Effective Parameters of Charged Spherical Particles in 1 : 1 Electrolyte Solutions
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ctive Parameters of Charged Spherical Particles in 1 : 1 Electrolyte Solutions A. I. Dolinnyi* Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, Moscow, 119071 Russia *e-mail: [email protected] Received June 22, 2020; revised July 1, 2020; accepted July 6, 2020
Abstract—In scope of the Poisson–Boltzmann theory, the electrostatic potential profiles in the vicinity of spherical particles immersed in 1 : 1 electrolyte solutions have been precisely calculated. Using the data on the behavior of the profiles at large distances from the particle surface, effective surface potential ψ eff , and its limiting value ψ sat eff , to which it tends upon an infinite growth of the surface charge, have been determined for a wide range of model parameters (surface charge density, particle radius, and electrolyte concentration). A universal curve has been plotted to represent the dependence of ψ sat eff on reduced particle radius κa (a is the radius and κ is the reciprocal Debye screening radius) and to evidently illustrate the existence of two known limiting laws of variations in the effective potential that corresponds to the saturation conditions. The energy criterion and the analysis of its sensitivity to the cutoff threshold have been employed to evaluate the thicknesses of the shells formed by immobilized counterions around the spherical particles. Dependences of the shell thickness on the surface charge density, particle radius, and 1 : 1 electrolyte concentration have been sat analyzed. It has been revealed that there is limiting thickness leff , which is reached upon the infinite growth sat of the surface charge density. A universal κleff (κa) curve is presented and compared with the ψ sat eff (κa) curve. DOI: 10.1134/S1061933X20060034
INTRODUCTION Previously [1], the modified Poisson–Boltzmann (PB) theory was used to study (under the conditions of a constant surface charge density) the distribution of the electrostatic potential and the concentration of ions near a spherical 15-nm particle immersed in a 0.1 M 1 : 1 electrolyte solution. The modified PB theory comprised limitations on maximum allowable concentration Cmax of ions in a solution, with this concentration being determined by their effective sizes. There are three following results of this work, which are of importance for understanding the structure of the electrical double layer (EDL) formed around a charged spherical particle. (1) When absolute value ϕ ( r ) of the electrostatic potential becomes lower than its thermal value kT ϕT = B (where kB is the Boltzmann constant, T is e the temperature, and e is the elementary charge) while moving away from the particle surface, the potential drop is, irrespective of ion sizes, described by the following equation characteristic of the linearized PB theory [2–8]:
ψ (r ) =
ϕ (r ) = ψ eff a exp ( −κ ( r − a )) . r ϕT
(1)
Here, r is the distance from the center of a particle, a the particle radius, κ the reciprocal Debye screening radius, and ψ eff = lim ψ ( r ) is the effective s
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