Penetrable-sphere model spherical surfaces

We have seen in 4.8 the difficulty of reconciling the arbitrariness of the surface of tension, z defined medianically (or quasi-thermodynamically) by (2.89) and the planar limit of the surface of tension of a drop, introduced through the thermodynamic arguments of 2.4—arguments which fix its position with respect to the equimolar surface, R. One way of analysing problems of this kind is by exact calculations for a model system. The mean-field treatment of the penetrable-sphere model is not exact, but, as we have seen, it becomes so in the two limits of (1) infinite dimensionality at all temperatures, and (2) zero temperature for all dimensions. Here we examine the three-dimensional spherical drop (and bubble) in the mean-field approximation and show that the results resolve some of the difficulties of 4.8. 

It is simplest to use the primitive form of the model, that is the system described by the pair potentials of (5.77), and to consider a drop rich in component a surrounded by a fluid rich in b. The transcription to the one-component version of the model is then made by using (5.129). 

Consider first, the two homogeneous phases a (inside the drop) and P (outside the drop) at points well removed from the interface. The pressures are given by  

SO that the pressure difference tr = tt - is known in terms of the four densities etc. The conditions of equilibrium between the two phases are, cf. (5.128) 

The physical state of the system can be specified by two independent variables for which we choose Air and fi,. From these ( pt, pg, pg, and pg can be found by solving the five equations formed from (5.149) and a subtraction of the two equations in (5.148). 

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