Investigation of the Electrical Double Layer with a Graphene Electrode

Article pubs.acs.org/JPCC

Investigation of the Electrical Double Layer with a Graphene Electrode by the Grand Canonical Monte Carlo Simulation Rafał Górniak and Stanisław Lamperski* Department of Physical Chemistry, Faculty of Chemistry, A. Mickiewicz University of Poznań, Umultowska 89b, 61-614 Poznań, Poland ABSTRACT: Structural and thermodynamic properties of the electric double layer with a graphene electrode are investigated by the grand canonical Monte Carlo simulations. The nonelectrostatic carbon−ion and ion−ion interactions are described by the Lennard-Jones potential. The results (the ion singlet distribution functions, the mean electrostatic potential, the integral, and the differential capacitance) for an explicit corpuscular structure are compared with those obtained for the structureless carbon sheet and hard surface electrodes. Simulations are carried out for 1.0 mol/dm3 1:1 electrolyte at T = 298.15 K and εr = 78.5 in the range of the electrode surface charge density from −0.9 to +0.9 C/m2. The surface density of carbon atoms in graphene is 5.03066 × 1019 m−2. The singlet distribution functions of ions show that the ion adsorption at the carbon electrodes is evidently stronger than that at the hard surface electrode. The profiles of the mean electrostatic potential near the positively charged carbon electrodes have a minimum that is characteristic of divalent anions near a hard surface electrode. For both carbon electrodes, the integral and differential capacitance curves have the bell shape with a broad maximum, while the curve for the hard surface electrode has a camel-like shape with two humps. The difference between the graphene and structureless carbon electrodes is manifested mainly in capacitance. In the range of a small magnitude of electrode charges, the capacitance results for the graphene electrode are smaller than those for the structureless carbon electrode.

1. INTRODUCTION The most common electrode materials used in the electrical double layer capacitors, also called super capacitors, are based on carbon. The materials like activated carbon,1 nanotubes,2 graphene, 3 and hybrid films like graphene/nanotubes4 are characterized by relatively low costs of production, good mechanical properties, and high-energy storage. Among the carbon-based electrodes, the most interesting are graphene ones and their modifications as they have not only good catalytic,5 electronic,6 and conductivity7 properties but, in particular, have a large surface area, approximately 2630 m2/g, and a high capacitance,8,9 as has been shown by numerous experimental and theoretical works.10−15 The experimental value of graphene capacitance exceeds 75 F/g.16,17 Although in some earlier theoretical papers on the electrical double layer (EDL) the soft interaction with electrodes has been studied, the main emphasis has been put on the electrolyte side. For example, Lanning and Madden18 have considered molten salt (KCl) near rigid charged walls, which consisted of soft spheres of lithium atoms. Fedorov and Kornyshev19 have analyzed ionic liquids near two parallel square lattices that were composed of soft-sphere carbon atoms. Trulsson et al.20 have used a charged linear pentamer composed of tethered Lennard-Jones monomers to model the interfacial properties of ionic liquids. They showed the importance of dispersion interactions at low permittivity by comparing their results with those obtained for hard interactions. Fedorov and Linden-Bell21 have been probing © 2014 American Chemical Society

the neutral graphene/1,3-dimethyimidazolium chloride (ionic liquid) interface by molecular dynamics simulation. The primary aim of this paper is to study the properties of the corpuscular graphene electrode/aqueous 1.0 mol/dm3 electrolyte interface by Monte Carlo simulations in the grand canonical ensemble. We also want to establish the effect of the explicit corpuscular structure of the graphene electrode on the properties of the EDL. To do this, we consider another carbon electrode that is a sheet of structureless carbon (SC) atoms. The surface density of carbon atoms is the same as that in the corpuscular graphene. Later, we will refer to this second electrode as the SC electrode. The nonelectrostatic interactions of these two carbon electrodes are modeled by a soft potential. The hard potential is commonly used in theoretical and simulation investigation of EDL. Thus, another objective of this study is to show how the replacement of a hard by a soft potential influences the properties of EDL.

