Potential of Average Force in Mixtures

In section 5.4 we defined the potential of average force between two tagged particles in a one-component system. This definition can be extended to any pair of species for example, for the AA pair, the potential of average force is defined by 

The first term on the rhs of (6.4.2) is simply the direct force exerted on the first T at R, by the second A at R . The average force exerted by the solvent now has two terms, instead of one in (5.4.11). The quantity V f/. .i(R, R ) is the force exerted by any A particle (other than the selected two Ts) located at R on the particle at the fixed position R and p(R /R, R ) is the conditional density of A particles at R, given two As at R and R . Integration over all locations of R gives the average force exerted by the A component on the A particle at R. Similarly the third term on the rhs of (6.4.2) is the average force exerted by the B component on the A particle at R. The combination of the two last terms can be referred to as the solvent induced force. (The term solvent is used here for all the particles in the system except the two selected tagged particles.) 

The second extreme case occurs when 0. Note, however, that we still have two As at fixed positions R, 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. This is similar to the situation discussed in Chapter 3 and in Appendix I. We can think of Waa R) in the limit p - 0 as the work required to bring two Ts from infinite separation to the distance R in a pure solvent B at constant T and V (or 7, P depending on the ensemble we use). 

Clearly, there are regions that are attractive for Xa 1 (say 2 R 2.5) but become repulsive at xa 0. Therefore when we change the composition of the system continuously, there are regions in which the two terms on the rhs of (6.4.2) produce forces in different directions. The result is a net diminishing of the overall solvent-induced force between the two tagged T-particles. This corresponds to the flattening of g R) or of W R) that we have observed in Fig. 6.4 at x. 0.65. 

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