Mesoporous Ordered Silica Films: From Self-Assembly to Order 3030895351, 9783030895358

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Advances in Sol-Gel Derived Materials and Technologies Series Editors: Michel A. Aegerter · Michel Prassas

Plinio Innocenzi

Mesoporous Ordered Silica Films From Self-Assembly to Order

Advances in Sol-Gel Derived Materials and Technologies Series Editors Michel A. Aegerter, Résidence Vert Pré, Bottens, Switzerland Michel Prassas, Avon, France

Sol-gel chemistry has been practiced for more than a century. However, the rapid growth of the field in the past 30 years has been due to the integration of sol-gel techniques into other areas of materials science such as polymers, soft matter, nanostructured materials, and biomaterials. The resulting multidisciplinary approach has allowed the emergence of highly complex materials and new functionalities. Today, sol-gel chemistry is used in a wide range of industrial sectors and applications from catalysis, sensors, and optics to biomedicine and energy to cite a few. The book series “Advances in Sol-Gel Derived Materials and Technologies,” which is published in collaboration with the International Sol-Gel Society, is devoted to fostering an integrative approach to materials and technologies that are prepared and applied using the latest developments in sol-gel methods. Each book is written by internationally recognized experts, and presents a state-of-the-art overview of the topic. The series will serve as an authoritative source for a broad audience of individuals involved in research and product development, and will be of value to advanced undergraduate and graduate students in materials science and engineering and solid-state chemistry.

More information about this series at https://link.springer.com/bookseries/8776

Plinio Innocenzi

Mesoporous Ordered Silica Films From Self-Assembly to Order

Plinio Innocenzi Department of Biomedical Sciences University of Sassari Sassari, Italy

ISSN 2364-0030 ISSN 2364-0049 (electronic) Advances in Sol-Gel Derived Materials and Technologies ISBN 978-3-030-89535-8 ISBN 978-3-030-89536-5 (eBook) https://doi.org/10.1007/978-3-030-89536-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The possibility of synthesizing materials with controlled porosity with pores in the mesoscale (5–10 nm) has aroused enormous interest, one due to the possible technological applications, the other from the aspects of basic science connected to the self-organization of materials at the nanoscale. The realization of materials with interconnected and organized porosity able to overcome the dimensional limit of the zeolites represents a great scientific and technological breakthrough. The need to have materials with a high surface area and controlled porosity available was the driving force that led researchers in the field of catalysis to synthesize mesoporous materials. However, it was clear from the beginning that such a family of materials opened new perspectives in many different areas, in biological applications such as drug delivery, optics and photonics, in sensors and electronics. One of the fascinating aspects of mesoporous materials is undoubtedly the ability to self-assembly. The self-organization process is typically bottom-up and exploits the intrinsic properties of supramolecular structures such as surfactants to create ordered and complex structures thanks to the weak forces operating in the colloidal scale. Block copolymers are an extraordinary example of this capacity for supramolecular organization and the formation of ordered mesostructures. The self-assembly process underlying the synthesis of mesoporous materials is flexible and can be used to obtain particles, powders and thin films. This last possibility has, in turn, opened up new application opportunities in high-tech sectors such as photonics and advanced sensors. Self-assembly in thin films is, however, much more difficult to control because the chemical–physical processes leading to the formation of ordered mesostructures are much faster. This process has been referred to as evaporation-induced self-assembly even if evaporation only produces the conditions so that the weak interactions can guide the organization. The study of the process of forming mesoporous films via self-assembly has known a great impulse both for the interest in technological applications and because it is a complex process whose understanding has undoubtedly allowed acquiring much more advanced knowledge of the self-assembly processes in the nanoscale.

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The research in the field of mesoporous films has focused on various objectives pursued by the different research groups; understanding the process of self-organization, control of the mesophases obtainable, the orientation of twodimensional mesostructures, synthesis of mesostructures with different types of oxides and finally hybrid mesoporous films in which the pore walls are organic– inorganic. The second phase of the research has been, instead, focused on the development and exploration of new applications and is still ongoing. The study of the self-assembly process of mesoporous materials needs two requirements, in-depth knowledge of sol–gel chemistry and that of surfactants, particularly of block copolymers. The formation of a mesostructure occurs thanks to a supramolecular template formed by a surfactant, but in the formation of a thin film, the process can be much more complex. The inorganic units (building blocks) can interact with the surfactant molecules cooperating to form an ordered structure. The sol–gel chemistry that governs the hydrolysis and condensation processes of inorganic precursors requires careful design of the reactions because small changes in the chemistry can produce structures with a different order or prevent self-assembly. The sol–gel reactions also strongly depend on the choice of the precursor; the final composition is the result of accurate control of the kinetics with which the reactions take place. Mesoporous films of silica and titania are undoubtedly the most studied, and self-organization processes have been well understood and explained. The sol– gel chemistry of silica, although the most studied, is particularly complex since different species can form in solution, thanks to the tetrahedral structure that favours the formation of a ring or cage-like structure with an order on the nanoscale. This book is devoted to describing the self-assembly process in the case of silica film. The choice to limit to this specific oxide has several reasons; one is that it is undoubtedly the oxide on which more studies have been devoted. The plenty of experimental data allow us to have a general and a rather precise picture of the process. It is also the mesoporous material for which more applications have been developed. Furthermore, focusing attention on a specific case, which in any case has a general value, makes it possible to describe the self-assembly process more effectively. Chapter 1 presents the introduction which is dedicated to a brief critical discussion of the concept of self-assembly, which is a ubiquitous phenomenon and on which there is still an open discussion to define it. Chapter 2 instead describes, in general, how the self-assembly of mesostructured materials takes place using supramolecular templates. It is a general description with a general value and where the formation of micelles and the phase diagrams of the surfactants are discussed with particular attention. Chapter 3 focuses on the sol–gel chemistry of silicon, of which a brief description is provided to understand the fundamental notions for the design of inorganic building blocks. Chapter 4 describes in detail the self-assembly in silica films through the so-called evaporation-induced self-assembly process. Chapter 5 shows the details of the evaporation process, broken down into different stages and how the system organizes itself according to evaporation. Instead, the last Chap. 6 is dedicated to the analysis of the order that can be obtained and how to control the alignment direction

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of the mesophase, which represents one of the most significant challenges for the development of applications. The book is focused on the description of the self-assembly process of silica films and does not have the ambition to represent an exhaustive discussion of mesoporous materials necessarily. Therefore, I had to make a very selective choice of topics and articles to mention. The time that has elapsed since the first articles published on the subject now allows a perspective and critical view, which I could take advantage of to write this book. A special thank goes to the colleagues and friends from all over the world with whom we shared the enthusiasm and excitement of discovering this fascinating aspect of materials science and nanotechnology. Sassari, Italy

Plinio Innocenzi

Contents

1 Mesoporous Materials and Self-assembly . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 5

2 The Chemical-Physical Processes Behind Self-assembly . . . . . . . . . . . . 2.1 The Thermodynamics of Self-assembly . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Formation Pathway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Cooperative Self-assembly Pathway . . . . . . . . . . . . . . . . . . . . 2.2.2 Liquid Crystal Templating Pathway . . . . . . . . . . . . . . . . . . . . . 2.2.3 True Liquid Crystal Templating . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Controlling the Kinetics in Silica Systems . . . . . . . . . . . . . . . 2.4 Interaction at the Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Surfactants and Micelle Templating Agents . . . . . . . . . . . . . . . . . . . . 2.5.1 Low Molecular Weight Charged Surfactants . . . . . . . . . . . . . 2.5.2 Mesostructures Templated by Amphiphilic Block Copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Self-assembly of Block Copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Self-assembled Micelles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 The Packing Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Effect of Solvent and Cosolvent . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 9 10 11 12 13 14 16 17 19 21 24 29 35 38

3 The Sol–Gel Chemistry of Silicon for Self-assembly . . . . . . . . . . . . . . . . 3.1 Sol–Gel Chemistry for Self-assembly . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Silica Hydrolysis and Condensation . . . . . . . . . . . . . . . . . . . . 3.1.2 Effect of Aging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Sol-Aging and Cyclization of Silica . . . . . . . . . . . . . . . . . . . . . 3.2 Interfacial Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 42 46 47 52 54

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4 Evaporation Induced Self-assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A General Overview of Evaporation Induced Self-assembly . . . . . . 4.2 Tunable Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 What Type of Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 2D Mesostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 3D Mesostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Tricontinuous 3D Architectures . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Mesophase Transition During Processing . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 59 63 64 64 65 68 74

5 Peering into Evaporation During Self-assembly . . . . . . . . . . . . . . . . . . . 5.1 Evaporation of Water and Ethanol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sol–Gel Reactions in a Silica Film During Evaporation . . . . . . . . . . 5.3 Micellization During EISA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Evaporation Kinetics During Self-assembly . . . . . . . . . . . . . . . . . . . . 5.5 Simultaneous Time-Resolved In-Situ Analysis . . . . . . . . . . . . . . . . . . 5.6 Evaporation and Processing Conditions . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 78 81 85 85 88 91 93

6 From Defects to Controlled Alignments . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Point Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Linear Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Disclinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Planar Defects, “Polycrystalline-Monocrystalline” Mesostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Orienting the Mesostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 99 99 102 103 107 111

Chapter 1

Mesoporous Materials and Self-assembly

Abstract Mesoporous materials are defined in general terms by their pore dimension which falls in the 2–10 nm range (Beck and Vartuli in Curr. Opin. Solid State Mater. Sci. 1:76–87, 1996 [1]; Beck et al. in J. Am. Chem. Soc. 114:10834–10843, 1992 [2]). This porosity is coupled with a very high surface area that could be as large as 1000 m2 g−1 .

