Kirkwood correlation sphere

We now turn to discussion of the computer dynamical procedure, which shows great promise for calculations of static permittivity. The model of Rahman and Stillinger contains sufficient information in the computer program to enable the mean dipole moment A of a sphere of known radius to be calculated at a known temperature. This moment is related to the Kirkwood correlation parameter g by equations (9) and (10), with g defined as in equation (10). 

As in previous Chapters, for practical use this infinite set (7.1.1) has to be decoupled by the Kirkwood - or any other superposition approximation, which permits to reduce a problem to the study of closed set of densities pm,m with indices (m + mr) 2. As earlier, this results in several equations for macroscopic concentrations and three joint correlation functions, for similar, X (r,t),X-s r,t), and dissimilar defects Y(r,t). However, unlike the kinetics of the concentration decay discussed in previous Chapters, for processes with particle sources direct use of Kirkwood s superposition approximation gives good results for small dimesionless concentration parameters Uy t) = nu(t)vo < 1 only (vq is d-dimensional sphere s volume, r0 is its radius). The accumulation kinetics predicted has a very simple form [30, 31]... 

To avoid computation of the periodic electric field arising from the (in any case artifidal) periodic boundary conditions, they coimt the interactions of each molecule with only those neighbours which lie within a sphere of radius about 9 A. ThQr show how to allow for this in order to estimate the Kirkwood static correlation factor g. . The factor involved is far from unity, being 9 o/[(co + 2)(2cq -b 1)] with Co 75, and it is not evident what (smaller) change will be needed in Nee and Zwanzig s analysis. 

Simple Liquids. The correlation parameters (151) and (152) are accessible to numerical calculation for simple models of radial interaction between the molecules. In a number of cases, evaluations for simple liquids can be usefully made using the Kirkwood model of hard spheres of diameter d and volume v = Trd lS, when... 

For polymer chains a complicating feature is the presence of cross-correlation terms due to the angular correlations of dipole vectors along a chain (1,6,8). The dielectric increment Ae is a time-averaged (equilibrium) property of a system and may be expressed in terms of the fluctuations of the macroscopic dipole moment TJ(t) of a sphere of volume V according to the Kirkwood-Frohlich relation (1). 

Evaluating the mean square moment of a sphere (Eq. (31)), and using the definition of the correlation factor above gives the anisotropic version of the Kirkwood-Froh-lich equation ... 

Tests of the validity of the Kirkwood-Riseman picture, inquiring directly if diffusing objects actually have cross-diffusion tensors that match their supposed hydrodynamic interactions, have recently been accomplished Crocker used videomicroscopy and optical tweezers to study the correlated Brownian motions of a pair of 0.9 xm polystyrene spheres, thereby determining their cross-diffusion ten-sors(3). Crocker found that the diffusion tensors are accurately described by the hydrodynamic interaction tensors, exactly as Kirkwood and Riseman had assumed. An optical trap experiment by Meiners and Quake observed the motions of two Brownian particles, further confirming the validity of the Oseen approximation for hydrodynamic interactions(4). 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

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