Sedimentation Equilibrium of Mixtures of Charged Colloids

with ν λ = ν 0 +λw, the free energy functional can be written, after integrationof equation (2.25),F [ρ] = F 0 [ρ] + 1 2∫ 10dλdrdr ′ ρ (2)λ (r, r′ )w(r, r ′ ), (2.26)where F 0 [ρ] is the free energy functional of the reference system with ν = ν 0 .Substitution of the two particle density by the pair correlation functionh (2) (r, r ′ ) = ρ(2) (r, r ′ )ρ(r)ρ(r ′ ) − 1 (2.27)splits the free energy functional into three partsF [ρ] = F 0 [ρ] + 1 ∫ 1 ∫ ∫dλ dr dr ′ ρ(r)w(r, r ′ )ρ(r ′ )2 0+ 1 ∫ 1 ∫ ∫dλ dr dr ′ h (2)λ2(r, r′ )ρ(r)w(r, r ′ )ρ(r ′ ), (2.28)0where the last part (2.28) is the correlation term of the free energy. Themean field approximation is made by neglecting this correlation term.Stated less formally and taking the electrostatic potential as an example:by using the MFA, one considers the long-range pair potential as a selfconsistentexternal field for which∫φ(r) = dr ′ Q(r ′ )|r − r ′ | , (2.29)where Q(r) is the total charge density.VThe local density approximation , also known as LDA, is based on theassumption that macroscopic thermodynamics applies locally. The intrinsicHelmholtz free energy may then be written as∫F [ρ] = drf(ρ(r)) (2.30)where the free energy density f(ρ) satisfiesVF (N, V, T )f(ρ) = , (2.31)Vwith ρ = N/V the homogeneous density. For systems with long-range interactions(like Coulomb interactions), the LDA cannot be expected to hold.For example, hydrostatic equilibrium (eq.density approximation.B.16) is derived using a local9