Tangent hard sphere chain model,

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

Thus, we first consider Eq. (8.10) for hard-core chain models, specifically tangent hard-sphere chain models (Dickman and Hall (1986) Yethiraj and Hall, 1991). Models and theories of the packing problems associated with hard-core molecules have been treated in Sections 4.3, 6.1, 7.5, and 7.6. We recall... 

The focus of this chapter is on an intermediate class of models, a picture of which is shown in Fig. 1. The polymer molecule is a string of beads that interact via simple site-site interaction potentials. The simplest model is the freely jointed hard-sphere chain model where each molecule consists of a pearl necklace of tangent hard spheres of diameter a. There are no additional bending or torsional potentials. The next level of complexity is when a stiffness is introduced that is a function of the bond angle. In the semiflexible chain model, each molecule consists of a string of hard spheres with an additional bending potential, EB = kBTe( 1 + cos 0), where kB is Boltzmann s constant, T is... 

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

As a first approximation " one can model the vinyl polymer as a freely jointed, tangent hard sphere chain as depicted on the second line of Figure 1. Thus each bond (of fixed length) is completely flexible with each site, including the side group site C, acting as a universal joint. Invoking the Flory ideality hypothesis, the intramolecular structure functions in Eq. (2.4) become "... 

Figure 8.1 The process of computing the incremental chemical potential involves adding one extra segment to an M - 1 segment chain moving in the solvent. The tangent hard sphere model of a (M — l)-mer (M = 5) is shown here. The dashed circles enclose the volume excluded to the centers of the solvent spheres.

Equation of State. A number of modeling studies of amorphous polymers have included PVT behavior, either as an end in itself, as a means to validate the potential ftinctions, or to investigate other aspects of polsrmer behavior (411). Polymer melts and glasses that have been studied in some detail include polyethylene (23,370,412), poly(dimethylsiloxane) (413), poly(ethylene oxide) (414), and poly(ethylene terephthalate) (78). Dendrimers having a variety of generations have been compared with linear chains in the respect of their equation of state by means of a tangent hard sphere model by Lue (415). 

The canonical model for hard chains has been the tangent-hard-sphere freely jointed chain model, and several equations of state are available for fluids composed of these molecules. There are two categories of hard chain equations of state (i) those based on the mean field ideas of Flory and Huggins, and (ii) those based on the theory of associating fluids. ... 

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