Electrostatics of Colloids within the Framework of the Poisson-Boltzmann Equation and Beyond
Pavel Dyshlovenko · AIP conference proceedings · 2003
The non‐linear Poisson‐Boltzmann equation describes, in some approximation, the electric potential and charge distribution in colloidal systems. Information on the free energy, forces and related quantities can be obtained from the solution of the equation. The adaptive numerical method for the Poisson‐Boltzmann equation is proposed. Then the method is applied to some problems of the electrostatic interaction of colloids. Special attention is given to the effects of non‐linearity and geometrical confinement. Lastly, a possible generalization of the Poisson‐Boltzmann approach is briefly discussed. The numerical method is a further development of the approach proposed in [1,2]. This is a two‐dimensional finite‐element method combined with the adaptive mesh refinement and a posteriori error evaluation. The method is shown to be flexible enough to solve the problems in which the areas of high gradient or high errors of the solution are not known a priori. It is well suited for two‐dimensional and three‐dimensional axially symmetric problems with sophisticated geometry and various boundary conditions. The proposed numerical method is applied to some problems of colloids’ interaction. These are different particle‐particle and particle‐wall problems: two free identical particles, two identical particles confined in a charged cylindrical pore, a particle near a charged plane and others. Two‐dimensional colloidal crystals are also investigated. Different electrical models of colloidal particles are considered and discussed. Special attention is given to the effects of the geometrical confinement and non‐linearity. In particular, the famous problem [3] of the long‐range attraction for two confined like‐charged colloidal particles is studied numerically. Some properties of colloidal interaction are not described by the Poisson‐Boltzmann equation. One possible generalization of the Poisson‐Boltzmann approach taking into account the ion correlation is briefly discussed.