Local structure of the networks–cross-linking regions

Most physical gels have complex multiple junctions. In Section 7.4, we studied thermoreversible gelation with junctions of variable multiplicity. In this section, we consider a new method to find the local structure of the networks, i.e., the structure of the network junctions.

Junction multiplicity k was defined by the number of chains connected to a single junction. Simple pairwise cross-links, for example, have multiplicity k = 2, whose sol–gel transition is detailed in Section 7.4 and in the classical literature [1]. For networks with junctions of multiplicity larger than two, the conventional Eldridge–Ferry procedure [2] to find the enthalpy of melting, which plots the logarithm of the gelation concentration ln c* against the inverse temperature, does not work because it assumes pairwise crosslinking.

What is a gel?

Definition of a gel

Gels are three-dimensional networks made up of molecules, polymers, particles, colloids, etc., that are connected with each other by the specific parts on them such as functional groups and associative groups. The connected parts are called cross-links. Gels usually contain many solvent molecules inside their networks, and hence they are close to liquid in composition, but show solid-like mechanical properties due to the existence of the cross-links [1–4].

Although this statement can be adopted as a formal definition of gels, there are many exceptional ones that do not fall neatly into this categorization. For instance, colloidal suspensions exhibit gel-like rheological behavior at high densities. Entanglements of long rigid fibrillar molecules or an assembly of molecules mutually hinder their motion, and lead to gel-like rheology due to jamming of the rigid segments. In such materials, there are no direct cross-links, but geometrical or topological constraints play similar roles to the cross-links, although they are delocalized. Therefore, to define gels by connectivity only is not sufficient to include these materials.

Dimer formation as associated block-copolymers

The first system we study is a mixture of R{A1} and R{B1} chains, each carrying a functional group A or B at one end. Diblock copolymers are formed by the end-to-end association (hetero-dimerization). End groups A and B are assumed to be capable of forming pairwise bonds A·B by thermoreversible hetero-association. The hydrogen bond between acid and base pair is the most important example of this category.

For such mixtures, composite diblock copolymers R{A1}-block-R{B1} with a temporal junction are formed (Figure 6.1). The system is made up of a mixture of diblock copolymers (1,1), and unassociated homopolymers of each species (1,0) and (0,1). It is similar to the mixture of chemically connected diblock copolymers dissolved in their homopolymer counterparts, but its phase behavior is much richer because the population of the block copolymers varies with both temperature and composition.

Networks with temporal junctions

In most polymer blends and solutions of practical interest, the polymer chains carry functional groups that interact with each other by associative forces capable of forming reversible bonds. These forces include hydrogen bonding, ionic association, stereo-complex formation, cross-linking by the crystalline segments, or solvent complexation. Because the bond energy is often comparable to the thermal energy, bond formation is reversible by a change in temperature or concentration.

Polymer science has expanded over the past few decades and shifted its centre of interest to encompass a whole new range of materials and phenomena. Fundamental investigations on the molecular structure of polymeric liquids, gels, various phase transitions, alloys and blends, molecular motion, flow properties, and many other interesting topics, now constitute a significant proportion of the activity of physical and chemical laboratories around the world.

But beneath the luxuriance of macromolecular materials and observable phenomena, there can be found a common basis of concepts, hypotheses, models, and mathematical deductions that are supposed to belong to only few theories.

One of the major problems in polymer physics which remain unsolved is that of calculating the materials properties of self-assembled supramolecules, gels, molecular complexes, etc., in solutions of associating polymers from first principles, utilizing only such fundamental properties as molecular dimensions, their functionality, and intermolecular associative forces (hydrogen bonding, hydrophobic force, electrostatic interaction, etc.).

Theoretical studies of polymer association had not been entirely neglected, but their achievements were fragmentary, phenomenological, and lacked mathematical depth and rigor. What I have tried to do, therefore, is to show how certain physically relevant phenomena derive from the defining characteristics of various simple theoretical model systems.

The goal of this book is thus to present polymer physics as generally as possible, striving to maintain the appropriate balance between theoretical descriptions and their practical applications.

Polymer–surfactant interaction

The problem of the interaction between polymers and surfactants was laid initially in the study of proteins associated with natural lipids, and later extended to their association with synthetic surfactants [1, 2]. More recently, the interaction of water-soluble synthetic polymers such as poly(ethylene oxide) with ionic and non-ionic surfactants [3–7] has attracted the interest of researchers because of its scientific and technological implications.

Adding surfactants to polymer solutions, or vice versa, followed by the formation of a polymer/surfactant complex, can substantially alter the original physical properties substances involved.

Thermodynamics of rubber elasticity

Elastic properties of rubbers and gels are markedly different from those of metals, ceramics, and glasses. The properties of rubbers may be summarized as follows:

Thus, rubbers seem to be peculiar materials. We can, however, understand these unique properties very naturally if we consider that the main cause of the elasticity comes not from the interaction energy of the constituent molecules but from the conformational entropy of the chain segments, which are free to move.

Conformation of polymers

Internal coordinates of a polymer chain and its hindered rotation

The complete set of space coordinates which specifies the conformation of a polymer in three-dimensional space is called its internal coordinates. To study the positions of the carbon atoms along the linear chain of a polymer, let us consider three contiguous atoms -C-C-C- along the chain (gray circles in Figure 1.1). Because they are connected by covalent bonds, the length l of a bond is fixed at l = 0.154 nm, and the angle θ between the successive bonds is fixed at θ = 70.53° (tetrahedral angle with cosθ = 1/3). The bond to the fourth carbon atom, however, can rotate around the axis of the second bond although its length and angle are fixed. Such freedom of rotational motion is called the internal rotation of the polymer chain [1–5].