Physics of Polymer Gels E-Book

• Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 485

Veröffentlichungsjahr: 2020

Beschreibung

Explains the correlation between the physical properties and structure of polymer gels

This book elucidates in detail the physics of polymer gels and reviews their unique properties that make them attractive for innumerable applications. Geared towards experienced researchers and entrants to the field, it covers rubber elasticity, swelling and shrinking, deformation and fracture of as well as mass transport in polymer gels, enabling the readers to purposefully design polymer gels fit for specific purposes.

Divided into two parts, Physics of Polymer Gels starts by explaining the statistical mechanics and scaling of a polymer chains, and that of polymer solutions. It then introduces the structure of polymer gels and explains the rubber elasticity, which predicts the solid-like nature of polymer gels. Next, it describes swelling/deswelling, which can be understood by combining the rubber elasticity and the osmotic pressure of a polymer solution. Large deformation and fracture, and the diffusion of substances in polymer gels, which are essential for practical applications, are also introduced. The last half of the book contains the authors' experimental results using Tetra-PEG gels and provides readers with the opportunity to examine and compare it with the first half in order to understand how to utilize the models to experiments. This title:

* Is the first book dedicated to the physics of polymer gels
* Describes in detail the properties of polymer gels and their underlying physics, facilitating the development of novel, polymer gel-based applications
* Serves as a reference for all relevant polymer gel properties and their underlying physics
* Provides a unified treatment of the subject, explaining the physical properties of polymer gels within a common nomenclature framework
Physics of Polymer Gels is a must-have book for experienced researchers, such as polymer chemists, materials scientists, organic chemists, physical chemists, and solid-state physicists, as well as for newcomers to the field.

Leseprobe

Physics of Polymer Gels

Edited by

Takamasa Sakai

Editor

Prof. Takamasa Sakai

The University of Tokyo

Graduate School of Engineering

Department of Bioengineering

7-3-1 Hongo, Bunkyo-ku

113-8656 Tokyo

Japan

Cover Images:

Polymer gel, @Kozlova/Shutterstock; Graph: courtesy of Prof. Takamasa Sakai, The University of Tokyo, Japan

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Print ISBN: 978-3-527-34641-7

ePDF ISBN: 978-3-527-34653-0

ePub ISBN: 978-3-527-34655-4

oBook ISBN: 978-3-527-34654-7

Preface

Polymer gels are defined as a three-dimensional polymer network swollen with a solvent. If properly designed, polymer gel can hold 1000 times as much solvent as polymer weight, have deformability more than 10 times, and can retain/release macromolecular substances like protein. Such high swellability, deformability, and permeability of polymer gel are unique characteristics not found in other materials. Due to these characteristics, hydrogels are used for many applications such as absorbent materials for paper diapers, soft contact lenses, drug reservoir, etc. Recently, considerable attention has been paid as a material for future medical care such as tissue-replacement material and scaffold for regenerative medicine. This book explains the correlation between the physical properties and structure of polymer gels and is prepared for university students learning polymer gels for the first time.

Unique properties of polymer gels stem from their unique structure; though the significant component is solvent, a polymer gel is solid due to the 3D polymer network with a few percent by weight. Notably, the polymer network and the solvent are not separated in two phases, but they exist together as a gel phase. Thus, a polymer gel has both a solid-like nature stemming from a three-dimensional network and a polymer solution-like nature arising from a solvent dissolved and retained in a polymer network. This duality is the feature of polymer gels not found in other materials and the source of the uniqueness. At the same time, however, this duality often causes difficulty in understanding polymer gels. The solid nature of polymer gel is discussed based on the theory of rubber elasticity, while the liquid-like nature is presented based on the theory of polymer solutions. Therefore, to understand the basic concept of polymer gel, fundamental understanding of both is indispensable, and the harmony of both is essential.

Both rubber elasticity and polymer solution theory are based on statistical mechanics. Under ideal assumptions, both theories bring rigorous prediction by mathematical formulas for various physical property values. On the other hand, the 3D polymer network is inherently heterogeneous; this is obvious from the fact that even synthetic linear polymer chains have heterogeneous distribution in length. It is impossible to synthesize a polymer network with uniform mesh size and number of branching. Since it is not possible to accurately define the structure, it is difficult to formulate the distribution function and to adapt the statistical mechanics approach practically.

