Sedimentation Equilibrium of Mixtures of Charged Colloids

Chapter 3Sedimentation of amonodisperse system3.1 The systemUsing the theory of last chapter, we will turn now to a more complicatedsystem: charged colloidal spheres in an ionic solution in the gravity field ofthe earth. The colloidal spheres have a charge −Ze, gravitational length L,and a diameter σ, and their number density as a function of height is denotedby ρ(x). Instead of the bare mass, the buoyant mass of the colloid is takenin L = k B T/mg, dependent on the density of the incompressible solvent.The solvent further provides a dielectric constant ɛ and temperature T . Themicro-ions have a charge ±e, and have yet-unknown number densities ρ ± (x).In addition we will describe the micro-ions grand canonically, the colloidalspheres canonically. Such a situation can be realised by putting the solutionin osmotic contact with a salt reservoir, from which the colloids are separatedby a semipermeable membrane, while the co- and counter-ions can flow freely.The equilibrium properties of such a system can be determined with twodifferent methods. We will call them the Donnan-method and the Poisson-Boltzmann-method inspired by [8], which treats both methods profoundly.Subsections 3.2 and 3.3 follow the steps of [8].3.2 Donnan-methodThe osmotic equilibrium with the reservoir is also called Donnan equilibrium,named after the person who described this system first. In this way thechemical potential of the ions in the suspension can be fixed, since it mustbe equal to that of the reservoir. The chemical potential of the ions in the11