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This chapter starts off with the classification of solids, then discusses the various types of bonding or interaction in solids, including ionic bonding, metallic bonding, covalent bonding (band theory), and van der Waals interaction. Numerous examples are given for each type of bonding.

Realistic molecular dynamics and self-assembly is represented in a lattice simulation where water, water-hydrocarbons, and water-amphiphilic systems are investigated. The details of the phase separation dynamics and the constructive self-assembly dynamics are discussed and compared to the corresponding experimental systems. The method used to represent the different molecular types can easily be expended to include additional molecules and thus allow the assembly of more complex structures. This molecular dynamics (MD) lattice gas fills a modeling gap between traditional MD and lattice gas methods. Both molecular objects and force fields are represented by propagating information particles and all microscopic interactions are reversible. Living systems, perhaps the ultimate constructive dynamical systems, is the motivation for this work and our focus is a study of the dynamics of molecular self-assembly and self-organization. In living systems, matter is organized such that it spontaneously constructs intricate functionalities at all levels from the molecules up to the organism and beyond. At the lower levels of description, chemical reactions, molecular selfassembly and self-organization are the drivers of this complexity. We shall, in this chapter, demonstrate how molecular self-assembly and selforganization processes can be represented in formal systems. The formal systems are to be denned as a special kind of lattice gas and they are in a form where an obvious correspondence exists between the observables in the lattice gases and the experimentally observed properties in the molecular self-assembly systems. This has the clear advantage that by using these formal systems, theory, simulation, and experiment can be conducted in concert and can mutually support each other. However, a disadvantage also exists because analytical results are difficult to obtain for these formal systems due to their inherent complexity dictated by their necessary realism. The key to novelt simpler molecules (from lower levels), dynamical hierarchies are formed [2, 3]. Dynamical hierarchies are characterized by distinct observable functionalities at multiple levels of description. Since these higher-order structures are generated spontaneously due to the physico-chemical properties of their building blocks, complexity can come for free in molecular self-assembly systems. Through such processes, matter apparently can program itself into structures that constitute living systems [11, 27, 30].

The classical theories of rubber elasticity presented in chapter 2 are based on a hypothetical chain which may pass freely through its neighbors as well as through itself. In a real chain, however, the volume of a segment is excluded to other segments belonging either to the same chain or to others in the network. Consequently, the uncrossability of chain contours by those occupying the same volume becomes an important factor. This chapter and the following one describe theoretical models treating departures from phantom-like behavior arising from the effect of entanglements, which result from this uncrossability of network chains. The chains in the un-cross-linked bulk polymer are highly entangled. These entanglements are permanently fixed once the chains are joined during formation of the network. The degree of entanglement, or degree of interpenetration, in a network is proportional to the number of chains sharing the volume occupied by a given chain. This is quite important, since the observed differences between experimental results on real networks and predictions of the phantom network theory may frequently be attributed to the effects of entanglements. The decrease in network modulus with increasing tensile strain or swelling is the best-known effect arising from deformation-dependent contributions from entanglements. The constrained-junction model presented in this chapter and the slip-link model presented in chapter 4 are both based on the postulate that, upon stretching, the space available to a chain along the direction of stretch is increased, thus resulting in an increase in the freedom of the chain to fluctuate. Similarly, swelling with a suitable diluent separates the chains from one another, decreasing their correlations with neighboring chains. Experimental data presented in figure 3.1 show that the modulus of a network does indeed decrease with both swelling and elongation, finally becoming independent of deformation, as should be the case for the modulus of a phantom network. Rigorous derivation of the modulus of a network from the elastic free energy for this case will be given in chapter 5. The starting point of the constrained-junction model presented in this chapter is the elastic free energy.

The classical theories of rubber elasticity are based on the Gaussian chain model. The only molecular parameter that enters these theories is the mean-square end-to-end separation of the chains constituting the network. However, there are various areas of interest that require characterization of molecular quantities beyond the Gaussian description. Examples are segmental orientation, birefringence, rotational isomerization, and finite extensibility, and we will address these properties in the following chapters. One often needs a more realistic distribution function for the end-to-end vector, as well as for averages of the products of several vectorial quantities, as will be evident in these chapters. The foundations for such characterizations, and several examples of their applications, are given in this chapter. Several aspects of rubber elasticity (such as the dependence of the elastic free energy on network topology, number of effective junctions, and contributions from entanglements) are successfully explained by theories based on the freely jointed chain and the Gaussian approximation. Details of the real chemical structure are not required at the length scales describing these phenomena. On the other hand, studies of birefringence, thermoelasticity, rotational isomerization upon stretching, strain dichroism, local segmental orientation and mobility, and characterization of networks with short chains require the use of more realistic network chain models. In this section, properties of rotational isomeric state models for the chains are discussed. The notation is based largely on the Flory book, Statistical Mechanics of Chain Molecules. More recent information is readily found in the literature. Due to the simplicity of its structure, a polyethylene-like chain serves as a convenient model for discussing the statistical properties of real chains. This simplicity can be seen in figure 8.1, which shows the planar form of a small portion of a polyethylene chain. Bond lengths and bond angles may be regarded as fixed in the study of rubber elasticity because their rapid fluctuations are usually in the range of only ±0.05 A and ±5°, respectively. The chain changes its configuration only through torsional rotations about the backbone bonds, shown, for example, by the angle for the ith bond in figure 8.1.

This chapter is a brief overview of the topics treated in the book. It is aimed, in particular, at providing some qualitative information on rubber elasticity theories and their relationships to experimental studies, and at putting this material into context. The following chapter describes in detail the classical theories of rubber elasticity, that is, the phantom and affine network theories. The network chains in the phantom model are assumed not to experience the effects of the surrounding chains and entanglements, and thus to move as “phantoms.” Although this seems to be a very severe approximation, many experimental results are not in startling disagreement with theories based on this highly idealized assumption. These theories associate the total Helmholtz free energy of a deformed network with the sum of the free energies of the individual chains—an important assumption adopted throughout the book. They treat the single chain in its maximum simplicity, as a Gaussian chain, which is a type of “structureless” chain (where the only chemical constitution specified is the number of bonds in the network chain). In this respect, the classical theories focus on ideal networks and, in fact, are also referred to as “kinetic” theories because of their resemblance to ideal gas theories. Chain flexibility and mobility are the essential features of these models, according to which the network chains can experience all possible conformations or spatial arrangements subject to the network’s connectivity. One of the predictions of the classical theories is that the elastic modulus of the network is independent of strain. This results from the assumption that only the entropy at the chain level contributes to the Helmholtz free energy. Experimental evidence, on the other hand, indicates that the modulus decreases significantly with increasing tension or compression, implicating interchain interactions, such as entanglements of some type or other. This has led to the more modern theories of rubber elasticity, such as the constrained-junction or the slip-link theories, which go beyond the single-chain length scale and introduce additional entropy to the Helmholtz free energy at the subchain level.