Mean-field description of ionic size effects with nonuniform ionic sizes: A numerical approach

Ionic size effects are significant in many biological systems. Mean-field descriptions of such effects can be efficient but also challenging. When ionic sizes are different, explicit formulas in such descriptions are not available for the dependence of the ionic concentrations on the electrostatic potential, that is, there is no explicit Boltzmann-type distributions. This work begins with a variational formulation of the continuum electrostatics of an ionic solution with such nonuniform ionic sizes as well as multiple ionic valences. An augmented Lagrange multiplier method is then developed and implemented to numerically solve the underlying constrained optimization problem. The method is shown to be accurate and efficient, and is applied to ionic systems with nonuniform ionic sizes such as the sodium chloride solution. Extensive numerical tests demonstrate that the mean-field model and numerical method capture qualitatively some significant ionic size effects, particularly those for multivalent ionic solutions, such as the stratification of multivalent counterions near a charged surface. The ionic valence-to-volume ratio is found to be the key physical parameter in the stratification of concentrations. All these are not well described by the classical Poisson–Boltzmann theory, or the generalized Poisson–Boltzmann theory that treats uniform ionic sizes. Finally, various issues such as the close packing, limitation of the continuum model, and generalization of this work to molecular solvation are discussed.

Figure 1

Ionic concentrations in the midplane $z=40\hspace{0.5em}\AA$.

Figure 2

Convergence histories.

Figure 3

Log-log plot of the CPU time vs the number of grids for both SMPBmove and AugLagMulti.

Figure 4

Counterion concentrations vs distance in $\AA$ngstroms to the charged surface with different ionic sizes and with no ionic sizes (the classical PB theory). The linear size of counterions is $a_1.$ The counterion concentration at the charged surface is $1/a_1^3=1.666$ M when $a_1=10\hspace{0.5em}\AA$ and is $1/a_1^3=3.254$ M when $a_1=8\hspace{0.5em}\AA .$

Figure 5

Counterion concentration vs the distance in $\AA$ngstroms to the charged surface with different values of the surface charge density $\sigma$.

Figure 6

Concentrations of multivalent counterions vs the distance in $\AA$ngstroms to the charged surface. In the symbols, $+i$ means the concentration of the counterion with the valence $+i$ $(i=1,2,3).$ (a) The classical PB solution: without the size effect. (b) The size effect included with the linear sizes of group I. (c) The size effect included with the linear sizes of group II. (d) The size effect included with the linear sizes of group III.

Figure 7

Concentrations of multivalent counterions vs the distance in Ångstroms to the charged surface. In (a) and (b), ${\alpha }_{+2}>{\alpha }_{+3}>{\alpha }_{+1}$. In (c) and (d), ${\alpha }_{+1}>{\alpha }_{+2}>{\alpha }_{+3}$.