Sedimentation Equilibrium of Mixtures of Charged Colloids

Appendix CThe block modelThe model described in the section 4.6.4 can be simplified by a crude approximationof the form of the packing fraction. Although the expressionsfollowing from this model are even less reliable than the slope model, it canbe used to determine the order of the layer widths quickly, and by hand. Thecondition that the total width ξ should be smaller than the total height His not needed either.The equations (4.27) and (4.28) become easier to obtain by assuming thepacking fraction to be constant η i (x) = b i . Then the electric field needs noapproximation, since by ρ ′ i(x) = ddx a i exp(−x/L + Zφ(x)) = 0,φ ′ (x) = 1Z i L i, for i ∈ I i . (C.1)From the equation of state the same solution is found as in equations (4.22),but now also((η i (x) − η j (x) = b i − b j =σ3 1) 2 ( 1) 2 )− . (C.2)48λ B Z i L i Z j L jbecause η i (x) is constant. Only one b i is needed to solve all b’s by recursion.For ξ < H, the value of b n is taken to be b n = √ 8ρ s σ 3 H ¯η/3Z 2 nL n (equation(4.24)), so(b i = η(x i ) = σ3 1( ) 2 1)− ( ) 2 + b n48λ B Z i L i Z n L nξ i = ¯η b i(C.3)In the case where the colloids reach the top of the system, ξ = H, theexpressions are slightly different. Two conditions then determine the layer69