Molecular Models of Viscoelastic Polymers

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

The Doi-Edwords equation [Eq. (9.20)] is the first attempt at a molecular model for viscoelasticity of entangled polymers. It ignores tube length fluctuation modes that relax some stress on shorter time scales. These modes significantly modify dynamics of entangled polymers, as described in Section 9.4.5. 

The objectives of the present research were (i) to develop a solvent transport model accounting for diffusional and relaxational mechanisms, in addition to effects of the viscoelastic properties of the polymer on the dissolution behavior (ii) to perform a molecular analysis of the polymer chain disentanglement mechanism, and study the influence of various molecular parameters like the reptation diffusion coefficient, the disentanglement rate and die gel layer thickness on the phenomenon and (iii) to experimentally characterize the dissolution phenomenon by measuring the temporal evolution of the various fronts in the problem. 

It appears, then, that the mechanical degradation process is intimately connected with the molecular structure of the macromolecule and the resulting fluid rheology that arises from this structure. For a flexible coil macromolecule, such as HPAM or polyethylene oxide, the polymer solutions are known to display viscoelastic behaviour (see Chapter 3) and thus a liquid relaxation time, may be defined as the time for the fluid to respond to the changing flow field in the porous medium. It may be computed from several possible models (Rouse, 1953 Warner, 1972 Durst et al, 1982 Haas and Durst, 1982 Bird et al. 1987). The finite extendible non-linear elastic (FENE) (Warner, 1972 Bird et al, 1987a Haas and Durst, 1982 Durst et al, 1982) dumbbell model of the polymer molecule may be used to find the relaxation time, tg, as it is known that this model provides a good description of HPAM flow in porous media (Durst et al, 1982 Haas and Durst, 1982) the expression for fe is ... 

Since molecular theories of viscoelasticity are available only to describe the behavior of isolated polymer molecules at infinite dilution, efforts have been made over the years for measurements at progressively lower concentrations and it has been finally possible to extrapolate data to zero concentration. The behavior of linear flexible macromolecules is well described by the Rouse-Zimm theory based on a bead-spring model, except at high frequencies . Effects of branching can be taken into account, at least for starshaped molecules. At low and intermediate frequencies, the molec-... 

With these generally accepted, but not necessarily accurate, conceptual models in hand, major efforts are going into molecular modeling of more complex real behavior. This is the state of the art. Some important areas of current work include nonlinear viscoelasticity, branched polymers, blends of different molecular weights, and chemical composition. E)eep problems remain, such as the definitive explanation of the 3.4 power law for the molecular weight dependence of melt viscosity and proper description of concentrated solution rheology. 

The controversy concerning the universality of the equation is connected, in fact, with the need of choice of a reference temperature that has been desired to be universal. Both Tg and To were proposed as reference temperatures (61). While the choice of Tg is supported by the Doolittle hole model of viscoelastic fiow (62), To actually is arbitrary reference temperature at all. Data processing with the use of WLF equation is widely used for the description of mechanical properties and molecular motion in polymers in the aforementioned temperature region. However, this information should be considered as approximate since it is obtained by extrapolation to the region of low temperatures and based on the analogy between mechanical and dielectric relaxations that are not always controlled by one and the same mechanism. 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

Contents Chain Configuration in Amorphous Polymer Systems. Material Properties of Viscoelastic Liquids. Molecular Models in Polymer Rheology. Experimental Results on Linear Viscoelastic Behavior. Molecular Entan-lement Theories of Linear iscoelastic Behavior. Entanglement in Cross-linked Systems. Non-linear Viscoelastic-Properties. 

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