Critical behavior of modified hypernetted-chain equations: Universal quantities

For a double Yukawa interparticle potential which well approximates the Lennard-Jones form, two modified hypernetted-chain equations, corresponding to two chosen model bridge functions, are solved numerically near their respective critical points. For the first, the use of a hard-sphere bridge function yields an isothermal compressibility $K_T$ that deviates from power-law behavior. In contrast to this, if the bridge function is chosen to cross over from a hard-sphere to mean-spherical form, then power-law behavior in $K_T$ is recovered with the mean-spherical result γ=2. This striking difference is traced to the presence of correspondingly long-ranged behavior in the crossover bridge function itself.