Model of large volumetric capacitance in graphene supercapacitors based on ion clustering

Brian Skinner, M. M. Fogler, and B. I. Shklovskii

Phys. Rev. B 84, 235133 – Published 21 December 2011

Abstract

Electric double-layer supercapacitors (SCs) are promising devices for high-power energy storage based on the reversible absorption of ions into porous conducting electrodes. Graphene is a particularly good candidate for the electrode material in SCs due to its high conductivity and large surface area. In this paper, we consider SC electrodes made from a stack of graphene sheets with randomly inserted spacer molecules. We show that the large volumetric capacitances $C ≳ 100 F / {cm}^{3}$ observed experimentally can be understood as a result of collective intercalation of ions into the graphene stack and the accompanying nonlinear screening by graphene electrons that renormalizes the charge of the ion clusters.

Received 27 September 2011

DOI:https://doi.org/10.1103/PhysRevB.84.235133

Authors & Affiliations

Brian Skinner¹, M. M. Fogler², and B. I. Shklovskii¹

¹Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
²Department of Physics, University of California-San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA

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Vol. 84, Iss. 23 — 15 December 2011

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Images

Figure 1
A schematic of a graphene SC electrode. Graphene sheets are arranged in a stack that is placed in contact with a reservoir of ionic liquid. As a voltage is applied between this electrode (which we take to be the negative electrode) and the opposite electrode (not shown), cations (red circles with +'s) are driven to intercalate between graphene layers to neutralize the electrode's negative electronic charge.Reuse & Permissions
Figure 2
A schematic of a portion of the GS electrode, shown in a side view. The stack of graphene sheets (black lines), arranged with a separation $c$ between layers, has some concentration of spacer molecules (tan-colored circles) in it. When a voltage is applied, ions (red circles with +'s) enter the electrode collectively in the form of disks whose typical size is determined by the average distance $R$ between spacers in a given plane.Reuse & Permissions
Figure 3
The capacitance as a function of voltage for the hypothetical case where $α = 1$ and the disk charge density $σ_{0} = e / c^{2}$ , plotted for different values of disk size $R$ . At $R / a \sim 1$ , the elastic energy-mediated attraction between ions is eliminated, and the capacitance follows the mean-field quantum capacitance given by Eq. (14) (thin black line). At $R \to \infty$ , ions enter the GS in stages, and the capacitance follows the result of Eq. (24) (dashed curve). The behavior for finite $R$ , derived in Sec. 5, is shown schematically by the red and blue thick lines for two different values of $R$ . Proper numerical coefficients are shown on the axes for the thin black and dashed curves, while for the thick curves, the location and height of the peak capacitance are indicated only by an approximate scaling relation.Reuse & Permissions
Figure 4
A schematic of the screening atmosphere surrounding a disk of ionic charge (rectangle) with diameter $R$ inside the graphene stack. The limit of TF screening, which corresponds to a distance $z_{TF}$ from the disk, is shown by the dashed line. The region outside the TF region corresponds to linear screening. The characteristic decay length $z_{0}$ of the nonlinear potential is indicated by the dotted line. The total amount of charge inside the TF screening atmosphere is denoted as $q$ and is much smaller than the bare charge $σ_{0} R^{2} \sim e {(R / c)}^{2}$ of the disk of ions.Reuse & Permissions
Figure 5
The volumetric capacitance $C$ (thick black curve, left axis) and the stage number $s$ (thin blue curve, right axis) as a function of voltage in the TF approximation for turbostratic graphite ( $R = \infty$ ) with $σ_{0} = e / {(1 nm)}^{2}$ and $ɛ_{r} = 3$ . The capacitance in this plot is normalized to the volume of stage-1 graphite with intercalated ions of diameter $a = 1 nm$ . The dashed black line is an approximation for $C (V)$ at large voltage, given by Eq. (29).Reuse & Permissions

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