occupy the same spot in space, for example, if two adjacent monomer segments would have a bonding angle of φ = 0°, which is, nevertheless, allowed to be possible in the random chain model. This is why this model is also referred to as the phantom chain model, as in this framework, the chain segments can penetrate through each other like phantoms. With that simplifying assumption, the random chain model relates the number of monomer segments, N, and their length, l, to the end-to-end distance of the polymer chain, ~ r, in a very simple fashion, as we will show now by simply estimating the end-to-end distance through the bond vector sum. For a reason that we will detail below, we use the square value of the end-to-end vector, ~ r2 . With that, we calculate the square of the bond vector sum. This must be done using a scalar product term, which gives us a sum of squared lengths of the bonds vectors plus a cosine term of their angles: ~ r=

X ~li

(2:3a)

 2 ~ r2 = ~l1 +~l2 +~l3 +      X = l1 2 + l2 2 + l3 2 +    + li lj cos φij

(2:3b)

With this calculation, so far, we have only estimated the momentary-squared end-to-end distance of one single polymer coil. This, however, is not representative, as the coil conformation is subject to constant dynamic change, and as in a sample with many chains, even in a momentary view, they won’t all exhibit the same end-toend distance, but a distribution of it. So, what we are actually interested in is the mean of this distribution. If we would calculate it for ~ r directly, however, the mean would always be zero. This is because ~ r is a vector quantity that has a direction. In a sample with many representants of this quantity, there will always be pairs of same length that point into opposite directions, such that the mean of all these would cancel out to zero. To get rid of this circumstance, we take the square of the

38

2 Ideal polymer chains

end-to-end vector first before averaging. That way, we eliminate the directional dependence and obtain the mean-square end-to-end distance:  X  2  2 ~ li lj hcos φij i r = l1 + l2 2 + l3 2 +    +  2  2 2 = l1 + l2 + l3 +    = Nl2

(2:4)

In the latter form of the scalar product, the average of the angular dependence, hcos φij i, is zero. This is because we assume that the bonding angle, φ, is free and exhibits a random distribution, and so the cosine of that random angle in eq. (2.3b) delivers random values in an interval of –1 and +1 around 0, which all occur with same likelihood. So, each realization in the positive direction has an equally large counterpart in the negative direction, thereby all cancelling to zero upon averaging in eq. (2.4). The square of the end-to-end distance therefore only depends on the number of monomer segments, N, and their length, l. With this calculation, we have derived a simple power law for the chain meansquare end-to-end distance as a function of the number of chain segments. To relinearize the physical dimension of the mean-square end-to-end distance, we can take the square root and obtain the root-mean-square (rms) end-to-end distance,  2 1=2 . Such rms values can be found all over the field of physical chemistry, as R= ~ r they are a practical tool to handle vector quantities that are subject to a distribution. rms values eliminate the vectors’ directional dependencies and handle their distributed nature by using the arithmetic mean of their square values. From this, the square root is then formed to re-linearize the mean square quantity to its truly meaningful physical dimensions. Following eq. (2.4), the rms end-to-end distance of our phantom polymer chain scales with N1/2: R = h~ r 2 i1=2 ⁓ N 1=2

(2:5)

Note that from eq. (2.5), it follows that a polymer that has twice as many segments as another one is not twice as large as that other one in space, but only 21/2 ≈ 1.4 times as large. To get a polymer that is twice as large as another one in space, it must have 22 = 4 times as many segments! A very similar scaling law to eq. (2.5) can be found for freely diffusing particles that move by a random walk. Here, the rms displacement, h~ x2 i1=2 , scales with the square root of the number of diffusion steps, and therefore also with the square root of time, t1=2 , if we assume that each elementary step takes a defined period. This scaling is based on the Einstein–Smoluchowski equation: h~ x2 i1=2 ⁓ t1=2

(2:6)

The similarity of eq. (2.5) and eq. (2.6) is striking. Upon closer scrutiny, however, it is reasonable. The shape of a random phantom coil composed of bonds with fixed

2.2 Simple chain models

39

length l and free random bonding angles φ is the same as that of a path of diffusion steps of fixed length l and free random directional change from step to step, as shown in Figure 16.

Figure 16: Trajectory of a two-dimensional random walk.

2.2.2 The freely rotating chain So far, we have assumed no angular dependence in our calculations. However, we actually know how such a dependence should be described, because we know the bonding angle, φ, from the chemical specificity of each monomer. Thus, we can assume that each bond vector~lj projects a component l · cos φ onto the next bond vector ~lj + 1 . Accounting for that, eq. (2.4) can be refined to  2 1 − cos φ ~ r = Nl2 1 + cos φ

(2:7)

This equation is derived through a series expansion, which is explained in detail in Rubinstein’s and Colby’s book Polymer Physics. When we calculate ð1 − cos φÞ=ð1 + cos φÞ for an sp3-hybridized carbon bond with a bonding angle of φ = 109.6°, we get a value of 2. Consequently, we see that we underestimate such a polymer coil’s size by a factor of √2 if we disregard its fixed bonding angle, as we do in the phantom chain model.

40

2 Ideal polymer chains

2.2.3 The chain with hindered rotation We can further take into account the bond’s torsional angle, θ. We have seen in Section 2.1 that the torsional angle is not fixed, but prefers specific conformations due to their lower energies relative to other conformations. Accordingly, we can average over all possible bond conformations per time and chain ensemble to determine the average torsion angle hcos θi:   R +π ΔEðθÞ exp − cos θ dθ −π RT   (2:8) hcos θi = R +π ΔEðθÞ exp − dθ −π RT Equation (2.8) is based on the integral of a Boltzmann term of the torsional-angledependent potential energy, ΔE(θ), from Figure 14. With this value, we can append a term for the torsional angle dependence to our chain model:  2 1 − cos φ < 1 + cos θ > ~ · r = Nl2 1 + cos φ < 1 − cos θ >

(2:9)

We have now developed a model that takes into account both the universal physics of a polymer coil, which is expressed by its universal scaling according to h~ r 2 i1=2 ⁓ N 1=2 , and that also contains terms that account for the chemical specificity of a given type of monomer or structural unit, characterized by its bonding angle and its average torsional angle. These specific values are constant for each kind of repeating unit, which is why they are often summed up into polymer-specific parameters. The torsional-angle-dependence h1 + cos θi=h1 − cos θi is often called the obstruction parameter, σ2. It is determined by the bulkiness of the side groups of a repeating unit: the more energy is needed for these units to change their conformation, the higher this value. More often, the characteristic ratio, C∞ , is used; it includes all chemistry-specific parameters in the form of C∞ =

1 − cos φ h1 + cos θi · 1 + cos φ h1 − cos θi

(2:10)

With that, we get the general scaling law for ideal polymer chains: D 2E ! = C∞ Nl2 r The characteristic ratio is large for stiff polymer chains with bulky or like-charged substituents along the chain backbone, such as PS, poly(methylmethacrylate), or poly(sodium acrylate), whereas it is low for flexible chains with uncharged and unbulky substituents, such as PE. Note that for short chains, the characteristic ratio shows a dependence on the number of repeating units, N, as depicted in Figure 17; it is then abbreviated by the symbol CN . From about 80–100 repeating units on,

2.2 Simple chain models

41

Figure 17: Characteristic ratio, CN, as a function of the number of repeating units, N. Initially, CN rises with N, but after about 80–100 repeating units, it reaches a plateau for each polymer. That is why the characteristic ratio is often referred to as C∞ . Note how its value rises from about 3 to about 10 if we go from flexible chains with unbulky substituents such as polyethylene to chains with bulky substituents such as poly(methylmethacrylate). Picture redrawn from H. G. Elias: Makromoleküle, Bd. 2: Physikalische Strukturen und Eigenschaften (6. Ed.), Wiley VCH, 2001.

however, it reaches an N-independent plateau, as also shown in Figure 17; it is then abbreviated by the symbol C∞ .

2.2.4 The Kuhn model

Figure 18: The Kuhn model sums up several actual polymer chain segments of length l to create a new conceptual polymer chain made up of N=C∞ Kuhn segments of conceptual length lK = C∞ l, the Kuhn length. The end-to-end distance of the original chain, ~ r, is retained. The bonding angles of the Kuhn segments, however, are now random, whereas the bonding angles of the original chain were fixed. As a consequence, the conceptual chain can be assessed as a random chain.

42

2 Ideal polymer chains

When we recapitulate what we have done in the preceding sections, we see that we have taken up the most simplistic model and modified it with further terms to achieve a closer resemblance to reality, thereby making it more complex. But wouldn’t it be magnificent to have a model at hand that would retain the original simplicity of the random chain and, at the same time, still capture chemical specificity? This can be done on the basis of an ingenious approach by Werner Kuhn, which is the Kuhn model. In this model, a number (not necessarily a natural number) of several segments of a polymer chain with length l are grouped such to create a new conceptual polymer chain made up of new conceptual segments, the socalled Kuhn segments of new conceptual length lK, the so-called Kuhn length (see Figure 18). The original chain end-to-end distance is retained for this new conceptual chain. The ingenious element of this approach is the following: the chemical specificity of the original polymer is incorporated into the model by normalizing the number N and length l of the repeating units to the characteristic ratio C∞. That way, each given polymer “forgets its chemical specificity” and can be assessed by a universal power law:  2 N ~ ðC∞ lÞ2 = NK · lK 2 r = C∞ Nl2 = C∞

(2:11)

The new conceptual chain is made from NK = N =C∞ Kuhn segments of length lK = C∞ l. The bonding angle φ between these segments is now random, whereas it was a fixed constant in the original chain (Figure 18). As a result, the Kuhn chain can be assessed like a random chain. In other words: on scales beyond the Kuhn-length, the chain chemical specificity is no longer seen, and instead, each chain, independent of what sort of polymer it actually is, displays universal, random-walk type statistics. On these scales, hence, specific monomeric chemical primary structures do no longer matter, which allows us to conceive and draw the chain not by its actual chemical composition, but as a universalized schematic of a wiggly line; this is what we actually do from now on throughout the whole remainder of this book. With the Kuhn-approach, we have traced our more and more complex models described in the previous sections back to their simple origin. Note that with R2 = C∞ Nl2 as well as lK = C∞ l we get an interesting formula: R2 = C∞ Nl2 = C∞ lNl = lK lcont , with lcont = Nl the contour length, that is, the chain length in an hypothetic fully decoiled conformation.

2.2 Simple chain models

Werner Kuhn (Figure ) was born on February ,  in Maur, Switzerland. He developed a lively interest in natural phenomena in childhood days already and was schooled and studied in Zürich, where he also obtained his PhD degree with research on the photochemical degradation of ammonia in . After two years in Copenhagen, where he co-worked with Niels Bohr and also found his wife, Kuhn obtained his habilitation in Zürich in . He then worked at Heidelberg University from  till , where he developed his famous theory for the optical activity of chiral molecules. Kuhn then moved to Karlsruhe, where he started his work on polymers, obtained a professorship at Kiel in , and happily accepted a position at Basel in . In the s, he researched on membranes and gels, before he became director of Basel University in . Kuhn died unexpectedly on August ,  in the middle of fruitful work. There is an anecdote in the polymer community about Kuhn’s time in Basel, which is very close to the place of Freiburg, where Staudinger acted at the same time: Kuhn, a physical chemist, and Staudinger, an organic chemist, are said to having had kind of a “loyal rivalry” on the viewpoint of how polymers are properly conceived. For example, Kuhn’s work “Über die Gestalt fadenformiger Moleküle in Lösung” started with the statement that Staudinger’s view on caoutchouc and cellulose to be elongated rods in solution would in fact be outruled by Staudinger’s very own experiments. The aftermath of that was present up to Staudinger’s th birthday, where it was expressed poetically as “die Kuhnschen Knäuel sind uns hier ein Greuel”.

43

Figure 19: Portrait of Werner Kuhn. Image reprinted with permission from Chemie in unserer Zeit 1985, 19(3), 86–94. Copyright 1985 Wiley VCH.

2.2.5 The rotational isomeric states model We can also do the opposite of the Kuhn model’s simplicity and try to find a maximally realistic expression for C∞ for a given specific chain; this is the approach of the rotational isomeric states (RIS) model. It explicitly considers the energies of the most relevant conformational states T, G+, and G–, denoted as ΔE(θ) in Figure 14, and then uses the Boltzmann distribution to quantitatively appraise their populations:

44

2 Ideal polymer chains



ngauche − ΔEðθÞ = 2 · exp ntrans RT

(2:12)

This equation determines the exact ratio of gauche and trans conformations in a given polymer, as shown in Figure 20. The factor of 2 before the exponential term is due to the existence of two gauche conformations, G+ and G–. In addition to the gauche-to-trans conformational ratio, the RIS model also appraises how the conformation of each bond i is depending on what the conformation of the preceding bond i – 1 is; in other words, the model quantifies how many dyads of TT, TG, and GG are present in the chain. This is done by quantifying the energies of these different combinations and translating them into relative population ratios, as summarized in Table 5.

Figure 20: The RIS model quantitatively appraises the populations of the low-energy conformational states T, G+, and G–, and it determines the population of each of these states with the Boltzmann distribution. The upper diagram is a replot of Figure 14 and shows the energies of monomer–monomer bond torsional angles for butane; the lower diagram displays the relative populations of these bond angles at a temperature of 300 K. Upper diagram redrawn from H. G. Elias: Makromoleküle, Bd. 1 – Chemische Struktur und Synthesen (6. Ed.), Wiley VCH, 1999; lower diagram redrawn from R. H. Boyd, P. J. Philips: The Science of Polymer Molecules, Cambridge Solid State Series, 1993.

2.2 Simple chain models

45

Table 5: Relative population numbers of T and G conformations of a bond i in a polymer chain depending on the conformation T or G of the preceding bond i – 1.

0° (T)

+120° (G+)

–120° (G–)

1

0.54

0.54

1

0.54

0.05

+120° (G+)

1

0.05

0.54

–120° (G–)

0°(T)

Bond i – 1

Bond i

2.2.6 Energetic versus entropic influence on the shape of a polymer A key part of eq. (2.12) is the ratio of ΔE/RT. If this ratio is high, then the chain primarily adopts the energetically most favorable shapes. In synthetic polymers, these are either all-trans or alternating TGTGTG or TTGGTTGG conformational shapes, depending on the bulkiness of the substituents and their optimal low-energy arrangement in space. The most well-ordered polymers in nature are proteins; they have very specific energetically highly favorable folded shapes, and it is this specificity that gives rise to their specific functions, because these specific outstanding ordered shapes make possible the lock-and-key principle that enables specific enzyme–substrate interactions. By contrast, at a low ratio of ΔE/RT, a polymer chain adopts the entropically most favorable conformation, which is that of a random coil. A related ratio is that of the energy of activation for torsional change, ΔE ‡TG/RT. This ratio corresponds to the height of the energy barrier between the T and G states and thereby describes the dynamics of a polymer chain. A high ratio of ΔE ‡TG/RT is encountered for stiff polymer chains that carry bulky side groups, such as PS. A low ratio, on the contrary, is encountered for flexible polymer chains that carry unbulky side groups, such as PE. This ratio is directly reflected in one of polymers’ most important property: the glass-transition temperature, which is the temperature needed for activation of main-chain dynamics. This temperature is high for PS, whereas it is low for PE due to the less easy activation of bond conformational changes in PS than in PE. (See: again, here we have a structure–property relation!) Both the latter relations are dependent on temperature, which is because ΔG = ΔH − TΔS

(2:13)

In general, the change of Gibb’s free energy, ΔG, needs to be negative for any process to proceed spontaneously. We can see on the right-hand side of eq. (2.13) that

46

2 Ideal polymer chains

the change of enthalpy, ΔH, and that of entropy, ΔS, are connected to each other by temperature, T. It directly follows that at low temperature, the entropy term has a low significance, so that the enthalpy term dominates. As a result, a polymer coil expands and adopts a more ordered macroconformation at low temperature. This can even lead to higher-order chain assemblies such as polymer crystals. On the opposite, at high temperature, the entropy term dominates; here, a polymer coils up to a disordered macroconformation.

2.2.7 The persistence length Due to the fixed bonding angles and somewhat preferred conformational arrangements of the bonds in a polymer chain, as captured by the latter three models above, each segment i imparts a directional preference onto the next following ones i + 1, i + 2, i + 3, . . . along the chain. It therefore takes a certain number of following segments until the directional “memory” of a given first segment is “forgotten”. This “directional memory” or persistence is captured in the form of a quantity named persistence length. Mathematically, it can be defined as the projection of all following bond vectors i + j onto the direction of a given first one i, as shown in Figure 21.

Figure 21: Projection of all bonds i + j following a given one i onto its direction.

This projection is expressed as lp =

1 X  li lj li j > i

(2:14)

Just as the characteristic ratio, C∞, and the Kuhn length, lK = C∞l, the persistence length captures the stiffness of a polymer chain. The stiffer the chain is, the longer is the length up to which the directional influence of a given segment is still recognizable. It is not surprising that the two quantities lp and C∞ are related to one another: C∞ = 2

lp −1 l

(2:15)

47

2.2 Simple chain models

From that, we also obtain a relation between the persistence length, lp, and the Kuhn length, lK: C∞ + 1 ≈ C∞ = 2

lp l

, C∞ l = lK = 2lp

(2:16)

As an example, consider PS. In this polymer, we have C∞ = 10.2 and l = 0.154 nm; hence, based on eq. (2.15), we get lP = 0.86 nm. This is still low! For comparison, consider double-stranded DNA, which has a persistence length of 63 nm. This extremely large value originates from the constrained primary constitution of the DNA double strand and from the multiple repulsive electrostatic interactions between the charges of its phosphate units. Chains with such high stiffness are no longer named flexible, but instead semiflexible or wormlike. In these, a different geometrical view on the persistence length (that reflects the same mathematics, of course) is commonly used, given by the intersection angle of two tangents on the chain, as shown in Figure 22.

Figure 22: Schematic of a semiflexible, wormlike chain with two tangents constructed to it that cross in a certain intersection angle.

In wormlike chains, the persistence length is the length that it takes to have the average cosine of the angle shown in Figure 22 to decay down to 1/e (=0.368).

48

2 Ideal polymer chains

2.2.8 Summary We have learned throughout this chapter that the most probable shape of polymer chain is that of a random coil. We have developed the simplest model to describe this coil, which is the random chain model. Based on that, we have continuously expanded this model to incorporate the chemical specificity of the monomer segments by adding contributions from the bonding angle, φ, in the freely rotating chain model, and the bond torsion angle, θ, for the chain with hindered rotation (see Table 6). This chemical specificity can be captured by the characteristic ratio C∞, which is taken in the Kuhn model to resimplify and universalize the chain model while simultaneously retaining the given specificity of each given polymer. Furthermore, we have seen that the macroconformation of the chain is dependent on temperature and on the specific chemical design of the monomeric units. Polymers with bulky side groups exhibit stiff chains at ambient temperatures, whereas polymers with small side groups are flexible at ambient temperatures. We have therefore created our first rational structure– property relation. Table 6: Summary of the characteristic simple chain models addressed in this chapter.  2 R2 = ~ r = C∞ Nl2

Random chain Freely rotating chain Chain with hindered rotation

RIS model

Bond length (l)

Fixed

Fixed

Fixed

Fixed

Bond angle (φ)

Free

Fixed

Fixed

Fixed

Free

Fixed average

Discrete: T, G+, G–

1 − cos φ 1 + cos φ

1 − cos φ h1 + cos θi Specific · 1 + cos φ h1 − cos θi

Torsion angle (θ) Free Char. ratio (C∞)



To get an impression on some numbers, consider PE with N = 20,000, l = 0.154 nm, and C∞ = 6.87. Whereas the length of the maximally elongated chain, r = Nl, would be 3080 nm, its rms end-to-end distance is R = 1/2 = (C∞ N l2)1/2 = 57 nm. This estimate clarifies two insights. First, the polymer is heavily coiled in its natural state. Second, in that state, it has a size that falls right into the colloidal domain.

Questions to Lesson Unit 2

49

Questions to Lesson Unit 2 (1) The most likely conformation of a polymer chain is ______ a. that of a random coil, since the bond torsion angles can be changed by small activation energies. b. that of a random coil, since the bonding angles are statistically distributed by the synthesis process of the polymer. c. that of a defined coil, since bond length, angle, and torsion angle are predetermined and fixed by the chemical structure. d. that of an elongated chain, since the bond torsion angles arrange themselves such that the segments are as far apart as possible. (2) The characteristic ratio ______ a. reflects the ratio of the actual coil’s mean-square end-to-end distance to that of a hypothetical ideal chain with the same segmental number and length. b. reflects the characteristic ratio of bonding angle and torsion angle for each polymer. c. allows the specific characteristics and chemical identities of polymers to be clearly specified. d. describes the characteristic behavior of a freely rotating chain. (3) The Kuhn model ______ a. considers a random chain and is therefore independent of the characteristic ratio. b. is based on normalizing the segmental length and number to the characteristic ratio. c. does not reflect chemical specificity but is universal in nature. d. divides the original chain into a number of arbitrarily small segments. (4) The persistence length ______ a. is a measure of the durability of a chain. b. agrees exactly with the Kuhn length. c. is smaller for larger substituents along the chain. d. is a measure of the stiffness of the chain’s backbone.

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2 Ideal polymer chains

(5) Consider two equally sized ensembles of one-dimensional random walks: ensemble “Walk A” with a number of four steps each and ensemble “Walk B” with a number of ten steps each. Which statement regarding the random walks is correct? a. RMS and mean end-to-end distance reflect the same magnitude and therefore always have the same value, but they are greater for Walk B than for Walk A. b. For Walk A and B, the mean end-to-end distance is the same, but the RMS end-to-end distance of Walk B is greater than that of Walk A. c. For Walk A and Walk B, the RMS end-to-end distance is the same, but the mean end-to-end distance of Walk B is greater than that of Walk A. d. None of the previous statements is correct. (6) Order the characteristic ratios of the following polymers from small to large: PE, PP, PS. a. PP – PE – PS b. PE – PS – PP c. PP – PS – PE d. PE – PP –PS (7) The scaling law R2 ¼ C∞ Nl2 ______ a. combines chemical specificity with the physical relationships that are universally valid for all polymers. b. does not apply to ideal chains, since they have no specificity. c. is a special case and cannot be applied to all polymers. d. results from the assumption of a freely rotating chain. (8) At high temperatures, a polymer takes the ______ a. enthalpically favored conformation of a rod. b. entropically favored conformation of a rod. c. enthalpically favored conformation of a coil. d. entropically favored conformation of a coil.

51

2.3 The Gaussian coil

2.3 The Gaussian coil LESSON 3: GAUSSIAN COILS AND BOLTZMANN SPRINGS The last lesson has taught you that an ideal chain has the shape of a random coil. The following lesson will further refine this insight and make you understand how exactly the segmental density is distributed inside the coil. It will also show how the most fundamental thermodynamic property of such a coil, its entropy, reacts on deformation of the coil, thereby introducing the most relevant property of flexible polymers: their entropy-based elasticity.

We now want to take a closer look at how exactly a polymer coil looks like. We start by tying into the number example from the end of the preceding section and again consider PE with N =20,000, l = 0.154 nm, and C∞ = 6.87, for which we have estimated an rms end-to-end distance of R = 1/2 = (C∞ N l2)1/2 = 57 nm. We now calculate the volume of just all the segments together, that is, the volume that a dense globule would have into which the coil might collapse: Vsegments = no. of segments × vol. persegment ≈ N l3. For comparison, we also consider the volume that the coil has in its natural state: Vcoil = 4/3 π R3 = 4/3 π (C∞ N l2)3/2 ≈ 4 C∞3/2 N3/2 l3. When we compare these two volumes by calculating their ratio, Vcoil/Vsegments = (4 C∞3/2 N3/2 l3)/(N l3) = 4 C∞3/2 N1/2, we realize that this ratio is more than 10,000! This means that most of the volume of the polymer coil is actually empty and not occupied by the polymer chain material itself. But how, then, is that chain material distributed within the coil? The polymer chain segments in a coil are distributed according to a Gaussian radial density profile: 3 cseg = N 2πRg 2

!3=2

− 3r2 · exp 2Rg 2

! (2:17)

From the graphical representation of this equation, in Figure 23, we can see that the highest polymer segmental density is at the coil’s center. For the blue curve in Figure 23, denoting a polymer with N = 20,000 segments, we have a segmental density of about 11 segments per nm3 in the coil center; this corresponds to a molar concentration of about 20 mol L–1. The further we move away from the coil center, the more does the density drop, which is especially true for short polymer chains with only a few segments, whereas longer chains have a less steep radial segmental density profile. Nevertheless, note that the degree of coiling, Q, gets more pronounced at high N, as shown in the following estimate: Q=

Lcont hr2 i1=2

=

Nl = N 1=2 N 1=2 l

(2:18)

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2 Ideal polymer chains

Figure 23: Concentration of polymer segments in a coil as a function of the coil radial coordinate, r, for two polyethylene chains with N = 20,000 and N = 80,000 repeating units. In both cases, the majority of segments is located at the coil center, which is more pronounced for the shorter of the two chains. In radial direction away from that center, the segmental density drops according to a Gaussian profile. The dashed lines indicate the radii of gyration of both coils. The upper-right sketch illustrates the radial segmental density profile in a schematic fashion.

Hence, longer chains are coiled more notably than shorter chains, but in terms of their segmental density profile, long chains are more “dilute” at the coil center than short chains but have a higher segmental density at the rim in turn.22 This last consideration is a good indication that we have now moved beyond average values to learn about the shape of a polymer coil. In the following, we will therefore look more closely at the coil statistics, with the aim to not only describe its average size, but the full distribution of it.

 This circumstance is a quite astonishing consequence of the fractal nature of polymers, which we will get to know in Section 2.7. For the moment, we can capture it only (but at least) from a mathematical viewpoint: According to Eq. (2.17), we get a local segmental density proportional to N–1/2 for r = 0 (where the exponential factor is exp(0) = 1) if we plug in Rg2 ⁓ N in the prefactor; we may also get to this notion by simply estimating N/V ≈ N/R3 ⁓ N/(N1/2)3 ⁓ N–1/2. This means: the higher the degree of polymerization, the lower is the segmental density in the core. The reason is because random Gaussian coils have a fractal dimension of 2, as captured in the scaling R2 ⁓ N, but when given 3 actual dimensions, we get terms with R3, which then leads to the above mathematical result. By contrast, this effect vanishes if we match the geometrical dimension with the fractal one, that means, if we consider a two-dimensional flat polymer on a surface. In this case, we get a segmental density that is independent of N, because then, our calculation reads N/A = N/R2 = N(N1/2l)2 = N0.

2.4 The distribution of end-to-end distances

53

2.4 The distribution of end-to-end distances 2.4.1 Random-walk statistics We have learned in Section 2.2 about the fundamental scaling law between the meansquare end-to-end distance and the number of monomer segments of an ideal polymer   chain, r2 ⁓ N. We have also noted the similarity to the Einstein–Smoluchowski law   of diffusion, x2 ⁓ t, and realized that this is due to very similar assumptions made in the two models: the path of a diffusing particle has no memory and can cross itself, and an ideal phantom polymer chain is assumed to be composed of volume-less segments without interactions, so it can cross itself, too. We now rely on this similarity to appraise the distribution of the chain end-to-end distance, r, by adopting randomwalk statistics. We start by considering a one-dimensional random walk composed of N steps on the x-axis:

N

x

Statistics

You may imagine this walk as follows: think that you place a marker at zero and flip a coin. Every time the coin lands on heads, you move the marker up in the x-direction, and every time the coin lands on tails, you move the marker down. After one flip, the marker can be at x = ±1 with a probability of 50% each. After two flips, the marker can either be back at x = 0, or it can be at x = ±2, with a probability of 50% for x = 0 and 25% for each x = ±2. In general, for a total number of steps N = N+ + N–, an end position of x = N+ – N– can be calculated. It is apparent from the statistics column in the table above, which resembles Pascal’s triangle, that the most

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2 Ideal polymer chains

probable outcome of any such random walk will be x = 0. This is because there’s many walks that we can take for a given number of steps, but most of them lead us back to zero, whereas just few of them lead us far apart from zero. Using the binomial statistics behind the Pascal triangle in the table above, we can appraise the number of walks reaching position x after N steps as N! W ðN, xÞ = N + x N − x 2 ! 2 !

(2:19)

Equation (2.19) can be understood by illustrating the sequence of N steps as an ordered row of lots that we randomly draw out of a container. The first one that we draw is one out of a total of N, the second is one out of a remaining total of (N – 1), the third is one of a remaining total of (N – 2), and so forth. This gives us N · (N – 1) · (N – 2) · . . . = N! possible permutations to arrange that sequence. Each lot has two possible values in a random walk: it can denote a step in the positive direction or a step in the negative direction. As a result, we can divide this sequence into two subensembles. One comprises all those steps that go into the positive direction; we have a total of N+ such steps. The other comprises all those steps that go into the negative direction; we have a total of N– such steps. The order of these N+ and N– steps is irrelevant, because we always end up at the same final position of x = N+ – N–. Hence, we have to divide our N! by the number of permutations of these N+ and N– steps, which is N+! and N–! This gives us N!/(N+! · N–!) possible walks with a given number of positive and negative steps. We have seen already that the majority of these walks will lead us close to the center at around x = 0, so we are particularly interested about those walks with N+ = N/2 + (x/2) and N– = N/2 – (x/2). Replacing the N+ and N– in the expression N!/(N+! · N–!) with these terms turns it into eq. (2.19). The total number of walks that we can take at all is 2N. Together with eq. (2.19), this gives us the probability of reaching position x after N steps: W ðN, xÞ 1 N! = N ·  N + x  N − x 2N 2 2 ! 2 !

(2:20)

As W(N, x) is a big number, we take the logarithm of it and apply Stirling’s approximation ln(N!) ≈ N (ln(N) – 1) to eliminate the factorials. We may then expand it into a Taylor series with x/N as the variable. This will give us a zeroth-order element that is independent of x/N, whereas we will have no first-order element, as the distribution is symmetrical. The second-order element will be of kind (x/N)2. If we interrupt the series after this element, it will be of type ln[W(N, x)] ≈ a – b·(x/N)2, so we have W(N, x) ≈ exp [a – b·(x/N)2] = exp [a] · exp [–b·(x/N)2]. The exact calculation yields rffiffiffiffiffiffiffi 2

W ðN, xÞ 2 −x (2:21a) · exp ffi 2N 2N πN

2.4 The distribution of end-to-end distances

55

If we normalize our coordinate system from a step width of 1, as we had it in the sketch above, to a step width of ½, we transform the integer distance on the x-axis in the walks in the table above from 2 to 1. With that, we get the probability distribution of the displacement x in a one-dimensional random walk: rffiffiffiffiffiffiffiffiffi 2

1 −x (2:21b) · exp p1d ðN, xÞ = 2N 2πN Equation (2.21b) differs from eq. (2.21a) just by a factor of 2, simply because the integer distance on the x-axis was 2 for the walks in the table above, whereas it has now been normalized to 1. So far in this book, we have considered the mean-square end-to-end distance, which corresponds to the mean square displacement in the random walk. In general, such mean values can be calculated from distribution functions by the use of mathematical operators. Applying such an operator to eq. (2.21b) gives us the mean square of the distribution: rffiffiffiffiffiffiffiffiffi +ð∞ +ð∞ 2

1 −x dx = N (2:22) x2 p1d ðN, xÞdx = x2 exp hx2 i = 2N 2πN −∞

−∞

This simple result provides a useful identity, as it connects N to . With that, we can replace one by the other in eq. (2.21b) and obtain a formula with just one variable (x): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 − x2 · exp (2:23) p1d ðxÞ = 2 2π We now want to move into the three-dimensional realm, so we need to consider all three spatial directions. Fortunately, a simple random walk does not have a preferred direction; in other words: we consider an isotropic case. In such a case, we may construct a three-dimensional random walk simply by the superposition of three independent components, one for each spatial dimension:   (2:24) rÞdrx dry drz = p1d ðrx Þdrx · p1d ry dry · p1d ðrz Þdrz P3d ð~ In this simple treatment, the three-dimensional end-to-end distance is additive of its three components:  2  2  2 Nl2 r = Nl2 = hrx i2 + ry + hrz i2 · hrx i2 = ry = hrz i2 = 3

(2:25)

With that, we can formulate an expression for a three-dimensional random walk as follows:

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2 Ideal polymer chains



3=2

nð~ rÞ 3 − 3r2 p3d ð~ · exp = rÞ = P ~ 2 2π r nð r Þ

(2:26)

In this equation, nð~ rÞ is the number of walks (or polymer chains) with a given disX placement (or chain length) ~ r, and nð~ rÞ is the total number of walks (or chains). r Let us now examine how a graphical representation of eq. (2.26) looks like. As an example, a three-dimensional random-walk-type probability distribution of the end-to -end distance for the example of a PE chain with N = 20,000 repeating units is shown by the green curve in Figure 24. It visualizes the probability to have an end-to-end vector ~ r with a given length j~ rj and a given direction. This corresponds to the probability of finding the second chain end in a certain specific volume element at a distance

Figure 24: Distribution of end-to-end distances of an ideal polyethylene coil composed of N = 20,000 repeating units. The green curve is the outcome of a three-dimensional random-walk statistical treatment (eq. (2.26)). Here, the most likely trajectory, or end-to-end vector, ~ r, is zero. This corresponds to the likelihood of finding the second chain end exactly in a specific small volume element in space if the first chain end is placed in the origin, as illustrated in the lower left sketch. The blue curve filters off all directional dependences by multiplication with a sphere surface area (eq. (2.27)), thereby reflecting the distribution of end-to-end vector lengths without caring about into which direction they point. This corresponds to the likelihood of finding the second chain end localized anywhere on a spherical orbit with radius j~ r j around the first end that is again placed in the origin, as illustrated in the lower right sketch. (Note that the green p(r) is actually of type pð~ r Þ, whereas the blue p(r) is actually of type pðj~ r jÞ. For common representation on the same simple r-axis, they are both denoted in a simplified form as p(r) on both ordinates. Their mathematical difference is apparent from the different physical units, though).

2.4 The distribution of end-to-end distances

57

and direction ~ r from the origin if the first chain end is placed right there. This probability is maximal at an end-to-end vector of zero, as the random-walk statistics has given us the highest likelihood of obtaining that very displacement,23 whereas the probability distribution for other end-to-end vectors or displacements drops in the form of a Gaussian bell curve.24 In elementary statistics, the likelihood of an event is proportional to the frequency of its occurrence in a large ensemble. Hence, the Gaussian probability distribution that we have just found is directly related to the Gaussian distribution of the monomer segmental density from Section 2.3, which is also highest at the center of the polymer coil. Due to that identity, ideal polymer chains are often named Gaussian chains. We can filter off the directional dependence of the random walk by multiplying it with the surface area of a sphere of radius r: rjÞ = 4πr p3d ðj~

2

3 2πhr2 i

3=2



− 3r2 · exp 2h r 2 i

(2:27)

That way, we generate a function that expresses the probability distribution of endto-end vector lengths, independent of their direction.25,26 The blue curve in Figure 24  Here we make a simplifying approximation. Actually, only the one-dimensional random walk has a most likely displacement of zero. Two-dimensional and three-dimensional random walks, by contrast, have not. Of course, we may assume a two- or three-dimensional random walk to be composed of a superposition of two or three independent one-dimensional walks, and of course, individually, those indeed have a most likely displacement of zero, but to have an allover displacement of zero also in the superposition, all of them must get to zero together. However, as the likelihood for other displacements is also not too small, especially for small displacements, chances are high that in two- or three-dimensions at least one of the two or three overlaid walks will actually not show a zero displacement. So, two- and three-dimensional random walks actually do not have a maximal likelihood of an overall displacement of zero. For the sake of keeping our treatment simple, though, we ignore that and simplify the three-dimensional displacement to be maximal at zero, too, like the one-dimensional displacement. 24 Equation (2.26) is similar to an equation that you (hopefully) know from elementary physical chemistry: the velocity distribution in the kinetic theory of gases, pð~ vÞ. That distribution has the form of a Gaussian bell curve as well, as it essentially reflects the Boltzmann distribution of the (kinetic) energies of the gas particles, which has its maximum at zero. 25 Again, something similar is known to you from the kinetic theory of gases, where pð~ vÞ turns into pðj~ vjÞ, the Maxwell–Boltzmann distribution, by the same mathematical operation.  Even more, you may have encountered a similar concept earlier in yet a different field of elementary physical chemistry already. In quantum mechanics, the square of the wave function of the s-orbital is considered as a measure of the likelihood of finding the electron in a certain point in space. This likelihood is maximal at a distance of zero from the nucleus and then drops when moving to larger distances. Just as with the ideal-polymer case treated here, in this problem, though, the directional dependence is not of so much interest than the radial-distance-dependence. To account for that, the square of the wave function is multiplied with a spherical surface area of radius r to obtain a function that gives the likelihood of finding the electron at a certain distance from the nucleus, independent of the direction. Whereas the first function (the square of the wave function)

58

2 Ideal polymer chains

reflects this relation for the same PE compound considered above. It visualizes the probability to find any end-to-end vector of length j~ rj independent of its direction, that is, any end-to-end vector on a spherical circumference with radius r.

2.5 The free energy of ideal chains We now have a formula at hand that reflects how many microconformations, characterized as three-dimensional random walks, lead to a certain chain macroconformation, characterized by its mean-square end-to-end distance. This is information about the structure, about the shape of the polymer chain. In this book, however, we have set out to build structure–property relations. So, is it possible to use the structural information we have obtained and extract from them information about the polymer’s properties? Would it be possible, for example, to calculate thermodynamic quantities from them? As it turns out, this is feasible: we can use statistical thermodynamics, which derives macroscopic thermodynamic quantities from the likelihood of microscopic states of a system. In statistical thermodynamics, Boltzmann’s entropy formula connects the entropy, S, to the number of possible microconfigurations, W, as S = kB ln(W). When we insert the probability distribution of the end-to-end distances into that formula, we can derive an expression for the entropy, S, of an ideal chain: rÞ ffi S0 − S = S0 + kB ln W ðN,~

3kB r2 2h r 2 i

(2:28)

where S0 is the entropy at an end-to-end distance of zero. We see that the total entropy, S, is maximal (with a value of S0) for an end-to-end distance of r = 0, because the probability of obtaining that distance in a random walk is maximal (green curve in Figure 24). From this, we can further formulate the free energy, F, of an ideal polymer chain: F = U − TS = F0 +

3kB Tr2 2h r 2 i

(2:29)

where U is the internal energy and F0 is the free energy at an end-to-end distance of zero. In the case of an ideal polymer chain, U is independent of the end-to-end distance, because we imagine the chain segments to have no interactions, so energy doesn’t care about their spatial arrangement and microconformations. This means that any change of the ideal-chain free energy stems from entropy alone.

drops with the radial coordinate, the second function (the sphere surface-area function) increases as a power law (with a power of 2). The product of both functions first raises, then reaches a maximum, and then drops with the radial coordinate, with the intermediate maximum denoting the atomic radius.

2.6 Deformation of ideal chains

59

2.6 Deformation of ideal chains 2.6.1 Entropy elasticity When we slightly stretch an ideal polymer chain, the most feasible microscopic process of accounting for that deformation is decoiling of the chain, that is, transformation of local gauche or cis conformations into trans conformations, as this does not cost any significant amount of energy (see Figure 14). With that, though, we reduce the number of possible microconformations that realize the coil’s macroconformation, thereby forcing it into an entropically more unfavorable state. As a result, as soon as the deformation cedes, the chain will relax back to the most probable coiled structure. This phenomenon is based on entropy and is therefore called entropy elasticity. When we differentiate the free energy with respect to rx, we can calculate the force needed for a deformation in the x-direction: ∂F ðN,~ rÞ 3kB T 3kB T ~ = 2 ·~ ·~ rx rx = fx = ∂rx Nl2 hr i

(2:30)

The latter formula tells us how much force is needed to stretch a chain by a distance rx. Its form is that of Hooke’s law (f = κ·x), which connects the force necessary to achieve a certain extent of deformation of a body, f ⁓ x, by its spring constant, κ, a fundamental material property expressing how good mechanical deformation energy can be stored and released, and hence, how much the material deforms upon application of a given force. In the case of an ideal polymer chain, we can consider  ð3kB T Þ Nl2 to be an entropic spring constant. As this quantity only contains structural information, namely, the number of segments and the segmental length, we have established another structure–property relation. Strictly speaking, eq. (2.30) and the concepts we have derived from it are only valid for Gaussian chains in terms of the Kuhn model, that is, chains with a free segmental bonding angle φ. How would this picture change when we would consider it for the freely rotating chain that has a fixed segmental bonding angle φ? This question can be answered on two levels. On a conceptual level, the freely rotating chain is already more ordered than a Gaussian chain. It therefore already has a lower entropy, such that a further loss of entropy would not have such a significant influence, which means that these chains should be easier to stretch. On a mathematical level, we would have to consider the characteristic ratio C∞ as a further factor in the denominator of the entropic spring constant. This would result in a smaller entropic spring constant, which would also mean that freely rotating chains are easier to stretch.  The entropic spring constant 3kB T Nl2 has a temperature dependence in its numerator. According to this dependence, it will rise with increasing temperature, thereby making it harder to deform an ideal polymer chain at higher temperature. This effect can be understood because entropy in thermodynamics always occurs

60

2 Ideal polymer chains

along with temperature in the form of TΔS. Hence, the reduction of entropy upon chain stretching is even more pronounced at higher temperature, making it more unfavorable and therefore harder to realize. So far, with eq. (2.30), we operate at a single-chain level. When we consider an ensemble of n chains, the essence of eq. (2.30) still holds for each single chain, such that the total force needed to deform all n of them is just n times that of a single chain, so a factor n will enter the right side of eq. (2.30). Furthermore, if we normalize the force to an area, this translates into a pressure, p, on the left side of the equation, whereas the length l in the denominator on the right side merges with that newly introduced area to a volume. We then end up with an expression strikingly close to the ideal gas law: p = nkBT/V. Once more, we have discovered astonishing similarity between the physics of the ideal polymer chain and the ideal gas. For our present discussion on entropy elasticity, this is straightforward to understand. According to the ideal gas law, the pressure of an ideal gas rises if we compress it at constant temperature. Why is that? Energetically, the point-like gas molecules do not care about their mutual distance, and with that, about the volume we give them, because they do not have interactions, neither attractive nor repulsive. Entropically, however, a reduction of the system’s volume reduces the number of possibilities for the molecules to arrange themselves in space. In short: a reduction of the volume reduces the freedom of the gas molecules. This comes along with an entropic penalty, which translates into an increase in free energy, and thus, to a backdriving force just like the one according to eq. (2.30), which translates into a pressure if we normalize it to area. The same is seen by a rise of the stress, σ = f/A, in a polymer sample that is subject to stretching, which is also caused by the loss of conformational freedom, and hence, the entropy penalty that comes along with that. In contrast to the entropic origin of elasticity of ideal polymer chains and ideal gases, the deformation of a classical solid such as a wire of metal is fundamentally different. Here, upon deformation, the metal atoms are lifted out of their equilibrium positions in the crystal lattice that correspond to a minimum in their (Lennard–Jonestype) interaction potential, and as a result, an energetic-based restoring force arises. Upon increase of temperature, such an energy-elastic body will expand, because the additional energy enables the atoms to oscillate more extensively around their energy-minimum positions, and due to the skew shape of the (Lennard–Jones-type) interaction potential well (which has a steeper incline at its left than at its right rim), this stronger wiggling corresponds to a shift of the average atom positions in the lattice to larger separations. This energetic excitation also allows the material to be deformed easier at higher temperatures. Ideal polymer chains, by contrast, will shrink at elevated temperatures, as an energetically favorable but entropically unfavorable excess of local trans conformations in the chains will get less dominant in this case, due to the fundamental coupling of temperature and entropy in the form of TΔS. As a result, the chain end-to-end distances in a rubber sample under slight load shrink upon heating in order to achieve their entropically most favorable value of zero, and yet in

2.6 Deformation of ideal chains

61

Figure 25: Overview of the different possible modes of deformation for classical solids (metal), polymers (rubber), and gases. Upon heating, a classical solid and a gas expand, whereas a polymer such as rubber shrinks. In a classical solid, this is because the atoms in a crystal lattice oscillate more heavily around their equilibrium positions at higher temperatures, and due to the skew shape of their Lennard–Jones-type interaction potential well (which has a steeper incline at its left than at its right rim), this stronger wiggling corresponds to a shift of the average atom positions in the lattice to larger separations. In a gas, even more simply, the basic equation of state pV = nRT explains expansion on rise of temperature. Ideal polymer chains, by contrast, shrink at elevated temperatures, as an energetically favorable but entropically unfavorable excess of local trans conformations in the chain will get less dominant in this case, due to the fundamental coupling of temperature and entropy in the form of TΔS; as a result, the chain end-to-end distances in a rubber sample under slight load shrink upon heating in order to achieve their entropically most favorable value of zero, and yet in turn, the whole sample specimen shrinks. Upon mechanical deformation, the atoms of a classical solid are moved out of their potential energy minima, thereby creating a restoring force based on energy. Deformation of a polymer and a gas, by contrast, reduces the number of conformations or freedom of arrangement of the molecules in space, respectively, thereby creating a restoring force based on entropy. Picture inspired by J. E. Mark, B. Erman: Rubberlike Elasticity: A Molecular Primer (2nd ed.), Cambridge University Press, 2007.

62

2 Ideal polymer chains

turn, the whole sample specimen shrinks. An ideal gas, yet in turn, does not contract when temperature is increased, but instead, it shows thermal expansion, because its volume is directly proportional to temperature (pV = nRT). All these various differences and similarities of the two fundamental examples of entropy-elastic materials, which are the ideal gas and ideal polymer chains, as well as the classical example of an energy-elastic material, which is a piece of metal wire, are illustrated in Figure 25.

2.6.2 A scaling argument for the deformation of ideal chains As an alternative to the lengthy statistical treatment of the entropic elasticity of ideal polymer chains that we have just discussed in the preceding sections, Rubinstein and Colby have introduced a clever treatment based on a blob concept (that originates from De Gennes). In this concept, the chain is regarded as a sequence of blobs of size ξ, each containing g segments. Up to the blob length scale, kBT is the most relevant energy. As a consequence, any external energy is smaller than kBT inside each blob, it is exactly kBT at the blob scale ξ, and it is larger than kBT above the blob scale. When the chain is stretched by an external force fx, the segments inside the blobs are unaffected by this deformation, because on scales up to the blob scale ξ, the deformation energy is weaker than – or, in other words, screened by – the ever-present thermal energy kBT. As a result, the deformation is effective only on larger length scales, where enough deformation energy is cumulated to outweigh kBT. In this view, the chain segments inside the blobs always obey the ideal scaling law ξ 2 = gl2

(2:31)

 2 which is a blob-scale variant of the basic law R2 = ~ r = Nl2 that we have discussed in terms of the ideal chain in Section 2.2.1 (eq. (2.4)). In contrast to these ideal statistics on small scales, the stretched length of the entire chain can be viewed as a unidirectional linear sequence of blobs, as shown in Figure 26. Mathematically, a unidirectional linear sequence of blobs is expressed as the number of blobs, (N/g), times the blob size ξ: rx =

N ξ g

(2:32a)

Replacing g in the denominator by eq. (2.31) transforms this into rx =

Nl2 ξ

(2:32b)

2.6 Deformation of ideal chains

63

Figure 26: Schematic representation of the blob concept. Here, a chain with length rx is viewed to be composed of conceptual units named blobs of a limiting length scale ξ, each containing g of the actual monomer segments. At scales smaller than the blob size, any external energy is smaller than kBT. This means that when the chain is stretched by a force fx, the deformation energy inside each blob is screened by kBT, and deformation is only effective on length scales longer than ξ. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

That can be rearranged to ξ=

Nl2 rx

(2:33)

g=

N 2 l2 rx 2

(2:34)

Plugging that into eq. (2.32) gives

These latter formulae show that at stronger deformation (larger rx), the blobs get smaller, meaning that the length scale up to which the deformation is screened by the thermal energy gets smaller: thereby, the blobs of course also get more numerous, because if they get smaller, then more of them are needed to build up the chain. At maximal deformation (rx = Nl), the blob size is ξ = l (and g = 1), which means at such extreme deformation, the blobs have shrunken down to the size of the actual monomer; then, the deformation is notable on all length scales. When the chain is conceived as such a sequence of blobs and then stretched in x-direction, the blob sequence gets ordered from being random to being aligned in that direction. This ordering comes along with a loss of one directional degree of freedom per blob, thereby raising the chain’s free energy by that very extent. Hence, the free energy of stretching in the x-direction, Fx, is one kBT increment per blob: Fx = kB T

N rx 2 = kB T 2 Nl g

(2:35)

This energy is small at weak extent of stretching, where the chain is conceptually composed of just few large blobs (meaning that the deformation energy is so small then that up to quite large blob-lengthscales, it is screened by the thermal noise kBT), whereas it is large at strong extent of stretching, where the chain is then conceptually

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composed of many small blobs (meaning that the deformation energy is then so large that only on very small blob-lengthscales, it is still screened by the thermal noise kBT). Differentiation of that free energy to the stretching distance yields the force for deformation, fx, in the limit of moderate stretching: fx =

∂F rx ≈ kB T 2 Nl ∂rx

(2:36a)

This result is qualitatively similar to that from the longer exact derivation in the preceding section, but with Rubinstein’s and Colby’s blob concept and scaling approach, it was obtained much quicker and easier. This is a great advantage of scaling discussions as the one just led: they give results in good semiquantitative agreement to those obtained from exact and often more complicated derivations, but they do so in a much simpler and quicker fashion. Another great advantage is that they make us see conceptual grounds. We may get such an insight by replacing  rx Nl2 in the last equation by eq. (2.33), yielding fx =

∂F rx kB T ≈ kB T 2 ≈ Nl ∂rx ξ

(2:36b)

Thus, we can conceive that the deformation energy is kBT per blob. As shown above, the blobs get smaller (and then in turn more numerous) as the deformation gets stronger (see eq. (2.33)). The greatest extent of deformation is if the chain is fully expanded to a rod-like object of length Nl. In that extreme, according to eq. (2.33), the blobs have shrunken down to the monomer segmental length of l (and accordingly, following eq. (2.34), the number of monomers per blob is then just 1). In that extreme, according to eq. (2.36b), the energy for deformation is kBT per monomer. The reason why the change in scale that we have just used is valid and can still be described by the same mathematical equations is the fact that polymers are fractal and self-similar objects, a topic that we will discuss in the following paragraph.

2.7 Self-similarity and fractal nature of polymers The relations between the mass and the characteristic size of any geometrical object can be described by scaling laws. A three-dimensional sphere, for example, exhibits scaling of m ⁓ r3 . A two-dimensional piece of paper exhibits scaling of m ⁓ r2 , and a one-dimensional piece of wire exhibits scaling of m ⁓ r1 . Generally, any object exhibits scaling according to m ⁓ rd where d is its geometrical dimension.

(2:37)

2.7 Self-similarity and fractal nature of polymers

65

The same principle is valid for ideal polymers; these obey the basic scaling law R ⁓ N 1=2 that we have developed in Section 2.2.1. N is proportional to their mass (m = N · mmonomer); thus, comparison to the general relation m ⁓ rd denotes ideal polymers to have a dimension of 2. This is a so-called fractal dimension, as it is different from the geometrical dimension, which is 3 for a polymer in our threedimensional world. In the form introduced here, the fractal dimension is that of a mass fractal, as it establishes a relation between the object’s mass and its size. This fractality is not limited to ideal polymer chains. As we will see in the next main chapter, the general scaling law of a real polymer chain is R ⁓ Nν

(2:38)

with ν = 1=dfractal , the Flory exponent. Table 7 compiles a glimpse on fractal dimensions of various polymer types. In general, a smaller fractal dimension denotes the object that it belongs to, to be less dense. If we compare the fractal dimensions of ideal polymer chains, which is 2, to that of real chains with short-range repulsion, which is 5/3, we see that the latter is smaller. This means that real chains are less dense than ideal chains. The reason for that is the short-range repulsion between the monomer segments in a real chain, which pushes them apart from one another, thereby resulting in coil expansion that comes along with a lower segmental density inside the coil. Table 7: Overview of the fractality of ideal and real polymer chains with linear or branched architecture. Architecture

Interactions

Spatial dimension

Linear Linear Linear Branched Branched Branched

None Short-range repulsion Short-range repulsion None Short-range repulsion Short-range repulsion

Any   Any  

Fractal dimension  / /  / 

Fractal objects are also self-similar. This concept can be illustrated by the example shown in Figure 27(A). A two-dimensional square with side length L and area A has a scaling law for its area of A = L2. A subsquare within that plane with a side length l and area a has the same basic shape, and it also exhibits an area scaling law of a = l2. Without information on the actual length scale that we look at, we can therefore not distinguish whether we look at the whole object or a subunit of it. This phenomenon is called self-similarity. Many objects in nature are self-similar: clouds, coastlines, the surface of broccoli, and more. For all these objects, when being shown a picture of them, you cannot tell whether you see a small or large section of the object, as it has the same appearance on different scales.

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Figure 27: Illustration of the concept of self-similarity. (A) Consider a two-dimensional square with side length L and area A, and within that, a subsquare with side length l and area a. Both squares exhibit the same scaling law for their area, but on different length scales: A = L2 and a = l2. These laws are self-similar. (B) The same principle holds true for polymers: The basic scaling law of an ideal chain is R ⁓ N1=2 , which has a variant for subsegments of the chain of ξ ⁓ g1=2 . Again, these laws are self-similar. Picture in (B) is redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

The shape of an ideal polymer coil is also self-similar, as shown in Figure 27(B). Although the shape of the whole chain shown in this figure is not exactly equal to that of the magnified chain subsection, on average the same random-walk-type sequence is seen on these different length scales. With that, also the scaling law of the ideal chain applies on both these (and further) different length scales. For the whole chain of N monomers, we have a scaling of R ⁓ N 1=2 . For a subunit of the chain with just g monomers, we have a similar scaling of ξ ⁓ g1=2 . Rearranging these two scaling equations yields: R = N 1=2 l , l = RN −1=2

(2:39a)

ξ = g1=2 l , l = ξg −1=2

(2:39b)

We can combine these latter two equations via their common variable l to obtain RN − 1=2 = ξg− 1=2 , R =

1=2 N ξ g

(2:40)

This equation has again the same form as the one we started with: it relates the polymer coil size, R, to the square root of a dimensionless number times an elementary unit size. In this notation, the chain can be viewed as a sequence of (N/g) segments of size ξ each, which was just the notation taken for the development of the blob concept in Section 2.6.1, where ξ was set to be the blob size.

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Eq. (2.40) describes our chain, which is actually a random walk of N segments of size l each, as a new conceptual random walk of (N/g) blobs of size ξ each. We have done something similar before in Section 2.2.4: in the Kuhn concept, we have also renormalized our actual chain of N segments of size l each as a new conceptual chain of NK segments of size lK each. Now, the same was done with blobs as the new conceptual segments. Due to the self-similarity of polymer chains, this kind of renormalization works with any new scale, as long as we stay on scales larger than the Kuhn length lK, because below that length, the polymer chain is no longer universal and self-similar but markedly exhibits its chemical specificity. In general, such kind of scale transformation works with any object that is self-similar and therefore scale invariant, meaning that it does not have a natural length scale that sets its further properties (like its mass or surface area). Scale invariance is a fundamental phenomenon in nature, similar to symmetry. As a summary, in both the Kuhn model and in the blob concept, we renormalize a chain by a sequence of new conceptual rather than the actual repeating units and thereby shift the segmental length and number to new scales. The only mathematical function that allows such renormalization to be performed is power laws, which are also named scaling laws for that reason. If another kind of function, for example, a transcendental function27 like R = R0 ln(N/N0), would describe the N-dependence of R, such rescaling would be mathematically impossible. Again, the reason for this is the absence of a natural scale (as it would be captured by N0 and R0 in the latter hypothetic equation) in objects that are scale invariant. Hence, the characteristics of self-similar and therefore scale-invariant objects such as polymers are always powerlaw type. This is the reason why power laws occur all over this textbook; they are inherent to polymers, as polymers are self-similar and fractal. [One further note: it is always single power-law terms that describe the relations between the characteristic quantities of self-similar objects (such as the N-dependence of R of a polymer), but not sums of them. Imagine, for example, that a sum of two power-law terms would describe the N-dependence of R in a form like R = A·(Nl) + B·(Nl)2. Then, there would be coefficients A and B with different physical dimensions; A would be dimensionless, while B would have the unit m–1. Their ratio A/B would then have a unit of m and therewith again reflect a natural length scale in the object under consideration.]

 In a transcendental function, the argument is not allowed to have a physical unit; typical examples are exp, ln, sin, cos, and so on.

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Questions to Lesson Unit 3 (1) Consider a coil A with 100 and a coil B with 300 chain segments. Which statement is correct? a. Coil B has a steeper drop in radial density than coil A, because the higher number of chain segments in coil B results in a more distinct spherical radial density profile. b. Coil A is less densely packed in the center than coil B, because shorter chains are generally less dense. c. Coil B is less densely packed in the center than coil A, because longer chains generally expand over a larger space and “pull on themselves.” d. Coil A shows a wider profile than coil B, because the longer a chain is, the more coiled it is; hence, B is more coiled than A. (2) Which statement about the random walk is wrong? For a one-dimensional random walk ______ a. the probabilities of positions after a certain number of steps are binomially distributed, that is, they can be predicted by Pascal’s triangle. b. the probability to come out at a position far away from the origin is the highest. c. it is not possible to predict for a single walk at which position it will end. d. the probability to come out at the origin position is the highest. (3) What happens to the internal energy when stretching an ideal coil? a. ΔU < 0 b. ΔU = 0 c. ΔU > 0 d. ΔU cannot be estimated as long as the type of interaction is not known. (4) The free energy of an ideal polymer chain ______ a. does not depend on the chain’s end-to-end distance. b. depends on the chain’s end-to-end distance via internal energy due to the distance-dependent segmental interaction potential. c. depends on the chain’s end-to-end distance by both a distance-dependent energy and a distance-dependent entropy term. d. depends on the end-to-end distance only by entropy due to the probability distribution of the random walk.

Questions to Lesson Unit 3

69

(5) The entropic spring constant ______ a. is independent of temperature. b. is dependent on temperature according to T. c. depends on temperature according to T 1 . d. depends on the temperature according to T 2 . (6) An ideal coil ______ a. is also named Gaussian coil, because the probability of the end-to-end distance distribution according to random walk statistics is Gaussian and therefore also the segmental density distribution of the coil has the shape of a Gaussian bell. b. is also named Gaussian coil, because the segmental density distribution of the coil is Gaussian. c. is also named Gaussian coil, because the probability of the end-to-end distribution is Gaussian due to random walk statistics. d. is something else than a Gaussian coil. (7) Which statement about the deformation behavior of ideal polymers is true? a. Ideal polymers can be deformed better at low temperatures, because the restoring force to bring the coil back to its initial position is then lower. b. Ideal polymers can be deformed less easily at low temperatures, because the thermal energy is then too low to dislocate the chain segments. c. Ideal polymers can be deformed better at high temperatures, because entropy generally increases with increasing temperature, and thus the coil requires more space. d. Ideal polymers behave analogously to metallic solids in their deformation behavior, and so does the temperature dependence. (8) What is a result of the self-similarity of polymers? a. The scaling factor of corresponding physical relations corresponds to the fractal dimension. b. Corresponding physical contexts always scale linearly in length. c. Corresponding physical relations are equally valid on different length scales, that is, they are scale invariant. Mathematically, this manifests itself in the form of power-law dependencies. d. Similar polymers can always be synthesized in similar ways.

3 Real polymer chains LESSON 4: REAL CHAINS In your elementary classes on physical chemistry, at some point, the ideal gas model reached its limitations, and it was necessary to expand it such to account for the finite volume as well as interactions of the gas particles. A similar point is reached when interactions and the own volume of polymer-chain segments can no longer be disregarded. This lesson accounts for both and thereby introduces an elementary quantity named excluded volume. This quantity, in turn, allows a spectrum of different types of solvents to be defined for polymers.

Now that we have examined and understood the ideal chain model, we can take the next step and model a polymer chain that resembles the real world more closely. This leads us to the real chain model. We approach it by readjusting our premises. So far, we have imagined a polymer chain to consist of monomer segments that have no volume and show no interactions, neither with one another nor with a solvent in the surrounding. Now, we overcome these simplifications and explicitly consider that the chain segments have a finite covolume and that they have interactions.28 Our first focus is on appraising how these two effects affect the shape of the real polymer chain. We may get a first notion on that by pure intuition. If the monomer segments of a real chain have a finite volume, then two of them cannot occupy the same spot in space, as it was hypothetically possible for an ideal phantom chain with a self-crossing random-walk-type coil shape, as shown in the left sketch of Figure 28. A real chain, by contrast, has the shape of a self-avoiding random walk, as shown in the right sketch of Figure 28. In such a chain, the blocking of space by each monomer unit to be no longer occupiable by other monomer units (i.e., the self-avoidance) causes part of the volume to be excluded volume. As a consequence, the chain has less freedom of arrangement, and the coil must expand. Indeed, the self-avoiding walk in Figure 28 has a larger end-to-end distance than the self-crossing one. On top of that, if we also consider that the monomer segments of a real chain display attractive and repulsive interactions with one another and with their surrounding environment,29 then the ratio of these monomer–monomer (M–M) and monomer–solvent (M–S) attractive and repulsive interactions will further

 This expansion of our simplistic model to a more realistic one is conceptually identical to the step from the ideal gas to the real gas in elementary physical chemistry. Here, the gas molecules are also considered to have a finite volume and interactions. This is done by inclusion of two parameters for these two effects, thereby transforming the ideal gas law to the van der Waals equation.  In that environment, we have either solvent molecules if we consider a polymer solution or segments of other chains if we consider a polymer melt. https://doi.org/10.1515/9783110713268-003

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Figure 28: Trajectories of a self-crossing random walk, which corresponds to the shape of an ideal polymer coil, and a self-avoiding walk, which corresponds to the shape of a real polymer coil, in which part of the volume around each monomer unit is excluded for occupation by other units.

determine the extent of coil expansion; it will also determine the polymer solubility or miscibility.

3.1 Interaction potentials and excluded volume As a start, let us examine the interaction potential between two monomer segments in a chain that have no direct chemical bond to each other (i.e., segments that are not direct neighbors along the chain). In fact, we do not even need to consider these monomers to be connected in the form of a chain; instead, it is sufficient to just consider them as molecular entities that have distance-dependent interactions through space, both attractive and repulsive. A suitable functional form to describe these interactions is the Lennard–Jones potential; it quantifies the distance-dependent interaction energy, U(r), of two molecules, in a form similar to the illustration in Figure 29. This function is generally called to be a 6–12 potential, because the energetic contribution of the attractive interactions scales with the intermolecular distance by r–6, whereas the energetic contribution of the repulsive interactions scales with r–12.30 As a consequence, the repulsive interactions are more influential

 The repulsive part of the Lennard–Jones potential reflects the strong energy penalty that arises if atoms or molecules are brought into contact so close that their occupied orbitals start to overlap; this is basically what the Pauli principle expresses, after which two electrons cannot be identical in all their quantum numbers (actually, this is expressed by a phenomenon named exchange interaction in quantum mechanics). That strongly repulsive energy has a distance dependence that is actually not necessarily of type r–12; we could also use r–11 or r–13 for it. The r–12 exponent is chosen

3.1 Interaction potentials and excluded volume

73

Figure 29: Effective Lennard–Jones interaction potential, U(r), between two molecules with attractive effective interaction. The optimal distance, at which we have the lowest U(r) value, depends on the interactions between the two molecules themselves (shaded circles) and with the surrounding medium (white circles). Picture inspired by M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

at short distances, where they contribute positive large U(r) values. At longer distances, by contrast, the attractive interactions dominate and contribute negative U(r) values (that have a greater absolute magnitude than the small positive ones

somewhat arbitrarily, because it is an even number that makes a steep distance dependence. The attractive part of the Lennard–Jones potential is more substantiated. It has its cause in electrostatic interactions. You know a particularly strong representative of this type of interaction from your elementary chemistry education: the ionic bond, held together by Coulomb interaction. This type of bond, though, is not meant here. But there are three kinds of related electrostatic interactions that form the attractive part of the Lennard–Jones potential, all invloving electrically neutral molecules subject to dipole moments. The first kind concernes molecules with a permanent dipole moment, which is present if their bonds are covalent but polarized; an example is HCl. This dipole–dipole interaction, also called Keesom interaction, scales with r–6. A second type of such interaction is given if a dipolar species induces a complementary one in a non-polar neighboring molecule, so that these dipole moments can then be paired. This interaction also scales with r–6; it is called Debye interaction. A third type concerns completely neutral molecules: even those can have electrostatically induced attractive interactions. This is due to so-called correlated quantum fluctuations. Fluctuations exist for almost all quantities, including - for example - the dipole moment of an argon atom. If these were purely random, the dipolar forces would cancel out on average. But this is not the case, because if two or more dipole moments are aligned in pairs or groups, this lowers the energy, and therefore such an alignment becomes favorable. This indirect mechanism leads to attraction of the molecules; it is called London dispersion interaction. The strength of this kind of interaction increases as the polarizability of the molecules increases, and the polarizability itself increases with the number of electrons in the shell. This is why argon has a higher boiling point than helium, although both are completely non-polar noble gases. The distance-dependence of the London dispersion interaction is again an r–6 law. The Keesom, the Debye, and the London interaction are summarized as van der Waals interactions.

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contributed by the repulsive interactions there). The overlay31 of both causes a potential well, in which a maximally negative U(r) value denotes the most favorable equilibrium distance. At infinite distances, the potential levels off toward zero, because there the molecules are too far apart to “see each other”. In a real polymer system, we need to consider the attractive and repulsive interactions for two species: the monomer (M) and the surrounding medium, that is, the solvent (S). We therefore have to take into account M–M and M–S interactions, both having attractive and repulsive contributions each. To simplify the discussion, we discuss effective M–M interactions, in which we incorporate both the M–M and M–S contributions. We do this by redefining repulsive M–S interactions to be just the same as attractive M–M interactions, because both have the same effect: the monomers prefer to stay in closer proximity to one another than to the solvent. The same works for the opposite case: attractive M–S interactions just act like effectively repulsive M–M interactions, as in both cases, the monomers prefer to be further apart from one another than from the solvent. The resulting distance-dependent effective M–M interaction potential has a Lennard–Jones-type appearance again, as depicted in Figure 29. Based on this premise, we can delimit three boundary cases: 1. In most cases, M–M contact is favored over M–S contact, so we have attractive effective M–M interactions and a minimum in U(r), as depicted in Figure 29. This is due to the perfect structural match of two monomer units M to one another, whereas that of M and S is not as perfect, but at most just similar. As a result, whatever kind of interactions M can undergo (be it hydrogen bonding, random-fluctuating, induced, or permanent, dipolar interaction, or whatever), it will find a perfect partner to do so in another M, and a less perfect (at most similarly good, but never as perfect) partner to do so in S, such that M–M is more preferable than M–S.32 In most polymers, the M–M interactions are indeed van-der-Waals-type dipolar forces, which comes along with effective M–M interaction energies in the order of kBT; this magnitude quantifies the depth of the well in the effective M–M interaction potential U(r). 2. In the second boundary case, the M–M interactions are equal to the M–S interactions; this situation is encountered if M and S have practically the same chemical structure, thereby allowing them to establish the same interactions equally fine with one another or with each other. Consequently, the effective M–M

31 This is done by adding up the two parts as a sum, whereby the attractive one comes along with a negative sign, as it lowers the total energy, whereas the repulsive one comes along with a positive  sign, as it increases the total energy. The exact formula is UðrÞ = U0 + ε ðrre Þ − 12 − 2ðrre Þ − 6 . In this equation, re is the equilibrium distance at which the potential has its minimum, and ε is the depth of the energy well at that separation. Note that the attractive term needs a numerical prefactor of 2, because only then will the potential have a value of –ε at the equilibrium distance re along with a potential minimum (i.e., a zero first derivative) at that distance.  This general principle is reflected in the proverb “birds of a feather flock together”.

3.1 Interaction potentials and excluded volume

3.