IUPAC (International Union of Pure and Applied Chemistry) classifies the mesopores as “pores of intermediate size”. The pores whose widths do not exceed about 2 nm size are micropores, whilst those larger than mesopores, about 50 nm, are classified as macropores [3, 4]. This means that the mesoporosity is in the range of 2–50 nm (Table 1.1). Zeolites fall within the microporous materials. The synthesis of mesoporous materials marked a significant turning point as it allowed breaking the barrier represented by the porosity of zeolites, which is generally limited to around 1.5 nm. The possibility of having materials with a high surface area and porosity in the meso dimension has paved the way for developing a real new class of materials. Porosity can also be described in general terms by being open or closed and possibly hierarchical, in which case pores of different dimensional scales can be found in the same material [5]. Another important property that defines the porosity is the presence of a possible order in the porous structure. This property is typical of mesoporous materials obtained through self-assembly using supramolecular templates. Through self-organization materials both in bulk and in the form of powders or thin films can be fabricated. The porosity on a mesoscale shows an ordered structure like conventional crystals. The shape of the pores is not limited to the high symmetry of a sphere but can assume other forms such as cylindrical channels, which can also be ordered into organized structures, while it is possible and indeed common to have domains with different ordered porous structures, including unorganized one. The use of the terms mesoporous and mesostructured in the scientific literature it is not always clear, and the two words are often employed without making a clear distinction. In general, mesoporous is used to identify the class of porosity, which should be within 2 and 50 nm. Mesostructured materials are, instead, characterized not only by their porosity but to have an ordered porosity in the mesoscale. Different © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Innocenzi, Mesoporous Ordered Silica Films, Advances in Sol-Gel Derived Materials and Technologies, https://doi.org/10.1007/978-3-030-89536-5_1

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Table 1.1 Porous domains defined by IUPAC

Porous domain

Pore size interval

Example

Microporous

50 nm

Ceramics

MESOPOROUS ORDERED MATERIALS POROSITY SCALE

ORDER AT NANOSCALE

Microporous: pore diameter < 2 nm

Organization of the porosity

Mesoporous: pore diameter 2 50 nm Self-assembly

Porous mesostructure

Macroporous: pore diameter > 50 nm

High surface area Possibility of surface functionalization Pore organization

(size, shape, orientation with respect to the subtrate)

Fig. 1.1 Mesoporous ordered materials are defined in terms of pore scale, pore topology and the process of self-assembly

types of porous ordered structures can be obtained whose topology depends on the ultimate templating organization. Another term that is also used by some authors is Periodically Ordered Mesoporous Materials. An ordered mesoporous material is, however, identified not only in terms of structure but also by the peculiar way it is obtained, the processing. Two properties, mesoporosity and ordered porosity, and one process, the self-assembly. The mesoporous materials are, therefore, identified in terms of pore scale (2–50 nm), pore topology and the process of self-assembly (Fig. 1.1). It is important to define also what is intended for self-assembly [6, 7], a general definition is: Spontaneous organization of materials through non-covalent interactions and without external intervention. The forces that lead to self-assembly are weak, such as hydrogen bonding, medium-long range electrostatic forces and van der Waals forces. This is very important to keep in mind. Strong covalent bonding is the glue that connects the inorganic

1 Mesoporous Materials and Self-assembly

3

or hybrid organic–inorganic backbone but does not intervene during self-assembly, which is governed by weak interactions. The spontaneous organization into mesoporous ordered materials is also defined as a process of co-organization of organic molecules (generally amphiphilic surfactants or block copolymers) and inorganic building blocks. This is not the only possible route because, as we will see in the next chapter, self-assembly could also be driven by the formation of a liquid crystal template and the condensation of building blocks at the hybrid interface. The three essential elements of the recipe for self-assembly mesoporous ordered materials are the surfactant (amphiphilic molecule), solvent and inorganic precursors. The interaction between these different components governs the self-assembly. Self-assembly and self-organization are generally used without making a clear distinction, but it is better to make some observations that can help to focus the idea of self-assembly [8]. Self-organization The concept of self-organization is linked to the idea that a dynamic system can evolve towards spontaneous order [9]. In general, self-organization requires an external source of energy and, therefore, the process has a dissipative nature. This implies that once the energy source that induced the process ceases, the self-organized order decays. It is very important to emphasize that self-organization must satisfy the conditions of non-equilibrium because this is precisely what distinguishes it from the concept of self-assembly. Self-assembly To define self-assembly, we can start from an initial consideration that in most chemical processes, the interactions or reactions give rise to a product that is not the result of the simple assembly of single components but is a distinct new entity. The selfassembly process, instead, requires that the individual components remain substantially unchanged throughout the process. This property distinguishes the process itself. Another important issue is that self-assembly occurs through secondary interactions, hydrogen bonds, ionic or electrostatic interactions, thus excluding covalent bonds. These interactions, in general, are weaker than covalent bonds, but above all, they are characterized by being reversible and the final structure is in thermodynamic equilibrium with its components. For the system to evolve towards order during self-assembly, it is crucial that the initial differences between the individual components already contain the instructions that would regulate the dynamic interactions to develop to order. The interactions occurring during any self-assembly process reflect, in fact, the specific properties of the parts interacting with each other in a directional and specific way. This occurs in contrast to self-organized systems in which the initial encoding of the single components is not required. The preconditions that a system must satisfies to evolve to order can be summarized as follows:

4

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2. 3. 4.

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The set of individual components must contain coded instructions that determine the interactions so that the information makes the system evolve in a specific and directional way. Examples of encoded information are surface properties, charge, mass, shape, dimension, polarizability, etc. The interaction between the individual components and the partially assembled substructures must be thermodynamically favourable. The final structure must be in thermodynamic equilibrium. The system must be reversible to explore spatial interactions that potentially lead to more favourable geometric configurations.

It is important to emphasize that the self-assembly process is spontaneous since the energy of the individual unassembled components is greater than the assembled structure, which is in static equilibrium. Thus, the order persists without the need to supply energy to the system. The thermodynamic stability characterizes the self-assembled materials. For selfassembly to occur without the intervention of external forces, the process must minimize the Gibbs free energy. Thus, this implies that self-assembled structures are thermodynamically more stable than the single unassembled components. The self-assembly produces a system with properties, such as order, that did not exist before the organization. The precursor solution is isotropic, while the final material obtained via self-assembly is the product of a spontaneous break in the symmetry [10]. The symmetry breaking produces new properties, such as organized porosity. Self-assembly is a ubiquitous process in nature, and several structures form via natural organization [11], such as liquid crystals [12], viral capsids [13], proteins [14], to cite some (Fig. 1.2). Sensitivity to perturbations Self-assembled systems, as we have seen, are characterized by weak interactions and thermodynamic stability. These characteristics must be remembered to explain another property often observed in self-organized systems: the sensitivity to perturbations that changes in the external environment can induce. Small fluctuations can lead to change in the thermodynamic variables and induce even very significant variations in the structure. These fluctuations can be so wide that they could be even able to compromise the self-assembly process itself. The very nature of weak interactions accounts for the flexibility of the structure and the possibility of its rearrangement in directions determined by thermodynamics. The evolution of the system towards geometrically more favourable configurations can also occur through stochastic thermal processes. If the fluctuations bring the thermodynamic variables back to their initial conditions, the structure itself could evolve to return to its initial configuration. This is a characteristic property of selfassembly, which is precisely reversibility. As we will see in the following chapters, especially in self-assembled mesostructured films, there is a particular state in which an as-deposited templated film can structurally rearrange by increasing or decreasing the degree of order or even changing the symmetry of the mesostructure. As we will

1 Mesoporous Materials and Self-assembly

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Fig. 1.2 Stylized view of the lipid-like and protein-like self-assembly [15]. a Lipid molecules form lamellar, tubular, and vesicular structures, the flexibility and fluidity of which are emphasized in the illustration. b Proteins form lamellar, helical tubular and regular icosahedral structures with rigidity and crystallinity (hexagonal lattice in this case). c SDS@2β-CD assembles, in a protein-mimetic way, into lamellar, helical tubular and rhombic dodecahedral structures with inherent rigidity and in-plane, rhombic crystalline nature. In the molecular view, SDS is a anionic surfactant with a hydrocarbon tail (yellow) and a –(SO4 )− headgroup (blue and red), while β-CD is a ring of heptasaccharides (green C and red O atoms). Reproduced under the CCBY 4.0 license. https://commons. wikimedia.org/wiki/File:Lipid-like_and_protein-like_self-assembly.jpg

see, the self-assembly process is strongly dependent on the processing parameters, which allows a pretty precise design of the structure and properties.

