Structure of multi-component/multi-Yukawa mixtures

Recent small angle scattering experiments reveal new peaks in the structure function S(k) of colloidal systems (Liu et al 2005 J. Chem. Phys. 122 044507), in a region that was inaccessible with older instruments. It has been increasingly evident that a single ( or double) Yukawa MSA-closure cannot account for these observations, and three or more terms are needed. On the other hand the MSA is not sufficiently accurate (Broccio et al 2005 Preprint); more accurate theories such as the HNC have been tried. But while the MSA is asymptotically exact at high densities(Rosenfield and Blum 1986 J. Chem. Phys. 85 1556), it does not satisfy the low density asymptotics. This has been corrected in the soft MSA ( Blum et al 1972 J. Chem. Phys. 56 5197, Narten et al 1974 J. Chem. Phys. 60 3378) by adding exponential type terms. The results compared to experiment and simulation for liquid sodium by Rahman and Paskin ( as shown in Blum et al 1972 J. Chem. Phys. 56 5197) are remarkably good. We use here a general closure of the Ornstein-Zernike equation, which is not necessarily the MSA closure ( Blum and Hernando 2001 Condensed Matter Theories vol 16 ed Hernandez and Clark ( New York: Nova) p 411). c(ij) (r) = Sigma(M)(n=1) k(ij)((n))e(-Znr)/r; K-ij((n)) = K-(n)delta((n))(i)delta((n))(j); r >= sigma(ij) (1) with the boundary condition for g(ij) (r) = 0 for r <= sigma(ij). This general closure of the Ornstein-Zernike equation will go well beyond the MSA since it has been tested by Monte Carlo simulation for tetrahedral water (Blum et al 1999 Physica A 265 396), toroidal ion channels (Enriquez and Blum 2005 Mol. Phys. 103 3201) and polyelectrolytes ( Blum and Bernard 2004 Proc. Int. School of Physics Enrico Fermi, Course CLV vol 155, ed Mallamace and Stanley ( Amsterdam: IOS Press) p 335). For this closure we get for the Laplace transform of the pair correlation function an explicitly symmetric result 2 pi(g) over tilde (ij)(s) = - e(-s sigma ij)/D-tau(s) {1/s(2) + 1/s Q(ij)' (sigma ij) + Sigma(M)(m=1) Z(m)(X) over tilde ((m))(i)f(j)((m))s + z(m)}. (2) This function is also easily transformed into S(k) by replacing s double right arrow ik. For low density situations ( dilute colloids) D-tau (s) similar to 1 + O(p) and S(k) is a sum of M Lorentzians. For hard sphere PY mixtures we get the simple ( compare Lebowitz 1964 Phys. Rev. 133 A895 and Blum and Stell 1979 J. Chem. Phys. 71 42) 2 pi(g) over tilde (ij)(s) = e-(s sigma)(ij)/s(2)D(tau)(s) {1 + s[(Q(HS)) (t)(ij) (sigma(ij))]} where D-tau (s) is a scalar function. For polydisperse electrolytes in the MSA a simpler expression is also obtained ( compare Blum and Hoye 1977 J. Phys. Chem. 81 1311). An explicit continued fraction solution of the one component multi-Yukawa case is also given.