Hypemetted chain theory

FIG. 17 Distribution functions P(r) for the systems with Manning parameter 4.2 and 10.5 from Figure 16. The solid lines are the results from the molecular dynamics simulations while the dashed lines are the predictions from hypemetted chain theory. (See Ref. 36.)... 

Eq. (5) is useful when analyzing different approximations in the theory of inhomogeneous fluids. In particular, if all the terms involving third- and higher-order correlations in the right-hand side of Eq. (5) are neglected, and if Pi(ro))P2( o)i )Pv( o) are chosen as the densities of species for a uniform system at temperature T and the chemical potentials p,, the singlet hypemetted chain equation (HNCl) [50] results... 

Having at our disposal accurate structural and thermodynamic quantities for HS fluid, the latter has been naturally considered as a RF. Although real molecules are not hard spheres, mapping their properties onto those of an equivalent HS fluid is a desirable goal and a standard procedure in the liquid-state theory, which is known as the modified hypemetted chain (MHNC) approximation. According to Rosenfeld and Ashcroft [27], it is possible to postulate that the bridge function of the actual system of density p reads... 

Indeed, f is found first to increase with t, but from a certain value on it decreases and would finally even become negative. Note that this would reverse the drift direction in electrophoresis measurements. While nonlinear and linearized PB theory coincide with the data and with each other for small Manning parameter, they completely fail to predict the back-bending, which already sets in at comparatively small values of t Hypemetted chain... 

A fundamental approach to liquids is provided by the integral equation methods (sometimes called distribution function methods), initiated by Kirkwood and Yvon in the 1930s. As we shall show below, one starts by writing down an exact equation for the molecular distribution function of interest, usually the pair function, and then introduces one or more approximations to solve the problem. These approximations are often motivated by considerations of mathematical simplicity, so that their validity depends on a posteriori agreement with computer simulation or experiment. The theories in question, called YBG (Yvon-Bom-Green), PY (Percus-Yevick), and the HNC (hypemetted chain) approximation, provide the distribution functions directly, and are thus applicable to a wide variety of properties. 

Computer simulations are not the only methods which can be used to calculate the dielectric constant of pure liquids. Other approaches are given by the use of integral equations, in particular, the hypemetted chain (HNC) molecular integral equation and the molecular Omstein-Zemike (OZ) theory (see Section 8.7.1 for details on such methodologies). 

There have been other attempts to apply integral equations methods derived from Equation 3.51 or other similar expressions. One of them is the hypemetted chain (Henderson 1983), which is a generalization of the MSA theory, applicable to higher charge/potential values. It gives results comparable to the MPB ones but requires more extensive numerical evaluations. Another proposal is so-called dressed-ion theory of Kjellander and Mitchell (1994, 1997). 

In the last two decades, new extended laws have been obtained for the concentration dependence of transport properties. It was possible [ 15,16] to use the Fuoss-Onsager theory together with new, more accurate equilibrium pair distribution functions as obtained with the help of the hypemetted chain (HNC) or mean spherical approximation (MSA). 

FH = Flory-Huggins GF = generalized Flory GFD = generalized Flory dimer HNC = hypemetted chain HTA = high temperature approximation IFJC = ideal freely joined chain ISM = interaction site model LCT = lattice cluster theory MS = Martynov-Sarkisov PMMA = polymethyl methacrylate PRISM = polymer reference interaction site model PVME = polyvinylmethylether PS = polystyrene PY = Percus-Yevick RMMSA = reference molecule mean spherical approximation RMPY = reference molecular Percus-Yevick SANS = small angle neutron scattering SFC = semiflexible chain TPT = thermodynamic perturbation theory. 

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