Hard core interaction

Lee Y S, Chae D G, Ree T and Ree F H 1981 Computer simulations of a continuum system of molecules with a hard-core interaction in the grand canonical ensemble J. Chem. Phys. 74 6881-7... 

In order to calculate the phase boundary concentrations for stiff polyelectrolyte solutions, we express the total intermolecular interaction u for the polyion as the sum of the hard-core interaction u0 and the electrostatic interaction wel, and assume Eq. (1) for u0 and the following for wd ... 

In the case of hard core interactions the treatment of the problem must be different. Again, finite difference equations are employed and the configuration is allowed to evolve until a collision takes place. The consequences of the collision are predicted according to the laws of classical mechanics and, again, the configuration is allowed to evolve until the next collision takes place. 

For hard-core interactions this variance is p — p)/m, with p familiar as the variance for the case of Bernoulli sampling that applies with hard-core insertions. How should the sample size be adjusted when the thermodynamic state, and hence p, is adjusted The interesting circumstance is when p is small. Intuitively, we expect the sample size must be larger than m 1 /p for credible results. 

The strategy for our derivation will be to insert this resolution of unity, Eq. (7.13), within the averaging brackets of the potential distribution theorem, then expand and order the contributions according to the number of factors of b (j) that appear. We emphasize that physical interactions are not addressed here and that the hard-core interactions associated with discontinuity in / (j) appear for counting purposes only. 

Ben-Naim et al. (1989a) provided a theoretical framework for separating the solvation thermodynamics into their several components (1) hard-core interactions, which depend on the volume of the solute and the cost of making a cavity in the solvent and (2) interactions of the... 

The potentials (7-1), (7-2), and (7-4a), when combined, form the basis of the celebrated DLVO (Derjaguin and Landau, 1941 Verwey and Overbeek, 1948) theory of colloid stability. This theory is useful in predicting the conditions of surface potential, ionic strength, and so on, under which flocculation will occur. But the theory has important limitations, in part because it only considers van der Waals, electrostatic, and hard-core interactions. 

This is very close to the HNC closure for hard core interactions, indeed the argument in the logarithm is the multiplicative mean of the two correlation functions g+ + and g+. If the multiplicative mean is replaced by the arithmetic mean, then this closure relation is the HNC approximation. Equation (5.1.2) can then be written as... 

Even at low density the nonelectrostatic contribution to the osmotic coefficient is larger than 1, particularly for the trivalent system. This must originate from the strong repulsive hard-core interactions between condensed ions and the rod, since at those densities interionic repulsions can no longer play a role. This strong repulsion is compensated by the electrostatic attraction of the condensed ions. However, since the latter also acts upon the ions that do not touch the rod, the total pressure drops below the ideal gas contribution. 

Although one might question the applicability of this kind of perturbation theory to hard-core interactions (where the potentials become infinite), we note here that the term gd(r) [V(r) — Vj(r)] never diverges since (i) do < d so divergences due to the actual hard core in V are subtracted off by the hard core of the effective potential, Vj, and (ii) the pair-correlation function, gd vanishes when r < d. 

A) A hard-core interaction with an exclusion radius a. 

For the formation of a lyotropic phase in solutions, the critical parameter is the ratio of Ip/D (the aspect ratio ). Predictions for semi-flexible cylinders with hard-core interaction have revealed that the ratio Ip/D should be of the order of 10 to introduce a lyotropic behavior at reasonable concentrations [151]. The driving force for ordering phenomena in the bmsh solutions is the excluded volume interactions. In contrast to flexible cyhnders with hard-core interactions, the CPBs - at least their side chains - begin to interpenetrate when the threshold concentration (which can be rather low for brushes with long side chains) is exceeded. With a further increase in the concentration above the threshold, the excluded volume interactions will gradually diminish, such that the lyotropic behavior would be expected to disappear again at somewhat higher concentrations [152]. 

We now proceed to reduce (292) for the case of hard-core interaction. The dynamical description of a collision between two hard spheres of diameter cr is depicted in Fig. 5. The initial relative separation and momentum are r and p. It is evident that there will be no collision unless r p < 0 and where b is the... 

However, the Df — 2 random walk fails in one important respect to produce a correct model of the coil formed by a linear polymer soaking in a good solvent (see Chap. 7 on polymers) there is nothing to stop the walker from coming back through a previously visited site. In a non-crosslinked molecule whose constituent monomers repel each other by hard core interactions, this would clearly be impossible. The structure of such a molecule could only be correctly described by a self-avoiding random walk, rather than the simple... 

In the EP theory one regards fluctuation of the composition but still invokes a mean-field approximation for fluctuations of the total density. This approach is accurate for dense mixtures of long molecules, because composition fluctuations decouple from fluctuations of the density. The energy per monomer due to composition fluctuations is typically on the order of UbT/N, and it is therefore much smaller than the energy of repulsive interactions (on the order of 1 8 ) in the polymer fluid that give rise to the incompressibility constraint. If there were a couphng between composition and density fluctuations, a better description of the repulsive hard-core interactions in the compressible mixture woidd be required in the first place. 

If go(r), g CrX and g (r) are known exactly, then all three routes should yield the same pressure. Since liquid state integral equation theories are approximate descriptions of pair correlation functions, and not of the effective Hamiltonian or partition function, it is well known that they are thermodynamically inconsistent [5]. This is understandable since each route is sensitive to different parts of the radial distribution function. In particular, g(r) in polymer fluids is controlled at large distance by the correlation hole which scales with the radius of gyration or /N. Thus it is perhaps surprising that the hard core equation-of-state computed from PRISM theory was recently found by Yethiraj et aL [38,39] to become more thermodynamically inconsistent as N increases from the diatomic to polyethylene. The uncertainty in the pressure is manifested in Fig. 7 where the insert shows the equation-of-state of polyethylene computed [38] from PRISM theory for hard core interactions between sites. In this calculation, the hard core diameter d was fixed at 3.90 A in order to maintain agreement with the experimental structure factor in Fig. 5. 

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