Distribution functions of binary solutions (exact analytic solution)

We show that the general solution of the Ornstein-Zernike system of equations for multicomponent solutions has the form h_αβ=∑A _αβ ^j exp(-λ_jr)/r, where λ_j are the roots of the transcendental equation 1-ρΔ(λ_j)=0 and the amplitudes A_αβ ^j can be calculated if the direct correlation functions are given. We investigate the properties of this solution including the behavior of the roots A _αβ ^j and amplitudes A_αβ ^j in both the low-density limit and the vicinity of the critical point. Several relations on A_αβ ^j and C_αβ are found. In the vicinity of the critical point, we find the state equation for a liquid, which confirms the Van der Waals similarity hypothesis. The expansion under consideration is asymptotic because we expand functions in series in eigenfunctions of the asymptotic Ornstein-Zernike equation valid at r→∞. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 500–515, June, 2000.