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Mixtures in OrnsteinZernike.jl

For a mixture with (N_s) species, the Ornstein–Zernike equation becomes matrix-valued:

[ h{ij}(r) = c{ij}(r) + \sum{k=1}^{Ns} \rhok \, (c{ik} * h_{kj})(r) ]

Outputs such as (g{ij}(r)), (S{ij}(k)), (c_{ij}(r)) are stored as 3D arrays:

  • gr[r, i, j] = pair distribution between species i and j.
  • Sk[k, i, j] = partial structure factor between species i and j.

Example: Binary Lennard–Jones Mixture

using OrnsteinZernike, Plots, StaticArrays

ρ = [0.3, 0.2]  # densities of species A and B

function lj_matrix(r, params)
    ϵ_AA, σ_AA, ϵ_BB, σ_BB, ϵ_AB, σ_AB = params
    return SMatrix{2,2, Float64, 4}(
        4*ϵ_AA*((σ_AA/r)^12 - (σ_AA/r)^6),  4*ϵ_AB*((σ_AB/r)^12 - (σ_AB/r)^6),
        4*ϵ_AB*((σ_AB/r)^12 - (σ_AB/r)^6),  4*ϵ_BB*((σ_BB/r)^12 - (σ_BB/r)^6)
    )
end

pot = CustomPotential(lj_matrix, (1.0,1.0, 1.2,0.8, 1.1,0.9))
system = SimpleMixture(3, ρ, 1.0, pot)

sol = solve(system, HypernettedChain())

plot(sol.r, sol.gr[:,1,1], label="g_AA(r)")
plot!(sol.r, sol.gr[:,1,2], label="g_AB(r)")
plot!(sol.r, sol.gr[:,2,2], label="g_BB(r)")
Example block output

Interpretation

  • (g_{AA}(r)): correlations within species A
  • (g_{BB}(r)): correlations within species B
  • (g_{AB}(r)): cross-correlations

These can be combined into total structure factors, or into Bhatia–Thornton components.