Sedimentation Equilibrium of Mixtures of Charged Colloids

with F [ρ] the intrinsic Helmholtz free energy functional obeying the Euler-Lagrange equationδ[ ∫]∣∣∣ρ(r)=ρeqF [ρ(r)]− drρ(r)ψ(r) = δF [ρ eq(r)]+φ(r)−µ = 0 (2.20)δρ(r)(r) δρ eq (r)Hohenberg, Kohn and Mermin proved two important theorems:Theorem 1: The intrinsic free energy functional F [ρ] is a unique functionalof the one-particle density ρ(r), i.e. for a given intermolecular potential energyV N , F [ρ] has the same functional form, whatever the external potentialφ(r).So, only one external potential φ(r) can be associated with a given ρ(r).Theorem 2: The auxiliary functional∫Ω V ext[˜ρ] = F [˜ρ] + dr˜ρ(r)(V ext (r) − µ) (2.21)reaches its minimum when the trial density ˜ρ(r) coincides with the equilibriumdensity ρ(r),δΩ V ext[˜ρ(r)]∣ = 0 (2.22)δ ˜ρ(r) ∣˜ρ(r)=ρ(r)and the minimum of Ω V ext coincides with the equilibrium grand potential Ω(defined in 2.7, and the first line of this section).For the proof see [16]. This formalism and its concepts are also referredto as density functional theory or ’DFT’.2.3.2 Mean Field Approximation and Local DensityApproximationThe mean field approximation is an approximation scheme, often usedfor Coulombic systems. For these systems the pair interaction ν(r) is writtenas ν 0 (r, r ′ ) + w(r, r ′ ). By viewing the long range-part w(r, r ′ ) as a perturbation,ν λ (r, r ′ ) = ν 0 (r, r ′ ) + λw(r, r ′ ). (2.23)In this thesis we use w to be a perturbation on the ideal gas, ν 0 (r, r ′ ) = 0).By introducing the two particle densityρ (2)λ (r, r′ ) = 〈ˆρ(r)ˆρ(r ′ )〉 λ − ρ(r)δ(r − r ′ ) (2.24)=δF2δν λ (r, r ′ ) , (2.25)8