2. MODEL AND METHODS We consider three models of electrodes, corpuscular graphene, SC, and hard surface. In the first model, the electrode is a honeycomb structured corpuscular graphene. The positions of the carbon atoms are fixed. Each carbon atom has a point partial charge. Ions are the soft spheres with point electric Received: November 28, 2013 Revised: January 15, 2014 Published: February 5, 2014 3156

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⎧∞ for rij < d ⎪ ⎪ uij = ⎨ zizje0 2 ⎪ for rij ≥ d ⎪ 4πε0εrrij ⎩

charge in the center. They are immersed in a continuous dielectric medium. The soft ion−ion and carbon−ion interactions are described by the Lennard-Jones (LJ) potential ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ r r uLJ = 4ε⎢⎢⎜⎜ 0 ⎟⎟ − ⎜⎜ 0 ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

The overall potential of electrode−ion interactions is given by23 (1)

where r0 and ε are the LJ potential parameters and rij is the intermolecular distance. The electrostatic potential between ions is given by

u iw

zizje 2

uij =

4πε0εrrij

zieq 4πε0εrrij

σ ρA

(3)

The diameter d of hard-sphere ions was equal to r0 of the softsphere ions and equal to 400 pm. The surface density of carbon atoms in graphene was ρA = 5.03066 × 1019 m−2. The simulations were carried out by the Monte Carlo technique in the grand canonical ensemble (GCMC). The length and width of the simulation box were equal to 6800 and 5600 pm, respectively. The periodic boundary conditions were applied to the directions parallel to the electrode surface. The long-range interactions were handled by the Torrie−Valleau25 technique. One electrode was charged, while the second was neutral. The surface charge density of the electrode, σ, varied from −0.9 to +0.9 C/m2. The number of configurations applied to equilibrate the system was 1 billion, while the ion density profiles were calculated from the subsequent 2 billion configurations. The ion density profiles were used to calculate the profiles of the mean electrostatic potential. The integral and differential capacitances were calculated from the electrode potential. The other details of the GCMC technique are described in the literature.25,26 The GCMC simulation requires the knowledge of ionic activity coefficients. They were calculated by the inverse grand canonical Monte Carlo technique.27 The values of individual activity coefficients were the same for both kinds of ions and reached 0.6262 for soft-sphere ions and 0.8050 for hard-sphere ions. We begin the discussion from the presentation of the ion singlet distribution functions for two representative surface charge densities. Figures 1 and 2 show, respectively, the anion and cation singlet distribution function g for the three models of electrodes at σ = 0.05 C/m2. The difference in the singlet distribution functions near the graphene and SC electrodes is subtle for both anions and cations. The distribution functions have two rounded peaks. The first peak is located close to x/r0

(4)

where ρA is the surface density of carbon atoms of the graphene electrode. The distance between carbon atoms is 142 pm. The SC electrode is modeled by a monolayer of carbon atoms uniformly distributed on a planar sheet. The surface density ρA of carbon atoms is the same as that of graphene. The electrode charge is uniformly distributed on a plane that goes through the centers of carbon atoms. The soft interaction between a SC electrode and an ion is described by the LJ 10−4 wall potential22 ⎡ ⎛ ⎞10 ⎛ ⎞4 ⎤ r 2 r u i = 2πρA r0 ε⎢ ⎜ 0 ⎟ − ⎜ 0 ⎟ ⎥ ⎢⎣ 5 ⎝ xi ⎠ ⎝ xi ⎠ ⎥⎦ 2

(5)

where xi is the distance between the ith ion and the electrode in the direction x perpendicular to the electrode surface. The electrostatic ion−SC electrode interaction is given by23 u iw =

zieσxi 2ε0εr

(8)

3. RESULTS AND DISCUSSION We have considered a 1:1 electrolyte with charge numbers z+ = 1 and z− = −1 and concentration c = 1 mol/dm3. The calculations were carried out for temperature T = 298.15 K and the relative permittivity εr = 78.5, which is that of pure water. The LJ parameters were r0 = 400 pm, ε = 400 J/mol for ions and r0 =340 pm, ε = 359.16 J/mol14 for carbon atoms of graphene. The LJ parameters for the ion−carbon atom interaction were calculated from the Lorentz−Berthold mixing rules24 1 εij = εiεj r0 = (r0i + r0j) (9) 2

where rij is the distance between the ith ion and jth carbon atom and q is the point electric charge located in the center of a carbon atom. This charge can be positive or negative depending on the surface charge density σ of the electrode. It is calculated from the formula

q=

⎧ d for xi < ⎪∞ ⎪ 2 =⎨ σ z e x d ⎪ i i for x ≥ i ⎪ 2ε ε 2 ⎩ 0 r

The effect of images is not considered in this investigation, which means that the permittivity of an electrode material is the same as that of a solvent.

