Hard sphere chains

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 quantity of primary interest in the study of nonuniform fluids is the density profile of the fluid at a surface. Dickman and Hall [28] reported the density profiles of freely jointed hard-sphere chains at hard walls. Their focus was on the equation of state of melts of hard-chain polymers, and they performed simulations of polymers at hard walls because the bulk pressure, P, can be calculated from the value of the density profile at the surface using the wall sum rule ... 

Figure 4. Density profiles of fused-hard-sphere chains at a hard wall for (a) A (b) N = 16. Also shown are predictions of the density functional theory of Yethiraj [39]-...

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. 

Yethiraj and Hall [94] studied the density profiles, surface forces, and partition coefficient of freely jointed tangent hard-sphere chains between hard walls. The theory was able to capture the depletion of chain sites at the surface at low densities and the enhancement of chain sites at the surface at high densities. This theory is in qualitative agreement with simulations for the density profiles and partitioning of 4 and 20 bead chains, although several quantitative deficiencies are present. At low densities the theory overestimates the value of the density profile near the surface. Furthermore, it predicts a quadratic variation of density with distance near the surface, whereas in reality the density profile should be linear in distance, for long chains. At high densities the theory underestimates the value of the density near the surface. The theory is quite accurate, however, for the partition coefficient for hard chains in slit-like pores. 

The density functional theories are also accurate for the density profiles of fused-sphere chains. Figures 4(a) and 4(b) compare the theory of Yethiraj [39] (which is a DFT with the Curtin-Ashcroft weighting function) to Monte Carlo simulations of fused-hard-sphere chains at hard walls for N = 4 and 16, respectively. For both chain lengths the theory is in quantitative agreement with the simulation results and appears to get more accurate as the chain length is increased. Similarly good results were also found by SCMC who compared... 

Fig. 32. Theoretically predicted molecular-weight dependence of the mean-square radius of gyration for star-molecules with rays grafted onto a large nucleus. Full line nucleus is a hard sphere chain curve ABC nucleus dotted line A3 nucleus1145...

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... 

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. 

Kalyuzhnyi, Yu.V., and Cummings, P.T. Multicomponent mixture of charged hard-sphere chain molecules in the polymer mean-spherical approximation. Journal of Chemical Physics, 2001, 115, p. 540-551. 

Jog, P.K. and Chapman, W.G., Application of Wertheim s thermodynamic perturbation theory to dipolar hard sphere chains, Mol. Phys., 97(3), 307-319, 1999. 

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... 

The PHCT pressure is the sum of the hard-sphere-chain pressure of Equation (4.172) and the attractive pressure of the chain of spheres of Equation (4.179),... 

Shimizu S, Ikeguchi M, Nakamura S, Shimizu K. Size dependence of transfer free energies a hard-sphere-chain-based formalism. J Chem Phys 1999 110 2971-2982. 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

FEN Feng, W., Wen, H., Xu, Z., and Wang, W., Comparison of perturbed hard-sphere-chain theory with statistical associating fluid theory for square-well fluids, Ind. Eng. Chem. Res., 39, 2559, 2000. 

The perturbed-hard-sphere-chain (PHSC) equation of state is a hard-sphere-chain theory that is somewhat different to SAFT. It is based on a hard-sphere chain reference system and a van der Waals-type perturbation term using a temperature-dependent attractive parameter a(T) and a temperature-dependent co-volume parameter b(T). Song et ap-... 

The best-known model of this kind is the Statistical Associated Fluid Theory (SAFT) model [58-61]. Here, a non-spherical molecule (solvent or polymer) is assumed to be a chain of identical spherical segments. Starting from a reference system of m hard spheres (A ), this model considers three perturbation contributions, which are assumed to effect independently attractive interactions of the (non-bonded) segments (A ), hard-sphere chain formation (A ), and association (A ° ) ... 

Subsequently, various perturbation theories were developed which are also based on Eq. (3) but differ in the use of specific expressions for the different types of perturbations. Examples are the Perturbed Hard-Sphere-Chain Theory (PHSC) [64], as well as the models proposed by Chang and Sandler [65], Gil-ViUegas et al. [66], and Hino and Prausnitz [67]. 

The above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

FIG. 6 Simulated adsorption isotherms for hard-sphere chains of 4, 16, and 100 sites in a slit-like pore. Reference refers to [24]. 

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