For this reason, it is difficult to understand the correlation between the structure and physical properties of a real polymer gel, and the contribution of theory to material design may be limited. However, we believe that understanding the fundamentals of polymer gels is still relevant, because the hurdle for practical application of polymer gels is high, and it is difficult to go beyond the difficulty relying only on experience. Material design based on fundamentals is indispensable to overcome this situation. Understanding the phenomenon in the form of a mathematical expression is extremely meaningful, even though we cannot observe quantitative agreement between the theory and the experimental result. Useful information is often obtained by comparing experimental results with theoretical values obtained under certain exact assumptions. It is also essential to change the degree of coarse-graining and watch over the rough sketch of physical properties using scaling theory. In this way, it is crucial to handle the heterogeneity of polymer gels. On the other hand, it is vital to deepen the fundamental understanding of polymer gels by experimentally verifying the theories using polymer gels with a well-defined structure.

Therefore, in this book, as an introduction of basic knowledge, we first explain the statistical mechanics and scaling of a polymer chain in Chapter 1 and that of polymer solution in Chapter 2. In Chapter 3, we introduce the structure of polymer gels and explain the rubber elasticity, which predicts the solid-like nature of polymer gels. In Chapter 4, we describe the swelling/deswelling, which can be understood by combining the rubber elasticity (solid-like nature) and the osmotic pressure of a polymer solution (liquid-like nature). We introduce the large deformation and fracture in Chapter 5 and the diffusion of substances in polymer gels in Chapter 6, which are essential for practical applications.

The last half of this book contains our experimental results using Tetra-PEG gels, which is a near-ideal polymer gel developed by us. We briefly explain Tetra-PEG gels in Chapter 7 and experimental results in Chapters 8–17. We examined the validity of the most of the theories presented in the first half using Tetra-PEG gels. This book was designed to be an introduction to understand polymer gels. By comparing the first half and the last half, readers can learn how to examine the models and how to utilize the models to experiments. I hope this book will help you know polymer gels.

Takamasa Sakai

Tokyo

December 2019

Acknowledgements

I want to express my gratitude to all the students who have promoted research together. This textbook is a culmination of 12 years that I learned about polymer gels with you. Special thanks go to Takuya Katashima, Xiang Li, Takeshi Fujiyabu, Yuki Yoshikawa, and Yuki Akagi. Also, I would like to express my gratitude to Junsei Kishi, who helped to publish the original Japanese version.

I sincerely express my most substantial gratitude to Prof. Ung-il Chung, who proposed writing this book. Almost every day, he has encouraged me, “Is writing textbooks progressing? You must write every day, even little by little.” I can say that this book has never been realized unless Ung-il has encouraged me.

I also thank my family, Kanako, Chiho, and Itsuki with all of my love.

Part ITheories

1Single Polymer Chain

Takamasa Sakai

Graduate School of Engineering, The University of Tokyo, Tokyo, Japan

1.1 General Features

A polymer gel is a three-dimensional network of polymer chains containing a large amount of solvent (Figure 1.1). When a network structure is formed only by chemical bonding, all the polymer chains are included in a single molecule; one large macromolecule traps a large number of solvent molecules. Let us calculate the molecular weight of a polymer gel. For example, 100 g of a gel with a polymer concentration of 5% contains 5 g of polymer networks. In other words, one molecule has a weight of 5 g. Because the molecular weight is given by the sum of Avogadro's number of weights of individual molecules, the molecular weight of the polymer gel is 3 × 1024 g/mol, which is extremely large. When one stretches a piece of polymer gel, all the chains forming this extremely large macromolecule are stretched, which is why the mechanical properties of polymer gels are predicted based on the simple sum of the contributions of single polymers connected to neighboring chains via crosslinks. Thus, learning the characteristics of a single polymer chain is important for understanding polymer gels. This chapter introduces some methodologies for extracting the universal characteristics of a single polymer chain.

1.1.1 Conformation of a Polymer Chain

A polymer chain is a linear molecule containing a large number of atoms. Prior to considering the shape of a polymer chain, let us focus on the local structure of a polymer composed of four carbons (Figure 1.2). When a carbon–carbon single bond is present between the monomers, the distance between each monomer is approximately 1.5 Å. Additionally, if carbons are connected by a single bond, the bond angle θ is essentially constant at 109.5°. Even if the bond length and bonding angle are constant, rotation around the bond axis, represented by ψ, is allowed, resulting in conformational flexibility. In fact, the value of ψ takes the trans (ψ = 0°) or gauche (ψ = ±120°) stable angles due to steric hindrance.