(2)

where zi is the charge number of the ith ion, e is the elementary charge, ε0 is the vacuum permittivity, and εr is the relative permittivity of the continuous dielectric medium. The potential of the electrostatic interaction between a charged carbon atom of graphene and an ion is u iq =

(7)

(6)

In the third model, the ions are represented by hard spheres with point electric charge ze embedded in the center. The diameter of a hard sphere is d, and it is the same for anions and cations. The ions are immersed in a homogeneous medium of the relative electrical permittivity εr. This is the primitive model of an electrolyte. The electrode is modeled by a uniformly charged planar hard surface. The interionic potential includes the hard-sphere and electrostatic interactions 3157

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Figure 1. Anion singlet distribution function for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.05 C/m2.

Figure 3. Anion singlet distribution function for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.4 C/m2.

Figure 2. Cation singlet distribution function for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.05 C/m2.

Figure 4. Cation singlet distribution function for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.4 C/m2.

= 1, and the second one is formed near x/r0 = 2. Although the first peak assigned to cations is smaller than that of anions because of electrostatic repulsion, it is still high. This is a result of the attraction arising from the second term of the LJ potential. The second peaks seen in Figures 1 and 2 evidence the formation of the second layer of ions. It appears because the ions near the electrode surface are crowded. Similar behavior was observed by Fedorov and Lynden-Bell21 for the neutral graphene/ionic liquid interface. It is worth noting here that for the hard surface interface, the second layer of counterions is observed at much higher electrode charges.28 The qualitatively different shape of the singlet distribution function is observed for hard-sphere ions near a hard surface. The anion g function has one sharp peak at a contact distance x/r0 = 1/2. The contact value of g was obtained by extrapolating the g results to the contact distance.29 The anion maximum is smaller than the corresponding ones near the graphene and SC electrodes. There is no second maximum. Cations are repelled from the vicinity of the electrode surface. Figures 3 and 4 present the singlet distribution functions of anions and cations, respectively, at σ = 0.4 C/m2. An increase in the surface charge of the electrode leads to an increase in the

maximum of g functions for anions (Figure 3). The difference between these functions near the graphene and SC electrodes is again very small, but for cations, it is significant. Figure 4 shows that cations are strongly repelled from the surface of the graphene and SC electrodes, but repulsion of the former one is slightly weaker. For both carbon-based electrodes, we again observe formation of the second layer of cations at x/r0 = 2, which is higher than that at σ = 0.05 C/m2. For the hard surface electrode, the cations are totally removed from the closest vicinity of the electrode surface. No maximum is observed near x/r0 = 2. Figures 5 and 6 show the dependence of the mean electrostatic potential against the distance from the electrode surface for σ = 0.05 and 0.4 C/m2, respectively. For the graphene and SC electrodes, the potential minimum is seen at x/r0 ≈ 1. The minimum is slightly deeper for the SC electrode especially for σ = 0.4 C/m2, where the ion−electrode electrostatic interaction is stronger. After leaving the minimum, the potential tends to zero. For the hard surface electrode, the potential decreases monotonically to zero with increasing 3158

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Figure 5. Potential profile for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.05 C/m2.

Figure 7. Dependence of the electrode potential against surface charge for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes.

Figure 6. Potential profile for corpuscular graphene (black circles), SC (red triangles), and hard surface (green squares) electrodes at σ = 0.4 C/m2.

Figure 8. Dependence of the integral capacitance against surface charge for corpuscular graphene (black circles), SC (red triangle), and hard surface (green squares) electrodes.

distance from the electrode surface. No minimum is observed. It is worth mentioning that the potential profile of EDL with divalent anions near the positively charged hard surface electrode shows also the minimum.30,31 Dependencies of the electrode potential against the electrode charge density for the graphene, SC, and hard surface electrodes are shown in Figure 7. For clarity of the graph, we present here only the positive branch of the results. The electrode potential increases with increasing surface charge. The inflection point is at σ = 0. For the graphene and SC electrodes, the difference in the electrode potential is again very small. For the hard surface electrode, we observe two inflection points. The third is at the negative σ. These profiles of the electrode potential have impact on the shape of the integral and differential capacitance of EDL. Figure 8 shows the dependence of the integral capacitance on the surface charge density. For the hard surface electrode, the integral capacitance curve has two maxima near σ = −0.6 and +0.6 C/m2 and a minimum at σ = 0. It is called the camel-like shape with two humps. For the graphene and SC electrodes, the curves have a bell shape with a broad maximum at σ = 0. In the range of σ from −0.6 to +0.6

