Numerical Studies on Electrical Interaction Forces and Free Energy between Colloidal Plates of Finite Size

Relation of EDL Forces between Clay Particles Calculated by Different Methods

Calculation of the electrostatic double layer force (EDL force) between clay particles is relevant as it is closely related to important macroscopic mechanical behaviors of clays. The popular method to calculate the EDL force is to integrate the electric potential and Maxwell stress along the boundary enclosing a simply connected domain within which a clay particle resides. The EDL force has also been calculated by the integration of the electrostatic force density over the preceding domain. However, the subtle relation of the EDL forces calculated by the different existing methods has not yet been investigated. By means of theoretical analysis and finite element simulation, it was shown that the force calculated by the integration of Maxwell stress along the complete boundary enclosing a multiply connected domain in which the clay particle is excluded, and that along the partial boundary enclosing the preceding simply connected domain represents the electrical attractive force and osmotic repulsive force, respectively, while the integration of the potential along both the same complete and partial boundary denotes the osmotic force. Numerical results showed that the calculated EDL force deviates from its actual value significantly with the decrease in distance between the chosen integral boundary and particle surface, and the deviation varies with surface potential and angle between particles. Moreover, the recommended minimum distance was proposed to be 10 times the thickness of the particle based on the present simulation results.

Numerical Studies on Electrostatic Interaction Forces and the Free Energy between Parallel Colloidal Rods of Finite Size in Skewed Configurations

Formulas for interaction forces F(s) and the free energy G(s) between two parallel charged prismatic rods of various scaled values of d, ψ_s, and L in skewed configurations are obtained, where s is the lengthwise positional difference between the front-end faces of the respective rods, and d is the minimal distance between the opposing faces of the rods, ψ_s is the electric surface potential, L is the length of the rods. To obtain the free-energy function G(s), (i) 3D spatial distributions of the electric potential ψ around two rods were determined by numerically solving the nonlinear Poisson-Boltzmann equation with a finite element method, (ii) with the ψ distributions so determined, the lengthwise interaction electrostatic Maxwell stress tangential to the midplane between the rods was calculated to obtain the (discrete) s dependence of the stress, and (iii) by introducing two different fitting functions, the discrete s dependence was transformed into a continuous force function, F(s), which was then lengthwise integrated to derive G(s). It was found that the curves of G(s) linearly decreased with increasing s between 1 and L + 1 due to a localization of the stress. Although natural, it is of interest that the values of G(0) calculated for rods of various values of d, ψ_s, and L were in good agreement with those of the interaction free energy obtained in our preceding work by the widthwise integration of repulsive electrostatic forces normal to the midplane between the parallel rods in nonskewed configurations.