Sedimentation Equilibrium of Mixtures of Charged Colloids

One quick check shows ∫ ∞dx Q(x) = −σ, which reassures global charge0neutrality. From equation (B.10) the meaning of κ becomes obvious. Thetypical decay length of ρ ± (x) is κ −1 as is shown in figure (B.1) below.ρ −ρ sρ +κσ/2ρ sφ0 1 2 3 4κxFigure B.1: Density profiles ρ ± (x) of the ions near the charged plane at x = 0as well as the dimensionless potential (dashed line). Typical decay length isequal to κ −1 , the Debye screening length.B.2 The gravitational lengthThis section is about the gravitational length. This parameter is widelyused in models that consider sedimentation of gases in gravitational fields.The grand potential functional for a system of uncharged colloids under theexternal potential of the Earth’s gravity field is of the formβΩ[ρ] = βF [ρ] +∫ H0dx ρ(x)( x ρ(x) − βµ),L (B.11)with x/L the external potential due to gravity. By the variational principlewe find the Euler-Lagrange equation, using an LDA (section 2.3.2)βf ′ (ρ) + x − βµ = 0.L (B.12)66