Potential of mean force in mixtures

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 

Two extreme cases of equation (2.155) are the following. If pB — 0, then p(RB/R, R ) — 0 also and the third term on the rhs of (2.155) vanishes. This is the case of a pure A. The solvent in this case will consist of all the A particles other than the two tagged particles at R1 and R . 

The second extreme case occurs when pA — 0. Note, however, that we still have two A s at fixed positions (R1, R ), but otherwise the solvent (here in the conventional sense) is pure B. We have the case of an extremely dilute solution of A in pure B. Note also that at the limit pA — 0, both the pair and the singlet distribution functions of A tend to zero, i.e., 

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). 

As in the case of pure liquids, the solvent-induced force can be attractive or repulsive even in regions where the direct force is negligible. An attractive force corresponds to a positive slope of W(R), or equivalently, to a negative slope of g(R). The locations of attractive and repulsive regions change when the composition of the system changes. Specifically, for xA lwe have the second peak of (R) at about 7 + 7 2. On the other hand, for 0, the second peak of g iR) is at ss 2.5 Clearly, there are regions that are 

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