Features of Electrical Double Layers Formed Around Strongly Charged Nanoparticles Immersed in an Electrolyte Solution. The Effect of Ion Sizes

Abstract

Using the modified Poisson–Boltzmann (PB) theory, which includes restrictions on the maximum attainable concentration of ionic species in a solution C_max determined by their effective sizes, the distributions of electrostatic potential φ(r) and ion concentration near a spherical nanoparticle with radius a immersed in a 1 : 1 electrolyte solution have been studied under the conditions of constant surface charge density σ_s. For weakly charged particles, the φ(r) profiles are almost independent of C_max and coincide with the profile obtained in terms of the classical PB model. Surface potential |φ_s| gradually increases with a rise in |σ_s|. Far from the particle, when the potential becomes lower than its thermal value ${{\varphi }_{{\text{T}}}} = \frac{{kT}}{e}$ (k is the Boltzmann constant, T is the temperature, and e is the elementary charge), the potential decreases exponentially irrespective of the counterion sizes: $\psi \left( r \right) = \frac{{\varphi \left( r \right)}}{{{{\varphi }_{{\text{T}}}}}} = {{\psi }_{{{\text{eff}}}}}\frac{a}{r}\exp \left( { - \kappa \left( {r - a} \right)} \right),$ where r is the distance from the particle center and κ is the reciprocal Debye radius. According to the classical PB theory, the growth of the surface charge leads to the saturation of $\left| {{{\psi }_{{{\text{eff}}}}}} \right|~\left( { \to 4} \right),$ with the value of $\left| {{{\psi }_{{\text{s}}}}} \right|$ obtained by solving the nonlinear PB equation being higher than $\left| {{{\psi }_{{{\text{eff}}}}}} \right|.$ In the modified PB theory, which takes into account the size of electrolyte ions in the simplest form, this effect of saturation is absent. Now, $\left| {{{\psi }_{{{\text{eff}}}}}} \right|$ depends on both the value of the surface charge and the sizes of counterions. Moreover, at a large size of counterions, $\left| {{{\psi }_{{{\text{eff}}}}}} \right|$ substantially exceeds the corresponding value obtained by solving the nonlinear modified PB equation. The difference between the electrical double layer properties obtained by solving the classical and modified PB equations directly follows from the fact that the modified theory predicts the appearance of a condensed layer at a particle surface, with the concentration of counterions in this layer being equal to C_max. Therewith, the thickness of the layer grows with increasing |σ_s| (at a constant size of the ions) and the size of the ions (at constant σ_s).