References 1. J.S. Beck, J.C. Vartuli, Recent advances in the synthesis, characterization, and applications of mesoporous molecular sieves. Curr. Opin. Solid State Mater. Sci. 1, 76–87 (1996) 2. J.S. Beck, J.C. Vartuli, W.J. Roth, M.E. Leonowicz, C.T. Kresge, K.D. Schmitt, C.T.W. Chu, D.H. Olson, E.W. Sheppard, S.B. McCullen et al., A new family of mesoporous molecular sieves prepared with liquid crystal templates. J. Am. Chem. Soc. 114, 10834–10843 (1992)

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3. IUPAC, Compendium of Chemical Terminology, 2nd edn. (the “Gold Book”), Compiled by A.D. McNaught, A. Wilkinson (Blackwell Scientific Publications, Oxford, 1997). Online version (2019) created by SJ Chalk. ISBN 0-9678550-9-8. https://doi.org/10.1351/goldbook 4. J. Rouquerol, D. Avnir, C.W. Fairbridge, D.H. Everett, J.M. Haynes, N. Pernicone, J.D.F. Ramsay, K.S.W. Sing, K.K. Unger, Recommendations for the characterization of porous solids (technical report). Pure Appl. Chem. 66, 1739–1758 (1994) 5. P. Innocenzi, L. Malfatti, G.J.A.A. Soler-Illia, Hierarchical mesoporous films: from selfassembly to porosity with different length scales. Chem. Mater. 23, 2501–2509 (2011) 6. G.M. Whitesides, B. Grzybowski, Self-assembly at all scales. Science 295, 2418–2421 (2002) 7. G.M. Whitesides, M. Boncheva, Beyond molecules: self-assembly of mesoscopic and macroscopic components. Proc. Natl. Acad. Sci. U.S.A. 99, 4769–4774 (2002) 8. J.D. Halley, D.A. Winkler, Consistent concepts of self-organization and self-assembly. Complexity 14, 10–17 (2008) 9. P.W. Anderson, D.L. Stein, Self-organizing Systems, ed. by I. Yates, F. Eugene (Plenum, New York, 1987) 10. S. Förster, T. Plantenberg, From self-organizing polymers to nanohybrid and biomaterials. Angew. Chem. Int. Ed. 41, 688–714 (2002) 11. A. Klug, From macromolecules to biological assemblies (Nobel lecture). Angew. Chem. Int. Ed. Engl. 22, 565–636 (1983) 12. L.M. Blinov, Structure and Properties of Liquid Crystals (Springer Science & Business Media, Dordrecht Heidelberg London New York, 2011) 13. A. Valbuena, S. Maity, M.G. Mateu, W.H. Roos, Visualization of single molecules building a viral capsid protein lattice through stochastic pathways. ACS Nano 14, 8724–8734 (2020) 14. J.J. McManus, P. Charbonneau, E. Zaccarelli, N. Asherie, The physics of protein self-assembly. Curr. Opin. Colloid Interface Sci. 22, 73–79 (2016) 15. S. Yang, Y. Yan, J. Huang, A.V. Petukhov, L.M.J. Kroon-Batenburg, M. Drechsler, C. Zhou, M. Tu, S. Granick, L. Jiang, Giant capsids from lattice self-assembly of cyclodextrin complexes. Nat. Commun. 8, 15856 (2017)

Chapter 2

The Chemical-Physical Processes Behind Self-assembly

Abstract This first chapter introduces the models and theories necessary for a basic understanding of mesoporous materials self-assembly. The self-assembly process that drives the formation of an organized mesoporous structure depends on several chemical-physical processes that must be properly designed and governed. In particular, the interaction between the supramolecular template and the inorganic silica building blocks is a critical parameter, such as the organization of the amphiphilic surfactant into micelles. Different organization pathways drive self-assembly, but the thermodynamic and the kinetics of the processes involved should follow specific relationships to achieve order.

2.1 The Thermodynamics of Self-assembly The formation of an ordered mesostructure through self-assembly is governed by the simultaneous occurrence of different chemical-physical phenomena which address the system towards order or disorder [1]. The thermodynamics of the entire self-assembly process depends on the free energy variation, Gmes , associated with the organization of the individual components, which contribute to the formation of an ordered structure of mesoscopic scale (2–50 nm). If Gmes < 0 the system evolves to self-organization [2]. The configuration of the ordered mesophase would represent a minimum of energy. The total free energy of self-assembly, in absence of external fields, is given by the contribution of four processes: Gmes = Gorg + Ginorg + Ginterf + Gsolv (1)

(2)

(2.1)

Gorg . The formation of micelles, that is, the supramolecular templates whose organization would determine the structure and topology of the pores, is associated with the free energy, Gorg . Organic micelles are formed through non-covalent bonds and will depend on the amphiphilic nature of the molecules. Ginorg . The formation of the supporting structure of the mesoporous material occurs through covalent bonds and the free energy associated with the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Innocenzi, Mesoporous Ordered Silica Films, Advances in Sol-Gel Derived Materials and Technologies, https://doi.org/10.1007/978-3-030-89536-5_2

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(4)

2 The Chemical-Physical Processes Behind Self-assembly

formation of these bonds is Ginorg . The bonds can be inorganic or organic– inorganic, through the formation of non-hydrolyzable Si–C bonds, in the case of silicon. Ginterf . The formation of the hybrid interface between the organic micelle and the inorganic species contributes to the free energy Ginterf . The interactions that govern the formation of the interface are of a weak type, van der Waals, hydrogen bonding or electrostatic interactions. Gsolv . The solvent in which self-assembly takes place also plays an essential role in changing the system’s free energy through the solvation and solvent co-assembly. The condensation of the precursor alkoxides produces water and alcohol as by-products. At the same time, the variation in concentration of the different species makes this term time-dependent and dependent on the stages of self-assembly.

The formation of supramolecular structures starting from amphiphilic molecules and inorganic clusters generates a general decrease in the system’s free energy and is a thermodynamically favoured process. A further variation of the free energy is associated with the formation of an interface formation between the organic and inorganic structures. The two variations in free energy represent, therefore, the terms that generally govern self-assembly. The spatial organization of hydrophobic and hydrophilic species creates a “hybrid” interface and an ordered mesostructure. The thermodynamics of self-assembly can be modified by applying an external field. Getting order beyond the mesoscale is difficult, and mesoporous materials, both bulk and film, exhibit order only in oriented domains, generally within the micron scale. The materials after self-assembly resemble the structure of a “polycrystalline” system whose pores are generally ordered only in a reduced scale of some microns. A possibility of aligning the mesopores is the application of an external field which would change the free energy by an additional term, Gext : Gmes = Gorg + Ginorg + Ginterf + Gext (5)

(2.2)

Gext . It expresses the difference in free energy between a random macroscopic alignment and an alignment by applying an external field.

The thermodynamics of self-assembly in most systems is mainly governed by the formation of the interface and the relative free energy Ginterf and condensation through covalent bonds of the inorganic part that will form the scaffold of the porous mesostructured, Ginorg . The term Ginorg takes into account both the electrostatic interactions that lead to the formation of non-covalently bonded inorganic clusters and the inorganic condensation. The formation of the organic–inorganic interface between the micelle and the inorganic species depends on their mutual charge.

2.2 The Formation Pathway

9

2.2 The Formation Pathway The formation of materials with controlled porosity on a mesoscale through selfassembly is explained using different models, particularly the cooperative and liquid crystal models [3]. The interaction between the inorganic building block and the surfactant governs the self-assembly, and it is where the different models diverge. In one case, the surfactant takes the shape of a pre-formed organized template which gives rise to a micelle array, while in the other case, the surfactant and the inorganic building block units cooperate to achieve self-assembly. Figure 2.1 shows the general strategy to obtain mesostructured porous materials. The dimension and shape of the surfactant dictate the ordering length scale and periodicity. The ordered mesoporous materials are finally formed upon removal of the supramolecular template and the pore structure is a replica of the ordered array of micelles.

2.2.1 Cooperative Self-assembly Pathway Organic surfactant–inorganic silicate interaction. The amphiphilic nature, hydrophobic-hydrophilic and charge, characterize the surfactant molecules (Fig. 2.2).