75

interaction is zero. The U(r)-potential then does not exhibit any potential well, but only the hard sphere repulsive upturn in the limit r ! 0 (where U(r) ! ∞). In the third boundary case, the M–M interactions are disfavored over the M–S interactions. This is a rare special-case scenario that we may have in polyelectrolytes in which each monomer unit carries a like charge, such that the monomer units repel each other.33 Again, no potential well exists in U(r), but by contrast, an additional repulsive term is added on top of the inherent hard-sphere repulsion.

In the next step, we calculate the probability p to find two monomers at a certain distance r. This is simply done by deriving a Boltzmann term from the interaction energy potential U(r):34

− U ðr Þ (3:1) p ⁓ exp kB T A graphical representation of this term is shown in Figure 30(A) for the case of attractive effective M–M interactions. We can see that the probability to find two monomers at very short distances is zero, which corresponds to the hard-sphere repulsion term of the interaction energy potential, where U(r) ! ∞. The highest probability can be found at the most favorable distance r that corresponds to the minimum in the potential well. In this region, U(r) has values of around –kBT, which translates to values of the Boltzmann term in the range of about e (=2.718). At very long distances, the probability is independent of r, because there are no operable interactions between the two monomers at these large separations. Here, U(r) has a plateau value of zero, such that the Boltzmann term has a plateau value of one. When we normalize the Boltzmann expression such that the limit for r ! ∞ equals zero rather than one, we generate an auxiliary function called the Mayer f-function, as shown in Figure 30(B):

− U ðr Þ −1 (3:2) f ðrÞ = exp kB T By integrating over the Mayer f-function, we can calculate the excluded volume, which corresponds to the area underneath the curve, shown in gray in Figure 30(B):

 We may also have such a case if there is specific attractive M–S interaction, which manifests itself as if there were repulsive M–M interaction. This can be the case if the monomer and the solvent are somehow mutually complementary, for example, if the one carries hydrogen-bonding donor sites, whereas the other carries hydrogen-bonding acceptor sites, thereby being able to form heterocomplementary bonding with one another that does not work among themselves alone. 34 In general, the likelihood (and therewith the population or frequency of occurrence) of a state or situation  of interest with a known energy U is estimated by the Boltzmann distribution: nU −U n = exp kB T ; the same principle holds for our situation of interest, which is the likelihood of having a certain distance r between two monomers if that comes along with an energy of U(r).

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Figure 30: (A) Probability to find two monomers at a specific distance, r, for the case of attractive effective monomer–monomer (M–M) interactions. At short distances, the repulsive M–M interactions are so strong that the probability to find two monomers at that separation is practically zero. In contrast, the probability is highest at the U(r)-potential minimum, while it is independent of r for r ! ∞. (B) Plot of the Mayer f-function; it normalizes the Boltzmann term shown in Panel (A) to a value of zero at r ! ∞. From that functional form, the excluded volume can be calculated by integrating the total area underneath the graph, shaded in gray. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

ð ð ve = − f ðrÞdr3 = − 4πr2 f ðrÞdr

(3:3)

The excluded volume quantifies the space that each chain segment blocks in its surrounding due to (i) its own volume and, on top of that, (ii) the M–M interactions; if these interactions are repulsive, the blocking of (i) is further exacerbated, whereas if these interactions are attractive, the blocking of (i) is attenuated. The repulsive term in the potential U(r) at distances shorter than the equilibrium distance at which the potential has its minimum, r < re, has a negative contribution to the integral over the Mayer f-function, which translates into a positive contribution to the excluded volume. By contrast, the attractive term at r > re has a positive contribution to the integral over the Mayer f-function, and hence, a negative contribution to the excluded volume. The example in Figure 30(B) shows a situation in which the attractive and repulsive parts largely balance each other, leading to an excluded volume close to zero. This is a very special state, named the Θ-state, in which the chain displays a quasi-ideal conformation. That special state is very important in the field of polymer physics, and we will treat it more deeply in the following.

3.2 Classification of solvents

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3.2 Classification of solvents Based on what we have learned about M–M and M–S interactions, we can compile a list that classifies solvents by the extent of the resulting excluded volume ve that a chain has in them. When the M–M interactions are equal to the M–S interactions, the solvent is called to be an athermal solvent. This terminology is because when the interactions are the same, so is their temperature dependence, such that any change of temperature will have no effect on the effective M–M interactions. Such a solvent is structurally identical to the monomeric repeating unit of the polymer; a prime example is ethylbenzene for polystyrene, which is practically equal to polystyrene’s repeating unit. In an athermal solvent, there are no attractive effective M–M interactions, leaving only the hard-sphere repulsion at short distances (r ! 0) in the effective M–M interaction potential U(r) (where U(r) ! ∞). As a result, there is no positive contribution to the integral over the Mayer f-function, and hence, no negative contribution to the excluded volume. The outcome is a maximally positive excluded volume; it is equal to the covolume of the monomer segments: ve = l3. In a good solvent, the M–M interactions are slightly more favorable than the M–S interactions. This leads to a small well in the effective M–M interaction potential U(r), which somewhat compensates the inherent M–M hard-sphere repulsion. The excluded volume, thus, will still be positive, but smaller than in the athermal case: 0 < ve < l3. A typical example for a good solvent is toluene for polystyrene. A very special case is the Θ-state, which is present in a Θ-solvent. In that state, there are quite attractive effective M–M interactions,35 leading to a pronounced well in the interaction potential U(r), which just exactly balances the hard-sphere M–M repulsion at short distances in the potential, as indicated in Figure 29(B). At such balance, the negative and positive contributions to the integral over the Mayer f-function cancel each other. For the coil, it then appears as if neither the hard-sphere repulsion, i.e., neither the monomer co-volume, nor the effective M–M interaction was present—just like it is in an ideal chain made of pointlike and interaction-less segments. Hence, at that constellation, the excluded volume is zero, ve = 0, and the chain is in a (pseudo-)ideal state36 and adopts the shape of a Gaussian coil with random-walk conformation. The Θ-state is therefore beloved by theorists, as it allows them to model the coil conformation by simple ideal Gaussian statistics. Experimentalists, by contrast, do not love the Θ-state as much, because it is highly temperature

 This means that there are quite repulsive M–S interactions, meaning that the solvent S is quite dissimilar to the monomer repeating unit M.  A truly ideal state is one without any interactions. A pseudo-ideal or quasi-ideal state is one with attractive and repulsive interactions at balance.

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dependent and can only be present at a special temperature, the Θ-temperature.37 This temperature marks the borderline to the nonsolvent state, such that even a slight change of temperature into the wrong direction will cause precipitation of the polymer, requiring tedious redissolution and re-equilibration before experiments can be conducted. A famous example of a Θ-state is polystyrene in cyclohexane at TΘ = 34.5 °C. In a bad solvent, the M–M interactions are much more favorable than the M–S interactions, leading to strong effective M–M attraction. This causes a strong minimum in the effective M–M interaction potential U(r) and a negative excluded volume in the range of –l3 < ve < 0. An example is ethanol for polystyrene. In an even more extreme case, the nonsolvent, the M–M interactions are so much more favorable than the M–S interactions that all solvent is expelled from the polymer coil and the excluded volume becomes maximally negative: ve = –l3. An example is water for polystyrene. The latter two cases cannot be realized in practice, because both actually lead to nondissolution and can therefore only be studied by computer simulations. They do, however, have practical applicability for preparatively working polymer chemists: turning a good or a Θ-solvent into a bad or nonsolvent, for example, by suitable change of temperature or by addition of a bad or nonsolvent excess, can serve to separate polymers from a mixture by precipitation, which is an easy means of polymer purification. To complete the listing above, note that another special case is the one where the M–M interactions are disfavored over the M–S interactions. We may have that situation in polyelectrolyte solutions in which each monomer unit carries a like charge, such that the monomer units repel each other. We may also have such a case if there is specific attractive M–S interaction, for example, mutual heterocomplementary hydrogen bonding, which manifests itself as if there were repulsive M–M interactions. In that situation, just like in the athermal case, no well exists in the interaction-energy potential U(r), and on top of that, there is an additional repulsive term added to the inherent hard-sphere repulsion. As a result, the excluded volume is even greater than in the athermal case: ve > l3.

 Again, we can find an analogy to gases: a real gas differs from an ideal one in a sense that there are attractive and repulsive interactions between the gas particles and that the gas particles have a finite covolume, which is in fact nothing else than the strongly repulsive low-distance branch of the interaction-energy potential. Both these interactions are quantified by two parameters, a and b, in the equation of state, through which the ideal gas law turns into the van der Waals equation. At a special temperature, the Boyle temperature TBoyle, however, these interactions are at balance, such that the van der Waals equation turns back into the ideal gas law. Hence, the Boyle temperature for a gas is analog to the Θ-temperature for a polymer–solvent system.

3.3 Omnipresence of the Θ-state in polymer melts

79

3.3 Omnipresence of the Θ-state in polymer melts The Θ-state cannot only be realized in solution at a specific temperature, but it is also always present in polymer melts, at any temperature. This can be understood by the following line of thought: consider a polymer melt in which one chain is somehow different in color, but structurally identical to the others. We can call this a “blue” chain in a matrix of “black” chains, as shown in Figure 31. Due to the chains’ structural identity, this is an athermal state: M–M interactions are the same as M–S interactions, as the segments of the black “solvent” polymer chains are of the same kind as those of our blue “dissolved” chain. The blue chain, thus, has a strong tendency to expand. However, the black chains all have the same tendency. Thus, all chains in the system want to expand at the same time, and as a result, no

Figure 31: In a polymer melt, the excluded volume interactions in a chain of interest, here shown in blue color, are screened by overlapping segments of surrounding chains, here drawn in black color. Picture redrawn from P. G. De Gennes: Scaling Concepts in Polymer Physics, Cornell University Press, 1979.

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3 Real polymer chains

chain can actually do so. The expansion tendencies all balance each other, and the chains therefore all exhibit their unperturbed ideal conformation. In other words, the excluded volume interactions of each coil are screened by overlapping segments of other coils.

Questions to Lesson Unit 4 (1) Which statement is not true for the Lennard–Jones potential? a. The attractive interactions scale with r6 . b. The attractive interactions are based on van der Waals forces. c. The repulsive interactions scale with þr12 . d. The repulsive interactions constitute a hard-sphere potential at short distances. (2) What is meant by “excluded volume”? a. The volume that is inaccessible to the segments of a real polymer due to a selfavoidance condition in the chain random-walk shape, which results from the repulsive part of the effective monomer–monomer interaction potential. b. The volume that is not occupied by the polymer due to the self-overlapping random walk of the polymer chain segments. c. The volume that is not occupied by the polymer due to the ideal random walk of the polymer chain segments. d. The volume that results from the sum of the volume of all monomers of the polymer chain. (3) Which statement is not true about effective monomer–monomer interactions? a. Effective monomer–monomer interactions include both the corresponding monomer–solvent interactions and the actual monomer–monomer interactions. b. Effective monomer–monomer interactions essentially include only the corresponding monomer–solvent interactions and “translate” them into something related only to one species, which is the monomer species. c. Actual attractive monomer–solvent interactions are understood as effectively repulsive monomer–monomer interactions. d. Actual repulsive monomer–solvent interactions are understood to be effectively attractive monomer–monomer interactions. (4) Rank the different solvent classes in ascending order according to the excluded volume of polymers dissolved in them. a. θ-solvent, good solvent, athermal solvent, poor solvent, nonsolvent. b. Athermal solvent, good solvent, θ-solvent, poor solvent, nonsolvent.

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81

c. Nonsolvent, bad solvent, athermal solvent, good solvent, θ-solvent. d. Nonsolvent, poor solvent, θ-solvent, good solvent, athermal solvent. (5) The excluded volume ______ a. leads exclusively to compression of the polymer coil. b. is larger the worse the solvent is. c. can be determined by integration over the Mayer f-function. d. is minimal in an athermal solvent. (6) Match the given sequences of solvent types to the given properties (from top to bottom). 1. The polymer shape is a Gaussian coil. 2. The integral over the Mayer f-function is positive but not maximally positive. 3. No effective attractive monomer–monomer interactions are present. 4. Monomer–monomer interactions are slightly more favorable than monomer–solvent interactions. 5. The state in this solvent is opposite to the athermal state. a. 1. θ-Solvent 2. Poor solvent 3. Athermal solvent 4. Good solvent 5. Nonsolvent b. 1. Athermal solvent 2. Bad solvent 3. Nonsolvent 4. θ-Solvent 5. Good solvent c. 1. θ-Solvent 2. Nonsolvent 3. Good solvent 4. Athermal solvent 5. Bad solvent d. 1. Athermal solvent 2. Good solvent 3. θ-Solvent 4. Bad solvent 5. Nonsolvent (7) Which statement regarding solvent and resulting state of a polymer solution is correct? a. The athermal solvent represents a quasi-ideal state, since here the monomer–monomer interactions are equal to the monomer–solvent interactions.

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3 Real polymer chains

b. The athermal solvent represents the ideal state since it is independent of temperature. c. The θ-solvent represents a quasi-ideal state, since repulsive and attractive parts of the interaction potential balance each other here. d. The θ-solvent represents an ideal state since there are no effective interactions. (8) Which of the following is not a requirement for always having a θ-state rather than an athermal state in polymer melts? a. The polymer is surrounded by its own kind allover. b. The polymer chains mutually block each other from expanding. c. The monomer–monomer interactions are equal to the monomer–solvent interactions. d. The polymer chains all have the same structure.

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3.4 The conformation of real chains

3.4 The conformation of real chains LESSON 5: FLORY EXPONENT The excluded volume introduced in the last lesson makes polymer coils expand in good solvents. In this lesson, you will learn how severe this expansion is, and how the size of a coil can be captured by a universal scaling law that introduces one of the most elementary parameters in polymer science: the Flory exponent.

3.4.1 Coil expansion To quantify the size difference of a real polymer chain compared to an ideal one, an expansion factor α is introduced: 1=2

1=2

hRg 2 ireal ≈ α hRg 2 iideal

(3:4)

This factor has values above one, that is, α > 1, for athermal and good solvents, in which the polymer coil expands. It is exactly one, which means, α = 1, for a Θsolvent, in which the coil has the same size as an ideal polymer chain. For bad or nonsolvents, the expansion factor is smaller than one, that is, α < 1. But how severe is this expansion? Let us consider polyethylene with NK = 50 Kuhn-segments as an example. In Chapter 2.2.8, the characteristic ratio of this polymer was named to be C∞ = 6.87 (see also Figure 17), so we get a length of lK = C∞ l = 6.87 · 0.154 nm = 1.06 nm for each Kuhn-segment. At strong attractive effective M–M interactions, that is, at non-solvency conditions, the polymer is collapsed to a dense globule with volume V ≈ NK lK 3 = 60 nm3 . This corresponds to a linear dimension of R = V 1=3 = NK 1=3 lK = 3.9 nm, as shown in Figure 32(A). Hence, in this collapsed state, the globule’s across diameter is just about eight times the length of one of its Kuhn-segments (as also seen in Figure 32(A)). This shows how densely packed the material is in this entity. With balanced M–M interactions, or zero effective M–M interactions, we realize the Θ-state. As discussed above, the polymer then has the shape of a random Gaussian coil and obeys the ideal chain scaling law R = NK 1=2 lK = 7.5 nm, as shown in Figure 32 (B). Hence, in that state, compared to the nonsolvent situation discussed earlier, the random coil is expanded to be about twice as large as the collapsed globule. When we put the polymer chain into a good solvent, it has repulsive effective M–M interactions, which manifests itself in a positive excluded volume ve. The coil now has the shape of a self-avoiding walk and obeys scaling according to R = NK 3=5 lK = 11 nm, as shown in Figure 32(C). (This scaling law will be derived in the next section.) Now, the polymer has a size right in the typical colloidal domain.

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Figure 32: Visualization of the different stages of expansion for a polymer with 50 Kuhn-segments. (A) In its fully collapsed state, the polymer is a dense globule (R = 3.9 nm), whereas (B) at Θ-conditions, it is a coil with the shape of a random walk (R = 7.5 nm). (C) Effective repulsive M–M interactions cause the coil to expand to a self-avoiding walk shape (R = 11 nm). (D) At maximally strong repulsive M–M interactions, the polymer fully expands to a stiff rod (R = 53 nm). Volume and size dependencies on the number of Kuhn segments, NK, and their length lK, for each regime are shown underneath each image.

At maximally strong repulsive effective M–M interactions, meaning at a maximally large excluded volume ve, the chain is in its most expanded conformation, which is a rodlike object, as shown in Figure 32(D). Its size can be calculated according to R = NK lK = 53 nm, which is considerably larger than in the other states discussed before. We see from the above example that a better solvent quality leads to remarkable expansion of the polymer coil, from a few single to several tens of nanometers. This, however, comes at the cost of a lower segmental density in the coil interior. Throughout our discussion above, we have treated the same polymer chain; we have not added new segments to it, but only given it more space to arrange itself. When we do this, we get a greater coil expansion, but, in turn, we naturally reduce the number of monomer segments per volume. This is shown for the segmental density of polystyrene in either the θ-state or in a good-solvent state in Figure 33. Generally, we can appraise the polymer coil’s size for all expansion regimes by its root-mean-square end-to-end distance according to the following general scaling law: hr2 i1=2 ≈ N ν l

(3:5)

In this law, ν is the Flory exponent that varies between 1/3 and 1 in our above line of thought, depending on the solvent quality. It is this power-law exponent that causes the difference in the size of a polymer at different states of solvency, because in the

3.4 The conformation of real chains

85

Figure 33: Segmental density of a polystyrene coil in a good and a Θ-solvent. The good solvent expands the coil as compared to the Θ-solvent, leading to a greater occupancy of the volume far from the coil center, but in turn, to a lower segmental density in its core.

power law (3.5), a large ν empowers a given number of (Kuhn-)segments N to a greater extent than a small ν does. Due to the mathematical nature of power laws, this different empowerment is even more pronounced when the number of segments in the chain is large. For example, if we consider polyethylene again, but this time with 20,000 repeating units (just as we have done in Section 2.2.8), we have a polymer with NK = CN∞ = 2899 Kuhn-segments of length lK = C∞ l = 1.06 nm each. In a fully collapsed state, this polymer has a size of R = NK 1=3 lK = 15 nm, whereas at Θ-conditions, it has a size of R = NK 1=2 lK = 57 nm (just as already calculated in Section 2.2.8). In a good solvent, it is expanded even further, to a size of R = NK 3=5 lK = 127 nm, and when fully expanded to a rod, the polymer exhibits a size of R = NK 1 lK = 3080 nm (as also calculated already in Section 2.2.8). The size of this polymer differs even more markedly than the one in the example of Figure 32 depending on its solvency state, spanning the whole colloidal domain between single nanometers to a few micrometers.

3.4.2 Flory theory of a polymer in a good solvent Let us consider an expanded polymer coil composed of N monomer units and a size R bigger than that of the ideal Gaussian coil, R > R0 = N 1=2 l. We can estimate the number of monomers that are located inside the excluded volume of a given first one as follows:

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3 Real polymer chains

ve ·

N R3

(3:6)

Here, ve is the excluded volume of one monomer segment, which is the volume we want to consider here, and RN3 is the number of segments in the coil per volume of the coil, in other words, the general segmental density in the coil. In the optimal case, the expression of eq. (3.6) should equal one. Then, the excluded volume of a given monomer segment is only occupied by one such segment: by itself. Any additional monomers that intrude into the excluded volume impose an energy penalty that can be appraised to be kBT per such unfavorable M–M contact: Fexcl, per monomer = kB Tve

N R3

(3:7)

If there is only one monomer in the excluded volume (which is the optimal situation), the resulting energy is kBT, which can be easily “paid” by the ever-present thermal energy that is exactly kBT. Each additional monomer–monomer contact in the excluded volume, however, adds another kBT-increment to the penalty. Consequently, additional energy is required. This is generally unfavorable, such that the system has a tendency to avoid such M–M contacts, which can be achieved by coil expansion. Upon such an expansion, R in eq. (3.7) will get larger, such that Fexcl,per monomer will get lower, which means it is less unfavorable. In a chain with N monomers, the excluded-volume-interaction energy appraised by eq. (3.7) is present N times: Fexcl, per chain = kB Tve

N2 R3

(3:8)

Again, the only nonconstant parameter here that can be adjusted is the polymer coil’s size R. Because R enters the equation in the denominator, the chain can minimize this energy contribution by increasing its size, R ! ∞. The coil, thus, has an expansion tendency. On the other hand, coil expansion causes an entropy-based restoring force, corresponding to an energy Felast, due to the loss of microconformational freedom upon coil expansion, as we have learned in Section 2.6. This can be appraised as Felast = F0 +

3 kB TR2 · Nl2 2

(3:9)

Again, the size R is the only adjustable parameter that, in this case, enters the equation in the numerator. From this, it follows that the chain can minimize this energy contribution by decreasing its size, R ! 0. The coil, thus, has a contraction tendency. To calculate the total energy of the chain subject to the two preceding influences, we have to sum up both the excluded-volume-interaction energy Fexcl and the elastic energy Felast:

3.4 The conformation of real chains



N 2 3R2 F = Fexcl + Felast = kB T ve 3 + R 2Nl2

87

(3:10)

To find the coil size with the minimum total energy, we calculate the derivative and set it zero:

∂F N 2 3R ! (3:11) = kB T − 3ve 4 + 2 = 0 ) R ⁓ ve 1=5 l2=5 N 3=5 R ∂R Nl With that, we have shown that the N-dependent scaling of R of a coil in a good solvent is R ⁓ N 3=5

(3:12)

There is just one problem: so far, we always found a length-dependent scaling of R ⁓ l1 . This is in contradiction to eq. (3.11), where we have found R ⁓ l2=5 instead. The reason for this discrepancy is a mistake in the above derivation. In eq. (3.9), we have used ideal-chain scaling of hr2 i = Nl2 in the denominator, even though all our above line of argument in fact has the purpose to appraise nonideal, expanded coil dimensions. This mistake causes the erroneous finding of R ⁓ l2=5 . Nevertheless, our finding of R ⁓ N 3=5 is correct though. Why is that? It is because in view of the N-dependence, our mistake is compensated by another one. In eq. (3.6), we have appraised the segmental density in the coil to be uniform (in the form of N/R3), whereas we know from Section 2.3 that it actually has a Gaussian radial profile. This wrong estimate of the segmental density in eq. (3.6) compensates the incorrect scaling in the denominator of eq. (3.9) in view of the N-dependence of R, whereas it is not cancelled out in view of the l-dependence of R. The general scaling of R as a function of N in the form of eq. (3.12) can be written as R = h~ r2 i1=2 ⁓ N ν

(3:13)

with ν the Flory exponent. For a polymer in a good solvent, we have just derived it to be ν = 3/5. For an ideal chain, that is, a chain at θ-conditions, we have shown earlier (in Section 2.2, eq. (2.5)) that ν = 1/2. For a coil that is fully collapsed to a dense globule, that is, a chain in a nonsolvent, we have shown earlier (Section 3.4.1) that ν = 1/3, whereas for the other extreme, a fully expanded rodlike chain, we have shown that ν = 1. If we conduct the preceding estimate for the general case of a d-dimensional space, we have to use Rd in eqs. (3.6)–(3.8) and a numerical factor of d=2 in eq. (3.9). With that, we get R = h~ r2 i1=2 ⁓ N 3=ðdgeometrical + 2Þ

(3:14)

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  In this general form, the Flory exponent is ν = 3= dgeometrical + 2 , allowing us to illustrate the role of the geometrical dimension, dgeometrical, as follows. In earlier sections, we have considered the three-dimensional case, d = 3, where the Flory exponent is 3/5. In a two-dimensional situation, d = 2, this is little different. Here, according to eq. (3.14), the Flory exponent has a value of 3/4, which is larger than 3/5, indicating that a coil with given N has a greater R in two dimension than in three dimension. The reason is that there is less freedom for the polymer coil to arrange itself in a 2dspace than in a 3d-space, because it has only two spatial directions to occupy. Hence, the coil must expand more in 2d than in 3d to avoid unfavorable M–M contacts. This trend is even more extreme in a one-dimensional situation, d = 1. Now, according to eq. (3.14), the Flory exponent is 3/3 = 1. This is because in 1d, the coil has no other way to avoid M–M contacts than to fully expanding itself into a rodlike object with R = Nl. By extreme contrast, in a four-dimensional situation, d = 4, the coil has so much freedom to arrange itself that M–M contacts are generally very unlikely. The coil can therefore adopt its random Gaussian shape like an ideal chain or a real chain in the Θ-state. Here, consequently, the Flory exponent is 3/6 = 1/2. (If we further follow this line of thought, even higher geometrical dimensions such as d = 5, 6, . . . would denote smaller Flory exponents such as ν = 3/7, 3/8, . . ., indicating that a coil in such high dimensions would contract. This can be understood on the very same basis as the discussion just led: the higher the geometrical dimensions, the smaller can be the coil size without causing unfavorable M–M contacts. Higher dimensions, therefore, allow R to be small, which minimizes the elastic energy according to eq. (3.9) (where R enters in the form of R2, irrespective of d), without excessively increasing the excluded volume interaction energy according to eq. (3.8) (where R enters in the form of R–d, which means that if d is high, there is lesser need for large R to make that term small).) In addition to the preceding discussion of the geometrical dimension, we may also do so for the fractal dimension, a concept that we have introduced in Section 2.6.2. With this dimension, we get R = h~ r i1=2 ⁓ N 1=d fractal 2

(3:15)

Hence, the Flory exponent is nothing else than the inverse fractal dimension. When looking on the compilation of Flory exponents in one to four geometrical dimensions above, we see that the inverse of these exponents, that is, the fractal dimension, is getting lower and lower. In general, a smaller fractal dimension denotes the object that it belongs to be less dense. If we compare the fractal dimension in four geometrical dimensions, dfractal = 1/ν = 1/(1/2) = 2, to that in three geometrical dimensions, dfractal = 1/ν = 1/(3/5) = 5/3, and in two geometrical dimensions, dfractal = 1/ν = 1/(3/4) = 4/3, we obtain a series of smaller and smaller values. This means that chains in lower geometrical dimensions are less dense than in higher geometrical dimensions. The reason for that is the excluded-volume repulsion between the monomer segments, which pushes them apart from one another to avoid M–M

3.5 Deformation of real chains

89

contact; this does not need to be as pronounced in higher geometrical dimensions, as such contact is generally less likely there. A special case is the one of a densely collapsed globular polymer in a nonsolvent; here we have a Flory exponent of ν = 1/3, corresponding to a fractal dimension of dfractal = 1/ν = 1/(1/3) = 3. This fractal dimension matches the geometrical one, thereby denoting a nonfuzzy, dense object, which a collapsed polymer globule indeed is. Paul John Flory (Figure ) was born on June , , in Sterling, Illinois. He studied at Manchester College and obtained his Bachelor of Science degree during the time of the great depression, supporting himself with various jobs at the side. During that period, his interest in science, particularly chemistry, was inspired by Professor Carl W. Holl, who encouraged him to enter graduate school at Ohio State University in . Flory followed this advise, pursued his graduate studies at Ohio State, and obtained his PhD degree in . From  to , he worked in several industrial research laboratories for companies such as DuPont, Standard Oil, and Goodyear. He was offered a faculty position at Cornell University in , where he stayed until . Having transferred to and led the Mellon Institute in Pittsburgh until , he became a full professor at Stanford University until he retired in . One year prior to his retirement, he received the Nobel Prize in Chemistry “for his fundamental achievements, both theoretical and experimental, in the physical chemistry of macromolecules”. He died on September , , at the age of  in Big Sur, California.

Figure 34: Portrait of Paul J. Flory. Image reproduced with permission from Stanford University Libraries, Department of Special Collections and University Archives (SC0122, Stanford University News Service records, Box 90, Folder 48, Paul Flory).

3.5 Deformation of real chains Just like in Chapter 2, after having made ourselves a mind about the size and shape of a real chain, we now want to translate that structural information into information on properties, specifically, into the elastic properties upon deformation of the chain. Other than for the simple case of an ideal chain, as treated in Section 2.6, such a discussion for a real chain will be mathematically challenging. To simplify it, Rubinstein’s and Colby’s clever scaling argument that has been used already in Section 2.6.1 can be applied again to conceptually renormalize the polymer chain within a blob concept. This approach is valid for both ideal (see Section 2.6.1) and real chains, so both are treated simultaneously below.

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3 Real polymer chains

The root-mean-square end-to-end distance of a polymer chain is calculated as Ideal chain: hr2 i0

1=2

= R0 = N 1=2 l

(3:16a)

Real chain: hr2 i1=2 = RF = N 3=5 l

(3:16b)

Due to the self-similarity of polymer chains, the same scaling also applies to subsections of the chain that only encompass n monomers: Ideal chain: r0 = n1=2 l

(3:17a)

Real chain: rF = n3=5 l

(3:17b)

We now regard a special subsection scale, named ξ. On scales smaller than ξ, the external deformation energy is weaker that the ever-present thermal noise kBT; as a result, on scales smaller than ξ, the chain segments show unperturbed random-walk-type (in the case of an ideal chain) or excluded-volume-expanded (in the case of a real chain) conformations, but they do not “feel” any external deformation. By contrast, on scales larger than ξ, external deformation is effective, as its energy is stronger than kBT there. Hence, although the chain segments on scales smaller than ξ are not affected by external deformation, the whole chain is. It can therefore be conceived as an oriented sequence of deformation blobs of size ξ, inside of each being a chain subsegment of g monomers that is not affected by the deformation, as shown in Figure 35. In this way, the chain can maximize its entropy even under the external constraint of deformation. By adaption of eq. (3.17) to the blob scale, the blob size is expressed as Ideal chain: ξ = g1=2 l

(3:18a)

Real chain: ξ = g3=5 l

(3:18b)

On length scales larger than ξ, the chain is an oriented sequence of deformation blobs; as a result, its end-to-end distance can be approximated as the blob size, ξ, times the number of blobs, (N/g), which we may then rewrite further by using eq. (3.18) (rearranged to g and then replacing g in N/g) and eq. (3.16) (replacing the then occurring numerators of type Nl1/ν) Ideal chain: Rf ≈ ξ

N Nl2 R0 2 = = ξ ξ g

(3:19a)

Real chain: Rf ≈ ξ

N Nl5=3 RF 5=3 = 2=3 = 2=3 g ξ ξ

(3:19b)

Rearranging these equations yields an expression for the blob size ξ

3.5 Deformation of real chains

91

Figure 35: Modeling of a polymer chain, ideal (upper sketch) and real (lower sketch), subject to an external stretching force that leads to an end-to-end distance Rf as a conceptual object composed of blob elements with size ξ. At scales below ξ, the polymer chain does not experience deformation and displays ideal random-walk-type (ideal chain) or expanded (real chain) conformations. At scales above ξ, by contrast, in both cases, the polymer is an oriented sequence of blobs, as on these scales, the external deformation is effective. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

Ideal chain: ξ = Real chain: ξ =

R0 2 Rf

RF 5=2 Rf 3=2

(3:20a)

(3:20b)

As we have learned in Section 2.6.1, the free energy of deformation, F, is kBT per blob. By using eqs. (3.19) and (3.20) to further rewrite, we get 2 Rf Rf N = kB T Ideal chain: Fideal = kB T = kB T ξ R0 g Real chain: Freal = kB T

5=2 Rf Rf N = kB T = kB T ξ RF g

(3:21a)

(3:21b)

We have also learned that the force needed to deform the chain by a distance of Rf corresponds to the ratio of the thermal energy kBT and the blob size ξ, which we can express by eq. (3.20) to get Ideal chain: f =

kB T kB T kB T Rf · = 2 Rf = ξ R0 R0 R0

(3:22a)

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Real chain: f =

3=2 Rf kB T kB T kB T · = 5=2 Rf 3=2 = RF ξ RF RF

(3:22b)

From this expression we realize that the force needed to deform a real chain increases stronger with Rf than the force needed to deform an ideal chain. The absolute force values, however, are always smaller for a real chain due to their bigger swollen, or “prestretched”, original shape. This fact is visualized in Figure 36. A general equation for the deformation of a polymer chain can be expressed using the Flory exponent ν: F = kB T

1 Rf 1 − ν Nνl

(3:23)

To summarize, the preceding scaling discussion has shown that both ideal and real chains lose conformational freedom upon deformation, which is reflected by their entropic spring constant kB T =ξ . However, they do so in different ways: an ideal chain is deformed from its Gaussian coil dimensions, R0, whereas the real chain is already prestretched due to excluded volume interactions to a bigger size, RF. From this, it follows that the deformational force for real chains is smaller than that of their ideal counterparts, albeit it increases more steeply upon deformation.

Figure 36: Force, f, needed to deform an ideal and a real polymer chain by a distance Rf. For an ideal chain, the required force scales with the deformation distance Rf with a power-law exponent of one, as expressed by eq. (3.22a), whereas for a real chain, the required force scales with the deformation distance Rf with a power-law exponent of 3/2, as expressed by eq. (3.22b). Thus, the force needed to deform a real chain increases more steeply than its counterpart for an ideal chain. The absolute force values, however, are always smaller for a real chain due to their bigger swollen, or “prestretched”, original shape. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

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93

Questions to Lesson Unit 5 (1) Match the following scaling laws with the appropriate polymer–solvent conditions under which they apply (from left to right): 1

R ¼ ðNK Þ2 lK

3

R ¼ ðNK Þ5 lK

R ¼ ðNK Þ1 lK

1

R ¼ ðNK Þ3 lK

a. No effective monomer–monomer interactions Repulsive effective monomer–monomer interactions Maximally repulsive effective monomer–monomer interactions Maximally attractive effective monomer–monomer interactions b. No effective monomer–monomer interactions Maximally attractive effective monomer–monomer interactions Maximally repulsive effective monomer–monomer interactions Repulsive effective monomer–monomer interactions c. Maximally attractive effective monomer–monomer interactions Repulsive effective monomer–monomer interactions Maximally repulsive effective monomer–monomer interactions No effective monomer–monomer interactions d. Maximally repulsive effective monomer–monomer interactions Repulsive effective monomer–monomer interactions Maximally attractive effective monomer–monomer interactions No effective monomer–monomer interactions. (2) By what factor is a polymer with a Kuhn segment number of 30,000 in a θ-solvent larger than a polymer with a Kuhn segment number of 10,000 and the same Kuhn length under the same conditions? pffiffiffi a. 3 b. 31 c. 3 d. 32 (3) By approximately what factor is a polymer with a Kuhn segment number of 30,000 in a good solvent larger than a polymer with the same Kuhn segment number and length in a θ-solvent? 3 a. 35 1 b. 30; 00010 1 c. 32 5 d. 33

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(4) Which statement regarding the energies that are decisive for the resulting expansion of a real coil is correct? a. The energy for the occupation of the excluded volume and that resulting from the entropy-elastic restoring force upon expansion are both positive, which is why the polymer shows a tendency to contract to reach a minimum total energy. b. The energy for the occupation of the excluded volume as well as that resulting from the entropy-elastic restoring force upon expansion are both positive, which is why the polymer shows an expansion tendency to reach a maximum total energy. c. The energy for the occupation of the excluded volume results in a contraction tendency of the polymer, whereas the energy resulting from the entropy-elastic restoring force upon expansion results in an expansion tendency. The overall energy minimum gives the optimal combination and therefore the optimal coil size. d. The energy for the occupation of the excluded volume results in an expansion tendency of the polymer, whereas the energy resulting from the entropyelastic restoring force upon expansion results in a contraction tendency. The overall energy minimum gives the optimal combination and therefore the optimal coil size. (5) The Flory theory provides a correct description of the coil size R of a real chain ______ a. only with respect to the Kuhn segment number NK , since the theory assumes an ideal chain entropy elasticity and a Gaussian segmental density profile, and these assumptions provide a correct description with respect to NK , but not with respect to the Kuhn segment length lK . b. only with respect to the Kuhn segment number NK , because the theory assumes an ideal chain entropy elasticity and a homogeneous segmental density, and these two simplifying approximations luckily compensate each other with respect to NK , but not with respect to the Kuhn segment length lK . c. with respect to the Kuhn segment number NK and the Kuhn segment length lK , because it assumes an ideal chain entropy elasticity and a homogeneous segmental density, and these two simplifying approximations luckily compensate each other completely in the resulting exponents. d. with respect to the Kuhn segment number NK and the Kuhn segment length lK , since the dependence of NK compensates the simplifying approximations from the assumption of an ideal chain entropy elasticity and a homogeneous segmental density profile, while additionally the dependence of lK results only from the assumption of an ideal chain and yields a correct exponent 52.

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(6) The Flory theory is also applicable to polyelectrolytes, that is, charged polymers. For this purpose, an additional energy term must be added to the Flory theory approach. What is valid for this term? a. The term corresponds to the Coulomb interaction, which contributes to the contraction tendency of the coil. b. The term corresponds to the Coulomb interaction, which contributes to the expansion tendency of the coil. c. The term corresponds to the van der Waals interaction, which contributes to the expansion tendency of the coil. d. The term corresponds to the van der Waals interaction, which contributes to the contraction tendency of the coil. (7) Which statement regarding the fractal dimension of polymers is not correct? a. The Flory exponent is the reciprocal of the fractal dimension. b. A higher fractal dimension results in a more densely packed coil. c. The higher the geometric dimension, the lower the fractal dimension. d. At maximum contraction, the fractal dimension equals the geometric one. (8) Which statement regarding the deformation of a real chain is correct? a. The force required for deformation of a real chain is always greater than that required for the same deformation of an ideal chain. b. The force required for deformation of a real chain is always smaller than that required for the same deformation of an ideal chain. c. A real chain with the same Kuhn length and number of segments is de3 formed by a factor N 5 more than the corresponding ideal chain if the same force f is applied. d. The force required to deform a real chain increases exponentially with the Flory exponent.

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3.6 Chain dynamics LESSON 6: CHAIN DYNAMICS So far, polymer chains were viewed to be static in this book. This chapter will go beyond that view and introduce two fundamental frameworks to model and quantify chain dynamics: the Rouse and the Zimm model. Both will show that it takes a certain time before a chain can move as a whole, whereas below, only parts of it move. This time delimits the scale on which a polymer is a viscous fluid from the scale where it is a viscoelastic body.

3.6.1 Diffusion In the last chapters, we have looked at the shape of ideal and real polymer chains; we have also discussed how their shape changes when they are deformed. In this chapter, we will focus on the motion, the dynamics, of polymer chains. As a fundament for that, we first recapitulate some elementary physical chemistry. The basic description of the thermal diffusive translational motion of a molecular or colloidal object is that of a random walk,38 a concept that we have already discussed in Section 2.4.1. The trajectory of such a random walk is displayed in Figure 37. A characteristic quantity for random walks is their mean-square displacement, , which is expressed by the Einstein–Smoluchowki equation:39

 Often, the diffusive molecular motion is synonymized as Brownian motion. But note that this is actually not the same. The Scottish botanist Robert Brown observed that plant pollen floating on a surface of water show seemingly self-driven random motion. This Brownian motion, however, is actually a result of the diffusive motion of the molecules in the surrounding medium, as is was correctly interpreted later on by Albert Einstein and Marian Smoluchowski. In Brown’s experiment, the plant pollen are hit by the surrounding water molecules, but as the hitting is random, the number of hits from different directions are not equal at a time per particle. As a consequence, the overall momentum transfer at each timepoint pushes the pollen particles into randomly changing directions, causing their seemingly self-driven random wiggling motion that Brown saw.  In Figure 37, the displacement of the moving particle corresponds to the distance of the last arrow of the sequence to the starting point of the walk. The mean-square of it is obtained by averaging over many of such walks, whereby in that averaging, each displacement is first squared to get rid of the directional dependence. Otherwise, if that was not done, the average would always be zero, as each displacement would be cancelled by one showing exactly into the opposite direction in a great ensemble of walks. As a result, the quantity of interest is the mean-square displacement – the mean over many individual displacements in a squared form. Often, to relinearize the physical dimension, people further take the square-root, thereby obtaining the root-mean-square displacement. We also do so all over Chapters 2 and 3 when we talk about the root-mean-square end-to-enddistance of polymer chains.

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Figure 37: Trajectory of a (two-dimensional) random walk. 2 h~ R2 i = hð~ rðtÞ −~ rð0ÞÞ i = 2 d D t

(3:24)

Here, d denotes the geometrical dimension, and D is the translational diffusion coefficient, a quantity that expresses how mobile the moving molecule or particle is. According to eq. (3.24), D has the unit m2·s–1, thereby quantifying what meansquare distance the moving object passes per time. The diffusion coefficient can be calculated by the Einstein equation: D=

kB T f

(3:25)

This equation relates D to the ratio of the thermal energy that drives the diffusion, kBT, and the friction that drags the diffusion; the latter is expressed by a friction coefficient, f, that connects the frictional force to the velocity of the moving object, ~ f = f~ v. The friction coefficient of a spherical object is given by Stokes’ law as f = 6πηrh

(3:26)

Here, η expresses the viscosity of the surrounding medium (this is what actually exerts the friction) and rh denotes the hydrodynamic radius, which is the radius of the moving object itself plus its solvent shell (and potential swelling medium inside) that is dragged with it during the motion. Both the latter equations can be combined to give the Stokes–Einstein equation: D=

kB T 6πηrh

(3:27)

Equation (3.27) is fundamental in physical chemistry, as it relates the size of a moving molecule or particle, denoted by rh, to its mobility, denoted by D. Together with the Einstein–Smoluchowki equation (3.24), this allows us to determine how far a

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diffusing molecule or particle of given size rh can move in a prescribed time t, or vice versa, how long it takes to move a given distance R2. This is of elementary relevance in many scientific fields, for example, in the field of drug delivery, when the time shall be appraised that a drug will need to reach a receptor in a cell or tissue environment, or vice versa, when it shall be appraised how far at all the drug can move in a given timeframe. The same question is also relevant in fields like chemical technology when it comes to appraising the efficiency of heterogeneous reactions, where partners must find each other by diffusion across phase boundaries, or in chemical engineering, when it comes to appraising how far diffusive smearing will impair precision in micropatterning or 3d printing. In the field of polymer and colloid science, a specified form of eq. (3.24) is relevant; this is a form in which eq. (3.24) is rephrased such to express the timescale τ for the displacement of a moving (macro)molecule or (colloid)particle by exactly its own size, R: τ=

R2 R2 f = 2d · D 2dkB T

(3:28)

On timescales shorter than this characteristic time τ, the colloidal or polymeric building blocks of a material cannot move over distances at least corresponding to their own size, meaning they are practically static. By contrast, on timescales longer than τ, the material’s building blocks are macroscopically mobile. As a result, τ delimits the time domain on which a material is a solid from that on which it is a liquid. For polymeric and colloidal matter, this limiting time is often on experimentally relevant orders of magnitude, which causes these materials to exhibit both solidlike and liquidlike appearance, depending on the timescale of observation. So far, all the above discussion accounts for simple molecules or particles. When it comes to the dynamics of a flexible polymer coil, however, in addition to its global motion, we also must consider that it has multiple kinds of coil-internal dynamics, as it is a large multibody object. To account for this complexity, two different models have been developed: the Rouse model and the Zimm model.

3.6.2 The Rouse model The Rouse model conceptually describes a polymer chain as a number of N spherical beads, representing the monomeric units, connected through elastic springs of length l, representing the bonds between the monomer segments, as depicted in Figure 38. Each bead is assigned an individual segmental friction coefficient fsegment. Together, the beads constitute a freely drained coil, which means that only the beads feel friction with the surrounding medium, but the springs do not. As a result, the solvent can pass freely through the polymer coil and hit each bead, where it imparts a frictional increment fsegment. The total friction coefficient of the coil is therefore simply

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Figure 38: Spring-and-bead modeling of a chain in the Rouse model. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

the sum of all these individual frictions: ftotal = N · fsegment. This generates a system of coupled differential equations from the equations of motion of the beads subject to drag by the springs and friction by the solvent. If we substitute ftotal in the Einstein eq. (3.25) by N · fsegment, we get DRouse =

kB T kB T = ⁓ N −1 ftotal N · fsegment

(3:29)

We now formulate an Einstein–Smoluchowski analog expression for the timescale to achieve a displacement by exactly one coil size, the Rouse time: τRouse =

R2 fsegment 2 = NR 2d · DRouse 2dkB T

(3:30)

This time denotes the upper extreme of a whole spectrum of relaxation times that we will discuss in more detail below. On timescales longer than τRouse, the coil can migrate over distances further than its own size; macroscopically, this means that on these timescales the building blocks of a polymer material can effectively displace against each other, such that the material exhibits flow. The other extreme of the characteristic relaxation time spectrum is the time it takes for at least each monomeric segment to be displaced by a distance equal to its own size, l: τ0 =

fsegment · l2 2dkB T

(3:31)

On timescales shorter than τ0, no motion whatsoever is possible in the coil. On these short timescales, the material therefore is an (energy-)elastic solid with glassy appearance. A polymer chain is a fractal object with size R = N ν l (cf. Section 2.6.2). When we insert this into the above equation for the Rouse time, we can derive a generalized expression that relates τRouse to τ0:

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τRouse =

fsegment · l2 1 + 2ν = τ0 N 1 + 2ν N 2dkB T

(3:32)

The latter equation combines the two characteristic ends of the relaxation time spectrum, and thereby delimits three different domains of dynamics. On timescales shorter than τ0, that is, t < τ0, the polymer does not have sufficient time to move at all, neither the entire chain nor any of its constituent segments. On this timescale, the polymer forms an energy-elastic, glassy solid. On intermediate timescales, τ0 < t < τRouse, at least monomeric segments and sequences of them (i.e., subchain segments) can move over distances equal to their own size. The time is, however, not yet long enough for the entire chain to effectively displace itself. On this timescale, the polymer is a viscoelastic solid. Only on timescales longer than the Rouse time, t > τRouse, the whole polymer coil can move over distances greater than its own size. Now, the material exhibits flow and is a viscoelastic liquid.

3.6.3 The Zimm model So far, we have considered the polymer coil to be freely drained by solvent. This view, however, is not always true. M–S interactions can cause moving polymer chain segments to drag solvent molecules with them. These, in turn, do the same to the solvent molecules adjacent to them. This drag spreads from one solvent molecule to another until, eventually, it reaches other segments of the polymer chain. Hence, even distant segments are coupled to each other through space by hydrodynamic interactions. As a result, each coil drags the solvent in its pervaded volume with it on its motion; this trapped solvent is in diffusive exchange with the medium in the surrounding, but does not drain through the coil. As a result, the coils appear like solvent-filled nanogel particles, as visualized in Figure 39. The Zimm model expands on the Rouse model to incorporate such hydrodynamic interactions. It assumes that the polymer coil and the trapped solvent inside act together as a united object of size R = Nνl. Substitution of the hydrodynamic radius, rh, in the Stokes–Einstein equation with this expression yields DZimm =

kB T kB T ≈ ⁓ N −ν 6πηrh ηN ν l

(3:33)

Compared to the Rouse model, the power-law scaling is less steep here: it is just –ν compared to –1.40 Hence, in the Zimm model, the diffusion coefficient has a less

 Note: for stiff, rodlike chains, with ν = 1, the Zimm-type scaling matches the Rouse-type scaling, as for such a polymer, there is no trapped solvent in it, as it is fully decoiled. So, Zimm- and Rouse-type dynamics become indistinguishable for this kind of chain.

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Figure 39: Modeling of polymer coils as microgels entrapping the portion of solvent within their pervaded volume due to hydrodynamic interactions and dragging it with them on their motion through the free solvent in the surrounding medium. The trapped portion of solvent is in diffusive exchange with that in the surrounding, but the coils are not drained by the medium. Picture inspired by B. Vollmert: Grundriss der Makromolekularen Chemie, Springer, 1962.

pronounced dependence on N than in the Rouse model. The reason is the lack of solvent draining in the Zimm model. In the Rouse model, viscous drag is exerted by each bead that constitutes the coil, be it beads on the coil front or in the coil interior. In the Zimm model, this is done only by the beads (and the trapped solvent between them) on the coil’s frontal face. Thus, in the Rouse model, any change of the number of beads directly translates to the friction on the coil motion, and hence, results in an inverse proportionality of its diffusion coefficient on N. By contrast, in the Zimm model, friction is exerted by the coil frontal face only, and the size of that face scales with the size of the coil as a whole, that is, according to Nν, so that the diffusion coefficient just inversely scales with N–ν. As we have done in the Rouse model, we can formulate an Einstein–Smoluchowski analog expression for the timescale of displacement by exactly one coil size, the Zimm time: τZimm =

R2 η 3 ηl3 3ν ≈ R ≈ N ≈ τ0 N 3ν 2d · DZimm kB T kB T

(3:34)

Again, compared to the Rouse time, the power-law exponent is smaller here: 3ν compared to (1 + 2ν).41 Again, this means that there is a weaker dependence of the longest relaxation time τ on the number of monomer segments N in the Zimm

 Yet again, for stiff, rodlike chains, with ν = 1, the Zimm-type scaling matches the Rouse-type scaling.

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model than in the Rouse model. As a consequence, τZimm is shorter than τRouse. The reason is the absence of solvent draining, which comes along with less viscous drag, causing the time its takes for the coil to diffuse a given distance (such as that of its own size) to be shorter than in the case of more pronounced viscous drag in the Rouse scenario. Bruno Hasbrouck Zimm (Figure ) was born on October , , in Woodstock, New York. He studied at Columbia University, where he obtained his bachelor’s degree in , his master’s degree in , and his PhD degree under Joseph. E. Mayer in . He then moved across town for postdoctoral work with Herman Mark at the Polytechnic Institute of Brooklyn. In , he transferred to the University of California in Berkeley, where he became an assistant professor in –. After that, he was the head of the General Electric research labs in Schenectady. In , he became a full professor at the University of California in San Diego. He retired from research in . He is most famous for his work on light scattering, where he developed the Zimm plot, his extension of the Rouse model of polymer dynamics, and his groundbreaking work on the structure of proteins and DNA. He died on November , , at the age of  in La Jolla, California.

Figure 40: Portrait of Bruno H. Zimm. Image reprinted with permission from Macromolecules 1985, 18(11), 2095–2096. Copyright 1985 American Chemical Society.

3.6.4 Relaxation modes We have learned in Section 2.6.2 that polymers are self-similar and fractal objects. This self-similarity is also valid for chain dynamics: a subchain with g segments in a whole of N segments relaxes like an individual chain composed of just g segments in total, describable by the same Rouse and Zimm formalism as outlined earlier. These subchain relaxations are appraised by so-called relaxation modes numerated by an index p. The pth mode corresponds to the coherent motion of subchains with N/p segments in our whole chain with N segments. At p = 1, coherent motion of the entire chain is possible, meaning that the entire chain can relax and get displaced by a distance equal to its own size, whereas at p = 2, only each half of it can move coherently and relax and therefore get displaced by a distance equal to its own size, respectively. At p = 3, just only each third of the chain can move coherently and relax and therefore get displaced by a distance equal to its own size, and so forth. At p = N, only single monomeric units can relax and get displaced

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against each other by their own size. Figure 41 visualizes this hierarchy of relaxation modes. At a time τp after abrupt deformation, all modes with index above p are relaxed already, whereas all modes with index below p are still unrelaxed. In general, the energy storage upon deformation is of order N · kBT. In a deformed viscoelastic material, stress can be relaxed, whereby each mode relaxes a portion of kBT. This means that the stored energy drops from a total of kBT per segment at τ0 to kBT per chain at τRouse or τZimm. After an intermediate time of τp, only p · kBT remains stored. From this notion, as well as from an expression for the time dependence of the mode index – an equation that tells us up to which number p modes are already relaxed at a time of interest τp – we may derive quantitative expressions for the time-dependent energy storage and relaxation capabilities of polymer materials, which we will do in Section 5.8.2.

Figure 41: Relaxation modes, indexed by a number p, of a schematic polymer chain. The first mode, p = 1, relates to relaxation of the entire chain. In the second mode, p = 2, subchain segments with length of just half of the chain can relax. The third mode, p = 3, corresponds to the relaxation of subchain segments with length of only a third of the chain, and so on. In the last mode, p = N, only single monomeric units can relax (not sketched here). Picture modified from H. G. Elias: Makromoleküle, Bd. 2: Physikalische Strukturen und Eigenschaften (6. Ed.), Wiley VCH, 2001.

Table 8 summarizes the characteristic times that we have determined for the Rouse and the Zimm model as well as the time dependence of the mode index p. Table 8: Characteristic parameters of the Rouse and Zimm model. This table serves as a toolbox to derive analytical expressions for the time-dependent mechanical spectra of polymer solutions and melts in Chapter 5. Rouse model Longest relaxation time Relaxation time of the pth mode Shortest relaxation time Time dependence of the mode index)

τ 1 = τ Rouse = τ 0 N  1 + 2ν τ p = τ 0 Np τN = τ0   −1 τ 1 + 2ν p = τp ·N 0

Zimm model 1 + 2ν

τ 1 = τ Zimm = τ 0 N3ν  3ν τ p = τ 0 Np τN = τ0  −1 τ 3ν p = τp ·N 0

Plugging in a value for τp allows us to calculate the mode down to which relaxation has already proceeded after a time of interest τp. These equations are obtained from the ones in the second row by simply rearranging those for p. 1)

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3.6.5 Subdiffusion Let us get back to the Einstein–Smoluchowski equation we have introduced in Section 3.6.1. Actually, it is not yet fully complete. To make it generally applicable, we must extend it by an exponent α: hð~ r ðt Þ − ~ rð0ÞÞ i = 2 d D tα 2

(3:35)

When α = 1, we get the regular Einstein–Smoluchowski equation that describes normal Fickian diffusion. There are, however, many cases in which α ≠ 1. If α < 1, the diffusion is constrained; this situation is called subdiffusion. A typical cause for subdiffusion is if there is temporary trapping of the diffusing molecules, which can be the case when they encounter binding sites on their way of motion.42 By contrast, if α > 1, the diffusion is promoted; this situation is called superdiffusion. A typical cause for superdiffusion is if the diffusing molecules can ride on the back of carriers for some time and thereby quickly span large distances on their way.43 Let us now consider a polymer chain with N segments, and within that, a sub  rj ð0Þ chain with N/p segments. This subchain gets displaced by a distance R =~ rj τp −~ of length l · (N/p)ν, which corresponds to its own size, during the time τp. Now imagine that we somehow label one monomer segment j on the subchain and follow its displacement. After τp it is 2v

D 2 E 2 N 2v 2 τp 1 + 2v rj ð0Þ =l =l according to the Rouse model: ~ rj ðτp Þ −~ τ0 p according to the Zimm model:





D 2 E 2 N 2v 2 τp 2=3 ~ rj ðτp Þ −~ rj ð0Þ =l =l τ0 p

(3:36a)

(3:36b)

In the latter two equations, we have first written down a squared variant of the dis  tance R =~ rj τp −~ rj ð0Þ of length l · (N/p)ν that we consider here, and then we have substituted the time dependence of the mode index p in the denominator by plugging in the Rouse- or Zimm-model-related expression from Table 8. The resulting exponents for the time dependence of the squared displacements that we obtain with this approach are 1 +2ν2ν for the Rouse model (this is ½ in an ideal state with

 As an analogy, consider someone on a wine fest in Mainz who has enjoyed the wine too much and therefore conducts a random walk home. The mean-square displacement of that person scales linearly with time according to the Einstein–Smoluchowski equation. If, however, that person is trapped on the way, for example, by meeting other people to chat with (if this is still possible in that state) or by being attracted by more wine stands, the mean-square progress will be less than proportional to time.  As an analogy, consider our drunken friend at the wine fest again. This person may accelerate the random-walk home by riding a bus on the way.

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ν = 1/2) and 2/3 for the Zimm model, as shown in Figure 42. Both these exponents are smaller than one. Thus, the segmental motion at times shorter than τRouse and τZimm is subdiffusive. This is due to the hindrance imparted on the movement of each monomer segment (such as our labeled one) by its neighbors, to whom it is bound by chemical bonds. Our selected labeled monomer segment can move into a given direction only when the neighboring monomers do so as well. By contrast, when the neighbors do not move into the same direction, our selected monomer is dragged on its motion, thereby forcing its time-dependent displacement to be subdiffusive rather than freely diffusive. As an analogy, remember our conceptual picture of a polymer as a chain of people holding hands in Section 1.2. In such a chain, if one individual wants to take some steps into a desired direction, this is possible only if the neighbors join that motion. If they don’t, then that motion is constrained. Once more, the motion is slower in the Rouse than in the Zimm scenario, as also seen in Figure 42, where the line with slope 1/2 leads to smaller displacements in a given time than the line with slope 2/3. Once more, this is due to the greater viscous drag imparted by the freely draining solvent in the Rouse situation as compared to the solvent drag only on the frontal surface of the solvent-filled coil in the Zimm situation.

Figure 42: Time dependence of the mean-square displacement of a “labeled” monomer unit (named j) in the Rouse and Zimm scenario. At times smaller than the Rouse- or Zimm-time, the power-law exponent in the two scaling laws is smaller than one, indicating subdiffusion of the labeled segment. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

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3.6.6 Validity of the models Now that we have treated two different models that describe the dynamics of a polymer chain, a natural question is which of the two is able to make more accurate predictions. As it turns out, both do, but their validity depends on the polymer’s surroundings. In dilute solution, where the polymer concentration is low, hydrodynamic interactions between the segments in a coil are strong. In this case, the Zimm model is the more valid one. We can see evidence for that in the left half of Figure 43, which shows the molarmass-dependent scaling of the diffusion coefficient of three types of polymers in dilute solution. Each exhibits scaling to the power of the negative Flory exponent that applies to that specific polymer–solvent combination, in perfect agreement to eq. (3.33). By contrast, in the semi-dulite concentration regime or in a melt, both characterized by marked mutual interpenetration of the coils, the hydrodynamic interactions are screened by overlapping segments of other polymer chains. This effect is similar to the screening of excluded volume interactions in polymer melts that therefore always show θ-type coil conformation. In these regimes, the Rouse model is better suited to describe the polymer chain dynamics.44 We can see evidence for that in the right half

Figure 43: Scaling of (A) the polymer-chain translational diffusion coefficient, D, in dilute solutions and of (B) the polymer-melt viscosity, η, both as a function of the polymer chain length, here assessed by the polymer molar mass, M. Although dilute solutions show scaling of D ⁓ M–ν, as predicted by the Zimm model, melts of chains with a length shorter than a certain critical one exhibit η ⁓ M1, as predicted by the Rouse model (for a detailed, full explanation of this specific scaling see Section 5.10.1).

 Actually, in the semidilute regime, polymers display both Rouse- and Zimm-type dynamics, depending on the length scale under consideration.

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of Figure 43, which shows the molar-mass-dependent viscosity of different polymer melts. In the low molar-mass regime, this quantity rises linearly with the molar mass, as will be proven to be Rouse-model based in Section 5.10.2. (In the high molar-mass regime, the scaling is significantly steeper, indicating a different mechanism of chain motion: the reptation mechanism, which we will also discuss in Section 5.10.2.)

Questions to Lesson Unit 6 (1) How does the root-mean-square displacement of a diffusing molecule or particle differ if we give it twice as much time? a. It is larger by a factor of 2. pffiffiffi b. It is larger by a factor of 2. c. It is larger by a factor of 22 . d. This requires a more extensive calculation. (2) On which basic assumption is the Rouse model not based? a. The coil is freely drained by the surrounding medium. b. The chain is modeled as a series of beads connected by springs. c. The total friction coefficient is the sum of the individual segmental friction coefficients fSegment . d. Both beads and springs experience friction together as they flow through the surrounding medium. (3) Match the appropriate timescale to the material behavior. 1. Individual segments or small chain sections can move; the material has the properties of a viscoelastic solid. 2. Not even single segments can move; the material has the property of a glass. 3. Time is sufficient for the entire polymer chain to move, such that displacement beyond the chain size occurs; the material has the property of a viscoelastic fluid. a. 1. τ0 < t < τRouse 2. t < τ0 3. t > τRouse b. 1. τ0 < t < τRouse 2. t > τRouse 3. t < τ0 c. 1. t < τ0 2. τ0 < t < τRouse 3. t > τRouse

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d. 1. t > τRouse 2. τ0 < t < τRouse 3. t < τ0 (4) On which basic assumption is the Zimm model not based? a. The chains can be considered as nanogel particles filled with solvent. b. The moving chain segments exert viscous drag on neighboring solvent molecules. c. The entrapped solvent remains in the coil. d. Because of hydrodynamic interactions, even distant chain segments can affect each other. (5) Which statement concerning relaxation modes is correct? a. At a time τp ; all relaxation modes with index smaller than p are already relaxed, whereas all modes with index higher than p are still unrelaxed. b. The zeroth relaxation mode (index p ¼ 0) corresponds to the relaxation of the entire chain. c. The concept of relaxation modes is applicable only within the Rouse-model framework but not within the Zimm model, as it does not account for hydrodynamic coupling. d. The relaxation of the chain can be conceived as a sequential relaxation of chain subsegments, which become larger with increasing time index. (6) Why does the relaxation timescale have a higher exponent on N in the Rouse model than in the Zimm model? a. In the Zimm model, due to the non-draining condition, only the segments on the frontal surface of the coil experience friction, whereas in the Rouse model, all segments are affected because the solvent drains through the coil. b. This is a pure result of the derivation of the corresponding formula and has no trivial physical meaning. c. In the Zimm model, the relaxation time is shortened by hydrodynamic interactions, because the segments can interact with each other even at a higher distance. d. The hydrodynamic interactions in the Zimm model compensate the friction due to the solvent, which makes the friction on the individual segments smaller compared to the Rouse model.

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(7) Assign the mentioned properties to the phenomena of subdiffusion or superdiffusion; choose the correct assignment order. 1. The exponent in the generalized Einstein–Smoluchowski equation is greater than 1. 2. Diffusing particles are temporarily slowed down when they encounter binding sites. 3. Diffusing particles are temporarily transported in a directed manner, for example, by docking to transport systems in a living cell. 4. The exponent in the generalized Einstein–Smoluchowski equation is less than 1. a. 1. Subdiffusion 2. Subdiffusion 3. Superdiffusion 4. Superdiffusion b. 1. Superdiffusion 2. Subdiffusion 3. Superdiffusion 4. Subdiffusion c. 1. Superdiffusion 2. Superdiffusion 3. Subdiffusion 4. Subdiffusion d. 1. Subdiffusion 2. Superdiffusion 3. Subdiffusion 4. Superdiffusion (8) The motion of individual segments within a polymer chain is subdiffusive ______ a. since this motion is inhibited by attractive interactions with the solvent. b. for timescales smaller than τ0 (time for relaxation of a monomer). c. since individual segments restrict each other’s free mobility due to their binding to each other. d. if good-solvent conditions are fulfilled.

4 Polymer thermodynamics LESSON 7: FLORY–HUGGINS THEORY All the previous discussion on polymers in this book was limited to single chains. The following lesson moves beyond and introduces a concept to model and quantify the thermodynamics of mixing of multiple chains with either a low-molecular solvent or with another polymer species. You will see from this modeling that the mixing entropy, which is majorly responsible for the ease of mixing of many substances in classical chemistry, plays a minor role only in polymer science, such that polymers only mix when there is very high enthalpic compatibility. This is captured in a fundamental quantity: the Flory–Huggins interaction parameter.

Throughout this textbook, so far, we have investigated how a polymer chain is shaped, we have learned how it interacts with itself and with a solvent, and we have obtained a picture on how it moves. The common factor in all of this is that, so far, we have focused only on single chains. In the following, we want to expand our view to multi-chain systems. We now ask ourselves: how do many chains interact with one another and with a solvent? Answering that question will be a complex endeavor, because it will require the appraisal of a huge number of interactions, and is, thus, mathematically hard to describe. In addition, the entire multichain system is subject to constant dynamic change, making it even more difficult for us to describe it adequately. So, how do we solve this? We approach this challenge by just considering average monomer–monomer and monomer–solvent interactions. In short, rather than trying to appraise how many M–M and how many M–S interactions we have at a time in the system, and rather than trying to appraise where exactly in the system they occur, we are satisfied with the knowledge of how many of such interactions we have on average (over space and time). A very convenient aspect of that simplification is that this number simply scales in proportion to the respective volume fractions of M and S in the system. This approach is called a mean-field treatment.

4.1 The Flory–Huggins mean-field theory Using a mean-field approach, we are able to conceptualize the thermodynamics of polymer–solvent and polymer–polymer mixtures. Said mean-field approach was independently developed by Maurice L. Huggins and Paul J. Flory and is called the

https://doi.org/10.1515/9783110713268-004

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Flory–Huggins mean-field theory.45 It appraises the change of the Gibbs free energy of mixing, ΔGmix, which must be negative for two components to readily mix with each other: ΔGmix = GAB − ðGA + GB Þ = ΔHmix − TΔSmix

(4:1)

Here, GAB is the free energy of the mixture, whereas GA and GB are the free energies of the demixed components. ΔGmix can also be expressed in terms of the enthalpy and entropy of mixing, ΔHmix and ΔSmix, respectively. In general, entropy always favors mixing, as that decreases the order of the system, whereas enthalpy usually disfavors mixing, because that would force the components to interact with each other rather than with themselves, which is a less perfect mutual match in most cases. Maurice Loyal Huggins (Figure ) was born on September , , in Berkeley, California. He studied chemistry at the University of California in Berkeley, where he obtained his Master’s degree in . He obtained his PhD degree two years later, in , under Charles M. Porter for work on the structure of benzene. He then worked at different institutes, including Stanford Research Institute, Johns Hopkins University (Baltimore, MD), before he joined Eastman Kodak (Rochester, NY) in . From , he returned to Stanford Research Institute and eventually retired from research in . During his active time, he independently conceived the idea of hydrogen bonding in , was an early advocate for its role in stabilizing protein secondary structures, and produced a model of the α-helix in , roughly eight years ahead of the modern model of Linus Pauling, Robert Corey, and Herman Branson. He also developed the mean-field theory for polymer solutions, the Flory–Huggins theory. He died on December , , in Woodside, California.

Figure 44: Portrait of Maurice L. Huggins. Image reproduced with permission from Oregon State University Libraries Special Collections & Archives Research Center.

 Flory himself suggested naming Huggins first (see J. Chem. Phys. 1942, 10(1), 51–61), as Huggins was in fact the erstwhile of the two scientists to publish the mean-field theory (Huggins’ paper was printed in the May issue of J. Chem. Phys. 1941, whereas Flory’s paper appeared in the August issue of that journal). By the time of Flory’s suggestion, though, the former name was already established in the scientific community, such that it is still used in the order of Flory–Huggins today.

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In classical mixtures of low-molar-mass compounds, the inherently favorable entropy influence in eq. (4.1) is powerful enough to outweigh the usually unfavorable enthalpy influence, such that mixing occurs. (And if that does not occur at room temperature, it often helps to heat up, as this will give the entropy term in eq. (4.1) more emphasis.) However, the favorable entropy term is much less powerful for polymer systems. This is because polymers are long chain-like molecules that bring a large number of monomer units into quite some preorder, simply by tying them together in the form of chains. In that state, these monomers cannot arrange themselves in a solvent with that much freedom as they could if they were not polymerized, and as a result, there is much less entropy gain in the polymeric state than it could be in a nonpolymeric state upon mixing with a solvent. Consequently, dissolution and intermixing of polymers is mostly dependent on the enthalpic term, as we have learned already in Section 3.2. The Flory–Huggins mean-field theory now aims at independently estimating the entropic and the enthalpic contributors to eq. (4.1). It is based on a lattice model, as shown in Figure 45. Component A is represented by black beads, whereas component B is represented by white beads. These beads can be placed on lattices, either separate ones for each species (left-hand side of Figure 45), which represents the demixed state, or a common lattice for both species (right-hand side of Figure 45), which represents the mixed state. We assume that all molecules involved, be it the solvent (white beads) or the dissolved component (black beads; note: if that compound is a polymer, the beads are its monomeric units!), have the same volume.

