Sedimentation Equilibrium of Mixtures of Charged Colloids

x-axis, only τ xx = −(ɛ/8π)E 2 x is non-vanishing. The total force Π = P + τ xxtogether with E x = −ψ ′ (x) becomesβΠ(ρ c (x), φ(x)) = ρ c (x) + 2ρ s (cosh φ(x) − 1) − 18πλ B(φ ′ (x)) 2 . (3.10)We again use hydrostatic equilibrium (B.16), with P replaced by ΠdβΠ(ρ c (x), φ(x))dx= ρ c ′ (x) + 2ρ s φ ′ (x) sinh φ(x) − 14πλ Bφ ′ (x)φ ′′ (x) = − ρ c(x)L .To solve the force balance, we use the Poisson-Boltzmann equation 1 .(3.11)φ ′′ (x) = κ 2 sinh φ(x) + 4πλ B Zρ c (x). (3.12)The differential equation now simplifies todβΠ(ρ c (x), φ(x))ρ ′ c (x) − Zρ c (x) = − ρ c(x)dxL . (3.13)After integration of equation (3.13) the Boltzmann distributionρ c (x) = ρ 0 exp[− x + Zφ(x)] (3.14)Lis found, where ρ 0 is determined by the total amount of colloids in the suspension.Equation (3.14) is a direct generalization of the barometric distribution(B.17) to include electrostatic interactions. While (B.17) is an explicit formfor the density profile, (3.14) is an equation for two unknown profiles ρ c (x)and φ(x), and needs at least a second equation to be solved, the Poisson-Boltzmann equation (3.12).The great benefit of the Poisson-Boltzmann-method is that it does not containthe inconsistencies of the Donnan-method. Only global charge neutralityis demanded, and the solutions of ρ c (x) and φ(x) are not restricted to specifieddensity or spatial regimes (with discontinuous jumps). The disadvantageis that profiles can only be found numerically. Some properties however canstill be seen analytically, but that will remain a secret until the next chapter.. .1 combination of Poisson’s equation (B.7) with the Boltzmann distributions of the salt,equation (B.6)14