A templating strategy • The final material should mantain the shape and dimension of the supramolecular template • Stability of the template during the inorganic condensation • Easy removal of the template without degradation of the inorganic framework Mesoporous material

SELF-ASSEMBLY

Supramolecular template

Inorganic building block

Template removal

Fig. 2.1 A general overview of the templating strategy to mesoporous materials

10

2 The Chemical-Physical Processes Behind Self-assembly

Inorganic nano building-block

Amphiphilic molecule

Weak interacon Hydrophobic

Hydrophilic

Fig. 2.2 The amphiphilic molecule (surfactant) interaction with a soluble silicate species

At the beginning of the process, the silicon precursors hydrolyze to form negatively charged species that electrostatically interact with the surfactant. Thus, the surfactantsilica species form a hybrid precursor molecule which becomes the structural unit necessary to build an ordered mesostructure. The process is defined as cooperative because both the surfactant and the silicate are essential parts to achieve self-assembly [2, 4]. The formation energy is, as we have seen, Ginterf , and is a critical parameter to address self-assembly at the initial stage. The first stage can be considered cooperative nucleation in solution. The formation energy is, as we have seen, Ginterf , and is a critical parameter to address the self-assembly at the initial stage. Therefore, the first stage can be considered as cooperative nucleation in solution. A very important aspect of cooperative self-assembly is that since surfactantinorganic building block species form, they affect the type of mesophase that the surfactant will form organizing in micelles. Thus, at the end of the process, selfassembly will be observed, but it is not always possible to establish a direct correlation between the type of mesophase formed and the specific concentration of the surfactant used in the synthesis.

2.2.2 Liquid Crystal Templating Pathway The Liquid Crystal Templating (LCTA) is another self-assembly pathway used to explain the formation of ordered mesoporous materials [5]. The LCT mechanism hypothesises that the liquid crystal determines the final mesostructure while the silicates condense around the pre-formed mesophases. The liquid crystalline phase [6] can form even in absence of the inorganic silicates and when the organic surfactant concentration is high enough the formation of the micelles is the dominant process [7]. The free energy of the formation of the organic phase, Gorg , is triggering self-assembly and Gorg > Ginterf . The condensation of the inorganic species occurs around the surface of the micelles, which template the mesostructure. The inorganic backbone forms upon

2.2 The Formation Pathway

11

condensation on the liquid crystalline mesophase. This pathway is the one used to explain the self-assembly of mesoporous ordered silica films.

2.2.3 True Liquid Crystal Templating Although the method has been defined as Liquid Crystal Templating (TLCT), no liquid crystalline phase is actually formed during the self-assembly process. In fact, the mass fractions of the surfactant used for self-assembly are generally of the order of 2.5% in volume or smaller, a value far below the concentrations necessary to form lyotropic liquid crystals. The self-assembly process is dynamic involving numerous chemical and physical parameters that govern the formation of organized structures. During the process, phases can be formed that may have a liquid crystal character but are not thermodynamically stable [8]. They are only an intermediate phase of the dynamic self-assembly process, which is substantially guided by the progress of the silica condensation. However, it is possible to use another approach for self-assembly which is defined as “direct” or True Liquid Crystal Templating [9]. In this case the surfactant concentrations in the starting solution are very high and can reach up 30– 60% in volume, thus allowing the formation of crystalline liquid phases in binary water-surfactant systems. In TLCT, the mesophase is pre-formed as a liquid crystal; immediately after, the inorganic precursors are infiltrated between the micelles, and the hydrolysis and condensation reactions start. In this sense, we can speak of True Liquid Crystal Templating because the mould is formed by a specific liquid phase, formed by the micelles above the critical concentration. Thus, the process can be seen as a kind of nano casting technique like the macroscopic process of casting molten metal into a mould. This approach’s advantage is obtaining a mesophase with the desired symmetry precisely because it can be determined a priori based on the surfactant phase diagram However, the formation of the inorganic network can lead to an uncontrolled phase separation in the lyotropic liquid. In general, it is not easy to obtain powders and films through TLCT, which is normally used to produce monoliths. The first group to introduce the TLCT technique was Attard’s one in 1995, who managed to obtain mesoporous silica using tetramethylorthosilicate (TMOS) as a precursor and alkyl-polyethylene oxide as a surfactant [7]. The surfactant in a concentration of 50% by weight in slightly acidic water forms a hexagonal crystalline liquid phase (H1 ). Immediately after the addition of TMOS, the system turns into an isotropic fluid. In fact, during the hydrolysis of TMOS in acidic conditions, small quantities of methanol are formed as a by-product of the reaction. Methanol temporarily destroys the liquid crystalline phase, which forms again only after the alcohol evaporates under vacuum (Scheme 1.1).

12

2 The Chemical-Physical Processes Behind Self-assembly

Si(OCH3)4 + 4 H2O

Si(OH)4 + 4 CH3OH - CH3OH

- H2O SiO2

Si(OH)4-x (O)x/2 + x/2 H2O

Scheme 1.1 Chemical reactions in TTLC. In the first step, TMOS hydrolyzes as soon as is added to the acid water—surfactant solution which forms a hexagonal liquid crystal. The hydrolysis of TMOS produces methanol as a by-product that must be removed while the silica polycondensation reactions proceed. In the next stage, the water is evaporated at 80 °C until complete polycondensation of the silica is obtained

2.3 The Kinetics To produce an ordered mesoporous structure via self-assembly the different chemicalphysical chemical processes which concur must be carefully controlled. One is the formation of a hybrid interface, the second one is the micellization and finally the inorganic condensation. These processes must be controlled in terms of kinetics, because if one is too fast phase separation can easily occur and the order given by the liquid crystalline template is lost. The three main rate constants that must be kept under control are: (a) (b) (c)

kinterf . The rate constant of formation of a hybrid inorganic–organic interface between the organic surfactants and the silica precursors. korg . The rate constant for the assembly of the organic molecules into ordered supramolecular arrays. kinorg . The rate constant which considers the reaction kinetics of two inorganic species to form a covalent bond.

To get order and being able to form an organized assembly of micelles which form a template for the condensation of the inorganic species without disruption of the array, the kinetics of the different processes must follow a precise order. If the condensation is too fast, in fact, the silica inorganic phase would separate into a distinct phase, the kinetics constants must follow, therefore, the right order: kinterf > korg > kinorg

(2.3)

The relationship shows that the formation of the mesostructured is controlled by the creation of the interface between the organic micelles and the inorganic building blocks. Clearly if kinorg > kinter or kinorg > korg the formation of the hybrid interface is inhibited and self-assembly does not occur [10]. Even if the creation of an ordered mesostructure is thermodynamically favoured, the inorganic polycondensation should follow the right relationship with the other

2.3 The Kinetics

13

kinetics constants otherwise no order will be achieved. The thermodynamic of selfassembly a mesostructure must be combined with kinetic considerations. In inorganic sol–gel systems with transition metal alkoxides the reactivity is so fast that the reaction needs to be adjusted and slowed down to get an ordered mesostructured.

2.3.1 Controlling the Kinetics in Silica Systems If silica systems adjusting the kinetics of silicate condensation is relatively easier with respect to other oxides. This has allowed obtaining a large variety of mesoporous structures especially in the case of thin films when the fast evaporation of the solvent can induce several phase transitions which depends on the capability of the system to comply with a rearrangement of the micelles. The most common precursor employed for the synthesis of silica porous mesostructures is a silicon alkoxide, tetraethyl orthosilicate (TEOS), but also SiCl4 has been largely employed. Water–ethanol mixtures in basic or acidic conditions also containing the amphiphilic surfactant, is the most common precursor. The pH of the solution largely controls the surface charge of the silica species and the hydrolysis and condensation rates. At low pH, in acidic conditions, when pH < 2, the silica chemistry is characterized by a fast hydrolysis rate and a slow condensation, and the silica species are negatively charged. At low pH the silica hydrolyzed species are very stable because the condensation is hindered. In highly basic conditions, pH > 12, the silica species instead are negatively charged and very stable. To get order when the pH has intermediate values is much more difficult because, for instance, at neutral conditions the silicon alkoxides exhibit a fast condensation rate and slow hydrolysis. The presence of unhydrolyzed species, such as the ethoxy groups in TEOS, creates a charge unbalance at the interface weakening the electrostatic interaction. In this condition order is difficult to achieve. Some tricks are also possible to get order in silica systems in a larger range of pH. The silica hydrolysis rate can be boosted by fluoride species and SBA-15 silica mesoporous materials have been obtained in the 0–9 pH interval [11]. The next chapter is dedicated to get a better in sight in the sol–gel chemistry of silica. The competition between order and disorder during the deposition of a thin mesoporous silica film can well described by the concept of “race towards order” (Fig. 2.3). The condensation of the inorganic phase, which forms the interconnected silica backbone, should be adjusted to avoid that the increase in viscosity of the system may “freeze” the disordered intermediate mesophase. Once defined the thermodynamics and the kinetics in self-assembly we can go back to the self-assembly pathways to give a general description (Fig. 2.4).