Let us increase the number of carbons to 100 and consider the shape of the resulting polymer chain. For example, if all the bonds take trans conformations, the polymer chain takes an elongated form with an end-to-end distance of approximately 25 nm. Conversely, if all bonds are in gauche conformations, the polymer chain takes a helical structure, and the end-to-end distance becomes very short. Although these structures can be realized by some specific macromolecules or under specific conditions, conventional polymers contain both trans and gauche forms and have highly complicated structures. However, by applying coarse-graining concepts, sufficiently long polymer chains can be approximated to a model chain regardless of the details of the monomer unit.

Figure 1.1

Schematic diagram of polymer gels. The polymer network consists of polymer chains connected to neighboring chains via crosslinks.

Figure 1.2

Conformation of local structures containing four carbon atoms (a) and the energy landscape (b).

1.1.2 Coarse-Graining of a Polymer Chain

Here, we introduce “coarse-graining,” which is an important concept in discussing polymers. Coarse-graining is one methodology for extracting the universal characteristics of a phenomenon. Roughly speaking, coarse-graining methods intentionally shift focus away from the trivial matters for the characteristics of interest, simplify the problem, and provide the universal characteristics. Let us see an example of coarse-graining for polymer chains. The first coarse-graining is “setting the bond length as constant and the bond angle as freely rotational.” This assumption represents a considerable “jump” from the discussion earlier. In principle, the bonding angle should be constant at approximately 109.5°, and the local conformation should be trans or gauche. However, one simple idea justifies this coarse-graining. The idea is to combine some monomers together and to make a “segment.” Figure 1.3 shows a schematic of making a segment from three monomers; as a result, a polymer chain can be visualized as a sequence of segments. As shown in Figure 1.3, the bonds between neighboring segments can take various angles relative to the bonds between monomers, and the individual properties of each monomer can be masked. Masking the individual properties of each monomer is of great importance in polymer physics, because only under such conditions can we extract the universal properties of the polymer chain. The length of the smallest segment that has freely rotating bonds is called the segment length, which is intrinsic to each monomer unit. Conversely, by taking the appropriate segment with the segment length, the end-to-end distance of a polymer chain is determined by considering a series of segments connected by freely rotating bonds. For simplicity, this book considers polymer chains as consisting of monomers that act as segments with freely rotating bonds, following the method of de Gennes [1]. In other words, the monomer length is the same as the segment length, and the degree of polymerization is the same as the number of segments.

Figure 1.3

Coarse-graining of monomeric units in a polymer chain.

1.1.3 Free Rotation Model

Chains consisting of segments with free rotation can be addressed using the free rotation chain model. Assuming that a polymer chain consists of N vectors (ai) of size a, the end-to-end distance (r) of the chain is written as follows:

(1.1)

Since it may be difficult to start with a three-dimensional problem, let us first consider the problem in one dimension. The one-dimensional version of this problem is actually given by the familiar problem as follows:

A point proceeds +a or −a with equal probabilities in one step. How far is the point from the origin after N steps?

This problem is equivalent to tossing coins in high school mathematics. In this case, the displacement, r, can be calculated as an expected value as follows:

(1.2)

The result of r = 0 is not essential. This answer is obvious from the expression of Eq. (1.2); the situations in which a point reaches −r and r have equal probabilities and cancel each other. In both cases, the end-to-end distance should be considered, r. The absolute value of the displacement must be considered to correctly evaluate the size. In general, the absolute value of the displacement is obtained by the square root of the root mean square of r (〈r2〉1/2). Let us return to the three-dimensional problem from here. For a general three-dimensional vector r, 〈r2〉 is calculated as follows:

(1.3)

Here, aiak = 0 (if i ≠ k) since each jump vector is uncorrelated (〈cos θ〉 = 0 because the average value of bond angle is 90°). Given that the polymer chains are isotropic, the polymer chains are considered spheres of diameter aN1/2. In a one-dimensional problem, some people may feel uncomfortable that vectors can overlap each other. Although the overlap is highly reduced in the three-dimensional space, overlap between the monomer units is permitted under this model. This polymer chain is called an ideal chain [2–4]. This concept is analogous to an ideal gas having no volume. Of course, the overlapping of monomers is not allowed in real polymers; this model is incorrect except in special cases. Despite this assumption being unrealistic, it is the foundation for many theoretical models because the end-to-end distance of an ideal chain follows the Gaussian distribution. The Gaussian distribution is a simple and useful statistical model and thus provides physical quantities in simple forms with less difficulty than other methods. Section 1.2 shows that the Gaussian distribution successfully describes the end-to-end distance of an ideal chain.