Figure 45: Mixing of black and white beads with the same volume on a lattice. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

Based on this premise, the volumes of the individual demixed lattices, VA and VB, are additive upon mixing and sum up to the volume of the combined lattice, Vmix: Vmix = VA + VB

(4:2)

When the mixing is thermodynamically favored (which we aim to appraise here), all beads will randomly occupy new positions within the new combined volume

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Vmix. The resulting volume fractions46 occupied by component A and B, ϕA and ϕB, are given as follows: VA VA + VB

(4:3a)

VB = 1 − ϕA VA + VB

(4:3b)

ϕA = ϕB =

We define the unit volume of each single lattice site to be ν0. Then, the total number of lattice sites, n, is n=

VA + VB ν0

(4:4)

Component A occupies nϕA of the lattice sites, whereas component B occupies nϕB of the lattice sites. We can also use ν0 to calculate the chain volumes, νA and νB, of both components: νA = NA ν0

(4:5a)

νB = NB ν0

(4:5b)

Here, NA and NB are the numbers of adjacent lattice sites occupied by each molecule of component A or B, respectively. For a polymeric sample, NA and NB correspond to the degrees of polymerization of the polymer chains (which scales to their sizes as RA ⁓ NAν and RB ⁓ NBν). We can delimit three different cases, depending on the relative size of NA and NB, all three of which are shown in Figure 46. In the first scenario, depicted in Panel (A), both NA and NB are one. In this case, we talk about a regular solution of low-molar-mass compounds. Such a mixing process is highly entropy driven, as also seen by the quite messy appearance of Panel (A). This is very different for scenario two, shown in Panel (B). Here, NA >> 1 (in the figure, it is NA = 10), corresponding to long polymer chains, and NB = 1, corresponding to solvent molecules. This is the scenario of a polymer solution. Due to their connectivity to chains, the monomer segments now have much less possibilities to arrange themselves than in the first scenario, as seen by the more tidy distribution of the black beads in Panel (B) as compared to Panel (A); only the solvent molecules still can arrange themselves quite freely in that kind of system. In scenario three, shown in Panel (C), both NA and NB are >> 1 (in the figure, NA = NB = 10). This is the case for a mixture of two different polymers, a polymer blend. In that case, the freedom of arrangement of

46 The volume fraction is a common measure of concentration in polymer science. It denotes what fraction of the volume of a system under consideration is occupied by a species of interest. You know a related quantity from elementary physical chemistry: the mole fraction. If there’s two comn n ponents A and B, it calculates as xA = n +A n and xB = n +Bn = 1 − xA . A

B

A

B

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Figure 46: Arrangement of 50 black and 50 white beads on a 10 × 10 lattice with elementary-unit volume v0 to model the binary mixing of two components in (A) a regular solution of two low-molar-mass components A and B, both consisting of molecules with size equal to the lattice-site size, (B) a polymer solution of a low-molar-mass component A that is built of monomer units (here: 10 per chain) that each occupy one lattice site, such that each macromolecule of the whole species occupies NA adjacent sites, dissolved in a low-molar-mass solvent B that consists of molecules equal to the latticesite size, and (C) a polymer blend of one high-molar-mass species A mixed with another high-molarmass species B, both occupying multiple adjacent lattice sites, NA and NB (here: both 10 per chain). Pictures redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

both the black and the white beads is quite limited, as both are constrained by their connectivity to chains.

4.1.1 The entropy of mixing The entropy of mixing, ΔSmix, can be estimated by the number of microstates that realize the two macrostates mixed and demixed. This is a statistical approach, through which we can calculate the entropy by Boltzmann’s formula, S = kB ln(W). In a homogeneous A–B mixture, each bead has n possible positions where it can be placed on the common lattice. In other words, each bead can sit anywhere on the lattice. In the demixed phases, component A only has nϕA possible positions in the A phase; the rest is occupied by component B in the B phase. From this, we get the entropy of mixing for a single A-bead, ΔSA, as47 ΔSA = kB ln n − kB ln nϕA = kB ln

1 = − kB ln ϕA ϕA

(4:6a)

Here, kB ln n relates to the mixed, and kB ln nϕA to the demixed state.

 Do not confuse the index B at kB, where it denotes Boltzmann, to the index B at nB, NB, and ϕ B, where it denotes species B.

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Analogously, the entropy of mixing for a single B-bead is ΔSB = − kB ln ϕB

(4:6b)

With both these expression, we can calculate the total entropy of mixing, ΔSmix ΔSmix = nA ΔSA + nB ΔSB = − kB ðnA ln ϕA + nB ln ϕB Þ

(4:7)

If we express the number of A-molecules, which are single beads in the case of a low molar-mass substance or chains built up of multiple beads in the case of a polymer, nA, as n · ϕA =NA as well as the number of B-molecules, nB, as n · ϕB =NB in eq. (4.7) and then divide by n, we get the entropy of mixing per lattice site:

ΔSmix ϕ ϕ (4:8) = − kB A ln ϕA + B ln ϕB ΔSmix = n NA NB In this equation, note again that ϕA and ϕB are the volume fractions of the components A and B, and NA and NB are their degrees of polymerization, meaning how many adjacent lattice sites are occupied per A and B (macro)molecule. If we apply this formula to a regular mixture, with NA = NB = 1, and with equal molar volumes of the A- and Bbeads (which we assume allover the present discussion anyways), we may replace the volume fractions in eq. (4.8) by mole fractions, thereby yielding a form of the equation that you know from elementary physical chemistry: ΔSmix = − kB ðxA ln xA + xB ln xB Þ. When inspecting eq. (4.8), we can get some conceptual insights. As the numerical values for ϕA and ϕB are both 0. . .1, the logarithms in eq. (4.8) will always be negative (or zero at most). Together with the minus sign in front of the right-hand side of the equation, this will always lead to a positive mixing entropy. That makes sense, as entropy always favors mixing. The extent of positivity, however, depends on the denominators NA and NB. The larger they are, the less positive will be the mixing entropy. This means that, even though entropy always favors mixing, it does so with less power at high chain length. This insight matches the qualitative discussion that we have just led on the preceding pages. Let us deepen our conceptual insight by some quantitative calculation and estimate the entropy of mixing for each of the three cases displayed in Figure 46. Here, the volume fractions of component A and B are equal, ϕA = ϕB = 0.5. In our first scenario, Panel (A), we have 50 white and 50 black beads that we can arrange freely on the lattice. This means that there are 100! / (50! 50!) = 1029 different possible microstates to realize the macrostate “mixed”. The entropy of mixing per lattice site normalized by Boltzmann’s constant, ΔSmix =kB , is 0.69 in this case. In Panel (B), our second scenario, we retain the 50 white solvent beads, but change the structure of component A, represented by the black beads, such that it is now a 10-segment polymer, thereby now represented by five 10-segment chains of black beads. Because the black beads are connected to each other in this fashion, the normalized entropy of mixing is markedly reduced, ΔSmix =kB = 0.38. In the last scenario, Panel

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(C), this trend continues. Now, the white beads are also connected together in five polymer chains with 10 segments each. As a result, the number of possible arrangements on the lattice for both components drops to only about 103. Consequently, the normalized entropy of mixing is further reduced, ΔSmix =kB = 0.069. To visualize these findings, the normalized entropy of mixing is plotted as a function of the volume fraction of component A in Figure 47. Again, we realize from this plot that the entropic gain from mixing a polymer with a solvent is drastically lower than that of a regular mixture of low-molar-mass compounds. Furthermore, we see that the entropic gain of the polymer blend is almost negligible. What we also realize from this plot is that the extent of the entropic gain also depends on the volume fraction; it is greatest at a 50:50 mixture (ϕA = 0.5) if the mixing components have equal sizes, which is the case for our scenario A with NA = NB = 1 and for our scenario C with NA = NB = 10, whereas the entropy maximum is at a different ϕA (of about 0.65) if the mixture consists of compounds with different sizes, as it is the case in scenario B, where we have NA = 10 but NB = 1.

Figure 47: Entropy of mixing for the three different cases shown in Figure 46.

The just minor gain of entropy upon mixing polymers with solvents or other polymers is one of two reasons for nature to rely the build-up of complex structures on polymeric building blocks: since their mess-up would bring just minor entropic gain, their tiding to ordered structures does not cost much either.48

 The second reason is that, due to their interaction energies of just a few couple RT, polymeric building blocks can serve to build entities that are stable enough to persist, but also labile enough to be de- and rearrangeable with not too much energy input.

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4.1.2 The enthalpy of mixing Let us now focus on the energetic contributions to the mixing of two components A and B. To do this, we have to quantify all possible interactions, which are A–A, B–B, and A–B. This, however, cannot be done analytically. Remember that a regular 1:1 mixture of even just 50 A-beads and 50 B-beads gives 1029 possible combinations. To address this challenge, we use a mean-field approach. In that approach, we conceptually “hydrolyze” our polymer to unconnected segments that have the same size as their surrounding molecules, and distribute them randomly on the lattice. We then only discuss interactions between two direct neighbors on this lattice. This gives us three different interaction energies, u, that we must discuss: uAA, uBB, and uAB. The average pairwise interaction energy of a monomer A with one of its neighboring lattice sites is given as UA = uAA ϕA + uAB ϕB

(4:9a)

Here, uAA is the interaction energy in case of an A–A contact and ϕA the probability for the neighboring site of the A-monomer under consideration to be indeed occupied by another A. This value is equal to the volume fraction of A in the system. The second term denotes the same for the case of an A–B contact. Analogously, the average pairwise interaction energy of component B with one of its lattice sites is UB = uAB ϕA + uBB ϕB

(4:9b)

Arguing this way exposes the simplicity and advantage of a mean-field approach. Normally, the interaction of one molecule (or bead) of component A is either exactly uAA or exactly uAB, depending on what type of molecule (or bead) sits next to it. However, we do not know which of the two interactions is present at a given time and spot on the lattice, so we assume an average mean interaction of both based on their frequency of occurrence in the system, ϕA and ϕB. We know that each of the n lattice sites has z nearest neighbors to interact with. From the single component average pairwise interaction energies, we can calculate the total average interaction energy in the mixed state: U=

z·n ðUA ϕA + UB ϕB Þ 2

(4:10a)

with UA and UB the average pairwise interaction energies given by eq. (4.9a) and (4.9b). The factor of ½ rows back double-counting of the pairwise interactions. The average interaction energy in the demixed states can also be determined. In this case, we use the individual interaction energies, uAA and uBB, in an expression just like eq. (4.10a):

4.1 The Flory–Huggins mean-field theory

U0 =

z·n ðuAA ϕA + uBB ϕB Þ 2

119

(4:10b)

With the interaction energies in the mixed and demixed states, we can calculate the energy change upon mixing. We do this by subtracting the energies and normalize the resulting difference by dividing it by the number of lattice sites, n:  mix = ΔU

U − U0 z = ð2uAB − uAA − uBB Þ ϕA ϕB = kB TχϕA ϕB n 2

(4:11)

This equation introduces a very important quantity for the discussion of polymer thermodynamics: the Flory–Huggins interaction parameter: χ=

z 2uAB − uAA − uBB · kB T 2

(4:12)

This parameter is a dimensionless measure of the difference of the pairwise interaction energies before mixing (uAA and uBB) and after mixing (2uAB), normalized to the lattice geometry (z/2) and the elementary thermal-energy increment (kBT). During mixing of A and B, we break A–A and B–B contacts but in turn establish two new A–B contacts per such breaking. This means that we “loose” uAA and uBB, but we gain 2uAB. Hence, according to eq. (4.12), when χ < 0, this means that there is more energy gained than lost upon this exchange. In that case, 2uAB is smaller, that means either less positive (=thermodynamically less unfavorable) or (even better) more negative (= thermodynamically more favorable), than uAA and uBB are together; this corresponds to an exothermic mixing process. When χ > 0, by contrast, there is more energy lost than gained upon mixing. In that case, 2uAB is larger, that means either more positive (=thermodynamically more unfavorable) or less negative (= thermodynamically less favorable), than uAA and uBB together; this corresponds to an endothermic mixing process. A Flory–Huggins interaction parameter of χ = 0 indicates no energy change upon mixing. The mixing process is then driven by its entropic contribution only, a situation commonly referred to as ideal mixture.49 Most usually, mixing of two components A and B is disfavorable, as two

 Do not confuse such an ideal mixture with what we have referred to as an ideal state (or pseudo-ideal state to be precise) in Section 3.2, when introducing the Θ-state. In an ideal mixture (sometimes synonymized as “ideal solution”), interactions of kind A–A, B–B, and A–B are all the same; we refer to that as an “ideal mixture”, as mixing is driven only by entropy in that case. In polymer systems, such a case corresponds to a system where M–M, S–S, and M–S interactions are all the same; we have referred to this as an “athermal solution” in Section 3.2. An ideal state, by contrast, is one without any interactions, which is merely theoretical. A truly existing variant is a pseudo- or quasi-ideal state which shows rather strong attractive effective M–M interactions, so strong that they just balance the M–M hard-sphere repulsion; we name this state a pseudo- or quasi-ideal one, as it is not free of interactions, but displays a delicate balance of effective M–M attraction and M–M hard-sphere repulsion (just like a real gas at the Boyle temperature shows

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A–B contacts just cannot be as perfect matches as an A–A and a B–B contact.50 This corresponds to an endothermic mixing process, characterized by a positive χ. If we assume, for the sake of simplicity, that our mixing process is isochoric, which means that the total volume does not change upon mixing, ΔVmix = 0, then we can substitute the enthalpic term, ΔHmix, in eq. (4.1) by ΔUmix, that is, by eq. (4.11). This yields the change of the Gibbs free energy upon mixing:

ϕA ϕB  (4:13) ln ϕA + ln ϕB + χϕA ϕB ΔGmix = kB T NA NB We have already stated that ΔGmix must be negative for mixing to occur, so let us look a little bit closer at the latter equation. The combinatorial entropy term always contributes something negative, as the numerical values for ϕA and ϕB are both 0 . . . 1, so the logarithms in eq. (4.13) will always be negative (or zero at most). This makes sense, as entropy always favors mixing. For the enthalpic term, however, it depends on the difference of the pairwise interaction energies before and after mixing, indicated by the Flory–Huggins parameter χ. As just said above, in most cases, χ > 0, because the components rather want to interact with themselves than with one another. The enthalpic term, then, counteracts the entropic contribution. It therefore depends on how strong the entropic contribution is, and how large the counteracting enthalpic term is, which is most dominantly regulated by the magnitude of χ· As we have discussed earlier in this chapter, the entropy of mixing is very small for polymer solutions and blends (see Figure 47). As a result, χ must be lower than a critical value (which is 0.5 in the case of polymer solutions and close to zero in the case of polymer blends, as will be demonstrated in Section 4.2) for mixing still to happen. Just in very rare cases, χ can be smaller than 0. For that to be, there must be a mutually complementary favorable interaction between A and B, such that there will be an additional energy gain when both components interact with each other rather than with themselves. A clever realization of that premise was presented in Freiburg and Mainz in the 1980s by Reimund Stadler. He equipped polymer chains with complementary hydrogen bonding motifs, as shown in Figure 48. The energy gained from these transient interactions very much fuels the mixing of these polymers.

balanced effective A–A attraction and hard-sphere A–A repulsion). Be aware of the risk of confusion of the terms ideal state (which is a polymer in a Θ-solvent at Θ-temperature) and ideal mixture (which is a polymer in an athermal solvent).  “Birds of a feather flock together.”

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Figure 48: Reimund Stadler’s approach of equipping polymers with self-complementary hydrogen bonding motifs to fuel a favorable enthalpic contribution to the free energy of mixing. Pictures reprinted with permission from R. Stadler, L. L. de Lucca Freitas, Coll. Polym. Sci. 1986, 264(9), 773–778, copyright 1986 Steinkopff Verlag (now Springer Nature), and R. Stadler, L. L. de Lucca Freitas, Macromolecules 1987, 20(10), 2478–2485, copyright 1987 American Chemical Society.

Reimund Stadler (Figure ) was born on October , , in Stühlingen, Germany. He studied chemistry at the Albert Ludwigs University Freiburg, where he received his diploma and PhD degree, with a thesis on “Viscoelasticity and Crystal Melting of Thermoplastic Elastomers”. He then became a postdoc at Porto Alegre in Brazil before his habilitation in  under Hans-Joachim Cantow back in Freiburg. Stadler was appointed as full professor at the Johannes Gutenberg University Mainz, and then switched to the University of Bayreuth, where he died suddenly just one year later, on June , . To his honor, the German Chemical Society gives out an award named after him to rising-star researchers in the field of polymer science every other year.

Figure 49: Portrait of Reimund Stadler. Image reproduced with permission from Designed Monomers and Polymers 1999, 2(2), 109–110. Copyright 1999 Taylor & Francis.

4.1.3 The Flory–Huggins parameter as a function The Flory–Huggins theory is a simplified approach to polymer thermodynamics: it appraises the entropy of mixing based on a combinatorial argument, and the energy of mixing based on mutual average interactions in a mean-field treatment. So far, so good. There are, however, several aspects that are not at all or at least not adequately captured by that theory, for example, additional entropic effects such as the need for favorable mutual orientation of the molecules to each other to interact, or mistakes in the enthalpic part of the theory that come from the mean-field treatment, ignoring the rather inhomogeneous distribution of the two species in the system, where we have high densities of the black species inside the polymer coils, but none of it between the coils. All these deviations from reality are lumped into the

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Flory–Huggins parameter χ. It is therefore not a given and fix number, but a function that itself is composed of an entropic part, χS, and a temperature-weighted enthalpic part, χH. χ = χS +

χH T

(4:14a)

It is therefore the interplay of both contributions, χS and χH, and temperature that determines the miscibility or immiscibility of a polymer and a solvent. The role of the entropy part and the enthalpy part of χ can be understood based on the following thought: Let us consider all contributions to the mixing free enthalpy, that means, not only the simple configurational entropy term and the simple pairwiseinteraction enthalpy term in the Flory–Huggins eq. (4.13), but also all further specific entropic and enthalpic effects that the mean-field approach doesn’t account for; we name this full set of contributions ΔGmix, excess . We may then write the following equation to connect this to the (then phenomenological) parameter χ:  χ  ΔGmix, excess = ΔHmix, excess − T · ΔSmix, excess = kB T · χS + H T

(4:14b)

Based on that, we identify − kB · χS as an excess mixing entropy ΔSmix, excess that is not based on simple statistical-combinatorial considerations, but lumps all the specificity of the molecules acting as the solvent and the solute polymer. kB · χH = ΔHmix, excess still reflects the mixing enthalpy based on mean-field average pairwise interactions in the form of χH = ðz=2kB Þð2uAB − uAA − uBB Þ, so that ΔHmix, excess = ðz=2Þð2uAB − uAA − uBB Þ. (Note that in this notion, χH has a physical unit [K], whereas χS has no unit.) Let us examine the enthalpic contribution to the Flory–Huggins parameter, χH, a little more closely. In unpolar systems, it is commonly positive, χH > 0. This is due to mostly attractive intramolecular interactions such as van der Waals interactions in those systems, which oppose mixing as they are better built between each component itself but not so perfectly between the components each other.51 These interactions, however, can be overcome at high temperatures, such that mixing is promoted at high temperatures. Mathematically, this situation is reflected by the temperature dependence in the denominator of eq. (4.14a). If χH > 0, then we add something positive to χs, such that our total χ gets larger; this is bad for mixing, as χ should be small for that, smaller than a certain critical limit (which we will discuss in the next section). As T reduces the χH contribution by its position in the denominator, however, this unfavorable part of χ is attenuated if T is high. In the opposite case, usually encountered in polar, protic systems, χH can be negative, that is, χH < 0. This situation is encountered if we have specific intercomponent interactions, such as hydrogen

 A quantitative justification for χH > 0 in systems that prefer intra- over intercomponent contact is given in Footnote No. 58 in Section 4.2.1.

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bonds or dipole–dipole interactions. These transient bonds favor the mixing, but they are broken at high temperature. Mathematically, we can see this again from the temperature dependence in the denominator of eq. (4.14a). If χH < 0, then we add something negative to χS, such that our total χ gets smaller; this is good for mixing. As T reduces the χH contribution by its position in the denominator, however, this favorable part of χ is attenuated at high T. Figure 50: Poly(N-isopropylacrylamide) is a polymer with a largely unpolar backbone and an unpolar side group that both dislike water, whereas a polar amide group in between can promote mixing with water due to hydrogen bonding. The hydrogen bonds, however, are broken at high temperature, and the polymer then precipitates from the solution.

A prime example of the latter case is an aqueous solution of poly(N-isopropylacrylamide), pNIPAAm, whose structure is depicted in Figure 50. This polymer has a largely unpolar backbone and an unpolar side group that both disfavor interactions with water, but it also has a polar amide group that can form hydrogen bonds with water molecules. This promotes mixing overall, but only up to a temperature of 32 °C. At higher temperatures, the hydrogen bonds break and the polymer precipitates, because then, the so-called hydrophobic effect comes into play: Water molecules that find themselves in close proximity to hydrophobic domains, such as the isopropyl moieties in pNIPAAm, form clusters and therefore assume a more ordered intermolecular arrangement than they would have in their natural state, because in that ordered arrangement they avoid unfavorable hydrophobic–hydrophilic contact. This effect causes an additional unfavorable excess mixing entropy ΔSmix, excess < 0. With increasing temperature, we find that the contribution of this excess entropy to ΔGmix, excess , which is − T · ΔSmix, excess , increases linearly, and therefore the excess free enthalpy of mixing ΔGmix, excess can compensate the usual configurational Flory–Huggins free enthalpy of mixing, eventually leading to ΔGmix, total > 0, thereby inducing phase separation. Such a solubility-temperature barrier is called Lower Critical Solution Temperature, LCST. There is a whole class of LCST-type polymers such as pNIPAAm, all of which exhibiting LCSTs reasonably close to the human body temperature.52 This makes these polymers interesting for biomedical applications, possibly in targeted

 This is because hydrogen-bonding interactions have a strength of some few singles to tens of RT, which is easily activated to open in the temperature window of 30–50 °C. Nature uses the same principle for immune reactions (e.g., for fever, which denaturates hostile proteins by breaking their hydrogen-bonding interactions) or to bind and unbind functional entities on demand, such as the double strands of DNA.

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drug-release systems. For example, consider that inflammatory tissue has a little higher temperature than healthy tissue. If an LCST polymer is designed such to display a coil-to-globule (i.e., swollen-to-deswollen) transition right in the range of that temperature difference, a nanocapsule system may be made from that polymer such that it can collapse and thereby release an anti-inflammatory drug only in the inflammatory tissue regions, but not in the healthy ones. Furthermore, we may expect that protonation or deprotonation of these polymers may further drastically affect their LCST, and with that, their regions of solubility and nonsolubility in water. As a result, we may also use these polymers for pH-dependent active systems, such as nanocapsules that release drugs only in cancer tissue but not in healthy tissue, making use of the circumstance that there is a pH difference between these two kinds of tissues. As a closing remark, note that in addition to the temperature dependence, the Flory–Huggins parameter also often exhibits a concentration dependence that may be captured in a virial-series form: χ = χS +

χH + χ 1 ϕ + χ 2 ϕ2 +    T

(4:15)

This does even more underline our above statement that χ is not a unique number, but a phenomenological parameter that itself depends on multiple influences, the most relevant being temperature (variable T in eq. (4.15)), composition of the system (variable ϕ in eq. (4.15)), and chain length (not included in eq. (4.15); there could be further terms in it to account for that variable as well).

4.1.4 Microscopic demixing We have seen that the entropic gain of a polymer–polymer blend is negligible. As a consequence, it is almost impossible to dissolve one polymer in another. Even when the polymers are almost chemically identical, the enthalpic penalty is still high enough to cause demixing.53 This immiscibility has severe consequences for systems where two polymers are chemically connected to one another, as it is the case in block copolymers. In these systems, the two blocks usually want to demix, but they can do so only on nanoscopic scales, because they are tied together. As a result, the block-copolymer system will show microphase separation, with a morphology that depends on the block lengths, as shown in Figure 51. At equal block length, shown in the center part of Figure 51, both blocks will arrange themselves in lamellar phases, whereas if one of the blocks is shorter, it will undergo coiling while the  Polystyrene and deuterated polystyrene are a prime example of this. Both are chemically almost the same (they just differ by exchange of hydrogen for deuterium), but they still have a positive, albeit very small, Flory–Huggins parameter, χ = 10–4. This causes both polymers to be immiscible if their molar mass is greater than about M = 3 × 106 g·mol–1.

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125

segments of the longer block will be less coiled, thereby changing the lamellae thickness. If the block-length mismatch gets too severe, different other morphologies will result, as shown in the left and right outer parts of Figure 51. These have astonishing order on both microscopic and mesoscopic scales, which is determined by the mutual pinning of locally microseparated phases by chain-blocks that partition in each of them. The overall phase microstructure therefore results from the length, degree of coiling, and extent of partition of either chain block. The astonishing order of the different morphologies shown in Figure 51 is therefore not due to spontaneous selfassembly of the polymers to such states, but instead, due to the hindered disassembly of the immiscible polymer blocks that results from their connectivity.

Figure 51: Various block copolymer phases that can form depending on the fraction of the constituent blocks A and B.

Note that a stark contrast to block copolymers is random or alternating copolymers. These copolymers manage to incorporate two chemically different building-block species along their chains. As such, they can serve to incorporate both these components in one polymeric system, without chain–chain immiscibility issues. This circumstance makes random or alternating copolymerization so attractive, as it is often the only way to realize such a material combination, due to the foresaid extreme difficulty of realizing polymer blends. So to speak, in a random or alternating copolymer, the mixing of the different chemical species is done “along the chain backbone”.

4.1.5 Solubility parameters The Flory–Huggins parameter quantifies polymer–solvent interactions; it does so, however, for pairs of these only, such that it is somewhat unhandy to tabulate it. More practical would be pairs of parameters, one for the polymer and one for the

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solvent, that can be tabulated independently and then taken together to appraise the miscibility of both compounds. Such parameters have been introduced by Hildebrandt and Scott. They are based on quantifying the attractive interactions of the solvent molecules or the monomer units of a polymer chain amongst themselves via their cohesive energy, ΔEA, which can be determined by measurement of the heat of combustion. With that, Hildebrandt and Scott defined rffiffiffiffiffiffiffiffi ΔEA δA = (4:16) vA Here, δA is the solubility parameter for compound A, ΔEA is its cohesive energy, and vA its molecular volume. In the lattice model that we have discussed above, the interaction energy for a z ΔEA and thus, given lattice site in a plain-A system is uAA , which is equal to − v0 vA 2 according to eq. (4.16), also equal to = − v0 δA 2 : z ΔEA = − v0 δA 2 uAA = − v0 vA 2

(4:17a)

The interaction energy of the second component in a plain-B system is calculated analogously z ΔEB = − v0 δB 2 uBB = − v0 vB 2

(4:17b)

By the geometric mean of both parameters, δA and δB, we can estimate the mutual interaction energy, uAB z uAB = − v0 δA δB 2

(4:17c)

Strictly speaking, the solubility parameter as defined above exists only for liquid substances, that is, only for solvents, but not for polymers, as the latter do not exhibit combustion. It can be presumed, though, that this is only due to the covalent connectivity of the monomers, such that if this connection would be lost, they would combust just as an own independent substance composed of molecules with a structure that matches the one of the repeating units in the polymer. Thus, the solubility parameter of the polymer matches the one of such a hypothetic fluid.54 In other words, although the solubility parameter of a solvent can be measured directly from its heat of combustion, the one for a polymer can be appraised by finding a fluid with a best identical structure to the monomer units in the polymer and

 Based on our classification in Section 3.2, that fluid would be an athermal solvent for the polymer.

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then estimate the solubility parameter of that fluid. As a practical alternative, we may also just seek a fluid that best dissolves the polymer, as assessed by maximum coil expansion in that solvent, which we may determine by light scattering or viscometry, and then presume this polymer–solvent mixture to be an athermal one with interaction energies of kind uAA = uBB = uAB, such that we can set the solubility parameter of the polymer practically equal to that of the solvent. The Flory–Huggins parameter, χ, is related to the solubility parameters by χ=

v0 ðδA − δB Þ2 kB T

(4:18)

With that equation, χ can be directly calculated from the δ-parameters of a polymer and solvent of interest. Solubility parameters therefore have the advantage that they can be used singularly for these two components and do not have to be specified for each possible pair, as it is the case with the Flory–Huggins parameter. In this way, lists such as Table 9 can be compiled that collect many different solubility parameters, and these collections can then be used to select a good solvent for a given polymer. For this purpose, we just have to choose a solvent with a solubility parameter that is similar to that of the polymer. This even works for solvent mixtures, because the net-δ value is the average of its constituents. In that way, solvent mixtures tailored to the polymer’s solubility parameter can easily be prepared.55 Table 9: Solubility parameters according to Hildebrandt and Scott for some typical solvents and polymers. Solvent Cyclohexane Benzene Chloroform Acetone Methanol Water

δ (cal·cm–)/ . . . . . .

Polymer

δ (cal·cm–)/

Polyethylene Poly(vinyl chloride) Polystyrene Poly(methyl methacrylate) Polyamine  Polyacrylonitrile

. . . . . .

The unit for the solubility parameter, (cal·cm–3)1/2, is commonly named 1 Hildebrandt.

We see that χ can only be positive according to eq. (4.18). That implies perfect solubility of the two components when δA = δB, and thus χ = 0, according to the proverb “similia similibus solvuntur”, meaning “like dissolves like”. Note that δ-values are large for polar solvents, because these have high cohesive energies ΔEA. Such polar

 Most interestingly, even mixtures of two nonsolvents can constitute a solvent for a polymer with a δ in the middle range of Table 9, namely if the mixture brings together one nonsolvent with a too large δ and another nonsolvent with a too small δ, whose average will then lie in the middle and thereby match the δ of the polymer.

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solvents actually require a three-dimensional solubility parameter that takes into account possible secondary interactions:  1=2 δ3d = δvdW 2 + δDipole − Dipole 2 + δH − Bonds 2

(4:19)

Questions to Lesson Unit 7 (1) Which one of the following is true for an ideal mixture? a. The mixing process is entropically controlled, since the mixing of two components A and B increases entropy, whereas in view of energy, the interactions of type A–B are inferior to those of types A–A and B–B. b. The mixing process is purely entropically controlled, since the mixing of two components A and B increases entropy, whereas in view of energy, the interactions of type A–B are equal to those of types A–A and B–B. c. The mixing process is entropically controlled, since the mixing of two components A and B increases entropy, whereas in view of energy, the interactions of type A–B, despite being favorable, contribute less than the entropy. d. The mixing process is enthalpically controlled, since the interaction energies of two miscible species are significantly larger in magnitude than the extent of entropy increase that results from their mixing. (2) What is meant by a “regular solution”? a. A solution in which particles of component A and particles of component B are not randomly distributed and the interaction energies between A and B differ. b. A solution for which H E ≠ 0 and SE ¼ 0 holds. c. A solution for which V E ≠ 0 holds, that is, there is an excess volume when mixing. d. A solution under thermodynamically regular conditions (standard pressure and standard temperature). (3) What are the differences between the mixing process of polymers and the mixing process of low-molecular-weight compounds? a. There are no differences; both mixing processes are exclusively entropically driven. b. There are no differences; both mixing processes are mainly enthalpically driven.

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129

c. The mixing process in polymers is more entropically driven than that in low-molecular-weight compounds, because the larger molar mass means that entropy is more strongly involved in the mixing process. d. The mixing process in polymers is less entropically driven than that in lowmolecular-weight compounds, because the given arrangement of the monomers in chains limits their freedom of arrangement in a mixture. (4) Which of the following is not true for the entropy of mixing? a. It can be calculated with the help of the Boltzmann formula. b. Its values do not differ for a polymer solution on the one hand and a regular solution on the other hand; the decisive factor is the presence of a solution. c. Its value is always positive, that is, the entropy is always in favor of a mixture. d. For polymers, it is very low, which is one way to explain why many ordered structures in nature are based on polymers. (5) What does the Flory–Huggins interaction parameter χ indicate? a. The difference of the pairwise interaction energies before and after mixing, normalized to kB T and the lattice geometry. b. The ratio of the pairwise interaction energies before and after mixing, normalized to kB T and the lattice geometry. c. The difference of the pairwise interaction energies before and after mixing, normalized to RT and the number of all pairs. d. The ratio of the pairwise interaction energies before and after mixing, normalized to RT and the number of all pairs. (6) Which of the following is not true for the Flory–Huggins interaction parameter χ? a. χ describes the energy balance of a mixing process, comparable to the energy balance of a chemical reaction. b. χ can take values below, above, and of exactly zero c. χ usually has a value of zero. d. A value of χ ¼ 0 is achieved only in an ideal mixture. (7) The Flory–Huggins theory does not pick up all entropic and enthalpic aspects. If one includes additional entropic and enthalpic aspects, the Flory–Huggins parameter results as a function, which can be split into an entropic part χS and an enthalpic part χH : χ ¼ χS þ ðχH =T Þ. What are the consequences with respect to χH for a nonpolar system and a polar-protic system, respectively? a. In the nonpolar system, intramolecular attractive interactions dominate, which can be formed more strongly at higher temperatures because of the better mobility of the molecules, and thus mixing is inhibited.

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b. In the nonpolar system, intramolecular attractive interactions dominate; these are temperature-independent, in contrast to the entropic part, so that miscibility depends only on the entropic part. c. In the polar-protic system, intercomponent interactions dominate, favoring or inhibiting mixing depending on the interaction type. d. In the polar-protic system, intercomponent interactions dominate; the resulting bonds are broken at high temperatures and thus mixing is favored only at low temperatures. (8) The effect outlined in the previous question for nonpolar polymers in polarprotic solvents such as water is also called the hydrophobic effect. This can also be viewed from the entropic side. Which statement correctly describes this? a. Since there is no interaction partner for the water molecules in case of dissolution of a hydrophobic species, they distribute themselves as widely as possible to maximize their entropy. This makes the dissolution process of the existing polymer more difficult – especially at higher temperatures. b. The water molecules near the contact surfaces of the polymer are more ordered than “free” water molecules. To minimize these contacts, entropy causes the formation of polymer globules, which becomes more pronounced at increasing temperature. c. At high temperatures, the system strongly tends to maximize its entropy. In principle, this is only possible in the state where polymer and solvent are separate. d. To maximize entropy, existing hydrate shells are preferentially broken at high temperatures to obtain more possibilities in the arrangement of water molecules and thus higher entropy.

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131

4.2 Phase diagrams LESSON 8: PHASE DIAGRAMS Based upon the Flory–Huggins theory introduced in the last lesson, the following will show at what combinations of relevant variables, mostly temperature and composition, a polymer will or will not mix with a solvent or with another polymer. A plot of these parameters gives a phase diagram, of which the following lesson will introduce two fundamentally different types for polymer systems.

The Flory–Huggins theory has given us conceptual and quantitative insight about the miscibility of polymers with solvents or with other polymers. In the following, we want to expand this insight to make even more detailed quantitative predictions at what conditions polymer–polymer and polymer–solvent systems are miscible or immiscible. In other words, we want to construct phase diagrams. A phase diagram is a plot of two practically relevant variables against each other such to create a map with regions that show at what combination of these variables the system is in what state. You may remember the most simple case of a phase diagram from your elementary physical chemistry classes: a one-component phase diagram, which is a map of pressure versus temperature that delimits at which p–T pair the substance is a solid, a liquid, or a gas. This map also delimits in which regions we have coexistence of two or even three of these phases. In general, the construction of a phase diagram is based on seeking the minimum of the free energy for each region in the map. In the following, we want to derive such a map for two-component polymer mixtures; thus, one of our two practically relevant variables will be the system’s composition, that is, the volume fraction of the polymer of interest in the mixture, ϕ. The other variable will be a measure of the energetic interactions between the components. Naturally, this variable is χ but as we will see later, that can actually be translated easily to an even more practical variable, T.

4.2.1 Equilibrium and stability The mixing and demixing of a system with a component A at the initial composition ϕ0 is determined by the free energy of its mixed phase, Fmix(ϕ0), and the free energy of the separated phases α and β, Fαβ(ϕ0). When Fmix(ϕ0) < Fαβ(ϕ0), then the mixture is stable; in this case, when brought together, both components will mix spontaneously and stay mixed. By contrast, when Fmix(ϕ0) > Fαβ(ϕ0), then the mixture is unstable and will phase-separate, or the components would not even mix in the

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first place. To determine this ratio, we need to determine functions that describe Fmix(ϕ0) and Fαβ(ϕ0). In the phase-separated state, an initial system composition ϕ0 is constituted by ϕ0 = fα ϕα + fβ ϕβ

(4:20)

Here, fα is the relative fraction of phase α, and ϕα is the volume fraction of component A in that phase α. Analogously, fβ is the relative fraction of phase β, and ϕβ is the volume fraction of component A in that phase β. The energy of the phase-separated state, Fαβ(ϕ0), is a simple sum of similar kind: Fαβ = fα Fα + fβ Fβ . By combination with eq. (4.20), we get     ϕ β − ϕ 0 F α + ϕ 0 − ϕ α Fβ Fαβ = fα Fα + fβ Fβ = (4:21) ϕβ − ϕα This expression has the form of a linear equation and gives a straight line when plotted with ϕ0 not as a single composition point but as variable in the interval [ϕα; ϕβ]. The energy of the mixed state, Fmix(ϕ0), can be estimated based on the energy change upon mixing, as expressed by the Flory–Huggins formula from the last chapter (eq. 4.13). We substitute ϕA = ϕ and ϕB = 1 – ϕ and get

ϕ ð1 − ϕÞ  ΔFmix = kB T ln ϕ + lnð1 − ϕÞ + χϕð1 − ϕÞ (4:22) NA NB This expression has the form of a curve when plotted.56 Both functions, Fmix(ϕ)57 and Fαβ(ϕ), are shown in Figure 52 for the case of a system where demixing is more favorable (Panel A) and one where mixing is more favorable (Panel B). We can see for the phase-separated system in Panel (A) that the curve of the mixed state at composition ϕ0, where we have a value of Fmix, lies above the straight line of the demixed state, where we have a value of Fαβ. Thus, when mixed initially, the system can spontaneously lower its energy by phase separation and thereby drop from Fmix to Fαβ at the composition ϕ0. We will then have two separate phases, α and β, with separate free energies Fα and Fβ, which have an

 The energy change expressed by eq. (4.22) adds upon the initial state, which is the phaseseparated one at an energy of Fαβ(ϕ0), when mixing occurs, with either positive or negative sign depending on whether the mixing is disfavorable (positive sign) or favorable (negative sign). After the mixing, we therefore have a free energy of Fmix(ϕ) = Fαβ(ϕ) + ΔFmix(ϕ).  This is the function just “derived” in the latter footnote: Fmix(ϕ) = Fαβ(ϕ) + ΔFmix(ϕ). It consists of a summand ΔFmix(ϕ), which is the Flory–Huggins formula (4.22), which is added to (and therefore “sits upon”) the straight line of the phase-separated state, Fαβ(ϕ) (4.21). At conditions for favorable mixing, the curve lies below the straight line as the ΔFmix(ϕ)-summand has negative values, and vice versa.

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133

Figure 52: Free energy, F, as a function of the volume fraction of a component of interest, ϕ, in mixed or phase-separated polymer systems. In both plots, the straight line corresponds to the phase-separated state, whereas the curve corresponds to the mixed state (it represents the Flory–Huggins equation). In (A), the concave curve lies above the straight line in a region between compositions ϕα and ϕβ, meaning that the mixed state is less favorable than the phase-separated state there. As a result, a mixed system that has a composition ϕ0 will spontaneously decompose into separated phases α and β, in which the component of interest is present in volume fractions ϕα and ϕβ, respectively. In (B), the convex curve lies below the straight line in a region between compositions ϕα and ϕβ, meaning that the mixed state is more favorable that the phase-separated state. Pictures redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

average value of Fαβ. The straight line that denotes the demixed state in Figure 52(A) connects these points, whose abscissa values denote the volume fractions of our component of interest A in the two phases, which are ϕα in the α-phase and ϕβ in the β-phase. The relative fractions of the two phases are given by the length of the two parts of the straight line left and right of the point at (ϕ0;Fαβ), commonly referred to as the lever rule. In full contrast to all that, we can see for the mixed system in Panel (B) that the curve of the mixed state at composition ϕ0, where we have an energy of Fmix, lies below the straight line of the demixed state, where we have an average energy of Fαβ at the composition ϕ0, which is actually a mean of two separated phases α and β with separate free energies Fα and Fβ and volume fractions of our component A of ϕα in the α-phase and ϕβ in the β-phase. In that situation, the system can spontaneously lower its energy by mixing and drop from the Fαβ(ϕ0) line to the Fmix(ϕ0) curve. We can delimit the two different scenarios in Figure 52 by analyzing the curvature of the ΔFmix-curve. This can be calculated by the second derivative of ΔFmix. In case of the phase-separated system, the curve has a concave curvature. This means  that the second derivative of ΔFmix is negative, ∂2 ΔFmix ∂ϕ2 < 0. The mixed state, by contrast, exhibits a convex curvature, and the second derivative of ΔFmix is positive,  ∂2 ΔFmix ∂ϕ2 > 0.

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Let us look at some boundary-case examples to understand this better. For an ideal mixture, the enthalpic contribution is zero, that is, ΔHmix = 0 or ΔUmix = 0. In that case, only the temperature-dependent entropic contribution remains (see eq. 4.1). The second derivative of ΔFmix then is

 mix ∂2 ΔF ∂2 ΔSmix 1 1 >0 = −T = kB T + NA ϕ NB ð1 − ϕÞ ∂ϕ2 ∂ϕ2

(4:23)

The outcome is a positive value at any ϕ, which means that ΔFmix(ϕ) has a convex curvature. Hence, mixing is always favorable, which is reasonable: entropy always favors mixing independent of the mixture’s composition. In an opposite example, let us ignore the entropy term and only look at the energetic contributions. The second derivative of ΔFmix then is  mix ∂2 ΔF ∂ϕ

2

=

 mix ∂2 ΔU ∂ϕ2

= − 2χkB T

(4:24)

This represents the findings that we have derived in the last chapter. When χ < 0, which according to our discussion in Section 4.1.2 denotes that mixing is favored, then the curvature is convex, which also denotes mixing to be favored according to what we have just said above. When χ > 0, which according to our discussion in Section 4.1.2 denotes that mixing is disfavored, then the curvature is concave, which also denotes mixing to be disfavored according to what we have just said above.58 In an actual mixture, both the energetic and the entropic term are of course relevant:  mix ∂2 ΔF ∂ϕ2

=

 mix ∂2 ΔU ∂ϕ2

−T

∂2 ΔSmix ∂ϕ2



1 − 2χkB T = kB T + NA ϕ NB ð1 − ϕÞ 1

(4:25)

Let us look at how the curves in an F(ϕ) diagram look like in that situation.

58 We may get a further insight when we write out the temperature-dependent form of χ, namely χ = χS + χH/T, in eq. (4.24):  mix ∂2 ΔF ∂ϕ2

=

 mix ∂2 ΔU ∂ϕ2

 χ  = − 2χkB T = − 2 χS + H kB T = − 2kB TχS − 2kB χH T

 mix ∂2 ΔU  mix ∂2 ΔF For T → 0, this leads to = = − 2kB χH def − z · ð2uAB − uAA − uBB Þ 2 = ∂ϕ ∂ϕ2 From this, we see that uAA + uBB , which corresponds to an unstable mixture, then we have χH > 0 – if uAB > 2 uAA + uBB – if uAB < , which corresponds to a stable mixture, then we have χH < 0 2

4.2 Phase diagrams

135

Figure 53 displays an example of an unsymmetric polymer blend (NA ≠ NB). Shown is the change of the free energy of mixing, ΔFmix, as a function of the volume fraction of component A, ϕ, at different temperatures. At high temperature, the entropy term dominates and creates a global minimum. The curve is convex allover, which means that mixing is favored at all compositions. At low temperature, by contrast, the energy term that usually disfavors mixing creates a miscibility gap. In this region, the curve is locally concave, and the system can decrease F by phase separation down to a value represented by the straight line that connects the energy minima. This line is the common tangent, because that kind of line is the deepest possible one touching the curve such that ∂F/∂ϕ is the same for the curve and the straight line in the two phase-coexistence points, respectively. This must be the case, as ∂F/∂ϕ corresponds to the chemical potential, which must be the same in the coexistence points ϕ′ and ϕ″ to have equilibrium there.

Figure 53: Free energy of mixing, ΔFmix, as a function of the volume fraction of a component of interest, ϕ, in a polymer system at different temperatures. At high temperature, dominance of the entropy term creates a distinct minimum, thereby favoring mixing at all compositions. At low temperature, the energy term that usually disfavors mixing creates a miscibility gap at intermediate compositions.

According to this latter thought, the common tangent delimits the stable “rim-ϕ regions” from the unstable “mid-ϕ region”. However, a “bad” concave curvature is present actually only in the innermost-ϕ region, between the inflection points. Hence, we can delimit three different regimes. First, we have the rim-ϕ regime beyond the common-tangent touching points. This corresponds to two regions, one at the left ϕ-rim and the other at the right ϕ-rim, that both enable stable mixtures. In contrast, the second, innermost-ϕ regime between the inflection points delimits the region in which the mixture is truly unstable. The third domain lies between these two extremes, and is called the metastable domain. Here, we have a convex curvature, indicating favorable mixing, but we are already above the deepest possible straight line, the common tangent, which indicates unfavorable mixing.

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Figure 54: The miscibility gap is further subdivided into unstable and metastable domains. (A) At the edges of the miscibility-gap region, the curvature of the F(ϕ)-curve is convex, which would correspond to a stable mixture, but we still have Fmix > Fphase-sep. there. As a result, this is a metastable domain. (B) The separation between the single phase and the metastable regime is called the binodal line, whereas the separation between the metastable and the unstable regime is called the spinodal line. Both these lines coincide at the critical point (χc;ϕc).

We can take a closer look at the miscibility gap by considering a symmetric polymer blend with two components that have the same N, as shown in Figure 54(A). Between the inflection points, that is, for the volume fractions ϕsp1–ϕsp2, the curva   ture is concave ∂2 ΔFmix ∂ϕ2 < 0 . The mixture is unstable, and even smallest fluctuations induce phase separation, which is called spinodal decomposition. Between the inflection points and the common-tangent touching points,59 for the volume frac   tions ϕ′–ϕsp1 and ϕsp2–ϕ″, the curvature is convex ∂2 ΔFmix ∂ϕ2 > 0 , but still Fmix (ϕ0) > Fαβ(ϕ0). In this regime, the mixture is metastable, which means that it is locally stable against small fluctuations, whereas phase separation will set in at larger fluctuations. In other words, phase separation needs nucleation and growth here. Figure 54(B) shows the same system in a plot of the Flory–Huggins parameter, χ, as a function of the polymer volume fraction, ϕ. This is a phase diagram. At low χ, the components like or at least tolerate each other, and mixing is possible at all compositions. To the contrary, at high χ, the components do not like each other,

 In the case of a symmetric blend, the common tangent is a horizontal line with slope zero, so its touching points with the ΔFmix(ϕ)-curve are the minima of that curve.

4.2 Phase diagrams

137

such that demixing will occur at certain compositions, delimited by the same boundaries as just discussed for the F(ϕ)-plot above. The point where this can first happen is called the critical point at χc;ϕc.60 The regions of nonstability are delimited by two lines. The binodal line separates the stable from the nonstable domain. The nonstable domain itself is composed of the unstable and the metastable regime, which are delimited from each other by the spinodal line.

4.2.2 Construction of the phase diagram So far, we have derived a diagram of F versus ϕ with a set of T-dependent curves to estimate at what composition, at a given temperature, the system is mixed or demixed. More intuitive, though, would be a diagram with two practical variables, T and ϕ, in which we can directly map domains of miscibility and immiscibility. To construct this, we take two steps. First, we derive a χ–ϕ diagram as plotted in Figure 54(B). Second, we derive a T–ϕ diagram from the χ–ϕ diagram by taking into account the temperature dependence of the Flory–Huggins parameter χ. We begin by calculating the binodal and spinodal lines of the first diagram. As a toolkit, we need the energy of mixing given by

mix = kB T ϕ ln ϕ + ð1 − ϕÞ lnð1 − ϕÞ + χϕð1 − ϕÞ (4:26a) ΔF NA NB its first derivative given by

 mix ∂ΔF ln ϕ 1 lnð1 − ϕÞ 1 + − − + χð1 − 2ϕÞ = kB T ∂ϕ NA NA NB NB

(4:26b)

and its second derivative given by

 mix ∂2 ΔF 1 1 = k T + − 2χ B NA ϕ NB ð1 − ϕÞ ∂ϕ2

(4:26c)

The phase boundary, or the binodal line, is given by the common tangent. For simplicity, we consider a symmetrical blend with NA = NB = N. In this case, the common tangent is a horizontal line with slope zero. This can be calculated by setting the first derivative of Fmix to zero:

 This point is conceptually identical to the critical point in a one-component phase diagram and the related van der Waals equation at (pc;Tc;Vc), at which liquefaction of a gas can first occur.

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∂ΔFmix ln ϕ lnð1 − ϕÞ ! = k T − + χ ð 1 − 2ϕ Þ =0 B ∂ϕ ϕϕ == ϕϕ′′′ N N

(4:27)

Rearrangement for χ yields the binodal χ(ϕ) curve:

χBinodal =



 ln



ϕ 1−ϕ

1 ln ϕ lnð1 − ϕÞ = − ð2ϕ − 1ÞN 2ϕ − 1 N N

(4:28)

The inflection points delimit the metastable from the unstable regime. They can be calculated by setting the second derivative of Fmix to zero:

 mix ∂2 ΔF 1 1 ! = kB T + − 2χ = 0 2 Nϕ N ð 1 − ϕ Þ ∂ϕ Rearrangement for χ yields the spinodal χ (ϕ) curve:

1 1 1 χSpinodal = + 2 Nϕ N ð1 − ϕÞ

(4:29)

(4:30)

The minimum of both the latter curves denotes the critical point, (χc;ϕc). This point marks the very first possibility for demixing. Below χc, mixing is possible for all volume fractions. We can calculate the critical point by setting the first derivative of the latter equation for the spinodal line to zero to estimate its minimum: ! ∂χSpinodal 1 1 1 ! + (4:31a) =0 = − ∂ϕ 2 Nϕ2 N ð1 − ϕÞ2 At this point of our discussion, we can actually drop our above simplification and consider the general case of a nonsymmetric blend again. This is mathematically imprecise, but rather based on a logical argument: the N in the first summand belongs to the volume fraction ϕ of compound A (note that in this section, we write ϕ for what we have earlier denoted as ϕA), whereas the N in the second summand belongs to the volume fraction 1 – ϕ (i.e., 1 – ϕ), which is that of compound B (because 1 – ϕA = ϕB). So, we get ! ∂χSpinodal 1 1 1 ! + (4:31b) =0 = − ∂ϕ 2 NA ϕ2 NB ð1 − ϕÞ2 Solving this equation for ϕ yields the critical composition, ϕc: pffiffiffiffiffiffi NB ϕc = pffiffiffiffiffiffi pffiffiffiffiffiffi NA + NB

(4:32)

4.2 Phase diagrams

139

According to that latter formula, a symmetric blend with NA = NB has a critical composition of ϕc = 0.5. This value is understandable, because at such a 50:50 composition, we have the greatest extent of disfavorable hetero contacts in our system, and so it should be that composition at which demixing is most likely to happen first. Precisely, this happens if χ is too unfavorable, meaning if the components are not compatible enough; and this is the case if χ lays below χc. That critical threshold χc can in turn be quantified by inserting ϕc for ϕ in eq. (4.30): χc =



2 1 1 1 pffiffiffiffiffiffi + pffiffiffiffiffiffi 2 NA NB

(4:33)

Let us examine the latter equation in some detail. In a regular small-molecule solution, where NA = NB = 1, the critical interaction parameter is χc = 2. As a result, everything with χ < χc = 2 will mix, whereas everything with χ > χc = 2 may demix if ϕ is around ϕc. In a polymer solution, with NA >> 1 and NB = 1, the picture is different. Here, χc = ½, which is much lower than in the small-molecule scenario. Here, everything with χ < χc = ½ will mix, whereas everything with χ > χc = ½ may demix if ϕ is around ϕc. A polymer solution in which χ is exactly ½ is in the Θ-state, which means at the borderline between miscibility and immiscibility. In a polymer blend, where both NA >> 1 and NB >> 1, the critical Flory–Huggins parameter is basically zero, χc ≈ 0. As a result, mixing is hardly possible at all in this case, as for mixing to occur, χ must be smaller than χc, which is hard to achieve when χc ≈ 0 already. Mixing two polymers therefore practically only works for χ = 0, which is very rare! We therefore have to resort to other strategies than mixing if the properties of different polymers shall be combined. A common way to achieve that is by copolymerization of their different monomers in one copolymer species (“mixing within the chain”), or by incorporation of attractive side groups in the to-bemixed polymers that aid to overcome the enthalpy penalty of mixing. As a numerical example, we calculate the volume fractions at the phase boundaries of the spinodal curve of a polymer solution at a degree of polymerization of NA = 1000 with a Flory– Huggins interaction parameter of χ = 1.5; hence, we consider a system beyond χc that already exhibits a decent two-phase regime. The spinodal curve separates the two-phase regime from the metastable coexisting regime. Mathematically, it is determined by the set of inflection points of the function ΔFmix(ϕA); these points can be calculated by setting the second derivative of this function to zero. The resulting polymer volume fractions then correspond to the polymer concentrations of the two separate phases that arise from the phase separation.61 For our example, the calculation goes as follows:

 Note that these volume fractions are temperature dependent due to the temperature dependence of the Flory–Huggins parameter χ(T), as will be discussed in the next paragraph.

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ΔFmix ϕA ϕ · ln ϕA + ϕB · ln ϕB + χϕA ϕB = A · ln ϕA + ð1 − ϕA Þ · lnð1 − ϕA Þ + χϕA ð1 − ϕA Þ = kB T NA NA The first derivate of that function is: ∂



ΔFmix kB T

∂ϕA

 =

1 ϕA 1 −1 · + · ln ϕA + ð1 − ϕA Þ · − lnð1 − ϕA Þ + χ − 2χϕA ð 1 − ϕA Þ NA ϕA NA

=

1 1 + · ln ϕA − 1 − lnð1 − ϕA Þ + χ − 2χϕA NA NA

We further calculate the second derivative, plug in the numbers from the example and set the entire equation equal to zero: ∂2



ΔFmix kB T ∂ϕA 2

 =

1 1 −1 1 1 1 ! · − + − 2χ = − 2 · 1.5 = 0 · NA ϕA ð1 − ϕA Þ 1000 ϕA ð1 − ϕA Þ

Further rearrangement yields:

1 1 1 · ð1 − ϕA Þ + ϕA − 3ð1 − ϕA ÞϕA = 0 , 3ϕA 2 + 1 − − 3 ϕA + =0 1000 1000 1000 , ϕA 2 − 0.667ϕA + 0.0003333 = 0 We have now generated a quadratic equation in its reduced form and can solve for its two solutions using the p–q formula: 0.667 ϕA = ± 2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi

0.667 2 0.667 − 0.0003333 = ± 0.333 = 0.6665 and 0.0005 2 2

As we can see from the resulting volume fractions, spinodal decomposition does in fact not produce pure solvent and pure polymer phases, but instead, a polymer-rich phase (here: 67% polymer) and a polymer-lean phase (here: 0.5‰ polymer) are obtained. This means that the phase containing most of the polymer still contains a lot of solvent (here: 33%), and thus needs to be properly dried to obtain the pure polymer.

We have learned in Section 4.1.3 that the Flory–Huggins parameter is not a constant, but rather a temperature-dependent function according to eq. (4.14a). This allows us to translate the χ–ϕ representations we have discussed so far (Figure 54 (B)) into more practical T–ϕ diagrams, as shown in Figure 55. These can have two very different appearances, depending on the sign of the χH-contributor, corresponding to two types of miscibility. In the first case, depicted on the left-hand side of Figure 55, χH > 0, which means that the overall χ parameter decreases as the temperature rises. These systems display a volume-fraction-dependent upper critical solution temperature (UCST), above which mixing is possible. In the second case, χH < 0, and the overall χ parameter increases with rising T. Such mixtures exhibit a volume-fraction-dependent lower critical solution temperature (LCST), above which mixing is impossible!

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141

Figure 55: The sign of the enthalpy-component of the T-dependent Flory–Huggins parameter determines whether mixing is favorable or disfavorable at high or low T, which are two opposite scenarios assessed by either an upper critical solution temperature (UCST), as shown on the left, or a lower critical solution temperature (LCST), as shown on the right. In both schematics, we see that there is also an impact of the molar mass of the chains.

A UCST is the usual case for many polymer–solvent mixtures without specific intercomponent interactions such as mutual hydrogen bonding, as we have discussed in Section 4.1.3. The exact value of the UCST is also molar-mass dependent, as can also be seen in Figure 55. In the UCST graph, we see that upon decrease of temperature, an accompanying decrease of the solvent quality such that χ exceeds ½, and with that, an onset of precipitation is encountered for long chains sooner than for short chains. Analogously, in the LCST graph, we see that upon increase of temperature, an accompanying decrease of the solvent quality such that χ exceeds ½, and with that, an onset of precipitation is encountered for long chains sooner than for short chains. In both cases, this circumstance can be used for fractionation of polydisperse samples, simply by dissolving such a sample in a solvent and then gradually worsening the solvent quality, either by suitable change of temperature or by successive addition of a nonsolvent. Upon that, first the longest chains precipitate, then the next shorter fraction, and so forth. Isolating these fractions before precipitating the next thereby allows the broadly polydisperse sample to be fractionated into several less polydisperse ones. This may either be used preparatively or at least analytically to determine a coarse-grained molar-mass distribution by analyzing the average

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molar mass and the amount of each fraction through means like light scattering, osmometry, and gravimetry. More complex phase diagrams may arise in special cases due to the sometimes complex temperature dependence of the Flory–Huggins interaction parameter that might change its sign multiple times. Figure 56 shows some of these special cases. Panel (A) is for a mixture with both a UCST and an LCST, Panel (B) is for a mixture with a closed miscibility gap, and Panel (C) displays an hourglass-shaped phase diagram. The first case has a rather simple explanation, shown in a χ–T representation in Panel (D). Initially, we see a regular Flory–Huggins-type UCST upon increase of temperature. On top of that, however, comes another effect: an increase of temperature often leads to thermal expansion of both the solvent and the polymer. This expansion is usually more pronounced for the solvent than for the polymer. Hence, in our lattice picture, we have to increase the lattice-site volume for the solvent more than for the polymer’s repeating units upon increase of temperature, meaning that we need to distort our lattice to fit them both on, which entails an entropy penalty. If this penalty is too marked, the system will phase-separate again, creating an LCST in the process. Hence, we actually always have both an LCST and a UCST, and depending on where they and their two binodal curves lie, we may have phase diagrams of kind (A), (B), or (C). The reason why we often nevertheless (seemingly) observe simple phase diagrams such as those shown in Figure 55, with just one critical temperature (LCST or UCST, but not both of them) is that many polymer–solvent systems actually exhibit both, but one of them is usually “hidden”, either in the form of a UCST that lies

Figure 56: Complex shapes of phase diagrams that result from the interplay of the T-dependent χparameter in the enthalpy term and the T-dependent TΔS term in the entropy term of the Flory–Huggins equation. Case (A) has an illustrative explanation, as visualized in (D): the usual UCST-type T-dependence of χ leads to a decrease of it upon raise of T (making mixing more favorable, or better saying “less disfavorable” at high T), whereas the different thermal expansion of the solvent and the polymer that comes along with a temperature increase causes energy and entropy penalties that oppose this; together, this creates a minimum of χ (T), corresponding to a closed miscibility gap in the phase diagram. The sketch in Panel (D) is inspired by J.M.G. Cowie: Chemie und Physik der synthetischen Polymeren, Vieweg, 1991.

4.2 Phase diagrams

143

below the melting point of the solvent, or in the form of an LCST that is located above the boiling point of the solvent.

4.2.3 Mechanisms of phase separation 4.2.3.1 Spinodal decomposition In the unstable two-phase regime (see Figure 54(B)), any random concentration fluctuation self-amplifies and eventually leads to macroscopic phase separation. This is because the gradient of the chemical potential μ is opposite to the gradient of concentration in this regime. In very general, concentration fluctuations in a system are equilibrated by a diffusive flux of matter along the gradient of the chemical potential μ, which is defined as μ = ð∂G=∂nÞ = ð∂G=∂ϕÞ and exhibits a monotonic concentration dependence in mixed phases: μmix = μ0 + RT ln ϕ. The volume fraction ϕ in the second equation necessitates a high chemical potential at high concentrations, and conversely, a low chemical potential at low concentrations. If     there is a gradient in the chemical potential, ∂μ=∂ϕÞ = ∂2 G ∂ϕ2 > 0, diffusion leads to an overall flux of matter from the higher to the lower potential until the gradient is equilibrated. Normally, such a chemical-potential gradient is in line with the concentration gradient, so the diffusive flux goes from high to low concentrations, referred to as downhill diffusion. In the two-phase spinodal decomposition regime, by contrast, the gradient of the chemical potential is reverse to that in     concentration, ∂μ=∂ϕÞ = ∂2 G ∂ϕ2 < 0. Diffusion still takes place along the chemical-potential gradient, but now this means it occurs against the concentration gradient. The resulting self-amplification of concentration differences is called uphill diffusion and eventually leads to a macroscopic phase separation, the so-called spinodal decomposition. This phenomenon occurs on the characteristic lengthscale of the concentration fluctuations in the system. If that lengthscale is large, diffusion has to cover large distances, which is unlikely; if the lengthscale is short, many new polymer–solvent interfaces must be created, which is disfavored, too. Therefore, spinodal decomposition occurs on an intermediate lengthscale that constitutes the best compromise between the two effects. This lengthscale remains constant at first and is amplified only in its amplitude. During later stages of the process, the lengthscale diverges, and the phase separation actually becomes macroscopic. This last phase can be observed, for example, by optical microscopy, if one of the components is colored. Earlier stages can be observed by complementary methods such as neutron or X-ray scattering. 4.2.3.2 Nucleation and growth In the metastable regime (again, see Figure 54(B)), the system is able to tolerate small concentration fluctuations, and thus, spinodal decomposition is prohibited. Here, a

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different phase-separation mechanism is in action: nucleation and growth. Instead of a macroscopic phase separation, small polymer-rich phase clusters form locally. As can be seen in Figure 57, these clusters need to grow to a specific minimal size with a radius r* for the nucleation process to be energetically favorable. This is because the unfavorable interfacial energy of the nuclei first is stronger than their favorable volume energy, but this ratio reverses from a cluster minimal size r* on. If the nuclei do not make it to that size, nucleation does not surpass the minimum size for macroscopic phase separation, so the system can remain two-phased. By contrast, once the minimal size r* is reached, macroscopic phase separation sets in. Therefore, impurities with a radius bigger than r*, such as dust particles, often function as a seed for nucleation. Alternatively, artificial nucleation seeds can be added on purpose to facilitate the process. This is called heteronucleation. Once the critical nucleation size is exceeded and a nucleus is formed, more material adds to it easily, and the new phase grows.

Figure 57: Graphical representation of the free energy of nucleation. Two factors contribute to the free energy: the interface and the volume of the nucleus. Nucleation is energetically favorable only after the cluster has grown to a minimal size r*.

Figure 58 summarizes the two polymer phase-separation mechanisms in view of the system’s state of stability or instability. In the unstable regime, spinodal decomposition is the mode of phase separation. In the metastable regimes, phase separation happens through nucleation and growth.

Questions to Lesson Unit 8

145

Figure 58: Schematic representation of phase separation regimes of polymer–solvent or polymer–polymer mixtures. Shown is the mixing free energy as a function of the polymer volume fraction, like in Figure 53. In the unstable regime, spinodal decomposition takes place. In the metastable regimes, phase separation occurs by nucleation and growth.

Questions to Lesson Unit 8 (1) What applies to phase diagrams of polymer mixtures or solutions? (In the following, both are referred to as a mixture) a. Like the phase diagram of a single component, in a two-component system, a phase diagram represents the state of aggregation, that is, whether it is a solid, liquid, or gas phase, in the form of a pressure versus temperature diagram. b. Like the phase diagram of a single component, in a two-component system, a phase diagram represents the mixing state, that is, whether it is a mixed or demixed system, in the form of a pressure versus temperature diagram. c. Unlike the phase diagram of a single component, in the case of mixtures, instead of pressure versus temperature, we plot the interaction energy in terms of χ, which can also be reformulated as a temperature dependence versus the composition of the mixture in terms of mole fractions xi. d. Unlike the phase diagram of a single component, in the case of mixtures, instead of pressure versus temperature, we plot the interaction energy in terms of χ, which can also be reformulated as a temperature dependence versus the composition of the mixture in terms of volume fractions ϕi .

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(2) Which statement applies to the state of a mixture in view of the free energy of mix ? mixing Δ F a. A concave curvature of the mixing energy curve always implies miscibility of the components. b. A concave curvature of the mixing energy curve always means nonmiscibility of the components. c. A convex curvature of the mixing energy curve always means miscibility of the components. d. A convex curvature of the mixing energy curve always means nonmiscibility of the components. (3) A phase diagram of a two-component system is classically represented in the form of a χ–ϕ diagram. This can be derived from the F–ϕ diagram. The χ–ϕ diagram then consists of ______ a. a curve connecting all minima of the curve in the F–ϕ diagram and a curve connecting all inflection points of the curve in the F–ϕ diagram. b. a curve connecting all maxima of the curve in the F–ϕ diagram and a curve connecting all inflection points of the curve in the F–ϕ diagram. c. a curve connecting all minima of the curve in the F–ϕ diagram and a curve connecting all maxima of the curve in the F–ϕ diagram. d. a curve connecting all extrema of the curve in the F–ϕ diagram and a curve connecting all inflection points of the curve in the F–ϕ diagram. (4) What is the critical point? a. The point in the phase diagram with values χc and ϕc at which segregation is first possible. b. The point in the phase diagram with values χc and ϕc at which the temperature is so high that mixing is no longer possible. c. The point in the phase diagram with values χc and ϕc at which it is no longer possible to distinguish whether it is a mixture or two demixed phases. d. The point in the phase diagram with the values χc and ϕc at which the metastable state transitions into the unstable state; accordingly, there are two critical points.

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147

(5) What do the terms LCST and UCST denote? a. The LCST is the temperature below which mixing is impossible; UCST is the temperature above which mixing is possible. b. The LCST is the temperature above which mixing is impossible; the UCST is the temperature above which mixing is possible. c. The LCST is the temperature below which mixing is possible; the UCST is the temperature above which mixing is impossible. d. The LCST is the temperature below which mixing is impossible; the UCST is the temperature below which mixing is possible. (6) Which statement regarding the metastability of a mixture is correct? a. A mixture that shows a greater tendency to phase separation at high temperatures is called metastable. b. A mixture is metastable if it remains mixed for only a limited time after mixing and then segregates. c. If the composition of a mixture is above a critical value ϕc , spontaneous segregation can always occur, and the mixture is metastable. d. If the free energy of mixing exhibits a convex curvature in its course but is energetically above the total free energy of the system, this region is called metastable, since phase separation can only occur through nucleation here. (7) What is the critical value of the Flory–Huggins parameter χc of a polymer solution? a. It is always 2. b. It is always 21. c. It is always 0. d. It depends on the system at hand. (8) For a polymer mixture, mixing is possible only when χ ≈ 0. To achieve this, what is true for the individual components? a. They must be as different as possible. b. They must be as similar as possible. c. Their χ-values must be as large as possible. d. Their χ-values must be as small as possible.

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4.3 Osmotic pressure LESSON 9: OSMOTIC PRESSURE You have got to know the Flory–Huggins parameter as an elementary tool to appraise whether or not polymers are miscible with solvents or other polymers. What you are lacking, though, is an experimental technique to determine this parameter. Such a technique will be introduced in the following lesson. You will see that in addition to quantifying interactions between a polymer and its surrounding, and therewith determining the Flory–Huggins parameter, this method also allows you to determine the molar mass of a polymer. Furthermore, you will again see the striking similarity of the physical-chemical concepts behind real polymer chains and real gases.

