Potential of average force

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

The thennodynamic properties are calculated from the ion-ion pair correlation fimctions by generalizing the expressions derived earlier for one-component systems to multicomponent ionic mixtures. For ionic solutions it is also necessary to note that the interionic potentials are solvent averaged ionic potentials of average force ... 

C. Defect Distribution Functions and Potentials of Average Force... 

Meeron60 62 first pointed out how the terms in S(Jt> in the solution theory can be arranged in a form much more compact than that above, which is of the form of a virial expansion in which the coefficients involve the Debye Hiickel potential of average force rather than the unscreened potential. Similar manipulations can be made in the present case, but we shall omit the details, which are very simple, and quote only the final result. It is found using Meeron s form of S that the activity coefficient of defect number s can be written... 

The discussion of the defect distribution functions and potentials of average force follows along rather similar fines to that for the activity coefficient. The formal cluster expansions, Eqs. (90)-(91), individual terms of which diverge, must be transformed into another series of closed terms. This can clearly be done by... 

In Eq. (154), we assume indeed that only the ions (Z 0) interact with each other and that the resulting interaction is simply the Coulomb potential modified by the zero-frequency dielectric constant e of the solvent. Of course, in an exact theory, we would have to take explicitly into account the interactions with the solvent, and the dielectric constant itself should come out of the calculation. The proper way of attacking this problem is based on the theory of the potential of average forces and is carefully analyzed in H. L. Friedman s monograph.11 However, the explicit calculations always become exceedingly complicated and, in one way or another, one always has to have recourse to an approximation of the type (154). It amounts to assuming ... 

Figure 9.3. Schematic comparison of the direct pair potential U(R) and the potential of average force as a function of the ligand-ligand separation.

The potential I speak of is usually called the potential of average force. Insofar as it is to be identified to a thermodynamic potential it is a local Helmholtz free energy as a function of the coordinate positions of all the atoms (or radicals) that must change relative positions in the reaction it may be defined by... 

In the first instance, and as a first approximation valid for very dilute solutions, one may ignore all types of ion-ion interactions except those deriving from simple Coulombic" forces. Thus, short-range interactions (e.g dispersion interactions) are excluded. This is a fundamental assumption of the Debye-Hiickel theory. Then the potential of average force U simply becomes the Coulombic potential energy of an ion of charge z, q in the volume element dV, i.e., the charge on the ion times the electrostatic potential in the volume element dV. That is,... 

Theoretical discussion of the osmotic pressure of polyelectrolytes has been made by two methods, one using the Donnan equilibrium and the other the McMillan and Mayer theory. Both methods are equivalent but in order to obatin explicitly the osmotic pressure we should know in the former case the activities of component systems and in the latter case the potential of average force between the solute molecules. 

We shall consider in this section a rigid spherical polyelectrolyte. According to the Debye-Huckel theory, the potential of average force between the polyelectrolytes may be approximated as follows ... 

In tile following we shall consider solutions of flexible linear potymer molecules and assume that the interactions between the pol3uner segments have short ranges. The distribution functions of the polymer molecules are determined by the potentials of average force as in Eq. (2.7). Assuming an addivity in the potentials we may write down Fj(l, 2) as follows ... 

Here (s) is the potential of average force between two segments at a distance s. 

As first remarked by Flory and Keigbaum (32) A vanishes at the temperature at which becomes zero. Since 6g is determined by the potential of average force between the s ments, this temperature is indepaident of the molecular weight of the polymer molecule. Also, we expect that the slope of the curve of the second virial coefficeint plotted against tranperature is expected to be independent of the molecular weight at this point. 

In section 2.8, we defined the potential of mean force (PMF) between two tagged particles in a one-component system. This definition can be extended to any pair of species for example, for the A-A pair, the potential of average force is defined by... 

However, the pair correlation function as well as the potential of average force are finite at this limit. We can think of WAA(R) in the limit of pA — 0 as the work required to bring two A s from infinite separation to the distance R in a pure solvent B at constant Tand V(or T, P depending on the ensemble we use). 

Unfortunately, it is not easy to calculate the solute-solute pair correlation function in the cluster model. Nevertheless, we can predict the results that could have been obtained in the cluster model from the following considerations. Recall that the indirect part of the potential of average force is related to the solvation Gibbs energies of two monomers and dimer by... 

A function related to the pair correlation function is the potential of average force (and torque), defined by ... 

The form of the function W R), with R = R" — R for LJ particles, and its density dependence are depicted in Fig. 2.9. Clearly, at very low densities, the potential of average force is identical with the pair potential this follows from the negligible effect of all the other particles present in the system. At higher densities, the function W R) shows successive maxima and minima [corresponding exactly to the minima and maxima of g(R), by virtue of the definition (2.77)]. The interesting point worth noting is that the indirect force at, say R > a, can be either attractive or repulsive even in the region where the direct force is purely attractive. 

Fig. 2.10. The forms of W(R)lkT for hard spheres at low densities. The curves are computed for spheres of diameter (x = 1.0 and two densities, = 0.1 and q — 0.4. Note that for < R< 2, we have an attractive potential of average force. [Computed from Eq. (2.85) note that Eq. (2.85) is valid at very low densities, and may not hold for these particular densities.]...

Finally, we now relate the potential of average force to the Helmholtz free energy. This relation will also be found useful for the study of hydro-phobic interactions (Chapter 8). 

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