Sedimentation Equilibrium of Mixtures of Charged Colloids

The next constant that can be solved is b n−1 . From the relation between b n−1and b n , using equation (4.22),where ξ n−1 can be found by the normalisation1H∫ xnx n −ξ n−1dxb n−1 = b n + t n + c n−1 ξ n−1 (4.25)()b n + t n − c n−1 (x − x n ) = ¯η (4.26)we find an expression for b n−1 and thus, for all b i . By repeating the stepsfrom equation (4.24) we find expressions for all constants b n and layer widthsξ i√2Hρ s¯ηπση(x i ) = b n =3+ (b3ZnL 2 n+1 + t n+1 ) 2 (4.27)nξ i = 3Z2 i L iρ s πσ 3 (b i − b i+1 − t i+1 ) (4.28)Figure 4.17 shows a comparison with the full numerical solutions. The dottedlines represent the approximation. The left figure shows an approximationof figure 4.1. The right figure shows the packing fractions of a systemwith colloidal charge number Z = (200, 250, 300), and gravitational lengthL = (0.4, 0.7, 1)mm, and the usual system parameters (see tab. 4.1). Althoughthe slope model calculates the equilibrium properties in an instant,and can even be done by hand, the Poisson-Boltzmann calculations are consideredmore reliable, and do not need the condition of pure layers. Especiallythis last condition is not often met for the systems considered in this thesis.An additional problem with the slope model is the jump constant. Multiplecomparisons with PB-calculations show that it is indeed proportional tot n ∝ ((Z N L n ) −2 − (Z n−1 L n−1 ) −2 ), but the proportionality constant provednot always equal to σ 3 /48λ B .36