So far we have looked at polymer thermodynamics from a theoretical viewpoint and introduced the Flory–Huggins parameter χ as a simple means to quantify the polymer–solvent interaction and to predict the state of solvency of one in the other (meaning if it is a good-solution state, a θ-state, or a bad- or nonsolvent state). Earlier in this book, in Chapter 3, we have introduced a quantity that expresses pretty much the same: the excluded volume, ve. Now, we will take a look on how to determine χ and ve experimentally; this discussion will also show us how these two parameters are related to each other. Consider the experimental setup that is shown in Figure 59. This setup features a vessel containing a polymer solution immersed in a bath with excess solvent. Both are separated by a semipermeable membrane that allows for solvent interdiffusion but blocks polymer interdiffusion, simply because its pores are too tight for the large polymer coils to diffuse through, whereas the small solvent molecules can. As a result, since the polymer cannot diffuse out of its vessel to spread over the whole system (and thereby equilibrate its chemical-potential gradient), in turn, part of the solvent will diffuse into the polymer vessel such to dilute the polymer solution in there. This influx of solvent into the polymer chamber creates a mechanical pressure, in the present case by raise of a fluid column in a vertical tube. Once this mechanical pressure, p, is equal to the osmotic pressure of the solution, ∏, equilibrium is reached. At this state, the elevation of the fluid column in the vertical tube, Δh, has come to a final value. We may then write: Π=p=

m · g Vρ · g πr2 Δhρ · g = Δhρ · g = = Area Area πr2

(4:34)

4.3 Osmotic pressure

149

Figure 59: Experimental setup to determine the osmotic pressure of a polymer solution. The polymer cannot pass through the pores of a membrane separating it from excess surrounding solvent, whereas the solvent molecules can. As a result, influx of solvent into the solutioncontaining compartment will cause a mechanical pressure, for example, by rise of a fluid column in a tube. Equilibrium is reached when this mechanical pressure matches and balances the osmotic pressure of the solution. Picture inspired by B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

With eq. (4.34), we have a simple means to measure ∏, simply by measuring Δh. ∏, in turn, is a central parameter in the equation of state of an osmotic system, which is Van’t Hoff’s law for dilute solutions: Π · V = n · RT

(4:35)

Rearrangement yields a relation between the osmotic pressure, ∏, and the polymer mass-per-volume concentration, c: Π=

n · RT m RT c · RT RT · ) Mn = = = V Mn V Mn Π=c

(4:36)

With that, we can determine the number-average molar mass of our polymer simply by a concentration-dependent osmometric experimental series.62 Note that in the

62 In general, the number-average molar mass of a dissolved species can be determined by solutionbased effects whose magnitude depends on the number of dissolved molecules. Such effects are known as colligative properties (derived from Latin collegere = to collect), the prime examples of which being the osmotic pressure, the boiling-point elevation, and the freezing-point depression of a solution. All these effects have a magnitude that is proportional to the number of dissolved molecules, and therefore they allow the molar amount of the dissolved compound in the solution to be determined. If the solution has been made from a known mass of that dissolved compound, then dividing the one by the other gives the weight per mole, and therewith, the molar mass. We may express this general principle

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upper notation, the concentration c = m/V has the unit g·L–1, as it is common in polymer science. In classical physical chemistry, by contrast, concentration is commonly defined as n/V with the unit mol·L–1. In this form, Van’t Hoff’s law would read ∏ = c RT instead of the polymer-science variant noted in eq. (4.36). Van’t Hoff’s law is valid for dilute solutions. Nondilute solutions deviate from that simple relation and are best described by a virial series expansion of it

1 2 + A 2 c + A3 c + . . . (4:37) Π=c = RT Mn In the notation of eq. (4.37), the osmotic pressure is given in a reduced form, ∏/c.63 A key parameter in the above equation is the second virial coefficient, A2. It accounts for deviations from the ideal Van’t Hoff law due to interactions at higher concentration. Thus, A2 is an important parameter to quantify such interactions. Often, experimental data can be well fitted to eq. (4.37) in a linear form, where the series is expanded only up to the linear A2 term. Figure 60 sketches such a dataset and linear fit. From the intercept, RT/Mn, we get the polymer’s number-average molar mass, and from the slope, RT·A2, we get the second virial coefficient A2. The second virial coefficient quantifies the interaction between a polymer and a solvent. With that, it is closely connected to the Flory–Huggins parameter χ, as we will discuss in more detail below. Generally, a positive A2-value means good solvation quantitatively as follows. Let Y be a colligative property (i.e., one of the above three effects), whose extent is proportional to the number of dissolved molecules of species i per volume, Ni/V, with a conN

stant of proportionality K: Y = K Vi . If there are several different species in the solution, for example, P N several different molar masses of a polymer with a polydisperse distribution, then we get Y = K Vi i . In P P w NM polymer science, a typical measure of concentration is mass per volume, that is, c = Vi i = Ni Vi i . If A

we normalize our colligative property Y by the concentration c (see the next footnote for some backP N N V ground on this way of normalization), we obtain: Yc = K Vi i · P AN M . Together with the definition of i i i P NM i i i , this yields: Y = KNA . We see from that identity that all the number-average molar mass, Mn = P c Mn N i

colligative properties are connected to the number-average molar mass Mn, and therefore they all allow it to be determined experimentally. The extent of how strongly a colligative property Y depends on N is set by the constant of proportionality K. For the osmotic pressure, this constant is particularly high, so the effect is very strong for this kind of colligative property. This is the reason why it is favored as a method in polymer science: in a given mass-amount of polymer sample we only have a few molecules (but each one very large in turn), and so we need a colligative property where even such few (large) molecules have a strong effect that is well measurable. Osmotic pressure is such a favorable property. On top of that, there is another reason why this method is beloved by polymer scientists: in addition to Mn, it also gives quantitative information on the quality of polymer–solvent interactions. This is what the following discussion in this section is about.  A reduced quantity is one that is normalized by concentration. Other such normalized quantities are known to you from elementary physical chemistry already, for example, specific quantities (normalized by mass) or molar quantities (normalized by the molar number).

4.3 Osmotic pressure

151

Figure 60: Plot of the series expansion of the concentration-dependent osmotic pressure in its typical form.

of the polymer, whereas a negative A2-value means nonsolubility; in the Θ-state, A2 is zero.64 Figure 61 schematizes typical data for poly(methyl methacrylate) in various solvents as an example. The quality of these solvents can be determined by the absolute values of A2, and with that, by the slopes in the plot. m-Xylene turns out to be a Θ-solvent with a slope of zero. Dioxane has a positive linear slope and therefore positive A2, denoting it to be a good solvent. For the best solvent, chloroform, the concentration dependence of the osmotic pressure is even stronger, and it even deviates from the linear form, which makes a third nonlinear virial term necessary to fit the experimental data. Despite all these different slopes, however, the intercept is the same for all samples, as this is only dependent on the molar mass average, Mn, but not on the solvent quality.

4.3.1 Connection of the second virial coefficient, A2, to the Flory–Huggins parameter, χ, and the excluded volume, ve We have already hinted at the close interrelation between the second virial coefficient A2 to the Flory–Huggins interaction parameter χ, as both are a measure of polymer–solvent interactions. The following paragraph will examine this relationship and determine it quantitatively.

64 Again, we can make an analogy to a real gas, whose equation of state can be written by a virial B a series as p = RT  ð1 + V  + . . .Þ with the second virial coefficient B = b − RT , wherein a and b are the van V der Waals coefficients, with a accounting for attractive intermolecular interactions, and b accounting for the covolume and other repulsive interactions of the gas molecules. At the Boyle temperature, attractive and repulsive interactions are at balance, causing B to be zero and the virial series thereby to simplify to the ideal gas law. The real gas then displays pseudo-ideal behavior, analogous to polymers in a solvent at the Θ-temperature.

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Figure 61: Schematic of the concentration-dependent reduced osmotic pressure for poly(methyl methacrylate) in different solvents. It exhibits strongly differing slopes, corresponding to strongly differing second virial coefficients, which is due to the different solvent qualities for the polymer. Picture inspired by B. Vollmert: Grundriss der Makromolekularen Chemie, Springer, 1962.

As a starting point, we take the thermodynamic definition of the osmotic pressure:

 mix

∂ΔFmix n · ∂ΔF = − (4:38) Π≡ − ∂V ∂V n n A

A

We have seen in the last section that the second virial coefficient A2 is related to the reduced osmotic pressure, ∏/c. Hence, we must transform the above equation into a practical expression for the osmotic pressure as a function of concentration. To achieve that, we use the expression for ΔFmix that we have appraised with the lattice model of the Flory–Huggins mean-field theory in Section 4.1. This expression uses the volume fraction, ϕ, as a measure of concentration, which is connected to the total volume of the system, V, by ϕA =

nA NA v0 V

(4:39a)

Here, nA is the number of A-molecules on the lattice, NA the number of lattice sites per A-molecule, which is equal to the degree of polymerization of species A, and v0 the volume of a single lattice site. Rearranging eq. (4.39a) and expressing it in a differential form yields

1 nA NA v0 ¼ ∂ϕ (4:39b) ∂V ¼ nA NA v0 ∂ ϕ ϕ2 whereby the identity d(1/x) = –x–2 dx was used. We can insert this relation into the above expression for the osmotic pressure and further use the identity nϕA = nA NA => n = nA NA / ϕ to change the variable V to the variable ϕ:

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4.3 Osmotic pressure

Π≡

 1 0  ΔF

2 mix =ϕ ∂ n N ·  A A n · ∂ΔF mix ϕ A − =@ · ∂V n NA v0 ∂ϕ A n A

 1 0  2 ∂ ΔF mix =ϕ ϕ A =@ · v0 ∂ϕ

nA

nA

(4:40) The Flory–Huggins mean-field solution for the Free Energy of mixing is

ϕ ð1 − ϕÞ  ΔFmix = kB T ln ϕ + lnð1 − ϕÞ + χϕð1 − ϕÞ NA NB

(4:22)

We now have to differentiate this term. Unfortunately, this is difficult for the B component part of the equation, as our variable ϕ enters in a complicated form here. To simplify this point of our calculation, we use a series expansion according to ! ð 1 − ϕÞ 1 ϕ2 ϕ3 lnð1 − ϕÞ = −ϕ+ + + ... (4:41) 2 6 NB NB By plugging this into the term for ΔFmix, we generate !



2 3 ϕ 1 ϕ 1 ϕ mix = kB T ΔF + ln ϕ + ϕ χ − − 2χ + + ... 2 NB 6NB NA NB

(4:42)

This expression, in turn, can be inserted into our original formula for the osmotic pressure, eq. (4.40) !

k B T ϕ ϕ2 1 ϕ3 Π= + − 2χ + + ... (4:43) 2 NB 3NB v0 NA In the next step, we transform further by inserting the concentration as a variable according to c = ϕ/v0. This variable has the unit L–1; it expresses the number of chain segments per volume, that is, per liter:



c c2 1 + v0 − 2χ + . . . (4:44a) Π = kB T NA 2 NB Division of this formula by c generates the needed expression for the reduced osmotic pressure, ∏/c



1 c 1 + v0 − 2χ + . . . (4:44b) Π=c = kB T NA 2 NB When comparing the general form of the above formula to the one of eq. (4.37), we realize that they both are of kind

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Π=c = Thermal Energy

1 + Interaction Parameter · c Chain Length

(4:45)

Equation (4.44b) has a dimensionless first summand in the parentheses; this means that the second summand must be dimensionless, too. As a result, since c has the unit L–1, the interaction parameter in front of it must have a unit of liters, L. Hence, this parameter must be some kind of volume that quantifies the polymer–solvent interactions. A quantity doing this is the excluded volume, ve, which therefore serves as the second virial coefficient in eq. (4.44b):

1 − 2χ (4:46a) ve = v0 NB Here, NB is the degree of polymerization of the second component, which is either a solvent or a second polymer. In a polymer solution, we have NB = 1. When we now plug in the Flory–Huggins parameter for the Θ-state, χ = ½, then we end up with an excluded volume of ve = 0, exactly as the definition of the Θ-state demands. For the case of an athermal solvent, by contrast, χ = 0, and as a result, ve = v0. The excluded volume is then maximally positive and coincides with the monomer-segmental volume, as we have already seen in Section 3.2. In a mixture of two different polymers, a polymer blend, we have NB >> 1. A Flory–Huggins parameter close to zero, χ ≈ 0, which is needed to successfully blend two polymers, now leads to an excluded volume also close to zero, ve = v0/NB ≈ 0. This shows us that, even though the A–B interaction in a polymer blend is very good (it must be so to mix!), the chains adopt their ideal conformation, as they do in the Θ-state. We can further express the above equation as ve =

v0 l3 − 2v0 χ = − 2l3 χ NB NB

(4:46b)

Here, l3/NB is the volume of the polymer segments (normalized to the degree of polymerization of the solvent), which corresponds to the hard-sphere branch of the Mayer f-function. 2l3χ is a measure of the effective segment–segment attraction, which corresponds to the potential-well part of the Mayer f-function. The concert of both determines the excluded volume (remember that this volume is in fact calculated from the negative integral over the Mayer f-function). We now see once more that for the Θstate (ve = 0), coil expansion due to the co-volume of the segments just exactly balances the coil contraction due to the net-attractive segment–segment interaction. So far, we have expressed the reduced osmotic pressure on the molecular level, meaning we use kBT for the thermal energy and a concentration c = ϕ/v0 with the unit L–1 to account for the number of molecules per volume. To make this more practical, we can transform eq. (4.44b) to molar values by using RT for the thermal energy and c = ϕ/vspecific,polymer with the unit g·L–1 as the concentration. For a polymer solution (NB = 1), this yields

Questions to Lesson Unit 9

!

v2specific, polymer 1 1 Π=c = RT + −χ c+ ...  Solvent Mn 2 V When we compare this to the Van’t Hoff equation

1 + A2 c + . . . Π=c = RT Mn

155

(4:47)

(4:37)

we immediately recognize the connection between the second virial coefficient, A2, and the Flory–Huggins parameter, χ: A2 =



v2specific, polymer 1 − χ  Solvent 2 V

(4:48)

It is thanks to this relation that we can determine χ through an osmotic pressure experiment as described at the beginning of this chapter.

Questions to Lesson Unit 9 (1) Which molar-mass average can be determined by colligative phenomena? a. The number-average molar mass Mn , since colligative properties depend on the number of (macro)molecules in a solution. b. The viscosity-average molar mass Mη, since colligative properties depend on the viscosity of a (macro)molecular solution. c. The weight-average molar mass Mw , since colligative properties depend on the mass of the (macro)molecules in a solution. d. The centrifugal-average molar mass Mz , since colligative properties depend on the amount of (macro)molecules in a solution. (2) Which colligative property is preferred for studying polymer solutions? a. The depression of the freezing point, since this is most pronounced for polymer solutions. b. The elevation of the boiling point, since this is most pronounced for polymer solutions. c. The osmotic pressure, since this is most pronounced for polymer solutions. d. All three of the previously mentioned colligative properties can be used equally since all three depend on the number average of the polymer mass.

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(3) An osmometer consists of two chambers, one inside the other; the inner chamber contains a polymer solution at the beginning and the outer chamber contains only solvent. The chambers are separated by a semipermeable membrane. What does that semipermeable membrane do? a. It allows only the polymer to pass, whereas the solvent remains in the inner chamber. Pressure builds up from outside on the inner chamber. b. It allows only the solvent to pass, so the solvent flows out of the inner chamber in this case. Pressure builds up from the outside on the inner chamber. c. It only allows the solvent to pass, so the solvent flows out of the outer chamber into the inner chamber. The incoming solvent increases the pressure in the inner chamber. d. It allows the solvent and polymer to pass, but they can only escape from the inner chamber into the outer chamber and not vice versa. The solution therefore flows into the outer chamber and thereby exerts pressure onto the inner chamber. (4) The excluded volume ve , the second virial coefficient A2 of the modified Van’t Hoff equation, and the Flory–Huggins parameter χ are related because ______ a. all are linearly dependent on the concentration of the polymer. b. all are dimensionless quantities. c. all are a measure of polymer–solvent interactions. d. all are a measure of the degree of polymerization of the polymer. (5) What is the rising height in an osmosis cell as the one shown in Figure 59 for a (dilute) aqueous polymer solution at a concentration of 0.1 mol L−1 at normal conditions? a. 2.5 cm b. 25 cm c. 2.5 m d. 25 m (6) The Van’t Hoff equation can be modified for the undiluted, nonideal case by a virial series expansion – similar to the way the ideal gas equation can be transformed into a more general form by a related virial series expansion. At a special temperature, the Boyle temperature, the real gas equation simplifies to the ideal gas equation. Which case is analog to this for a polymer solution? a. Athermal solvent, A2 ¼ 0 b. θ -Solvent, A2 ¼ 0 c. Good solvent, A2 > 0 d. Nonsolvent, A2 < 0

Questions to Lesson Unit 9

157

(7) Particularly good solvents show a deviation of the commonly linear concentration dependence of the reduced osmotic pressure. To what extent can this course be described? a. The course cannot be described with the given theory; for this scenario, a new equation based on a more complex theory is needed. b. The course can only be described section by section; small sections correspond to a locally linear course. c. The course can be modeled by including another virial coefficient A3 , which represents a quadratic dependence of the reduced osmotic pressure on the concentration. d. The progression can be modeled by adding an appropriate constant C to the existing equation. (8) How can the θ-temperature for a dilute solution be determined via osmometry? a. The temperature at which the concentration-dependent profile of the reduced osmotic pressure passes through the origin is determined; this temperature is the θ-temperature. b. The temperature is determined at which the concentration-dependent course of the reduced osmotic pressure has a slope of zero; this temperature is the θtemperature. c. The temperature is determined at which the concentration-dependent course of the reduced osmotic pressure has a minimum; this temperature is the θtemperature. d. The temperature at which the concentration-dependent course of the reduced osmotic pressure deviates from the linear course is determined; this temperature is the θ-temperature.

5 Mechanics and rheology of polymer systems LESSON 10: RHEOLOGY The most relevant properties of polymers, on which ground they have taken dominance in our materials world, are their mechanical ones. In this lesson, you will be introduced to the fundamentals of a methodology to assess these mechanics: rheology. You will become acquainted with some elementary physical and working principles in this field and get to know the most relevant material property of polymers: the elastic modulus.

5.1 Fundamentals of rheology At the very beginning of this book, it has been pointed out that the prime goal of polymer physical chemistry is the understanding of the relations between the structure and properties of polymer-based materials, thereby bridging the fields of polymer chemistry and polymer engineering, as shown in Figure 1. To achieve this goal, so far, we have studied the structure and dynamics of single chains and the thermodynamics of multichain systems in the preceding chapters; with that, we have erected three fundamental pillars of that foresaid bridge. Now, the construction of the actual bridge will be the goal of the forthcoming chapter. This will help us to rationally and quantitatively understand why many of the polymeric materials that we encounter in our everyday lives behave the way they do. Our prime focus will be on those properties of polymers that have made greatest impact on our lives in the past decades, which is their mechanical properties. Hence, we will first deal with a fundamental introduction to the field of rheology. The term rheology derives from the Greek words rheos, meaning “flow”, and logos, meaning “understanding”. The name reminds of a famous aphorism voiced by the Greek philosopher Simplicus, who had stated that panta rhei, “everything flows”. As such, rheology is the study of the flow of matter, and therefore the branch of physics that deals with the deformation and flow of materials, both solids and liquids.

5.1.1 Elementary cases of mechanical response Two elementary extreme cases can be distinguished from one another in view of a material’s mechanical properties: it can behave either as an elastic solid, like a rubber band, or as a viscous fluid, like water or syrup. In the first case of an ideal elastic solid, the energy introduced into the material through exertion of an external force, f, which can also be expressed in a normalized https://doi.org/10.1515/9783110713268-005

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5 Mechanics and rheology of polymer systems

form as a stress, σ = f/A (with A being the cross-sectional area of the material specimen onto which the force acts), is stored elastically. In this case, the material is immediately deformed to an extent proportional to the applied stress, or vice versa, a counteracting stress builds up in it upon and proportional to deformation by a given extent. This proportionality of the extent of deformation, or strain, ε, to the stress, σ, is captured by Hooke’s law: σ=E·ε

(5:1)

In this law, the strain is the relative extent of deformation, ε = ΔL/L0, that is, the change of the material specimen length, ΔL, relative to its length before deformation, L0. The constant of proportionality is the elastic modulus, also referred to as Young’s modulus, E. In a mathematical view, from eq. (5.1), we realize that the elastic or Young’s modulus has a unit of N·m–2 = Pa, because the stress is described by the unit N·m–2 = Pa as well, whereas the strain, ε = ΔL/L0, is a dimensionless quantity. In a conceptual view, the modulus is an elementary material parameter that quantifies how much energy can be stored in the material upon deformation by a given extent. In this view, it is illustrative to note that the unit N·m–2 may also be written as J·m–3, which denotes the energy density in the material that we need to work against when we want to deform it; it originates from the energies and sizes of the material’s building blocks (e.g., polymer-network strands of energy kBT and size ξ in a rubber). As we already see from the latter note, the capability for energy storage is directly related to the internal microscopic structure and dynamics of the material. Hence, the modulus is nothing less than a quantitative expression of a structure–property relation. With knowledge of the modulus, we can calculate from eq. (5.1) how hard it is to achieve a certain extent of deformation upon application of a given stress, which is of direct relevance for applications. As soon as the stress cedes, the stored energy is released, and the material snaps back to its original shape. This means that elastic deformation is reversible. This type of mechanical response can be visualized illustratively in the form of an elastic spring, as shown in Figure 62(A). In the second case of an ideal viscous fluid, the exertion of an external stress also causes a deformation. Here, however, the stress, σ, is proportional not to the deformation itself but to its rate, dε/dt. It follows that the material will reside in its deformed state as soon as the stress cedes. There is no energy storage in this case; instead, the energy of deformation is dissipated to the environment as heat, which means that viscous deformation is irreversible. The proportionality between the rate of deformation, dε/dt, and the stress, σ, is captured by Newton’s law: σ=η·

dε = η · ε_ dt

(5:2)

5.1 Fundamentals of rheology

161

Figure 62: Elementary cases of mechanical response in rheology. (A) Hooke’s law, relating the stress, σ, imposed on an ideal elastic solid, to its strain, ε, via the Young’s modulus, E. This model can be visualized in the form of an elastic spring with a spring constant of E. (B) Newton’s law, relating the stress, σ, imposed on an ideal viscous fluid to its time-dependent rate of deformation, dε/dt, via the viscosity, η. This model can be visualized in the form of a dashpot filled with a fluid of viscosity η.

In this law, the constant of proportionality is the viscosity, η. Just like the modulus in the elastic case above, the viscosity is an elementary material parameter that quantifies how much resistance a material exerts on flow.65 As we will see later in this chapter, the capability for exerting resistance on flow can be directly related to the internal microscopic structure and dynamics of a material. Hence, just like the elastic modulus, the viscosity is a quantitative expression of a structure–property relation. The viscous type of mechanical response can be visualized illustratively in the form of a dashpot filled with a fluid, as illustrated in Figure 2(B). In view of eq. (5.2), we realize that the viscosity has the unit Pa·s, because the stress is described by the unit N·m–2 = Pa, and the rate of deformation has the unit s–1.

5.1.2 Different types of deformation in rheology Strain can be applied to a material in many directions, but we may simplify our view to two particularly relevant cases. To visualize these, consider a cube with edges A, B, and C. In this cube, A is the edge in and out-of-plane, B is the vertical edge, and C the horizontal one (see Figure 63). In the first case, a uniform stress exerted onto the A–B face causes uniaxial strain. This strain, ε, can be calculated from the ratio between the deformation, ΔC, and the original length of the edge C. For the two elementary cases of an elastic solid and a

 Later on, in Sections 5.6.1 and 5.7.1, we will see that the elastic modulus and the viscosity are connected to one another by one of the most central equations in the field of rheology: η = E τ.

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5 Mechanics and rheology of polymer systems

Figure 63: Two ways to deform a material: (A) a uniform strain exerted onto the A–B face is called uniaxial strain. (B) A strain exerted onto the A–C face is called shear. In both cases, the stress, σ, can be calculated with Hooke’s law for an ideal elastic material or with Newton’s law for an ideal viscous fluid.

viscous fluid, Hooke’s and Newton’s law can be applied as stated in Section 5.1.1 to calculate the stress, σ, related to such a uniaxial strain (see Figure 63(A)). In the second case, the stress is exerted onto the A–C face. This situation is called shear. The resulting shear strain, γ, is calculated from the ratio between the deformation, Δc, and the original length of edge B, which corresponds to the tangent of the shearing angle α (see Figure 63(B)). For an ideal viscous fluid, the stress, σ, is still calculated using Newton’s law as stated in Section 5.1.1. Hooke’s law, however, needs to be modified to incorporate a new proportionality constant: the shear modulus, G. Both the Young’s modulus, E, and the shear modulus, G, are connected to one another in the form of E = 2Gð1 + μÞ

(5:3)

In eq. (5.3.), µ is named Poisson’s ratio; it is a measure of the Poisson effect, a phenomenon according to which a material tends to expand in directions perpendicular to the direction of compression or, vice versa, tends to laterally contract upon extension; both is a consequence of the retention of the material body’s volume upon compression or extension. µ quantifies this effect by relating the relative lateral contraction of a material specimen, expressed by the change of its thickness relative to the one before deformation, Δd/d0, to its axial strain, ΔL/L0: μ=

Δd=d0 ΔL=L0

(5:4)

Incompressible objects such as fluids have a Poisson’s ratio of µ = 0.5. This value also commonly applies to polymers, because linear extension of a polymer sample

5.1 Fundamentals of rheology

163

is usually accompanied by a concurrent lateral contraction, meaning that it does not undergo a volume change upon stretching. In this case, E = 3G.

5.1.3 The tensor form of Hooke’s law The equations for Hooke’s and Newton’s laws named above are actually simplified forms for a stress acting normal to the sample and in one direction only. In their generalized forms, they are tensor equations that account for all possible spatial directions. In this generalized form, Hooke’s law reads τ=c·γ

(5:5)

with τ the stress tensor, c the elasticity tensor, and γ the strain tensor. With this nomenclature, we follow a common convention, in which – unfortunately – the tensor form of the stress is abbreviated with the symbol τ. This is inconvenient, because in polymer physics, τ is very commonly used to abbreviate the relaxation time as well.66 To avoid this confusion, we use the symbol σ for the special-case variant of normal stress in this textbook; this is the stress with both its directional indices being the same, that is σ11 = τ11, σ22 = τ22, and σ33 = τ33. To illustrate the relation of eq. (5.5), again consider a cube representing a volume section of an elastic body. If we impose a stress τik in the direction k, the cube will deform, but now, the resulting strain will not be limited to just one spatial direction, but instead, it will have contributions for all spatial directions (see Figure 64). This relation is reflected by the following equation: τik =

3 X 3 X

ciklm · γlm = cik11 · ε11 + cik12 · γ12 + cik13 · γ13 +

l=1 m=1

cik21 · γ21 + cik22 · ε22 + cik23 · γ23 + cik31 · γ31 + cik32 · γ32 + cik33 · ε33

(5:6)

The strength of each contribution is given by the elastic constant ciklm. Spatial directions for each value are indicated by the indices l and m. For a comprehensive consideration, we would have to evaluate all nine stress components (σ11 , τ12 , τ13 , τ21 , σ22 , τ23 , τ31 , τ32 , σ33 ), which are connected by nine elastic constants ciklm to nine strain components (ε11 , γ12 , γ13 , γ21 , ε22 , γ23 , γ31 , γ32 , ε33 ). This adds up to 81(!) single components to fully describe the deformation of a cubic volume. Fortunately, if we consider the symmetry of the body, this expression can be severely simplified. If the body is isotropic, meaning it is uniform in all orientations,

 If you are unsure whether a symbol τ in a textbook or a paper represents stress or (relaxation) time, check its physical unit: if it is Pascal, then τ represents a stress, and if it is seconds, then τ represents a time.

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5 Mechanics and rheology of polymer systems

Figure 64: Deformation of an elastic body taking into account all spatial directions. (A) A stress τik imposed in the direction k results in a (B) corresponding strain γik in the same direction. Picture redrawn from C. Wrana: Polymerphysik, Springer Verlag Berlin, Heidelberg, 2014.

the number of independent constants is reduced to two. Polymers exhibit homogeneous and amorphous structures and can therefore be considered to be isotropic. We can now apply this to the two cases of uniaxial strain and shear deformation, which leads to the following equations for Hooke’s law considering all three spatial direction x, y, and z: Uniaxial strain: σ xx = E · εxx ; σ yy = E · εyy ; σ zz = E · εzz

Shear deformation: τ xy = G · γxy ; τ xz = G · γxz ; τ yz = G · γyz

These are just the equations we introduced in Section 5.1.1. We can see that ε and γ are, in fact, special-case variants of γ, σ and τ are special-case variants of τ, and E and G are special-case variants of c.

Questions to Lesson Unit 10 (1) Which property of polymers cannot be probed by rheological experiments? a. Their compressibility b. Their elasticity c. Their flow rate d. Their viscosity (2) Which are the two extreme cases of a polymer’s ideal mechanical response? a. The elastic solid and the viscous fluid b. The elastic solid and the viscous solid c. The glassy solid and the elastic fluid d. The glassy solid and the viscous fluid

Questions to Lesson Unit 10

165

(3) Which statement does not hold for the case of an elastic solid? a. The strain of the material scales linearly with the stress applied to it. b. The deformation of the material is irreversible. c. An elastic solid can be simply modeled as a spring. d. The mechanical response is described by Hooke’s law with the elastic modulus as a proportionality factor. (4) What is the relation between the elastic modulus and the microscopic structure of an elastic solid? a. There is no trivial relation between them. b. As the elastic modulus is a parameter that quantifies the elasticity, it is directly related to the lifetime of the transient agglomerates formed due to deformation. c. As the elastic modulus is a parameter that quantifies the extent of deformation, it is directly related to the number of bonds between the building blocks that are broken during deformation. d. As the elastic modulus is a parameter that quantifies the stored energy, it is directly related to the coupling and sizes of the building blocks that are moved due to deformation. (5) What is stress (in mechanics)? a. Stress is a different denomination for pressure. They both have the same dimension with the unit “Pascal.” b. Stress is a measure of the internal forces in a continuous medium that builds up because of an external load. c. Stress is the deformation of a continuous medium caused by an applied external force. d. Stress is the internal tension in a continuous medium caused by an external force per volume segment of the medium. (6) Which statement about strain is true? a. Strain is the relative deformation of a continuous medium caused by an applied force or stress. b. Strain is the angular deformation of a continuous medium caused by an applied force or stress. c. A body always retains its volume when being strained. d. A body always retains its shape when being strained.

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5 Mechanics and rheology of polymer systems

(7) Which statement does not hold for the case of a viscous fluid? a. The strain of the material scales linearly with the stress applied to it. b. The deformation is irreversible. c. A viscous fluid can be simply modeled as a dashpot. d. The behavior is described by Newton’s law with the viscosity as a proportionality factor. (8) Which model describes the actual mechanical response of a polymer? a. That of an ideal elastic solid since the bonds between the building blocks work as springs which can snap back into shape after deformation. b. That of an ideal viscous fluid since the movement of the building blocks due to deformation is irreversible. c. It depends on the experimental parameters. Based on those, polymers display either fully elastic or fully viscous behavior. d. It depends on the experimental parameters. Based on those, polymers may display elastic or viscous or both characteristics. This is described as the term “viscoelasticity.”

5.2 Viscoelasticity

167

5.2 Viscoelasticity LESSON 11: VISCOELASTICITY The rich mechanical properties of polymers emerge because they are neither a true fluid nor a true solid; instead, they display characteristics of both, referred to as viscoelasticity. This lesson will show how this duality manifests itself in a polymer’s time- or frequencydependent stress relaxation or strain creep. It will also show how this is captured by complex-number material constants, such as the complex modulus.

Often, materials exhibit both elastic as well as viscous properties. These materials are named viscoelastic. Strictly speaking, all matter exhibits this dualism. For example, imagine watching a glacier for one day. You will not be able to observe any perceivable movement of the ice and could therefore say that it is a solid. Now imagine visiting the same glacier once a day over the course of an entire year. Every time you visit, you take a photograph, and in the end you edit all pictures together to create a time-lapse movie. The glacier will appear to be flowing like a fluid. Another example is the big, ornamental glass windows that can be found in old cathedrals: often, they are thicker at their bottom than at the top. This difference is mainly related to the process with which they are produced: in historic times, church windows were manufactured upright, which meant that the cooling glass melt would flow a little from top to bottom inside the manufacturing frame. In addition to this manufacture-related thickness gradient, however, the glass of these windows still flows today, and over very long timescales (longer than centuries in fact), it causes further thickening of the window bottoms compared to their tops. Viscoelasticity is a phenomenon most strikingly encountered in polymer systems. In contrast to glaciers and glass windows, polymer systems usually transition from a regime where their elastic properties dominate to one where the viscous properties are dominant on timescales that are right in the practically and experimentally relevant window of milliseconds to seconds, depending on the specific system at hand. This makes polymers highly relevant for practical applications, because their mechanical properties are versatile, dynamic, and, with the proper understanding of the underlying concepts that govern their mechanics, can be rationally designed to serve a specific purpose. Polymer’s mechanical properties are usually assessed and quantified in three types of rheological experiments, as will be discussed in the following subsection.

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5.2.1 Elementary types of rheological experiments 5.2.1.1 The relaxation experiment In a relaxation experiment, uniaxial or shear strain, ε or γ, is applied to a sample and the resulting stress in the material, σ, is recorded as a function of time, as shown in Figure 65. An elastic sample shows direct proportionality of both quantities according to Hooke’s law. The proportionality factor is the time-dependent relaxation modulus E(t). As long as strain is applied, an equivalent stress is recorded. An ideal viscous sample, instead, displays two spikes of the stress, one at the moment where the stress is abruptly increased and one at the moment where the stress is abruptly decreased. This is because according to Newton’s law, only the time-dependent rate of deformation is considered, and hence, at times where the strain is constant, no stress is recorded, whereas an infinite spike of stress occurs during an abrupt change of the strain. In the case of a viscoelastic sample, the initial response is similar to the elastic one: the stress increases proportionally to the strain. Once the strain is kept at a constant level, though, some of the stored energy of deformation is dissipated as heat, leading to stress relaxation over time. If the sample is a viscoelastic solid, the stress will reach a plateau value with a plateau modulus Eeq = E(t ! ∞).

Figure 65: In a relaxation experiment, strain, ε or γ, is applied to a sample, and the time-dependent stress, σ, that builds up in the sample is recorded. An ideal elastic sample shows direct proportionality of both quantities according to Hooke’s law. In an ideal viscous sample, the stress displays spikes at times of abrupt change of the strain. In a viscoelastic sample, the stress in the material relaxes to a plateau value for t ! ∞, because some of the energy of deformation is dissipated to heat over time.

5.2.1.2 The creep test In a creep test, stress, σ, is applied to a sample, and the resulting uniaxial or shear strain of the specimen, ε or γ, is recorded as a function of time (see Figure 66). An elastic sample shows direct proportionality of both quantities according to Hooke’s

5.2 Viscoelasticity

169

Figure 66: In a creep test, stress, σ, is applied to a sample, and the resulting time-dependent strain of it, ε or γ, is recorded. An ideal elastic sample shows direct proportionality of both quantities according to Hooke’s law. In an ideal viscous sample, the strain increases linearly with time as long as constant stress is applied; the material exhibits creep. In a viscoelastic sample, the response is a mix of both: the recorded strain value rises instantly, but not to same extent as in the elastic sample, and with time it rises further due to creep of the material. A viscoelastic solid reaches a plateau for t ! ∞, whereas a viscoelastic liquid proceeds creeping steadily. Once the application of stress is released, the sample creeps back with time.

law. The proportionality factor is called the time-dependent creep compliance, J(t). As long as stress is applied, an equivalent strain is recorded. An ideal viscous sample exhibits a strain that increases linearly as long as the stress is kept constant; the material exhibits creep. Once the test is finished by release of the stress, the strain does not return its original level, thereby showcasing the irreversibility of viscous flow. In case of a viscoelastic sample, the response is a mix of both: once a constant amount of stress is applied, the recorded strain value rises instantly, but not to the same extent as in the elastic sample. With time, however, it rises further due to creep of the material. If the sample has more elastic than viscous character, it would reach a plateau for t ! ∞, whereas it would proceed creeping steadily if the sample has more viscous than elastic character. Once the test is finished and the application of stress is released, the sample exhibits creep in the direction back. 5.2.1.3 The dynamic experiment Another method to probe a material’s mechanical properties is the dynamic experiment. Here, a sinusoidally modulated stress, σ = σ0 exp(iωt), is applied to the sample, and the time-dependent strain, ε or γ, is recorded, as illustrated in Figure 67. This can also be done vice versa, that is, a sinusoidally modulated strain is applied and the emerging time-dependent stress in the material is recorded. In the case of an ideal elastic sample, both stress and strain are in phase, as both quantities are directly proportional to each other at each time according to Hooke’s law; as a

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Figure 67: In a dynamic experiment, a sinusoidal stress, σ, is applied to a sample, and the timedependent strain, ε or γ, is recorded. Both the stress and strain curves are proportional and in phase for an ideal elastic sample. By contrast, the time-dependent strain curve of an ideal viscous sample exhibits a phase shift to the stress curve with a phase angle, δ, of 90° or π/2. In the intermediate case of a viscoelastic sample, the stress and strain curves display a phase angle between 0 and π/2. In this case, δ reflects the individual contribution of the elastic and viscous parts of the sample.

result, their phase angle, δ, is zero. In the case of an ideal viscous sample, according to Newton’s law, it is not the strain itself but its derivative that is proportional to the applied stress; hence, an original sinewave condition of strain causes a cosine answer of stress, or vice versa, an original sinewave condition of stress causes a minus-cosine answer of strain. In either case, as a result, stress and strain have a phase shift of 90°; this means that the phase angle, δ, is π/2. A viscoelastic sample will be intermediate between these two extreme cases and display a phase shift with phase angle, δ, in the range 0 < δ < π/2. From δ, the individual contribution of the elastic and viscous parts of the viscoelastic sample can be calculated. The closer δ is to zero, the more does the elastic contribution dominate, whereas the closer δ is to π/2, the more does the viscous contribution dominate.

5.3 Complex moduli In Hooke’s law, eq. (5.1), the elastic modulus reflects the ratio of stress, σ, to strain, ε or γ. In the case of a dynamic experiment as just discussed, this modulus is a complex quantity, E* or G*, connecting the ratio of the sinusoidally modulated stress and strain and the phase angle between them, δ, as follows: E* =

σ σ0 expðiωtÞ σ0 expðiδÞ = = ε ε0 expðiðωt − δÞÞ ε0

(5:7a)

5.3 Complex moduli

G* =

σ σ0 expðiωtÞ σ0 expðiδÞ = = γ γ0 expðiðωt − δÞÞ γ0

171

(5:7b)

With Euler’s formula, this can be rewritten as a trigonometric function with a real and an imaginary part; these two parts can be viewed to be two different parts of the complex modulus: E* =

σ0 ðcos δ + i sin δÞ ε0

= E′ + iE′′ G* =

(5:8a)

σ0 ðcos δ + i sin δÞ γ0

= G′ + iG′′

(5:8b)

The first part is the storage modulus, E′ or G′; it captures the sample’s elastic properties. The second part is the loss modulus, E″ or G″; it captures the sample’s viscous properties. Storage modulus: E′ =

σ0 cos δ ε0

(5:9a)

G′ =

σ0 cos δ γ0

(5:9b)

E′′ =

σ0 sin δ ε0

(5:10a)

σ0 sin δ G′′ = γ0

(5:10b)

Loss modulus:

Depending on the phase angle δ, the first or the second contribution will dominate the overall modulus E* or G*. If δ = 0, then E″ and G″ are zero, such that only the elastic parts E′ or G′ contribute to E* or G*. By contrast, if δ = π/2, then E′ and G′ are zero, such that only the viscous parts E″ and G″ contribute to E* or G*. In between these two extremes, both parts contribute to E* or G*, and δ determines with what relative magnitude they do. The extent of the storage and the loss modulus contributions to E* and G* can be judged from their ratio E″/E′ and G″/G′. Following the simple trigonometric evaluation shown in Figure 68, or alternatively, the simple trigonometric identities of eqs. (5.11), these ratios can be described as the tangent of δ. It is called the loss tangent, and its value describes the contributions of the elastic and the viscous parts to a sample’s viscoelastic properties in one simple number. tan δ > 1 means

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that the value of the loss modulus is larger than that of the storage modulus. The sample’s mechanical properties are then dominated by its viscous part, and it is therefore called a viscoelastic liquid. When tan δ < 1, by contrast, the sample’s mechanical properties are dominated by its elastic contribution. The value of its storage modulus is then larger than that of its loss modulus, and it is therefore called a viscoelastic solid. When tan δ = 1, both the elastic and the viscous parts contribute equally. σ0 ε sin δ E″ sin δ = tan δ = 0 = σ cos δ 0 cos δ E′ ε0

(5.11a)

σ0 γ0 sin δ G″ sin δ = = = tan δ σ G′ cos δ 0 cos δ γ0

(5.11b)

Figure 68: Visual representation of the complex modulus E* in an Argand diagram. The abscissa represents the real part of the complex modulus, E′, whereas the ordinate represents the imaginary part, E″. A vector connecting the value of E* and the origin exhibits a slope of tan δ, the loss tangent.

The contributions of the storage modulus, E′ or G′, and the loss modulus, E″ or G″, to the complex modulus E* or G* can be visualized in an experiment as shown in Figure 69. When a person drops a rubber ball from the hand to the floor, it bounces off the floor and shoots back up, but it does not reach its original height, because some of the energy is dissipated to heat during the bounce. This amount of energy can be directly related to the loss in height, and this is the contribution of the loss modulus, E″ or G″, to E* or G*. The height that the ball is able to gain back when bouncing is directly related to the energy stored elastically within the material. This is the contribution of the storage modulus, E′ or G′ to E* or G*. The latter example is a direct illustration of a structure–property relation. The extent of the rubber ball bounce is a direct resemblance of the loss tangent of its rubbery material, which is a ratio of the moduli E″ (or G″) and E′ (or G′). These moduli, in turn, have to do with the material’s structure. As we will see later, in Section 5.9, the ability of a rubbery polymer network to store mechanical energy is proportional to its polymer-chain crosslinking density, as this determines the tightness of the polymer-network meshes. The ability for energy dissipation, by contrast, has to do with structural motifs that are linked to the network with only one extremity, such as loops or dangling chains, as these may dissipate external deformation energy. All of the latter has to do with the network structure, and based on the aforesaid, this structure directly translates into properties. Although the example with a rubber ball may appear silly, the same holds for other rubbery products such

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Figure 69: Visualization of the contributions of the loss modulus, E″, and the storage modulus, E′, to the complex modulus E*. In an experiment, a rubber ball is dropped, but does not bounce back to its original height. The amount of energy dissipated to heat corresponds to the loss in height after the bounce; this is the loss modulus, E″. The residual height of the bounce corresponds to the energy stored elastically within the material; this is the storage modulus, E′. An ideal elastic ball would regain its original position, whereas an ideal viscous ball would not bounce at all.

as shoe soles. Tailoring their properties to either good damping (as requested by customers for leisure activities) or good bounce (as requested by customers for professional marathon running) can be achieved on the very same basis.67

 For this structure–property relationship to actually apply, however, the polymer must be in an entropy-elastic rubbery state, which means that (network-strand) chains must be capable of accounting for chain stretching by local monomer-segmental bond rotation, that is, conversion of local narrow cis or gauche into wide trans conformations. This is only possible at temperatures above the glass-transition temperature, T > Tg. At temperatures below the glass-transition temperature, T < Tg, by contrast, a completely different material state is present, namely an energy-elastic glassy state. Here the modulus is much higher, and the material is not elastic and ductile but hard and brittle. A tragic example of a material application where this was not taken into account is the Challenger disaster. On the morning of January 28, 1986, the Challenger space shuttle with seven astronauts on board was to be launched from the Kennedy Space Center in Florida on a 6-day space mission. On that morning, the temperature was low, and so rubber sealing rings in the fuel system were glassy and therefore not tight. This led to fuel leakage, causing a catastrophe that took all seven lives on board. We see from this tragic example: the ratio of the designated application temperature T to the glass-transition temperature Tg has a decisive, here literally life-decisive influence on the state and thus the mechanical properties of a polymer material. Tg, in turn, is correlated to how easy or difficult it is to rotate individual monomer bonds, which has to do with the local chemical substitution pattern, meaning the bulkiness and interactions of the chemical side groups on the polymer backbone. This is assessed by parameters such as the characteristic ratio, the persistence length, or the Kuhn length, that we have got to know in our second chapter. We see from all this, that the microscopic chemical and polymer topological structure is coupled to macroscopic properties on many micro- and even nanoscopic levels, and that we have to take all this into account in applications (and not lose sight of the operation window such as the temperature-range intended for an application). Expressed more positively, we also see that we actually can do this, and we see how certain desired material properties at foreseen defined application conditions can in fact be rationally translated into specific polymer chain structures.

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5.4 Viscous flow In addition to the elastic Young’s or shear modulus E* or G*, other quantities that we have introduced in the previous paragraphs, such as the creep compliance, J, and the viscosity, η, can also be determined in a dynamic experiment and therefore be expressed as complex quantities. The complex creep compliance, J*, which is the reciprocal of the complex modulus, 1/E* or 1/G*, reflects the ratio of strain, ε or γ, to stress, σ. For the case of oscillatory shear deformation, it calculates as follows: J* =

γ γ0 1 1 expð − iδÞ = = = σ σ0 G* G′ + iG′′ ′

(5:12a)

′′

We can expand this expression by G′ − iG′′ to gain G − iG

J* = 

G′ − iG′′ G′ − iG′′ G′ G′′  = 2 = 2 −i 2 = J ′ − iJ ′′ 2 2 2 ′ ′′ ′ ′′ G + iG G − iG G′ + G′′ G′ + G′′ G′ + G′′

(5:12b)

The complex viscosity, η*, is calculated analogously. It reflects the ratio of stress, σ, to the strain rate, dε/dt or dγ/dt. For an oscillatory shear deformation experiment, it calculates as follows: σ η* =   = dγ dt

σ0 expðiωtÞ d ð γ dt 0 expðiðωt − δÞÞÞ

Insertion of G* =

σ0 γ0

=

σ0 expðiωtÞ σ0 1 = expðiδÞ iωγ0 ðexpðiðωt − δÞÞÞ γ0 iω

(5:13a)

expðiδÞ transforms the equation to η* =

G* G′ + iG′′ G′ iG′′ = + = iω iω iω iω

(5:13b)

By expanding the first summand with i=i and cancelling out the i in the second summand, we can write iG′ G′′ iG′ G′′ G′′ G′ + = + = − i = η′ − iη′′ iiω ω − 1ω ω ω ω

(5:13c)

We can see that the complex viscosity, η*, is connected to the complex moduli, E* and G*, by the frequency, ω. It should be noted here that η′ is connected to E″ and G″, whereas η″ is connected to E′ and G′. This makes sense: the real part of the viscosity accounts for the dissipation of energy by a material, just as the imaginary part of E* and G* does. This viscosity of a polymer solution is often quite different than that of a solution of small molecules that exhibits Newtonian flow according to eq. (5.2). The latter’s viscosity is independent of the strain rate or shear rate dε/dt or dγ/dt. If we

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plot the stress, σ, as a function of that rate (such a plot is named flow curve), the viscosity is the slope of a straight line in this case, as this plot is a simple graphical representation of Newton’s law. In polymer solutions and melts, however, the relation of stress to the strain rate or shear rate is often nonlinear. Instead, polymer systems often exhibit shear-thinning, that is, a decrease of the viscosity at increasing shear rate. The reason for this is because shear forces destroy possible associate structures and orient the chains along the direction of flow, thereby enabling the polymer solution or melt to flow more freely by reducing the friction and mutual impairment between their chains. Due to this structural change, shear-thinning has been termed structural viscosity by Wolfgang Ostwald junior in 1825. Because their viscosity does not relate linearly to the shear rate, polymer systems are often discussed in terms of the zero shear viscosity, η0, which is obtained by extrapolating the shear-thinning flow curve in Figure 70(A) to a shear rate of zero. In this limit, the shear-thinning flow curve coincides with the ideal Newtonian one. Shear-thinning properties, nevertheless, are actually wanted for many applications. For example, wall paint is specifically engineered such to be shear-thinning, because in this way, it can be applied with ease onto a wall through the exertion of shear by brushing, whereas once this is done, it will stay on the wall and dry instead of drip. Shear-thinning polymers are also used in everyday-life products such

Figure 70: Plot of a flow curve, which is a graphical representation of stress, σ, as a function of shear rate, dγ/dt, of a flowing liquid. (A) In case of a Newtonian fluid, we obtain a straight line passing through the origin in which the slope is the viscosity, η, which is independent of the shear rate. This is different for the case of non-Newtonian flow, which can be observed in different manifestations. In the case of shear-thickening, the viscosity increases with the of shear rate, whereas in the case of shear-thinning, the viscosity decreases with the shear rate. The latter is often found in polymer systems, as shear breaks associates and orients polymer chains, thereby enabling them slide against each other more easily. (B) Further deviations from Newtonian flow are cases in which the flow curve exhibits an intercept different from zero. A Bingham fluid is solid at low stress, but starts to flow beyond a specific stress threshold. A Casson fluid is a special case of such a fluid that exhibits additional shear-thinning once flow sets in.

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as shrink foils. These foils are produced by extrusion of a shear-thinning polymer melt in which the chains orient themselves when squeezed through the extrusion nozzle. After quick cooling and solidification, this chain orientation is frozen in the foil material. When wrapping an object with this foil and heating it, the oriented chains will relax to their original random conformation, and thus, the foil shrinks. Resolidification then locks this state. This is especially useful for materials that require an airtight packaging such as consumables. A special case is the Bingham fluid: This is a material that has a flow threshold at a specific yield stress. It acts as a solid at zero and very low shear below the threshold, but flows like a Newtonian fluid when the shear rate is increased above it. This mechanical behavior is often attributed to secondary interactions such as Van der Waals forces or hydrogen bonds in the material, which build up a three-dimensional network structure that gives the material solid-like properties. These assemblies, however, can be broken by shear due to the low binding energies of the transient interactions involved. Bingham fluids known from everyday life include tooth paste, mayonnaise, or ketchup. Similar to that is a Casson fluid, which acts like a Bingham fluid but then furthermore exhibits shear-thinning. Melted chocolate is an example of a Casson fluid. The opposite type of shear-altering behavior is called shear-thickening, which is a common feature of particulate suspensions, whereas it is less common for polymer solutions and melts. Here, transient hydrodynamic clusters of particles can form and break spontaneously in the material. At high shear rates, the shear oscillation period gets shorter than the transient lifetime of these clusters, such that they become permanent on the timescale of the experiment, thereby imparting hindrance on flow. A prime example is a water–sand suspension, that means wet sand. When carefully burying your feet into it on the beach, you can easily remove them from the sand when imposing only low shear by moving your feet slowly. However, if you impose high shear and try to abruptly pull your feet out of the wet sand, they will be stuck as if they were cemented in concrete. Two further variants of complex rheological behavior are thixotropy and rheopexy. In a thixotropic case, the viscosity is observed to decrease over time during the experiment. An illustrative explanation for such a behavior is a time-dependent breakdown of a house-of-cards structure in the sample. In the opposite case, named rheopexy, the viscosity increases over time. There is no illustrative picture for that.

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Questions to Lesson Unit 11 (1) Which is not a typical rheological experiment? a. Creep test b. Relaxation experiment c. Dynamic oscillatory experiment d. Plateau test (2) Which statement describes the stress in a relaxation experiment when a boxprofiled strain is applied to a viscoelastic solid? a. The applied constant strain is followed by an instant response of the stress that slowly decreases with time until it reaches a plateau value; it does not return to zero. b. The applied constant strain is followed by an instant response of the stress that slowly decreases with time until the point where the applied strain is gone and it immediately returns to zero. c. The applied constant strain is followed by a delayed response of the stress that slowly decreases with time until it reaches an equilibrium value. d. The applied constant strain is followed by a delayed response with a sharp rise of stress which drops sharply back to zero till the applied strain ends, followed by another spike in the stress signal. (3) Choose the answer with the correctly assigned experimental setups that resulted in the following graphs:

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a. ① ② ③ ④

Dynamic experiment with an ideal viscous liquid Dynamic experiment with a viscoelastic material. Relaxation experiment with a viscoelastic solid. Creep test with a viscoelastic solid.

① ② ③ ④

Relaxation experiment with a viscoelastic solid. Creep test with a viscoelastic solid. Dynamic experiment with an ideal viscous liquid. Dynamic experiment with a viscoelastic material.

① ② ③ ④

Creep test with a viscoelastic solid. Relaxation experiment with a viscoelastic solid. Dynamic experiment with a viscoelastic material. Dynamic experiment with an ideal viscous liquid.

① ② ③ ④

Dynamic experiment with an ideal viscous liquid. Dynamic experiment with a viscoelastic material. Creep test with a viscoelastic solid. Relaxation experiment with a viscoelastic solid.

b.

c.

d.

(4) What is the benefit of expressing the complex moduli as a sum of sine and cosine? a. They are easier to handle mathematically compared to the exponential expression. b. The signal of a dynamic experiment is sinusoidal. c. The different summands capture the elastic or the viscous properties. d. One can identify the phase angle δ more easily. (5) What is the benefit of expressing the complex moduli as complex numbers? a. They are easier to handle mathematically compared to the trigonometrical expression. b. The expression eiωt has the form of an exponential build up. c. The ratio between the real and the imaginary parts is given by a simple sine. d. One can identify the elastic or the viscous properties more easily.

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(6) When plotted in a complex number plane, the ratio of E′′ to E′ is given by the tangent of the phase angle δ. Which is the correct value assignment to the following three characteristics (left to right): Viscoelastic solid; viscous and elastic contribute equally; viscoelastic liquid a. tan δ > 1; tan δ < 1; tan δ = 1 b. tan δ < 1; tan δ = 1; tan δ > 1 c. tan δ = 1; tan δ < 1; tan δ > 1 d. tan δ < 1; tan δ > 1; tan δ = 1 (7) Which statement about the complex viscosity is true? a. The complex viscosity is related to the imaginary part of the complex modulus because it represents the viscous properties. b. The complex viscosity is related to the reciprocal of the real part of the complex modulus which represents their elastic properties. c. The real part of the complex viscosity is related to the real part of the complex modulus and is the same for the imaginary parts; in short: real to real, imaginary to imaginary. d. The real part of the complex viscosity is related to the imaginary part of the complex modulus and vice versa since both quantities are quantifying contrary properties. (8) What is not a good and frequent reason for researchers’ interests in polymers as viscoelastic materials? a. The timescale on which their rather elastic behavior changes into a rather viscous behavior is in between ms and s, which are practically relevant timescales. b. The comparably long timescales of motion combined with the comparably big size of the polymer building blocks (compared to low molar mass molecules) make them easily accessible for experimental methods. c. The viscoelastic properties result in a very robust material, that is quite resistant to fluctuations in temperature. d. The viscoelastic properties are very dynamic, which results in a broad range of possible applications.

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5.5 Methodology in rheology LESSON 12: PRACTICE AND THEORY OF RHEOLOGY So far, you got a notion about the viscoelastic nature of polymers, how it manifests itself in relaxation and creep experiments, and how it is reflected in complex material parameters. The following lesson adds information on how these parameters are actually probed in experiments, and how the phenomenology of viscoelasticity can be treated in simplifying mechanical models: the Maxwell model and the Kelvin–Voigt model. We will see how these models are conceptually very simple but yet powerful in quantitatively capturing a polymer’s time- or frequency-dependent mechanics.

5.5.1 Oscillatory shear rheology The most common and experimentally easiest way to assess the viscoelasticity of a sample is the method of oscillatory shear rheology. Here, a sample is placed on a lower plate that is usually an even plane, and an upper plate is lowered onto it, as shown in Figure 71. A motor then turns the upper part in an oscillatory manner, which results in a torque that is exerted by the sample-medium’s resistance in between the two plates. From this torque, the values for the moduli are determined, and if that kind of estimate is performed at different oscillation frequencies, they are even obtained in a frequency-dependent fashion. Sometimes, a very shallow cone is used instead of a plate as the upper geometry for the following reason: the turning motion creates a shear-force gradient facing outward from the sample’s center. The cone’s angled geometry eliminates this gradient and ensures a constant shear force throughout the sample. This, however, only works at a very defined gap size, which is the distance between the upper and lower geometry. Polymer solutions are therefore

Figure 71: Schematic representation of a setup for a shear rheology experiment. The sample is applied onto the lower part of the measuring geometry, which is usually an even plane. Then, the upper part of the geometry is lowered onto the sample and sheared against the lower plate, either in a continuous or in an oscillating manner. This upper part is either also an even plate, thereby enabling the use of a user-defined gap size (A), or it is a shallow cone, thereby eliminating possible disruptive shear-force gradients throughout the sample, which, however, only works at a predetermined gap size (B).

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often measured with a cone–plate geometry, whereas polymer gels and melts are mostly measured using plate–plate geometries.

5.5.2 Microrheology Another way to capture a sample’s viscoelastic properties is the method of microrheology. Here, a population of nano- or micrometer-sized probes is immersed within the sample, which then explores it, as sketched in Figure 72. By quantifying the relative ease of these probes’ movement through the sample medium, the sample’s viscoelastic characteristics can be quantified. As the probes are small, this information can be obtained on a local microscopic scale. Furthermore, if methods of imaging microscopy are used to measure the probe motion, the information on the sample’s viscoelastic mechanics is even obtained in a spatially resolved manner. With that, potential heterogeneity of the sample composition or structure can be determined.

Figure 72: Schematic of a polymer sample loaded with nanoparticles that explore it by either random-thermal or externally directed motion. In the method of microrheology, observation of this probe motion is used to draw quantitative conclusions about the viscoelastic properties of the surrounding sample medium.

There are two variants of microrheology. In an active microrheology experiment, the probe particles are displaced by external forces. In a passive microrheology experiment, by contrast, only the ever-present thermal energy moves the probes. 5.5.2.1 Active microrheology A prime example of an active microrheological method is magnetic-bead rheometry. Here, the sample is loaded with spherical magnetic microparticles that can be moved by application of an external magnetic field. The magnetic force, ~ fmag , is re~ ðtÞ, and the lated to the product of the magnetic moment induced in the bead, M external field gradient, ∂~ BðtÞ=∂x, which can also be expressed as the magnetic

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susceptibility of the bead, χ, multiplied by its volume, V, and the field strength ~ times its gradient, ~ BðtÞ ∂B∂xðtÞ: ~ ∂~ BðtÞ ~ ~ ðtÞ ∂BðtÞ = χV~ fmag, x ðtÞ = M BðtÞ ∂x ∂x

(5:14)

The resulting displacement of the bead can be monitored by microscopy and then translated into the viscoelastic properties in a spatially resolved manner. In the following, we first detail on how this works for the elementary case of a purely viscous medium. In such a medium, the application of a constant magnetic force, ~ fmag , accelerates the probe particles, accompanied by emergence of a counteracting fricν, and the viscosity tional force, ~ ffrict . This force is related to the particle’s velocity, ~ of the medium, η, by Stoke’s law. In a steady state, these two forces are at balance, and the particles reach a constant velocity. At this state, knowledge of the external magnetic force on the basis of eq. (5.14) along with knowledge of the probe particle size, r, and measurement of the particle velocity in the sample, ~ ν, allows the viscosity, η, to be calculated: ~ ffrict = 6πηr~ ν fmag =~

(5:15)

When the sample is viscoelastic, its properties are best captured in a dynamic experiment, in which the magnetic field is sinusoidally modulated. The resulting timedependent displacement of the bead, x(t), is given by the following expression: xðtÞ = x0 expðiðωt − ’ÞÞ

(5:16)

As we have seen in Section 5.3, a complex exponential can be rewritten as a trigonometric function using Euler’s formula. Analogous to the dynamic macrorheological experiment, we can find the frequency-dependent storage and loss moduli, G′(ω) and G″(ω), as follows: G′ðωÞ =

f0 cos ’ 6πrjx0 ωj

(5:17a)

G′′ðωÞ =

f0 sin ’ 6πrjx0 ωj

(5:17b)

5.5.2.2 Passive microrheology Passive microrheology experiments are based on the same principles as active ones, but in contrast to those, the passive variant solely relies on thermal energy, kBT, to drive the probe-particle motion through the sample. Based on that premise, the probes’ mean-square displacement, Δx2 ðtÞ, is observed as a function of time to characterize the sample’s viscoelastic properties.

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In a purely viscous medium, only Fickian diffusion of the probes takes place. In this case, the probes’ mean-square displacement obeys the Einstein–Smulochowski equation: hΔx2 ðtÞi = 6Dt

(5:18)

Together with the Stokes–Einstein equation for the diffusion of spherical particles in a viscous medium, we can express the sample’s viscosity as follows:   kB T kB Tt η = ^ G′′ðtÞ = = 6πDr πrhΔx2 ðtÞi

(5:19)

In a purely elastic medium, the diffusion of the probes is constrained. They may move a certain short distance during which they do not yet realize the constraint exerted by the solid medium, but as soon as they feel the elastic trapping of their surrounding, they cannot move further. This means that their time-dependent mean-square displacement, hΔx2 ðtÞi, first exhibits free diffusion with Einstein–Smulochowski-type scaling, but then eventually reaches a plateau value, hΔx2 ip (Figure 73). Balancing the driving thermal energy to the dragging elastic energy, with the latter expressed in a Hooketype form with the material’s spring constant, κ, yields: 1   kB T = κ Δx2 p 2

(5:20)

Figure 73: Mean-square displacement of probe particles, hΔx 2 ðt Þi, in an elastic medium as a function of time, t. At short times, the particles travel so short distances that they do not yet realize the elastic trapping in their surrounding. At long times, by contrast, the trapping is fully operative and prevents motion of the probes further than a distance reflected by the balance of their thermal driving energy and the counteracting elastic energy of the medium.

The loss modulus, G″, can be neglected in a purely elastic environment; as a consequence, the storage modulus, G′, can be determined as ) G′ðtÞ ⁓

κ kB T ≈ r rhΔx2 ip

(5:21)

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The intermediate case of a viscoelastic medium relates to the intermediate part of the curve in Figure 73, where the probes partially but not yet fully experience elastic constraint on their diffusive motion. In this range, the Einstein–Smulochowski equation (5.18) is extended by an exponent α(t): hΔx2 ðtÞi ⁓ tα

(5:22)

In a purely viscous medium, α(t) = 1 and the probes can diffuse freely. In a purely elastic medium, by contrast, α(t) = 0, and we find a plateau value for the probes’ mean-square displacement, as described above. When 0 < α(t) < 1, the probe displacement is called subdiffusive. This is the case in a medium that has both viscous and elastic properties. The curve in Figure 73 shows that α(t) is time dependent. At short times, the potential elastic constraint of the surrounding medium is not yet felt by the moving particles, such that α(t) ≈ 1. At longer times, the particles more and more experience such constraint, such that α(t) < 1 (more and more as time proceeds), and eventually, in the long-time limit, the elastic constraint is fully felt and prevents particle motion further than the plateau value of hΔx2 ip , such that α(t) = 0. From α(t), the complex modulus, G*, with its two parts G′ and G″, can be determined based on a calculation by Mason and Weitz: h πi (5:23a) G′ðωÞ = GðωÞ cos αðωÞ 2 h πi G′′ðωÞ = GðωÞ sin αðωÞ (5:23b) 2 with GðωÞ =

πrhΔx2

k T  1 B ω iΓ½1 + αðωÞ

(5:23c)

5.6 Principles of viscoelasticity 5.6.1 Viscoelastic fluids: the Maxwell model Now that we have understood that many materials, especially the soft-matter type, display mechanical properties with both elastic and viscous contributions, we have to find a way to mathematically describe this viscoelastic mechanics. A suitable approach is to start from simple mechanical models that combine both the stereotype mechanical element of an elastic body, which is a spring, and the stereotype element of a viscous fluid, which is a dashpot. The easiest way to combine these two elements is to connect them in series, as shown in Figure 74. This is done in the Maxwell model. When stress is applied to this model by an external deforming force, the elastic spring

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Figure 74: A Maxwell element is composed of a purely viscous damper (a dashpot) and a purely elastic spring connected in series. When stress is applied, the elastic spring deforms instantly, followed by delayed irreversible deformation of the viscous damper. Once the stress is released, only the spring snaps back to its original position, whereas the dashpot remains deformed.

element deforms instantly, followed by a delayed irreversible deformation of the viscous damper element that relaxes the stress in the deformed spring as far as possible. If that relaxation is not allowed to come to completion, there is residual stress in the system, but once this stress is released externally, only the spring element will snap back to its original position, whereas the dashpot element will remain deformed irreversibly. The applied stress is equal in both parts, σ = σ1 = σ2 , whereas the total strain is the sum of the strain of the two separate parts, ε = ε1 + ε2 . The same holds true for its time derivative dε dε1 dε2 + = dt dt dt

(5:24)

The elastic contribution by the spring is given by Hooke’s law; in differential form it reads dσ dε1 =E dt dt

(5:25)

The viscous contribution from the dashpot is described by Newton’s law as σ=η

dε2 dt

(5:26)

We can now use the latter two equations and insert them in eq. (5.24) to generate dε 1 dσ σ = + dt E dt η

(5:27)

Solving this differential equation with the boundary condition of constant deformation, dε=dt = 0, yields



Et t = σ0 exp − (5:28) σðtÞ = σ0 exp − η τ The latter equation quantifies the time-dependent stress relaxation of the Maxwell element; it contains a parameter that is essential for that: the relaxation time, τ = η=E. This parameter connects two macroscopic quantities, the elastic modulus, E, and the

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viscosity, η, by a microscopic time constant, τ.68 In a viscoelastic medium, τ is the characteristic time of molecular rearrangement. On timescales longer than τ, the molecules (or macromolecules in case of a polymer sample) can fully rearrange themselves, such that viscous flow dominates the viscoelastic properties. By contrast, on timescales shorter than τ, the (macro)molecules cannot yet rearrange themselves, such that elastic solidity dominates the viscoelastic properties. At the timescale of τ, the one behavior transitions into the other, and we observe marked viscoelastic properties. For polymers, τ is in the range of microseconds to seconds. With that, it falls right into the domain that many experiments and practical applications cover. As a result, polymers display rich viscoelastic mechanics in practice. The exact value of τ, and with that, the extent of viscous or elastic dominance on their mechanics, depends on parameters such as the polymer size, shape, and interactions. From the preceding mathematical consideration, we see that the Maxwell element is good to model an experimental situation of constant strain, which leads to time-dependent stress relaxation, as shown on the left-hand side of Figure 75; this is

Figure 75: Relaxation experiment (left) and creep test (right) applied to a Maxwell element. During the relaxation experiment, the spring is first stretched, but then the damper follows and relieves the spring, such that stress can relax over time. In the creep experiment, the spring first shows an initial response that is then followed by the dashpot, which starts to exhibit normal Newtonian flow, such that the deformation steadily proceeds. Note that the latter behavior is not creep in a sense as we have defined it in Section 5.2.1.2, but instead, just flow. Hence, the Maxwell element is good at modeling stress relaxation in a viscoelastic fluid, but not creep of a viscoelastic solid.

 At this point, the identity η = E τ was introduced as a pure mathematical tool to replace η/E in eq. (5.28) by a time constant τ to turn the exponential decay function exp (–Et/η) into the mathematically general form exp (–t/τ). With that, we have obtained the identity η = E τ as a sort of “byproduct”. It is, however, one of the most central equations in the field of rheology, with very fundamental physical meaning. Later on, in Section 5.7.1, we will derive it again “properly”. Anyhow, we may already understand it conceptually here: viscosity (η) is composed (= a product) of the ability of a medium to initially store energy upon deformation (E) times how long it takes to dissipate that (τ).

