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Edited by: Yaser A. Abu-Lebdeh, National Research Council Canada (NRC-CNRC), Canada

Reviewed by: Ali Kachmar, Qatar Environment and Energy Research Institute, Qatar; Yufeng Zhao, Yanshan University, China

Specialty section: This article was submitted to Energy Storage, a section of the journal Frontiers in Energy Research

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Supercapacitors deliver higher power than batteries and find applications in grid integration and electric vehicles. Recent work by Chmiola et al. (

The renewable energy (RE) is gaining prominence over fossil fuels giving rise to the necessity of energy storage systems for RE integration. Electrochemical storage systems can be deployed at various scales unlike mechanical storage systems. They can be used in large grid-scale, small scale off-grid applications including electric vehicles and consumer electronics. The Li-ion battery provides the best electrochemical properties but is very expensive for large-scale deployment. Lead acid batteries, however, are cheaper but exhibit lower energy and power density (Parfomak,

In 2006, Chmiola et al. (

The classical double-layer theories have limitations in explaining the behavior at confined regions of double-layer supercapacitors. Continuum level theories for double-layer formation such as the Gouy–Chapman theory, Stern theory are limited to treatment of dilute or moderate concentrations of electrolytes (Kornyshev,

The main challenge in modeling EDLCs is to simulate the electrode polarizability, which directly affects ionic organization at its interface with the electrolyte. It may also be required to further calculate electron transfer processes between the electrode and electrolyte (Petersen et al.,

Molecular simulation techniques (Kiyohara and Asaka,

We have studied mechanism of EDLC using Monte Carlo simulations in Gibbs ensemble. The study consists of simulating the two electrodes in separate simulation boxes, with the positive electrode in box I and the negative electrode in box II (Figure

Schematic of two box EDLC model. A microscopic region of each electrode is simulated in separate simulation box. Periodic boundary conditions are applied in all the three directions to get macroscopic properties.

The partition function (Kiyohara and Asaka,

Each simulation box is electrically neutral, i.e.,

The Monte Carlo moves used in the simulations are explained below. Their acceptance criteria are derived by application of detailed balance. An electrode is chosen randomly for each step following Monte Carlo moves.

Translation move: in a translation move, an ion is displaced randomly (with random magnitude and direction). The acceptance criteria is given by the following equation:

Insertion move: in this move, an ion pair is inserted in the pores of an electrode. The acceptance criteria is given by the following equation:
^{2}e^{βμ} is the activity of an ion pair.

Deletion move: in this move, an ion pair is deleted from the pores of an electrode. The acceptance criteria is as follows:

Charge plus ion transfer: in this move, a randomly picked ion is transferred from one electrode to the other with simultaneous transfer of charge to maintain constant net charge in each box. Acceptance criteria for an anion transfer from box I to box II is given by following equation:

CV move: this move takes into account charge polarization on electrode atoms induced by electrolyte molecules. In such a move, two neighboring atoms are picked randomly. Random amount of charge is added in one atom and subtracted from the other, ensuring net change in total charge is 0. Since this change occurs within an electrode, no work is done against the voltage. The acceptance criteria is given by,

Our model for electrolyte is representative of a pure IL phase consisting of charged hard spheres. We use restricted primitive model (RPM) to simulate these IL. Both the cations and anions are monovalent, with same diameter “_{ij}_{i}_{j}_{ij}

The length scale is the hard sphere diameter of the ions,

The volume scale is volume of unit cell, ^{3}.

The charge scale is electronic charge, ^{−19} C.

The temperature scale is ε/_{B}.

The energy scale is ε = ^{2}/(4πε_{0}ε_{r}

The voltage scale is ε/_{0}ε_{r}

The capacitance scale is ^{2}/ε = (4πε_{0}ε_{r}

The interactions between atoms, which are beyond cutoff distance, are calculated as long range interaction energy. The long-range interactions are calculated using Ewald summation in Fourier space. The variable ε_{0} is the permittivity of free space, and ε_{r} is the dielectric constant of the solvent. The value of the dielectric constant is not known accurately for modeling the ILs by the RPM model. The value ε_{r} = 2 reflects electronic polarizability of simple ions. For complex IL molecules, ε_{r} is >2 due to other non-electronic internal degrees of freedom (Shim and Kim,

Electrodes are modeled as three-dimensional porous crystalline structures, which resemble a cubic wire mesh. Initial configuration consists only of electrode atoms, with equal charge on each atom. Positions of the atoms are fixed, but unlike previous work (Limmer et al.,

Schematic of three different structures of electrodes used in the simulations. Geometries

