Usually, the formula for the surface tension of a planar charged and polarized interface is obtained from that for a system involving only short-range forces, y = - dz [p - px(z)] by replacing the tangential pressure p , by p , + E2/8u. Problems with this include (a) p, is no longer explicitly defined, (b) the electrostatic stress term E2/8 pi is not correct in general but only if polarization is proportional to density of polarizable species, (c) the derivation of the formula in terms of p and p, involves calculating the work to expand a volume containing the interface, and this work cannot be written in terms of the pressure of the surroundings when there are long-range forces. To derive a formula free from these objections, we consider the spherical system contained between r = R, and r = R2 and containing charged and dipolar particles, the orientation of the latter giving rise to the electrical polarization. There is no electric field, electric polarization, or local charge density for r < R, or for r > R2. If this system is expanded keeping the ratios of all radii fixed, the work done by the surroundings is 4u1R,2bR-l p2R&3R2),which is set equal to the change in free energy, calculated from the canonical partition function. The surface tension is defined as (R,,/2)(pl -p2), where R,, is the surface of tension. When R, becomes infinite (plane interface), the value of R,, becomes irrelevant. Both long-range and short-range terms in the surface tension are shown to behave properly for R, - m, the long-range terms being proportional to l d r [-p(r) V(r) + 3P(r) E(?)] (P = polarization). If only charged particles are present (no polarization), correlations and short-range forces are neglected, and the distribution of each charged species ni follows the Boltzmann law with energy qiV, it is shown that kmini - E2/8r is independent of z. Using this fact with our surface tension formula, we prove the Lippmann equation. If dipolar particles are present as well as charged particles, the former must be included in Cini. Then the quantity k E i n , - E2/8u - EP is shown independent of z, and our surface tension formula again leads to the Lippmann equation.
Goodisman, J. (1992). Surface tension of a charged and polarized system. Journal of Physical Chemistry, 96(15), 6355-6360.
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