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typical for a viscoelastic fluid. By contrast, if we consider the opposite case of a constant stress, dσ=dt = 0, then eq. (5.27) just turns into Newton’s law. As a result, in this case, the Maxwell element just responds with some initial strain of the spring followed by normal Newtonian flow of the dashpot, as shown on the right-hand side of Figure 75. This is not realistic; we know from Section 5.2.1.2 that a viscoelastic solid exhibits time-dependent creep under the condition of constant stress instead. As an alternative to the purely mathematical discussion above, we can also interpret the mechanical responses of the Maxwell element shown in Figure 75 in a conceptual way. If we apply constant strain to the Maxwell element, as shown in the upper-left schematic in Figure 75, the resulting time-dependent stress initially rises sharply, but then drops back to its original value over time, as sketched in the lower left in Figure 75. This is because the spring is stretched instantly, but then the dashpot follows and relieves the strained spring. In this way, stress can relax over time, following an exponential function as in eq. (5.28):



t t = ε0 E0 exp − (5:29a) σðtÞ = σ0 exp − τ τ From that, we can calculate a time-dependent elastic modulus, E, as follows:

t EðtÞ = E0 exp − (5:29b) τ By contrast, if we apply constant stress to the Maxwell element, as sketched in the upper right of Figure 75, the resulting time-dependent strain initially rises to some extent, and then grows linearly as long as stress is applied, as shown in the lower right of Figure 75. Once the stress ceases, the strain drops by the exact same amount as it rose at the start, but retains the increase it has gained over time. Conceptually, the initial response can be attributed to the instant deformation of the spring. The dashpot follows and starts to flow, such that the deformation proceeds to rise. After the experiment, the spring returns to its original unstrained form, thus creating the drop in the strain signal. Note that this kind of behavior, however, is not creep in a sense as we have defined it in Section 5.2.1.2, but instead, just flow. Hence, the Maxwell element is good at modeling stress relaxation in a viscoelastic fluid, but not creep of a viscoelastic solid. For the latter experimental situation, as a complement to the time-dependent elastic modulus in the preceding case, a time-dependent creep compliance can be calculated:

σ0 1 t (5:30a) t = σ0 + εðtÞ = ε0 + η0 E0 η0 J ðt Þ = J0 +

t η0

(5:30b)

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In a dynamic experiment, we apply a sinusoidally modulated strain to the Maxwell element. We may imagine easily how the response should be in the limit of high and low frequencies. At high frequencies, the spring element can respond instantly even to a quickly modulad sinusoidal load, whereas the dashpot cannot respond tue to its inertia. As such, in that regime, the response of the Maxwell element is predominatly elastic. By contrast, at low frequencies, the dashpot can fully follow the motion of the then just slowly agitated spring to relieve it from the external strain. As such, in that regime, the response of the Maxwell element is predominantly viscous. In addition to that qualitative line of though, we may also treat the Maxwell element quantitatively. As we have learned from Section 5.3, the involved complex strain, ε*, and the complex stress resulting from it, σ*, can be expressed as exponential functions, of which we also need the derivatives in the following: ε* = ε0 expðiωtÞ ) σ* = σ0 expðiðωt + δÞÞ )

dε* = iωε0 expðiωtÞ dt

(5:31a)

dσ* = iωσ0 expðiðωt + δÞÞ dt

(5:31b)

The starting point for mathematical modeling of the Maxwell element under the condition of a dynamic experiment is again eq. (5.27), which we expand by a factor of 1 = η=η on the right side, as this allows us to rearrange the equation as follows:

dε 1 η dσ σ 1 η dσ dε η dσ = + = +σ , η = +σ (5:32) dt E η dt η η E dt dt E dt When we apply the fundamental equation η = E0 τ0 to the left-hand side of eq. (5.32) along with applying a rearranged form of it, τ0 = η=E0 , to the right-hand side of eq. (5.32), we get E0 τ0

dε dσ = τ0 +σ dt dt

(5:33)

Plugging in the exponential functions given in eq. (5.31a) and (5.31b) yields E0 τ0 iωε0 expðiωtÞ = τ0 iωσ0 expðiðωt + δÞÞ + σ0 expðiðωt + δÞÞ = σ0 expðiðωt + δÞÞðτ0 iω + 1Þ

(5:34)

That can be rearranged as follows: E0 τ0 iω σ0 expðiðωt + δÞÞ σ = = = E* τ0 iω + 1 ε0 expðiωtÞ ε

(5:35)

Expansion of the left-hand side of eq. (5.35) with the conjugate complex number allows us to write out the real and imaginary parts of the right-side, E* = E′ + iE′′:

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E′ =

E0 τ0 2 ω2 τ0 2 ω2 + 1

(5:36a)

E′′ =

E0 τ0 ω τ 0 2 ω2 + 1

(5:36b)

Analogous calculations can be made for the complex oscillatory shear modulus G* = G′ + iG″. Let us examine the latter two equations more closely by first focusing again on two extreme frequency regimes, a high and a low one. At low frequencies, the denominators of both eq. (5.36a) and (5.36b) have a value close to 1, because the τ0 2 ω2 term in them becomes very small then, and therefore negligible compared to the 1 in them. Then, the power-law form of the remaining numerators of eq. (5.36a) and (5.36b) gives straight-line graphs when plotted in a double-logarithmic representation, with a slope determined by the exponent of the variable ω. Figure 76 shows such a log–log plot, in which we find a slope of 2 for E′, and a slope of 1 for E″.

Figure 76: Frequency-dependent elastic modulus of a Maxwell element, E(ω), in a doublelogarithmic representation. The graphs can be separated into two frequency regimes, separated by the crossover point of E′ and E″. At low ω, the material has enough time to relax, and its viscoelastic properties are dominated by its viscous contributions, so E″ > E′. At high ω, the material cannot fully relax anymore, and its viscoelastic properties are dominated by its elastic contribution, so E″ < E′.

At high frequencies, by contrast, the τ0 2 ω2 term in the denominators of both eq. (5.36a) and (5.36b) is much larger than the 1 in them, such that the latter can be neglected. Many of the variables in the numerators and in the denominators cancel out then. For E′, no dependence of ω remains due to that cancelling, such that its graph is a

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flat line with a value of E0 at high frequencies. For E″, we obtain a frequency dependence of ω − 1 due to that cancelling, such that its graph is a straight line with slope –1 in a double-logarithmic representation such as the one shown in Figure 76. Both frequency regimes are separated by the terminal relaxation time, τ0, at a frequency of ω0 = 1/τ0. At that frequency, according to eq. (5.36a) and (5.36b), both E′ and E″ have values of ½E0, as also seen in Figure 76; hence, both the viscous and elastic part contribute equally here. Now let us do what we have done several times already in the course of this book and use this mathematical insight to obtain conceptual insight into the timedependent mechanical properties of a viscoelastic Maxwell-type fluid. Again, we discuss this in terms of a low- and a high-frequency regime. At low frequencies, there is much time for relaxation of the system. A deformed material then has enough time to rearrange its building blocks on a microscopic scale; for example, if it is a polymer system, the chains have enough time to relax back to their equilibrium conformations after deformation. Reconsidering the Maxwell element from Figure 74, the time is long enough for the damper to relax the energy stored in the spring. As a result, at such long timescales, the material’s mechanical properties are dominated by its viscous contribution. This becomes more and more dominant the longer the timescale is or, in turn, the lower the frequency is. We see that clearly in Figure 76: the lower the frequency, the more does the E″-curve outweigh the E′-curve. At high frequencies, the picture is to the contrary: here, there is not enough time for relaxation of the system. Visually speaking, the damper does not have time to be active. On these short timescales, the material therefore behaves as an elastic solid. This becomes more and more dominant the shorter the timescale is or, in turn, the higher the frequency is. Again, we see that clearly in Figure 76: the higher the frequency, the more does the E′-curve outweigh the E″-curve. The transition from the low-frequency viscous regime to the high-frequency elastic regime happens at the terminal relaxation time, τ0. It denotes the time it takes for the building blocks of any material, which in our case are polymer chains, to rearrange themselves over a distance equal to their own size. On timescales longer than τ0, such effective displacement of the microscopic building blocks against one another is possible, causing flow on the macroscale. On timescales shorter than τ0, such displacement is not possible, and the material responds elastically. When the time is exactly τ0, then both the viscous and the elastic parts of the sample’s mechanical spectrum contribute equally, and E′ and E″ have identical values of E0/2. We can observe this as the crossover point of the graphs in Figure 76. Polymers and colloids, which are composed of large building blocks with high molar masses, exhibit τ0 in the milliseconds to seconds range. This is advantageous, because it makes these relaxation processes observable. The same basic processes are also present in common low molar-mass materials, such as water, but in those materials, τ0 is in the nanosecond range. These timescales are much harder or even impossible to assess experimentally. Fortunately, however, the knowledge gained

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from observation of the slower polymeric or colloidal systems can be adapted to the hard-to-observe classical materials due to their common physical grounds. This insight of the French physicist Pierre-Gilles de Gennes was awarded with the Nobel Prize in Physics in 1991. In addition to leading to that scientific breakthrough, the millisecondto-second timescale of relaxation of polymeric and colloidal matter makes these rich in mechanical behavior on practically relevant timescales, opening a plethora of mechanical application possibilities. It is for these that polymers are in fact most famous.

5.6.2 Viscoelastic solids: the Kelvin–Voigt model The Maxwell model is not the only possible combination of a Hookian spring and a Newtonian dashpot. When the two are connected in parallel, we generate a system that is the basis of the Kelvin–Voigt model, as depicted in Figure 77.69 Upon application of a constant stress, the Kelvin–Voigt element exhibits creep and deforms at a decreasing rate, asymptotically approaching the steady-state strain σ0/E. When the stress is released, the material gradually relaxes to its undeformed state. A graphical representation of this experiment is shown in Figure 78. Since the deformation is reversible (though not suddenly), the Kelvin–Voigt model describes an elastic solid that also exhibits some viscous contribution.

Figure 77: Kelvin–Voight element composed of a purely viscous damper and a purely elastic spring connected in series.

By simple geometrical need, the total strain of the Kelvin–Voigt element is the same as the individual strain of each element: ε = ε1 = ε2 , whereas the total stress is the sum of the individual stresses: σ = σ1 + σ2 . We can express the elastic contribution to the Kelvin–Voigt body by Hooke’s law as

 The model was originally proposed by Oskar Emil Meyer in 1874 in his essay “Zur Theorie der inneren Reibung”, published in the Journal for Pure and Applied Mathematics. It was later independently “rediscovered” by the British physicist William Thomson, who later became the first Baron Kelvin and by the German physicist Woldemar Voigt in 1892.

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Figure 78: Creep test of a Kelvin–Voigt element. The response of the spring is delayed by the simultaneous but slow response of the damper. The final deformation will eventually approach that for the pure elastic part, σ0/E. Once the stress is released, the entire Kelvin–Voigt element slowly relaxes to its original position, which means that the deformation is reversible.

σ1 = Eε

(5:37)

and the viscous contribution by Newton’s law as σ2 = η

dε dt

(5:38)

We can now plug the latter two equations into σ = σ1 + σ2 to generate dε σ Eε = − dt η η

(5:39)

Solving this differential equation with the boundary condition of a constant applied stress of σ = σ0 yields



σ0 Et σ0 t = (5:40) εðtÞ = 1 − exp − 1 − exp − E E η τ Here, τ is the retardation time: it quantifies the delayed response to an applied stress and is a material-specific parameter. The Kelvin–Voigt model is good at predicting creep, because in the infinite σ time limit the strain approaches a constant value of limt!∞ ε = E0 , whereas the Maxwell model predicts an infinite linear relationship between strain and time. Thus, the Kelvin–Voigt model element is good to model a viscoelastic solid. By contrast, as we have seen above, the Maxwell model is good to model a viscoelastic liquid.

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5.6.3 More complex approaches By addition of further Hookian spring and Newtonian damper elements, a plethora of more complex models have been created to describe the viscoelastic properties of many substances. These models include but are not limited to the standard linear solid or Zehner model, the generalized Maxwell model, the Lethersich, the Jeffreys, and the Burgers model. The latter one, which is depicted in Figure 79, combines a Kelvin–Voigt with a Maxwell element by connecting them in series. It was developed by the Dutch physicist Johannes Martinus Burgers in 1935 to model the behavior of bitumen. It is sometimes also applied to polymer-based materials.

Figure 79: The Burgers model of viscoelastic media by a combination of a Kelvin–Voigt and a Maxwell element. It is tailored to accurately predict the viscoelastic properties of bitumen, but is sometimes applied to polymers as well.

In this model, the strain is the sum of the individual strains of the mechanical elements, just like in the Maxwell model: εð t Þ = ε 1 + ε 2 + ε 3 = ε 1 + ε 2 ð t Þ + ε 3 ð t Þ The stress is equal in the three parts, as also in the Maxwell model: σ = σ1 = σ2 = σ3 = σ0 According to the Kelvin–Voigt model, the total stress in the second part is the sum of those in the elastic and the viscous mechanical elements: σ0 = σ2 = σ2e + σ2v Plugging both the latter equations into the first one gives the time-dependent strain εðtÞ for the Burgers model:

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σ0 σ0 E2 t σ0 t + εðtÞ = ε1 + ε2 ðtÞ + ε3 ðtÞ = + · 1 − exp − E1 E2 η2 η3

Questions to Lesson Unit 12 (1) What is a major disadvantage of classical oscillatory shear rheology? a. It is only feasible for polymer gels or melts and does not work for low-viscous polymer solutions. b. The temperature cannot be kept at a constant level since the sample gets warmed up with time due to dissipation. c. The amount of sample needed is large compared to the amount one obtains in a typical laboratory-scale synthesis. d. After the measurement the sample cannot be used for further analytics, because the structure is damaged during the experiment. (2) What is the relaxation time? a. The time needed for the microscopic building blocks of a material to move distances of their own size. b. The time necessary for a building block to move distances of the size of a polymer chain. c. The time interval it takes for stress relaxation. d. The time that a sample needs to re-equilibrate after a rheological experiment.

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(3) Remember: The Maxwell model consists of a dashpot and a spring connected in series. Choose the correct stress response σðtÞ to the given applied strain signal εðtÞ for a viscoelastic fluid.

a. b. c. d.

① ② ③ ④

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(4) Remember: The Kelvin–Voigt model consists of a dashpot and a spring connected in parallel. Choose the correct strain response εðtÞ to the given applied stress signal σðtÞ for a viscoelastic solid.

a. b. c. d.

① ② ③ ④

(5) What is a major advantage of the small probe volume in microrheology? a. The measurement is a lot faster. b. The obtained data applies locally; even spatial resolution is possible. c. The embedded probe particles can be easily removed. The sample is available for further measurements after that. d. The experimental realization is easy.

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(6) What is the meaning of the exponent α in the adapted Einstein–Smoluchowskilike relation for the mean-square displacement of a particle hΔ x2 ðtÞi ⁓ tα ? a. α captures the elastic properties only; the higher the exponent, the more dominant is the elasticity of the medium. b. α captures the viscous properties only; the higher the exponent, the more dominant is the viscosity of the medium. c. α captures the elastic properties as well as the viscous ones; for α ¼ 1, the medium shows purely viscous characteristics, whereas for α ¼ 0, the medium is purely elastic with a plateau value Δ xp2 . d. α captures the elastic properties as well as the viscous ones; for α ¼ 1, the medium is purely viscous, whereas for α ¼ 2, it is purely elastic with a plateau value Δ x2p . (7) What happens to a particle in a not purely viscous medium? a. The particle is partly dragged, which leads to superdiffusion. b. The particle is partly trapped, which leads to superdiffusion. c. The particle is partly dragged, which leads to subdiffusion. d. The particle is partly trapped, which leads to subdiffusion. (8) A probe sample consists of a polymer with network structure and embedded probe particles. Which statements about the time-dependent displacement of the particles isare false? a. At long timescales, the particles are constrained by the polymer network; their diffusion is restrained. b. At long timescales, the particles are constrained more and more up to a value at which the elastic outnumbers the driving thermal energy, and the diffusion is stopped. c. At long timescales, the particles are constrained more and more until the energies of kB T and of the elastic constraint are at balance and the plateau value is reached. d. At short timescales, the particles do not “feel” any constraint, and they can diffuse freely.

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5.7 Superposition principles LESSON 13: SUPERPOSITION PRINCIPLES Παντα ρηει – everything flows. This fundamental aphorism of rheology holds for polymers in particular, because even if their momentary appearance may be that of an elastic solid, in many cases, waiting long enough will let them flow. In this lesson, you will see which fundamental physical principle lies behind this, and how it allows us to superimpose data collected in different experiments to generate mechanical spectra spanning multiple decades in time or frequency.

Up to this point, we have looked upon some simplified models to describe the mechanical response of a viscoelastic material. But what do we observe in a real experiment? Let us look at a typical amorphous polymer and consider its creep compliance, J, as a function of time, t, for a range of temperatures, T, in Figure 80. (Note that J and t are the inverse quantities of E and ω, respectively.) At high temperatures, J has high values and increases proportionally with time t, which means that the material exhibits purely viscous flow. We can understand that by making ourselves clear that in this limit of viscous dominance, J is dominated by its viscous part J”, and since it is the inverse of the elastic modulus, this part is connected to 1/E”, for which we have seen from the quantitative treatment of the Maxwell body in Figure 76 that in

Figure 80: Creep compliance, J, of an amorphous polymer probed as a function of time, t, at various temperatures. At high temperature, J is proportional to t, meaning that the material exhibits purely viscous flow. At low temperature, by contrast, J is fully independent of t, which means that the material shows a purely elastic response. At intermediate temperatures, J is first independent of t and then becomes proportional to it.

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the limit for which it dominates the mechanics (which is in the limit of low frequencies, corresponding to long times), it has a frequency-dependent scaling of E” ⁓ ω1, and so its inverse has a frequency-dependent scaling of J” ⁓ t1. At low temperatures, by contrast, J has low values and is independent of t, which means that the material shows a purely elastic response. Again, we can understand that by making ourselves clear that in the limit of elastic dominance, J is dominated by its elastic part J’, and, since it is the inverse of the elastic modulus, this part is connected to 1/E’, for which we have seen from the quantitative treatment of the Maxwell body in Figure 76 that in the limit for which it dominates the mechanics (which is in the limit of high frequencies, corresponding to short times), it is frequency- (and therefore also time-)independent and has high values (approaching E0), so as a result, its inverse, 1/E’ = J’, is also frequency- and time-independent and then has low values. At intermediate temperatures, J has intermediate values, and it is first independent of t but then becomes proportional to it. To sum up, we see that the polymer displays a rich mechanical behavior depending on both time, t, and temperature, T. In this section, we will see how the quantities t and T are related to each other in a dataset such as the one shown in Figure 80, and, while doing so, we will also learn some fundamental principles of the overlay, or superposition, of rheological data.

5.7.1 The Boltzmann superposition principle The Boltzmann superposition principle formulates that the current state of stress or deformation of a viscoelastic material is the result of its history.70 In mathematical terms, this means that the total stress or deformation applied to a material is the sum of the time-dependent evolution of every single stress or deformation increment that it has experienced. In other words, the system remembers any former stresses or deformations and continues to relax or creep from them even when new ones are applied on top, as depicted in Figure 81. Let us consider an example to illustrate the Boltzmann Superposition principle. The sum of all shear deformations applied to a material can be expressed as X γi (5:41) γ= i

According to Hooke’s law, the resulting sum of stresses can be expressed by the sum of the shear modulus G multiplied by the individual deformation γi:

 In a philosophical view, this is just like with us as individuals: our histories have made us, or “shaped us”, into the persons we are today.

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Figure 81: When multiple relaxation experiments (left) or creep tests (right) are performed in a sequence, the system remembers the stresses or deformations of each single one and continues to creep or relax from these even when new ones are applied on top.

σ=

X i

σi =

X

Gðt − ti Þγi

(5:42)

i

Here, t is the time of measurement, whereas ti is the time of application of the strain increment γi. We can derive an equation for the case of continuous change by integration: ð ðt ðt     dγ ′ σ ðt Þ = d σ = G t − t dγ = G t − t′ dt′ (5:43a) dt −∞ −∞ To simplify this expression, we use the method of substitution and write t − t′ ≡ u to generate ð0 σ ðt Þ = −

−∞

_ =+ GðuÞγdu

ð∞

GðuÞγ_ du

(5:43b)

0

This can be solved if we have an expression of the time-dependent modulus at hand. As a simple example, we take the relaxation function GðuÞ = G0 expð− uτÞ from the Maxwell model and insert it: ð∞  u exp − γ_ du σðtÞ = G0 (5:44) τ 0 In a continuous-shear scenario, the shear rate γ_ is constant. For a monodisperse polymer sample, this yields ð∞  u h  ui∞ _ _ exp − du = G0 γ_ − τ exp − = G0 γτ (5:45) σ ð t Þ = G0 γ τ τ 0 0

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We can rearrange that equation to derive an expression for the stress divided by the shear rate, which, according to Newton’s law, is the viscosity η: σ ≡ η = G0 τ γ_

(5:46)

Equation (5.46) is a very fundamental relation in the field of rheology, because it connects the macroscopic properties viscosity, η, and shear modulus, G, to the microscopic property of the relaxation time, τ, that contains information about the structure of the system’s building blocks (here: polymer coils).71 With that, we have a structure–property relation. This relation defines flow as a relaxation process, in which the viscosity is composed of the relaxation modulus and the relaxation time. We can understand that quite illustratively: viscosity (η) quantifies the ability of a fluid medium to initially store energy upon shear deformation (G) times how long it takes to dissipate that by subsequent relaxation (τ), that means, by positional change of the fluid’s building blocks (molecules or particles). The higher both these contributors are, the harder it is to stir or pump the fluid. In the more complex case of a polydisperse relaxation time spectrum, the viscosity is expressed as the sum of multiple relaxation contributions:

X X u σ ) G exp − Gτ (5:47) ≡ η = GðuÞ = i i i i i τi γ_ This is the case when either the system’s building blocks are polydisperse in size, or if monodisperse building blocks exhibit multiple, hierarchical relaxations, such as coil-internal Rouse modes. In summary, we see that relaxation occurs on the timescale τ, or on a time spectrum of multiple overlaying τi, and that these determine the viscosity by the elementary formula η = G0 τ. In the next section, we will take a closer look at the temperature dependence of that.

5.7.2 The thermal activation of relaxation processes Flow requires the molecules or particles that constitute a material to change their positions. This necessitates them to slip by each other, for which they have to overcome an energetic activation; it also necessitates some free volume around them to move into. Both these prerequisites are temperature dependent, which means that, conceptually, they can be treated together as an effective activation energy barrier,

 Here at this point, we have derived the fundamental identity η = G τ “properly”, whereas before, in Section 5.6.1, it was obtained more as a “mathematical by-product”.

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Ea,eff, that determines the temperature-dependent duration of positional changes τ, in the form of an Arrhenius-type expression:72





Ea;eff Ea V* · exp (5:48) ≈ exp τ ⁓ η ⁓ exp kB T kB T Vf In eq. (5.48), the first term features the activation energy, Ea, for each molecular or particulate positional change past other molecules or particles, whereas in the second term, V*=Vf reflects the probability to find enough free space in the vicinity during such an attempt of positional change, assessed by the free volume in the system, Vf, and the volume required for each molecular or particulate rearrangement, V*.73 The effective activation energy Ea,eff is more often overcome at high temperatures, which means that high temperatures facilitate positional change of the molecules. Thus, high temperatures lead to small (=short) relaxation times and therefore also small (=low) viscosities, whereas low temperatures lead to high (=long) relaxation times and therefore also high viscosities. The lowest possible temperature at which movement is still possible at all is the glass-transition temperature, Tg. Below this specific temperature, thermal activation is not possible anymore, all movement is frozen, and the relaxation time is infinite, τ ! ∞.

5.7.3 Time–temperature superposition The fundamental connection of temperature to the rate of chain relaxation, which – in turn – is connected to the viscosity according to eq. (5.46), allows us to shift and superimpose viscoelastic data that were recorded at different temperatures so as all superimpose on a common time axis; this is done by shifting these data along the time axis by application of a shift factor, aT:

t (5:49) Gðt, T Þ = bT G , T0 aT The left-hand side of eq. (5.49) features G(t,T), which is the shear modulus measured at a time t and temperature T. The right-hand side of the equation tells us that the 72 This is analogous to the Arrhenius relation for the rate constant k of a chemical reaction. This rate constant increases with temperature, because the activation energy is overcome more frequently at higher temperature. The rate constant, in turn, is inversely related to the reaction half-life time, τ, in the form of k ⁓ τ1, whereby the constant of proportionality depends on the order of the reaction. In that inverse form, the exponential term on the right side of the equation has a negative argument, which is indeed the common way ofnotation of the Arrhenius law in reaction kinetics.  Both these contributions have first been introduced independent of one another according to the free volume theory by Doolittle (J. Appl. Phys. 1951, 22(12), 1471–1475) and the theory of rate processes by Eyring; later on, however, it was recognized that they must both be accounted together by Macedo and Litovitz (J. Chem. Phys. 1965, 42(1), 245–256).

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same modulus will also be measured at a different temperature, T0, if the timescale is shifted by a factor aT. Often, a suitable standard reference temperature is taken for T0, for example, room temperature or the polymer’s glass-transition temperature, Tg. To also account for potential density differences of the material at the temperatures T  and T0 (due to thermal expansion), a further shift factor bT = ρT ρ0 T0 is often applied as well, as also seen in eq. (5.49). The shift factors are not just loose arbitrary quantities, but they can be conceptualized and understood to be part of further interrelations. One such relation is the Williams–Landel–Ferry (WLF) equation. It was originally discovered empirically, but it can also be derived from eq. (5.48) as follows: if two viscosities η1 and η2 are measured at two temperatures T1 and T2, their ratio according to eq. (5.48) is

η1 V* V* (5:50) = exp − η2 Vf1 Vf2 if we assume that the temperatures T1 and T2 are high enough such that the influence of the temperature on the fundamental positional change process can be neglected ðkEaT − k EaT ≈ 0Þ. Furthermore, the relationship between the free volumes at B 1 B 2 T1 and T2 is as follows: Vf2 = Vf 1 + ΔαVm ðT2 − T1 Þ

(5:51)

Here, Δα is the difference between the material’s thermal expansion coefficients at both temperatures, and Vm is the eigenvolume of the chain material. Logarithmization of eq. (5.50) as well as rearrangement and insertion of eq. (5.51) yields ln

η1 V* = · η2 Vf1

T2 − T1  ΔαVm + ðT2 − T1 Þ Vf

(5:52)

2

If T1 is replaced by the glass-transition temperature of the material under consideration, Tg, the equation for any second temperature T = T2 is     V* ηg Vf, g · T − Tg (5:53) = V   ln  f, T ηT + T − Tg ΔαVm

Based on temperature-dependent viscosity measurements, Williams, Landel, and Ferry estimated the volume V* required for a microscopic positional change as V* ≈ 40 · Vf, g and the free volume at the glass-transition temperature, Vf,g, as Vf, g ≈ 52 · ΔαVm . Furthermore, in a polymer melt, V* can be approximately equated with the eigenvolume of the chain material Vm. If we do that, then the stepwise change in the thermal expansion coefficient Δα is determined to be Δα = 4.8·10–4 K–1.

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By plugging in these values into eq. (5.53), we obtain the WLF equation that relates the shift factor aT at temperature T to the glass-transition reference state:     − c1 T − Tg − 17.44 T − Tg τ ηðT Þ  =   log aT = log ffi log   = (5:54) τg η Tg c2 + T − Tg 51.6K + T − Tg In the latter form, we have further transferred to decimal logarithms ðlog x = ln x · log eÞ. The numbers c1 = 17.44 and c2 = 51.6 K in eq. (5.54) are universal for many different polymers if Tg is taken as the reference temperature. The utility of the shift factors introduced in eq. (5.49), which we may either determine empirically or calculate with the WLF eq. (5.54), is that they allow us to create a master curve of many individual datasets of G(T, t) referenced to new variables G(T0, t/aT), as shown in Figure 82 (note that this plot has a frequency axis instead of a time axis, which is just the inverse). Such a master curve may then display the full rheological spectrum of a polymer, covering many orders of magnitude in G (from single pascals to gigapascals!) and t (or ω) (from nano- or milliseconds to decades!), which would be inaccessible by actual experimentation.

Figure 82: Construction of a rheological master curve according to eq. (5.49). Full lines denote the storage modulus, G′, whereas dashed lines denote the loss modulus, G″. The reference temperature in this example is the glass-transition temperature. By assembling the single curves obtained at various temperatures over a frequency span of about three decades, as collected in the right part of the figure, and shifting them along the frequency axis through application of suitable shift factors aT, along with potential additional shifting along the modulus axis to account for density differences at different temperatures through the use of shift factors bT, we can create a master curve spanning 12 (!) orders of magnitude on the time axis, as shown in the left part of the figure.

In such a full rheological spectrum, we can recognize several distinctly different regimes, depending on the timescale and on the temperature of observation. Figures 83 and 84 display these different regimes in a schematic fashion. Note that Figure 83 is flipped compared to both Figures 82 and 84, because it shows the moduli E and G as a function of time t rather than frequency ω. However, this is just

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Figure 83: Schematic representation of a polymer’s full mechanical spectrum. At low temperatures or on very short times, the polymer is a glassy, energy elastic solid. When the temperature is raised or the time extended, the polymer enters a leather-like regime in which it is a viscoelastic solid described by the Kelvin–Voigt model. For samples that are composed of long chains, we then observe an intermediate rubbery plateau regime. When the temperature is sufficiently high or the time sufficiently long, the material will eventually flow like a viscoelastic liquid described by the Maxwell model. The Pascal values on the ordinate are typical for polymers in the melt state.

transformation of the type of displaying, but the characteristic domains remain identical in either representation.

Questions to Lesson Unit 13 (1) Which statement about the creep compliance J is true? a. The creep compliance is the inverse property of the Young’s modulus, meaning that if the storage modulus E′ is described by a cosine, the corresponding part of the creep compliance J ′ has the form of a sine, and the sames the case for the imaginary parts. b. The creep compliance is the inverse property of the Young’s modulus, meaning that the real and imaginary parts of J, J ′, and J ′′ are given by the reciprocal value of the respective part of the elastic moduli E′ and E′′; −1 −1 J ′ = E′ , J ′′ = E′′ . c. The creep compliance is a contrary property to the Young’s modulus, meaning that the real and imaginary parts of J, J ′, and J ′′ are given by the opposite part of the elastic moduli E′′ and E′. d. The creep compliance is a contrary property to the Young’s modulus, meaning that the real and imaginary parts of J, J ′, and J ′′ are given by 1 minus the respective part of the elastic moduli E′ and E′′; J ′ = 1 − E′, J ′′ = 1 − E′′.

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(2) What is the meaning of the Boltzmann superposition principle? a. Stress is additive up to the maximal value of all applied single stresses, the “superposition threshold.” b. Under sequential deformations of equal magnitude, stress always increases linearly. c. Stress is additive; upon application of deformation to a viscoelastic sample, the resulting stress adds upon preexisting ones. The same holds for deformations. d. Stress is amplified by the already given stresses in the material. The same increment in the applied deformation leads to a comparably higher increment in stress if the material is pre-stressed. The same goes for deformations. (3) What statement about the relation η ¼ G0 τ is true? a. It is a structure–property relation because the macroscopic concept of flow is given by the viscosity, which is connected to the shear modulus that stands for the stored energy upon deformation and the microscopic property of the relaxation time of the building blocks. b. It is a structure–property relation that is simply based on the mathematical connection of macroscopic properties (η; G0 ) and microscopic ones (τ). c. It is not a structure–property relation, since all properties are macroscopic. There is no easy way to describe viscosity, the shear modulus, or the relaxation time on a microscopic scale. d. It is not a structure–property relation, because none of the properties can be derived through analyzing the sample’s structure. (4) The relaxation time of a material that exhibits flow can be estimated with the following expression:





Ea; eff Ea V · exp ≈ exp τ ⁓ η ⁓ exp kB T Vf kB T

Which answer describes the different contributions to the final rightmost expression correctly? a. The relaxation time depends on the ratio of the already relaxed building blocks to the total building blocks, which is reflected in a Boltzmann distribution-like expression times a volumetric term. b. The relaxation time is given by a ratio of still unrelaxed to relaxed states in a Boltzmann distribution-like expression. The temperature dependence comes from the thermal energy that is released during relaxation. c. Relaxation requires an activation energy, which is represented by a temperature-dependent Arrhenius-like term times a term that represents the ratio of the unrelaxed volume to the total volume after relaxation.

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d. Relaxation requires an activation energy as well as volume available to occupy. Both contributions can be compiled in a final Arrhenius-like temperature-dependent term. (5) The expression for the relaxation time of a flow-exhibiting material has an Arrhenius-like form with

Ea; eff τ ⁓ exp kB T

Hint: The Arrhenius equation from reaction kinetics has the following form:

Ea; eff k ⁓ exp − kB T

What statement describes the relation between both correctly? a. The Arrhenius equation gives an expression for the temperature dependence of the rate constant k of a chemical reaction. Since this rate constant is the reciprocal of the reaction’s half-life time it has basically the same form as the expression for the relaxation time. b. The Arrhenius equation gives an expression for the temperature dependence of the rate constant of a chemical reaction. Since this rate constant decreases during a reaction, there needs to be a negative sign in the exponent. Both expressions are not the same but very alike. c. The Arrhenius equation gives an expression for the temperature dependence of the rate constant k of a chemical reaction. Since both expressions describe a time-based quantity with the unit of seconds, they can be considered as similar expressions. d. There is no trivial relation between both expressions. (6) How can we describe the fundamental principle on which the time–temperature superposition is based? a. The same relaxation processes can happen at different relaxation times at different temperatures. b. Higher temperatures lead to a greater expansion, which means the relaxation is accelerated. c. Different temperatures cause a change in the way of relaxation. The different relaxation processes can be superimposed. d. Through different frequencies, relaxation processes can be recorded that need different activation energies, which results in a wide range of temperatures that can be measured.

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(7) Why is time–temperature superposition essential for measuring mechanical spectra in rheology? a. There is only a small range of timescales at which measuring is practical. Increasing the range would be at the expense of the measurement’s precision. By measuring at different temperatures and shifting the data one can obtain data for a wider range of timescales. b. Technically rheometers can only cover a time range of three decades. By measuring at different temperatures and shifting the obtained data it is possible to cover a time range of up to 12 decades. c. Probed materials are often very thermosensitive. The time–temperature superposition allows measuring at different timescales at the same temperature to generate data for different temperatures by shifting. d. Since the heating supplies of a rheometer are very limited in their temperature range, time–temperature superposition allows measuring at different timescales and shifting the obtained data to get a wider temperature range. (8) A common PhD thesis takes a time of 3 years. Which frequency regime would you be able to cover in that time in a regular oscillatory rheological measurement? a. 10–1 rad s –1 b. 10–3 rad s –1 c. 10–5 rad s –1 d. 10–7 rad s –1

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5.8 Viscoelastic states of a polymer system LESSON 14: MECHANICAL SPECTRA The principle of time–temperature superposition that you got to know in the last lesson allows master curves of the time- or frequency-dependent moduli of polymer samples to be composed. Such rheological spectra feature multiple different domains, depending on in which state of mobility or trapping the chains in the sample are. The following lesson will give an overview of these states and summarize the rich variety of viscoelastic states of polymers.

5.8.1 Qualitative discussion of the mechanical spectra Let us discuss the schematic of a polymer viscoelastic spectrum in Figure 83 from left to right along the time axis (that can also be viewed as a temperature axis due to the principle of time–temperature superposition), or alternatively, the variant schematic in Figure 84 from the right to the left on the frequency axis (which is nothing else than a flipped time axis on log scales). At very short timescales, corresponding to very high frequencies and to low temperatures, the elastic modulus has high values in the range of gigapascals, and its storage part, G′ or E′, exceeds its loss part, G″ or E″. This means that the material is a tough elastic solid in this regime. A molecular-scale interpretation is that on these very short timescales, or at that low temperatures, no motion is possible at all in the sample, neither of the polymer chains as a whole nor of just parts of them. This is because the temperature is either so low that any motion is frozen, or because we consider timescales so short that these motions can just not be achieved yet. As a result, we have a sample with amorphous structure and without any dynamics: a glass. This glassy state features energy-elastic mechanics with a high modulus. At a first characteristic time, τ0,74 the shortest possible relaxation time (which is typically around 1 ns at room temperature for a polymer melt), at least single monomeric units have enough time to relax in a sense that they can diffuse over distances equal to their own size. At times longer than that, more and more additional relaxation modes are activated, in a sense that now also sequences of pairs, triples, quadruples, and so on of monomers can be displaced over distances equal to their own size. This sequential activation of more and more relaxation modes more and more drops the modulus if we go from left to right on the time axis in Figure 83 or from right to left on the frequency axis in Figure 84. The polymer is then in a markedly  We hereby name the shortest possible relaxation time of a polymer τ0; this is commonly done so in many textbooks, referring to the “elementary” relaxation time of the units in the chain, which somehow appears to be well represented by the index zero. In the framework of relaxation modes treated in Chapter 3.6.4, we had named that time τN instead (see Table 8 in that chapter). Note here that τ0 in the context of the present nomenclature equals τN in the context of the nomenclature related to the relaxation-mode concept.

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viscoelastic leather-like regime, as we see in Figures 83 and 84, where both the storage and loss moduli, G′ and G″, have very similar values and very similar time- or frequency-dependent scaling. The mechanical properties in that state can be treated analytically by the Rouse model (for modeling the polymer subchain relaxation modes and the dropping of the time- or frequency-dependent modulus as more and more of these modes are activated) or phenomenologically by the Kelvin–Voigt model (to phenomenologically treat the viscoelastic solid-like appearance of the specimen).

Figure 84: Full rheological spectrum of a polymer melt at room temperature. At the shortest possible relaxation time, τ0, which lays on the right extreme of the spectrum shown in the present frequency-dependent form, only subchains of single monomeric units can relax; following upon that, at frequencies left of that extreme, comes a viscoelastic regime in which sequential activation of relaxation modes of pairs, triples, quadruples, and so on of monomer units leads to dropping of the moduli. At the entanglement time, τe, chain segments realize mutual entanglement, leading to an intermediate elastic plateau. From the Rouse time on, τR, there is sufficient time for the entire chain to be displaced, such that it could actually fully and freely relax and flow already if it would not be trapped by entanglements. Getting loose of that constraint is eventually possible on timescales longer than the reptation time, τrep. From that time on, on the very left end of the spectrum shown here, the material is a viscoelastic liquid phenomenologically described by the Maxwell model, with G′ ⁓ ω2 and G″ ⁓ ω.

At the end of the viscoelastic leather-like regime that features just subchain motion, intuitively, we would expect to observe a transition to a free-flowing regime in which relaxation and displacement of the whole chains can occur. From the Rouse (and also from the Zimm) model of polymer dynamics, we know that this should happen on times longer than the Rouse (or the Zimm) time, τR (or τZ), which is denoted to be about 1 ms in Figure 84. Such a transition will indeed be observed in the case of samples with short chains. In samples with long chains, by contrast, such as those shown in Figure 82–84, before that free-flow regime, we observe something else: an intermediate rubbery elastic plateau. This plateau starts at a second characteristic time: the entanglement time τe, which is around 1 μs in Figure 84. At that time, segments of long chains realize that they are trapped by mutual entanglement. This constraint hinders them from relaxing fully and freely,

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thereby leading to a regime in which deformation energy cannot be relaxed but is stored, with a modulus in the range of about a megapascal. As in that regime, motion of the chains is already activated but topological entanglement still hinders them to relax, deformation of the sample is accommodated by decoiling of the chains; as a result, the elasticity in this regime is of entropic origin. When the time is eventually long enough for the polymer chains to disentangle and move by a mechanism named reptation on timescales longer than the reptation time τrep, which is around 1 s in Figure 84, the polymer shows terminal viscous flow. In this regime, the material is a viscoelastic liquid phenomenologically described by the Maxwell model, with G′ scaling with ω2 and G″ scaling with ω. From then on, the more time we give the material, the more will G″ exceed G′, meaning that the more will the material mechanics be dominated by its viscous properties.

5.8.2 Quantitative discussion of the mechanical spectra Now that we are able to visualize the full mechanical spectrum of a polymer and to describe it phenomenologically, let us discuss these spectra quantitatively. For this purpose, we first focus on unentangled polymer melts or solutions that both do not display any intermediate rubbery plateau. We know from Section 5.6.1 that the mechanics of viscoelastic liquids is described by the Maxwell model, and we know that the dynamics of polymer chains is quantified by the Rouse and the Zimm model. We have also just learned that, in addition to the chain as a whole on timescales longer than the Rouse or Zimm time, smaller chain segments of various lengths can relax on shorter timescales. These subchain relaxations are appraised by so-called relaxation modes numerated by an index p. The pth mode corresponds to the coherent motion of a subchain with N/p segments if N is the total number of monomer segments in the whole chain. This means that at p = 1, the entire chain relaxes and can get displaced by a distance equal to its own size, whereas at p = 2, only each half of it relaxes and can get displaced by a distance equal to the halfchain size, respectively. At p = 3, just only each third of the chain relaxes and can get displaced by a distance equal to one-third of the whole chain size, and so forth. At p = N, only single monomeric units can relax and get displaced against each other by their own size. Figure 85 visualizes this hierarchy of relaxation modes. At a time τp after abrupt deformation, all modes with index above p are relaxed already, whereas all modes with index below p are still unrelaxed. Each of the p subchains that belong to the mode with index p relax like own independent chains of length N/p, with a relaxation time that can be appraised by the Rouse or Zimm formalism if this were applied to a chain with not N segments, as usual, but only N/p segments; this relaxation time may also be plugged into a Maxwell-type formalism to phenomenologically model the mechanical spectra of these subchains. Superposition of all these possible relaxation spectra yields

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G′ðωÞ = νkB T G′′ðωÞ = νkB T

N X τp 2 ω2 τ 2 ω2 + 1 p=1 p N X

τp ω 2 ω2 + 1 τ p p=1

(5:55a)

(5:55b)

Here, ν is the number-per-volume concentration of chains in the sample, whereas N is the number of monomer units per chain, and with that, also the total number of possible relaxation modes per chain.

Figure 85: Relaxation modes, indexed by a number p, of a schematic polymer chain. The first mode, p = 1, relates to relaxation of the entire chain. In the second mode, p = 2, subchain segments with a length of just half of the chain can relax. The third mode, p = 3, corresponds to the relaxation of subchain segments with a length of only a third of the chain, and so on. In the last mode, p = N, only single monomeric units can relax (not sketched here). Picture modified from H. G. Elias: Makromoleküle, Bd. 2: Physikalische Strukturen und Eigenschaften (6. Ed.), Wiley VCH, 2001.

We can see from this expression that at frequencies ω < 1/τ1, all coil-internal modes can relax. G′ then scales with ω2 and G″ scales with ω, because the denominators in eq. (5.55) can be assumed to be one for such small ω-values, as the τp 2 ω2 parts in them become negligible then. At frequencies ω > 1/τ1, by contrast, this changes: now, each unrelaxed mode contributes an energy-storage increment of kBT to the moduli. The moduli therefore increase from kBT per chain in the sample volume at τ1 (=τRouse or τZimm) to kBT per monomer in the sample volume at τN (=τ0). Note: by the normalization of these energies to the sample volume, which is done by the factor ν (number-per-volume concentration of chains in the sample) in eqs. (5.55), we obtain a quantity with unit Nm/m3 = N/m2 = Pa. In that regime, the time dependence of G′ and G″ (or inversely, the frequency dependence of these moduli) is therefore given by the time- (or frequency-) dependence of the mode index. This can be obtained from the Rouse model for polymer melts or the Zimm model for polymer solutions, as summarized in Table 8 in Section 3.6.4. According to eq. (5.55a) and (5.55b), we can calculate the frequency-dependent storage and loss moduli for the individual modes p, Gp′ and Gp″, and visualize how they contribute to the overall storage and loss moduli, G′ and G″. This is shown in Figure 86 for an ideal polymer with N Kuhn monomer units. We see that at times longer than τ1 (=τRouse or τZimm), there is enough time for the entire chain to relax by thermal motion, such that the energy-storage capability is just kBT per chain. At a shorter time, τ2, subsegments

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with a length of up to only half of the entire chain can relax, whereas the mode τ1 is still unrelaxed. The energy-storage contribution to the modulus is now kBT for each unrelaxed mode. This trend continues: the shorter the time, the more modes are still unrelaxed, with each unrelaxed mode contributing one kBT energy-storage increment to the modulus. The overlay of still-unrelaxed and already-relaxed modes leads to a scaling of G′ and G″ with ω½. At times shorter than τN (=τ0), not even single-monomeric segmental relaxation is possible anymore, and the polymer chain motion is practically frozen. All the unrelaxed modes now each contribute an energy-storage increment of kBT to the moduli, such that G′ has achieved its maximum value of G∞ = νN kBT.

Figure 86: Frequency-dependent storage (left) and loss (right) moduli for the individual modes p, Gp′(ω) and Gp″(ω), as well as the entire chain, G′(ω) and G″(ω), of a polymer with N Kuhn segments. Redrawn from C. Wrana: Polymerphysik, Springer, 2014.

Indeed, such mechanical spectra can be observed for unentangled polymer melts and solutions. The first type of sample (i.e., polymer melts) is better described by the Rouse model, whereas the second (i.e., polymer solutions) is better described by the Zimm model. Figure 87 shows schematics of the mechanical spectra of these types of samples. Again, at times below t = τRouse and t = τZimm, the polymer’s viscoelastic properties are those of a viscoelastic liquid, with scaling according to G′ ⁓ ω2 and G′′ ⁓ ω. Here, the chains have enough time to fully relax. At intermediate times, τRouse or τZimm < t < τN, only relaxation of chain segments is possible. The material is a viscoelastic solid and both moduli, G′ and G″, have similar values and both scale with ω according to power laws with exponents corresponding to the inverse time dependence (i.e., the frequency dependence) of the mode index from Table 8 in Section 3.6.4, giving values in the range of 0.5–0.6 if ν is set to be 0.5 (Θ-state) to 0.6 (good-solvent state). At times shorter than τN (=τ0), no relaxation is possible anymore, and the material is a glassy solid.

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Figure 87: Schematic viscoelastic spectra of unentangled polymers. Note that the names τ0 and τR correspond to the names τN and τ1 in the concept of relaxation modes in the Rouse and Zimm model, as discussed in Chapter 3.6.4 (see Table 8 there).

Questions to Lesson Unit 14 (1) Which answer assigns the right regimes in a full rheological spectrum to their following characteristics? ① G′ and G′′ have similar values and courses. ② G′ drops rapidly and G′′ outnumbers G′. ③ G′ and G′′ have both high values in the range of GPa. ④ G′ shows an intermediate plateau value. a. ① → Glassy, ② → viscous, ③ → leather-like, ④ → rubbery b. ① → Leather-like, ② → viscous, ③ → rubbery, ④ → glassy c. ① → Leather-like, ② → viscous, ③ → glassy, ④ → rubbery d. ① → Viscous, ② → rubbery, ③ → leather-like, ④ → glassy (2) What is the microscopic explanation for the glassy state? a. At very short timescales, no motion of the building blocks is possible in the sample, especially when it has a crystal structure. b. At very short timescales, no motion of the building blocks is possible in the sample, even if it has an amorphous structure. c. At very short timescales, only single building blocks can move distances of their own size, so the system flows like a glass. d. At very short timescales, the sample is fully transparent due to the arranged building blocks, looking like glass.

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(3) Which is the shortest possible relaxation time? a. τ0 , because at this time single monomeric units can move distances equal to their own size. b. τRouse , because at this time Rouse subsegments can move distances equal to their own size. c. τZimm , because at this time Zimm subsegments can move distances equal to their own size. d. τrep , because at this time the disentanglement of entangled chains is possible. (4) What properties does a material in the leather-like regime display? a. The material is viscoelastic since G′ and G′′ show very similar scaling as well as almost identical overall values of the modulus. b. The material is viscoelastic since G′ and G′′ have the same values despite different scaling with frequency. c. The material is a viscoelastic solid since the values of G′ are sufficiently higher than those of G′′. d. The material is a viscoelastic liquid since the values of G′′ are sufficiently higher than those of G′. (5) Which statement about the relaxation modes is true? a. At τp the relaxation modes with index < p are already relaxed, whereas the ones with index > p are still unrelaxed. b. The zeroth relaxation mode (p ¼ 0) means the relaxation of the whole chain. c. For a given chain, there is any number of relaxation modes with index p possible. d. The relaxation of the chain can be understood as sequential relaxation of the chain’s subsegments. Their sizes increase with longer timescales. (6) Why do G′ and G′′ scale with ω1=2 for intermediate timescales? a. The overlay of the still-unrelaxed modes and the relaxed modes leads to a scaling of ω1=2 . b. Considering the Zimm model and a θ-state, the scaling equals 1 + 2 1ð0:5Þ = 0:5. c. Since the time scaling is given by t2 ; the frequency scaling needs to be the reciprocal value. d. At intermediate timescales, the movement of the building blocks is subdiffusive. Therefore, the scaling is < 1.

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(7) Why do the moduli drop for intermediate timescales in the leather-like regime? a. The longer the time, the more chain entanglements can be disentangled, which leads to a drop of kB T per chain. b. The longer the time, the more modes can relax, and each mode leads to a drop of kB T in the modulus. c. The longer the time, the more chains can show, viscous flow leads to a significant drop, especially in the storage modulus. d. The longer the time, the more the chain dynamics can be described by the Maxwell model, which scales with a steeper slope in ω than the Kelvin– Voigt model. (8) Which statement about the terminal viscous flow is true? a. Terminal viscous flow is always possible when the time exceeds the Rouse time, respectively, the Zimm time τZimm . b. The reptation time marks the onset of terminal viscous flow, visible by an increase of the loss modulus with time behind that point. c. The mechanics of terminal viscoelastic flow can be described by the Maxwell model, scaling with ω2 for the storage modulus and with ω1=2 for the loss modulus. d. When the loss modulus exceeds the storage modulus, terminal viscous flow sets in. Behind that point, the dominance of the loss modulus further increases with time.

5.9 Rubber elasticity

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5.9 Rubber elasticity LESSON 15: RUBBER ELASTICITY In Lesson 3, you have got to know polymers as entropic springs. This principle causes polymer samples with entangled or crosslinked chains to show a characteristic plateau in the time- or frequency-dependent modulus. In this lesson, you will get to know how the height of this plateau can be conceptually and mathematically understood, and how this allows you to fundamentally connect the mesh structure of a polymer network (be it a permanent crosslinked or just a transiently entangled one) to its elastic modulus.

In the last lesson, we have examined the viscoelastic properties of short polymer chains. They transition directly from the viscoelastic leather-like regime at τ < τRouse to the viscous terminal-flow regime at t > τRouse, as shown in Figure 87. The picture, however, is different if the polymer chains are long enough such that they can form entanglements with one another and thereby mutually impair each other’s relaxation. As we will see later, this is possible once the chains are longer than a certain minimal length, which is connected to a minimal degree of polymerization, Ne, or analog, a minimal molar mass, Me, from which on chain entanglement can first occur. If chains are longer than Nel (and therewith heavier than Me), mutual entanglement

Figure 88: Mechanical spectra of chains with molar masses below (blue), at (green), and above (black) the entanglement molar mass, Me. The latter shows a distinct rubbery plateau between the 1 viscoelastic (G′ ⁓ ω2 Þ and the viscous (G′ ⁓ ω2 ) regimes. If an even higher molar mass were shown in addition, it would display an even more extended plateau. At the high-frequency end of the spectra, all data coincide, as here, subchain relaxation modes and transition to the glassy regime is covered, which is the same for a given type of polymer at given experimental conditions, independent of the total chain length of that material.

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impairs their relaxation, and only on long timescales, these entangled chains can escape this mutual constraint. On short timescales, by contrast, the entanglements act like crosslinking points between the chains, thereby allowing for elastic energy storage that manifests itself in the appearance of an intermediate plateau regime in the viscoelastic spectrum, the so-called rubbery plateau depicted in Figure 88. The longer the chains are, the longer does it take them to disentangle from one another, such that the extent of the rubbery plateau regime on the time or frequency axis is greater the higher the molar mass is. To quantitatively assess the impact of the entanglements on the mechanical properties of these long polymer chains, we regard them as permanent crosslinks and spend a focus on crosslinked polymer networks in this section.

5.9.1 Chemical thermodynamics of rubber elasticity In chemical thermodynamics, free energy, F, is given as the relation between internal energy, U, and temperature, T, multiplied by entropy, S: F = U − TS

(5:56)

The force of deformation of a material specimen f is the length derivative of this function:



∂F ∂U ∂S = −T (5:57) f= ∂l ∂l ∂l The entropy derivative in eq. (5.57) can be rewritten based on the identity S = − ð∂F =∂T Þ and the symmetry of second derivatives (Schwarz’ theorem) for total differentials:







∂S ∂ ∂F ∂ ∂F ∂f =− =− =− (5:58) ∂l ∂l ∂T ∂T ∂l ∂T Reinserting this into eq. (5.57) generates



∂U ∂f f= +T ∂l ∂T

(5:59)

This expression can be plotted as a linear equation with an intercept of ∂U =∂l and a slope of ∂f =∂T . Such a plot is shown for a typical rubber in Figure 89, in which the force has been normalized to the area, thereby yielding the more general quantity stress; the illustrated data can be imagined to be from an experiment in which the force to achieve a certain deformation is measured at various temperatures. At temperatures below Tg, where all chain dynamics is frozen, the material is energy elastic. In that state, deformation pulls the motionless monomer segments apart from each other against their attractive interaction potentials. This is easier to achieve at higher temperatures, leading to less stress in the material at higher

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Figure 89: Relation of stress to temperature for a typical rubbery material. Below the glass-transition temperature, Tg, the material is energy elastic, making it harder to deform the lower the temperature is. By contrast, above Tg, the material is entropy elastic, making it harder to deform the higher the temperature is. Picture inspired by B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

temperature upon deformation, which causes a negative slope in the left branch of the dataset in Figure 89. By contrast, above Tg, where chain dynamics is activated, the material is entropy elastic. In that state, deformation leads to uncoiling of the coiled chains by transition of local cis into local trans segmental conformations. This chain uncoiling reduces the chain entropy, thereby creating an entropy-based backdriving force. Due to the fundamental coupling of temperature and entropy, that kind of deformation gets harder at higher temperature, leading to a positive slope in the right branch of the dataset in Figure 89. Conversely, the material heats up upon stretching in that state. We can see this from the following expression:



∂T ∂T ∂S ∂T ∂H ∂S 1 ∂S = = =− T (5:60) ∂l ∂S ∂l ∂H ∂S ∂l cl ∂l When the entropy term ∂S=∂l is negative, which it is upon deformation, the temperature term ∂T =∂l is positive, meaning that the temperature increases upon deformation. We also see in Figure 89 that the extrapolated intercept of the positive-slope part of the dataset is small; this underlines once more that the energetic contribution to stress in a deformed rubbery sample is small.

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5.9.2 Statistical thermodynamics of rubber elasticity In addition to the chemical-thermodynamic argument above, a statistical-thermodynamic approach can give us particularly valuable quantitative information about the rubbery elastic state. In such an approach, we generally appraise the most central quantity in thermodynamics, entropy, as the degree to which a specific macrostate of interest, in our case the shape of the polymer coil, is realized by different possible microstates, in our case the local segmental cis or trans conformations. We have learned in Chapter 2 that the ideal shape of a single polymer chain is that of a Gaussian random coil, with an entropy given by random-walk statistics: S = S0 −

3kB r2 3kB r2 = S − 0 2hr2 i0 2Nl2

(5:61)

In eq. (5.61), hr2 i0 is the equilibrium mean-square end-to-end distance of the polymer chain, whereas r is the end-to-end distance in a given situation of interest. When we stretch the chain, we extend this latter distance, such that r2 > hr2 i0 . Because the variable r2 is in the numerator, whereas the constant hr2 i0 is in the denominator of a term with negative sign in front of it, we reduce the entropy S upon increase of r, that is, upon stretching. Note that the latter equation was actually derived for single uncrosslinked polymer chains; we now presume that it also applies to each network chain in a crosslinked rubber. Figure 90 displays the relevant parameters that are needed to treat the deformation of a polymer network sample, both on the macro- and the microscale. Due to the self-similar nature of polymers, we can express the entropy change ΔS of a single ideal network chain, which is a subchain ranging from one crosslink to another in the network, upon deformation as follows:75         3kB rx 2 + ry 2 + rz 2 3kB rx, 0 2 + ry, 0 2 + rz, 0 2 ~ ~ ΔS = S R − S R0 = − + = 2hr2 i0 2hr2 i0 =

3kB  2  2  2  2  2  2  λx − 1 rx, 0 + λy − 1 ry, 0 + λz − 1 rz, 0 2hr2 i0

(5:62)

The entropy change of the entire network is obtained simply by summation over all its n network chains.

n  X n  2  2  X 3kB  2  Xn 2  2 2 λx − 1 ðr Þ + λy − 1 r + λz − 1 ðrz, 0 Þi ΔS = − i = 1 x, 0 i i = 1 y, 0 i 2hr2 i0 i=1 (5:63)

 Note again that in this context, deformation means the decoiling of the polymer chain, but not stretching of the bonds between its monomeric units.

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Figure 90: When a polymer-network specimen is deformed, its new dimensions Lj can be described as the original ones Lj,0 multiplied by strain ratios λj (with j denoting the geometrical dimension x, y, and z). In an affine scenario, we presume the microscopic deformation of the chains that constitute the sample to be equal to the macroscopic deformation of the entire body. As a result, the strain ratios λj are identical on both microscopic and macroscopic scales. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

In an isotropic sample, the average end-to-end distances are equal in all directions: 1 Xn hr2 iu 2 2 2 2 ð r Þ = hr i = hr i = hr i = x, 0 x, 0 y, 0 z, 0 i i=1 3 n

(5:64)

Here, hr2 iu is the mean-square end-to-end distance of the network chains in the undeformed state. We can simply rearrange this to Xn

ðr Þ2 = i = 1 x, 0 i

Xn  i=1

ry, 0

2 i

=

Xn

ðr Þ2 = i = 1 z, 0 i

nhr2 iu 3

(5:65)

in which n is the number of network chains. We can insert this expression into eq. (5.63) to generate







 2  n 2  2  n 2 3kB  2  n 2 hr hr hr λ − 1 i + λ − 1 i + λ − 1 i ΔS = − x y z u u u = 2hr2 i0 3 3 3 =−

 nkB hr2 iu  2 λx + λy 2 + λz 2 − 3 2 hr2 i0

(5:66)

From that equation, it becomes clear that the entropy reduction upon stretching actually stems from two contributions. First, the entropy is reduced more if we stretch to a greater extent, which is expressed in the form of higher strain ratios λj in eq. (5.66).

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Second, the number of chains n that are stretched by that extent proportions the entropy reduction. Remember again in this context that we treat network chains, meaning chain segments ranging from one crosslinking junction to another. Consequently, a high crosslinking degree also amplifies the change of entropy, because it corresponds to a high number of (short) network chains. The proportionality factor hr2 iu =hr2 i0 in eq. (5.66) describes the ratio of the mean-square end-to-end distance of the network chains in the undeformed state to the mean-square end-to-end distance of an analog uncrosslinked ideal chain. This factor depends on the state of the network chains during the network formation. Often, the network is made in a state where the chains that are crosslinked (or formed in-situ during crosslinking) are in the same state (e.g., an ideal or θ-state) as they are during the network deformation experiment; in this case, the factor hr2 iu =hr2 i0 in eq. (5.66) is one. It may deviate from one when the states of network preparation and network probing are different, for example, when the network was formed in a swollen state but is measured in a deswollen state, or vice versa. For the rest of our argumentation, we assume the usual case of hr2 iu = hr2 i0 . Now that we have derived an expression for the change of entropy, ΔS, we can insert it into the fundamental equation for the free energy, eq. (5.56), and obtain ΔF = ΔU − TΔS = +

 nkB T  2 λx + λy 2 + λz 2 − 3 2

(5:67)

The internal energy, U, is independent of the network-chain end-to-end distance, as we presume these chains to be ideal, such that ΔU = 0. Any change of the network chains’ free energy, ΔF, therefore stems from entropy alone. Let us consider a typical experimental situation: isochoric uniaxial strain. For that situation, the latter expression can be simplified further. Isochoric means that the total volume change upon deformation is zero, ΔV = 0, which means that the product of the strain ratios has to be one, λx λy λz = 1. Uniaxial means that the strain is only applied along one spatial dimension, x, leading to one defined strain ratio λx = λ, whereas the two other strain ratios must follow according to the isochoric pffiffiffi condition: λy = λz = 1= λ. Both considerations lead to a simplified expression:

nkB T 2 2 ΔF = λ + −3 (5:68) 2 λ The deformative force fx can again be calculated by the length derivate along the axis of deformation:

∂ΔF ∂ΔF nkB T 1 (5:69) = λ− 2 = fx = ∂Lx ∂ðλLx, 0 Þ Lx, 0 λ By normalizing it to the plane of deformation, we can formulate an expression for the stress σxx

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5.9 Rubber elasticity





fx nkB T 1 nkB T 1 σxx = = λ− 2 = λ− 2 Ly Lz Lx, 0 Ly Lz Lx, 0 λy Ly, 0 λz Lz, 0 λ λ

(5:70a)

pffiffiffi Due to the isochoric condition, λy = λz = 1= λ, we can eliminate the direction-specific strain ratios and simplify the expression to



nkB T 1 nkB T 2 1 (5:70b) λ λ− 2 = σxx = λ − Lx, 0 Ly, 0 Lz, 0 V λ λ In the very beginning of our discussion on rheology, in eq. (5.1), we have defined the extensional strain as ε = ΔL/L0, that is, the change of the material specimen length, ΔL, relative to its length before deformation, L0; for uniaxial strain, this is equal to ε = λ – 1, so λ2 − λ1 ≈ ε2 + 2ε + 1 − ð1 − εÞ = ε2 + 3ε,76 which for small ε is nearly 3ε. With that, we get σxx =

nkB T 3ε V

(5:70c)

This equation is strikingly similar to Hooke’s law, eq. (5.1), that connects stress to nk T strain by a proportionality factor. In eq. (5.70c), this factor is 3 VB ; it therefore corresponds to the elastic modulus: E=3

nkB T V

(5:71a)

From that, along with eq. (5.3), we also get the shear modulus: G=

E nkB T = 3 V

(5:71b)

The latter formula is quite an extraordinary expression, as it connects the microscopic structural information of the network chain concentration, n/V, to the macroscopic property of the shear modulus, G, thereby creating a quantitative structure– property relation. We now realize that upon deformation, each network chain stores an energy increment of kBT. The modulus increases with temperature due to the entropy elastic nature of rubber elasticity. We also realize that a higher degree of crosslinking causes a higher modulus, as the network chains are then more numerous (and shorter), leading to a higher n in eq. (5.71a). We can formulate the latter identity on a molar scale as well: G=

ρ RT Mx

(5:71c)

3

3

76 The calculation for this is as follows: λ2 − λ1 = ðε + 1Þ2 − ε +1 1 = ðεε++11Þ − ε +1 1 = ðε +ε1+Þ 1 − 1 = ε +ε3ε+ 1+ 3ε. Polynomial long division turns that into ε2 + 2ε + 1 − ε +1 1. Taylor series approximation of ε +1 1 ≈ 1 − ε (as first described by Isaac Newton) then gives ε2 + 2ε + 1 − ð1 − εÞ. 3

2

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5 Mechanics and rheology of polymer systems

This equation relates the network density or network concentration, ρ, in the unit g·L–1, to the molar mass of the network chains, Mx, in the unit g·mol–1. From this, we again realize that a higher degree of crosslinking causes a higher modulus, as the network chains are then shorter (and more numerous), leading to a lower Mx in the denominator of eq. (5.71b). Equation (5.71b) is strikingly similar to the ideal gas law in elementary physical chemistry. This is because rubber elasticity at all is very similar to the concept of pressure for an ideal gas. Both are based on the reduction of freedom of arrangement for the system’s building blocks (gas particles in space, or monomer units in a polymer chain) upon application of deformation such as compression of the ideal gas, which limits the freedom of spatial arrangement of the gas particles, or stretching of the polymer chain, which limits the freedom of arrangement of the monomers along the chain from a mix of cis and trans local conformation to more trans. We can actually use the same statistical thermodynamic approach that we have used above to appraise the pressure of an ideal gas: we appraise the probability, Ω, for a gas molecule to sit in a subvolume V of a container with total volume V0 to be Ω = V/V0. From this follows that the probability for n gas molecules all to sit in that subvolume at a time is Ωn = (V/V0)n. We estimate the system’s entropy, S, by inserting this into the Boltzmann formula: S = kB lnΩn = kBn ln(V/V0). From physical chemistry lectures, we know that the pressure, p, in thermodynamics is calculated as p = –(∂G/∂V)T = –(∂H/∂V)T + T(∂S/∂V)T. In our case, we are looking at an ideal gas, whose particles do not interact. This means that there is no energetic change upon change of volume, that is, (∂H/∂V) = 0. This simplifies the latter expression to p = T(∂S/∂V)T = (kBn·T)/V. This is exactly equal to eq. (5.71a).77

An alternative calculation can be made for an isotropic shear experiment. In this case, there is no deformation in the z-direction; hence λz = 1. The two other strain ratios are then defined to be λx = λ and λy = 1=λ. For the free energy, this yields ΔF =

 nkB T  2 nkB T 2 nkB T  2 λ + λ−2 − 2 = λ − λ−1 = γ 2 2 2

(5:72)

Subsequent calculation of the stress gives σ=

nkB T γ V

(5:73)

Again, we end up with an expression that has the form of Hooke’s law. In this case, we identify the proportionality factor nkB T =V to be the shear modulus G: G=

nkB T V

(5:74a)

 Note from this identity that pressure and the elastic modulus also have the same unit: Pascal.

5.9 Rubber elasticity

225

Once again, we can formulate the latter identity on a molar scale as well: G=

ρ RT Mx

(5:74b)

So far, we have discussed rubber elasticity on the basis of permanent crosslinks. Another form of hindering polymer chains from relaxing, at least temporarily, is mutual entanglement, as shown in Figure 91(A). Hence, on intermediate timescales, such entanglements act like permanent crosslinks and create a rubbery plateau in the time- or frequency-dependent elastic modulus, as shown in Figure 91(B).

Figure 91: (A) Schematic representation of a polymer network composed of permanent crosslinks and chain entanglements. Both hinder the chains from relaxing and therefore contribute to the ability of elastic energy storage, thereby both contributing to the plateau modulus in the form of Gp ffi Gx + Gent . For permanently crosslinked networks, the plateau in the mechanical spectra, shown in (B), is infinite and reaches out to t ! ∞, whereas if no permanent crosslinks are present, the polymer chains can untie their mutual entanglement at long times. The extent of the rubbery plateau is defined by the polymer molar mass, which must exceed a specific minimum molar mass, Me, to show mutual entanglement at all.

A crosslinked network in fact usually has both modes of polymer connection: crosslinking and (trapped) entanglements, as shown in Figure 91(A). We can therefore split the modulus G into two contributions, one from actual permanent crosslinks, Gx, and one from (trapped) entanglements, Gent:

1 1 (5:75) G ffi Gx + Gent = ρRT + Mx Ment This means that even uncrosslinked polymer systems, in which Gx = 0, are elastic on short timescales solely due their mutual entanglements, whereby Gent typically is in the range of a megapascal:

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5 Mechanics and rheology of polymer systems

Gent =

ρRT ≈ 106 Pa Ment

(5:76)

There is just one prerequisite for this: to entangle with one another, chains must have a certain minimum length, which translates to a certain minimum molar mass, named the entanglement molar mass, Me. This quantity is related to the one found in the denominator of the last equations, named Ment there. Typically, Me ≈ 2 Ment, because Me denotes the minimum molar mass (and therewith also the minimum degree of polymerization Ne, and yet in turn, the minimum contour length Nel) that chains must have to form entanglements, whereas Ment denotes the length of a strand segment between two entanglement points in a well-entangled network. A simple geometrical thought leads us to the notion that Me must be twice as large as Ment, because to form entanglements at all, the chain must be minimally twice as long as one segment between two entanglement nodes in an entangled network. The reason for the existence of a minimum molar mass for entanglement is because the chains have a certain stiffness or persistence, which must be overcome by sufficient chain length to form entanglements. There is an analogy to everyday life. In large canteens, for example, in the Mensa, when spaghetti are cooked and served, they are short. This is because the kitchen personnel wants the noodles to be shorter than their entanglement length, as entangled noodles are much harder to process than unentangled ones. The stiffer the main chain is (which, on the nanoscale, translates to the ease or non-ease of monomer-bond rotatability), the longer must it be to be able to form entanglements. For polystyrene, an entanglement molar mass of Me ≈ 20 kg mol–1 is practically relevant. As a closing note, it shall be mentioned that all the above appraisal is based on the entropy of an ideal chain, which we have obtained in Section 2.6.1 by applying Boltzmann’s formula to the distribution of end-to-end distances that we have derived by random-walk statistics before. Hence, all the above applies to chains only with these statistics. This is valid only in the limit of low deformation, whereas at large deformation, we actually disturb the random-walk-type distribution of end-toend distances considerably, such that it is no longer valid. In this limit, other statistics have to be used, for example, the Langevin model.78 In addition, in polymers with high main-chain regularity and tacticity, crystallization is possible. Stretching of the network chains may then bring them in so close distance and good order that crystallization may occur, leading to crystalline nodes that act as further crosslinks. This phenomenon is named strain-induced hardening.79

 Again, there is an analogy to ideal gases here: at large pressure, they deviate from ideality and must be treated by a different model: the van der Waals equation.  And once more, there is an analogy to gases, which may liquefy at high pressure.

