J.J. López-García

Numerical study of colloidal suspensions of soft spherical particles using the network method. 1. DC electrophoretic mobility

The electrophoretic mobility of a spherical particle coated with a uniformly charged permeable membrane and suspended in a general electrolyte solution is calculated numerically. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as the membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of the ionic species, and electrolyte concentration. The theoretical model used is similar to the one presented by Ohshima (H. Ohshima, J. Colloid Interface Sci. 228 (2000) 190), except for the inclusion in the force balance equation of an additional term corresponding to the force exerted by the liquid on the core of the moving particle. This inclusion is theoretically proven in the limiting case of a nonconducting suspending medium, in which the equation system can be analytically solved. The results obtained coincide with existing analytical expressions when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high.

Poisson-Boltzmann description of the electrical double layer including ion size effects.

The electrical double layer is examined using a generalized Poisson-Boltzmann equation that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which imposes only a restriction on the ability of ions to approach one another. This amplification should allow for an interpretation of the experimental results using reasonable effective ionic radii (close to their well-known hydrated values).

Numerical study of colloidal suspensions of soft spherical particles using the network method: 2. AC electrokinetic and dielectric properties

The network simulation method is used to solve numerically the equation system that determines the dynamic electrophoretic mobility and the dielectric response of dilute suspensions of soft particles. This system was extensively studied theoretically by Ohshima (H. Ohshima, J. Colloid Interface Sci. 233 (2001) 142-152), who obtained analytical expressions for the static and dynamic electrophoretic mobility. However, the validity of his analytical result is restricted to relatively thick membranes with high drag coefficient and to relatively high electrolyte concentrations. As for the dielectric properties, there are only a few works dealing with particles without a core (ion exchange resins) and, to our knowledge, no numerical studies. Our theoretical model is basically similar to Ohshimas, except that we take into account the mechanical force acting on the surface of the core, which he neglects. The inclusion of this term is crucial when the general problem including arbitrary values of the parameters is analyzed. However, it has little bearing when the membrane is thick and the drag coefficient is high, so that our results for the electrophoretic mobility generally confirm Ohshimas equation when all the required conditions are met.

Excluded volume effect on the electrophoretic mobility of colloidal particles.

In a recent work [J. Colloid Interface Sci. 316 (2007) 196] we studied the influence of the excluded volume effect on spatial distributions of ionic species and electrostatic potential in the neighborhood of a suspended spherical particle. It was shown that the excluded volume effect considerably increases the surface potential (for a given value of the particle charge) as compared to the case when ideal ion behavior is assumed. In the present work we extend our previous equilibrium results to the perturbed/nonequilibrium problem and analyze the effect of ion size constraints on the electrophoretic mobility of a rigid spherical particle immersed in a general electrolyte solution. We find that the electrophoretic mobility always increases with the excluded volume effect, which might broaden the range of experimental data that can be interpreted, including those cases where the measured mobility exceeded the theoretical maximum value predicted by the standard model.

Equilibrium electric double layer of charged spherical colloidal particles: effect of different distances of minimum ion approach to the particle surface.

A study of the equilibrium double layer surrounding charged spherical particles is presented, considering that ions in the suspending medium have a finite size. It is assumed that each ionic species has a different minimum approach distance to the particle surface, while the distance of minimum approach between ions in the bulk has the same value for all ion species. Numerical calculations made using the network simulation method and including all the features of the considered model are presented, together with rigorous analytical results valid for a flat interface and point ions in the bulk electrolyte solution. It is shown that the double-layer parameters are very sensitive to the difference between the minimum approach distances of co-ions and counterions. For negative particles and greater approach distances for co-ions than for counterions, the potential always increases with this difference and, under appropriate circumstances, attains positive values leading to charge reversal. This phenomenon is favored by a high electrolyte concentration, high counterion valences, and low surface charge (in modulus). An analytical expression relating these parameters to the threshold value of the difference between the minimum approach distances of co-ions and counterions to the particle surface is presented.