14

2 The Chemical-Physical Processes Behind Self-assembly

Fig. 2.3 Scheme of the “race towards order” concept. Reproduced with permission from [1]

2.4 Interaction at the Interface The interaction between the inorganic species and the organic surfactant could be of different nature, depending on the combination of charges and the possible presence of counterions. This interaction is at the ground of the different synthesis routes for self-assembly. The chemistry of the inorganic species must be adjusted considering the nature of the surfactant which could be charged, such as in the case of ionic surfactants, or neutral as for the block copolymers. In the first case the interaction is direct through an electrostatic Coulomb force and in the other case is depending on the hydrogen bonding between the inorganic species and the non-ionic surfactant (Fig. 2.5). The charge of the silicates can be controlled by the pH of the solution and in the case of ionic surfactants the interaction can be both with positively and negatively charged silica species. The electrostatic interaction between the charge of the surfactant head and the charge of the silica species has a primary role in driving self-assembly. In the case of a synthesis employing an ionic surfactant the following cases could be observed: 1. 2.

S+ I− = (S+ ) surfactant cations—(I− ) negatively charged inorganic species S− I+ = (S− ) surfactant anions—(I+ ) positively charged inorganic species.

In both the cases the charges of the surfactant (S) and inorganic species (I) are opposite and S+ I− and S− I+ represent the two possible interaction pathways, which are defined as direct.

2.4 Interaction at the Interface

15

Cooperative Self-Assembly Pathway 1

2

3

4

kinterf > korg > kinorg Thermodinamics is dominated by ΔGinterf

Liquid Crystal Pathway 1

2

3

kinterf > kinorg

4

korg > kinorg

Thermodinamics is dominated by ΔGorg Fig. 2.4 Cooperative self-assembly pathway (top). 1. The surfactant and the precursors are put together in an aqueous solution. 2. The inorganic clusters electrostatically interact with the amphiphilic molecules. 3. Micelles form giving rise to a well-defined hybrid interface. 4. The inorganic phase condenses and the final mesostructured is stabilized. The process is governed by Ginterf . liquid–crystal pathway (bottom). 1. The surfactant is dissolved in the liquid phase. 2. The surfactant forms micelles. 3. The inorganic precursor is added. 4 The inorganic precursor condenses around the micelles. The process is governed by Gorg . Rearranged with permission from [44] Fig. 2.5 Direct interaction at the hybrid interface

Direct interaction Electrostatic Coulomb Forces S+IBasic

pH

H-Bonding

S-I+

I-

I+

+

-

S0I0 Acid

I+, I-: charged inorganic species S+, S-: ionic surfactant

I0 S0

I0: silicate species S0: nonionic surfactant

16

2 The Chemical-Physical Processes Behind Self-assembly

Fig. 2.6 Indirect interaction at the hybrid interface

Indirect interaction Electrostatic Coulomb Forces Double layer H bonding

S-X+I-

S+X-I+

I+

I-

X-

X+

+

-

Basic conditions X+ = Na+, K+ Acidic conditions X- = Cl-, Br-

I+, I-: charged inorganic species S+, S-: ionic surfactant X+, X-: counterions

If the surfactant and inorganic species are not charged the interaction is due to hydrogen bonding: 3.

S0 I0 = (S0 ) nonionic surfactant—(I0 ) silica species

This is defined as the neutral route, S0 I0 . The interaction at the hybrid interface can be also indirect, mediated by a charged counterion, X+ , cationic or X− anionic. In the synthesis of mesoporous silica using ionic surfactants the counterions mediate the interaction at the interface (Fig. 2.6). Two different combinations are possible as a function of the relative charges: 4. 5.

S+ X− I+ = (S+ ) surfactant cations—(X− ) negative counterion—(I+ ) positively charged inorganic species S− X+ I− = (S− ) surfactant anions—(X+ ) positive counterion—(I− ) negatively charged inorganic species.

The S+ X− I+ generally takes place in acidic conditions with halogenide anions as counterions, (X− = Cl− , Br− ). In turn, the S− X+ I− route is active in basic conditions in presence of alkaline cations (X+ = Na+ , K+ ).

2.5 Surfactants and Micelle Templating Agents The surfactants play, as we have seen, a fundamental role in self-assembly. The dual nature at molecular level, hydrophilic and hydrophobic, favours phase separation

2.5 Surfactants and Micelle Templating Agents

17

and formation of micellar mesophases in aqueous solutions. The micelles are the templates of the pores and their dimension and ordered mesophase is controlled through concentration and temperature, as shown in the relative phase diagrams [12, 13]. The self-assembly of the inorganic (or hybrid) building blocks is governed by the structure directing supramolecular template [14]. It plays the multiple roles of forming a specific ordered mesostructured, creating a hybrid interface and cooperating in self-organization with the inorganic species. The structure-directing agents are basically supramolecular structures formed by the aggregation of amphiphilic molecules, which are molecules containing both hydrophilic and hydrophobic groups and beyond critical concentration form micelles of different geometry and shape. The first studies on self-assembly have been realized using ionic, cationic, and anionic [15] surfactants as supramolecular templates. Later studies, instead, have shown that block copolymers could represent an alternative that offers greater possibilities to control some fundamental parameters of the final mesoporous material [16]. Block copolymers, in fact, allow obtaining a wider variation of the pore radius and better control of the mesostructure topology. Furthermore, the thicker pore walls which form using block copolymers increase the stability of the final mesostructure. The self-assembly capabilities of the supramolecular structures govern the properties of the mesostructure on a nanometric scale, and the amphiphilic surfactants allow for guiding and optimizing the geometry, shape and size of the pores. The selfassembly process is, however, governed by a complex intermixing of simultaneous chemical-physical processes which drive to order [17].

2.5.1 Low Molecular Weight Charged Surfactants Ionic surfactants, such as the cationic alkyltrimethylammonium (Cn TA+ , n = 8–18), anionic alkylsulfonates (Cn SO32− , n = 12–18) or anionic alkylphosphates are the first class of supramolecular templates employed in the synthesis of mesoporous materials. The synthesis has been carried out in extreme, acid or alkaline, pH conditions and well-organized materials with controlled pore geometries and size have been obtained. Cetyltrimethylammonium bromide and cetyltriethylammonium bromide are typical examples of cationic surfactants, which contain quaternary ammonium ions (Table 2.1). The first mesoporous material synthesised is the so called M41S [18, 19], belonging to the family called by the Mobil researchers, who were the first to obtain it, as MCM (Mobil Composition of Matter). To produce silica materials with an ordered porosity larger than in zeolites, the Mobil researchers employed an aqueous solution of an aluminosilicate or a silica precursor (TEOS, Ludox, fumed silica, sodium silicate), an ionic surfactant (an alkyl trimethylammonium halogenide Cn TA+ X− , such as cetyltrimethylammonium bromide, CTAB (C16 H33 (CH3 )3 N+ Br− ), and a base, such as NaOH. A hydrothermal process by heating the precursors to a temperature above 100 °C for 24–144 h was applied. The final solid product was then washed,

18

2 The Chemical-Physical Processes Behind Self-assembly

Table 2.1 Some of the most common ionic surfactants employed in the synthesis of mesoporous materials Surfactant

Chemical formula

Alkyl ammonium salts (e.g. CTAB)

Cn H2n+1 (C2 H3 )3 N+ X− , Cn H2n+1 (C2 H5 )3 N (n = 8, 9, 10, 12, 14, 16, 18, 20, 22, X = OH, Cl, Br)

Gemini surfactants

[Cm H2m+1 (CH3 )2 N-Cs H2s -N(CH3 )2 Cm H2m+1 ]Br2 m = 16, s = 2–12

Sulphates

Cn H2n+1 OSO3 (n = 12, 14, 16, 18)

Sulphonates

C17 H33 SO3 H, C12 H25 C4 H4 SO3 − Na+

Phosphates

C12 H25 OPO3 H2 , C14 H29 OPO3 − K+

Carboxylic acids

C17 H35 COOH (stearic acid), C14 H29 COOH

filtered, and thermally treated at 500 °C to stabilize the silica structure. The final material has an ordered porosity in the 1.8–10 nm range, with a pore topology of parallel channels, stacked with a two-dimensional hexagonal (2D-Hex) cross section. The alkyl chain length can tune the pore dimensions, of syntheses done using surfactants of different alkyl chain lengths Cn H2n+1 (n = 8, 9, 19, 12, 14, 16) have shown that a direct proportion between n and the d-spacing values of the hexagonal mesostructured exists. Figure 2.7 shows an example of phase diagram temperature-concentration formed by an ionic surfactant, cetyltrimethylammonium bromide (C15 TAB) and water [20– 22]. With the increase of the concentration four different phases are observed: I, isotropic solution including both spherical micelles and micellar rods; H, hexagonal; C, bicontinuous cubic phase; L, lamellar.

