Liquid-gas interface in the model of attracting hard spheres

We continue here to treat the model of attracting hard spheres in mean-field approximation, but we no longer assume the fli to be homogeneous. Taking the more general (5.6) instead of (5.7), and using the inhomogpneous form of (5.1), we shall find a functional equation for the density profile of the liquid-gas interface. 

The uniform value of the chemical potential in the two-phase system 

Then from (S.1S)-(S.17), the functional equation that must be satisfied by p(r) takes the form 

In any of the forms (5.14), (5.15), or (5.18), the functional equation is non-local, in that it expresses p(rt) in terms of flie density p(t at all points t2 in the fluid outside a sphere of radius b centred at r,. When the gradients are small—as we have already assumed in obtaining (5.14)— we may expand p(i2) about i] and truncate the expansion at some finite order, llie result is a differential equation for p(i,) which is of order equal to that of the last term retained in the expansion, and whidi is then local, for it expresses p(r ) only in terms of its derivatives of finite order evaluated at ii. If we truncate after the term of second order (recognizing that the term of first order vanishes by symmetry), and at the same time specialize the discussion to allow non-uniformity only in the z-diiectkm, perpendicular to the plane of the interface, (5.18) becomes 

With the three-dimensional fh (p) there are important quantitative differences from (5.21), but the associated p.(p, T) or M(p) from (5.17) remaiis qualitatively the same and is still like that in Fig. 3.4. We know from Chapter 3 that (5.19) then yields a density profile p(z) of the expected form, like that shown as (c) in Fig. 3.3. 

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