Molecular theories entanglement models

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

The theories and models of molecular structure and chemical bonding are closely entangled and based on a set of deceptively simple ideas first popularized by the pioneering works of Linus Pauling. The intuitive appeal of these models is such that very little effort has gone into the assessment of their scientific merit and validity although the uncritical handling of chemical concepts related to quantum theory has been clearly documented [168]. A brief critical review follows. 

Molecular Theory of Polymer Viscoelasticity — Entanglement and the Doi-Edwards (Reptation) Model... 

The primary normal stress coefficient at low shear rates according to equation 74 of Chapter 3 is given by i,o = 2i . At high shear rates, relations between SJ i(7) and t] y) are provided by some phenomenological theories " from a qualitative molecular standpoint, it may be inferred that the decrease in with increasing y is related to the drop in rj and hence, in the framework of the entanglement model, to a decrease in the steady-state concentration of entanglements. 

In this work, we discuss briefly some aspects of the molecular theory for diffusion in entangled polymer systems proposed by de Gennes, and we review the studies on diffusion of PDMS chains in melts and in networks, not only considering the influence of free volume or polydispersity effects but also focusing attention on data analysis procedures. Since most of the data available for self-diffusion of PDMS has been obtained using NMR techniques, some considerations of the NMR methods and models used for analysis of the experimental results are included as well. 

Fig. 4. The zero shear viscosity, no as a function of molecular weight, M, of polybutadiene as reported by Colby, Fetters and Grasseley.The data in this plot has not been corrected to the iso-free-volume state. The viscosities are reduced by M in order to remove any pure reptation component. Therefore, any non-zero slope represents a departure from reptation theory. The negative slope of about -2 for the first dozen points corresponds to the Rouse regime (M

In this section, we present the molecular theory for the linear dynamic viscoelasticity of miscible polymer blends by Han and Kim (1989a, 1989b), which is based on the concept of the tube model presented in Chapter 4. Specifically, the reptation of two primitive chains of dissimilar chemical structures under an external potential will be considered, and the expressions for the linear viscoelastic properties of miscible polymer blends will be presented. We will first present the expressions for zero-shear viscosity ob. dynamic storage and loss moduli G co) and G " co), and steady-state compliance J° for binary miscible blends of monodisperse, entangled flexible homopolymers and then consider the effect of polydispersity. There are a few other molecular theories reported... 

As will be shown in Chapter 6 there are molecular theories of polymer behavior that lead to relaxation moduli of the form of Eq. 4.16 with N=oo, where the parameters are precisely specified in terms of a few measurable parameters. Small et al. [4] used these theories to derive a discrete spectrum for a linear, monodisperse, entangled polymer by considering three relaxation processes, each described by a different molecular model. 

Equation (2.61) predicts a 3.5-power dependence of viscosity on molecular weight, amazingly close to the observed 3.4-power dependence. In this respect the model is a success. Unfortunately, there are other mechanical properties of highly entangled molecules in which the agreement between the Bueche theory and experiment are less satisfactory. Since we have not established the basis for these other criteria, we shall not go into specific details. It is informative to recognize that Eq. (2.61) contains many of the same factors as Eq. (2.56), the Debye expression for viscosity, which we symbolize t . If we factor the Bueche expression so as to separate the Debye terms, we obtain... 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

One tool for working toward this objective is molecular mechanics. In this approach, the bonds in a molecule are treated as classical objects, with continuous interaction potentials (sometimes called force fields) that can be developed empirically or calculated by quantum theory. This is a powerful method that allows the application of predictive theory to much larger systems if sufficiently accurate and robust force fields can be developed. Predicting the structures of proteins and polymers is an important objective, but at present this often requires prohibitively large calculations. Molecular mechanics with classical interaction potentials has been the principal tool in the development of molecular models of polymer dynamics. The ability to model isolated polymer molecules (in dilute solution) is well developed, but fundamental molecular mechanics models of dense systems of entangled polymers remains an important goal. 

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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