5.9 Rubber elasticity

227

5.9.3 Swelling of rubber networks In addition to stretching or shearing, there is another way to deform a polymer network: swelling. In the process of swelling, a fluid medium enters the network and expands it from the interior. The basis of this process is that upon contact of solvent molecules with a dry network, its chains want to dissolve. Their mutual connectivity, however, hinders them to do so freely. Hence, rather than separating the chains from one another completely, as it would be the case in an uncrosslinked sample, solvent can only penetrate into the network and stretch the polymer network chains in it. The degree of stretching, however, is limited by the entropy-elastic backdriving force that comes along with that, as we have just quantified above. At equilibrium, the polymer–solvent mixing energy is at balance with the entropy-elastic energy. The point where this equilibrium lies, and with that, the swelling capacity of the network, depends on the ratio of the network chain length (or in other words, the density of crosslinking points in the network) and the thermodynamic quality of the solvent (athermal, good, or Θ). To quantify the swelling equilibrium, we introduce the degree of swelling q as q=

Vuptaken solvent V =1+ Vunswollen network V0

(5:77)

To estimate this value, we need to know both contributors: the free energy of mixing and the entropy-elastic backdriving energy. This is best gauged via the chemical potentials related to these energies. The contribution of the free energy of mixing is given by the Flory–Huggins formula:     (5:78) Δμmix = RT ln 1 − q − 1 + q − 1 + χq − 2 The potential of the entropy-elastic backdriving force is calculated by





∂ΔFelast ∂ΔFelast ∂q = · Δμelast = ∂n ∂q ∂n

(5:79)

The elastic energy is given by the rubber elasticity formula:   3 ΔFelast = NkB T q2=3 − 1 2

(5:80)

Here, N denotes the total number of network chains. Assuming that both the volumes of the dry network and the solvent are additive, we can formulate the degree of swelling as follows:

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5 Mechanics and rheology of polymer systems

   solvent + V0 ) q = V = nsolvent V solvent + 1 ) ∂q = V solvent V = nsolvent V V0 V0 V0 ∂n



 solvent ∂ΔFelast ∂q 3 2 V  solvent (5:81) · = NkB T · q − 1=3 · ) Δμelast = = vRTq − 1=3 · V ∂q V0 ∂n 2 3 Here, v denotes the concentration of network chains, which is inversely related to the network chain length. This is because in a network with a given amount of polymer material in it, if the network chains are long, then there is just few of them and vice versa. Hence, v can be considered to be both the molar number of network chains n per volume unit V [mol·L– 1], or the network density ρ [g·L–1] divided by the network chain molar mass Mx [g·mol–1]: v=

n ρ = V Mx

(5:82)

At the swelling equilibrium, both the free energy of polymer–solvent mixing as well as the entropy-elastic backdriving energy are identical: Δμmix = Δμelast

(5:83)

This allows us to calculate the molar concentration of network chains as v=

ρ lnð1 − q − 1 Þ + q − 1 + χq − 2 =−  solvent q − 1=3 Mx V

(5:84)

We can now understand the value of doing a swelling experiment. It allows us to determine the degree of crosslinking at a known Flory–Huggins parameter χ, or, vice versa, the determination of the Flory–Huggins parameter at a known degree of crosslinking. The swelling ability and often soft appearance of polymer gels give rise to various areas of application. Particularly relevant in this context are hydrogels, which are polymer networks swollen with up to 99% (w/w) of water. This is, in principle, the same ratio of water and solid material as in biological matter, which is why hydrogels are often used as matrixes in the field of life science for procedures such as electrophoresis or chromatography. Dried hydrogels can also be found in hygiene products as super absorbers due to their superior ability to absorb moisture – such as urin in diapers. In more sophisticated applications, the polymer network can be designed to react to external stimuli that trigger either swelling or shrinking. Such a stimulussensitive reversible swelling is sketched in Figure 92. Stimuli-responsive gels have potential uses as artificial muscles, chemo-mechanical actuators, or as “intelligent” capsules for drug release. As an example for the latter, consider the following: cancer tissue has a marginally lower pH than the surrounding healthy tissue. In a

Questions to Lesson Unit 15

229

Figure 92: Schematical representation of a stimulus-sensitive polymer network that can undergo swelling or deswelling upon external triggers. This stimuli-responsiveness renders such materials useful for potential applications such as chemo-mechanical switches, artificial muscles, or “smart” drug-release carriers.

specifically tailored polymer network, the pH change would trigger a deswelling or a swelling process, thus initiating controlled release of an cytostatic drug (either by squeezing it out of the gel in the case of deswelling or by letting it diffuse out of the gel upon opening diffusive paths through the widened network meshes in the case of swelling). The same principle might be used for the controlled release of antiinflammatory agents. Inflamed tissue has a higher local temperature than its environment; hence, in that case, the temperature difference could provide a stimulus for the hydrogel to shrink or swell.

Questions to Lesson Unit 15 (1) Which relation to the chain length holds true for the elastic rubbery plateau? a. The elastic rubbery plateau has higher values of the storage modulus the longer the chains are. b. The elastic rubbery plateau expands over a larger frequency/time/temperature range the longer the chains are. c. The elastic rubbery plateau is shifted toward shorter timescales/higher frequencies the longer the chains are. d. The elastic rubbery plateau shows a more significant drop in the loss modulus the longer the chains are.

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5 Mechanics and rheology of polymer systems

(2) Chain entanglements in the elastic rubbery regime can be treated as _____ a. defect structures in the polymer chain, analog to dangling ends. b. cross-links between the polymer chains. c. sequential relaxation modes. d. interactions between the polymer chains in the magnitude of van der Waals interactions. (3) What does the glass-transition temperature for a rubbery material not denote? a. The change of a crystalline to an amorphous structure. b. The change of energy-based to entropy-based elasticity c. The change of easier to harder deformation with increasing temperature. d. The change of a negative slope of the stress-to-temperature relation to a positive one. (4) The affine network model ______ a. describes the shrinking of a rubbery material that has been stretched to a size that is smaller than its initial one. b. is only applicable for cross-linked polymer networks. c. is based upon the assumption that when deformed, the strands between two cross-link points of the network are stretched to the same extend as the network in total. d. states that entanglements in the chains attract each other, so that the chains get even more entangled. (5) What holds when comparing the shear modulus G for the case of applying isochoric uniaxial strain with the one when an isotropic shear deformation is applied? a. They have completely different form, because they are based on different conditions. b. They both are connected to the number of chains n, temperature, and volume but have a different scaling. c. They are proportional to the number of chains n, temperature, and volume. Just the values of the proportionality factors are different due to the different geometrics in shear vs. uniaxial deformation. d. They both are the same.

Questions to Lesson Unit 15

231

(6) Why do chains need to have a minimum molar mass to show a rubbery elastic plateau of the modulus? a. The chains need a minimum molar mass to be long enough to form van der Waals interactions, which get stronger with increasing molar mass. b. The chains need a certain length to overcome their own stiffness such as to be flexible enough to form entanglements. c. Only chains that are sufficiently long can form ties to others. d. The height of the plateau depends on the molar mass. If the molar mass is too low, the plateau is not visible in the mechanical spectrum. (7) What does not affect the swelling equilibrium of a polymer network? a. Its cross-linking density b. Its molar mass c. The pH value of the solution d. The quality of the solvent (8) We can also use the statistical approach that we used to access the concept of rubber elasticity, to derive the ideal gas law. Which are the basic thoughts that lead to the final formulation of the law? a. The Boltzmann formula and the thermodynamic expression for pressure. b. The Boltzmann formula and the thermodynamic expression for entropy. c. The thermodynamic expression for entropy and the statistical formula of pressure. d. The thermodynamic expression for entropy and the statistical formula of occupation probabilities.

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5 Mechanics and rheology of polymer systems

5.10 Terminal flow and reptation LESSON 16: REPTATION When a noncrosslinked polymer sample is given enough time, it will display viscous flow, even if the chains are long and entangled. On a microscopic scale, such flow requires the chains to change their positions against each other. Describing this microscopic motion is an essence to describe macroscopic material properties such as the polymer fluid viscosity. In this lesson, you will get to know a simple but powerful approach to do so: the reptation model. This model simplifies the direct environment of a chain in an entangled surrounding to be a rigid tube, out of which the chain can creep only by curvilinear motion.

So far, our knowledge on the viscoelastic spectra of polymers has two sides: a phenomenological macroscopic one and a conceptual microscopic one. On the one side, phenomenologically, we can describe the macroscopic viscoelastic solid-like mechanics of the short-time or high-frequency regime with the Kelvin–Voigt model, which gives us equations to approximate the leather-like creep properties of a sample in that domain. As a complement to that, we can use the Maxwell model to describe the viscoelastic liquid-like stress relaxation of the sample in the long-time or low-frequency domain. On the other side, conceptually and microscopically, we can understand how the viscoelastic solid-like appearance of the sample in the first domain originates from viscoelastic relaxation modes, that is, motion of only parts of the chains but not yet the chains as a whole, which we can quantify based on the Rouse or the Zimm model. We also understand how chain entanglement may introduce a further domain into the mechanical spectra, namely, that of a rubbery plateau. What we are lacking, though, is a conceptual microscopic understanding of the relaxation and flow in the terminal regime. Delivering that is the intent of the following section. We have already seen that the transition between the rubbery plateau and the terminal flow regime depends on something called the entanglement molar mass, Me (see Figure 91(B)), without further discussing this quantity; we will do so in this subchapter. Describing the long-term relaxation of an entangled multichain system is a challenging proposition. Many chains move simultaneously through the sample volume, part in concert with each other, part independently. They may also interact with each other or, if present, with a solvent. This is a very complex multibody phenomenon that can hardly be described analytically. The solution to this challenge is as easy as it is clever: we focus only on one test chain, and then simplify its environment drastically.

5.10 Terminal flow and reptation

233

5.10.1 The tube concept Consider a test chain in an ensemble of overlapping and entangled chains that we have somehow modified so we can identify it from the background and be able to follow its path of motion, as sketched in Figure 93(A). To describe its surroundings, we neglect every other chain that does not have contact with our test chain. Even the ones that have contact only do so with small portions of their own length, such that we must only regard these sections of them, as illustrated in Figure 93(B). These contacts form an array of obstacles in the direct environment of our test chain. Now, a smart approach by Samuel Frederick Edwards comes up. Edwards modeled the direct constraining environment of the test chain by a solid tube, as shown in Figure 93(C). The test chain can move only along this tube in a curvilinear manner, whereas perpendicular movement to the tube is prohibited. With this kind of motion, the chain can creep out of its confining tube, as illustrated in Figure 93(D).

Figure 93: Tube concept to simplify the complex multibody surrounding of a polymer chain in a system of many other mutually entangled chains. (A) We consider a test chain, here labeled in blue color, which is somehow distinguishable from its background into which it is embedded and with which it is entangled. (B) To simplify our view, we only consider the direct environment of the test chain. (C) The directly surrounding matrix chain segments around the test chain can be modeled as a confining solid tube that constrains the motion of the test chain to be possible only along the tube contour. (D) Over time, the test chain can creep out of this confining tube by curvilinear motion, which is named reptation. Note: as the surrounding matrix is still there, the test chain then finds itself entrapped in a new tube (not shown here). The overall diffusion of the test chain through the surrounding matrix can be viewed as a sequence of steps from one tube to another, each of which occurring by curvilinear creep along the tube contour. Pictures inspired by W. W. Graessley: Entangled linear, branched and network polymer systems – Molecular theories, Adv. Polym. Sci. 1982, 47 (Synthesis and Degradation Rheology and Extrusion), 67–117.

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5 Mechanics and rheology of polymer systems

This approach is called the tube concept. Mathematically, Edwards assumed that each neighboring segment imparts a parabolic confining potential on the test chain. The energetically most favorable way for the test chain to arrange itself in the tube and also to move along it is through the path of the potential minima through space, which is called the primitive path. The test chain fluctuates around this primitive path, but only up to an extent that the ever-present thermal energy kBT can activate, because anything further would require additional energy input. The tube diameter, a, is therefore given by the transverse distance at which the confining potential is exactly kBT. In other words, segments of the test chain fluctuate with an average displacement of a.

5.10.2 Rouse relaxation and reptation In the tube concept, the motion of a chain is largely restricted only to curvilinear creep along the tube contour, whereas migration in directions perpendicular to that is impossible. Pierre-Gilles de Gennes used this premise to mathematically describe the test chain motion along the tube, which, as it resembles the creep of a reptile through the woods, is referred to as reptation. De Gennes based his approach upon the Rouse model for polymer dynamics in the melt, in which the diffusion along the tube would be described by a one-dimensional Rouse-type diffusion coefficient: DTube =

kB T Nfseg

(5:85)

The diffusion of a polymer chain is related to its mean-square displacement by the Einstein–Smoluchowski equation that we have encountered numerous times in Section 3.6: hx2 i = 2dDt, with d the geometrical dimension of the motion. We can now adopt this for the reptating polymer: the tube has a contour length equal to the total length of the polymer test chain, L = Nl, which will leave the tube after a specific time τrep, the reptation time: τrep =

hLTube 2 i N 2 l2 = ⁓ N3 2 · DTube 2kB T=Nfseg

(5:86)

We see that the degree of polymerization, and with that, the molar mass of the test chain, has a paramount influence on the reptation time, as it scales with a powerlaw exponent of 3(!). Furthermore, we have learned about the fundamental connection between the shear modulus, G, and the viscosity, η, by the relaxation time, τ: η = G0τ (eq. 5.46). To estimate the viscosity, we only need to plug in the relevant values:

5.10 Terminal flow and reptation

235

In nonentangled melts, the relaxation is delimited by the Rouse time, τRouse. Under melt conditions, the chain conformation is ideal, and in that case, the Flory exponent is ν = ½, such that we get from the Rouse model (cf. Section 3.6.2): τRouse = τ0 N 1 + 2ν = τ0 N 2

(5:87)

The latter equation features two characteristic timescales: τ0 , which denotes the lower limit for any chain motion at all (below that timescale, not even single monomer segments can get displaced by a distance at least equal to their own size), and τRouse , which denotes the upper time limit for complete chain motion (above that timescale, the whole coil can get displaced by distances equal to or even greater than its own size). In the time domain between these two characteristic limits, the modulus G relaxes from an energy-storage capacity of kBT per monomer segment at τ0 to an energy-storage capacity of kBT per chain at τRouse (see Section 5.8.2). The shear modulus at the Rouse time is therefore given by GðτRouse Þ = kB T

ϕ Nl3

(5:88)

Here, ϕ is the polymer volume fraction, and Nl3 is the volume per chain in a fully  collapsed state, such that ϕ Nl3 is an expression for the number of chains per volume in the sample, denoted ν in Section 5.8.2. With that, the modulus at τRouse is kBT per chain in the sample volume. According to eq. (5.46), the viscosity at the Rouse time can then be calculated as η = GðτRouse Þ · τRouse ⁓ N − 1 N 2 ⁓ N 1

(5:89)

We realize that it scales linearly (power-law scaling exponent of 1) with the degree of polymerization, and with that, with the molar mass. The picture is very different for entangled melts. Here, the relaxation is delimited by the reptation time, τrep τrep ⁓ N 3

(5:90)

At τrep, we find ourselves just at the end of the rubbery plateau of the shear modulus G. Here, the modulus just depends on the entanglement molar mass, Me, (see eq. (5.76)), but not on he overall molar mass, Mtotal. This leads to a different dependence of the viscosity on the molar mass, and with that, on the degree of polymerization N, than that of eq. (5.89):   (5:91) η = G τrep · τrep ⁓ const · N 3 In eqs. (5.89) and (5.91), we have again connected the macroscopic properties viscosity, η, and shear modulus, G, to the microscopic property relaxation time, τ, that contains information about the molecular-scale characteristics of the system’s building

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5 Mechanics and rheology of polymer systems

blocks. In particular, we were able to delimit two very different viscosity–molar-mass scaling laws, depending on whether a specific molar mass, the entanglement molar mass, is surpassed. These dependencies are visualized in Figure 94.

Figure 94: Plot of a polymer melt sample’s viscosity as a function of the chains’ molar mass. Up to certain molar mass, Me, the dependence is linear. Once this molar mass is exceeded, however, the chains entangle with each other. This imparts severe topological constraint on the polymer chains’ motion, which is now to be described by P.G. de Gennes’ reptation model. Here, the viscosity has a severe molar-mass dependence with a power-law exponent of 3.

But how come that we need a minimal Me for chain entanglement? The reason is the polymer chain stiffness, as quantified by parameters such as the characteristic ratio, the Kuhn length, or the persistence length, as all introduced in Chapter 2. The stiffness of the chains requires them to have at least a certain length to bend enough to form entanglements, which is longer if the chains are stiffer. An every-day life analogy for the phenomenon of a minimum length for entanglement is spaghetti: when being served in a proper Italian restaurant, they are long and entangle with one another quite markedly. By contrast, when being served in a large canteen, for example, a university Mensa, they are short such that they can not entangle. The kitchen personnel does that to avoid processing problems of entangled noodles. So, even spaghetti have a minimum length (or “molar mass”) for entanglement. Getting back to Figure 94: note that experimental data actually suggest an η(M) scaling-law exponent of 3.4 rather than 3 for the initial reptation regime. This is due to fluctuations of the length of the confining tube. The chain ends fluctuate on

5.10 Terminal flow and reptation

237

timescales t < τrep. Because of this fluctuation, the effective tube length is actually shorter, which decreases τrep. This effect, however, gets less significant when the molar mass M increases. Hence, in addition to the primary effect of an M-increase on η, as captured by eq. (5.90), comes a secondary one by the no-longer-effectiveness of tube-length fluctuation. Together, this leads to the observed stronger molar-massdependence of a power of 3.4. In the very high molar-mass regime, the effect of such tube length fluctuation looses significance, such that the original reptation powerlaw exponent of 3 is recovered there.

5.10.3 Reptation and diffusion In addition to discussing the macroscopic viscosity of our sample, we may also quantify a microscopic characteristic parameter of it: the translational diffusion coefficient of the chains. On timescales shorter than τrep, the chains exhibit a one-dimensional tube diffusion coefficient, DTube, as captured by eq. (5.85). On timescales longer than τrep, by contrast, each chain diffuses three-dimensionally through space with a Fickian diffusion coefficient Drep. We can imagine this diffusion to be a hopping process from tube to tube, each of these hops taking a time increment of τrep. Based on that picture, the macroscopic chain displacement in a three-dimensional space can again be quantified by an Einstein–Smoluchowski equation: Drep ≈

R2 Nl2 ⁓ 3 ⁓ N −2 τrep N

(5:92)

This power-law scaling has indeed been well confirmed by multiple sorts of experiments. When we focus on the microscopic diffusion of polymer chains in a reptation scenario, as we have just started to do above by quantifying the two relevant chain diffusion coefficients, Dtube and Drep, we can also take an even more detailed view. In that detailed view, we consider one monomer segment on a reptating chain to be somehow labeled, for example, by radioactive marking or fluorescent tagging. When we follow the mean-square displacement, hΔr2 i, over time, t, of that labeled segment, we can distinguish four different regimes, as shown in Figure 95(B). 1) On timescales between the segmental relaxation time, τ0, below which no segmental motion at all is possible, up to the entanglement time, τe, from which on the segments first realize that they are trapped in a tube environment, the segmental displacement is impaired by the connectivity to neighboring segments. This constraint gives us a time dependence of the mean-square displacement according to the Rouse model: hΔr2 i ⁓ t1=2

(5:93)

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5 Mechanics and rheology of polymer systems

Figure 95: (A) Visualization of a chain entrapped in a confining tube with end-to-end distance r and contour length s. (B) Subdiffusive displacement of a labeled chain segment as a function of time on different timescales during a reptation process. The mean-square displacement, , of the labeled segment is proportional to time, t, only on scales longer than τrep, because only on these long timescales, the labeled segment follows the three-dimensional Fickian diffusion of the chain through space (which we may imagine as a hopping process from one tube to another with an overall three-dimensional diffusion coefficient DRep). On shorter scales, by contrast, the motion of the chain segment under consideration is sub-diffusive. First, on very short timescales, we have subdiffusive spreading of the labeled segment due to the constraint imparted by its connectivity to neighbor segments, which gives a time dependence of ⁓ t½ according to the Rouse model (which applies in polymer melts). Second, on longer timescales, from the entanglement time τe on, additional constraint comes into play, as our labeled segment now realizes that it is also entrapped in a tube and can only move along the tube contour, which in 3D space is a random walk with scaling of r ⁓ N½ such that the overall scaling exponent is ½ · ½ = ¼. Third, on even longer timescales, from the Rouse time τR on, the first constraint is lost, but the second constraint is still active, such that we still have subdiffusive scaling, now again according to ⁓ t½. Fourth, only on timescales longer than the reptation time, τRep, also the latter constraint is lost and the chain moves freely in space, and so does the labeled segment on it.

2)

At the entanglement time, τe, the chain segment also takes notice of the additional confining tube. Up to the Rouse time, τRouse, the segmental displacement along the tube is still Rouse type, but in three dimensions it is pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi (5:94) hΔr2 i ⁓ hs2 i ⁓ t1=2 ⁓ t1=4 This is because the tube itself has a random walk shape with r ⁓ s1=2 . In other words, in the time window τe–τRouse, our labeled segment is constrained by both its connectivity to other segments, giving us one scaling-exponent contribution of ½ according to eq. (5.29), and on top of that, it now also realizes that it is entrapped in a tube and can only move along the tube contour, which in pffiffiffiffiffiffiffiffi three-dimensional space is a random walk with scaling of hr2 i ⁓ hs2 i, giving us another scaling-exponent contribution of ½, such that the overall scaling exponent is ½ · ½ = ¼.

5.10 Terminal flow and reptation

239

3) At times longer than the Rouse time, τRouse, we now discuss the motion of the entire coil along the tube. Its mechanism of motion is still a one-dimensional Rouse-type diffusion, but in three dimensions it is pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi (5:95) hΔr2 i ⁓ hs2 i ⁓ DTube t ⁓ t1=2 4) On timescales longer than the reptation time, t > τrep, we have macroscopic three-dimensional diffusion of the entire coil according to the Einstein– Smoluchowski equation: hΔr2 i ⁓ 6Drep t

(5:96)

The polymer chain now has enough time to “hop” from one confining tube to another and thereby move freely through space.

5.10.4 Constraint release In the years after development of the reptation model, several refinements have been added to it to better match to experimental data. An impactful refinement was the discovery of the so-called constraint release mechanism. It takes into account that, in addition to the test chain, all other chains of the system are in motion as well. As a consequence, a topological constraint resulting from an entanglement point might resolve itself by diffusion of the constraining chain, as depicted in Figure 96. The effective diffusion is, thus, determined by reptation and the finite lifetime of the momentary tube:

Figure 96: Constraint release as an additional mechanism for chain relaxation in entangled systems. (A) Schematic of two entangled chains entrapped in their respective tubes. (B) Reptation of one chain opens a degree of freedom for the other chain to rearrange itself not only by curvilinear motion along its tube contour, but also lateral to it. Picture redrawn from M. Rubinstein, R. H. Colby: Polymer Physics, Oxford University Press, 2003.

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5 Mechanics and rheology of polymer systems

Deff ≈

R2 R2 + τrep τTube

(5:97)

Depending on which process is faster, the overall diffusion is dominated by reptational motion or the constraint release mechanism. As a consequence, both the tracer and matrix molar mass are important.

Questions to Lesson Unit 16 (1) Why can an entangled chain be described by the tube concept? a. Because the entangled chain tries to achieve a preferably high end-to-end distance, whereby a maximal value is achieved in the shape of a tube. b. Because the surrounding chain segments overlap and form a conceptual tube that wraps around the entangled chain. c. Because the only shape that the entangled chain can take is that of a tube. d. Because the tube shape shows the entropically most favorable conformation. (2) Which potential describes the constraint of the test chain by its neighboring segments? a. A Coulomb potential, because of the electrostatic repulsion of the surrounding segments and the test chain. b. A Lennard–Jones potential, because of the repulsive and attractive interactions between the surrounding segments and the test chain. c. A hard-sphere potential, because of the hard-sphere repulsion of the test chain and the surrounding segments when colliding during movement. d. A parabolic potential, which is just a simple and pragmatic concept without explicitly referring to any type of interactions. (3) How does the total molar mass of a polymer affect the rubbery elastic plateau in its mechanical spectrum? a. A higher molar mass leads to an overall increase of the shear modulus, meaning a higher plateau value. b. A higher molar mass leads to a greater extent of the plateau over a wider range of time/frequency. c. A higher molar mass leads to both a higher plateau value of the modulus and a more extended plateau. d. A higher molar mass does not affect the plateau in any manner.

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241

(4) What holds for the scaling of the viscosity η with the degree of polymerization N for the Rouse and the reptation time? a. At the Rouse time, η scales with N, whereas at the reptation time it does with N 3 . b. At the Rouse time, η scales with N 3 , whereas at the reptation time it does with N. c. At the Rouse time and the reptation time, η scales with N 3 . d. At the Rouse time and the reptation time, η scales with N. (5) How do you derive the macroscopic property viscosity η from the reptation model which describes a microscopic process? a. The viscosity is directly implemented in the mathematical description of the reptation model. b. The viscosity can be calculated through the reptation time, which is derived from the reptation model, by simply multiplying with the storage modulus G′. c. The viscosity can be calculated through the reptation time, which is derived from the reptation model, by simply multiplying with the loss modulus G′′. d. The calculation of the viscosity from the reptation time must be done numerically by computer simulation. (6) Why do experimental data show a higher scaling law exponent of 3:4 for the scaling of η with M (for molar masses above the minimal entanglement molar mass Me ) compared to the theoretical value of 3? a. With higher molar mass, the length of the polymer chains increases, which leads to an overall increase in the relaxation time or viscosity, respectively. b. An increase in the molar mass results in a higher pronounced constraint of the test chain and therefore a smaller diameter of the tube according to the tube concept, which leads to an increase in viscosity since the motion of the test chain is more confined. c. The term for the frictional coefficient only scales linearly in N if you consider comparably shorter chains. With increasing molar mass, meaning with increasing degree pffiffi of polymerization, the friction is described by a model that scales with N 2 ≈ N 1:4 leading to an overall power-law exponent of 3:4. d. The reptation time is effectively lower at the tube ends due to fluctuations of the tube length, which are especially pronounced at the tube ends. This effect decreases in magnitude with increasing molar mass.

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5 Mechanics and rheology of polymer systems

(7) The diffusion coefficient can be derived from the reptation time by using an Einstein–Smoluchowski approach. How does the diffusion coefficient scale in N? a. N 2 b. N 2 c. N 1=2 d. N 3 (8) Which times mark the range in which the constraint by the tube is effective? a. τ0 and τRouse b. τe and τRouse c. τRouse and τrep d. τe and τrep

6 Scattering analysis of polymer systems LESSON 17: SCATTERING METHODS IN POLYMER SCIENCE The overall goal of polymer physical chemistry is to bridge between polymer structures and properties. This requires both to be assessed. In this lesson, you will get to know scattering methods as a prime experimental approach to assess polymer structures on different scales. You will learn about the scattering vector as the “magnifying glass” in scattering, and you will see how the scattering intensity as a function of this vector is connected to microstructural aspects.

6.1 Basics of scattering The basic target of this book – and of the field of physical chemistry of polymers in general – is to unravel relations between structure and properties of polymer systems. In Chapter 5, we have learned that the most relevant class of properties of polymers are their mechanical characteristics, and we have learned about a class of methods to quantify these: rheology. What we are lacking still is an equivalent class of methods to characterize polymer structures. We know that the size of polymers and their superstructures is in the colloidal domain of 10. . .1000 nm, so the methodology for structural characterization that we seek for must be able to resolve that. Most polymer systems are amorphous, while only a few can show regular crystalline structures. Hence, we also need a methodology that is generally able to assess both in the colloidal size range. A class of methods that serves all that is based on scattering. The physics behind scattering has its ground in the interaction between quantum objects and matter. The most prominent quantum objects in our focus are neutrons, x-rays, and light; they all obey the particle–wave dualism, which is connected by the De Broglie wavelength λ=

h mν

(6:1)

If a quantum object hits a particle or scattering center in a sample, it is re-irradiated into space, and the overlay of multiple such scattered quantum objects that are reirradiated by multiple adjacent scattering centers in a sample causes a scattering intensity pattern. Analysis of this pattern allows us to draw conclusions on the spatial distribution of the scattering centers in the sample, and with that, about the structure of the sample that we investigate. Depending on the wavelength of the scattered quantum objects, we can do so on different length scales. The three mentioned kinds of quantum objects can therefore all be used as “magnifying glasses” to provide structural information about the sample, with different extent of magnification depending on their https://doi.org/10.1515/9783110713268-006

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6 Scattering analysis of polymer systems

wavelength. However, they do so from slightly different perspectives. This is caused by the different ways in which contrast between the primary beam and the sample is generated during the scattering experiment. When light is used, the scattering contrast stems from spatial variation of the refractive index in the sample, which is correlated to spatial variation of the polarizability. For x-rays, contrast is caused by spatial distribution of electron densities, while for neutrons, contrast is caused from differences in the so-called scattering lengths of certain atoms in the sample, which is most pronounced for the isotope pair of hydrogen and deuterium. All these different sources of scattering eventually trace down to the spatial distribution of matter in the sample, or in other words, to the sample nanostructure, which can therefore be explored by scattering. To properly account for the different origins of contrasting in light, neutron, and x-ray scattering, special sample preparation is necessary in some of these methods. In the case of neutron scattering, we need to tag interesting parts of the samples by exchange of hydrogen atoms by deuterium atoms.80 For x-ray scattering, we must have atoms with a high electron density in the sample. While this might seem tedious at first glance, it offers the chance to selectively label certain areas of a polymer, for example, its side chains or chain ends, and then to selectively investigate them. Often, combinations of the three scattering methods are used to generate a complimentary dataset. A simple scattering experiment is the Bragg diffraction on a crystal lattice; it is schematically shown in Figure 97. An incoming beam causes each scattering center

Figure 97: Bragg diffraction of a neutron or x-ray beam on crystal layers with a lattice constant of d. The part of the wave front scattered at the lower lattice plane has to travel further to reach the detector than the part scattered at the upper lattice plane. The extra distance traveled is highlighted in blue color. Following the geometry, it is 2d sin θ. Note that due to the reflection at an interface of an optically thin to an optically thick medium, the phase of the wave is shifted by 180° at the reflection point.

 The most simple way to realize that is to exchange the normal solvent by a deuterated one, as also used in NMR spectroscopy.

6.1 Basics of scattering

245

(here: atoms in the sample) to re-irradiate incident x-rays as an isotropic spherical wave. When looking from a specific scattering angle, θ, we detect an overlay of all the radiation that is deflected into this specific direction. The sketch in Figure 97 shows that the part of the wave front scattered at the lower lattice plane has to travel further than the part scattered at the upper lattice plane. Following the geometry, the extra distance traveled is 2d sin θ. This causes an interference that is positive and creates scattering peaks at the detector if that extra distance matches an integer multiple of the wavelength. This criterion is expressed in the Bragg equation 2d sin θ = nλ,

with n = 1, 2, 3, . . .

(6:2)

An alternative way to sketch this scattering experiment is with the concept of a wave vector, as shown in Figure 98. Here, ~ ke is the wave vector of the entering beam and ~ ks the wave vector of the scattered beam. The difference of both vectors is called the scattering vector ~ q. The absolute values of both vectors are equal to 2π ~ ~ ke = ks = λ

(6:3)

Figure 98: Vector representation of a scattering process. The entering wave vector is marked ~ ke , q. whereas ~ ks denotes the scattered vector. The difference of both generates the scattering vector ~ Picture redrawn from J. S. Higgins, H. C. Benoît: Polymers and Neutron Scattering, Clarendon Press, Oxford, 1994.

From geometry (see Figure 98), we obtain the scattering vector ~ q as follows: qj = q = j~

4π sinðθ=2Þ λ

(6:4)

With that, the Bragg condition for positive interference reads 2π d = q n

(6:5)

The equation shows how the structural feature d relates to the scattering vector q: the smaller the structures, the larger must q be to probe them. In other words, a low q-value means a low magnification, and vice versa.

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6 Scattering analysis of polymer systems

6.2 Scattering regimes The scattering vector, our “magnifying glass”, can be varied by the wavelength of the scattered quantum objects. Neutrons and x-rays have short wavelengths (λ = 0.01–10 nm) and thereby realize high scattering vectors, whereas light has long wavelengths (green laser λ = 543 nm, red laser λ = 633 nm), and its scattering therefore covers small q-values. Further fine adjustments can be made by the scattering angle θ. This allows the investigation of various structural characteristics of polymeric systems, ranging from the entire polymer coil down to its local chemical structure. The accessible structural motifs of a polymer system are schematically shown in Figure 99. Light scattering probes the first two domains. In the first, our zoom level is so low that the polymer coils appear as pointlike masses, which allows for determination of their molar mass (“zero-angle domain”). In the second regime, zoomed in a little more, the polymer coil dimensions become visible (Guinier domain). To reach even more magnification, we have to switch from light to smaller wavelength quantum objects. By using either x-rays or neutrons, we zoom in a lot further and reach the third regime, in which parts of the chain become visible (if we observe at small scattering angles). Here, no more information on the chain size is gathered, and hence, the degree of

Figure 99: Scheme of the accessible scattering regimes. Light scattering can reach the first two domains, in which the polymer chains appear either pointlike (A) or, in the Guinier domain, as objects whose dimensions, shape, and mass can be determined (provided their radius of gyration is larger than about 50 nm) (B). Small-angle neutron or x-ray scattering allows for further magnification. Here, we lose information about degree of polymerization and polydispersity, but we gain knowledge about substructural characteristics such as the persistence length in the case of linear chains in a dilute solution (C‘) or the correlation length in the case of cross-linked or overlapping chains in a network at semi-dilute concentration (C”). At further magnification, we start to observe single chain segments only, until we finally resolve the local chemical structure of the polymer segmental units (D). Picture redrawn from J. S. Higgins, H. C. Benoît: Polymers and Neutron Scattering, Clarendon Press, Oxford, 1994.

6.3 Structure and form factor

247

polymerization and the polydispersity of it becomes irrelevant. Instead, substructural characteristics such as the persistence length or, in solution, the solution-network correlation length are revealed. Even further magnification, as achieved at higher scattering angles, gives us a glimpse on single chain segments that may appear rodlike already if their persistence length is larger than the zoom level. Having reached the final regime, at even higher scattering angles, we can resolve the local chemical structure and gain information about chain tacticity and other (side-)chain orientations.

6.3 Structure and form factor If we take Figures 97 and 99 together, we can guess already that there must be a general connection between the q-dependence of the scattering intensity, I(q), and the structure of the sample. This is indeed the case, and that connection is given by the so-called structure factor, S(q). It is defined as I ðqÞ = Δb2 SðqÞ

(6:6)

The factor Δb is a measure of the difference in contrast to the species of interest and its surroundings. For light scattering, this is connected to relative polarizability differences. In small-angle x-ray scattering (SAXS), this is connected to relative electron-density differences that are especially strong if parts of the specimen are labeled with heavy atoms. For small-angle neutron scattering (SANS), contrast is generated by the so-called scattering-length density difference that is particularly pronounced between deuterium and hydrogen atoms. Hydrogen substitution for deuterium therefore allows for custom contrasting in SANS, but yet does not alter the polymer too much in its other properties. The reduced form of the above equation describes the differential scattering cross section per unit volume: 1 ∂σ 1 lð~ qÞr2 · = · I0 V ∂Ω V

(6:7)

where σ denotes the scattering cross section, or number of scattered quantum objects per detector area, and Ω the solid angle under which the sample is observed. The exðlð~qÞr2 Þ pression on the right side of the equation, V1 · I , is also called the Rayleigh ratio. 0 The structure factor, S(q), can itself be split into two contributors. For neutron scattering, we have SðqÞ = Nz2 PðqÞ + N 2 z2 QðqÞ

(6:8a)

where N corresponds to the number of molecules of the species of interest in the scattering volume (volume cross section of beam and detector view), where each molecule contains z scattering centers. P(q) is the form factor that accounts for

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6 Scattering analysis of polymer systems

intramolecular interferences happening when scattering waves originate from different scattering centers of the same (macro)molecule. It is directly connected to the shape of that molecule, and is therefore called form factor. Q(q) accounts for intermolecular interferences. In light scattering, by contrast, we have SðqÞ = Nz2 PðqÞ · Q′ðqÞ

(6:8b)

where Qʹ(q) accounts for interferences of light scattered from the centers of mass of different molecules. In the limit of a dilute solution, Q(q) → 0 and Qʹ(q) → 1, which simplifies both expressions to SðqÞ = Nz2 PðqÞ

(6:8c)

The crux now lies in determining an analytical expression for P(q) that matches the recorded dataset from the experiment. Note that at a concentration of c ! 0, when no intermolecular interference is possible, P(q)! 1 for q ! 0. Hence, in the zeroangle limit, the scattering intensity becomes independent of the scattering vector q and reflects the number of molecules in the scattering volume as well as the number of scattering centers per molecule. This can then be translated to a molar mass, and this quantity can therefore be determined from the scattering intensity in the low-q limit. The mathematical definition of the form factor, P(q), is given by PðqÞ =

z X z   1X hexp − i~ q~ rij i 2 z i=1 j=1

(6:9)

This equation defines a pair correlation function. We can understand what this is and why it is the basis to estimate a form factor (and therewith a scattering intensity pattern in the end) as follows: to calculate a spatially-dependent (that means, a q-dependent) scattering intensity, we would actually need to have precise information about the specific spatial distribution of the scattering centers in the detection volume (for example, the monomer units in a polymer coil if our detection volume shall be such that it just fits this one coil). What we only have, though, is average information on their density in space (such as the Gaussian segmental density profile in a polymer coil). A correlation function is some kind of “compromise” beween the needed rich and the given poor information. The most simple type of correlation function is the pair correlation: it estimates the probability to find two scattering centers at a directed (that means, vectorial) distance ~ rij from each other. So, in this defi~ nition rij is a vector that joins two scattering centers i and j in the (macro)molecule under consideration. The operator h. . .i indicates averaging over all orientations and conformations. The orientational average of a function f(θ,ϕ) is

6.3 Structure and form factor

1 hf ðθ, ϕÞi = 4π

2ðπ

249

ðπ f ðθ, ϕÞ sin θdϕdθ

(6:10)

ϕ=0 θ=0

When we apply this to eq. (6.9), we generate z X z 1X PðqÞ = 2 z i=1 j=1

sin qrij qrij



For continuous bodies, the following alternative expression is also valid:

ð sin qrij 1 dr PðqÞ = Ð nð r Þ qrij nðrÞ dr

(6:11)

(6:12)

where n(r) relates to the radial density function of the object of interest.

Figure 100: Sketch of the mean-square-average scattering intensity, , as a function of the reduced scattering vector, qR for spherical particles. In the low Guinier region (qR 1/R, denoted as the Mie regime.

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6 Scattering analysis of polymer systems

is proportional to the extent of the surface in the scattering volume. For the surface S we can express R ⁓ S0.5 and Nz2 = NV 2 = NS3 . Plugging this into eq. (6.8b), again under assumption of an asymptotic power-law tail, yields   −α (6:21) SðqÞ = Nz2 PðqRÞ = NS3 qS0.5 Thus, the scattering intensity must be proportional to the total surface of the sample in the scattering volume, which is proportional to NS. That can only be fulfilled at α = 4. SðqÞ ⁓ q − 4

(6:22)

This is called Porod’s law and visualized in Figure 100 by the envelope slope of –4.

Questions to Lesson Unit 17 (1) Depending on the quantum objects used, three relevant types of scattering experiments for polymer analysis can be differentiated: light, x-ray, and neutron scattering. What holds for the suitability of these three variants? a. Different variants are suitable for different types of polymer samples. Whereas fully swollen polymers can be investigated using light, fully collapsed polymers require the use of neutrons. b. Whether or not a quantum object is suitable to probe a given polymer sample depends on the size of the polymers, which should be in the range of the De Broglie wavelength of the quantum object. c. Each variant can be used on the same type of polymer sample. The only difference lies in the different structural features that are imaged in the different variants, whereas light scattering images single coils, neutron scattering images the structure along the polymer chain. d. Each variant can be used on the same type of polymer sample, resulting in different insights into the features of polymers, such as their molar masses or persistence lengths. (2) What is the criterion for a suitable wavelength to perform scattering analysis of a given material? a. The wavelength should match the distances of the scattering centers in the sample. b. The wavelength should be as different as possible from the distances of the scattering centers in the sample. c. The wavelength should be smaller than the distances of the scattering centers in the sample.

Questions to Lesson Unit 17

253

d. The wavelength should be bigger than the distances of the scattering centers in the sample. (3) What is the role of data obtained from scattering analysis? a. The data can be used to predict a microscopic structure. b. The data can be used to determine the microscopic structure. c. The data can be used to support a predicted image of the microscopic structure. d. The data can be used for reaction monitoring. (4) What holds for the relation of the interlattice distance d and the scattering vector q? a. The smaller the distance d the lower must q be to achieve a smaller magnification. b. The smaller the distance d the lower must q be to achieve a higher magnification. c. The smaller the distance d the higher must q be to achieve a smaller magnification. d. The smaller the distance d the higher must q be to achieve a higher magnification. (5) Besides the wavelength of the scattered quantum objects, which is a parameter that can be varied to achieve the desired magnification? a. The temperature b. The detection angle c. The sample thickness d. The intensity of the incident beam (6) What does the form factor PðqÞ account for? a. The form factor summarizes the contributions of intermolecular and intramolecular interferences. b. The form factor accounts for intermolecular interferences. c. The form factor accounts for intramolecular interferences. d. The form factor accounts for the number of intermolecular interferences, namely the number of scattering centers. (7) Why does the mean-square-average scattering intensity of solid regular spheres show characteristic peaks in the higher q Porod region but levels off to a constant value at the lower q Guinier region? a. At higher q, the magnification is high enough to detect single spheres, whereas at lower q, this differentiation is gone. The boundary line for the two cases is determined by the sphere’s radius.

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6 Scattering analysis of polymer systems

b. At higher q, only parts of the sphere have an impact on the scattering intensity, whereas at lower q, the whole sphere creates a scattering intensity that levels off in a plateau. c. At lower q, scattering is not pronounced, leading to a plateau of the intensity. Only at higher q values, the scattering is pronounced enough to show characteristic decreases in the intensity. d. At lower q, the scattering vector does not suit the size of the spheres, leading to no scattering response. Only at higher q values, the scattering vector and the sphere size are sufficiently similar, resulting in a noticeable scattering signal. (8) What is the common procedure when planning and analyzing scattering measurements on samples? Choose the answer that sorts the following steps in the right order. ① Use pair correlation function. ② Assume structure of the sample. ③ Compare experimental and theoretical data. ④ Conduct the type of scattering experiment. ⑤ Predict angular dependence of the sample. a. ③ → ① → ⑤ → ④ → ② b. ③ → ⑤ → ① → ④ → ② c. ② → ⑤ → ① → ④ → ③ d. ② → ① → ⑤ → ④ → ③

6.4 Light scattering

255

6.4 Light scattering LESSON 18: LIGHT SCATTERING ON POLYMERS The structure of many polymer systems can be assessed by different scattering methods, each with its specific pros and cons. The most widespread technique is light scattering. The following lesson sheds a special view to this method and shows how light scattering can serve to assess a polymer’s molar mass, shape, and interactions with the surrounding.

Based on the above general thoughts about scattering, we now look on how this is reflected in actual experiments. We focus on light scattering, as this method is much easier to realize than neutron scattering or x-ray scattering, and as it is less demanding in view of the sample composition, in a sense that no heavy element or isotope labeling is necessary. According to the discussion of the scattering vector q in the last lesson, light scattering covers the zero-angle and Guinier regimes. This means that information about the molar mass and the size of the polymer coils is accessible.

6.4.1 Static light scattering As we have learned earlier, light scattering generates its contrast from the relative differences of the refractive indices of the sample and its environment. But how does the light scattering process itself work? It starts with an incident, polarized light beam that hits a polarizable object, that is, a scattering center. There, the light induces a dipole moment μ according to μind = α · Eincident = α · E0 cosðωtÞ

(6:23)

where α is the polarizability of the object under investigation. Due to that induced dipole moment, the scatterer starts to vibrate, and as a result, it emits radiation into all directions. The scattered electrical field can be estimated according to Es ⁓

1 ∂2 μind 1 ⁓ sin ’ · E0 αω2 · −cosðωtÞ sin ’ · ∂t2 r r

(6:24)

 where 1r sin ’ · E0 αω2 is the amplitude of the scattered light that depends on the distance, angle, and oscillation frequency of the source; it corresponds to E0,s. The expression − cosðωtÞ denotes the periodicity of the scattered light that is also the periodicity of the incident light. The intensity of the scattered light scales with the square of the electrical field, I ⁓ E2 . This yields

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6 Scattering analysis of polymer systems



E0, s 2 Is = ⁓ α2 ω4 E0 I0

(6:25)

We realize that the scattering intensity has a strong dependence on the frequency. Blue light is scattered much stronger than other colors. This is the reason why we see that part of the sunlight majorly when looking at the sky, which therefore appears blue to us82 (while the sun appears yellow, because we see the remaining spectrum without the blue when looking at it).83 The exact expression is given by Is π2 · α2 · sin2 ’ = I0 ε0 2 · r2 · λ4

(6:26)

where ε0 is the vacuum dielectric constant and λ the wavelength of the utilized light. Now that we know how to treat a single particle, we can expand our view to multiple-particle systems. The easiest such system is that of a dilute ideal gas, because it contains N particles of the same nature as discussed above (spheres). Consequently, we only have to extend the last equation by the number of particles, N: Is, total π2 · α2 · sin2 ’ =N · I0 ε0 2 · r 2 · λ 4

(6:27)

The opposite to a dilute ideal gas is an ideal crystal. Such a material, however, cannot be investigated by light scattering. This is because the distance of the lattice planes is in the range of some picometers, whereas the wavelength λ of the utilized visible light is a couple of hundreds of nanometers. This causes each beam scattered in any direction to always find a counterpart point source that causes destructive interference, and therefore extinction of the scattered light. This is the reason why crystals are transparent. They might at most be colored, but this is caused by light absorption rather than scattering. At a first glance, the latter conceptual picture is also similar for liquids: although there is no perfect order like in a crystal, the molecules in a liquid are still so densely packed that their average distances from one another are still much smaller than the light’s wavelength. Again, as a result, the light is extinguished and the liquid has a transparent appearance. In contrast to a crystal, however, density fluctuations can occur due to the thermal motion of the molecules. As a result, not all of the scattered rays interfere with an extincting counterpart at a given time, and

 The oceans appear blue, too, because they reflect the blue color of the sky above them.  To be precise, the sky actually appears blue to us only when the light passes a short distance and is therefore scattered not too often, which is the case at the height of the day. In the morning and evening, by contrast, the light path is longer, so the blue components are scattered away completely, and we see the remaining red-orange parts at dawn and dusk.

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6.4 Light scattering

so we do observe some scattering. As an example, consider liquid water: it scatters 200 times stronger than the gaseous water. It should be kept in mind, however, that the particle-number density in liquid water is 1200 times higher than in gaseous water. The “normalized” scattering therefore is only 1/6 times higher, so there is in fact still notable extinction due to destructive interference. The polarizability of a molecule is usually anisotropic. If the molecule rotates and vibrates, this manifests itself such as if the polarizability α would oscillate: α = α0 + αk cosðωk tÞ

(6:28)

where ωk is the eigenfrequency of the molecule that is closely connected to its rotational and vibrational eigenfrequencies. We can insert the above expression into eq. (6.23) to generate μind = α · Eincident = α0 · Eincident + αk · cosðωk tÞ · Eincident

j Eincident = E0 · cosðωtÞ

= α0 · E0 · cosðωtÞ + αk · E0 · cosðωtÞ · cosðωk tÞ 1 = α0 · E0 · cosðωtÞ + αk · E0 · ½cosðω − ωk Þt + cosðω + ωk Þt 2

(6:29)

where the first term α0 · E0 · cosðωtÞ accounts for the Rayleigh scattering that is a predominantly elastic scattering by particles much smaller than the wavelength of the radiation. Elastic scattering means that the frequency of the scattered light is the same as that of the incident light, which means that Rayleigh scattering is not accompanied by any energy exchange between the photon and the scatterer. The other two terms account for Raman scattering, which encompasses the fraction of the light that is scattered inelastically, whereby the scattered photons have a frequency and energy different from those of the incident photons. When the scatterer absorbs energy and the scattered photon has less energy than the incident one, we have Stokes Raman scattering, whereas an energy transfer to the scattered photon so that its energy is higher than that of the incident one is called anti-Stokes Raman scattering. More important for static light scattering of liquids or solutions is the Rayleigh scattering part of the process, which is why we limit our further discussion to this type. Let us first examine the case of a dilute polymer solution. Its refractive index, nsolution, is given by nsolution = nsolvent +

dn c dc

square

)

n2solution = n2solvent + 2nsolvent



dn dn 2 c+ c dc dc

(6:30)

Due to its negligible influence, we will disregard the final term of this expression,

dn 2 c . dc

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6 Scattering analysis of polymer systems

The refractive indices n are connected to the polarizability α via the following expression: n2 − 1 =

ðN=V Þ α ε

(6:31)

Applied to the refractive index of the solution, nsolution, we generate       Nsolvent Nsolute Nsolute − V V V n2solution − 1 = αsolvent + αsolute ε0 ε0

(6:32)

The corresponding expression for the refractive index of the solvent, nsolvent, is   n2solvent

−1=

Nsolvent V

ε0

αsolvent

(6:33)

By subtracting the former two expressions from one another and taking eq. (6.30) into account, we can derive a formula that connects the refractive index differences to the incremental refractive index change with concentration, dn dc :   Nsolute V dn n2solution − n2solvent = ðαsolute − αsolvent Þ = 2nsolvent · c (6:34) ε0 dc Solving this for an equation describing the polarizability difference, Δα = αsolute − αsolvent , yields Δα = 2nsolvent · ε0

dn c dn M , = 2nsolvent · ε0 dc N=V dc NA

with

N c · NA = M V

(6:35)

We can now insert this expression into the Rayleigh formula from above (eq. 6.27):84  2 2 π2 · 4n2solvent · ε0 2 dn Is, total dc M =N · · sin2 ’ 2 2 2 I0 NA · ε0 · r · λ4

jN=

cNA V M

 2 2 cNA V π2 · 4n2solvent · ε0 2 dn dc M · sin2 ’ = · 2 2 2 M NA · ε0 · r · λ4 =c·M·V ·

2 4π2 · n2solvent · sin2 ’ dn · dc NA · r2 · λ4

(6:36)

 Equation (6.27) has α in it, because it holds for gases, where refractive index differences occur because we either have a gas molecule (with polarizability α) or nothing in a spot; in the case of solutions, we need to work with Δα instead, because then, refractive index differences occur because we either have a solvent molecule (with polarizability αsolvent) or a solute molecule (with polarizability αsolute) in a spot.

6.4 Light scattering

259

While this equation might seem complicated at first glance, all parameters except for the concentration c and the molar mass M are constant for a given system and setup. This allows us to determine the molar mass M of a compound by performing concentration-dependent static light scattering measurements. As light is scattered into all directions, there is a trivial sample–detector distance dependence of r–2. Furthermore, the scattering volume V has a trivial influence on the scattering intensity detected: the larger the volume, the larger is the I relative intensity, s, Itotal . To get experimental setup-independent results, we normal0 ize to these two factors and introduce the Rayleigh ratio, Rθ, that has the unit m–1: Rθ =



Iθ r2 4π2 n2 dn 2 2 sin ’ · cM = I0 V NA λ4 dc

(6:37a)

We can simplify this expression greatly by combining all the constant values into a constant K. This yields Rθ =

Iθ r 2 =K·c·M I0 V

(6:37b)

In a polydisperse sample, all components with molar mass Mi and the corresponding concentration ci scatter independently, yielding: X cM (6:38) Rθ = K i i i We know from Section 1.3.2 that the weight-average molar mass of a polymer is defined as P P Ni Mi 2 i ci Mi Mw = Pi = P (6:39) N M i i i i ci Inserting this into the expression for the Rayleigh ratio, Rθ, yields Kc 1 = Rθ Mw

(6:40)

With this equation, we can directly calculate the weight-average molar mass from the Rayleigh ratio measured during a scattering experiment. An alternative viewpoint can be gained from fluctuation theory, which combines thermodynamics with light scattering. To account for and quantify concentration fluctuations, which are the actual cause of scattering from solutions, we regard our system to be composed of many elementary cells δV, that are large compared to λ 3 Þ . Concentration fluctuations occur due to momolecular scales but smaller than ð20 lecular exchange between these cells, and are described by  + δα c = c + δc ) α = α

(6:41)

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6 Scattering analysis of polymer systems

Note in this context that strictly speaking, there are fluctuations of the polarizability in each elementary cell, α, not only due to concentration fluctuations, but also due to other types of fluctuations in the system, such as temperature and pressure:





∂α ∂α ∂α ∂c + ∂p + ∂T (6:42) δα = ∂c p, T ∂p c, T ∂T c, p However, these fluctuations affect both the solvent and the solute equally, so we may disregard them. Going back to the original scattering equation (6.27), we need an expression for α2. It is calculated from eq. (6.41) as  + δαÞ2 = α  2 + 2α δα + ðδαÞ2 α2 = ðα

(6:43)

2 , is Let us examine the three terms of this expression a little closer. The first one, α the same for all volume elements very much like in an ideal crystal, so there is no δα, can either have a positive contribution due to interference. The second term, 2α or a negative value, but it will be zero upon averaging. This only leaves the third term ðδαÞ2 to actually cause the observed scattering intensity. By inserting this last term into the Rayleigh equation, we generate Is, total π2 · ðδαÞ2 · sin2 ’ =N · I0 ε0 2 · r2 · λ4

(6:44)

Note that the sin2 ’ term is due to light polarization in this case. The mean-square fluctuation of the polarizability α is connected to the refractive indices n as follows: 2 dn hδc2 i hδα i ⁓ n dc 2

2

(6:45)

The mean-square fluctuation of the concentration, hδc2 i, is given by statistical thermodynamics as RTc hδc2 i = ∂π

(6:46)

∂c T

Plugging this into our above formula and using the virial series expansion of the concentration-dependent osmotic pressure, we get: Kc 1 = + 2A2 c +    Rθ Mw

(6:47)

where A2 is the second virial coefficient of the osmotic pressure, obtained as a z-average.

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261

λ For molecules that are larger than a twentieth of the used wavelength, 20 , or roughly 20 nm, there will be additional intramolecular interference due to optical path differences as sketched in Figure 101. This leads to an additional angular dependence of the scattered light, and necessitates to account for an angular-dependent particle form factor, P(θ):

Kc 1 = + 2A2 c +    Rθ Mw PðθÞ

(6:48)

Figure 101: Schematic of the optical path difference, Δλ, of two interfering beams scattered at different points of a polymer coil with different angles of observation. The lower left pair of beams has a bigger scattering angle, resulting in a bigger optical path difference, whereas the lower right pair of beams has a smaller scattering angle and therefore a smaller optical path difference.

We can use the formula for the form factor that we derived in the last section (eq. 6.17) and insert into it the expression for the scattering vector q (eq. 6.4) to generate 1 16π2 n2 θ hRg 2 isin2 PðθÞ = 1 − q2 hRg 2 i = 1 − 3 2 3λ2

(6:49)

Note that sin2 θ here is not due to light polarization, but comes from the definition of the scattering vector q. Inserting the above expression into eq. (6.48) yields

Kc 1 16π2 n2 2θ 2 + 2A2 c = 1+ hR isin g Rθ M w 2 3λ2

(6:50)

Measuring the Rayleigh ratio Rθ as a function of the scattering angle θ and the concentration c allows three parameters to be determined simultaneously: the weightaverage molar mass Mw, the radius of gyration Rg, and the second virial coefficient A2. This determination is often done graphically using a Zimm plot. For this purpose,

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6 Scattering analysis of polymer systems

experiments are performed at several angles that all satisfy the condition qRg < 1, and at least at four concentrations. Instead of creating two graphs for the angle- and the concentration-dependent series, both can be plotted in a single graph, as shown in Figure 102. From the linear extrapolation of the angle-dependent series as well as the concentration-dependent series to θ ! 0 and c ! 0, respectively, the weight-average molar mass can be determined. The radius of gyration can be calculated from the slope of the angle series at the zero concentration, and the second virial coefficient can be calculated from the slope of the concentration series at the zero angle.85 In statistical thermodynamics, the mean-square concentration fluctuation may also be given as hδc2 i = 

RTc 

∂2 g ∂c2 p, T

with g =

∂G ðGibbs free energy per volume elementÞ ∂V

(6:51)

From this expression, we see that Is RTc ⁓2  ∂ g I0

(6:52)

∂c2 p, T

At a certain critical demixing point, we have a horizontal slope of G(c), meaning 2 that ð∂∂cG2 Þ = 0, as shown in Figure 103. At this point we observe very strong scattering, the so-called critical opalescence.

85 Note, in this context, that the quantities Rg and A2 are obtained as z-averages from the Zimm plot, whereas the molar mass is obtained as a w-average, Mw. This is due to the following reasoning. In general, the electrical field amplitude, E, of a single light-scattering particle (or of a polymer coil in the limit of θ ! 0, where intramolecular interferences are irrelevant) is proportional to its mass: E ⁓ m (or, if you wish, to its molar mass: E ⁓ M). The intensity of the scattered light scales with the square of that electrical field amplitude: I ⁓ E2 . This means that I ⁓ m2 (or I ⁓ M2 if you like that better). For an ensemble of N particles, we get I ⁓ N·m2 (or, if we express it in molar numbers, I ⁓ (n/NA)·M2, with NA the Avogadro number), which already resembles the basic definition of a quantity’s z-average (whereas that of a w-average is related to n·M1 and that of an n-average relates to n·M0). As a result, any quantity obtained from light scattering is generally received as a z-average, including the hydrodynamic radius RH (that is determined from dynamic light scattering, as detailed in the following section), the radius of gyration Rg, and the second virial coefficient A2. The molar mass, however, is determined from the Zimm equation, RKc = M1w (6.40) (which θ reflectsPthe Zimm plot in the limit of θ ! 0 and c ! 0). The concentration in the numerator equals P ni Mi , whereas the Rayleigh ratio in the denominator equals to Rθ = K i ci Mi according to to c = i V P P nMM nM2 i i i i = K i i i . Plugging both into the Zimm formula eq. (6.38). Together, this gives Rθ = K V V P nM gives P in iM i2 = M1w , which directly reflects the definition of a w-average for the molar mass accordi i i

ing to eq. (6.39).

6.4 Light scattering

263

 Figure 102: Plot of RKc as a function of sin2 θ2 + kc, the so-called Zimm plot. The constant k can be θ chosen arbitrarily to make the plot look clear in a sense that the data points are spread equally in the x- and y-directions. From the common y-intercept of the extrapolations for θ ! 0 and c ! 0, the weight-average molar mass, Mw, can be determined. On top of that, the slope of the θ ! 0 extrapolated line gives the second virial coefficient, A2, whereas the slope of the c ! 0 extrapolated line gives the radius of gyration, Rg. Picture redrawn from B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

Figure 103: At the critical demixing point, the slope of G(c) is horizontal, such that very intense scattering, called critical opalescence, occurs.

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6 Scattering analysis of polymer systems

6.4.2 Dynamic light scattering

Figure 104: Data processing in dynamic light scattering. (A) Time-dependent intensity fluctuations as recorded by dynamic light scattering. (B) Intensity autocorrelation as a function of τ, shown both in a linear and in a (C) semilogarithmic plot. The characteristic time τ0 corresponds to the decay of the autocorrelation function to 1/e of its original value and can be easily spotted in the semilogarithmic representation as the inflection point of the function.

When light scattering measurements are carried out in a time-dependent fashion, we talk about dynamic light scattering. With this method, as its name implies, we can obtain information about the dynamics of a sample. This is because scattering centers in the probe volume are subject to thermal motion: they constantly change their mutual distances, thereby causing temporal fluctuation of the scattering intensity due to temporal fluctuation of the intermolecular interference. A time-dependent graph of these fluctuations is shown in Figure 104(A). The fluctuating scattering intensity can be mathematically treated by autocorrelation: g2 ðτÞ =

hI ðtÞI ðt + τÞi hI ðtÞi2

(6:53)

The index 2 indicates that this is an intensity autocorrelation function. An index of 1 denotes the electrical field autocorrelation function, with E instead of I. In such an autocorrelation analysis, we shift a duplicate of the fluctuation intensity signal along a conceptual time axis by a lag time τ. The further we shift, the less similar is the duplicated signal to the original. As a result, the autocorrelation function g2(τ) decays, as shown in Figure 104(B). A characteristic point is the decay to 1/e of the original at the characteristic time τ0, as also indicated in Figure 104(B). It is easily determined as the inflection point in a semilogarithmic plot, as shown in Figure 104(C). At this time, the scattering centers have displaced themselves by a distance of 1/q. This results in the following Einstein–Smoluchowski analog equation: 1 1 ≈ Dτ0 , = Γ ≈ Dq2 q2 τ0

(6:54)

Measuring τ0 at different scattering vectors q, that is, at different scattering angles, shows whether the motion in the sample is diffusive or not. For diffusive motion, we

Questions to Lesson Unit 18

265

need to find a linear interdependence of 1/τ0 and q2. If this is the case, then we can calculate the prime resulting parameter obtained by DLS: the diffusion coefficient D. Using the Stokes–Einstein equation, we can translate this diffusion coefficient into a hydrodynamic radius, Rh. This is not, however, the precise hydrodynamic radius of the sample, but that of an imaginary, perfectly spherical particle that diffuses ideally and has the same diffusion coefficient as the one obtained from the experiment. For polymers, this assumption is valid most of the time, because the shape of a Gaussian coil can be assumed to be spherical on timescales longer than the Rouse or Zimm time.

Questions to Lesson Unit 18 (1) Why does the sky appear blue? a. Because the other parts of the spectrum are absorbed by molecules in the atmosphere, only the blue parts remain. b. Pollution in the atmosphere scatters mainly the blue parts of the spectrum. c. At certain altitudes, molecules in the air appear blue. d. Due to its higher frequency, blue light is scattered stronger than the other parts of the visible spectrum. (2) Why can an ideal crystal not be investigated by light scattering? a. The lattice sites are so close that light with a shorter wavelength (UV at least) would be necessary. b. The lattice planes are so close that their distance is way smaller than the used wavelength. Every scattered beam always finds a counterpart and interferes destructively. c. The distances of the lattice planes are all the same. A scattered beam immediately gets absorbed by the next scattering center that it meets, which then scatters the beam again. The crystal becomes impermeable for light. d. The distances in the crystal are well defined. That is why monochromatic beams are necessary, which cannot be achieved in the visible light spectrum. (3) What do Rayleigh and Raman scattering correspond to? a. Inelastic scattering with energy loss and energy gain. b. Elastic and inelastic scattering. c. Elastic scattering based on visible and UV light. d. Inelastic scattering of particles smaller and bigger than the used wavelength.

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6 Scattering analysis of polymer systems

(4) What is the benefit of introducing the Rayleigh ratio? a. By normalization of the relative intensity with V and r2 , we get a setupindependent variable. b. The Rayleigh ratio is an intensity normalized to the concentration, meaning that it is independent of the sample’s concentration. c. The Rayleigh ratio is directly related to the molar mass of a polydisperse sample, leading to an easy experimental determination of this parameter. d. Since the unit of the Rayleigh ratio is m1 , it is directly related to the wavenumber of the scattered light. (5) Which parameter has no influence on the optical constant K? a. The wavelength b. The polarizability c. The refractive index d. The vacuum permittivity (6) Why is a virial-series expansion of the Zimm equation necessary? a. It is not possible to determine the radius of gyration with the Zimm equation. This parameter can be introduced by a virial-series expansion. b. For real polymer solutions that are not infinitely dilute, additional interactions must be considered. This is pragmatically accounted for by a virial-series expansion. c. By a virial-series expansion, more complicated special cases like multiple scattering can be included. d. With increasing polymer size, the form factor cannot be neglected anymore; it can be identified as one coefficient of the virial-series expansion. (7) Which parameters can be derived from the Zimm plot? a. Radius of gyration, second virial coefficient, weight-average molar mass b. Hydrodynamic radius, second virial coefficient, weight-average molar mass c. Radius of gyration, second virial coefficient, Z-average molar mass d. Weight-average molar mass, form factor, radius of gyration (8) What is the outcome of a DLS experiment? a. Dref b. RH c. τcorrel d. η

6.4 Light scattering

267

LESSON 19: LIGHT SCATTERING ON POLYMER GELS Polymer gels consist of three-dimensional networks of interlinked chains swollen by solvent; they are a prime representative of polymer-based soft matter, with diverse applications relying on their viscoelastic soft mechanics and their ability to take up and be penetrated by solvents and solutes. These applications have a strong dependence on the gel microscopic structures, given by the polymer network mesh sizes in a range of 1–10 nm and by spatial inhomogeneities in the gel cross-linking density in a range of 10–100 nm. To quantitatively assess these structural features, scattering techniques are the method of choice. The following excursion presents the principle and prospect of static and dynamic light scattering to characterize the structure of polymer gels.

6.4.3 Light scattering on polymer gels 6.4.3.1 Polymer networks and gels Polymer gels are three-dimensional networks loaded with solvent, whose chains are cross-linked physically (that means, through the action of coordinative or hydrogen bonds, ion pairs, or other weak interactions) or chemically (that means, by covalent bonds). In principle, they can be formed by any of the following mechanisms: (1) Copolymerization of mono- and multifunctional monomers (2) Polymer analogous reaction of pre-polymerized linear chains with cross-linkable groups (3) End-linking of linear polymer chains with multifunctional cross-linkers (4) Cross-linking of star-shaped macromonomers as precursors with suitable end-groups (5) Cross-linking of linear chains by a (typically harsh) transfer reaction In an idealized view, a polymer gel consists of a single network with homogeneous distribution of cross-linking points on scales of some few nanometers. In a realistic view, by contrast, the structure of polymer gels is usually more complex due to micro- and/or nanostructural inhomogeneities and irregularities, whose magnitude depends on which of the above ways of gel formation has been used.86 Shibayama and Norisuye suggested the following classification: (1) Spatial inhomogeneities due to a spatially inhomogeneous cross-linking density (2) Topological inhomogeneities due to local defects such as loops, cross-linker– cross-linker shortcuts, or loose dangling chain ends in the mesh structure of the network (3) Connectivity defects during the process of gel percolation

 S. Seiffert: “Origin of Nanostructural Inhomogeneity in Polymer-Network Gels.” Polymer Chem. 2017, 8, 4472–4487.

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6 Scattering analysis of polymer systems

The last two points can be summarized as local defects and in turn cause the spatial inhomogeneities on larger scales that are listed in the first point. On top of that classification, Okay further delimited the term “inhomogeneity,” referring to “fluctuation of the cross-link density in space” from the term “heterogeneity,” referring to “the existence of phase-separated domains in polymer gels.”87 Many of the most important gel properties, including the viscoelastic soft mechanics, the optical clarity, and the ability to take up and be penetrated by solvents and solutes, are decisively influenced by the (inhomogeneous) structure of the polymer network. As a result, the quantitative assessment of that structure is a key to rational materials design of polymer gels. 6.4.3.2 Static light scattering on gels If the scattering intensity of a polymer gel is compared with that of an equally prepared non-cross-linked polymer solution of the same concentration (in the semi-dilute regime, that means, in a range where chains overlap already), it turns out that the scattering intensity of the gel is higher. This is because light scattering in a solution is caused only by temporal local random concentration fluctuations, whereas in a gel, there are two types of concentration fluctuations: (1) dynamic thermal concentration fluctuations caused by molecular motion, rÞ δcF ð~ (2) immobile “frozen” concentration fluctuations by an inhomogeneous distrirÞ bution of cross-linking points across the gel, δcC ð~ The total concentration fluctuation δcð~ rÞ at a position ~ r is composed of both these contributions: rÞ rÞ þ δcC ð~ δcð~ rÞ ¼ δcF ð~

(6:55)

Figure 105 schematizes this overlay. Analogous to that, the scattering intensity, and with that also the Rayleigh ratio of a gel, RGel , is a superposition of two independent scattering contributions: the fluid scattering intensity RF by thermal fluctuations, which is largely equal to the scattering intensity of an alike uncross-linked polymer solution, and the excess scattering REx caused by the frozen spatial inhomogeneities of the gel: RGel ¼ RF þ REx

(6:56)

Just like the position-dependent “frozen” concentration fluctuation, δcC ð~ rÞ, the scattering intensity of a gel is also position-dependent, as shown in Figure 106. Because

 This view is in line with an earlier perspective by Dušek and Prins, who denoted “heterogeneous” gels to be formed by microsyneresis during their gelation.

