  We investigate generalized potentials for a mean-field density functional
theory of a three-phase contact line. Compared to the symmetrical potential
introduced in our previous article [1], the three minima of these potentials
form a small triangle located arbitrarily within the Gibbs triangle, which is
more realistic for ternary fluid systems. We multiply linear functions that
vanish at edges and vertices of the small triangle, yielding potentials in the
form of quartic polynomials. We find that a subset of such potentials has
simple analytic far-field solutions, and is a linear transformation of our
original potential. By scaling, we can relate their solutions to those of our
original potential. For special cases, the lengths of the sides of the small
triangle are proportional to the corresponding interfacial tensions. For the
case of equal interfacial tensions, we calculate a line tension that is
proportional to the area of the small triangle.
