Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions

Daniel L. Z. Caetano, Sidney J. de Carvalho, Guilherme V. Bossa, and Sylvio May

Phys. Rev. E 104, 034609 – Published 23 September 2021

Abstract

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

Received 28 May 2021
Accepted 1 September 2021

DOI:https://doi.org/10.1103/PhysRevE.104.034609

Physics Subject Headings (PhySH)

Capacitance Elasticity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Daniel L. Z. Caetano

Institute of Chemistry, State University of Campinas (UNICAMP), Campinas, São Paulo 13083-970, Brazil and Center for Computational Engineering and Sciences, State University of Campinas (UNICAMP), Campinas, São Paulo 13083-970, Brazil

Sidney J. de Carvalho and Guilherme V. Bossa ^*

Department of Physics, São Paulo State University (UNESP), Institute of Biosciences, Humanities and Exact Sciences, São José do Rio Preto, São Paulo 15054-000, Brazil

Sylvio May

Department of Physics, North Dakota State University, Fargo, North Dakota 58108, USA

^*guilherme.vbossa@outlook.com

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Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
Schematic illustration of a spherical macroion of radius $R_{m}$ immersed in a symmetric 1:1 electrolyte: an aqueous solution containing monovalent cations and anions of bulk concentration $n_{0}$ each. The macroion carries a uniform surface charge density $σ$ .
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Figure 2
Predictions of the classical Poisson-Boltzmann model: (a) shows the scaled surface charge density $p = 2 π l_{B} l_{D} σ / e$ as a function of the surface potential $Ψ_{0}$ calculated according to Eq. (17). In (b), we show the scaled differential capacitance $C_{diff} l_{D} / ε_{w} ε_{0}$ as a function of $p = 2 π l_{B} l_{D} σ / e$ . In both plots, different colors indicate different values of the scaled macroion curvature: $l_{D} c = - 0.4$ (black), $l_{D} c = - 0.2$ (red), $l_{D} c = 0$ (green), $l_{D} c = 0.2$ (blue), and $l_{D} c = 0.4$ (gray).
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Figure 3
(a) and (c) Scaled differential capacitance $C_{diff} l_{D} / ε_{w} ε_{0}$ as a function of scaled surface charge density $p = 2 π l_{B} l_{D} σ / e$ . Different colors indicate different macroion radii: $R_{m} = 2 nm$ (green), $R_{m} = 3 nm$ (blue), $R_{m} = 4 nm$ (red), $R_{m} = 5 nm$ (black), and the limiting case $R_{m} \to \infty$ (gray). (a) is for the LG model, and (c) is for the CS model. (b) and (d) Scaled differential capacitance $C_{diff} l_{D} / ε_{w} ε_{0}$ vs macroion curvature $1 / R_{m}$ . Different colors correspond to different values of the surface charge density: $σ = 0.01 e / {nm}^{2}$ (black), $σ = 0.05 e / {nm}^{2}$ (red), $σ = 0.2 e / {nm}^{2}$ (blue), and $σ = 0.4 e / {nm}^{2}$ (green). (b) is for the LG model, and (d) is for the CS model. In all four plots, circles are predictions obtained from Monte Carlo simulations, and solid lines indicate the results from numerically solving the self-consistency relation in Eq. (11). The dashed lines in (b) and (d) are results from the small-curvature expansion according to Eqs. (14, 15, 16). The four choices of $σ$ displayed in (b) and (d) are indicated at the bottom of (a) and (c) by small color-matching arrowheads.
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Figure 4
Scaled differential capacitance $C_{diff} l_{D} / ε_{w} ε_{0}$ vs scaled surface charge density $| p | = 2 π l_{B} l_{D} | σ | / e$ for $R_{m} = 2 nm$ (green), $R_{m} = 3 nm$ (blue), $R_{m} = 4 nm$ (red), $R_{m} = 5 nm$ (black), and the limiting case $R_{m} \to \infty$ (gray). Solid lines correspond to the LG model, and dashed lines correspond to the CS model, both calculated according to Eq. (11). The dotted gray line is the prediction of the classical Poisson-Boltzmann model for flat geometry ( $R_{m} \to \infty$ ) according to Eq. (18).
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Figure 5
The constants $τ$ (a) and $β$ (b) as a function of the scaled surface charge density $| p | = 2 π l_{B} l_{D} | σ | / e$ . Symbols (black circles) are data from Monte Carlo simulations, and dashed lines correspond to mean-field results obtained according to the classical Poisson-Boltzmann theory (red), the LG model (blue), and the CS model (green). The solid lines connecting the black circles serve as a guide to the eye.
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