500 450

Temperature / K

Fig. 2.7 Binary phase diagram water-CTAB. Redrawn with permission from [20]

L

400

I

H

C

350 300 250 0

20

40

60

80

CTAB concentration / wt %

100

2.5 Surfactants and Micelle Templating Agents

19

2.5.2 Mesostructures Templated by Amphiphilic Block Copolymers Block copolymers are an alternative supramolecular template to ionic surfactants [23, 24]. The copolymers (or heteropolymers) are composed by two or more different species of monomers that are covalently bonded to form a macromolecule. They differ from the homopolymers that are instead formed by only one type of molecular unit. Block copolymers can be synthesised using different arrangements of the units (Fig. 2.8). They can be classified, as a function of the distribution of the units along the chain, in two main groups: linear and branched copolymers. In the linear copolymers the single units can arrange to form alternating, random and block copolymers. In branched copolymers one or more polymeric chains are connected to the main one and can assume different architectures such as star or branched shapes. In the case of a diblock copolymer, formed by two different molecular units (circles and of different colors in the Fig. 2.8), several structures can form: copolymers where the two different units form two interconnected blocks in a linear chain, random copolymers with randomly interconnected small blocks, alternating copolymers with alternation in the chain of the two monomers and finally grafted copolymers where the chains are interconnected through branching. Block copolymers can be built using hydrophilic-hydrophobic segments to form amphiphilic block copolymers (ABCs). ABCs are part of a large family of surfactants widely used in industrial and household applications, such as foaming, dispersing agents, coatings and detergents. Besides these commercial applications, ABCs have also been largely employed in materials science. Some examples are the synthesis of nanoparticles and nanorods, hierarchical porous structures, biomaterials, and drug release. There are several well-established methods to prepare highly tailored ABCs

a) b) c) d) e)

Fig. 2.8 Homopolymers (a) and block copolymers (diblock) schematic structures: block (b), random (c), alternating (d) and grafted e. The two monomeric units are represented by circles of different colors

20

2 The Chemical-Physical Processes Behind Self-assembly

BRIJ

H(CH2)x

Diblock

(CH2CH2O)yH

CxEOy x = 12; 16; 18 y = 2; 4; 10; 20

PLURONIC

HO(CH2CH2O)x

EOxPOyEOx x and y = 20 - 100

CH3

(CH2CH2O)xH

Triblock

(OCH2CH2)xOH

HO(CH2CH2O)w

TWEEN

(CH2CH2O)y

O

CH(OCH2CH2)yOH

O

Star

CH2O(CH2CH2O)z-1 CH2CH2O

C

(CH2)7CH

CH(CH2)7CH3

With x + y + z + w = 20

Fig. 2.9 Some examples of diblock (Brij), triblock (Pluronic) and star (Tween) block copolymers

with specific geometries, chemical structure, degree of polymerisation and polydispersity. Diblock, triblock and multiblock linear copolymers or arranged as stars or grafted structures are some possible architectures [25]. Block copolymers have been successfully employed as structure directing agents in the synthesis of mesoporous materials and they have shown several advantages with respect to ionic surfactants of lower molecular weight [26, 27]. In general, ionic surfactants used as templates give mesopores in the order of few nanometers and produce small thickness walls of the inorganic frame. Aggregates of block copolymers are bigger in comparison to ionic surfactants and allow obtaining pores with a size of several nanometers. The chemical composition of the block copolymer, together with the solvent and cosolvent, give a better control of the mesophase organization which could be finely tuned. Another advantage is that several ABCs with different compositions are commercially available or can be synthesised ad hoc. Figure 2.9 shows some examples of typical block copolymers used in mesoporous self-assembly. Even if block copolymers and ionic surfactants can be both employed as structure directing agents in self-assembly, the choice affects the final properties of the mesoporous material. The main differences between ionic and block copolymer surfactants as structure directing agents during the self-assembly of mesoporous materials are listed in Table 2.2. The choice of the block copolymer for self-assembly, which is specifically defined in terms of the number of blocks, their relative length an chemical composition, is critical. The block copolymer drives the system to the organization if the proper conditions for aggregation in ordered phases are fulfilled and controls the final properties to a large extent [28]. The pore dimension is an example because its accurate design is difficult to achieve and largely depends on the surfactant. The choice of the blocks in the copolymers can direct the pore dimension. For instance, a stable caged cubic

2.5 Surfactants and Micelle Templating Agents

21

Table 2.2 Main differences in employing ionic or block copolymers surfactants as structure directing agents in self-assembly [26] Solution and mesophase behaviour

Use in the design of mesostructured materials

Ionic surfactants

Block copolymers

Molecular–monodisperse

Polymeric–polydisperse

Head + chain structure

Large range of architectures: linear, branched, star …

Object shape controlled by the g packing parameters

Shape controlled by N and fA , fB …

Simple micelle-like or bicontinuous mesostructures

Possibility of complex multiscale mesostructures

Micellization driven by hydrophilic-hydrophobic character

Micellization driven by hydrophilic-hydrophobic character, block size and conformation

Hard well defined hybrid interface

Blurry interface, swollen by the inorganic phase

Thin walls (1–1.5 nm)

Thick walls (2–10 nm)

Walls not entangled with the template

Walls entangled with the template (multiphase)

Pores size limited by micelle size

Pore size tailorable by modifying the polimerization degree, monomer nature or polymer fraction

mesoporous silica structure, Im3m, with s large and uniform pore size of 12 nm, has been obtained by using as the structure-directing agent a triblock copolymer PEO–PBO–PEO, which contains a more hydrophobic poly(butylene oxide) block [29]. The pore size is very dependent on hydrophobic domains, and a copolymer containing a highly hydrophobic PBO block gives mesoporous materials with larger pores.

2.6 Self-assembly of Block Copolymers One of the theories which gives a simple description of phase separation of block copolymers in solution is the mean-field theory (MFT) [30]. The theory considers thermodynamic and structural parameters that consider the solubility of the different blocks and the polymerization degree. A block copolymer is composed by flexible and chemically different blocks that can separate in microphases assuming a variety of different morphologies. The covalent bonding between block prevents phase separation, while the self-assembly is driven by an unfavourable mixing enthalpy and mixing entropy. When amphiphilic molecules form micelles in water, the hydrophilic head groups point to the outer surface and the hydrophobic tails is

22

2 The Chemical-Physical Processes Behind Self-assembly

the hydrophobic core. This arrangement is favourable because minimizes the repulsion of the hydrophobic tail with water. The charged head groups, however, that occupy the micelle’s surface experience a charge repulsion. It is the balance of these competing interactions that controls the micelle’s stability. The entropy per unit volume of mixtures composed by high molecular weight An and Bn homopolymers of different composition is generally small, and it scales inversely with the molecular weight. In mixtures of low molecular weight homopolymers is observed the opposite. Therefore, a small structural difference between the homopolymers A and B increases the free energy and leads to phase separation. The immiscibility of two homopolymers can be evaluated using the so-called Flory–Huggins interaction parameter, χ AB , (segment-segment interaction parameter), which gives a measure of the incompatibility between the two blocks, A and B:     Z 1 1 (2.4) ε AB − ε AB − (ε A A + ε B B ) χ AB = kB T 2 2 It expresses the increase of Gibbs free energy when two monomers A and B come in contact. Z is the number of nearest neighbour monomers, εAB is the interaction energy between A and B monomers, εAA and εBB the interaction energy per monomer between the same monomers A or B, with kB T the thermal energy in units. A positive χAB corresponds to a repulsion between the A and B monomers, which gives phase separation. On the contrary a negative εAB indicates that the entropy of the system favours miscibility of unlike monomers. In the case that A and B do not have any significant interaction, such as in the case of hydrogen or electrostatic interactions, εAB is assumed as positive with a typical value of 0.1. εAB depends also on the temperatures and higher temperatures favour mixing. These general observations can be extended to the case of amphiphilic block copolymers. The A and B units correspond to blocks of different chemical composition which are covalently bonded. In this case long-range hydrophobic-hydrophilic repulsion and short-range covalent attraction coexist. These competitive forces drive the formation of self-assembled aggregates. The total degree of polymerization, N, is another parameter that has a strong influence on the block copolymer. To quantify the driving force which leads to ordered phases is used the parameter Nχ, given by N, the polymerisation degree and χ, which represents the sum of χAB for all the A and B monomers. Nχ, the segregation product, allows getting an evaluation of the degree of microphase separation of the block copolymers. If Nχ ≤ 10 the entropy governs the system and only a disordered phase can be observed. Otherwise, when Nχ > 10, is the enthalpy the main term and a disorder-to-order transition occurs [31]. The geometrical factor has a primary role in self-assembly of diblock copolymers. In fact, in the case of a diblock copolymer of general composition A-B, the relative extension of the polymeric chains A and B is very important because would change

