The Ornstein-Zernike relation

The original derivation of the Ornstein-Zernike relation (Ornstein and Zernike 1914) employs arguments on local density fluctuations in the fluid. We present here a different derivation based on the method of functional derivatives (Appendix B). A very thorough discussion of this topic is given by Munster (1969), and by Gray and Gubbins (1984). 

Consider the grand partition function of a system of spherical particles exposed to an external potential i/r  

U -(RN) is the total potential energy due to interactions among the particles N, and the second term on the rhs of (C.2) is due to interaction of the system at configuration RN with the external potential. As in Appendix B, we use the symbol if to designate the whole function whose components are i/(Ri). The functional derivative of In S with respect to the component t/ (J ) is 

In the last step on the rhs, we used the definition of the singlet molecular distribution function of a system in an external potential if 

consider the second functional derivative of In E with respect to (// (R ), which can be obtained from (C.4)  

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Here, we obtained c(R) as a result of the first-order expansion. When (D.18) is used in the Ornstein-Zernike relation, we get the PY equation. Alternatively, one can assume relation (D.18) rewritten as... 

The relation derived in this Appendix is an extension of the Ornstein-Zernike relation for the compressibility by using the cell-pair correlation function rather than the usual particle-pair correlation function. 

The Percus-Yevick (PY) equation for spherical particles in three dimensions was found to be very useful for the study of the pair correlation function. For the purposes of the present 2-D system, the corresponding PY equation may be obtained most directly from the Ornstein-Zernike relation ... 

Finally, we mention an interesting recent study by Chandler that extended the Gaussian field-theoretic model of Li and Kardar to treat atomic and polymeric fluids. Remarkably, the atomic PY and MSA theories were derived from a Gaussian field-theoretic formalism without explicit use of the Ornstein-Zernike relation or direct correlation function concept. In addition, based on an additional preaveraging approximation, analytic PRISM theory was recovered for hard-core thread chain model fluids. Nonperturbative applications of this field-theoretic approach to polymer liquids where the chains have nonzero thickness and/or attractive forces requires numerical work that, to the best of our knowledge, has not yet been pursued. 

Allnatt (1964) showed that an expression equivalent to the Ornstein-Zernike relation can be written in matrix form as... 

FIG. 7 Radial distribution function of a typical suspension of charged spheres with screened Coulomb interaction. The exact results (open circles) from Monte Carlo computer simulations are compared with the theoretical predictions of the Ornstein-Zernike equation and different closure relations (lines). 

Now, let us look at Fig. 13. Here, the static structure factor of a three-dimensional homogeneous suspension of polystyrene spheres of diameter 94 nm is shown. The particles volume fraction is 0 = 2.0 x 10 4. Experimental data from static light scattering (closed circles) are compared with computer simulation (Monte Carlo) results (symbol x) and theoretical predictions (lines) obtained from the Ornstein-Zernike equation and different closure relations. The computer simulations and the theoretical calculations where carried out assuming that the interaction between the... 

Assuming the pair potential known, the radial distribution function for two-dimensional systems can be calculated using the two-dimensional version of the Ornstein-Zernike equation, Eq. (22), and one of the closure relations. Although Eq. (22) does not relate one to one the radial distribution function with the pair potential, one might attempt to invert the procedure to get u(r) from the experimental values for g(r). Thus, by taking the Fourier-Bessel (FB) transform [43,44] of Eq. (22) an expression for c(k) is obtained in terms of the FB transform of the measured total correlation function, i.e. 

FIG. 16 Effective pair potential between the colloidal particles in the systems of Fig. 15. In (a) and (b) are shown the cases with n = 0.023 and n = 0.48, respectively. The lines are the results of deconvoluting the radial distribution function using the Ornstein-Zernike equation and three different closure relations HNC, MSA and PY. The closed circles represent the potential of the mean force, which coincides with u(r) at low concentrations. Adapted from Carbajal-Tinoco et al. [42]. 

As is briefly described in the Introduction, an exact equation referred to as the Ornstein-Zernike equation, which relates h(r, r ) with another correlation function called the direct correlation function c(r, r/), can be derived from the grand canonical partition function by means of the functional derivatives. Our theory to describe the molecular recognition starts from the Ornstein-Zernike equation generalized to a solution of polyatomic molecules, or the molecular Ornstein-Zernike (MOZ) equation [12],... 

