Hard chain models

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. 

DFT has been extended to SFE in chain molecules by Yethiraj et al. [250] and applied to the freezing of polyethylene. Now that simulation results are available for molecular models of chain molecules, it will be interesting to investigate the performance of DFT for such model systems. The cell theory has recently been applied to the SFE in the hard chain models considered by Malanoski and Monson [62] with results comparable in accuracy to those achieved for hard spheres and hard dumbbells [251]. 

Anotlier model system consists of polymetliylmetliacrylate (PMMA) latex, stabilized in organic solvents by a comb polymer, consisting of a PMMA backbone witli poly-12-hydroxystearic acid (PHSA) chains attached to it [10]. The PHSA chains fonn a steric stabilization layer at tire surface (see section C2.6.4). Such particles can approach tire hard-sphere model very well [111. 

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 density functional theory is at least as accurate for other models of polymers as it is for freely jointed hard chains, but this is not the case with the... 

In order to further experimentally probe the hard-core model of polysilane aggregate chirality, a series of poly-[(alkyl)alkylphenylsilylene]s, differing in phenyl ring-substituent position and chain length were synthesized, designed such that the polymer chain diameter, d, and helical pitch, p, in the hard-core model were varied.343... 

Extension of the Peturbed Hard Chain Correlation (Statistical Mechanical Theory of Fluids)" (2, 5). Extend the PHC program under development to include additional compounds including water. This work is an attempt to combine good correlations for phase equilibrium, enthalpy, entropy, and density into a single model. 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

H. Meirovitch and H. A. Scheraga,/. Chem. Phys., 84, 6369 (1986). Computer Simulation of the Entropy of Continuum Chain Models The Two-Dimensional Freely Jointed Chain of Hard Disks. 

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.

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 freely jointed chain model may also be used with MC simulations. The excluded volume effect is taken into account by putting a hard sphere on... 

The constraint of a collision in a given sequence in our simple chain model means that there is a shock front propagating through the system, a front which reverses its direction every time an end atom collides with the hard walls. When a perfectly ordered crystal hits a hard wall, one can understand how a dispersion-free propagation of a shock wave is possible. The new feature is that such a shock front was seen in full MD simulations of impact heated clusters, using realistic forces, and has been recently studied in more detail. ... 

Jiang, J.W., Blum, L., Bernard, O., and Prausnitz, J.M. Thermodynamic properties and phase equilibria of charged hard sphere chain model for polyelectrolyte solutions. Molecular Physics, 2001, 99, p. 1121-1128. 

In the perturbed hard-chain theory (PHCT) of Beret and Prausnitz [14] and Donohue and Prausnitz [15], the reference fluid is modeled as chains of tangential hard spheres. Since the fluid is stiU composed of hard spheres, albeit bonded, the CS eos is applied with modification to account for the bonding. The hard-sphere-chain equation is... 

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