1.2 Statistics of a Single Polymer Chain

1.2.1 End-to-End Distance of a 1D Random Walk

InSection 1.1.3, the average end-to-end distance of an ideal chain was determined based on the distribution of end-to-end distances. This section considers the probability that an ideal chain has a specific distance of x. Again, let us start with a one-dimensional problem. Assuming that the number of steps the point proceeded in the + direction is N+ and that in the − direction is N− in the previously mentioned one-dimensional problem, the following equations are obtained:

(1.4)

(1.5)

For simplicity, we can assume that the length of a step is unity and estimate the number of situations (W(N, x)) in the case that the point reaches x after N steps. Because sets of N+ and N− for arriving at x are uniquely determined from Eqs. (1.4) and (1.5), W(N, x) is estimated as the number of arrangements of N+ pieces of “+” and N− pieces of “−” (Figure 1.4):

(1.6)

On the other hand, the total number of possible paths in N steps is 2N, which is calculated as the total number of situations that can occur when selecting one of the two choices N times. Thus, the probability of reaching x after N steps is expressed as follows:

(1.7)

Calculating the exact value for all N is a very painful task; however, if we make a proper approximation at a sufficiently large limit of N, this equation leads to a Gaussian distribution. Let us calculate this value following the method of Rubinstein and Colby [4]. First, the natural logarithm is taken of both sides of the equation:

Figure 1.4

Number of situations that reach

x

in

N

steps (

N

 = 10,

x

 = +2).

(1.8)

The last two terms are reduced to the following:

(1.9)

(1.10)

By substituting Eqs. (1.9) and (1.10) into Eq. (1.8), one obtains the following:

(1.11)

The fourth term in Eq. (1.11) can be rewritten as the following:

(1.12)

Here, we apply an important approximation of the relationship between s and N. The maximum value of s is N/2, and the number of situations corresponding to this case is only 1. In most cases, s stays close to the origin (see one-dimensional walks), making it sufficiently smaller than N. Here, by ignoring the case of large s, which is unlikely, and only considering the case where s ≪ N, the expression can be further transformed using a Taylor expansion (ln(1 + y) ≈ y).

(1.13)

UsingEq. (1.13), Eq. (1.11) can be transformed to the following:

(1.14)

Equation (1.14) can be reduced using the following Starling approximation:

(1.15)

(1.16)

As a result, the probability is given by the following:

(1.17)

If we consider x to be a continuous value and this function to be a continuous function, Eq. (1.17) corresponds to a probability density distribution function. To investigate the function, let us integrate it from −∞ to ∞:

(1.18)

Since this calculation corresponds to calculating “the sum of probabilities,” it is natural that the value of the integral is 1. The doubled integral value comes from the procedure of converting discrete x to continuous x. As shown in Table 1.1, in the lattice space, when N is an even number, the probability that x becomes odd is 0. On the other hand, if N is an odd number, the probability that x will be even is 0. Therefore, for any case, as x is changed to 1, 2, 3, …, the probability alternates between a finite value and 0 (Table 1.1). The integral value of 2 comes from simply changing the discontinuous function to a continuous function.

Table 1.1 Number of situations reaching x in N steps.

x

−4

−3

−2

−1

0

1

2

3

4

W

(

N, x

)

N

 = 3

0

1

0

3

0

3

0

1

0

N

 = 4

1

0

4

0

6

0

4

0

1

Figure 1.5

Probability density distribution function of the one-dimensional Gaussian distribution (

P

1D

with

a

 = 1,

N

 = 100).

By standardizing Eq. (1.17) by 2, the probability density function of a one-dimensional random walk (P1D(N, x)) is obtained.

(1.19)

This equation is the same as the Gaussian distribution with an average (〈x〉) of 0 and a variance (〈x2〉) of N (Figure 1.5). The general Gaussian distribution is expressed as follows:

(1.20)

At the end of the one-dimensional problem, let Eq. (1.20) be expanded to an arbitrary step length. When the step length is a, 〈x〉 = 0 and 〈x2〉 = a2N, resulting in the following:

(1.21)

1.2.2 End-to-End Distance of a 3D Random Walk

Let us expand the 1D discussion to three dimensions. In 3D space, the probability that one end is at the origin and the other at r = (rx, ry, rz) is expressed as follows:

(1.22)

By obtaining the root mean square of r from Eq. (1.3) and assuming the spatial isotropy, the following equation is obtained:

(1.23)

Here, we focus on the x-axis component. From Eqs. (1.21) and (1.23), the following equation is obtained:

(1.24)