Pair distribution function hard sphere

Now let us add the possibility of collisions. Before we proceed, we make the following two assumptions (1) only binary collisions occur, i.e. we rule out situations in which three or more hard-spheres simultaneously come together (which is a physically reasonable assumption provided that the gas is sufficiently dilute), and (2) Boltzman s Stosszahlansatz, or his molecular chaos assumption that the motion of the hard-spheres is effectively pairwise uncorrelated i.e. that the pair-distribution function is the product of individual distribution functions ... 

Figure 5.7 The pair distribution function g(r) for hard spheres as a function of the dimensionless centre-to-centre separation. This was calculated using the algorithm from McQuarrie1 at a range of volume fractions...

R. A. Marcus In Chem. Phys. Lett. 244, 10 (1995), a very rough approximate hard-sphere model used for liquids was mentioned to relate the frictional coefficient to the pair distribution function in the cluster. 

The microstructure of a semidilute suspension of hard spheres will become distorted with the application of flow. This is manifested by anisotropy in the spatial position of the centers of mass of the spheres, which is described the pair distribution function, P2 (r). This... 

To evaluate the volume integrals in (84), the radial distribution function must be known. The pair distribution function affected by the Brownian motion and the relative electrophoretic velocity between a pair of particles is generally nonuniform and nonisotropic. When the particles are sufficiently small so that Brownian motion dominates, one can use a simple distribution function based on hard-sphere potential... 

Figure 1. Variation of with density (full lines) fordilTerent values of R a using Chesnoy. The strong variation ofy (i ) with R at a given density leads to the conclusion that ifcannot be precisely determined, then the test of the binary interaction equation [Eq. (66)] is not critical. Dotted lines show that a hard-sphere pair distribution function fi(us(

Figure 7. Matrix-matrix unlike distribution function for poa3 = 0.2 for the low temperature (solid line and circles) and high temperature (dotted line and squares) matrix. In the latter case like and unlike correlations are identical to the uncharged hard sphere pair distribution function.

The former is obtained through a relation with the cavity pair distribution function, yAA Hs- between two non-polar solutes, assuming hard-sphere interaction among them and also between solute and water molecules. We discuss the evaluation of cavity distribution functions in Appendix 15.A. 

We begin by using the fact that if the gas were in equilibrium then x is given by the pair distribution function g(ri, f2) evaluated at ri — r2 = a. Consider now the density expansion of g(ri, r2) in terms of Mayer /-functions. This expansion for general Fi, T2 is given for hard spheres byt... 

The main problem in extending the microstructural theories to high Peclet number and volume fraction is related to the formulation of the many-body interactions. Recently, based on the Smoluchowski equation, Nazockdast and Morris (2012) developed a theory for concentrated hard-sphere suspensions under shear. The theory resulted in an integro-differential equation for the pair distribution function. It was used to capture the main features of the hard sphere structure and to predict the rheology of the suspension, over a wide range of volume fraction (<0.55) for 0 < Pe < 100 (Nazockdast and Morris 2012). 

The interpolation between the low and high density limits, which is inherent to this variational approach, leads in a very natural way to the scaled particle theory for the structure and thermodynamics of isotropic fluids of hard particles. This unifies, for the first time the Percus Yevick theory, which is based on diagram expansions, and the scaled particle theory of Reiss, Frisch and Lebowitz, and, at the same time yields the analytical expressions of the dcf conformal to those of the hard spheres. It provides an unified derivation of the most comprehensive analytic description available of the hard sphere thermodynamics and pair distribution functions as given by the Percus Yevick and scaled particle theories, and yields simple explicit expressions for the higher direct direct correlation functions of the uniform fluid. 

Figure 11.3 The pair distribution function determined by Stokesian dynamics simulation for hard spheres at ( ) = 0.45 and Pe = 1000 shown (a) in the full shear plane and (b) in spherical average as a function of pair separation. In (b), a comparison is made to the radial dependence of the isotropic equilibrium structure, which was shown in a planar view in Figure 11.2. Shear flow is as in Figure 11.2, left to right and increasing velocity up the page.

In the second step, the electrolyte is put in contact with a single big sphere (extra component at infinite dilution). As long as the radius R of this sphere is large compared to the characteristic distances in the electrolyte, its precise value is irrelevant and the sphere-electrolyte interface mimics a flat air-water interface. In practice, R = 50A is sufficient. The HNC equation is solved for the pair distribution functions, (z) between the big sphere and the ions and gives the local ionic profiles Pi z) = Pi liiz) where z = r — R is the normal distance to the interface. The interface-ion pair potential Ui z) contains the hard-sphere contribution, the dispersion contribution... 

Now, we would like to investigate adsorption of another fluid of species / in the pore filled by the matrix. The fluid/ outside the pore has the chemical potential at equilibrium the adsorbed fluid / reaches the density distribution pf z). The pair distribution of / particles is characterized by the inhomogeneous correlation function /z (l,2). The matrix and fluid species are denoted by 0 and 1. We assume the simplest form of the interactions between particles and between particles and pore walls, choosing both species as hard spheres of unit diameter... 

Here, we report some basic results that are necessary for further developments in this presentation. The merging process of a test particle is based on the concept of cavity function (first adopted to interpret the pair correlation function of a hard-sphere system [75]), and on the potential distribution theorem (PDT) used to determine the excess chemical potential of uniform and nonuniform fluids [73, 74]. The obtaining of the PDT is done with the test-particle method for nonuniform systems assuming that the presence of a test particle is equivalent to placing the fluid in an external field [36]. 

Likewise, the function, X2) may be interpreted as the relative probability of finding the two solvatons at Xi, X2. This function is sometimes referred to as a cavity-cavity distribution function. This term might be misleading for two reasons. First, our solvatons exert an external field of force which has both attractive and repulsive parts. It is only for hard-sphere solvatons that the external field is equivalent to a cavity. Second, even for hard-sphere particles that are in some sense equivalent to suitable cavities, one must be careful when reference is made to the probability of finding a single or a pair of cavities. The reason for that has been discussed in section 5.10. 

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