Sedimentation Equilibrium of Mixtures of Charged Colloids

The derivative to x results inAgain we use the LDA inf ′′ (ρ)ρ ′ (x) = −mg.(B.13)∂F (N, V, T )p = − = −f(ρ) + ρf ′ (ρ),∂V(B.14)sodpdρ = ρf ′′ (ρ).(B.15)Substitution of equation B.15 into B.13 leads to the equationdP (ρ(x))= −mgρ(x), (B.16)dxcalled hydrostatic equilibrium, and describes the relation between the pressureof the bulk fluid P (ρ) and the number density ρ at sedimentation equilibrium.The height above the surface is denoted by x, the mass of the particles by m,and the gravitational acceleration by g. For an ideal gas, P (ρ) = k B T ρ, thesolution of equation (B.16) is the so-called barometric height distribution:ρ(x) = ρ 0 exp(− x L ),(B.17)where ρ 0 is determined by the total number of particles in the system, andthe gravitational length.L = k B T/mg(B.18)B.3 The colloidal diameter, σIn this thesis the particles interact as point particles, as an ideal gas. Thediameter of the colloids will only be a scaling factor for the packing fractionη = πσ 3 ρ/6. In this way, the size can be used to check if the ideal gas approximationis applicable. For packing fractions higher than η > 0.1, packingeffects are expected to become important. For the theory, the diameter willtherefore play a minor role, although it can lead to confusing results whencomponents have strongly different diameters (section 4.6.1). For the MonteCarlo simulations the size does matter. Comparisons between theory andsimulations will show that for diluted systems (η < 0.05), packing effects arevery small.The diameter will be taken σ = 150nm as standard.67