Equilibrium properties of polymers

It seems possible to rule out finite extensibility as an explanation of shear rate dependence in the viscosity, based simply on the equilibrium properties of polymer chains and the parallel between t] and t] in their departures from >/0. Experimentally, the mean square end-to-end vector obeys Gaussian statistics in 0-solvents spring constant K in FENE dumbbells is chosen to match this condition, then K - 3kT/(r2. The parameter b is therefore given by... 

The Gaussian subchain model and its possible generalisations allows one to calculate, in a coarse-grained approximation, the different characteristics of a macromolecule and systems of macromolecules, playing a fundamental role in the theory of equilibrium and non-equilibrium properties of polymers. The model does not describe the local structure of the macromolecule in detail, but describes correctly the properties on a large-length scale. 

The PRISM approach for modeling homopolymer melts, based on the pioneering work of Curro and Schweizer,ii -i 9 g continuous space, liquid state methodology suited for the study of equilibrium properties of polymer chains. The technique is based on integral equation methods that have been generalized to deal with macromolecules. 

We have discussed only the equilibrium properties of polymers. Of course, in many real systems, the time scales for equilibriation can be very large. It is thus of interest to study non-equilibrium properties of statistical mechanical systems on fractals. A simple prototype is the study of kinetic Ising model on fractals. Closer to our interests here, one can study, say, the reptation motion of a polymer on the fractal substrate. This seems to be a rather good first model of motion of a polymer in gels. 

The work described in this chapter is concerned with static (equilibrium) properties of polymers in random media. There is a lot of theoretical work stiU to be done related to the dynamics, and especially nonequilibrium properties of polymers in random media. This is also of practical importance, for example for the separation of chain of different length or mass like DNA molecules under the effect of an applied force when embedded in a random medium like a gel [38]. 

The second virial coefficient of the macromolecular coil B T) depends not only on temperature but on the nature of the solvent. If one can find a solvent such that B T) =0 at a given temperature, then the solvent is called the 6-solvent. In such solvents, roughly speaking, the dimensions of the macromolecular coil are equal to those of an ideal macromolecular coil, that is the coil without particle interactions, so that relations of Sects. 2.1 - 2.3 can be applied in this case. However, it is a simplified description of the phenomenon. The fuller review of the theory of equilibrium properties of polymer solutions can be found in the monographs by des Cloizeaux and Jannink [29] and by Gross-berg and Khokhlov [27]. 

This chapter gives a concise overview of the equilibrium properties of polymer solutions and of frictional behavior, insofar as it elucidates molecular conformational properties. Significant results that bear upon the coherent picture we try to present are stated, but many details of derivations have had to be omitted. In view of the subject matter of Chapter 1 of this volume, scaling and renormalization ideas are treated much less fully than would otherwise be appropriate. The discussion of experimental work is in no way comprehensive the studies selected for consideration are intended as representative of correlations among data and of data with theory. 

The mechanical properties of polymers are of interest in all applications where they are used as structural materials. The analysis of the mechanical behavior involves the deformation of a material under the influence of applied forces, and the most important and characteristic mechanical property is the modulus. A modulus is the ratio between the applied stress and the corresponding deformation, the nature of the modulus depending on that of the deformation. Polymers are viscoelastic materials and the high frequencies of most adiabatic techniques do not allow equilibrium to be reached in viscoelastic materials. Therefore, values of moduli obtained by different techniques do not always agree in the literature. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

The equilibrium (relaxed) elastic properties of polymers in the rubbery state display two very important features ... 

The model polymer chain in an array of obstacles (PCAO) (see Fig. 4) combines the geometrical clarity of its image with the possibility to investigate the influence of entanglements on equilibrium and dynamic properties of polymers quite precisely. 

In 1981 Bruns and Bansal [94] used a Lennard-Jones (LJ) model of polymer and solvent and analyzed the structural properties of the chain. In contrast to previous reports [92,93], a significant solvent effect was observed by Bruns and Bansal [94], The study of solvent effect on the static properties of polymer was succeeded by Khalatur et al. [95] on the static properties of a 16-bead polymer chain in a solvent. All potentials used were of LJ type. There are many other reports in the literature to understand the equilibrium size and shape of polymer as a function of solvent quality [96,97,98], Most of these studies are exploratory rather than quantitative, probably due to the computational expense, since the large relaxation times of the polymer chains as well as the large system sizes imply powerful computer resources. However, with the availability of high-performance computers, the problem has been addressed in earnest [99],... 

Equilibrium properties of the collapse transition and the relationship between polymer size and solvent quality are studied widely, and are addressed in the previous section, but up to now less is known about the collapse dynamics. There has been a... 

The static and dynamic properties of polymer-layered silicate nanocomposites are discussed, in the context of polymers in confined spaces and polymer brushes. A wide range of experimental techniques as applied to these systems are reviewed, and the salient results from these are compared with a mean field thermodynamic model and non-equilibrium molecular dynamics simulations. 

The problem of the relationship between the equilibrium and the kinetic rigidity of the chain is of paramount importance since both the behavior of a chain molecule in solution and the main properties of polymer materials are related to these molecular characteristics. 

The material considered in the preceding sections shows that the FB and EB methods can be successfully applied to the solution of these problems providing information not only on the equilibrium but also on the kinetic properties of polymer chains. This is particularly valuable in those cases in which the application of other more common methods is difficult or even impossible (e.g. sedimentation measurements of polymer solutions in sulfuric acid). 

Just as the equilibrium conformational properties of macromolecules, the theory of which has been developed in well-known classical works by Kuhn, Flory, Volken-stein and others the kinetic properties of polymer chains can be determined by two main mechanisms of intramolecular mobility. First, it is the discrete rotational isomeric (rotameric) mechanism of mobility caused by the jump of small-chain segments (kinetic units) from certain energically stable allowed conformers into others is4-i6S) gg ond it is the continuous mechanism of motion deter-... 

B. Lindman and G. Karlstrom, Polymer-surfactant systems, in D.M. Bloor and E. Wyn-Jones (Eds.), The Structure, Dynamics and Equilibrium properties of Colloidal Systems. NATO ASI Series C 234,1989, pp. 131-147. 

This work reviews experimental results on the equilibrium properties of interfaces created by polymer mixtures confined in thin films. It confronts experimental data with theoretical expectations based mainly on mean field models. Some of these theoretical descriptions have been surveyed recently by Binder [6,7]. 

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