## Generalized Potentials for a Mean-field Density Functional Theory of a Three-Phase Contact Line

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 [Phys. Rev. E 85, 011120 (2012)], 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.