A simulation consists of 10^{4} equilibrium and 10^{5} production cycles, wherein each cycle consists of 100 or _{A} + _{C}) × ^{3}/

Values of the activity,

10^{−1} |
3.0 | 0.114291 |

10^{0} |
3.0 | 0.168531 |

10^{1} |
3.0 | 0.218011 |

10^{2} |
3.0 | 0.258951 |

10^{3} |
3.0 | 0.296231 |

The activity value of 10^{3} led to a close match between our simulations and the experiments reported in Largeot et al. (^{2}), which are unfavorable for EDLC performance. Hence, they need to be used at high temperature (400 K). The parameters used in the simulations are, electrolyte ion diameter; _{r} = 2, and voltage in the range 0.1–1.75 V. Each simulation is run five times. The averages of these five independent simulations with errors bars (95% confidence interval) are reported in the results section.

We modeled the electrode polarizability with Monte Carlo simulations in Gibbs ensemble by implementing the CV move (Punnathanam, ^{−3} for all the three electrode geometries. The results can be analyzed into three parts as, the effects of confinement/pore size, voltage, and CV moves on the overall capacitance (Figures

Charge density of electrodes versus applied voltage for geometry (a). The simulations were performed for ^{−3},

Specific capacitance of electrodes versus applied voltage for geometry (a). The simulations were performed for ^{−3},

Charge density of electrodes versus applied voltage for geometry (b). The simulations were performed for ^{−3},

Specific capacitance of electrodes versus applied voltage for geometry (b). The simulations were performed for ^{−3},

Charge density of electrodes versus applied voltage for geometry (c). The simulations were performed for ^{−3},

Specific capacitance of electrodes versus applied voltage for geometry (c). The simulations were performed for ^{−3},

Comparison of specific capacitance versus voltage for all the three geometries. The blue, red, and black points represent the results for geometry (a), (b), and (c), respectively.

Geometry (a) is the most confined structure studied. Charge density is expressed as the ratio of reduced charge and the number of electrode atoms and directly affects the capacitance. Figure

We simulated the system at moderately high packing fraction. This resulted in relatively low acceptance of insertion/deletion moves. An increase in capacitance with electrode pore size (for pores >2 nm in diameter) is usually seen in the literature, due to higher adsorption in larger pores owing to their increased surface area. In our work, the geometries (b) and (c) exhibit this behavior. In these pores, the diffuse layer of the electrolyte ions adjacent to the electrode surface accounted for most of the potential drop. However, in subnanometer pores, in pores whose dimensions are closer to that of the ionic diameters, the ions distort their solvation shells while entering the pores. This effectively reduces the screening due to the solvation shell and increases the net interaction between electrolyte and the electrode, thereby leading to an increased capacitance (Figure

Geometries (b) and (c) have higher pore width (and volume) than geometry (a). It can be seen from the Figures

Monte Carlo simulations of EDLC have been carried out in constant voltage Gibbs ensemble with simple RPM. Values of differential capacitance (with different pore widths) are obtained at different voltages. The decreasing trend of capacitance with increasing voltage in geometry (a) (Figure

The larger pores; geometries (b) and (c) have more space to accommodate additional ions leading to higher capacitance. The electrode structures are basically toy models and hence contribute only qualitatively in the study. However, with the same simulation methodology, one can build more realistic electrode/electrolyte atomic models, by simply incorporating additional Monte Carlo moves for the molecular degrees of freedom. Therefore, the methodology is easily transferable to model complicated systems. Future studies involving such aspects would contribute toward better understanding and design of high performance materials for supercapacitors.

GP, KR and VK have contributed equally to the analysis. PK has helped in drafting the article and critical revision of the work.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge Dr. Mridula Bhardwaj (CSTEP, Bangalore, India) and Prof. Sudeep Punnathanam (IISc, Bangalore, India) for the exciting discussions on this subject, constant support, and encouragement. The financial support has been provided by the grants obtained under the IRPHA program of DST, GoI, and the US-India Partnership to Advance Clean Energy-Research (PACE-R) for the Solar Energy Research Institute for India and the United States (SERIIUS), funded jointly by the U.S. Department of Energy (Office of Science, Office of Basic Energy Sciences, and Energy Efficiency and Renewable Energy, Solar Energy Technology Program, under Subcontract DE-AC36-08GO28308 to the National Renewable Energy Laboratory, Golden, CO, USA) and the Government of India, through the Department of Science and Technology under Subcontract IUSSTF/JCERDCSERIIUS/2012 dated 22nd November 2012.