2.6 Self-assembly of Block Copolymers

23

the geometry of the aggregates. The relative composition is described by fA and fB , defined as volume fractions of the constituent blocks (with NA or NB ) with respect N: fA = NA /N, fB = NB /N with fA + fB = 1

(2.5)

If the building blocks A and B have the same molecular weight, they would form lamellar phases (lam). If the symmetry is only slightly lost still some layered structures would form through which the high molecular weight components are interconnected (perforated layer, pl). Further asymmetries will lead to bicontinuous phases, where both blocks form interconnected phases with a cubic Ia3d symmetry group (gyroid phase, gyr). Block copolymers characterised by a low block symmetry form instead, 2D-Hexagonal (p6mm) and cubic (Im3m) phases, associated with cylindrical and spherical micelles, respectively. More asymmetric compositions will give a hexagonal phase with hexagonally packed cylinders while the arrangement of spheres in in a body-centered cubic (BCC) lattice gives a spherical phase. Another question is if on the base of geometrical assumptions would be possible, at least in principle, to predict the topology of the self-assembled mesophase phase. To obtain a prediction it is necessary to consider also the chemical and structural factors. They are expressed by the terms χij , N, fij , where i and j change in accordance with the number of blocks (for a diblock copolymer i = A, j = B). In general, there is another parameter that give quite good predictive indications on the type of mesophase that could form. This is the packing factor, g, which will be later introduced. For a predictive theory, other effects must be also considered. For instance, the driving force to phase segregation is countered by entropic forces within each single macromolecule. To maximise the distance between incompatible blocks, the organic chains adopt stretched configurations, which generate a restoring force, somehow analogous to Hooke’s law but entropic in origin. In the case of a block formed by N monomers in a stretched configuration and spaced by R, the restoring force can be written as: Fe =

3k B T R 2 2N a 2

(2.6)

with the characteristic length of the monomer which depends on the local chain structure. The number of macromolecules that participate to form a micelle define the aggregation number, Z, through the empirical law: −β

Z = Z 0 N Aα N B

(2.7)

with Z0 a constant depending on the type of block copolymer, NA and NB , the polymerisation degrees for the different blocks A and B (in case of a deblock copolymer), the exponential factor, α and β, are equal to 2 e 0.8, with small changes between the different systems. This empirical law describes the aggregation number in systems

24

2 The Chemical-Physical Processes Behind Self-assembly

containing di- and triblock, graft and star copolymers. Furthermore, it works also quite well for ionic and non-ionic surfactants of small molecular weight. Block copolymers in solution can form different aggregate structures whose stability depends on enthalpic, interaction of incompatible blocks, and entropic, chain stretching, spatial frustration, contributions to the Gibbs free energy. These contributions can be mathematically expressed by the parameters χN, which expresses the tendency towards block segregation, and f, the chemical composition. The number of blocks in the copolymer contribute to the increase of the degree of complexity of the structures, which is dramatically enhanced by the increases in the number of the different blocks N. In diblock copolymers, the parameters χAB , N and f are generally enough to define the phase diagram univocally. In the case of triblock copolymers, instead, a larger number of parameters must be known, at least three interaction parameters, χAB , χBC and χAC , and two compositional parameters, fA and fB . Other parameters that depend on the block copolymer architecture also need to be used, such as branching and side groups. More and more exotic symmetries are experimentally observed, increasing the complexity of the structure, for instance, from a diblock to a triblock copolymer.

2.7 Self-assembled Micelles The amphiphilic molecules concentration in an aqueous solution is the parameter that controls the aggregation state. At low concentrations, surfactants are in the form of single molecules whose polar head is solvated. As the concentration rises to reach a critical threshold, defined as the critical micelle concentration, cmc, the molecules self-organize, forming micelles. These supramolecular structures are held together by weak bonds between the single molecules and take on a well-defined shape. The aggregation number, N, defined as the average number of molecules per micelle, identifies the supramolecular aggregate. Once the cmc is exceeded, the micelles can still change shape as a function of increasing concentrations at a constant temperature. The phase transformations relate to the radius of curvature of the micelle, which changes with increasing concentration and follows a well-defined sequence: direct spheres, direct cylinders, lamellae, inverse cylinders, and inverse spheres (Fig. 2.10). It should be observed that amphiphilic block copolymers do not form real micelles a critical micelle concentration (cmc). Aggregation, in fact, occurs over a broad concentration range that is indicated as aggregation concentration range (ACR). When the surfactant reaches saturation, this point is defined as the limiting aggregation concentration (LAC). LAC would correspond to the conventional cmc. In the scientific literature regarding self-assembly mesoporous materials cmc is generally used also for block copolymers, and the same term has been, therefore, used in the present discussions, keeping in mind however about the difference. The sequence of micellar structures follows a monotonous variation of the curvature radius at the interface. The models used to explain the dependence of the micelle

2.7 Self-assembled Micelles

“Direct” spheres

Liposomes

25

“Direct” cylinders

Bicontinuos

Lamellae

“Reverse” micelle

Fig. 2.10 Examples of micellar structures: direct sphere, direct cylinder, planar bilayer (lamellae), reverse micelle, bicontinuous phase and liposomes. The order corresponds to a monotonical variation of the interfacial curvature

shape on the concentration consider several parameters such as hydrophobic interactions between organic chains, geometric constraints due to molecular packing, molecular exchange between aggregates, repulsion between polar heads. The formation of a supramolecular template represented by the micelles of amphiphilic molecules represents a critical aspect of self-assembly [32]. Studying the micelle formation process is, therefore, extremely important for governing and understanding the process. We will now briefly see the theory behind the formation of micelles, particularly the parameters that govern the formation of aggregates starting from monomers [33]. An extensive discussion of this topic can be found in Israelachvili’s book [12], “Intermolecular and Surface Forces: With Applications to Colloidal and Biological Systems”, whose reading is recommended and covers the whole theory of the formation of micelles extensively. Let us consider a starting solution formed by the same type of amphiphilic molecules, which, depending on the concentration, can give rise to the formation of aggregates. The chemical potential, μ, of all identical molecules that form different aggregates must be the same to ensure thermodynamic equilibrium. This means that a molecule will always have the same chemical potential when isolated or connected to form dimers, trimers, or larger aggregates. This assumption can be expressed in the following terms:

monomer

dimer

trimer

(2.8) where:

26

2 The Chemical-Physical Processes Behind Self-assembly

N N μ0N XN kB T

aggregation number (mean number of molecules per aggregate) 1, 2, 3, 4, … standard part of the chemical potential or mean interaction free energy per molecule in aggregates of aggregation number N concentration of molecules in aggregates of aggregation number N (the maximum value is 1) Boltzmann constant Temperature

Therefore in the case of a monomer, N will be equal to 1, and so we have: μ01 + k B T log X 1 which corresponds to the case of single molecules in solution. Since the chemical potential is constant, it can be expressed as: μ=μ = N

μN

μ0N

  XN kB T log = constant + N N

(2.9)

mean chemical potential relative to the aggregate of aggregation number N

while the energy per aggregate is: N μ0N . The single molecules dynamically associate and dissociate with the micelles at a high rate (10−5 –10−3 s) (Fig. 2.11); these rate can be expressed as:

Fig. 2.11 Association of N monomers into an aggregate (e.g. a micelle). The mean lifetime of an amphiphilic molecule in a small micelle is very short, typically 10−5 –10−3 s. Reprinted with permission from [12]

2.7 Self-assembled Micelles

27

 k1 X 1N = association rate k N

XN N

 = dissociation rate

The ratio between the two-reaction rate gives the equilibrium constant, K; this is a dynamic equilibrium because the molecules that form the aggregate with aggregation number N continuously exchange with the free molecules. K is expressed as: K =

   k1 = ex p −N μ0N − μ01 /k B T kN

(2.10)

This expression can be rewritten in the equivalent way:

   N /M  X N = N (X M /M)ex p M μ0M − μoN /k B T

(2.11)

and if we consider a monomer as a reference state, M, which expresses any reference state of aggregates, for M = 1 we have:

  N  X N = N X 1 ex p μ01 − μ0N /k B T

(2.12)

This equation is extremely helpful because tell us that there is a difference in the free energy between the molecules which are in an aggretated or dispersed state (N = 1). The functional variation of μ0N with N governs several important physical properties of the aggregates such as the mean size and polidispersity. Aggregation occurs only if the free molecule and the molecule in the aggregate display a different cohesive energy. We can now consider the case of molecules in aggregates of different dimensions, including the monomers, which have the same interactions with the surrounding aggregates and also the other monomers. This implies that μ0N must remain constant, because the mean free energy does not change, in aggregates of different sizes and also in single molecules. The previous equation becomes: X N = N X 1N for μ01 = μ02 = μ03 = . . . = μ0N

(2.13)

If μ0N − μo1 is negative the formation of aggregates is energetically favoured. Therefore, the necessary condition to form large and stable aggregates is: μ0N < μ1o for some values of N. Aggregates (micelles) form if μ0N descreases with the increase of the aggregation number N and the dependence of μ0N on N is, in general, dependent on the geometry of the aggregates. The free energy which defines the interaction between molecules, in the case of the simplest structures (rods, sheet and spheres), is expressed by: μ0N = μ0∞ + αk B T /N p

(2.14)

28

2 The Chemical-Physical Processes Behind Self-assembly

where: μ0N μ0∞ α p

standard part of the chemical potential or mean interaction free energy per molecule in aggregates of aggregation number N the “bulk” energy of a molecule in an infinite aggregate positive constant dependent on the strength of the intermolecular interactions number that depends on the shape or dimension of the aggregates.

