Integral equation approximation

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Integral equation approximations for the distribution functions of simple atomic fluids are discussed in the following. 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Watts R C 1972 Integral equation approximations in the theory of fluids Specialist Periodical Report vol 1 (London Chemical... 

Lupkowski and Monson have also discussed the connection between the approximations described above and the integral equation approximations based upon the CSL equation described in Section III.B. They show that the ISF-ORPA is the same as a reference mean spherical approximation (RMSA) to the CSL integral equation, following the work of Madden " in connection with the corresponding approximations for simple fluids. This relationship provides the basis of the computational methods that have been used so far. The approach has been applied to the calculation of the influence... 

D — and D oo the smooth and generic dimension-dependence of the integrals enables one to interpolate reasonably accurate D = 3 values (rms error 1%) from the dimensional limit results. The interpolated integrals can be used either on their ovm, or in conjunction with an integral equation approximation which sums some subset of the required integrals exactly (such as the hypemetted-chain or Percus-Yevick methods) the combination methods are invariably better than either dimensional interpolation or integral equations alone. Interpolation-corrected Percus-Yevick values can be computed quite easily at arbitrary order however, errors in higher-order values are... 

As an example, consider B4 for hard spheres. Using the combinatorial factors and numerical integrals in Table 2, the virial coefficient and the two integral equation approximations to it are... 

Two of the classic integral equation approximations for atomic liquids are the PY (Percus-Yevick) and the HNC (hypemetted chain) approximations that use the following closures... 

Cummings, P. T. (1994) Introduction to Integral Equation Approximations with Applications to Near-Critical and Supercritical Fluids. In Supercritical Fluids. Fundamentals for Application, E. Kiran and J. M. H. Levelt Sengers, Ed. Kluwer Academic Publishers Dordrecht, Vol. E 273 pp 287-312. 

In the present paper the fracture is treated as a real crack with infinite small distance between sides. For the solution of the elasticity problem with the cavity and the fracture, the modification of the Dual BEM with discontinuous elements is built [4]. It is the most optimal method with regard to the computational costs and the convenience of the integral equations approximation. Near the crack front special elements are used. They account the singularity of the elasticity problem solution. To improve the accuracy of the Stress Intensity Factors calculation, the special boundary elements near the crack front are accounted in the interpolation formulae. 

THE DIRECT-CORRELATION-FUNCTION AND INTEGRAL-EQUATION APPROXIMATIONS... 

Two Integral-equation approximations, which are useful in electrolyte theory, are the following ... 

The solutions to the integral equation approximations for electrolytes are discussed in the section entitled SOME THEORIES OF ELECTROLYTES. The numerical solutions to the HNC and PY approximations are usually obtained with Fast Fourier transform routines while the... 

The convergence of the Mayer expansion and the Stell-Lebowitz expansions for the free energy is slow, and accurate estimates of the thermodynamic properties for a model electrolyte at concentrations near 1 M are difficult to obtain. A way out of this difficulty is to consider approximations for the radial distribution functions which correspond to the summation of a certain class of terms which contribute to all of the virial coefficients. The integral-equation approximations, such as the HNC, PY, and MS approximations, attempt to do just this. They also provide information on the structure of the solutions to varying degrees... 

The solutions to the HNC approximation discussed in the section on DIRECT CORRELATION FUNCTIONS AND INTEGRAL EQUATION APPROXIMATIONS are the same as those of its analog when applied to the same ionic system, but the... 

The results were analysed using HNC calculations described in another chapter of this book. The ion-ion correlations in the electrolyte and the ionic profiles in the vicinity of the water-air interface were calculated within the HNC integral equation approximation at the Primitive Model level of description (ionic spheres immersed in a continuous dielectric solvent). The (solvent-averaged) ion-ion interaction potential y(r) is the sum of a hard-sphere contribution (radii ), a generic Coulombic Contribution ZiZje / 47T oer) (valency Z, dielectric constant e = 78) and a specific dispersion contribution. ... 

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