As new energy technologies are designed and implemented, there is a rising demand for improved energy storage devices. At present the most promising class of these devices is the electric double-layer capacitor (EDLC), also known as the supercapacitor. A number of recently created supercapacitors have been shown to produce remarkably large capacitance, but the microscopic mechanisms that underlie their operation remain largely mysterious. In this thesis we present an analytical, microscopic-level theory of supercapacitors, and we explain how such large capacitance can result.
Specifically, we focus on four types of devices that have been shown to produce large capacitance. The first is a capacitor composed of a clean, low-temperature two-dimensional electron gas adjacent to a metal gate electrode. Recent experiments have shown that such a device can produce capacitance as much as 40% larger than that of a conventional plane capacitor. We show that this enhanced capacitance can be understood as the result of positional correlations between electrons and screening by the gate electrode in the form of image charges. Thus, the enhancement of the capacitance can be understood primarily as a classical, electrostatic phenomenon. Accounting for the quantum mechanical properties of the electron gas provides corrections to the classical theory, and these are discussed. We also present a detailed numerical calculation of the capacitance of the system based on a calculation of the system's ground state energy using the variational principle. The variational technique that we develop is broadly applicable, and we use it here to make an accurate comparison to experiment and to discuss quantitatively the behavior of the electrons' correlation function.
The second device discussed in this thesis is a simple EDLC composed of an ionic liquid between two metal electrodes. We adopt a simple description of the ionic liquid and show that for realistic parameter values the capacitance can be as much as three times larger than that of a plane capacitor with thickness equal to the ion diameter. As in the previous system, this large capacitance is the result of image charge formation in the metal electrode and positional correlations between discrete ions that comprise the electric double-layer. We show that the maximum capacitance scales with the temperature to the power -1/3, and that at moderately large voltage the capacitance also decays as the inverse one third power of voltage. These results are confirmed by a Monte Carlo simulation.
The third type of device we consider is that of a porous supercapacitor, where the electrode is made from a conducting material with a dense arrangement of narrow, planar pores into which ionic liquid can enter when a voltage is applied. In this case we show that when the electrode is metallic the narrow pores aggressively screen the interaction between neighboring ions in a pore, leading to an interaction energy between ions that decays exponentially. This exponential interaction between ions allows the capacitance to be nearly an order of magnitude larger than what is predicted by mean-field theories. This result is confirmed by a Monte Carlo simulation. We also present a theory for the capacitance when the electrode is not a perfect metal, but has a finite electronic screening radius. When this screening radius is larger than the distance between pores, ions begin to interact across multiple pores and the capacitance is determined by the Yukawa-like interaction of a three-dimensional, correlated arrangement of ions.
Finally, we consider the case of supercapacitor electrodes made from a stack of graphene sheets with randomly-inserted "spacer" molecules. For such devices, experiments have produced very large capacitance despite the small density of states of the electrode material, which would seem to imply poor screening of the ionic charge. We show that these large capacitance values can be understood as the result of collective entrance of ions into the graphene stack (GS) and the renormalization of the ionic charge produced by nonlinear screening. The collective behavior of ions results from the strong elastic energy associated with intercalated ions deforming the GS, which creates an effective attraction between them. The result is the formation of "disks" of charge that enter the electrode collectively and have their charge renormalized by the strong, nonlinear screening of the surrounding graphene layers. This renormalization leads to a capacitance that at small voltages increases linearly with voltage and is enhanced over mean-field predictions by a large factor proportional to the number of ions within the disk to the power 9/4. At large voltages, the capacitance is dictated by the physics of graphite intercalation compounds and is proportional to the voltage raised to the power -4/5. We also examine theoretically the case where the effective fine structure constant of the GS is a small parameter, and we uncover a wealth of scaling regimes.