6.4 Light scattering

269

Figure 105: Scattering intensity of a polymer gel as a function of the local nanometer-scale spatial position r due to thermal fluctuations, “frozen” spatial inhomogeneities, and both.

of that permanent position dependence of these gel properties, a polymer gel is a non-ergodic system. Therefore, light-scattering measurements must be conducted at several different sample positions, and a proper statistical treatment of the resulting data is necessary to characterize the structure in detail, as discussed further. In practice, this can be achieved through the use of a rotating cuvette unit in combination with computer-based repetitive measurement protocols.

Figure 106: Speckle pattern of the time-averaged scattering intensity of a gel, hIit;p , probed at different positions p, along with a denotation of the ensemble average hIiE and the ensemble average of a hypothetic fluid-only reference, IFE .

Figure 106 displays a typical speckle pattern of the position-dependent scattering intensity in a gel, obtained by measurement of the time-averaged scattering intensity hIit;p at different positions p. We see that the scattering intensity differs from position to position. By averaging over several positions, the ensemble average hIiE can be determined. With the assumption that the minimum of the scattering intensity

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6 Scattering analysis of polymer systems

hIF iE corresponds to a plain fluid reference, the Rayleigh ratio of the excess scattering hREx i can be calculated: hREx i ¼ hRiE  hRFE i

(6:57)

Analogous to the scattering intensity of large particles in solution (which is subject to intraparticulate interferences), the excess scattering hREx ðqÞi is angular dependent. In a simple approach, the angular-dependent excess scattering hREx ðqÞi is conceived as a product of the zero-angle scattering intensity REx ð0Þ and a q-dependent structure factor, SEx ðqÞ, which can be treated analogous to the form factor of particle solutions: REx ðqÞ = REx ð0Þ · SEx ðqÞ = 4π · K · Ξ 3 n2 · SEx ðqÞ

(6:58)

In these equations, Ξ is the static correlation length, which is a measure of the average distance over which spatial concentration fluctuations occur in the gel; it typi  cally has values of some tens to hundreds of nanometers. n2 is the mean-square variation of the refractive index, which is a measure of how strong the concentration varies over that distance; it can be further translated to concentration fluctuations by knowledge of the refractive index increment, dn/dc, which usually turns out to be several single to tens of percent of that of the overall polymer concentra   tion in the gel. K ¼ 8π2 n2solvent λ4 is an optical parameter. Several empirical theories deliver expressions for the structure factor of gels, as summarized in Table 10. With the linearized equations in the lowermost row of Table 10, the static correlation length Ξ and the mean-square variation of the re  fractive index n2 are accessible by plotting and linear fitting (to obtain the slope m and intercept b) of the dataset, thereby delivering information on the magnitude and lengthscale of spatial inhomogeneities in the gel. 6.4.3.3 Dynamic light scattering on gels Analogous to the previous consideration for static light scattering, the temporally fluctuating intensity of a gel in dynamic light scattering is composed of two contributions: IGel ðtÞ ¼ IF ðtÞ þ IC

(6:59)

(1) Time-dependent fluctuating intensity IF ðtÞ caused by diffusive motion of polymer chain segments (2) Constant scattering contribution IC caused by immobilized static inhomogeneities in the gel Analytically, a temporal intensity autocorrelation is performed by multiplying the intensity I ðq; tÞ at a time t with the intensity I ðq; t þ τÞ shifted by a lag time τ. A

Æn2 æ

 1 4π · K · m3=2 b1=2

1 4πKΞ OZ 3 n2

1 1þq2 Ξ OZ 2

OZ n

þ 4πKΞ1

 1 b1=2 4π · K · m3=2

b

m1=2

b

m1=2

Ξ

1 REx ðqÞ =

qffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1=2 Ξ DB 2 + 21 πKn REx ðqÞ = 2 πKΞ DB 3 n2 2q

SOZ ðqÞ =

Linearized form

1

ð1þq2 Ξ DB 2 Þ2 2

q2

One species distributed in another

Ornstein–Zernike

SDB ðqÞ =

Two-phase system with sharp boundaries

Debye–Bueche

Structure factor, SEx ðqÞ

Explanation

Theory

Table 10: Prominent theoretical approaches to model the structure factor of polymer gels.

 1 expðbÞ · 4π · K · jmj3=2

jmj1=2

  lnðREx ðqÞÞ = ln 4π · K · Ξ GU 3 n2  Ξ 2Gu q2

Randomly distributed domains with different densities   SGU ðqÞ = exp q2 Ξ GU 2

Guinier

6.4 Light scattering

271

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6 Scattering analysis of polymer systems

normalized form of the intensity autocorrelation function is obtained from the intensity measurement signal by hardware correlators: gð2Þ ðq; τÞ ¼

hI ðq; tÞ · I ðq; t þ τÞi hI ðq; tÞÞi 2

(6:60)

For dilute solutions, the field correlation gð1Þ ðq; τÞ is accessible from the intensity correlation gð2Þ ðq; τÞ by the Siegert relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   (6:61) gð1Þ ðq; τÞ ¼ gð2Þ ðq; τÞ  1 ¼ exp Dq2 τ If, as in solutions, the light is only scattered by the (cooperative) diffusive motion of polymer network-mesh chain segments, then this is called homodyne scattering. In polymer networks, though, there are also constant scattering contributions caused by static inhomogeneities, causing partial heterodyning. Both are illustrated in Figure 107.

Figure 107: Homodyne and heterodyne scattering in gels. Picture redrawn from M. Shibayama, Bull. Chem. Soc. Jpn 2006 79(12), 1799–1819.

The field correlation function gð1Þ for one segment subject to thermal motion decreases exponentially according to expðDq2 τÞ, while the field, due to contributions of static inhomogeneities, is constant over time. Since the measured intensity corresponds to the product of two scattering fields, two cases can occur for the intensity correlation function:       ðhomodyneÞ (6:62) gð2Þ ðq; τÞ  1 ⁓ exp Dq2 τ · exp Dq2 τ ¼ exp 2Dq2 τ     ðheterodyneÞ (6:63) gð2Þ ðq; τÞ  1 ⁓ 1 · exp Dq2 τ ¼ exp Dq2 τ This means that in the case of plain heterodyne (HD) scattering, the field correlation is measured directly (cf. eqs. (6.61) and (6.63)).

6.4 Light scattering

273

As shown in Figure 105, the constant scattering contribution varies for each measurement position; the system is inhomogeneous and non-ergodic. This means that measurements at many different positions p are required to fully characterize a gel by dynamic light scattering. In 1973, Tanaka, Hocker, and Benedek (THB) were the first providing the physical principles for the interpretation of experimental results obtained from a polyacrylamide hydrogel probed by dynamic light scattering. The THB theory treats a gel as a viscoelastic polymer network of mesh sizes from 1 to 100 nm dispersed in a continuum of solvent. The hydrodynamic coupling of density fluctuations from solvent and polymer segments located between two junctions allows to distinguish longitudinal from transversal modes of fluctuations. With the assumption of a random distribution of cross-linking points, the scattering is approximately homodyne. According to the viscoelastic properties of a gel, the following formalism can be found for the collective diffusion coefficient: D ¼ Kos þ

4 G=ζ 3

(6:64)

where Kos is the osmotic bulk modulus, G is the shear modulus, and ζ is the polymer–solvent friction coefficient. D is further related to the density correlation function according to 1 D¼ 3N

∞ ð

gðrÞ 0

kT dr 6πξ h

(6:65)

with N being the degree of polymerization of the network strands. With the spatial correlation function for semi-dilute polymer solutions given by

ξ r (6:66) gðrÞ ¼ h exp  r ξh we obtain a Stokes–Einstein analog equation for the hydrodynamic correlation length ξh ¼

kT 6πηsolvent D

(6:67)

A different approach by Joosten considers the time-averaged intensity fluctuation at a certain measuring point p and takes into account the partial HD scattering by use of the following equation for the intensity correlation function:  2  i   h ð1Þ ð2Þ ð1Þ þ 2Xp 1  Xp gF ðq; τÞ (6:68) gHD;p ðq; τÞ ¼ 1 þ Xp2 gF ðq; τÞ homodyne

heterodyne

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6 Scattering analysis of polymer systems

The homodyne scattering ratio Xp as a scaling factor between both terms is defined as the ratio between the time-averaged fluid scattering intensity hIF it and the timeaveraged total intensity hIit;p at the position p: Xp ¼

h IF i t hI it;p

(6:69)

In the case of plain homodyne scattering, we obtain Xp ¼ 1. With increasing heterodyne (HD) scattering contribution, the total intensity hIit;p increases and Xp gets smaller. Also, the amplitude of the intensity correlation σ2 gets smaller with decreasing homodyne scattering ratio:   ð2Þ σ2 ¼ gHD;p ð0Þ  1 ¼ Xp 2  Xp

(6:70)

While the intensity correlation function varies from position to position, the fluid corð1Þ relation function gF ðq; τÞ can be assumed as a simple position-independent exponential function, since it now only captures the dynamic properties of the network:   ð1Þ gF ðq; τÞ ¼ exp DHD q2 τ

(6:71)

Shibayama took up the approach of Joosten by assuming only homodyne scattering ð2Þ for the intensity correlation gHD;p ðq; τÞ and taking into account the Siegert relation, eq. (6.61), which is strictly not valid, to determine an apparent diffusion coefficient for different measurement positions Dapp;p . By comparison to the approach of Joosten, a relation between the apparent diffusion coefficient Dapp;p and the diffusion coefficient DHD concerning partial heterodyning can be established:   DHD ¼ 2  Xp Dapp;p (6:72) With eq. (6.69), we obtain hIit;p hIF it;p 2 ¼ · hIit;p  Dapp;p DHD DHD

(6:73)

Linearized plotting and fitting of a dataset according to eq. (eq. 6.73), which is then based on simply experimentally accessible quantities only, enables one to determine the diffusion coefficient DHD and the fluid scattering intensity hI iFt;p . With the help of eq. (6.67), the dynamic correlation length ξ h can be determined from the diffusion coefficient DHD , which in turn serves as a measure of the mesh size, that is, the spatial distance between two cross-linking points in a polymer network gel.

Questions to Lesson Unit 19

Toyoichi Tanaka (Figure ) was born in Nagaoka city, Niigata Prefecture, Japan, in . He received his BSc (), MSc (), and doctorate degree () in physics from the University of Tokyo. Tanaka joined the physics faculty at M.I.T., Cambridge, MA, in , where he applied the method of dynamic light scattering to study biomedical phenomena. At the same time, he also discovered and detailed upon the volume phase transition of environmentally responsive polymer gels. In the s, Tanaka further evolved that concept to create “smart gels” that mimic functions of proteins. His idea was to create gels from a mixture of different monomers (playing the role of amino acids) with sequences designed through a process of “imprinting.” In the middle of that work, Tanaka unexpectedly died of heart failure on May , , while playing tennis at age .

275

Figure 108: Portrait of Toyoichi Tanaka. Image by Len Irish.

Questions to Lesson Unit 19 (1) What holds true when comparing the scattering intensity of a polymer gel and a polymer solution of the same polymer and same concentration? a. The scattering intensity of a polymer solution is higher than that of a polymer gel, because the light can transmit easier between the polymer coils, allowing multiple scattering to occur. b. The scattering intensity of a polymer solution is higher than that of a polymer gel, because the polymer gel absorbs parts of the light that can no longer be scattered, leading to a diminished scattering intensity. c. The scattering intensity of a polymer gel is higher than that of a polymer solution, because on top of the temporal concentration fluctuations there are permanent fluctuations due to spatial inhomogeneities in the cross-linking density. d. The scattering intensity of a polymer gel is higher than that of a polymer solution, because the density of the scattering centers in general is higher, leading to a higher probability for an incident photon to be scattered. (2) What are the consequences of the different contributions to the scattering intensity for the Rayleigh ratio in a polymer gel? a. The Rayleigh ratio of the gel is the sum of the Rayleigh ratios of the different scattering contributions. b. The Rayleigh ratio of the gel is the difference of the Rayleigh ratios of the different scattering contributions.

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6 Scattering analysis of polymer systems

c. The Rayleigh ratio of the gel is the mean of the Rayleigh ratios of the different scattering contributions. d. The Rayleigh ratio of the gel is the product of the Rayleigh ratios of the different scattering contributions. (3) What is ergodicity? a. When a process is aperiodic, it is referred to as ergodic in statistics. b. Ergodicity is fulfilled when a system shows the same averages over time and ensemble. c. A system with correlation decay, that is, loss of its memory after a specific amount of time, is called an ergodic system. d. Ergodicity is the characteristic of a system in which every parameter at a given time t is determined by the parameters that were present a timestep earlier, at t – τ. (4) Which of the following is an example for a non-ergodic system? a. A microparticle in a liquid b. An ideal polymer coil c. An ideal gas d. A polymer gel (5) What holds for the Rayleigh ratio of excess scattering? a. It depends on the time and the position, which is why averaging over time and positions is necessary. b. It depends on the time and the position, which is why averaging over time and positions as well as normalizing to a minimum value is necessary. c. It depends on the time and the position, which is why averaging over time and positions as well as correcting by a minimum value is necessary. d. It depends on the time and the position, which is why averaging over either time or positions is necessary, but only possible if the system is ergodic. (6) Because it is so complex, the structure factor of a polymer gel cannot be solved analytically. Therefore, there are multiple theoretical approaches which have been derived from empirical data. Which of the following approaches is not suitable to theoretically model a polymer gel? a. A system of two phases with clear boundaries b. A given ensemble, whose molecular dynamics is simulated by the given potentials c. Variable densities, which are present in randomly distributed domains d. Considering an overall main species in which another species is distributed

Questions to Lesson Unit 19

277

(7) How can you derive the intensity correlation from the field correlation function? a. gð2Þ ðq; τÞ ¼ gð1Þ ðq; τÞ2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b. gð2Þ ðq; τÞ ¼ gð1Þ ðq; τÞ þ 1 c. gð2Þ ðq; τÞ ¼ gð1Þ ðq; τÞ2  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d. gð2Þ ðq; τÞ ¼ gð1Þ ðq; τÞ  1 (8) What causes a heterodyne scattering contribution in polymer gels? a. Static light scattering b. Dynamic light scattering c. Static inhomogeneities d. Dynamic inhomogeneities

7 States of polymer systems LESSON 20: DILUTE POLYMER SOLUTIONS Polymer solutions are a prime state of appearance during many polymer syntheses and in almost all polymer-analytical investigations; also, many biopolymers find themselves in a solution state in natural environments. This lesson unit treats the most simple concentration regime of such solutions: the dilute regime. In that state, the polymer volume fraction is so low that the chains do not touch or overlap. Hence, the solution is alike a colloidal suspension of solvent-filled nanogel particles. This consideration allows the viscosity of such dilute solutions to be appraised conceptually, and as it turns out, that delivers a basis to determine the size and shape, and with that the molar mass and solution state of polymers by the simple yet precise experimental method of dilute-solution viscometry.

7.1 Polymer solutions So far, in the course of this textbook, we have learned about the fundamentals of polymer structures, dynamics, and properties, the latter particularly focused on mechanical features. In the following, we go ahead and deepen our knowledge by shedding a specific view to the characteristic and application-relevant states of appearance of polymer systems, discussing their structure, dynamics, and (mechanical) properties separately and specifically. We start by taking a closer look at polymer solutions, before the following subchapters will do the same for networks and gels and for polymers in glassy and crystalline solid states. The melt state, by contrast, is not considered specifically, as we have in fact said quite everything that is relevant for melts already in our preceding chapters, in a sense that in this state, the coils display ideal conformations, have Rouse-type dynamics, and exhibit viscoelastic properties as treated in our large Chapter 5.

7.1.1 Formation In Chapter 4, we have learned that the dissolution of a polymer in a solvent has just a little entropy contribution, and hence, the process of dissolution is mostly governed by the change in enthalpy, ΔH, that comes along with the establishment of monomer–solvent (M–S) contacts in a polymer solution replacing former monomer–monomer (M–M) contacts in the solid polymer body and solvent–solvent (S–S) contacts in the former plain liquid medium. From classical physical chemistry, we also know that a dissolution process is generally an equilibrium with a back-and-forth “reaction,” in a sense that each molecule in a solute species is subject to a solution–dissolution

https://doi.org/10.1515/9783110713268-007

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7 States of polymer systems

equilibrium. However, in a polymer, these molecules are connected to each other as segments in a chain, and so this dynamic equilibrium has to be on the dissolved side for all these segments until a whole chain can get dissolved. On top of that, the chains may be entangled with one another in the solid polymer specimen; in this case, it is not even enough to bring just one chain fully into the dissolved state for it to separate from the solid body, but other chains that are entangled with it must get dissolved, too. Therefore, dissolution of a solid polymer specimen usually takes a long time, often hours or days, sometimes even weeks! During such a long dissolution process, the small and therefore highly mobile solvent molecules will first penetrate into the (usually largely amorphous) polymer solid and fill the free volume between the individual chains, causing the material to swell. Hence, we usually observe swelling of the solid polymer body before it eventually dissolves. If the solvent is not a good one, the dissolution equilibrium of the monomer segments is less favorable, such that it will never be on the dissolved side for all monomers at a time. If that is the case, then only swelling occurs, but no free dissolution of the polymer chains is achieved. Practical note: In contrast to small-molecule solutions, sonication of a polymer solution is not a recommended method to speed up dissolution, as it will break the chemical bonds of the polymer chains! In fact, if you sonicate a mixture of a solid polymer specimen and a solvent, dissolution is indeed apparently fast, but this is because the sonication breaks the chains, and so their fragments dissolve quickly. The fragmentation of the chains is easily recognized by the observation that the final solution has a low viscosity, whereas that of a solution made by patient waiting rather than sonication is usually high. This is because η ⁓ M3, so a low M of fragmented short chains causes a much lower η than a high M of intact long chains.

7.1.2 Concentration regimes A polymer solution can be prepared in different characteristic concentration regimes, as schematically shown in Figure 109. In a dilute solution, each coil is dissolved and surrounded by solvent only but has no contact to other coils. In other words: the solution is so dilute that all the coils are far apart from each other.88 If we increase the polymer concentration, we add more coils to the given fixed volume of our system, such that the spacing between the individual coils shrinks. At the overlap concentration, c*, the polymer coils first start to have physical contact with one another. From this point on, they stop acting as single entities, but instead, they interpenetrate. This is possible because each coil is a loose fuzzy object

 Nevertheless, remember what we have learned about the Gaussian coil: the segmental density inside each single coil is comparably high, corresponding to a local molar concentration of some tens of moles per liter (depending on the degree of coil expansion, and with that, on the solvent quality). In a dilute solution, however, the segmental density between the coils is zero, as they are well separated by solvent.

7.1 Polymer solutions

281

with a lot of free volume inside89 that can be occupied by either solvent (as it is in a dilute solution) or by segments of other coils (as it starts to be the case beyond the overlap concentration). In such a state, we speak about a semi-dilute solution. This name refers to the duality of this concentration regime: on the one hand, the concentration is still comparably low (only a few wt%), so that the solution is still kind of “dilute,” but on the other hand, we have a situation qualitatively different from the true dilute one (where coils are isolated from one another). If we further increase the concentration by adding more polymer, this must somehow be accommodated in the given volume of the system. The only way how this is achievable is by more interpenetration of the coils. By that, the excluded-volume interactions inside each coil become progressively screened by overlapping segments of other coils, such that effectively, the good-solvent situation that each coil has found itself in at the dilute concentration turns into a less good one. As a result, all the coils shrink. At a special point, they have shrunken down to their unperturbed size as they would have in a state without any excluded-volume interactions, that is, in the θ-state. From this point on, our solution has turned into a melt-like fluid. We then speak about a concentrated solution. The overlap concentration, c* or ϕ*, is reached when the volume fraction of the chain segments in solution is equal to the volume fraction of chain segments in each single coil, which we appraise as the volume per segment, l3, times the number of segments, N, over the coil volume, R3: ϕ* =

Nl3 ⁓ N 1 − 3ν R3

(7:1)

We can readily see that the overlap concentration decreases with increasing degree of polymerization N. In other words, long chains (i.e., larger coils according to R ⁓ Nν) overlap at lower concentrations already. Often, an alternate expression is used for the overlap concentration:90   c* g · L − 1 =

3M 4πNA Rg 3

(7:2)

It calculates c* in the very common unit of g·L−1. Let us quantify c* for an exemplary polymer compound: polystyrene with a molar mass of M = 1,000,000 g·mol−1. In a good solvent, its c* is only 3.6 g·L−1, or 0.36 wt%, whereas in a Θ-solvent, its c* is 14.7 g·L−1, or 1.47 wt%. The first value is quite low,

 Again, remember that the ratio of the coil volume to the volume of the actual polymer segmental mass in it is about 10,000:1!  There is actually a whole lot of different expressions for the overlap concentration, as summarized by Ying and Chu in Macromolecules 1987, 20(2), 362–366.

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7 States of polymer systems

Figure 109: Schematic representation of the concentration regimes of a polymer solution. A key parameter is the overlap concentration, c*, at which the polymer chains start to have physical contact with one another. The concentration regime below c* is called the dilute regime, whereas the one above c* is called the semi-dilute regime. Provided that the polymer chains are sufficiently long, another parameter, the entanglement concentration, ce*, becomes relevant. At ce*, the polymer chains are so overlapped that they start to form mutual entanglements. The semi-dilute regime is therefore subdivided into a non-entangled and an entangled one. Commonly, ce* is the twofold to tenfold of c*. At very high concentrations, the solution resembles a polymer melt, albeit residual solvent is still present. This concentration regime is called the concentrated regime.

illustrating how much the polymer coil expands in a good solution. Thus, even a low amount of polymer already creates an overlapping polymer solution.

7.1.3 Dilute solutions 7.1.3.1 Structure The dilute solution regime is the easiest concentration regime to be captured conceptually, as the individual polymer coils are not in contact with one another in that regime. The overall coil conformation depends on the solvent quality according to the general scaling law R ⁓ N ν . Note, however, that on scales below the thermal blob size, a concept introduced in Section 2.6.2, there might be a potential change to ideal statistics. Up to this specific lengthscale ξT, the most relevant energy is the thermal energy kBT, outweighing any other energy such as the excluded-volume interaction. As a result, we can separate two lengthscales: at scales smaller than the thermal blob size, r < ξT, the excluded-volume interaction energy is smaller than kBT, and the polymer chain segments on these small scales have ideal conformation according to a three-dimensional random walk, whereas at scales larger than the thermal blob size, r > ξT, the excluded-volume interaction energy is larger than kBT, and the polymer chain segments on these scales have real coil conformation according to a three-dimensional self-avoiding walk. The structure, thus, is dependent on the lengthscale of observation.

7.1 Polymer solutions

283

7.1.3.2 Dynamics In the dilute regime, the Zimm model that we have encountered in Section 3.6.3 best describes the polymer chain dynamics. Here, M–S interactions cause moving polymer chain segments to drag solvent molecules with them. These, in turn, do the same to the solvent molecules adjacent to them. This drag spreads from one solvent molecule to another until, eventually, it reaches other segments of the same polymer chain. Hence, even distant segments couple to each other by these hydrodynamic interactions through space. As a result, each coil drags the solvent in its pervaded volume with it. The trapped solvent is in diffusive exchange with the surrounding solvent molecules, but it does not drain through the coil. As a result, the coils appear like solvent-filled nanogel particles, as visualized in Figure 110. Given that shape, it is the polymer coil size and degree of expansion in the solvent medium that makes the main effect of the solution viscosity. Hence, in turn, experimental estimation of dilute-solution viscosities can actually serve as a means to gain information about the size and shape of the coils, that is, on their average molar mass and on the extent of swelling of the coils, which further corresponds to the solvent quality. Based on this premise, the method of solution viscometry is a precise and easy means to characterize polymers, with outcome information similar to those obtained from osmometry, namely, a molar-mass information and a solvent-quality information.

Figure 110: Polymer coils entrapping solvent within their pervaded volume and dragging it with them on their motion due to hydrodynamic interactions. This portion of solvent is in diffusive exchange with the surrounding solvent molecules, but the coils are not drained by the medium on their motion through it. Picture inspired by B. Vollmert: Grundriss der Makromolekularen Chemie, Springer, 1962.

The viscosity of a dilute polymer solution can be assessed in multiple variant forms, all tracing back to the relative viscosity: ηrel =

ηsolution ηsolvent

(7:3)

It quantifies the relative extent of viscosity elevation of a solvent due to the presence of polymer coils in it.

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7 States of polymer systems

A variant form is the specific viscosity: ηsp = ηrel − 1 =

ηsolution η − ηsolvent ηpolymer − 1 = solution = ηsolvent ηsolvent ηsolvent

(7:4)

It “isolates out” only the effect of the polymer to the relative viscosity elevation. We may guess already that the extent of viscosity elevation of a fluid due to the presence of a polymer in it will certainly be concentration-dependent. Whenever we have to do with such dependencies, it is helpful to introduce so-called reduced quantities, in which we normalize to concentration; we know such a quantity already from Section 4.3, where we introduced and discussed the reduced osmotic pressure. Following that line of thought, we introduce the reduced viscosity: ηsp c

ηred =

(7:5)

As even small concentrations of polymer will already have a drastic effect on the viscosity, which is due to the large size of the polymer coils, a universal measure is introduced by extrapolation of the latter quantity to zero concentration; this limit is named limiting viscosity number91 or Staudinger index: ½η = lim ηred = lim c!0

1 ηsolution − ηsolvent ηsolvent

c!0 c

(7:6)

Let us get back to the concentration dependence we have touched upon already. For ηred, this is given by a formula introduced by Huggins: ηred = ½η + kHuggins ½η2 c

(7:7)

  ηsolution = ηsolvent 1 + ½ηc + kHuggins ½η2 c2

(7:8)

We may rearrange that to:

This is a form similar to the virial-series expansion of the osmotic pressure, in which the first virial coefficient has to do with the polymer molar mass, while the

 Actually, we should speak about a number only when a quantity has no physical unit. All the viscosity values we have introduced above have no unit, as they are all based on the relative viscosity introduced in eq. (7.3), in which the same unit appears in the numerator and the denominator and therefore cancels out. This means that the concentration in the denominator of eq. (7.5) must have no unit as well to give a dimensionless quantity [η]; this is only fulfilled if we use a dimensionless measure of concentration, such as the volume fraction. If we use a measure for the concentration that has a unit, though, such as the mass-per-volume concentration that is quite common in polymer science, then [η] has a unit that is the inverse of that (i.e., L g–1 or so). This is fine, but strictly speaking, we should not name it limiting viscosity number then, as it is not a number in that case.

7.1 Polymer solutions

285

second virial coefficient accounts for polymer–solvent interactions. The first role is taken by [η] in the latter equation, whereas kHuggins plays the second role.92 To understand the relation between [η] and the polymer molar mass, we have to look into the field of colloid science.93 In that field, the relative viscosity of a suspension of hard colloidal spheres (i.e., noninteracting spherical particles with sizes in the range of some tens to hundreds of nanometers, large enough to no longer show molecular characteristics, but small enough to still show no gravitational sedimentation) at a volume fraction ϕ is given by Einstein’s law:94 ηrel = 1 + 5=2ϕ

(7:9)

Application of that relation to calculate the limiting viscosity in the form of  .     ½η = lim ηred = lim ηsp ϕ = lim ð1=ϕÞ ηrel − 1 yields a number value of 5/2 for ϕ!0

ϕ!0

ϕ!0

spherical colloids.95 Other geometries like ellipsoids, rods, platelets, or whatever will lead to different numbers. With that, we see that [η] is a quantity that has to do with the shape of the solute colloids. We may transfer this to polymers by the following line of thought: First, even with the simple Einstein law, we might actually be able to distinguish between monomeric and dimeric species in the solution. If we assume the monomeric species to be spherically shaped, it will deliver a value of 5/2 for [η], whereas if we assume the dimeric species to be dumbbell-shaped with an aspect ratio of 2:1, it will deliver a value of 3 for [η], and thereby raise the viscosity more than spheres at same volume fractions would do. This trend persists: trimeric dumbbells with an aspect

 Strictly speaking, the parameter [η] already includes information on polymer–solvent interactions, as it is actually a factor that reflects the size and shape but not strictly the molar mass of the solute polymer, which of course depends on the solvent quality and therefore on polymer–solvent interactions. This actually explains why that factor is also present in the second virial-series term of eq. (7.8). Here, however, that parameter alone would be “a bit too much,” such that it gets “attenuated” by the additional factor kHuggins in the second virial-series term. This factor has numerical values of typically kHuggins = 0.38 in a good solvent, kHuggins = 0.5–0.8 in a thetasolvent, and kHuggins = 1–1.3 in a bad solvent.  The field of colloid science is actually a good neighboring discipline to polymer science, as polymers are in fact a (very versatile and interesting) type of colloidal soft matter. Before Staudinger’s notion about their macromolecular chain-like architecture, they were in fact believed to be colloidal clusters of small molecules. With Staudinger’s insight, the fields of colloid and polymer science separated, but with the merger of supramolecular and polymer chemistry in the most recent up to present times, also the fields of polymer and colloid science reapproached each other.  Note: that law is only valid for small volume fractions. This gets clear to us if we think about the limiting case of a dense-packed suspension; this has a viscosity of infinity, that is, much higher than what the Einstein law predicts.  Here we get a number, as we use the dimensionless volume fraction as the measure of concentration; we may then rightfully speak about the limiting viscosity number in that case.

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7 States of polymer systems

ratio of 3:1 would lead to a value of 3.6 for [η]. From that we see clearly that the limiting viscosity is a shape parameter. Second, we now assume a solution of polymer coils to be a simple limiting case by presuming that the polymer coils are spheres. In that case, the volume fraction of the coils in the solution can be appraised as ϕH ⁓ n · rH 3

(7:10)

where n is the number of coils and rH is the hydrodynamic radius of each coil,96 which makes rH3 to be a kind of measure of the coil volume, and hence, n·rH3 to be a measure of the volume of all coils in the sample. Division of that volume by the whole sample volume gives the polymer coil volume fraction in the sample, as expressed in a simplified proportionality form in eq. (7.10). This volume fraction is denoted with an index H in eq. (7.10) to strike out that it is a hydrodynamic volume fraction in a sense that it is not equal to just the volume fraction of chain segments inside each coil, but instead, also accounts for potentially additional hydrodynamically trapped solvent in the coil.97 This is indeed very relevant in dilute solutions, such that ϕH in eq. (7.10) does not simply denote the polymer volume fraction but the polymer coil volume fraction, which is composed of the chain material itself plus the entrapped solvent. In a hypothetic case where the polymer coils are fully collapsed to solid spheres without any entrapped solvent, we have a poor-solvent scaling of rH ⁓ Rg ⁓ hr2 i1=2 ⁓ N 1=3

(7:11)

At a given polymer volume fraction and therefore also a given polymer mass-pervolume concentration, the number of coils n is inversely proportional to the size and therewith also the weight per coil, which both scales inversely to its degree of polymerization N. With that, we get:  3 ηrel ⁓ ϕ ⁓ n · rH 3 ⁓ N − 1 · N 1=3 ⁓ N 0

(7:12)

This is exactly what the Einstein law expresses: the relative viscosity of a suspension of spheres is in fact proportional to the sphere volume fraction, but not explicitly to the size of the spheres! At the more common condition of Gaussian coils with some solvent inside, we have a theta-state scaling of

 We know that quantity from Section 3.6.1: it quantifies the radius of a hypothetic ideal sphere with the same diffusive properties as our object of interest, here: as our polymer coils.  This is the scenario treated in the Zimm model of polymer dynamics.

7.1 Polymer solutions

287

rH ⁓ Rg ⁓ hr2 i1=2 ⁓ N 1=2

(7:13)

 3 ηrel ⁓ ϕ ⁓ n · rH 3 ⁓ N − 1 · N 1=2 ⁓ N 1=2

(7:14)

and hence we get

In the case of a good solvent, the coils are expanded and carry a lot of solvent with them on their motion. Such coils exhibit good-solvent scaling according to rH ⁓ Rg ⁓ hr2 i1=2 ⁓ N 3=5

(7:15)

 3 ηrel ⁓ ϕ ⁓ n · rH 3 ⁓ N − 1 · N 3=5 ⁓ N 4=5

(7:16)

such that we get

A similar general form of a relation has also been found experimentally by Mark, Houwink, and Sakurada: ½η = K · Mη a in which Mη is the so-called viscosity-average molar mass sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P Ni Mi a + 1 a i P Mη = i Ni Mi

(7:17)

(7:18)

It commonly lies between the number average and the weight average.98 K is a constant of proportionality, and a is a scaling exponent that has to do with the extent of swelling of the polymer coils in the solvent of interest. Comparison of eq. (7.17) to eqs. (7.6) and (7.4), along with eqs. (7.12), (7.14), (7.16) indicates a = 3ν − 1

(7:19)

whereby ν is the Flory exponent from Section 3.4. The Mark–Houwink–Sakurada equation99 is therefore an important basis for experimental determination of the polymer molar mass, hereby obtained as the viscosity average. Such kind of estimation requires knowledge of the values of K and a

 In a good solvent, Mη is closer to Mw than to Mn, which you see by plugging in the good-solvent value of a = 0.8 (see eq. (7.16)), which turns eq. (7.18) into a form close to the definition of the weight-average molar mass Mw in Chapter 1.  This equation is encountered with numerous names in the literature by featuring all the three we name here or only two of them (most often Mark–Houwink only), or by additionally or alternatively featuring the names Staudinger and/or Kuhn.

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7 States of polymer systems

for the given polymer–solvent pair, which are tabulated for many different polymer–solvent mixtures. Herman Franz (Francis) Mark (Figure ) was born on May , , in Vienna. Already at the age of , he frequently visited a scientific laboratory at the University of Vienna. Mark started to study chemistry during World War I and obtained his PhD in  and then became an assistant of Wilhelm Schlenk at the University of Berlin in the same year. Just one year later, Mark was invited by Fritz Haber to work at the newly established Kaiser Wilhelm Institute of fiber chemistry in Berlin. In , he accepted an offer from I.G. Farben, which later became BASF at Ludwigshafen, to become a vice director of research; his scientific focus there was on x-ray analytics of fiber-forming polymers. In those times, Mark still opposed Staudinger’s theory of chain-like macromolecules, and not before  did Mark start to accept that insight. Apart from fundamental studies on rubber elasticity together with Guth, Mark explored the commercialization of polystyrene, polyvinyl chloride, and polyvinyl alcohol. After fascism took power in Germany, Mark moved to Vienna and later to New York, where he established the Institute of Polymer Research at Polytechnic Institute, Brooklyn, in , which he led until his retirement in . Mark died on April , , in Austin, TX.

Figure 111: Portrait of Herman F. Mark. Image reprinted with permission from Chemical Heritage Foundation/Science Photo Library.

The experimental determination of η is based on a capillary viscometer, the most famous one being the Ubbelohde viscometer, as sketched in Figure 112. In this apparatus, we determine the time Δt that a given definite volume ΔV of a polymer solution with density ρ needs to flow through a thin capillary of radius rc and length lc, measured at different polymer concentrations including concentration zero, that is, including the pure solvent. We may then convert these efflux times into the viscosity using the Hagen–Poiseuille equation: η=

ρghπrc 4 Δ t = const · ρΔ t 8lc Δ V

(7:20)

The risk in this kind of evaluation is the high power of 4 that the capillary radius rc has in eq. (7.20). Any imprecision in measuring that quantity (and any variation that it may have over the length lc of the capillary) will therefore cause a strong error. However, fortunately, our whole set of equations that leads to the limiting viscosity [η] is based on a relative viscosity, which is simply determined as

7.1 Polymer solutions

ηrel =

ρsolution Δ tsolution Δ tsolution ≈ ρsolvent Δ tsolvent Δ tsolvent

289

(7:21)

So, any error in rc appears in both the estimates of the solutions and the estimates of the pure solvent, and therefore cancels out. From ηrel, we can calculate the reduced quantity ηred and then plot it as a function of concentration, which gives a Huggins plot according to eq. (7.8), as sketched in Figure 113. From the slope, we can determine the Huggins constant, which assesses polymer–solvent interactions, and from the intercept, we can determine the limiting viscosity [η], which will then lead to the viscosity-average molar mass according to the Mark–Houwink–Sakurada equation. In addition, [η] can also serve to appraise the chain overlap concentration in the solvent considered, which is roughly given as c* = [η]−1.

Figure 112: Schematic of an Ubbelohde viscometer to determine the quantities treated in this section. A sample solution (typically about 20 mL is needed) is filled into the right tube. Then, the left thin venting tube is closed (as can be done manually by placing a finger onto it), and part of the fluid is pumped up into the upper reservoir by application of pressure to the right filling tube (as can be exerted manually by hand bellows). Removing the pressure and reopening the venting tube (by lifting off the finger from it, for example) then lets the fluid flow down the middle tube, through the thin capillary. Measurement of the time it takes for the fluid meniscus to pass from the upper to the lower light barrier gives the characteristic efflux time featured in eqs. (7.20) and (7.21).

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7 States of polymer systems

Figure 113: Schematic of a Huggins plot to determine the limiting viscosity [η] and the Huggins constant from a concentration-dependent series of estimates of the reduced viscosity ηred.

Questions to Lesson Unit 20 (1) Why is sonication not suitable to speed up the dissolution process of a polymer? a. Sonication reinforces the movement of the chains, leading to more entanglement and therefore to a hindered dissolution process. b. Sonication may break the bonds in the polymer chains, resulting in quickly dissolving but shorter polymer chains. c. Sonication leads to a forced, nonrandom distribution of the polymer chains with an entropy loss, which disfavors the dissolution process. d. Sonication speeds up the drainage of the solvent, which is trapped inside the polymer coils, leading to deswelling/shrinking and therefore to a hindered dissolution. (2) What does the term “overlap” in the overlap concentration c* refer to? a. The overlap of the dilute and semi-dilute regime b. The overlap of chain segments of the same polymer chain c. The overlap of chain segments of different polymer chains d. The overlap of entire polymer chains (3) The boundary between which concentration regimes does the overlap concentration denote? a. The dilute and the semi-dilute solution regime b. The semi-dilute and the concentrated solution regime c. The dilute and the concentrated solution regime d. The concentrated and the saturated solution regime

Questions to Lesson Unit 20

291

(4) Which energy determines the size of a blob according to the blob concept? a. The potential energy of the chain-internal interactions b. The energy of the chemical bonds connecting the chain segments c. The thermal energy kB T d. The internal energy U (5) The thermal blob size ______ a. marks the lengthscale that draws the line between an ideal coil conformation and a real coil conformation. b. gives the size of the swollen polymer coils in thermal equilibrium. c. is the upper critical size of a blob for which the Zimm model approach needs to be changed to a Rouse model one. d. describes the size of a conceptual blob in which the polymer chain moves due to thermal excitation. (6) Which assumption is not part of the Zimm model? a. The coils can be considered as nanogel particles filled with solvent. b. The moving chain segments drag solvent along with them. c. The trapped solvent remains within the coil. d. The hydrodynamic interactions allow interactions of chain segments that are far apart from each other. (7) Which properties can be determined via viscosity measurements? a. The molar mass and the polymer size b. The molar mass and the solvent quality c. The polymer size and the excluded volume d. The polymer size and the solvent quality (8) Because of the very thin capillary, a measurement in a capillary viscometer is very sensitive to errors in the capillary’s radius. Why do these errors nevertheless not affect the viscosity? a. The derived viscosity is a relative quantity, meaning errors in the radii cancel out. b. Since we perform a measurement series at different concentrations and determine the viscosity by the slope of the linear fit of the data, the error, which only leads to an offset of the data points, does not affect the slope. c. The errors in radii are from a similar magnitude than the common measurement errors in time. These errors compensate each other in the calculation of the viscosity. d. Deviations in the radius of the capillary are as much common as deviations in the capillary’s length for manufactory reasons. In the viscosity calculation, these errors cancel out.

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7 States of polymer systems

LESSON 21: SEMI-DILUTE POLYMER SOLUTIONS Other than in the dilute regime, at concentrations beyond the onset of coil overlap, polymer chains interpenetrate and perhaps even entangle with one another in a solution. This overlap drastically alters the structure and especially the dynamics of the coils in the solution, heavily impacting the solution viscosity and chain diffusivity. A simple yet powerful approach to treat that with the aim of deriving equations for the chain length and concentration dependence of these quantities is a scaling concept introduced by De Gennes.

7.1.4 Semi-dilute solutions 7.1.4.1 Peculiarity Polymer solutions with concentrations above c* are somewhat peculiar, as their structure and dynamics are dependent on the lengthscale under consideration. In such a semi-dilute solution, the system can be viewed to be a space-filling array of correlation blobs, which are conceptual spherical entities with a size of ξ, the correlation length. An illustrative representation of ξ is shown in Figure 114. At distances smaller than ξ, a randomly chosen monomer unit on a chain is surrounded only by solvent molecules or by other monomers from the same chain, whereas the presence of monomer units on other chains is not noticed. On scales larger than ξ, by contrast, overlapping segments of other chains can be “seen” by our chosen monomer, and these other segments screen its interactions such as excluded-volume or hydrodynamic interactions. 7.1.4.2 Structure Up to the correlation length, r < ξ, a chosen monomer can only ”see” solvent and monomers on its same chain – precisely a number of g of them that sit on a chain segment which occupies and defines the blob around the chosen monomer. The conformation of that chain segment within the blob is that of a real chain with a scaling of ξ = gν · l

(7:22)

Beyond the correlation length, r > ξ, the chain conformation is an ideal random walk of correlation blobs, each of which with a size ξ and of which we have a number of N/g. That gives a scaling of 1=2 N R=ξ · g

(7:23)

These correlation blobs are space filling, as is indicated in Figure 114. Hence, the polymer volume fraction of a single blob, given by the ratio of the segmental

7.1 Polymer solutions

293

Figure 114: Illustrative representation of the correlation length in a semi-dilute solution of overlapping polymer chains. In such a solution, the system can be viewed to be composed of a space-filling array of correlation blobs with size ξ each. Inside each blob sits on segments of one and the same chain only, whereas overlapping segments of other chains are present only on scales larger than the correlation-blob size. As a result, at small distances, a selected monomer unit is surrounded only by solvent or by other monomers from the same chain, whereas the presence of monomer units from other chains is noticed not until scales larger than ξ are viewed. Picture redrawn from to P. G. De Gennes: Scaling Concepts in Polymer Physics, Cornell University Press, 1979.

volume inside the blob, gl3, and the blob volume, ξ3, is equal to the polymer volume fraction of the entire system: ϕtotal system = ϕper blob ≈

gl3 ξ3

(7:24)

This is basically a blob-scale variant of the overlap concentration formulated in eq. (7.1). From the latter equation, along with some further scaling arguments detailed in Rubinstein’s and Colby’s textbook on polymer physics (Section 5.3 in there), one can derive a scaling-law dependence for the correlation length ξ as a function of the volume fraction ϕ: ξ ⁓ ϕ−

ν 3ν1

(7:25)

and for the dependence of the polymer coil size R on the volume fraction ϕ R ⁓ ϕ−

ν0:5 3ν1

(7:26)

In a good solvent, with ν = 3/5, the latter scaling laws are ξ ⁓ ϕ − 3=4 and R ⁓ ϕ − 1=8 , whereas in a Θ-solvent, with ν = 1/2, they are ξ ⁓ ϕ − 1 and R ⁓ ϕ0 , as shown in Figure 115. We see that both the correlation length ξ and the coil size R decrease upon increase of the polymer volume fraction ϕ or concentration c beyond the overlap

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7 States of polymer systems

threshold. This is reasonable for both parameters: as we add more polymer to a semi-dilute solution, there is more coil interpenetration, such that we force a randomly chosen monomer on a coil segment to see monomers from other segments at smaller distances already, or in other words, we decrease the correlation length ξ. A similar argument is valid for the polymer size R. The polymer coil is swollen in a dilute solution of a good solvent. Above the point of overlap, however, the coil expansion is mutually hindered because the space for each coil to expand into is restricted by the presence and overlap of other coils, and because coil-expanding excluded-volume interactions are screened by overlapping segments of other coils. Consequently, each coil shrinks in size. This shrinking goes on upon further increase of the concentration, until it has proceeded down to the coil’s ideal Gaussian

Figure 115: Dependence of the correlation length ξ and polymer coil size R on the polymer volume fraction ϕ, plotted in a log–log representation. Below the overlap concentration, ϕ < ϕ*, the chains are separate entities with Flory radius R = RF that do not exhibit any concentration dependence, as in this region, they do not yet realize each other’s presence. As soon as they start to overlap, at ϕ = ϕ*, a randomly chosen monomer on a chain can for the first time see monomers from other chains at distances greater than the correlation length, ξ, which is exactly the Flory radius RF at that point, as coils just first touch each other here. At higher concentrations, ϕ > ϕ*, screening of coil-expanding excluded-volume interactions sets in due to the overlap of interpenetrating segments from other coils, such that the polymer coil is reduced in size according to eq. (7.26), which gives a scaling of R ⁓ ϕ − 1=8 if ν = 3/5 (good solvent value); along with that, the more and more pronounced coil interpenetration causes the length until which a monomer can see only solvent or monomers from its same chain to decrease according to eq. (7.25), which gives a scaling of ξ ⁓ ϕ − 3=4 if ν = 3/5 (good solvent value). In the concentrated regime, at ϕ > ϕ**, the coil has shrunken down to its ideal dimensions with the ideal Gaussian radius of R0, and R does then no longer display any further concentration dependence. The screening of excluded-volume interactions inside each coil is then maximally operative, such that the solvent quality has turned from that of a good solvent (with strong excluded-volume interactions) to that of a θ-solvent (with no excludedvolume interactions); hence, the scaling of eq. (7.25) then turns out to be ξ ⁓ ϕ − 1 , with ν = 1/2 (θ-solvent value). The correlation length itself is then as small as the thermal blob size, ξT, below which residual excluded-volume interactions are weaker than the ever-present thermal energy anyways.

7.1 Polymer solutions

295

dimensions. This point is called ϕ** or c**, and the regime above, ϕ > ϕ** or c > c**, is called the concentrated regime. When we reach it, the coil size stays constant at its ideal Gaussian value of R0 , and the correlation length has decreased down to the thermal blob size, ξ = ξT, and then persists decreasing down to its absolute lowermost limit, which is the monomer size (named l in Figure 115); at that limit, we have reached the state of a polymer melt with a volume fraction of ϕ = 1. Note: the first characteristic threshold ϕ* is usually just a few wt% in many polymer–solvent systems, whereas the second threshold ϕ** is commonly very high. Hence, the largest portion of the ϕ-scale in many systems actually corresponds to the semi-dilute regime. This is why we spend so much emphasis on it here.

The scaling laws for R(ϕ) and ξ(ϕ) given earlier were alternatively and very cleverly derived by Pierre-Gilles de Gennes as well, who realized that the power-law-type nature of almost all relations in polymer science allows the implementation of the following strategy: De Gennes first a priori assumed some kind of power-law dependence to be operative, then included reliable boundary conditions for the upper and lower limits of applicability of that power law, and finally solved for the numerical value of the unknown power-law exponent. As it turned out, his method is as accurate as simple. Based on this premise, the derivation of R(ϕ) relies on the following argument: we know that up to the overlap concentration, ϕ*, a polymer coil has a conformation as in a dilute solution according to RF ⁓ N ν . We now assume a power-law dex pendence for concentrations higher than ϕ* according to R ≈ RF ðϕ=ϕ Þ , which we can formulate because we know by definition that, at the onset of the overlap, ϕ = ϕ*, the polymer coil has its expanded size, R = RF. Taking into account both RF ⁓ N ν and ϕ ⁓ N 1 − 3ν , x ϕ (7:27) R ≈ RF  ⁓ N ν + 3νx − x ϕx ϕ With the knowledge that the polymer coil in a semi-dilute solution is a random walk of correlation blobs according to R ⁓ N 1=2 , we must have ν + 3νx − x =

1 0:5 − ν ) x= 2 3ν − 1

(7:28)

The result is the same scaling law that we have previously derived in eq. (7.26). A similar derivation can be made for ξ(ϕ). Again, we assume a power-law dependence to be operative, using the same boundary condition as we did above (RF ⁓ N ν and ϕ ⁓ N 1 − 3ν ) ξ ≈ ϕ−

ν 3ν1

⁓ N ν + 3νy − y ϕy

(7:29)

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7 States of polymer systems

We know that the correlation length is independent of the total chain length beyond the chain overlap threshold, so the numerical value of the power-law exponent at the N must be zero. With that we get ν + 3νy − y = 0 ) y =

−ν 3ν − 1

(7:30)

Again, we arrive at the same expression we have derived previously in eq. (7.25). 7.1.4.3 Dynamics To discuss the chain dynamics in semi-dilute solutions, we need to further subdivide the concentration range into the unentangled (ϕ* < ϕ < ϕe*) and the entangled (ϕ > ϕe*) semi-dilute domain. Unentangled semi-dilute dynamics (ϕ* ≤ ϕ ≤ ϕe*) Let us start our discussion with the less complex scenario of an unentangled semidilute polymer solution; in that regime, we can use the same model(s) we have used for dilute polymer solutions before. Remember that the Zimm model describes a dilute solution with hydrodynamic interactions (see Section 3.6.3), whereas without these interactions, the Rouse model is applicable. So well then, which of these models shall we use now? This depends on the lengthscale of observation. On lengthscales below the correlation length ξ, hydrodynamic interactions are operative, so the Zimm model is the appropriate choice. On lengthscales above ξ, by contrast, hydrodynamic interactions are screened by overlapping segments of other polymer coils, so the Rouse model is correct here. For a quantitative treatment, we again imagine a polymer chain in the semi-dilute concentration regime to be a sequence of correlation blobs with size ξ. Within each blob, excluded-volume interactions lead to a scaling of ξ ⁓ gν , and hydrodynamic interactions are active as well. This causes a Zimm-type relaxation of the chain segments of length inside ξ each blob, described by τξ ≈

ηξ 3 kB T

(7:31a)

As we know the relation between the correlation length ξ and the volume fraction ϕ (eq. 6.25), we can reformulate eq. (7.31a) to τξ ⁓

η − ϕ kB T

3ν 3ν1

(7:31b)

For the entire chain that consists of N=g blobs, by contrast, excluded-volume and hydrodynamic interactions are screened. This leads to a Rouse-type relaxation of

7.1 Polymer solutions

297

the whole polymer chain, which we conceptualize to be a random-walk sequence of correlation blobs that relaxes according to τRouse = τξ

1 + 2ν N g

(7:32a)

On scales above ξ we observe ideal statistics, and the Flory exponent is ν = 1/2 (as just said: the whole chain can be seen as a random walk of blobs). This simplifies the last expression to τRouse = τξ

2 N g

(7:32b)

Just like in Section 5.8.2 (in the context of Figure 87), we can now plot the frequency-dependent relaxation spectrum of a semi-dilute unentangled polymer solution, which consists of four regimes: (i) a Maxwell-type full relaxation regime on timescales longer than τRouse , (ii) a Rouse-type leather-like domain with relaxation of chain segments longer than ξ on times between τRouse and τξ , (iii) a Zimm-type leather-like domain with relaxation of chain segments shorter than ξ on times shorter than τξ , and (iv) a glassy domain with no relaxation at all on times shorter than τ0 , as sketched in Figure 116.

Figure 116: Relaxation spectrum of an unentangled semi-dilute polymer solution. On times longer than the Rouse time, τRouse, the chains can diffuse freely, such that the viscoelastic response is described by the Maxwell model. The leather-like regime on timescales shorter than that (that is, on frequencies higher than the inverse of τRouse) is subdivided into two parts. In the Rouse domain, chain segmental motion takes place on scales larger than the correlation length ξ, such that hydrodynamic as well as excluded-volume interactions are screened. This is followed by the Zimm domain: here, chain segmental motion takes place on scales smaller than ξ, where hydrodynamic and excluded-volume interactions are present.

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7 States of polymer systems

We have already seen that the correlation blobs are space filling, as is indicated in Figure 114. From this, we have concluded: ϕtotal system = ϕper blob ≈

gl3

Together with eq. (7.25), specified to a form of ξ ≈ l ϕ − pendence of g ⁓ ϕ−

(7:24)

ξ3 ν 3ν − 1 ,

1 3ν1

we get a scaling-law de-

(7:33)

Taking eq. (7.32b) and replacing τξ by eq. (7.31b) as well as g by the latter eq. (7.33) then yields 2 − 3ν

τRouse ⁓ N 2 ϕ 3ν1

(7:34)

We can specify this for different solvent qualities as follows: in a good solvent, with ν = 3/5, the relaxation time τRouse scales with τRouse ⁓ ϕ1=4 , whereas in a Θ-solvent, with ν = 1/2, it scales with τRouse ϕ~1 . Another illustrative and practically measurable quantity that captures chain relaxation is the translational chain diffusion coefficient D. We estimate it based on a simplified Einstein–Smoluchowski equation D≈

R2

(7:35)

τRouse

Here we can insert what we have learned about the relations of the coil size and relaxation time to its degree of polymerization and volume fraction, basically expressed by eqs. (7.26) and (7.34), to obtain R ⁓ N 1=2 ϕ −

ν0:5 3ν1

) R2 ⁓ N 1 ϕ − 2 − 3ν

τRouse ⁓ N 2 ϕ 3ν1

2ν1 3ν1

(7:26) (7:34)

Plugging that into eq. (7.35) generates D ⁓ N −1 ϕ−

1−ν 3ν1

(7:36)

We can specify this for different solvent qualities as follows: in a good solvent, with ν = 3/5, we get D ⁓ N − 1 ϕ − 0:5 , whereas in a Θ-solvent, with ν = 1/2, we get D ⁓ N − 1 ϕ − 1 . As a further practically relevant quantity, we estimate the viscosity of our unentangled polymer solution. For this purpose, we can again use the scaling approach according to De Gennes that we have already encountered when discussing the concentration dependence of the coil and correlation blob size in the preceding section:

7.1 Polymer solutions

η = ηϕ = ϕ

ϕ ϕ

299

x (7:37)

Using eq. (7.1), we can specify to η ⁓ ηϕ = ϕ N ð3ν1Þx ϕx

(7:38)

Long-term relaxation modes are Rouse type with η ⁓ N 1 . Thus, we can solve for the power-law exponent: ð3ν − 1Þx = 1 ) x =

1 3ν − 1

(7:39)

Again, we can determine scaling laws for different solvent qualities: in a good solvent, the viscosity scales with η ⁓ N 1 ϕ2 , whereas in a Θ-solvent it scales with η ⁓ N 1 ϕ5=4 . Entangled semi-dilute dynamics (ϕ ≥ ϕe*) The dynamics of an entangled semi-dilute polymer solution is similar to that of an unentangled one on both long and short timescales, but on intermediate scales, a further domain is added to the mechanical spectrum; this domain is dominated by chain entanglements. By assuming a reptation-like chain motion in this domain, we can estimate the concentration dependence of the reptation time, τRep, by a scaling argument, again according to De Gennes. For this purpose, two boundary conditions are introduced. First, for ϕ > ϕe*, we expect a scaling of τRep ⁓ N 3 . Second, for ϕ < ϕ*, there is no reptation, and the longest chain relaxation time is the Zimm time τZimm. We assume a power law for the range between these two domains and specify it through the use of eqs. (3.34) and (7.1): m ϕ ⁓ τ0 N 3ν ϕm N − ð1 − 3νÞm ⁓ ϕm N 3νð1 − 3νÞm (7:40) τRep = τZimm  ϕ Satisfying the first boundary condition lets us solve for the power-law exponent: 3ν − ð1 − 3νÞm = 3 ) m =

3 − 3ν 3ν − 1

(7:41)

Thus, we can formulate a scaling law for the reptation time according to 3 − 3ν

τRep ⁓ N 3 ϕ 3ν1

(7:42)

Once again, just like above, we can plot the frequency-dependent relaxation spectrum of a semi-dilute entangled polymer solution, which now consists of five regimes: (i) a Maxwell-type full relaxation regime on timescales longer than τRep ; (ii) a plateau region on timescales between the reptation time τRep and the entanglement time τe , on

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7 States of polymer systems

Figure 117: Relaxation time spectrum of an entangled semi-dilute polymer solution. As an addition to the Rouse and Zimm domain that splits the leather-like regime in the case of semi-dilute unentangled solutions, a plateau region appears when entanglements are present. These entanglements dominate the mechanics of the system and lead to an entropy-elastic plateau that is delimited by the reptation time, τRep, after which free relaxation and Maxwell-type flow can set in.

which entanglements impair the chain relaxation, thereby leading to a sample with the ability for entropy-elastic energy storage. Beyond that, just like in the unentangled solution case, we have (iii) a Rouse-type leather-like domain with relaxation of chain segments longer than ξ on times between τe and τξ ; (iv) a Zimm-type leather-like domain with relaxation of chain segments shorter than ξ on times shorter than τξ , and (v) a glassy domain with no relaxation at all on times shorter than τ0 , as sketched in Figure 117. Yet again, as an illustrative and meaningful microscopic quantity, we calculate the translational chain diffusion coefficient as follows: D≈

R2 τRep

(7:43)

We again take what we have learned about the size and time relations: Size: R ⁓ N 1=2 ϕ −

ν0:5 3ν1

) R2 ⁓ N 1 ϕ −

2ν1 3ν1

(7:26)

Time: 3 − 3ν

τRep ⁓ N 3 ϕ 3ν1

(7:42)

and insert that into the expression for the diffusion coefficient to generate the following scaling law:

7.1 Polymer solutions

301

Figure 118: Set of scaling laws for the translational chain diffusion coefficient, D, as a function of the chain degree of polymerization, N, and the polymer volume fraction in the system, ϕ. ν2

D ⁓ N − 2 ϕ3ν1

(7:44)

When we specify that for different solvent qualities, we get D ⁓ N − 2 ϕ − 1:75 for a good solvent and D ⁓ N − 2 ϕ − 3 for a θ-solvent.100 To sum up, the dynamics of a polymer solution depends on many factors, such as the concentration regime and the lengthscale observed. We have derived a variety of scaling laws that all use the Flory exponent ν to take into account different solvent qualities. Figure 118 lists all these scaling laws for the chain diffusion coefficient and gives the numerical values of the power-law exponents for each concentration regime. We see that the scaling generally steepens for both parameters N and ϕ as we go from one concentration regime to the next on the ϕ-axis.

 Note: the correct literature reference to the derivation of these scaling laws by De Gennes (that we have recapitulated here) is Macromolecules 1976, 9(4), 587–593. Many experimental studies, instead, refer to De Gennes’ book Scaling Concepts in Polymer Physics (Cornell University Press, 1979) in that context. This is wrong, as this book actually only features a (rather carefully expressed) derivation of a formula for the reptation time in athermal solvents, whereas expressions for the concentration dependence of the diffusion coefficient are in fact not given in that book; by quite contrast, the book rather states that “Reptation studies on solutions at variable c have been started recently [. . .] – but precise exponents are not yet known.”

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Questions to Lesson Unit 21 (1) What are correlation blobs? a. Conceptual blobs that are in the size of the correlation length. b. Conceptual blobs that are present for the timespan of the correlation time. c. Conceptual blobs that move distances of their own size within the correlation time. d. Conceptual blobs for which the correlation function has a value of 1. (2) What holds for the chain conformation on lengthscales bigger than the correlation length ξ? a. The conformation is that of a real chain of chain segments. b. The conformation is that of a real chain of correlation blobs. c. The conformation is that of an ideal chain of chain segments. d. The conformation is that of an ideal chain of correlation blobs. (3) The polymer volume fraction of a total sample is _____ a. the same as that of a single correlation blob, because within the correlation blob the conformation of the chain is the same as for lengths bigger than the correlation length. b. the same as that of a single correlation blob, because the correlation blobs are space filling. c. the same as that of a single correlation blob, because the correlation blobs are subvolumes of arbitrary sizes of the whole polymer solution. d. not the same as that of a single correlation blob. (4) What happens to the polymer coil size when the polymer volume fraction increases? a. The coil expands due to more screening of excluded-volume interactions. b. The coil expands due to an increase in the correlation length upon increase of the overall number of polymer chains in the solution c. The coil shrinks due to mutual hindrance in the overlap with other coils. d. The coil shrinks due to coil internal interactions that are enforced with increasing polymer concentration. (5) Which coil size is present at the concentrated regime? a. Real coil b. Random coil c. Fully collapsed coil d. Random coil that decreases to a fully collapsed state with increasing concentration.

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(6) Choose the answer that assigns the relevant frequencies in the following relaxation spectrum of an unentangled semi-dilute polymer solution correctly.

a. b. c. d.

−1 ① = τRouse , ② = τξ− 1 , ③ = τ0− 1 −1 −1 ① = τRouse , ② = τZimm , ③ = τ0− 1 −1 −1 ① = τ0 , ② = τRep , ③ = τξ− 1 −1 ① = τ0− 1 , ② = τξ− 1 , ③ = τZimm

(7) The frequency-dependent moduli of an unentangled and an entangled semi-dilute solution are ______ a. completely the same, just the values of the relevant timescales differ. b. similar for shorter and longer timescales but not for intermediate ones. c. similar for shorter timescales but different for longer ones. d. completely different and therefore need different theoretical approaches. (8) Comparing the overall chain diffusion coefficients, what holds for the scaling of N and ϕ with increasing concentration and solvent quality? a. The scaling of N and ϕ flattens. b. The scaling of N steepens, whereas the scaling of ϕ flattens. c. The scaling of N flattens, whereas the scaling of ϕ steepens. d. The scaling of N and ϕ steepens.

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LESSON 22: POLYMER NETWORKS AND GELS A system of cross-linked polymer chains is named a network, and when immersed in (and thereby swollen by) a solvent, it is named a gel. Networks and gels are used in a myriad of applications, be it in vehicle tires, in hygiene products, or in medicine. This lesson unit sheds a spotlight on how we can conceive the formation of networks from a statistical perspective in the framework of the percolation theory, and what the most relevant structural and dynamic properties of networks and gels are, with a specific view to nanostructural inhomogeneities in them.

7.2 Polymer networks and gels 7.2.1 Fundamentals This chapter investigates the ability of polymers to form network structures that when being formed or immersed in a solvent medium turn out to be gels. These materials have valuable properties in our everyday life, be it as hygiene products, in food, or in the life sciences and medicine. In fact, life-forms themselves can be viewed to be gels: whereas animals are protein-based hydrogels, plants are polysugar-based hydrogels.101 It is therefore surprising to realize that this class of materials is lacking a basic common definition. This dilemma was first recognized by the British scientist Dorothy Jordan Lloyd in 1926, who stated that “the colloidal state, the gel, is one which is easier to recognize than to define.” In 1946, Dutch scientist P. H. Hermans defined a gel as “a coherent system of at least two components, which exhibits mechanical properties characteristic of a solid, where both the dispersed component and the dispersion medium extend themselves continuously throughout the whole system.”102 It was famous American polymer scientist and Nobel laureate Paul John Flory who eventually formulated a comprehensive definition of a gel in 1974,103 one that also comprises the above definitions. He delimited four classes of gels: 1. Well-ordered lamellar structures (e.g., soap gels and clays) 2. Covalently cross-linked polymer networks filled with a swelling medium (i.e., polymer gels) 3. Fibrillar networks formed by dynamic arrest of stiff (bio)polymers (e.g., actin fibers) 4. Particulate disordered structures (i.e., colloidal gels)  Freely quoted from Vollmert: Grundriss der Makromolekularen Chemie, Springer, 1962.  Colloid Science, Volume II: Reversible Systems, ed. H. R. Kruyt, Elsevier Publishing Company, New York, 1949, pp. 483–494.  Faraday Discuss. Chem. Soc., 1974, 57, 7–18.

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A further scheme of classification introduced by Paul S. Russo distinguishes between “fishnet gels,” that is, cross-linked network structures, and “lattice gels,” that is, all gel-type systems with a space-filling structure somehow different than a network mesh-type one.104 In the following paragraphs, we take a closer look at the fishnet type, corresponding to Flory’s second class of gels, made from polymeric chains. It should be noted here that samples of such fishnet gels comprised of a polymer network and a solvent almost always also incorporate unbound linear or branched polymer chains. These are part of the sample, but by definition not part of the gel.

7.2.2 Gelation Gelation describes the process of forming a gel. During this process, monomers or polymer chains bond to each other, thereby building up and branching larger and larger molecules up to a specific point, the gel point. At this point, a first “infinite molecule,” meaning a molecule that spans the entire system, appears, and we can speak of a gel for the first time. Further gelation increases this infinite molecule into an infinite cluster, the so-called gel fraction of the sample. Polymer chains that are not (yet) connected to the network structure, by contrast, belong to the sol fraction. These might also form non-system-spanning clusters, which might later be connected to the gel and thereby eventually become incorporated into the gel fraction. Gelation happens due to a multitude of reasons, but they all share one underlying principle, which is the creation of cross-linking points in a polymer network. The various ways of creating such points are listed in Figure 119. Based on the nature of the junction, the broad area of polymer gels (of the fishnet type) can be subdivided into the two classes of physical and chemical gels. In physical gels, the cross-linking points are based on physical (often transient) interactions such as ionic or van der Waals interactions between the chains or between certain functional groups along their backbones, or local helical or crystalline domains in which multiple chains (at least two) participate. This class can be further subdivided into weak and strong physical gels, depending on the strength of that interaction, determining on what timescales the gel is stable. In chemical gels, by contrast, the cross-linking process occurs by formation of chemical bonds between the chains. Chemical polymer gels can be produced via two different routes. They can either be made from cross-linking reactions of monomers during polyaddition, polycondensation, or radial polymerization, or they can be built from already preformed polymer precursors that are end- or side-group functionalized

 P. S. Russo in ACS Symposium Series, Vol. 350, ed. P. S. Russo, ACS, Washington DC, 1987, pp. 1–21.

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Figure 119: Classification and examples of modes of gel formation.

with binding motifs. Alternatively, precursor polymer chains can also be randomly cross-linked along their backbones in a direct way even without specific pre-functionalization by the action of suitable linkers, as done in the production of rubbers via vulcanization. The process of gel formation is a percolation phenomenon. This describes the buildup of branched and interconnected structures of growing size, eventually forming the aforementioned system-spanning molecule that turns the material into a gel at the gel point, corresponding to the percolation threshold, pc. While this might seem abstract at first glance, percolation phenomena are quite generic, and we are familiar to the concept from our everyday life. For example, percolation might happen during a deluge (that may be a result of severe global overheating). First, before the process, we have a land mass that is interspersed with separated individual bodies of water. Addition of water will eventually lead to a reversed system of a continuous lake or ocean that contains some islands of land. In this example, the percolation threshold is the point at which the first map-spanning lake is realized. In another example, a wildfire might or might not spread over a whole forest, depending on the density and dryness of the trees. If both parameters are too unfortunate (again perhaps as a result of global heating), the fire can jump from one tree to another, and at a critical point, the percolation threshold, a first aisle of fire spans over the whole forest. A third example is the one that the whole world witnessed in the early twenty-twenties: the spreading of a virus. First, there are local outbreaks that may actually still be confined locally, but if that chance is missed, at a critical stage, the virus is everywhere, and no local means of containment are effective anymore. For polymer gels, percolation can be assessed by statistical modeling of the gelation process based on a lattice model, very much alike the Flory–Huggins mean-

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field theory of polymer thermodynamics. The mean-field approach for percolation is called the Flory–Stockmayer theory, and we start examining it in detail with the simplest case: one-dimensional bond percolation. 7.2.2.1 One-dimensional bond percolation When limited to only one dimension, the only way that monomers or polymer precursors carrying complementary binding motifs A and B can connect to each other is in a linear fashion, as shown in the upper sketch of Figure 120. To treat this process statistically, we define the fraction of reacted A-groups as p. Consequently, the fraction of unreacted A-groups is (1 – p). This directly corresponds to the number density of molecules in the system, ntot(p). We can understand that by the following thought: as long as no bonds have been formed, the number density of molecules is unity, which indeed corresponds to ntot(p = 0) = (1 – 0) = 1. During gelation, each newly formed connection between the monomers or precursor polymers reduces the number of molecules in the system by 1. Because there is exactly one unreacted A-group per resulting molecule, the number density of the resulting molecules is identical to the fraction of unreacted A-groups, so we indeed get ntot ð pÞ = 1 − p

(7:45)

From that, the number-average degree of polymerization, Nn(p), can be determined as Nn =

1 1 = ntot ð pÞ 1 − p

(7:46)

This value diverges at a percolation threshold of pc = 1. Thus, a one-dimensional system can reach the percolation threshold by connecting all molecules in the system to one single system-spanning macromolecule, but it cannot exceed it. 7.2.2.2 Two-dimensional bond percolation To add another dimension to the percolation process, we need a multifunctional molecule that has the possibility to connect to more than to just one other molecule. Let us consider the condensation (or addition) of monomers with a single A-group and (f – 1) complementary B-groups, ABf–1. An exemplary structure with f = 3 is shown in the lower sketch of Figure 120. In a statistical treatment, we define p to be the fraction of reacted B-groups, so then p(f – 1) is the fraction of reacted A-groups, whereas the fraction of unreacted A-groups is 1 – p(f – 1). Again, this latter fraction is identical to the number density of molecules, ntot(p), as each resulting N-mer has one unreacted A-group: ntot ð pÞ = 1 − pðf − 1Þ Thus, the number-average degree of polymerization can be expressed as

(7:47)

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Nn ð pÞ =

1 1 = ntot ð pÞ 1 − pðf − 1Þ

(7:48)

It diverges at a percolation threshold of pc =

1 f −1

(7:49)

Considering the example shown in Figure 120 with a functionality of f = 3, the percolation threshold is pc = 0.5. This means that half of all molecules have to be connected to each other to realize a system-spanning molecule and, thus, to reach the gel point. These simple examples illustrate to us how bond percolation can be discussed in a statistical view, and how two key parameters, the percolation threshold, pc, and the number-average degree of polymerization, Nn(p), can be obtained from that. In the following, we generalize this discussion.