C/m2, the integral capacitance for the graphene electrode is smaller than that of the SC electrode. Here, the question arises whether the inhomogeneous distribution of electrostatic or of soft interactions is responsible for this difference. To answer this question, we compared the integral capacitance of the EDL, whose electrode was the hard surface with the electric charge distribution: (i) inhomogeneous similar as on the graphene electrode, and (ii) homogeneous similar as on the SC electrode. The simulation results show that there is no noticeable difference in the capacities between the homogeneous and inhomogeneous charge distributions on the electrode. This suggests that the different distribution of carbon atoms is responsible for the difference in the integral capacitance in the range of σ from −0.6 to +0.6 C/m2. This conclusion is supported also by the following evidence. The LJ potential is independent of σ; therefore, at high σ, where the electrostatic interactions are dominant, the capacitance curve is driven by electrostatic interactions, and a subtle difference between corpuscular and uniform distributions of carbon atoms is not manifested. On the other hand, near σ = 0, the soft interactions 3159

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dominate over the electrostatic, and thus, the capacitance difference is due to different models of soft interactions. It was found earlier26 for a hard wall that the capacitance of the EDL around zero surface charge undergoes a transition from having a minimum surrounded by two symmetric maxima at low electrolyte concentrations to having a maximum at higher concentrations. Perhaps the best explanation of this effect has been given by Henderson et al.32 We quote this explanation as it is relevant in some extention to our results. As the electrolyte concentration increases to sufficiently high values, the ion saturation near the electrode occurs even at small electrode charges. When the charge increases, the counterions are accumulated near the electrode surface. However, because of strong electrostatic repulsion between counterions, part of them are located at some distance from the electrode surface. This increases the width of the EDL and decreases the capacitance. The effect starts immediately as the electrode is charged. It results in the capacitance curve having a maximum at σ = 0. For the hard-sphere model of an electrolyte, the minimum−maximum transition is observed at the electrolyte concentration c ≈ 3 mol/dm3. The bulk concentration of our electrolyte is evidently lower, but as follows from Figure 1, there are two layers of counterions near the electrode. The counterion concentration in the first layer is about 25 times higher than that in the bulk, and in the second layer, it is 1.732 times higher. The first layer is responsible for the saturation and repulsion effects, while the second, whose height increases with increasing σ, broadens the EDL and lowers the capacitance. The formation of the second layer is responsible for the bell shape of the capacitance curve with a maximum at σ = 0. In contrast to the results of the earlier study,26 now the presence of the nonelectrostatic attraction between the electrode and ions is responsible for the appearance of the capacitance maximum. This case is a good example supporting the earlier hypothesis explaining the minimum−maximum transition effect. The differential capacitance Cd of EDL is the property that can be directly compared with experiment. The differential capacitance is defined as the derivative of the electrode surface charge density over the electrode potential. This derivative can be calculated numerically, but the results are usually scattered. Thus, we calculated Cd by the interpolation polynomial technique.31 Figure 9 shows the Cd results as a function of σ. The shape of these curves is similar to that of the integral capacitance curves. Similarly as before, the capacitance curve for the SC electrode goes above that for the graphene electrode in the range of small σ, and we observe again a bell-like curve with a maximum at σ = 0. For the hard surface electrode, we observe that the humps of a camel-like curve are located closer to each other and occur at around σ = −0.4 and +0.4 C/m2.

Figure 9. Dependence of the differential capacitance against surface charge for corpuscular graphene (black circles), SC (red triangle), and hard wall (green squares) electrodes.

transition will be observed at a lower concentration for the graphene electrode. Generally, the properties of EDL with the corpuscular graphene and SC electrodes are similar. Some difference occurs in the range of small electrode charges, where the capacitance results for the graphene electrode are smaller than those for the SC one.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are very grateful to Professor C. W. Outhwaite, University of Sheffield for his comments and suggestions. The financial support of Adam Mickiewicz University of Poznań, Faculty of Chemistry, is appreciated.

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REFERENCES

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4. CONCLUSIONS In this paper, we have shown that the properties of the electrical double layer with the graphene electrode are substantially different than those observed for the hard surface electrode. The nonelectrostatic carbon−ion attraction brings an additional contribution to ion adsorption on and near the electrode surface. Strong ion adsorption, even at small electrode charges, is responsible for the curve of the charge dependence of the integral/differential capacitance having a maximum at σ = 0 for the electrolyte concentration c = 1.0 mol/dm3. This is not the case for the hard surface electrode, which has a minimum at σ = 0. We can expect that the minimum−maximum capacitance 3160

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