Equation (5) is the familiar Ornstein-Zernike relation and hereafter we denote the direct correlation function C2(r) as c(r). Based on (1) and (4) with (2), one can discuss freezing transition of one-component liquids. " For an S-component mixture, which is specified by S density fields, nj(r) j = l,- -,5), we have, instead of (4),... 

This is the Ornstein-Zernike equation. It is an exact integral equation relating the two 2-particle correlation functions li2(l,2) and C2(l,2). It is possible to motivate this equation form purely physical arguments the idea is to interpret the total correlation function li2(l,2) as the sum of all possible direct correlations, thus C2(l, 2) is termed the direct correlation function. We imagine that 112(1,2) is the sum of the direct correlation between 1 and 2 (that is 2(1,2)), and all chains of direct correlations via a third, fourth etc., particle. The weakness of this heuristic derivation is that we do not know how to write down an expression for 2(1,2). The great advantage of the formal... 

We are now in position to derive a second equation—one that relates C2(1,2) to /i2(l,2) and other known functions. This second equation is normally called a closure relation, and when combined with the Ornstein-Zernike equation, we have a closed system of equations to solve (two coupled equations in two unknowns /ij and C2). In principle, the graphical expansion of 2(1,2) in Eq. (2.1.29) is the exact closure relation, and if we could calculate all of the graphs we would have the exact solution. In practice, this has not been possible and all closure relations involve some type of approximation. 

The SSOZ equation is an exact integral equation relating the two site-site correlation functions h y(r) and c y(r). Strictly speaking, it is simply a definition of the site-site direct correlation function c,y(r). This fact is made clearest in the derivation by H iye and Stell. Here they consider the Ornstein-Zernike equation for an equimolar mixture of two species a and y. For each atom... 

As we discussed in Section II.B, site-site correlation functions provide a very useful formalism for describing the structure of fluids modeled with interaction site potentials. In this formalism, information equivalent to g l,2) is obtained from the set of site-site correlation functions and intramolecular correlation functions. For this reason, a great deal of effort has been put into the development of integral equation theories for these correlation functions. The seminal contribution in this area was made by Chandler and Andersen, who sought to write an integral equation of the Ornstein-Zernike form in which the set of site-site total correlation functions were related to a set of site-site direct correlation functions. Their equation has the form... 

To sketch briefly the integral equation methodology,178181 we again focus on a one-component atomic system. The direct correlation function, c(r), can be related to the function g(r) — 1 = h(r) by the Ornstein-Zernike equation170 178... 

B2) depending on whether the virial theorem or the Ornstein-Zernike compressibility relation is used in the calculations of the exact Monte Carlo value, lies between the two values, (b) The third coefficient in the... 

A third approach is to inject particles based on a grand canonical ensemble distribution. At each predetermined molecular dynamics time step, the probability to create or destroy a particle is calculated and a random number is used to determine whether the update is accepted (the probability for both the creation and the destruction of a particle must be equal to ensure reversibility). The probability function depends on the excess chemical potential and must be calculated in a way that is consistent with the microscopic model used to describe the system. In the work of Im et al., a primitive water model is used, and the chemical potential is determined through an analytic solution to the Ornstein-Zernike equation using the hypemetted chain as a closure relation. This method is very accurate from the physical viewpoint, but it has a poorer CPU performance compared with simpler schemes based on... 

Equation (8) is a type of Dyson equation known as the Chandler-Andersen (oh RISM — reference interaction site method") equation (Chandler, 1982 Chandler and Andersen, 1972). It is a generalization of the Ornstein-Zernike integral equation for simple atomic fluids (Hansen and McDonald, 1986). It is an integral equation which relates the unknown h ( r-r ) to the equally unknown c ( r-r ). Indeed, Eq. (8) is essentially a definition of c yir). Another relationship is required to close the equation, and to construct a closure a field theoretic perspective can be suggestive. We turn to such a perspective now. 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

However, the partials h j, and the partials of the direet eorrelation funetion Cij, the latter defined as the subsets of graphs in without bridge points, are related via a Wertheim-type multidensity Ornstein-Zernike equation... 

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

At the core of any integral equation approach we have the (exact) Ornstein-Zernike (OZ) equation [300] relating the total correlation function(s) of a given fluid to the so-called direct correlation function(s). For the replicated system at hand, the OZ equation is that of a multicomponent mixture [30],... 

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