This expression tell us that in general for all the structures, μ0N decreases with N and this appears to be a necessary but not sufficient condition for the formation of aggregates. The amphiphilic molecules decrease their free energies only by forming large aggregates. It remains to define what is the minimum concentration of molecules to observe aggregation. Inserting this expression into 2.12 we obtain: 

  N X N = N X 1 ex p μ01 − μ0N /k B T

(2.15)

If the monomer concentration, X1 , is low enough, it means that:   X 1 ex p μ01 − μ0N /k B T  1

(2.16)

and therefore X1 > X2 > X3 … for any intermolecular interaction constant α. This corresponds to the case where most of the molecules are present as isolated monomers. If the   concentration of the monomers increases and X1 is approaching X 1 ex p μ01 − μ0N /k B T , this represents the limit value because XN can never exceed the unity. The monomer concentration (X1 )crit at which this occurs is exactly the critical micellar concentration (cmc) (or critical aggregation concentration): 

(X 1 )crit

μ0 − μ0N = cmc = ex p − 1 kT



≈ e−α for every p

(2.17)

At this critical concentration value the addition of further monomers into the solution would result in the formation of more aggregates while the monomer concentration does not change (Fig. 2.12). Above cmc the free molecules form aggregates whose shape and aggregation number depend on the properties and structure of that specific molecule. Definition: Critical micelle concentration (cmc). The concentration at which further addition of solute molecules to a solvent makes them go into finite sized micelles (aggregates) while the monomer concentration remains unchanged at the cmc [12].

2.7 Self-assembled Micelles

29

Fig. 2.12 Concentration of monomers and aggregates as a function of the total concentration. The larger the aggregation number N, which in general means a larger aggregate, the sharper in the transition at cmc. Reproduced with permission from [12]

2.7.1 The Packing Factor The previous equations allow us to comprehensively describe the interactions between the single amphiphilic molecules with an aggregate and determine the necessary and sufficient conditions for micelles to form after a particular critical concentration. However, these aggregates can take on different forms. Therefore, which of the possible structures can be formed as a function of the surfactant concentration needs, therefore, to be assessed. Self-assembly is governed by different competing interactions which direct the formation of specific structures, such as micelles and bilayers. These forces can summarized into two main groups that compete at the interfacial region: 1. 2.

Hydrophobic attractive interactions at the hydrocarbon-water interface that drives molecular aggregation. Hydrophilic, ionic or steric repulsion of the headgroups which imposes that headgroups remain in contact with water.

These two forces are in opposition and mainly act at the interface. In an aqueous phase the hydrophobic interaction would tend to decrease the interfacial area, a, while the other one at the opposite would increase (Fig. 2.13). The competition between these attractive-repulsive interactions determines the stability interval of the micelles with a certain geometric shape. Thus, based on simple mathematical and geometric considerations, it is possible to obtain a general formula that describes the total interfacial free energy for molecules in an aggregate:

30

2 The Chemical-Physical Processes Behind Self-assembly

Fig. 2.13 The hydrocarbon interiors in both micelles and bilayers are normally in the fluid state at room temperature. Repulsive headgroup forces and attractive hydrophobic interfacial forces determine the optimum headgroup area a0 at which moN is a minimum (see Fig. 20.2). The chain volume v and chain length ‘c setlimits on how the fluid chains can pack together, on average, inside an aggregate. Thus, the preferred molecular conformation depends on a0 , v, and ‘c. In stressed micelles or bilayers, the headgroup area a is larger or smaller than a0. Such stresses can come from compressing a monolayer or bilayer either normally or laterally, stretching it, or bending it. Reproduced with permission from [12]

K a

μ0N = γ a +

(2.18)

where: N γ a K

average number of molecules in a micelle surface energy of the hydrophobic chain surface area per molecule constant To find the minimum energy is enough to put: dμ0N =0 da

(2.19)

which gives the value a = Kγ , that is assumed to be the optimal surface area per molecule at the hydrocarbon-water interface and is indicated as a0 . The unknown constant in 2.16 can now be eliminated rewriting the formula as: μ0N (min) = 2γa0 +

γ (a − a0 )2 a

(2.20)

Fig. 2.14 The opposing forces, headgroup repulsion and interfacial attraction, are balanced in correspondence of an optimal headgroup area a0 . Redrawn with permission from [12]

31

Interaction free energy,

2.7 Self-assembled Micelles

Minimum energy at a0

Repulsive energy

Surface area per molecule, a is now expressed using two measurable parameters which are the surface energy of the hydrophobic chain, γ, and the optimal surface area per molecule at the hydrocarbonwater interface, a0 . This parameter is connected to both the size and charge of the surfactant headgroup; it also changes with variations in the electrostatic interactions of the surfactant headgroup [34]. Introducing the opposing forces in the model has the consequence of an optimal area per headgroup at which the total interaction energy per molecule has a minimum (Fig. 2.14). The optimum surface leads to the formation of micellar aggregates. It is possible now through simple geometric parameters to derive a description of the shape of the aggregates. Besides ao , we introduce: v lc

volume of the hydrophobic tail (considered incompressible for simplicity) plus any cosolvent organic molecules between the chains maximum length of the hydrophobic tail.

lc is a limite value for the tail because a further increase could cause a net increase in energy and can be considered a little smaller than the fully extended chain. If we know the three parameters, ao , lc and v, it is possible to make a prevision about the structure would form via self-assembly of amphiphilic molecules by using the dimensionless packing factor or packing parameter, g. It correlates the structure of the surfactant with the resulting aggregate morphology (2.21): (2.21)

therefore, the value of g increases as: a0 decreases; v increases and l c decreases. Experimental results have shown that that amphiphilic molecules with large polar headgroups tend to form high-curvature spherical aggregates and they pack following

32

2 The Chemical-Physical Processes Behind Self-assembly

ao = effective optimal surface of the polar head v = chain volume l = chain length lc = length of the fully extended chain

Polar head surface Hydrophobic volume (g: packing parameter) g = v/(llc . a0) ao

v

l Conical shape (icecream cone)

Inverse conical (champagne cork)

Fig. 2.15 The packing parameter g and two possible geometrical shapes

a cubic symmetry. Otherwise, amphiphilic molecules with longer hydrocarbon tails or smaller polar headgroups will give cylindrical structures packed in a 2D-Hex or lamellar structure. Vesicles or bilayers are preferentially formed by surfactants having two hydrocarbon tails because have a larger volume, v (Fig. 2.15). An example is the selfassembly of phospholipids that have two hydrophobic tails and naturally form cell membranes. If the hydrophilic headgroup points inwards, g > 1 and inverse micelles form (Fig. 2.16). The g value governs what geometrical structure is assumed by the surfactant and therefore the type of micelle it will form as a function of the packaging efficiency and the micelle’s radius of curvature. g < 1/3. In general, if the headgroups of the surfactant is large this reflects in a small packing factor, g < 1/3, and a conical shape (Fig. 2.17). The cones are a favourable geometrical form to pack efficiently by giving spherical micellar structures with high interfacial curvatures. 1/3 > g > ½. At larger g values, between 1/3 and ½, the surfactant volume assumes the shape of a truncated cone. This structure can also pack very efficiently producing micelles with a lower curvature, such as cylindrical micelles and rod-like structures. In turn, the cylindrical micelles can become the building blocks for the formation of 2D-Hex structures (p6mm). g = 1. At this g value the surfactant forms aggregates of cylindrical molecular shape and the cylinders pack into bilayer structures that give rise to lamellar micelles.

2.7 Self-assembled Micelles

33

Fig. 2.16 The different types of structures and “mesophases” formed by amphiphiles in aqueous solutions depending on their packing parameter, g = v/a0 lc . Figure 20.9 shows the somewhat different structures formed in the presence of oil (hydrocarbon), although both systems have similar designations: I for isotropic solutions of monomers, M for micellar, H for hexagonal (cylindrical), C for cubic (usually isotropically interconnected), L for lamellar, and subscripts I and II for normal and inverted (aqueous core) structures. There are many types of cubic and lamellar structures and phases that depend on the concentration and interactions between the aggregates and also on the sign and magnitude of the curvature energy. Reproduced with permission from [12]

g

Micelle

Mesostructure

g < 1/3 Spherical 1/3