Electrokinetics of suspended charged particles taking into account the excluded volume effect

In two recent works [López-García et al., J. Colloid Interface Sci. 316 (2007) 196; López-García et al., J. Colloid Interface Sci. 323 (2008) 146] we presented a simple modification of the standard electrokinetic model that takes into account the finite size of ions in the electrolyte solution. In the first we presented numerical results for the equilibrium properties while, in the second, we calculated the effect of the excluded ion volume on the electrophoretic mobility. In the present work we first extend our previous results incorporating a distance of closest approach of the ions to the particle surface. We then calculate the conductivity increment and present a detailed interpretation of the mobility and conductivity increment results, based on the analysis of the equilibrium and field-induced ion concentrations and of the convective fluid flow in the neighborhood of the particle surface. We show that the inclusion of the ion size effect generally improves the predictions of the standard electrokinetic model: both the electrophoretic mobility and the conductivity increment increase. We also show that, largely due to the above-noted extension of considering a minimum approach distance between the ions and the particle surface, the excluded volume effect is not negligible even for weakly charged particles.

Suspended particles surrounded by an inhomogeneously charged permeable membrane. Solution of the Poisson–Boltzmann equation by means of the network method

The Poisson-Boltzmann equation is numerically solved for a suspended spherical particle surrounded by a permeable membrane that contains an inhomogeneous distribution of fixed charges. The calculations are carried out using the network simulation method, which makes it possible to solve the problem in the most general case, extending previous results (J.P. Hsu, Y.C. Kuo, J. Membrane Sci. 108 (1995) 107). Approximate analytical expressions for the counterion concentration and the electric potential in the membrane are also presented, together with criteria that determine their ranges of validity. The limiting case of a distribution of fixed charges in the membrane that reduces to a surface charge is also analyzed. It is shown that the solution for this case, considering a vanishingly small radius of the core, reduces to a superposition of solutions corresponding to a charged impermeable particle suspended in an electrolyte solution and to a cavity filled with a charged electrolyte solution.

Equilibrium properties of charged spherical colloidal particles suspended in aqueous electrolytes: finite ion size and effective ion permittivity effects.

The equilibrium properties of a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined using an extension of the Standard Electrokinetic Model that takes into account the finite ion size by modeling the aqueous electrolyte solution as a suspension of polarizable insulating spheres in water. We find that this model greatly amplifies the steric effects predicted by the usual modified Poisson-Boltzmann equation, which only imposes a restriction on the ability of ions to approach one another. This suggests that a solution of the presented model under nonequilibrium conditions could have important consequences in the interpretation of dielectric and electrokinetic data in colloidal suspensions.

Influence of the finite ion size on the predictions of the standard electrokinetic model: frequency response.

An extension into the frequency domain of our previous static and stationary works that modify the standard electrokinetic model taking into account the finite size of ions in the electrolyte solution [J.J. López-García, M.J. Aranda-Rascón, J. Horno, J. Colloid Interface Sci. 316 (2007) 196; J.J. López-García, M.J. Aranda-Rascón, J. Horno, J. Colloid Interface Sci. 323 (2008) 146; M.J. Aranda-Rascón, C. Grosse, J.J. López-García, J. Horno, J. Colloid Interface Sci., in press] is presented. It is shown that the excluded volume effect can be quite substantial in some cases and is not negligible even for weakly charged particles. Furthermore, it generally improves on the predictions of the standard electrokinetic model since the low-frequency dielectric and conductivity increments as well as the electrophoretic mobility increase with the ion size.

Numerical solution of the Poisson–Boltzmann equation for suspended charged particles surrounded by a charged permeable membrane

The equilibrium properties of a charged spherical particle surrounded by a charged permeable membrane and immersed in electrolyte solutions have been investigated using the network simulation method. A network model for the nonlinear Poisson–Boltzmann equation describing the electrostatic potential distribution for this system has been proposed. With this model and an electric circuit simulation program, a series of functions such as the electric potential, the ion concentrations, and the charge densities inside the membrane and in the electrolyte solution can be easily obtained for arbitrary values of the particle radius, particle charge, membrane thickness, membrane charge, and ionic concentrations and charge number. The method proves to be quite general and extremely efficient, as well as applicable to a great variety of double layer compositions.