Figure 120: Schematic representation of one-dimensional (upper sketch) and two-dimensional (lower sketch) interconnection of mutually reactive AB-type monomers to a linear (upper sketch) and a branched (lower sketch) macromolecule.

7.2.2.3 Mean-field model of gelation In a mean-field approach to describe a bond percolation process, f-functional monomers are placed on an f-functional lattice, quite similar to the conceptual approach of the Flory–Huggins mean-field theory for polymer thermodynamics. In a very simple approach, this may be a regular lattice such as one with a square lattice geometry; in a more complicated approach, we may also work with an irregular arrangement of the monomers, as shown in Figure 121. On such geometries, however, possible ring formation (“backbiting”) would lead to high mathematical complexity. To avoid this, a dendritic type of lattice, a so-called Bethe lattice, on which ring formation is precluded by geometry, is used instead. It is displayed for a functionality of f = 3 in Figure 122. On such a lattice, the gelation process can be conceptualized as follows: at a conversion p, a bond between two randomly selected

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adjacent lattice sites has been formed with a probability p. At the percolation threshold, p = pc, the first fully system-spanning molecule shows up. From this point on, the system contains two fractions: the gel fraction that is composed of all molecules that are already part of the system-spanning “infinite cluster,” and the sol fraction that is comprised of all molecules that have either not yet reacted at all or that are part of a non-system-spanning “finite cluster” only.

Figure 121: Bond percolation concept in a lattice-based mean-field view. Random formation of bonds between adjacent lattice sites corresponds to interconnections of the monomers on them; this occurs with a conversion of p = 0–1. This conversion also directly corresponds to the likelihood that a bond has already been formed between two randomly chosen adjacent lattice sites at the extent of reaction p. At the critical percolation threshold, also named gel point, pc, a first system-spanning “infinite” molecule occurs.

Figure 122: A dendritic type of lattice, referred to as Bethe lattice or Cayley tree, is specifically simple to be treated mathematically in a bond-percolation approach. The sketch shows such a lattice with a functionality of f = 3.

I. The gel point To determine the gel point, we randomly select a lattice site that hosts a “parent” molecule. It bonds to one of its neighboring molecules, its “grandparent,” and further bonds to each of the remaining f – 1 adjacent sites (“potential children”) with a probability of p. This means that there are p(f – 1) “parent–children bonds.” Similarly, there are p(f – 1) bonds from each of the “children” to potential “grandchildren,” and so on. From this, two scenarios can be constructed: In the first, the average number of bonds p(f – 1) is smaller than 1, meaning that the bond probability p is smaller than 1/(f – 1). In this case, the “family dynasty” would not survive due to a lack of sufficient offspring. In the second scenario, the opposite is the case. The

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average number of bonds p is bigger than 1, and so the bond probability p is bigger than 1/(f – 1). Here, each generation produces more children than the one that came before, resulting in an “infinite family tree.” This corresponds to the creation of a system-spanning “infinite” molecule. The transition between both scenarios is the percolation threshold pc, or gel point, that can now be expressed as pc =

1 f −1

(7:49)

II. The sol and gel fraction There are two distinctive fractions beyond the gel point: the sol and the gel fraction. To determine the sol fraction, we define Q to be the probability for a molecule on a randomly chosen lattice site A to be not connected to the gel by neither of its f potential bonds. This is the case when it is either not connected to its neighbor B itself, which has a probability of (1 – p), or if it is connected to its neighbor B (probability of p), but B is not connected to the gel by any of its f – 1 further neighbors. This case would have a probability of Q f−1. This yields the recursive formula Q = 1 − p + pQ f − 1

(7:50)

The sol fraction is defined as the probability for a randomly chosen lattice site not to be connected to the gel by any of its f neighbors, neither directly nor indirectly. It can be formulated as Psol = Q f

(7:51a)

The corresponding gel fraction can then be easily calculated from Psol as Pgel = 1 − Psol

(7:51b)

Below the percolation threshold, the sol fraction always has a value of Psol = 1, and the gel fraction is always zero, Pgel = 0. Above the gel point, both fractions can be calculated by eqs. (7.51). This relation is schematically shown in the left schematic of Figure 123. Pierre-Gilles de Gennes was the first to realize that the gel fraction evolves like the order parameter of a second-order phase transition, such as the phase transition between ferro- and paramagnetic states in metallic solids. The similarity of its graphical representation, shown in the right schematic of Figure 123, and the evolution of the gel fraction after the gel point, shown in the left schematic of Figure 123, are striking. III. The number-average degree of polymerization at the gel point A key parameter to define the gelation process is the degree of polymerization at the gel point. To calculate it, we assume the following: without any bonds present

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Figure 123: The bond percolation yields the sol and gel fraction as characteristic parameters of a gelation process. At the percolation threshold, also named gel point, pc, a first system-spanning “infinite” molecule occurs. From that point on, addition of further monomers to this infinite cluster gradually increases the gel fraction in the system, whereas all residual monomeric or oligomeric (= non-space-filling) material is part of the sol fraction that steadily decreases in turn. The gel fraction can be taken as an order parameter to quantify the progress of the gelation process, similar to the conceptual treatment of second-order phase transitions like the ferromagnetic– paramagnetic transition in metallic solids.

in the system, the number density of molecules is defined to be unity, ntot(p = 0) = 1. Each newly formed bond then reduces the total number of molecules by 1. Consequently, the maximum number of bonds that can be formed is f/2. All intermediate states are calculated according to ntot ð pÞ = 1 − p

f 2

(7:52)

This yields the number-average degree of polymerization as Nn ð pÞ =

1 1 = ntot ð pÞ 1 − p 2f

(7:53)

At the percolation threshold, it is characterized by Nn ðpc Þ =

1 2ðf − 1Þ = 1 − f =2ðf − 1Þ f −2

(7:54)

We see that the number-average degree of polymerization does not diverge at the gel point, in contrast to the one- and two-dimensional percolation processes treated earlier. For example, a molecule with a functionality of 3 has a number-average degree of polymerization of 4 at the percolation threshold, Nn(pc) = 4.

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IV. The weight-average degree of polymerization at the gel point The approach to determine the weight-average degree of polymerization at the gel point is quite similar to the discussion of the sol and gel fraction. Let us define μ as the average number of molecules that are connected to a randomly chosen lattice site A by one of its f neighbors. μ is composed of two contributions: first, the direct neighbor site B, which is connected to A with a probability of p, and second, the remaining f – 1 neighbors of B, each of which connected to an average number of monomers μ in turn. This yields the following recursive formula: μ = pð1 + ðf − 1ÞμÞ ) μ =

p 1 − ðf − 1Þp

(7:55)

The weight-average degree of polymerization, Nw, corresponds to the average number of monomers that belong to a randomly chosen lattice site. This number is composed of 1 for the chosen lattice site itself and μ for each of its f neighbors. Thus, Nw can be calculated by Nw ð pÞ = 1 + f μ =

1+p 1 − ðf − 1Þp

(7:56)

In contrast to the number-average degree of polymerization, it does diverge at the gel point, because it puts more emphasis on large clusters/molecules and, consequently, “gets overwhelmed by the infinite one” that arises at the gel point. In the above discussion, we have (again) successfully simplified a complex multibody situation, bond percolation during a gelation process, using a mean-field approach to calculate statistical averages. We have seen how to estimate the percolation threshold, pc, and the number-average as well as the weight-average degrees of polymerization at this crucial point during the percolation process. Beyond the gel point, we have calculated the sol and gel fractions, and realized that they evolve much like in a second-order phase transition as often seen in magnetically active solids. Such understanding that soft-matter phenomena are fundamentally analogous to seemingly very different hard-matter phenomena have earned Pierre-Gilles de Gennes the Nobel Prize in physics in 1991. 7.2.2.4 The gel point The percolation process and the gel point can be monitored using rheology. This is because rheology is able to distinguish between the viscous, fluid-like and the elastic, solid-like component of a sample, which largely correspond to the sol and gel fraction. Furthermore, during a gelation process, the continuous buildup of longer and longer chains can be easily monitored with that method by measuring the raise of the sample viscosity, η. Most typically, the starting reaction mixture has a low initial viscosity value, η0, especially when it is made up of low-molar-mass monomers instead of precursor polymer molecules. As soon as the reaction starts, however,

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longer aggregates are formed, and the mixture becomes more viscous. The viscosity will rise even faster and in a nonlinear fashion as soon as these aggregates become sufficiently long to entangle with one another, especially as they are also branched. Once the first system-spanning molecule occurs, at the percolation threshold pc, the viscosity diverges. Figure 124 summarizes this mechanical signature of a gelation process. Having exceeded the percolation threshold, continuation of the gelation process is now represented in the temporal evolution of the storage and loss moduli, G′ and G″. Beyond the gel point, the sample is a viscoelastic solid in which the storage modulus is higher than the loss modulus, G′ > G″; it is no longer able to flow, but instead, exhibits elasticity. The loss modulus mostly reflects the remaining sol fraction in the sample as well as energy-dissipating entities in the gel network, such as loops and dangling chains. Beyond the gel point, such structures are more and more interconnected to and within the gel, and so its value gradually decreases until it finally vanishes at the point of full cross-linking. The storage modulus reflects the gel fraction and the gel’s mechanical strength; it is composed of a

Figure 124: A gelation process, shown as a sequence of sketches at the bottom of the figure, can be followed by rheology. Continuous determination of the viscosity will reveal the percolation threshold, pc, where the viscosity diverges. Beyond the gel point, the evolution of the sol and gel fractions can be followed by the storage modulus, G’, which represents the gel fraction (and which is composed of a fraction allocated to cross-links, Gx, and to (trapped) entanglements in the network gel, Ge), and the loss modulus, G″, which represents the sol fraction. As the cross-linking process proceeds beyond the gel point, the storage modulus G’ increases due to formation of further cross-links (that means, it is mainly its Gx contributor which increases), whereas the fraction of energy-dissipating entities such as noninfinite clusters in the system as well as loops and dangling chains in the gel network decreases, thereby decreasing G″.

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contribution made by (fixed) chain entanglements, Ge, and a contribution made by interchain cross-links, Gx. In an even deeper view, frequency-dependent oscillatory shear rheology can reveal if the percolation threshold has already been crossed, using the Winter–Chambon criterion of a power-law dependence of the relaxation of G at the gel point given by Gð t Þ ⁓ t − n

(7:57)

At the gel point, both the elastic and the viscous parts of the complex modulus, G’ and G″, scale with that very exponent in a frequency-dependent plot: G’(ω) ⁓ G″ (ω) ⁓ ωn; for G’, and this is shown in Figure 125. The exponent n reflects the relative amount of cross-linking junctions to the amount of cross-linker. It is one half, n = 0.5, for stochiometrically balanced end-linking networks, meaning that each junction is connected to another one by a suitable cross-linker. An excess of crosslinker leads to a smaller value, n < 0.5, whereas a lack of cross-linker causes a bigger value, n > 0.5. Precisely, the storage and loss modulus are given by the following formulas: ð πωn (7:58a) G0 ðωÞ = ω GðtÞ sinðωtÞdt = ΓðnÞ sinð1=2 πnÞ ð πωn (7:58b) G0 ðωÞ = ω GðtÞ cosðωtÞdt = ΓðnÞ cosð1=2 πnÞ According to these formulas, the power-law exponent n reveals information about the relative values of both moduli: if n = 0.5, both the storage and loss modulus have the same value, G’ (ω) = G″ (ω). A value of n > 0.5 leads to a bigger loss than

Figure 125: According to the Winter–Chambon relation, Gðt Þ ⁓ t − n , the complex modulus exhibits an infinite power-law dependence on the measurement frequency at the gel point, with both its parts scaling according to G’(ω) ⁓ G″(ω) ⁓ ωn. Below the gel point, by contrast, G’ has different scaling in the Maxwellian free-flowing and in the Kelvin–Voigt leather-like regime, whereas above, it exhibits an elastic frequency-independent plateau.

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storage modulus, G’(ω) < G″(ω), whereas a value of n < 0.5 entails a bigger storage than loss modulus, G’(ω) > G″(ω).

7.2.3 Structure of polymer networks and gels We have seen earlier that polymer gels are basically semi-dilute solutions of polymer chains that are cross-linked together. As such, gels are essentially “shape-stable fluids”: they consist of more than 90% (often up to 99%) of a fluid medium supported and held in place by a flexible polymer-network skeleton. In the most widespread type of gels, that fluid medium is water; those gels are referred to as hydrogels. The simplified perception of such hydrogels as “shape-stable water” points to their two most relevant areas of application. First, their ability to take up large amounts of fluids (here: water) and to undergo swelling, as we have quantified in Section 5.9.3, makes gels useful as superabsorbers, with a prominent application in diapers and other hygiene products. Second, their constitution of more than 90% water renders them permeable for small substances, whereas their polymer-network skeleton blocks diffusants that are larger than the network mesh size. This combination makes gels useful for applications as separators and matrixes in energy-conversion devices, in the analytical sciences, and in the large field of biomedicine. For all that, it is crucial to get a notion of the gel-network nano- and microstructure and about the diffusive dynamics of guest substances that pass through the gel network; this shall be in the focus of the following two sections. Let us start with a view on gel structures.

Figure 126: Spatial inhomogeneities and connectivity defects in polymer networks and gels. On scales of about Ξ = 10–100 nm, polymer-network gels usually feature a rather inhomogeneous spatial distribution of their cross-linking and polymer-segmental density. Furthermore, on scales of about ξ = 1–10 nm, polymer networks commonly display local structural defects such as loose dangling chains and loops, along with a nonuniform distribution of the network mesh sizes.

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An intuitive imagination of a polymer-network gel is that of a regular, monodisperse, and uniform array of meshes with a size denoted ξ, typically spanning a range of some few nanometers. This imagination, however, is incorrect in almost all cases. Instead, gels usually display a rich variety of structural imperfections and irregularities, as illustrated in Figure 126. At the local scale of the gel-network meshes, in the range of about ξ = 1–10 nm, we first have to consider a polydisperse distribution of network mesh sizes, featuring small ones, large ones, and all kinds of sizes in between. On top of that, polymer networks commonly display structural defects on these scales, such as loose dangling chains connected to the network only with one extremity and loops connected to the network with two extremities but not connecting to further parts of it, thereby not being able to store elastic energy upon deformation. At larger scales, in the range of about Ξ = 10–100 nm, polymer-network gels usually feature quite pronounced spatial inhomogeneity of their cross-linking and polymer-segmental density. We have touched upon that already in Section 6.4.1, and also detailed about the experimental assessment of these structures by light scattering in Section 6.4.2. These inhomogeneous structures form due to two reasons. First, in the case of gel formation by random interconnection of preexisting precursor polymer chains, thermal fluctuations in the system will always cause momentary nonhomogeneous distribution of the chain material in the system, and upon cross-linking, these random temporal fluctuations are “frozen-in” into the network structure to becoming permanent spatial fluctuations. Second, in the case of gel formation by polymerization of multifunctional monomers (or of a mixture of bifunctional monomers and some multifunctional ones acting as cross-linkers), in the early stage of the reaction, multiple cyclization and self-cross-linking will take place, leading to formation and growth of local nanogel clusters. At a later stage of the process, these clusters will become interconnected to form a space-filling gel, which will then still resemble the inhomogeneous distribution of the material from the early stage of the process. Some gels even display further structural inhomogeneity on scales of 100– 1,000 nm and beyond, referred to as porosity. Such pores in gels can be formed due to multiple reasons, including gas bubbles in the system that emerge due to formation of a gaseous by-product, the intentional use of a porogen, or microphase separation of a thermally responsive gel due to the heat of reaction during the gelation process. The inhomogeneous structures of gels have multiple consequences on their properties. Most technically relevant is their effect on the gel mechanics. We have learned how to appraise the elastic modulus of a polymer network (and therewith also of a gel) in Section 5.9.2, in the context of the rubber elasticity model. That model, however, assumes a monodisperse and uniform network mesh topology. In a more realistic polydisperse array of meshes, though, the experimentally assessed modulus will be some kind of average. Even more, in the case of presence of spatial

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inhomogeneities, the locally densely cross-linked nanogel clusters essentially do not store elastic energy internally, but instead, just act like one large cross-linking node each. That means that all the cross-linking points “buried” inside these clusters are “lost” and do not contribute to elastic energy storage, which is why the elastic modulus of gels is often a lot smaller than (typically just a few percent of) what might be expected based on their pure chemical amount of cross-linker.

7.2.4 Dynamics of polymer networks and gels The polymer network mesh structure (and all the inhomogeneity that it exhibits) renders gels interesting as separator materials that let pass substances smaller than the mesh size while blocking diffusants larger than the meshes. Based on that premise, it is logical that depending on the probe size and flexibility, different regimes and mechanisms of diffusion through a polymer-network gel matrix can be identified and presumed. In one scenario, the diffusion of small molecules through the network mesh structure is regarded. For that type of small-probe diffusion, three classes of models have been developed, all predicting a marked obstruction exerted by the polymer network on the diffusion of the small molecules as compared to their diffusion in the absence of the polymer matrix, which is commonly addressed to (i) reduction of the free volume for the probe to diffuse into, (ii) hydrodynamic interactions between the probes and the gel-network matrixes, and (iii) an increased path length for the probe motion due to obstacles set by the gel-network matrix; in addition, combinations of these effects have been discussed. A much more complicated situation is encountered when larger probes, such as macromolecules or colloidal particles, shall diffuse through a gel network, which in fact requires the precise consideration of the probe flexibility, its fractal dimension, its size compared to the network mesh size, and the ratio of its characteristic diffusion time to the characteristic time for network rearrangement, as illustrated and summarized in Figure 127. We can simplify that distinction by narrowing it down to just two cases: rigid mesoscopic probes that can pass the network meshes only if these are still not too small for the probe, and flexible mesoscopic probes that may find mechanisms to move though the network meshes even if these are inherently smaller than the probe. The prime example of rigid mesoscopic probes are colloidal hard spheres. The diffusion of those in polymer matrixes has been appraised originally by De Gennes for the case of uncross-linked semi-dilute polymer solutions, which was later reported by Langevin and Rondelez,105 by considering the increase of the friction

 Polymer 1978, 19(8), 875–882.

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Figure 127: Diffusion of probe particles in polymer-network matrixes as a function of the relative size of the diffusing species (R) to the network mesh size (ξ) and the fractal dimension of the diffusing species, related to its rigidity or flexibility. Figure reprinted with permission from Macromolecules 2002, 35(21), 8111–8121. Copyright 2002 American Chemical Society.

coefficient in a form of f0/f ⁓ exp(–(r/ξ)δ), with r being the probe radius and ξ being the screening length that corresponds to the (here transient) network mesh size. This notion is based on a simple estimation of the energy of activation for slipping through the meshes. A further argument by Cukier suggested δ to be unity.106 The same type of formula has also been found experimentally and later been derived theoretically by George D. Phillies, who suggested a universal scaling equation of type D ⁓ exp(–(r)α) to describe the translational diffusion coefficient, D, of various types of probes such as globular particles, linear chains, or star-shaped polymers in macromolecular matrixes over the entire concentration range from dilute to concentrated solution. In an alternative assessment, the motion of free flexible polymer chains through a polymer network gel may also be quantified based on a reptation mechanism along with scaling arguments, as we have treated it in Section 7.1.4.3.

 Macromolecules 1984, 17(2), 252–255.

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Questions to Lesson Unit 22 (1) What holds for unbound polymer chains according to the definition of a fishnet-type gel? a. Per definition, a polymer fishnet-type gel consists of crosslinked network chains, unbound chains, and solvent. b. The definition of the gel includes all polymer chains, the ones included in the polymer network and unbound ones. c. Only branched polymer chains that are not part of the network but contribute to the sample’s elasticity are part of the definition. d. Unbound chains are not included in the definition of the gel. (2) What is the difference between the sol and the gel fraction? a. The gel fraction only includes the system-spanning network(s), whereas every network of smaller size is included in the sol fraction. Unbound chains are not included in neither of these. b. The gel fraction only includes the system-spanning network(s), whereas every network of smaller size is included in the sol fraction together with currently unbound chains. c. The gel fraction includes every network in the sample, whereas the sol fraction only includes the unbound polymer chains. d. The gel fraction only consists of networks that are permanently crosslinked, whereas the sol fraction consists of the reversibly cross-linked ones. (3) What does the phenomenon of percolation in polymer science refer to? a. The buildup of a system-spanning network b. The separation of the gelled part from the still-liquid part of a polymer solution c. The filtration of a polymer solution d. The formation of polymer coils in a polymer solution (5) What is the percolation threshold? a. The point at which percolation is completed. b. The point at which percolation reaches a critical value – the first system spanning network is formed. c. The point at which percolation is at its highest rate. d. The point at which percolation is possible due to a high-enough polymer concentration.

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(6) What is another popular simple model of percolation next to bond percolation? a. String percolation, where adjacent lattice sites can be connected by linear sting-like structures. b. Site percolation, where all lattice sites are unoccupied at first and network formation depends on whether the adjacent lattice sites get occupied or not. c. Branched percolation, where dendritic networks are formed starting from a single percolation center. d. Cluster percolation, where inhomogeneous clusters form due to inhomogeneous conversion of the network structure. (7) What does not happen when the percolation threshold pc is reached? a. A system spanning network is formed. b. The gel point is reached. c. The gel fraction reaches a value of 1. d. The viscosity diverges. (8) Organize the following polymer network inhomogeneities according to their size from large to small: ① Loose dangling chains, loops, and diverse mesh sizes ② Pores, formed by gaseous by-products or phase separation during gelation ③ Clusters due to multifunctional monomers a. ② > ① > ③ b. ③ > ② > ① c. ① > ② > ③ d. ② > ③ > ①

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LESSON 23: POLYMERS IN BULK SOLID STATES The former lesson unit has introduced an important solid state of polymers: the gel. In that state, the mechanical appearance of a specimen is form-stable and elastic, meaning that the storage part of the complex modulus is greater than the loss part. The composition of a sample in that state, however, is mostly constituted by a marked amount of immobilized lowmolar-mass swelling medium. In the following, we will focus on solid states that are composed of a polymer in its bulk only, thereby featuring much higher moduli than gels. Depending on the regularity of microscopic packing of the material in these bulk solid states, we speak about a glassy or a crystalline polymer. In this lesson unit, we learn what the differences and characteristics of these states are, how they form, and what properties they have.

7.3 Glassy and crystalline polymers 7.3.1 The glass transition 7.3.1.1 Fundamentals To this point, we have extensively considered polymers with flexible conformations, as realized in the states of a melt, a solution, or a gel. We now switch gears and consider the opposite: polymers in a fixed, nonflexible conformation, as realized in bulk solid states. Depending on whether the fixed microscopic chain conformation is disordered and amorphous or ordered and regular (at least in part), we distinguish between polymers in a glassy or in a crystalline state (to be precise, it is actually a partially crystalline state for polymers). Unlike low-molar-mass molecules that predominantly crystallize in an ordered fashion upon decrease of temperature, polymers often undergo a glassy solidification named vitrification instead. This is because their usually nonregular primary constitution as well as the polymer chain random coiling and potential chain entanglement prevent many polymers to build up a regular crystal structure. Instead, such polymers rather “freeze” as an amorphous structure as soon as the environmental energy is too low to allow for segmental mobility. This point is called the glass-transition temperature, Tg, which is actually more a temperature range than a defined single temperature. Figure 128 shows the position of the glass-transition regime on the modulus versus time-or-temperature representation that we have already encountered in Section 5.7.3 (see Figure 83), when we learned about the mechanical spectrum of a polymer. In that transition regime, a polymer exhibits sequential activation of Rouse or Zimm modes, thereby transitioning from a completely frozen state without any kind of chain-backbone dynamics to a state in which such dynamics is activated. In the former state, chain deformation can only be accommodated on an atomic scale in a sense that bonds are stretched or bonding angles are distorted, which comes along with an energy-elastic response captured by values of the elastic

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Figure 128: Schematic representation of a polymer’s full mechanical spectrum. Below the glasstransition temperature, Tg, the energy is insufficient even for motion of only single chain segments, and so the amorphous polymer structure is frozen in time. Often, the glass transition happens gradually over a glass-transition regime rather than at a specific sharp temperature.

modulus in the gigapascal range. In the latter state, by contrast, chain deformation can be accommodated by much easier bond torsion, which comes along with an entropy-elastic response captured by values of the elastic modulus in the megapascal range. In the transition range in between these fundamentally different states, the polymer sample has both already activated “fluid” and still deactivated “solid” chain dynamics, and as a result, both the storage and the loss part of the complex modulus are relevant, which ideally results in mechanical characteristics with G’ ≈ G″, that is, tan δ ≈ 1.107 Due to the fundamental change of the polymer mechanical characteristics in the glass-transition regime, the glass-transition temperature is in fact one of the most relevant material parameters. There is one remarkable and tragic example of material failure caused by ignorance of that fundamental change. On the morning of January 28, 1986, the spacecraft Challenger was supposed to launch from Kennedy Space Center to a 6-day space mission. Temperatures on that January morning were low, and so rubbery gaskets in the fuel system were in a glassy state instead of a

 To be precise, this line of argument actually only holds for materials that directly transition from a solid-like elastic glass, where tan δ < 1, to a liquid-like viscous melt, where tan δ > 1, so that the intermediate point with tan δ = 1 marks the transition. According to Figure 128, however, samples with entangled or cross-linked chains instead show a transition from one solid state to another solid state, namely between an energy-elastic glass to an entropy-elastic rubber. In those, tan δ is not necessarily 1 at the transition point, because then, G′ > G″ in both the glassy and the rubbery realm, that means, both below and above the glass transition. But at least, G″ approaches G′ during the transition in that case, as also shown in Figure 128. Hence, in that case, tan δ is always smaller than 1, but at least it is “minimally small” (i.e., maximal) at the point of closest proximity of G′ and G″, which thereby marks the glass-transition.

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rubbery one, thereby having lost their ability to seal. The result was fuel leakage, leading to an explosion that disrupted the vessel 73 s after liftoff at about 15 km altitude. Seven astronauts’ lives on board were lost on that day, including the one of a school teacher who had planned to give classes from space. This catastrophe shows us how (literally) vital it is to be aware of the fundamentally different properties of a polymer material in the glassy versus the rubbery state, and what role temperature plays in the transition between the two.

It is important to note that the glass transition is not a phase transition phenomenon, despite it is being called a transition. A phase transition is defined as a change in the microstructure of a sample that entails an observable change of the macroscopic properties. During the glass transition, however, the chain segmental dynamics is frozen, but the microstructure of the sample does not change. Hence, even though the glass transition has some phenomenological characteristics that also second-order phase transitions have, it is actually not a phase transition but a strictly kinetic phenomenon! That phenomenon may in fact be expected to happen for all matter, that means, not only for polymers, as all matter may generally be subject to the exact same principle of getting to a point where its dynamics is frozen while its structure is still amorphous. For that to happen, we must simply have a situation at hand where during the drop of temperature, crystallization is precluded to happen somehow, so that we can get to a point where the temperature eventually gets so low that all dynamics is frozen while the sample still has an amorphous structure. This is the case if the rate of cooling is so fast that the sample has no time to crystallize underway, meaning that the molecules do not manage to arrange themselves on a regular crystalline lattice before their dynamics is already frozen. As classical molecular matter is commonly composed of rather small molecules, though, these in fact can do so in many cases, such that no glass transition but instead normal crystallization is seen usually, unless we cool down really fast, for example, by shock-freezing in liquid nitrogen, helium, or argon. In soft matter, by contrast, the large size and slow dynamics of the building blocks is a great drag or even impairment for crystallization to happen, such that glassy solidification happens in many cases even at moderate cooling rates. Here, we simply pass the point of potential crystallization too quickly and thereby then go to a state of frozen dynamics but still amorphous structure. This is especially pronounced for polymers, as in those, it is not the coils as a whole that must be arranged on lattice positions, but the segments in them, and due to their mutual connectivity, which is often even irregular, such regular arrangement on crystalline lattice positions is either pretty impaired or even completely impossible. This is why the glass transition is observed quite frequently (and why crystallization is observed quite rarely) in polymer materials.

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7.3.1.2 Conceptual grounds of the glass transition We can get a conceptual microscopic understanding of the glass-transition phenomenon from two viewpoints. The first viewpoint has a kinetic and energetic focus, as we have also partly argued just above already. Segmental dynamics in polymer chains takes time; we have quantified that in the concept of relaxation modes in Section 3.6.4. The higher the mode index, the more segments move in a cooperative fashion, and the longer does it take for that subsection of a chain to get displaced over a distance at least equal to its own size. And this is even more true when the temperature is low, because dynamics is generally thermally activated. In this conception, we may view the glass-transition temperature to be the one below which even the displacement of just single monomer units in a chain over a distance equal to their own size requires infinitely long time. In an alternative conception, we may also address an energy argument for the same matter: chain segmental motion requires bond torsion along the polymer backbone, and this has energies of activation in the range of some few multiples of RT (or kBT) at room temperature (see Figure 14 in Section 2.1.1). Even though these are low activation energies, at very low temperatures even those cannot be overcome, so that any sort of chain dynamics is frozen – or takes infinity long times if we translate the activation energy into a rate constant based on an Arrhenius-type equation. A second viewpoint on the glass transition has a focus on the sample volume. Most of the volume of a material at high temperature is constituted by free volume, that is, by empty space between its molecules. Now imagine a polymer melt that is cooled down. The cooling will reduce the volume, and according to what we have just said, this is first predominantly constituted by reduction of the free volume between the chain segments. At some point, the free volume is so low that the chain segments cannot move anymore, as they do no longer have enough empty space around to move into. This point is the glass transition, and the temperature at which it happens is the glass-transition temperature. At further cooling, further shrinkage of the sample does then only occur due to thermal contraction of the chain material itself, but no longer of the residual small free volume in between, which then stays constant. The latter conception of the glass transition is visualized in Figure 129, which displays the relation between a polymer compound’s specific volume and temperature. The volume of the chains themselves, Vp, increases constantly with temperature according to the thermal expansion coefficient, α. Above the glass-transition temperature Tg, there is an additional free volume Vf that also shows thermal expansion, and that expansion is actually a lot more pronounced than the underlying one of the chain material itself. Below the glass-transition temperature Tg, however, the free volume Vf,g is only minimal and constant, so that here, thermal expansion is constituted only by the contribution of the chain material itself, Vp. We may express this matter via the fraction of the free volume in the system, f = Vf /V, as follows:

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Figure 129: The temperature-dependent specific volume of a polymer sample is a sum of the inherent volume of the polymer chain segmental material itself, Vp, and the free volume in between the chains and segments, Vf. Vp steadily increases by the usual thermal expansion of the chain material. Vf adds upon that a much greater increase of the total sum volume at high temperatures. By contrast, at low temperatures, below the glass-transition temperature, Tg, the free volume in the glassy state Vf,g stays constant, such that thermal expansion (upon heating) or contraction (upon cooling) in the low-T range is due to expansion/contraction of the polymer chain material itself only. Picture redrawn from B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

  f = fg + T − Tg αf

(7:59)

Here, f is the just introduced free volume fraction, fg is the constant free volume fraction in the glassy state, and αf is the thermal expansion coefficient of the free volume. We can use these relations to again derive the Williams–Landel–Ferry (WLF) equation for time–temperature superposition of rheology data that we have already derived from a slightly different perspective in Section 5.7.3. Here, we start from the Doolittle equation for the sample viscosity:

V (7:60) ln η = ln A + B Vf This expression was found empirically, but it can also be regarded illustratively: V refers to the volume of the macromolecules or particles in a soft matter sample, whereas Vf is that of a corresponding void where these can move if dynamics is not frozen. If this void volume, Vf, is larger than the molecule/particle volume V, then this motion is easy and the viscosity η is low, and vice versa. In this picture, the Doolittle equation is like an Arrhenius or Eyring equation that relates the thermal energy in the system to that required to overcome a certain barrier. Use of the free volume fraction f = Vf/V turns this into ln ηðT Þ = ln A + Bð1=f Þ At the glass-transition temperature, we can write

(7:61a)

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    ln η Tg = ln A + B 1=fg

(7:61b)

Taking the ratio of these two expressions and replacing f according to eq. (7.59) yields ! ηðT Þ 1 1  − (7:62) ln   = B fg η Tg fg + αf T − Tg We can substitute the viscosity η by the relaxation time τ according to τ = η/G to generate    − B=2:303fg T − Tg ηðT Þ τðT Þ τðT Þ    ln   = ln   ≈ log   = log aT =  (7:63) η Tg τ Tg τ Tg fg =αf + T − Tg Here, aT is the shift factor for time–temperature superposition as encountered in Section 5.7.3. Empirical findings show that most polymers exhibit universal values of fg = 0.025 and αf = 4.8 × 10−4·K−1. This means that universally 2.5% of the available volume is free in the glassy state, whereas 97.5% of the volume is occupied by the polymer chain material. The latter finding stands in contrast to a different approach and prediction by Simha and Boyer. In their perception, both the glass and melt are amorphous and should therefore have the same volume at the absolute zero point, meaning their thermal expansion lines should meet there, as shown by the dashed lines in Figure 130. This perception then implies a free volume of 11.3% in the glassy state, as also shown geometrically in Figure 130, which is about the fivefold of the prediction in the WLF treatment. An explanation for this deviation might be that the WLF treatment has a sole focus on volume effects, whereas there is in fact also an energy of activation contribution that is lumped into volume effects in that notion; as a result, the volume confinement in the glassy state is overestimated in the WLF assessment, whereas in fact, the glassy arrest is only in part due to lack of free volume, and in other part due to lack of thermal activation of dynamics. The latter thought is in line with our initial conceptual consideration about the two factors contributing to the glass transition. The first is the temperature dependence of the free volume, Vf, as just discussed. The second is a temperature dependence of the rate of overcoming the energy of activation, Ea, of the chain segmental motion, as discussed further. Both are operative, and as a result, an Arrhenius plot for a glass transition may deviate from linearity, as is illustrated in Figure 131. This figure shows the logarithmic inverse time of occurrence of a glass transition, 1/τg = ωg, and an additional potential secondary transition in plots of G versus t, recorded at many different temperatures. The secondary transition, which is due to potential additional freezing of chain side-group rotations in the glassy state upon further cooling, exhibits a simple Arrhenius relation with a shallow slope due to its single

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Figure 130: Temperature-dependent specific volume of a polymer sample (solid line) and estimates of its composition of a 2.5% free volume, as it results from a WLF-based conception (dotted line), or alternatively, an 11.3% free volume, as it results from the Simha–Boyer expectation that the volumes of the glass and melt should coincide at the absolute zero point (dashed lines), since both these states share the same amorphous structure.

contribution of the energy of activation, Ea. The glass transition deviates from this linearity, especially at low temperatures, where it displays a much steeper slope. This is because in this regime, both the free volume, Vf, and the energy of activation, Ea, contribute. At high temperatures (T ! ∞) both converge, as there is enough free volume then, such that only the energy-of-activation contribution matters. Glass transitions are nonequilibrium kinetic processes. This can be readily visualized by the two examples shown in Figure 132. The two panels show plots of the specific volume as a function of temperature for two different phenomena. Panel (A) visualizes an aging process: here, a sample is cooled down from an initial temperature that is higher than Tg to a temperature below Tg. The glass transition is clearly visible from the bend in the graph. However, even after the cooling has been stopped, the sample further reduces its specific volume – a phenomenon named aging. The occurrence of that aging shows that the sample has not been at equilibrium before, even though the glassy state was already reached. Panel (B) displays the dependence of Tg on the rate of cooling. For different cooling rates, denoted q1 and q2, different glass-transition temperatures are obtained. Again, this ambiguity shows that the glass transition is not an equilibrium transition. Both the phenomena shown in Figure 132 therefore demonstrate that the glass transition has various kinds of time dependences; this proves it to be a kinetic phenomenon and not an equilibrium phase transition.

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Figure 131: Schematic representation of the logarithmic inverse time of occurrence of a glass transition, 1/τg = ωg, and a potential additional secondary transition in plots of G versus t, recorded at many different temperatures, T. The secondary transition exhibits a single contribution of the energy of activation, Ea, whereas the glass transition has a second contribution stemming from the free volume, Vf, that is exacerbated at low temperature (right end of the 1/T axis). At high temperature (left end of the 1/T axis), both graphs converge, as Vf is then large enough.

Figure 132: The kinetic nature of the glass transition is seen by aging processes of glassy samples (A) and by the dependence of the glass transition and the temperature of its occurrence on the cooling/heating rate, q (B). Picture redrawn from J. M. G. Cowie: Chemie und Physik der synthetischen Polymeren, Verlag Vieweg, 1996.

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7.3.1.3 Structure–property relations for Tg The glass-transition temperature is connected to the polymer sample’s structure by a set of structure–property relations. For example, bulky or polar substituents hinder the main chain rotation. These types of polymers are therefore more easily frozen in motion than their more flexible counterparts. Indeed, polyethylene (PE) exhibits a glass-transition temperature as low as Tg(PE) = 188 K, whereas by adding a bulky benzene side group to each monomeric unit, the glass-transition temperature of the resulting polymer polystyrene (PS) is raised by a factor of 2, to Tg (PS) = 373 K. Polypropylene (PP) exhibits a glass-transition temperature in between, at Tg(PP) = 253 K. Substitution of the unpolar methyl groups by polar nitrile groups yields polyacrylonitrile (PAN); the glass-transition temperature is then raised to Tg (PAN) = 378 K. Cross-linking of polymer chains also makes the system more rigid, thereby increasing Tg, whereas branching decreases Tg due to a lowered packing density and a resulting higher free volume as compared to a non-branched sample. Other means of artificially lowering the glass-transition temperature include the use of so-called softeners. These can either be chemical, through the inclusion of flexible comonomers, or structural, by using other architectures like side chains. 7.3.1.4 Experimental assessment of Tg All the preceding discussion shows us how central the glass-transition temperature is. But how can it be assessed experimentally? There are many possible answers. At the beginning of this section, we have assessed the glass-transition regime based on the mechanical spectrum of a polymer sample in Figure 128, wherein which the glass-transition temperature was in fact also marked on the abscissa. If the sample under consideration shows a rubbery elastic regime behind the glass transition, as it is sketched in Figure 128, the parameter tan δ is a proper quantity to capture this numerically: it is low in both the energy-elastic glassy state below Tg and in the entropy-elastic rubbery state above Tg, whereas it is high in the glass-transition range at and around Tg, because in that range, both still-frozen and already-activated main-chain dynamics is present in the sample, such that both G’ and G″ are relevant. The point where their ratio, tan δ, is maximal is therefore reliably seen to be an assessment of the glass-transition temperature Tg.108 Alternatively, a little later in our discussion, we have assessed the glass transition by the temperature-dependent change of the sample (specific) volume, as displayed in Figures 129, 130, and 132, wherein which the glass-transition temperature was also marked on the abscissa at the point where the volume change had a kink or bend. In addition to these two methods, there are several further ways to assess that temperature. A

 This assessment is actually only applicable if we deal with a plain polymer material. In practice, however, materials are often modified (e.g., enforced) by fillers. Such additives drastically change tan δ, thereby potentially rendering it impractical to serve as a marker of the glass transition.

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practical problem, however, is that all these different assessments lead to different values of Tg even for the absolute same sample. This ambiguity shows us again that the glass transition is not a sharp and well-defined phase transition, and so in practice, giving a number for Tg should always come along with telling how exactly it was determined.

7.3.2 Polymer crystallization 7.3.2.1 Fundamentals As an exemption to the general rule that polymers do not crystallize, some polymers do in fact show partial crystallinity before the glass-transition regime. Here, we can observe a true phase transition. However, polymers only crystallize under certain preconditions. First, they have to be composed of strictly linear chains to be able to align themselves in a regular fashion. Side chains, branching, or other complex primary architectures prevent this. Second, the monomer sequence needs to be well defined in view of the monomer-to-monomer connectivity and even in view of its stereochemistry, meaning that crystallization is only possible for isotactic or syndiotactic homopolymers. Third, intermolecular interactions that stabilize the polymer chain against the entropically unfavorable packing in a crystal aid the crystallization process as well. Polymer crystallinity is best characterized by x-ray diffraction. Here, a dry sample is exposed to x-rays that are reflected when they hit a crystalline scattering center. If these centers are arranged symmetrically with a separation d, part of the incoming beam is deflected by an angle of 2θ, producing a reflection spot in the diffraction pattern. A sketch of an x-ray diffraction experiment is shown in Figure 97 in Section 6.1. The Bragg equation quantifies the relation between the lattice constant, d, the scattering angle, θ, and the wavelength, λ: 2d sin Θ = nλ with n = 1; 2; 3; . . .

(7:64)

A typical result for a polymeric sample reveals two types of x-ray refractions: a diffuse halo and discrete reflexes on top. This means that the sample is composed of both amorphous and crystalline domains, a phenomenon that is called partial crystallinity. 7.3.2.2 Partial crystalline microstructures An initial description of a polymer’s partially crystalline microstructure was that of a fringed micelle structure. In this picture, crystalline domains are imagined to be randomly distributed throughout the sample and connected by amorphous areas in between, as shown in Figure 133. It was soon realized, however, that this is often not the case. A noticeable example is native cellulose, whose partial crystallinity

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Figure 133: Initial conceptual picture of a partially crystalline polymer morphology named micellar fringes; this is, however, very often untrue. Picture redrawn from B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

matches the structure of a fringed micelle. Most other crystalline polymer structures, though, are different. How can we imagine polymer crystalline structures to look like then? To answer this question, let us orient ourselves on a related kind of matter: long alkane oligomers. How do these crystallize? They do so in the form of unfolded all-trans chains arranged next to each other in lamellar platelets, whereby the oligomer-chain end-groups sit on the frontal faces. By simple analogy, we may imagine a similar kind of crystalline structures for longer polymers such as PE as well. However, in that case, to fully unfold a polymer to an all-trans conformation in order to arrange it into such a hypothetic lamellar assembly, we need to overcome an extremely high entropy barrier. In addition, irregularities along the chain backbone as well as entanglements with other chains prevent such perfect lamellar arrangements to be formed over the full chain contour. Hence, we may presume that only parts of the chain backbones are arranged in lamellar assemblies, whereas other parts find themselves on the frontal faces, just like the end-groups of alkane oligomers do in their lamellar crystalline phases. In experiments, it is commonly found that the chain-folded lamellae of many kinds of polymers are roughly 10 nm thick. The reason is a free-energy balance, as shown in Figure 134. The straightening of a polymer chain from a random coil in a polymer melt carries an entropy penalty, TΔS, that needs to be overcome. The attachment of the straightened chain segment to the crystalline domain, by contrast, carries an energy gain, ΔH, that is comparable to the lattice energy that low-mass molecules gain when they crystallize. Once the latter energy gain outweighs the first cost, crystallization is possible. If the growing polymer crystal is too thick, however, the entropy penalty is larger than the subsequent energy gain so that the total free-energy

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Figure 134: Conceptual growth mechanism of a polymer crystal. First, a polymer chain segment must be straightened and elongated from a random coil, which carries an entropy penalty of TΔS. In a second step, the straightened chain segment can attach to a crystalline domain, which carries an energy gain, ΔH. The overall free-energy balance is most favorable for lamellae that are 10 nm tall. Picture redrawn from R. A. L. Jones: Soft Condensed Matter, Oxford University Press, 2002.

balance is unfavorable, and crystallization cannot happen. On the other hand, a crystal that is too thin causes a too low energy gain in the second step, thereby again producing a disfavorable free-energy balance. The optimum crystal size that leads to an overall favorable energy balance is around 10 nm. This corresponds to roughly 50 all-trans monomeric units. The chain-folded lamellae constitute the basic building block for further higher order crystal morphologies. The simplest kind of structure are platelets of about 10–20 μm width and, due to the aforementioned free-energy reasons, 10 nm height. These structures are usually formed when crystals are grown from dilute solution, and they may be seen as the closest match to a hypothetic “single crystal” of a polymer. A schematic representation of such a polymer crystal platelet is shown in Figure 135A. In such platelets, the chain lamellae arrange themselves next to each other, which requires backfolding of the polymer chains on the surface. For that, we need about five gauche C–C bonds, creating a small amorphous domain. Depending on the regularity of the 50 all-trans and 5 gauche bond sequences, the platelets have a more or less ordered structure, as shown in Figure 135B. A very regular sequence leads to strong adjacent backfolding, whereas a less regular sequence produces weak adjacent backfolding. If the ratio of 50:5 is missed by a bigger margin, then a chain might not fold back to an adjacent other chain, but a random other one further away, which is referred to as the switchboard model. This arrangement can still lead to crystalline platelets, but the amorphous domain is much bigger in this case. When multiple platelets find themselves stacked on top of each other and separated by amorphous regions, they constitute stapled lamellae, as shown in Figure 136. Such structures are commonly obtained upon crystallization from concentrated

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Figure 135: Chain-folded lamellae can form crystalline platelets. Panel (A) shows a schematic representation of this structure. (B) Backfolding on the platelet surface requires about 5 gauche C– C bonds for every 50 all-trans monomeric units. Depending on the regularity of this alternating sequence, the adjacent backfolding can be either strong, weak, or irregular. Pictures redrawn from B. Tieke: Makromolekulare Chemie, Wiley VCH, 1997.

Figure 136: Further higher order assembly of chain-folded lamellar subunits in the form of stapled lamellae. Picture redrawn from J. M. G. Cowie: Chemie und Physik der synthetischen Polymeren, Verlag Vieweg, 1996.

solution or melt. Their formation can be explained by the solidification model by Erhard W. Fischer and Manfred Stamm, which presumes that crystallization happens by partial alignment of sections of the coils, thereby forming planes of mutually close and regular arrangement bridged and linked by amorphous chain sections in between. The advantage of that solidification mechanism is that it does not rely on far-distant diffusive motion of the material. Alternatively, the chain-folded lamellae can also be arranged in a radial fibrillar fashion. The resulting structure is called a spherulite, whose fibrils are again composed of chain-folded lamellae separated by layers of amorphous domains, as depicted in Figure 137. Such structures are also obtained upon crystallization from concentrated solution or melt, with a growth mechanism as shown in Figure 138. All the above examples show that polymer crystals are just partial, and all these crystalline samples still contain substantial degrees of amorphous parts. The

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Figure 137: Further higher order assembly of chain-folded lamellar subunits in the form of fibrillar spherulites. Picture redrawn from J. M. G. Cowie: Chemie und Physik der synthetischen Polymeren, Verlag Vieweg, 1996.

Figure 138: Schematic of the growth of a fibrillar spherulite.

Figure 139: (A) Fast cooling rates lead to kinetically trapped polymer coils that cannot rearrange anymore. This effect massively disfavors a high degree of crystallinity. (B) Slow cooling rates that allow for sufficient time for the polymer chains to arrange themselves into ordered structures allow for a high degree of crystallinity. Picture redrawn from B. Vollmert: Grundriss der Makromolekularen Chemie, Springer, 1962.

7.3 Glassy and crystalline polymers

335

overall degree of crystallinity within a solid polymer is highly dependent on the rate of crystallization, which experimentally corresponds to the rate of cooling of the sample. We have learned already that also the position of the glass-transition temperature can be influenced this way (see Figure 132). During crystallization, the polymer chains need a lot of time to arrange themselves in lamellae, and even longer to accomplish backfolding. When the rate of cooling is too fast, the glass-transition temperature is undercut before such an arrangement can come to pass. The coil shape is then kinetically trapped in the glassy state with no or just little crystalline domains, as shown in Figure 139A. Below Tg there is not enough energy for even single segmental motion, prohibiting the formation of further ordered structures. By contrast, hypothetic polymer single-crystal lamellae can only be assumed to grow by slowly cooling down to, but not under, the glass-transition temperature. That way, the polymer chains have enough time as well as enough energy to arrange themselves into chain-folded lamellae and/or higher order crystal morphologies, as shown in Figure 139B. In general, the kinetic signature of polymer crystallization is so important that the dominant crystal structure is not the one that has the most favorable thermodynamic signature (i.e., the lowest free energy), but the most favorable kinetic signature (i.e., the highest rate of formation). 7.3.2.3 Experimental assessment of the degree of crystallinity Apart from that conceptual understanding, how can we actually measure the degree of crystallinity experimentally? There are two common ways. The first way is based on the notion that the densities of the tightly packed crystalline and the loosely packed amorphous parts of a partially crystalline polymer are different. We may state that V = Vcryst + Vamorph

and m = mcryst + mamorph

(7:65)

so that Vρ = Vcryst ρcryst + Vamorph ρamorph

(7:66)

This yields ϕcryst =

ρ − ρamorph Vcryst = V ρcryst − ρamorph

(7:67)

and wcryst =

ρcryst ρ − ρamorph mcryst Vcryst ρcryst ρcryst Vcryst ρcryst = = · = · ϕcryst = · m Vρ ρ V ρ ρ ρcryst − ρamorph (7:68)

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The latter pair of formulae allows us to determine the degree of crystallinity by measurement of the sample density and knowledge of the densities in a purely crystalline and a purely amorphous reference. The first one can be calculated by crystallographic means if we know the elementary cell, which we can determine by knowledge of the chemical primary structure of the regular crystalline parts. The second one can be assessed by measurement of the density of a glassy reference sample in which crystallization is suppressed, for example, by fast shockfreezing. A second way of experimental assessment is x-ray scattering. The scattering profile of a partially crystalline polymer sample is composed of both distinct peaks (caused by the crystalline domains) on top of a diffuse halo (caused by the amorphous parts). The degree of crystallinity is then simply determined from the areas underneath these parts of the scattering curve: wcryst: =

Acryst: Acryst + Aamorph

(7:69)

7.3.2.4 Structure–property relations for Tm In addition to the abovementioned rate of cooling, the degree of crystallinity and also the temperature where crystallization happens are also affected by chemical factors, meaning by the type and constitution of the polymer backbone. In general, on the one hand, a backbone with side groups that undergo strong interactions favors crystallization and entails a high Tm, but on the other hand, if that backbone is not flexible enough (which it may be if the side groups interact too strongly), crystallization is hampered, because then the polymer has a hard time to arrange itself into the above-described chain lamellae. Yet on the other hand, if the chain backbone is too flexible, there are many different conformations possible in a melt state, and giving up that liberty upon crystallization comes along with high-entropy cost that hampers crystallization and enforces very low temperatures for it to eventually become possible. Hence, a specific intermediate chain flexibility – not too little but also not too high – is best for good crystallizability. Further factors affecting the ability for polymer crystallization, and with that also the temperature point where that happens, are alike those that have the same effect on the glass transition, as we have already discussed in Section 7.3.1.3. This might be the presence of comonomer units or specifically branching, because branches act like “a solvent chemically connected to the polymer.” As in fact both the glass-transition temperature and the crystallization temperature are equally affected by these (and further) factors, they are actually coupled, as expressed by the empirical Beaman–Bayer rule: Tg ≈ 2/3 Tm.

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337

7.3.3 Comparison of the glass and the crystallization transition To close this chapter, let us have a comparative look on the thermodynamic characteristics of the glass transition and the crystal–melt transition of polymers, as summarized in Figure 140. The uppermost row of schematic plots displays the temperature course of the central state function, Gibbs’ free energy, G, during these two processes. We see that whereas it exhibits a kink during the crystal– melt transition at the melting temperature Tm, thereby denoting it to be a true first-order phase transition, it is steady and smooth at the glass-transition temperature Tg, similar to what second-order phase transitions show. For both, actual discontinuities are seen in the first and/or second derivatives of the free energy, as assessed by the temperature-dependent volume V (⁓ ∂G/∂p) and heat capacity cp (⁓ ∂2G/∂T2) shown in the middle and lowermost rows. The curves that we have discussed most in this chapter (and that are the easiest to be determined experimentally) are the V(T) representations in the middle row, which actually show the difference between the crystal–melt and the glass transition most strikingly. In the crystal–melt transition, the system transitions from an ordered and perfectly packed state at temperatures below Tm to an unordered and less packed one at temperatures above Tm, thereby causing a distinct jump in V. With that, we have a discontinuity in a first derivative of the thermodynamic potential G, as V ⁓ ∂G/∂p, thereby indicating a first-order phase transition to be present. By contrast, during the glass transition, our system transitions from a state with frozen dynamics at temperatures below Tg to one with activated dynamics at temperatures above Tg, which comes along with different thermal expansion coefficients on these two sides, thereby only causing a kink or bend in the course of V(T). In this case, an actual jump is seen only in one of the second derivatives of G, like in the heat capacity cp (⁓ ∂2G/∂T2).109 Although this signature is typical for second-order phase transitions and might therefore indicate such one to be present here, the glass transition does not show any microstructural change along with that, with a fundamental principle of any phase transition; it is therefore not an actual phase transition but a kinetic phenomenon only. For further comparison, Figure 141 summarizes the characteristic change of the sample volume (which may also be assessed by its specific volume) of a purely glassy, a partially crystalline, and a hypothetic purely crystalline polymer upon

 In practice, datasets like those shown in the lowermost row of Figure 140 are probed by differential scanning calorimetry, DSC. Like any technique, of course, that method has practical limitations in resolution, and in a real experiment, we are also bound to practical constraints like the need of conducting an experiment in a finite time, with no possibility to perform an extensively slow heating or cooling ramp that allows for long thermal equilibration of the sample all the way on its course. Hence, the dataset on the left of the lowest row in Figure 140 practically more looks like the Greek letter lambda, whereas the dataset on the right practically looks like a step only.

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Figure 140: Characteristics of the glass and crystallization transitions of polymers. Whereas a crystallization process is a true first-order phase transition, accompanied by corresponding discontinuities in the derivatives of the thermodynamic potential G, such as the volume, V (⁓ ∂G/∂p) and the specific heat capacity, cp (⁓ ∂2G/∂T2), the glass transition is only a “pseudo”-second-order phase transition. It exhibits some characteristics that also second-order phase transitions have, but is actually a purely kinetic phenomenon.

heating. The glassy polymer shows the transition from a glassy to a melt state that we have discussed in Section 7.3.1.2. The hypothetic purely crystalline polymer would show a sharp first-order melting-phase transition as also sketched in the middle left schematic of Figure 140; it would also show no glass transition below the melting point, as its hypothetic perfect crystalline order in that domain would not leave any amorphous material that could undergo a glass transition. An actual sample, which has both crystalline and amorphous parts, first shows a glass-tomelt transition of its amorphous parts followed by melting of its crystalline parts.

7.3 Glassy and crystalline polymers

339

Figure 141: Temperature-dependent (specific) volume of a fully amorphous polymer that only shows a glassy state in the cold (blue curve), an hypothetic fully regular polymer that shows a perfect crystalline state in the cold (green curve), and an actual polymer that shows both partial crystallization upon cooling, followed by further vitrification of its remaining amorphous parts upon further cooling (black curve).

This melting is not sharp but spans a range, as the crystalline domains in the sample are diverse in size and morphology, each showing an own melting temperature, such that the overlay smears to a broad melting range. In the range between Tg and Tm, the sample exhibits particularly interesting and favorable properties, as it has both solid domains (these are the still unmolten crystalline domains) and liquid domains (these are the already dynamically activated amorphous domains). As a result, the sample specimen is form-stable like a solid but also dissipative like a fluid, thereby exhibiting good impact resistance.

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Questions to Lesson Unit 23 (1) What is the difference between the glassy and crystalline states? a. A sample in a glassy state is always transparent, whereas in a crystalline state it is opaque. b. A glassy state consists of amorphous structures, whereas a crystalline state is ordered. c. Glassy samples are often more brittle, whereas crystalline samples have better mechanical durability. d. A glassy sample has a lower melting point than a crystalline sample. (2) How do polymers usually solidify? a. They usually solidify in a crystalline state because it is the energetically most favorable structure. b. They usually solidify in a crystalline state because it allows the closest packing of chain segments. c. They usually solidify in a glassy state because the disordered chains “freeze” when lowering the temperature, which is called vitrification. d. They usually solidify in a glassy state because it is the entropically most favorable structure. (3) What holds for the glass transition of a sample with entanglements and crosslinks? a. The glass transition is the transition from a glassy state with elastic properties to a viscous melt. b. The glass transition is the transition from a glassy state with energy-elastic properties to an entropic-elastic rubber state. c. The glass transition is the transition from a less-ordered glassy to a higher ordered crystalline state. d. The glass transition does not take place in entangled polymers. (4) Which statement regarding the glass-transition temperature is correct? The glass-transition temperature ______ a. is actually a temperature range in most cases. b. marks the point of transition from an amorphous glassy state to an ordered crystalline one. c. is only noticeable in spectra when a slow cooling process is provided during measurement. d. only exists for highly cross-linked polymer samples.

Questions to Lesson Unit 23

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(5) Is the glass transition a phase transition? a. Yes, it is a phase transition of first order. b. Yes, it is a phase transition of second order. c. No, but it has similar characteristics to a second-order phase transition. d. None of the above statements is applicable. (6) What is not an important parameter when talking about the concept of glass transition? a. The different contributions to the sample’s volume. b. The relaxation dynamics of chains and their segments. c. The activation energies for chain segment’s displacements. d. The entanglement molar mass. (7) How does the free volume change with increasing temperature? a. For T < Tg the free volume is constant, and for T > Tg it increases linearly. b. For T < Tg the free volume is constant, and for T > Tg it increases exponentially. c. For T < Tg the free volume decreases exponentially, and for T > Tg it is constant. d. For T < Tg the free volume decreases linearly, and for T > Tg it is constant. (8) What shows that the glass transition is a kinetic process far away from equilibrium? a. Even beyond the glass-transition temperature, the specific volume of the sample further decreases with time even without further cooling; this is called aging. b. The glass-transition temperature is independent of the rate of cooling and just depends on the type of polymer. c. As soon as the glass-transition temperature is reached, the sample remains in the glassy state unless the temperature is changed. d. The vitrification actually starts at temperatures higher than the glass-transition temperature and is completed when the glass-transition temperature is reached.

8 Closing remarks The target of this book is to deliver an understanding of the relations between structure and properties of polymers, thereby bridging the fields of polymer chemistry, which focuses on the primary molecular-scale structure of individual polymer chains, and polymer engineering, which focuses on the macroscopic properties of polymer materials. With that arc, molecular parameters are rationally connected to macroscopic function. As a fundament for this bridge, the first chapters of this book have considered the shape of a single ideal polymer coil using models such as the phantom chain model and the Kuhn model. From that, we have recognized that the statistics of the ideal chain are those of a random walk, and we have used this notion to quantify the free energy and the mechanical characteristics upon deformation of such ideal chains. We have then moved further to incorporate chain–chain and chain–solvent interactions into our model, thereby focusing on real polymer chains; this enabled us to categorize solvents based on their ability to dissolve and swell a polymer. We have realized that when all interactions exactly balance each other, the real chain acts like an ideal chain, and we have called this quasi-ideal state the Θ-state. In a next step, we have developed a thermodynamic theory for polymer solutions, the Flory–Huggins theory, based on a mean-field approach, and with it we were able to construct a full phase diagram of a polymer solution or a polymer blend. Furthermore, we have learned about two conceptual approaches to model the dynamics and motion of polymer chains: the Rouse and the Zimm model. While doing all of the above, we have often encountered striking similarities of the concepts that we have employed to those known from elementary physical chemistry. These similarities appeared both on the conceptual and on the mathematical side of our argumentations. Conceptually, for example, the description of an ideal polymer chain and an ideal gas was based on the same assumptions. The same resemblance was also retained for real polymer chains and real gases, which are both subject to mutual and environmental interactions. As a consequence, both materials display quasi-ideal states at a very specific temperature, the Θ-temperature for polymer chains and the Boyle temperature for gases. Mathematically, we have seen similarities between the end-to-end distance distribution within a Gaussian polymer coil, the Boltzmann and the Maxwell–Boltzmann distributions of kinetics of gas particles, and the electron density distribution in an s-orbital. We have also seen how randomwalk statistics that describes the diffusive motion of a particle or molecule also applies to the statistical treatment of the shape of ideal polymer chains. Methodically, we have touched upon two fundamental approaches that are used in polymer physical chemistry. First, we have learned how a multibodied complex system can be simplified by only considering average values. This is the mean-field approach. Second, we are now familiar with the concept of scaling laws. Realizing https://doi.org/10.1515/9783110713268-008

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that polymers are self-similar objects, we can rescale our conception of them to different length-scales without having to change the mathematical description of them, as it is done in the blob concept. When we reconsider the conceptualized goal of this book, to build a bridge between polymer chemistry and polymer engineering, we may conclude that the first four chapters of this book have laid its pillars. Based upon that, the actual construction of the bridge has been outlined in the fifth chapter of this book. In this central chapter, we could rationally and quantitatively understand why many of the polymeric materials that we encounter in our everyday lives behave the way they do. Our prime focus in this consideration was on those properties of polymers that have made greatest impact on our lives in the past decades, which is their mechanical properties, spanning from viscous flow to elastic snap, along with their time and temperature dependence. To complete our picture, this book has then added a sixth chapter on scattering methods as a prime experimental platform to characterize nano- and microstructures of polymer systems. A final seventh chapter eventually showed us how the conceptual grounds of the preceding ones apply to some relevant states of appearance of polymer samples and systems. With that, we have closed the circle of our endeavor of deriving relations between the structure of a polymer system and its most relevant properties.

Index active microrheology 181 adaptivity 26 anionic polymerization 13 atactic 8 athermal solvent 77 attractive interactions 72 autocorrelation 264 bad solvent 78 Bingham fluid 176 binodal line 137 biopolymers 3 blob concept 62 blob size 62 block copolymer 124 Boltzmann distribution 43 Boltzmann superposition principle 199 bonding angle 32 Bragg diffraction 244 Bragg equation 245 caoutchouc 23 Casson fluid 176 cellulose 23 centrifugal-average molar mass 9 chain-growth 11 chain-transfer agent 13 characteristic ratio 40 cohesive energy 21, 126 coil expansion 83 colligative properties 10 colloidal 20 constraint release 239 creep 20 creep compliance 169 creep test 168 De Broglie wavelentgh 243 deformation blob 90 degree of swelling 227 demixing 131 diffusion coefficient 97 dynamic experiment 169 dynamic light scattering 264

https://doi.org/10.1515/9783110713268-009

effective interactions 74 Einstein–Smoluchowski equation 96 elastic modulus 160 elastic scattering 257 elastic solid 159 endothermic mixing 119 end-to-end distance 35 energy of mixing 20 entanglement 20 entanglement molar mass 226 entanglement time 210 entropic spring constant 59 entropy elasticity 59 entropy of mixing 20 excluded volume 71 exothermic mixing 119 Flory exponent 65 Flory theory 85 flow 20 flow curve 175 form factor 247 fractal dimension 65 free volume 201 freedom 60 freely-jointed chain 37 free-radical polymerization 11 Gaussian chain 57 glass transition temperature 1 good solvent 77 Guinier function 251 Guinier region 250 hard-sphere repulsion 75 Hooke’s law 160 hydrodynamic interactions 100, 283 hydrodynamic radius 97 ideal chain 31 ideal mixture 119 intensity autocorrelation function 264 interaction parameter 119

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Index

irregularity 7 isotactic 7 Kelvin–Voigt model 191 Kuhn length 42 lattice model 113 Lennard–Jones potential 72 living polymerization 13 loss modulus 171 loss tangent 171 lower critical solution temperature 123 macroconformation 32 macromolecule 3 Mark–Houwink–Sakurada equation 287 Maxwell model 184 Mayer f-function 75 mean-field theory 112 mean-square 38 mean-square displacement 96 metastable 135 microconformation 32 microphase separation 124 monomer 3 Newtonian flow 174 Newtonian fluid 20 Newton’s law 160 nonsolvent 78 nucleation and growth 136 number-average molar mass 9 osmotic pressure 10, 148 passive microrheology 181 persistence length 46 phantom chain 37 phase diagram 131 Poisson distribution 13 Poisson effect 162 Poisson’s ratio 162 poly(methylmethacrylate) 40 poly(sodium acrylate) 40 polyaddition 11 polyamide 5 polycondensation 6 polydispersity 9

polydispersity index 11 polyethylene 5 polymer blend 114 polymer solution 114 polymerization 5 polypropylene 32 polystyrene 32 polyvinylchloride 32 Porod’s law 252 power law 1 precipitation 19, 78 primitive path 234 properties 1 quasi-living polymerization 13 radius of gyration 35 Raman scattering 257 random chain 37 random coil 32 random walk 38 rational design 1 Rayleigh ratio 247 Rayleigh scattering 257 real chain 71 reference temperature 203 regular solution 114 relaxation 20 relaxation experiment 168 relaxation modes 102 relaxation modulus 168 relaxation time 21, 185 relaxation time spectrum 100 reptation 211, 234 reptation time 211 repulsive interactions 72 responsiveness 26 retardation time 192 rheology 159 root-mean-square 38 rotational isomeric states 43 Rouse model 98 Rouse time 99 rubbery plateau 218 rubbery-elastic plateau 210 scaling discussion 64 scaling law 1

Index

scattering intensity 243 scattering vector 245 Schulz–Flory distribution 12 Schulz–Zimm distribution 11 screening 106 second virial coefficient 150 segmental lentgh 32 self-assembly 26 self-avoiding walk 71 self-similar 64 sensitivity 26 shear modulus 162 shear rheology 180 shear-thickening 176 shear-thinning 175 shift factor 202 soft condensed matter 1 soft matter 20 specificity 1 spinodal decomposition 136 spinodal line 137 static light scattering 10, 255 Staudinger index 284 step-growth 11 Stokes–Einstein equation 97 storage modulus 171 strain 160 strain-induced hardening 226 stress 160 stress relaxation 168 structural viscosity 175 structure 1 structure factor 247 structure–property relation 34, 45, 48, 58 subdiffusion 104 superdiffusion 104

swelling 227 syndiotactic 7 tacticity 7 theta-solvent 77 theta-state 76 time–temperature superposition 202 torsion angle 32 tube concept 234 ultracentrifugal analysis 11 universality 1 uphill diffusion 143 upper critical solution temperature 140 Van’t Hoff’s law 149 vinyl 7 virial series expansion 150 viscoelastic liquid 172 viscoelastic solid 172 viscoelasticity 21 viscosity 161 viscous liquid 159 weight-average molar mass 9 Williams–Landel–Ferry equation 203 yield stress 176 Young’s modulus 160 zero shear viscosity 145 Zimm model 98 Zimm plot 251 Zimm time 101

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