Asymmetric coexistence curves in an exactly-solvable model of binary liquid mixtures
An exactly-solvable lattice-gas model for binary liquid mixtures is proposed which describes not only complex miscibility effects, such as closed-loop phase diagrams, but also asymmetry in the coexistence curves. A key feature of the model is the inclusion of lattice vacancies, allowing density fluctuations in the system. In fact, density fluctuations are found to be essential if the asymmetries in the underlying microscopic interactions to be expressed in the coexistence curves. The phase diagrams that result from this model are in accord with the qualitative features observed experimentally, and also indicate that other more novel forms of the coexistence curves await discovery. i where Si = fl, the first sum is over all nearest neighbors on the lattice, and the second sum is over all lattice sites. With the usual mixture-magnet correspondence the two values of Si are identified with the two components, A and B, of the mixture, KI represents the difference in van der Waals interaction between like and unlike molecules, and HI is directly related to the chemical potential difference between the components. Note that with this model the energy of interaction for an AA nearest-neighbor pair (Si = +1, s' - +1) is the same as that for a BB nearest-neighbor pair (Si = - 1, Sj = -l), so that the two types of molecules are handled symmetrically. To produce an asymmetric model it would seem only natural that one simply let the two energies mentioned above be different; that is, define EM to be different from EBB. In terms of the model, this is accomplished by including an additional magnetic field, H, with a nonzero value. The problem is that to obtain coexistence it is necessary that the total magnetic field in the system be zero, thus one must set HI = -H. This completely undoes the asymmetry inherent originally in EM # EBB, and the resultant phase diagram is symmetric as before. The net effect has simply been a shift in the coexistence condition from HI = 0 to HI = -H.