The pair correlation function

We now introduce the most important and most useful function in the theory of liquids the pair correlation function. Consider the two elements of volume dX and dX and the intersection of the two events  

The combined event written in (2.36) means that the first and the second events occur i.e., this is the intersection of the two events. 

Two events are called independent whenever the probability of their intersection is equal to the product of the probabilities of the two events. In general, the two separate events given in (2.36) are not independent the occurrence of one of them may influence the likelihood, or the probability, of the occurrence of the other. For instance, if the separation R= fi — R between the two elements is very small (compared to the molecular diameter of the particles), then the occurrence of one event strongly affects the chances of the occurrence of the second. 

In a fluid, we expect that if the separation R between two particles is very large, then the two events in (2.36) become independent. Therefore, we can write for the probability of their intersection 

The last equality is valid for a homogeneous and isotropic fluid. If (2.38) holds, it is often said that the local densities at X and X are uncorrelated. (The limit R— oo should be understood as large enough compared with the molecular diameter, but still within the boundaries of the system.) 

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In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

It follows that the exact expression for the pair correlation function is... 

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

Although stratification, according to the plot in Fig. 10, occurs continuously as increases, it is accompanied by a curious structural reorganization in transverse directions (i.e., parallel to the planar substrate). A suitable measure of transverse structure is the pair correlation function defined in Eq. (62). However, for simplicity we are concerned only with the in-plane pair correlation function defined as [see Eq. (62)]... 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

Finally, we relate the gradient of the local density ViPq(fi) to the pair correlation functions. For this purpose we take the gradient of the expansion (5)... 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

Equation 3 is exact for fluids obeying Equations 1 and 2. However, in order to compute the density n(r) from the YBG equation one must know the relationship between density distribution and the pair correlation function of Inhomogeneous fluid. Such a relationship Is not available in general. However, an approximation introduced by Fischer and Methfessel (1.) has been shown to give fairly accurate predictions of the density... 

To Anally complete the model a formula for the contact value of the pair correlation function g must be given. We choose the Carnahan formula... 

Local Average Density Model (LADM) of Transt)ort. In the spirit of the Flscher-Methfessel local average density model. Equation 4, for the pair correlation function of Inhomogeneous fluid, a local average density model (LADM) of transport coefficients has been proposed ( ) whereby the local value of the transport coefficient, X(r), Is approximated by... 

The prerequisite for an experimental test of a molecular model by quasi-elastic neutron scattering is the calculation of the dynamic structure factors resulting from it. As outlined in Section 2 two different correlation functions may be determined by means of neutron scattering. In the case of coherent scattering, all partial waves emanating from different scattering centers are capable of interference the Fourier transform of the pair-correlation function is measured Eq. (4a). In contrast, incoherent scattering, where the interferences from partial waves of different scatterers are destructive, measures the self-correlation function [Eq. (4b)]. 

In the case of coherent scattering, which observes the pair-correlation function, interference from scattering waves emanating from various segments complicates the scattering function. Here, we shall explicitly calculate S(Q,t) for the Rouse model for the limiting cases (1) QRe -4 1 and (2) QRe > 1 where R2 = /2N is the end-to-end distance of the polymer chain. 

The pair-correlation function for the segmental dynamics of a chain is observed if some protonated chains are dissolved in a deuterated matrix. The scattering experiment then observes the result of the interfering partial waves originating from the different monomers of the same chain. The lower part of Fig. 4 displays results of the pair-correlation function on a PDMS melt (Mw = 1.5 x 105, Mw/Mw = 1.1) containing 12% protonated polymers of the same molecular weight. Again, the data are plotted versus the Rouse variable. 

As mentioned in Section 3.1, the incoherent dynamic structure is easily calculated by inserting the expression for the mean square displacements [Eqs. (42), (43)] into Eq. (4b). On the other hand, for reptational motion, calculation of the pair-correlation function is rather difficult. We must bear in mind the problem on the basis of Fig. 19, presenting a diagrammatic representation of the reptation process during various characteristic time intervals. 

The pair correlation function of the velocities and the pair correlation functions of some time derivatives of the velocity are sometimes taken into account.75 However, the validity of this description in the nonadiabaticity regions also has to be proved. The dynamic description or the description using the differentiable random process is more rigorous in this region.76... 

A quantity of central importance in the study of uniform liquids is the pair correlation function, g r), which is the probability (relative to an ideal gas) of finding a particle at position r given that there is a particle at the origin. All other structural and thermodynamic properties can be obtained from a knowledge of g r). The calculation of g r) for various fluids is one of the long-standing problems in liquid state theory, and several accurate approaches exist. These theories can also be used to obtain the density profile of a fluid at a surface. 

Figure 31. The domain growth during the phase separation process reflected by the shift of the first zero in the pair correlation function (a) and by the surface area reduction (b). Although the surface area and first zero of the pair-correlation functions are equivalent lengthscales, the time dependence of the surface area is less affected by the finite lattice size affects.

It is thus entirely expressed in terms of the zero wave number Fourier coefficient pQ(p t). Similarly, the pair correlation function in a spatially homogeneous system is defined by17... 

In particular, in this monograph, we shall be mainly concerned with the calculation of the pair correlation function (46) for equilibrium situations, from which all other thermodynamic quantities may be calculated, and with the consideration of the electrical current out of equilibrium. This latter is given by ... 

Finally, in the calculation of the pair correlation function, we shall also need the so-called creation fragmentit is defined by ... 

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