Sedimentation Equilibrium of Mixtures of Charged Colloids

is denoted by ρ i . The salt ions have a charge ±e and number density ρ ± .The medium is responsible for a Bjerrum length λ B , the height of the systemis H. The suspension is in Donnan-equilibrium with a neutral salt reservoir,with a total salt concentration of 2ρ s . The system must sound veryfamiliar, after section 3. With the named section in mind, we can approachthe system in two ways, using either the Donnan-method or the Poisson-Boltzmann-method. The first method proved to be, in the one componentcase, a fast and clear way to obtain an explicit form for the number density.For mixtures, this method did not prove very successful, therefore we willstart with a PB 1 -theory. The system parameters that will be standard in thefollowing sections are denoted in table 4.1 below.Table 4.1, standard system parametersParameter Number Units Commentσ i 150 nm colloidal diameterλ B 2.3 nm ethanol at room temperatureH 20 cm hand size systemρ s 3 µM interesting region (see 3.2)¯η tot 0.005 low packing fraction4.2 The functional and minimum conditionsSince the Donnan-approach did not prove very successful to us in solving theequilibrium properties of the system, neither numerically nor analytically,another more fundamental approach is chosen. The potential and densitieswill now follow from a mean field grand potential functionalβΩ[{˜ρ(x)}] = ∑++n∑i=1n∑i=1− ∑α=+,−1 =Poisson-Boltzmann∫ H0∫ H0∫ H0α=+,−∫ H0dx ˜ρ α (x)(ln ˜ρ α (x)Λ 3 α − 1) +dx ˜ρ i (x)(ln ˜ρ i (x)Λ 3 i − 1)dx x L i˜ρ i (x) + λ B2dx βµ α ˜ρ α (x) −∫ H0n∑i=1dx Q(x)φ(x)∫ H0dx βµ i ˜ρ i (x) (4.1)16