Sum rules for the pair-correlation functions of inhomogeneous fluids: results for the hard-sphere–hard-wall system

1987 ◽  
Vol 410 (1839) ◽  
pp. 409-420 ◽  
Keyword(s):  
Sum Rules ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Distribution Functions ◽  
Hard Wall ◽  
Chain Equation ◽  
Hypernetted Chain ◽  
Chain Approximation ◽  
Derivatives Of

Starting from well-known relations for the derivatives of the radial distribution functions of a mixture of fluids, and allowing the diameter of one particle to become exceedingly large, three sum rules for a fluid with density inhomogeneities are obtained. None of these sum rules are new. However, the relation between the Lovett–Mou–Buff–Wertheim and the Born–Green hierarchy of equations seems not well known. The accuracy of a recent parametrization of the pair correlation of hard spheres near a hard wall and of the solutions of the Percus–Yevick and hypernetted-chain equation for this same function are examined by determination of how well these functions satisfy these sum rules and the accuracy of their surface tension, calculated from the sum rule of Triezenberg and Zwanzig. Generally speaking, the Percus–Yevick theory gives the best results and the hypernetted-chain approximation gives the worst results with the parametrization being intermediate.

Pair correlation function in a fluid with density inhomogeneities: results of the Percus‒Yevick and hypernetted chain approximations for hard spheres near a hard wall

1986 ◽  
Vol 404 (1827) ◽  
pp. 323-337 ◽  
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Bulk Fluid ◽  
Pair Correlations ◽  
Hypernetted Chain ◽  
Density Inhomogeneities ◽  
Bulk Case

Solutions for the pair correlation function and density profile of a system of hard spheres near a hard wall are obtained by using the Percus‒Yevick and hypernetted chain approximations, generalized for inhomogeneous fluids. The Percus‒Yevick (PY) results are similar in accuracy to those obtained for the bulk fluid. The PY pair correlation function is generally too small near contact but quite good overall. The hypernetted chain (h. n. c.) results are difficult to obtain numerically and are poorer than in the bulk. Often the h. n. c. pair correlations are too small at contact, in contrast to the bulk case where they are too large, although there are configurations where the contact values of the pair correlation function are too large. Nearly always the error in the h. n. c. results is much worse than is the case for the bulk. Both approximations are qualitatively satisfactory in that they predict the correct asymmetries between the values of the pair correlation functions for pairs of hard spheres whose line of centres is parallel or normal to the surface of the wall.

Radial Distribution Function in the Hard Sphere Mixtures

2008 ◽  
Vol 73 (3) ◽  
pp. 388-400 ◽  
Author(s):  
Tomáš Boublík
Keyword(s):  
Aspect Ratio ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Heterogeneous Systems ◽  
Distribution Functions ◽  
Ternary Mixtures ◽  
Hard Wall ◽  
Density Profiles ◽  
Equilibrium Structures

Equilibrium structures of both homogeneous and heterogeneous systems are, within statistical thermodynamics, characterized by distribution functions. Using the approach proposed recently - based on the determination of the cavity functions for the pair of hard spheres (HS) and the combined body - we studied the effect of different choices of the probe HS (which determines the shape of the combined hard body - enlarged dumbbell) on the prediction of the distribution functions in binary mixtures of HS with the aspect ratio 0.9, ternary mixtures with diameter ratios 1, 0.6 and 0.3, and density profiles of HS mixture with the aspect ratio 2 near a hard wall. It was found that the method, that uses the average geometric functionals determined for the probe HS with individual diameters multiplied by the respective mole fractions yields better results than the approaches based on average probe diameters.

Comparison of Hypernetted Chain Equation and Monte Carlo Results for a System of Charged Hard Spheres

1970 ◽  
Vol 52 (2) ◽  
pp. 1013-1014 ◽  
Author(s):  
P. N. Vorontsov‐Veliaminov ◽  
A. M. Eliashevich ◽  
J. C. Rasaiah ◽  
H. L. Friedman
Keyword(s):  
Monte Carlo ◽  
Hard Spheres ◽  
Chain Equation ◽  
Hypernetted Chain ◽  
Charged Hard Spheres

scholarly journals “Thought experiments” as dry-runs for “tough experiments”: novel approaches to the hydration behavior of oxyanions

2016 ◽  
Vol 88 (3) ◽  
pp. 163-176 ◽  
Author(s):  
Ariel A. Chialvo ◽  
Lukas Vlcek
Keyword(s):  
Neutron Diffraction ◽  
Pair Correlation ◽  
Local Environment ◽  
Pair Formation ◽  
Distribution Functions ◽  
Ambient Conditions ◽  
Coordination Numbers ◽  
Anion Coordination ◽  
The Individual

AbstractWe explore the deconvolution of correlations for the interpretation of the microstructural behavior of aqueous electrolytes according to the neutron diffraction with isotopic substitution (NDIS) approach toward the experimental determination of ion coordination numbers of systems involving oxyanions, in particular, sulfate anions. We discuss the alluded interplay in the title of this presentation, emphasized the expectations, and highlight the significance of tackling the challenging NDIS experiments. Specifically, we focus on the potential occurrence of $N{i^{2 + }} \cdots SO_4^{2 - }$ pair formation, identify its signature, suggest novel ways either for the direct probe of the contact ion pair (CIP) strength and the subsequent correction of its effects on the measured coordination numbers, or for the determination of anion coordination numbers free of CIP contributions through the implementation of null-cation environments. For that purpose we perform simulations of NiSO4 aqueous solutions at ambient conditions to generate the distribution functions required in the analysis (a) to identify the individual partial contributions to the total neutron-weighted distribution function, (b) to isolate and assess the contribution of $N{i^{2 + }} \cdots SO_4^{2 - }$ pair formation, (c) to test the accuracy of the neutron diffraction with isotope substitution based coordination calculations and X-ray diffraction based assumptions, and (d) to describe the water coordination around both the sulfur and oxygen sites of the sulfate anion. We finally discuss the strength of this interplay on the basis of the inherent molecular simulation ability to provide all pair correlation functions that fully characterize the system microstructure and allows us to “reconstruct” the eventual NDIS output, i.e., to take an atomistic “peek” (e.g., see Figure 1) at the local environment around the isotopically-labeled species before any experiment is ever attempted, and ultimately, to test the accuracy of the “measured” NDIS-based coordination numbers against the actual values by the “direct” counting.

Pair correlation function in a fluid with density inhomogeneities: parametrization for hard spheres near a hard wall

1985 ◽  
Vol 400 (1818) ◽  
pp. 163-174 ◽  
Keyword(s):  
Surface Tension ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Sum Rule ◽  
Density Inhomogeneities ◽  
Semi Empirical ◽  
Good Agreement

The pair correlation function, g( z 1 , z 2 , r 12 ), of a system of hard spheres near a hard wall is approximated by a simple function that interpolates in a reasonable manner between the Percus shielding approximation for the case when the line of centres of the spheres is normal to the surface of the wall and the bulk pair correlation function, with a semi-empirical modification, for the case when the spheres are equidistant from the wall. This modification is chosen so that the surface tension, calculated from the pair correlation function, is given correctly. The resulting pair correlation function is in good agreement with computer simulations and satisfies a sum rule for the slope of the density profile fairly well.

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