On the Gouy–Chapman–Stern model of the electrical double-layer structure with a generalized Boltzmann factor
Introduction
The electrical double-layer (EDL) is more than a century-old fundamental topic of electrochemistry and physical chemistry [1], [2], [3], [4], [5]. It is found in many applications such as in membrane science and technology [6], [7], water desalination by capacitive ion removal [5], [8], [9], capacitive mixing [10], [11], [12], energy storage in porous electrodes and fractional-order capacitors [4], [13], [14], [15], [16], [17], colloidal suspensions [18], [19], etc. It refers to the general phenomenon of charge separation across an interfacial surface, which can be imagined as a plane that separates two adjacent and distinct phases (e.g. metal and electrolyte) having excess electronic or ionic charges (see Fig. 1). At such an electrified interface, the electrode charge is not necessarily constant, but on the contrary fluctuates in response to the stochastic thermal motions in the adjacent electrolyte. These fluctuations are related to a set of configuration variables including the number, positions and momenta of mobile ions of each species near the electrode, and become important as the correlation between species increases [20], [21]. The extent of these correlations is related to the nature of the interfacial liquid, and is expected to increase with the increase of ions concentration as well as in nanometer-sized confined geometries at the electrode surface. As a result the electrical properties of within small scales of the electrode, such as the differential capacitance, electrostatic potential or accumulated charge density for instance, are expected to be affected by the statistics of these microscopic fluctuations [21], [22].
Several computational models such as Monte-Carlo (MC), molecular dynamics (MD) and density functional theory (DFT) simulations are now very apt to provide good descriptions of the EDL structure [9], [12], [22], [23]. However, because of the computational resources needed for the explicit microscopic analysis of large numbers of interacting solvent and mobile ions, mean-field calculations remain popular due their simplicity and comparatively good accuracy [4], [5], [13]. The classical Gouy–Chapman–Stern (GCS) approach in which the Poisson–Boltzmann (PB) equations are used for the treatment of the diffuse part of the solvent (reviewed in Section 1) is widely used because it is verified to be asymptotically correct in the weak-coupling regime and it provides the essential of the EDL properties [1], [2], [5], [7], [24], [25], [26]. It assumes the equilibrium ion concentrations to decay exponentially with the distance from the interface following a Boltzmann distribution function with respect to the mean electrostatic potential energy in the EDL. This statistical representation is considered and widely used as the generic case for thermal equilibrium of many uncorrelated or weakly-correlated systems. However, when effects such as ion–ion correlations, ion polarizability, finite size of ions, electrostriction and dielectric saturation of the solvent cannot be ignored, the GCS model becomes less reliable in describing the features of the EDL [2], [20], [27]. Furthermore, if the electrode surface is fractal and/or consisting of small confined geometries, the stochastic nature of state variables associated with the mobile ions for instance is also expected to deviate from Boltzmann’s distribution profile [28].
The purpose of this study is to incorporate into the mean-field GCS model the Tsallis -deformed exponential distribution [28], [29], [30], [31], [32], [33], [34] for the ion concentrations in the interfacial liquid (Section 2). The same analysis can be applied in principle to other approaches dealing with the EDL structure. The -exponential function we are considering allows the presence of random fluctuations of ion concentrations around their mean values, which in turn provides a sort of a generalized Boltzmann factor. With this approach, one can still analyze a microscopic system in which correlations are expected as if they were absent [28]. We provide analytical solutions for the charge density and differential capacitance from the boundary-value PB problem parameterized with the value of that can be attributed to the strength of elementary noise sources in the EDL structure. A similar approach that came to our attention during the writing of this paper was reported by Garcia–Morales et al. [28], in which the focus was mainly on the -dependence of the PB equation for describing counterion concentration around a charged surface. However, this study is further concerned with the overall capacitive performance of the EDL structure by taking into account the series combination of the Helmholtz capacitance and the diffuse layer capacitance for a symmetric electrolyte, which was not provided in Ref. [28]. We also evaluate the effect of the values of (taken between −1 to 1) on the EDL capacitance vs. voltage profiles which allowed to capture both the traditional inverse bell-shaped curves for aqueous solutions and bell-like curves measured for ionic liquids.
Section snippets
The Gouy–Chapman–Stern model
The standard model to describe the equilibrium EDL structure at the junction metal/electrolyte (dilute solutions) is the GCS mean-field model which can quantitatively explain most experimental results. It consists of subdividing the EDL into (i) a first sheet of uniform constant electronic charges on the metal surface, (ii) a charge-free inner compact layer or Helmholtz layer (also known as Stern layer) of a few angstrom in width and constant charge, and (iii) an outer, semi-infinite diffuse
The generalized Boltzmann factor
The Boltzmann distribution function, as mentioned above, provides a valid approximation only under certain assumptions for equilibrium thermodynamic [25], [39], [40]. While it allows a relatively accurate description of macroscopic systems in which a very large number of stochastic events take place, when one goes to smaller scales for instance, such a description breaks down [41]. It is expected that at small dimensions, fluctuations of some intensive quantity such as temperature or pressure
The extended Gouy–Chapman–Stern model
By applying generically the generalized Boltzmann factor given by Eq. (13) to the concentrations of charged ions in the EDL structure (instead of Eq. (1)), we write the -exponential relation [30]: Again, this means that the fluctuations of the mean-field value of the ion concentrations are assumed to follow a gamma distribution with the parameter representing the extent of these fluctuations. The adequate determination of the statistical distribution of such ions
Discussion
From Fig. 4, it is evident that incorporating the generalized Boltzmann factor for the ions concentrations into the mean-field GCS model affects greatly the capacitive performance of the EDL structure. It is understood that other electrical characteristics are also impacted, but the focus of the discussion here is on the differential EDL capacitance, and thus energy storage applications. The strength of the fluctuations increases as the value of deviates further from unity at which the
Conclusion
In this paper, we analyzed the effect of superposing fluctuations onto the mean-value Boltzmann distribution of ion concentrations in the GCS model on the capacitive properties of the EDL structure. This was done via the -exponential function which allows to embrace a spectrum of empirical processes connected to the degree of fluctuations in the intensive inverse temperature parameter. This compact and efficient approach provides an extended version to the classical GCS model specifically for
CRediT authorship contribution statement
Anis Allagui: Conceptualization, Methodology, Analysis, Software, Simulation & Validation, Writing - review & editing. Hachemi Benaoum: Analysis, Reviewing & editing. Oleg Olendski: Reviewing & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
We thank Mathijs Janssen for useful discussions and comments on an earlier version of the manuscript.
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