4.1. Equivalent Circuit Models

Equivalent circuit models employ mathematical or computer models of fundamental electric circuit components, such as resistors and capacitors, to model complex electrochemical processes. Simple equivalent circuits have long been used to

predict the performance characteristics of porous electrodes [39-43]. These equivalent circuits primarily have been applied to attempt to capture the behavior of the double-layer at the interface between the electrode pores and electrolyte solution.

More recently, equivalent circuits have been developed to capture additional Faradaic effects observed in pseudocapacitors [23-24]. The hierarchy of equivalent circuits used to model porous electrodes is presented in Figure 6. This hierarchy begins with a simple capacitor (6a) and adds components one at a time to arrive at the complete equivalent circuit for a porous electrode (6e). In this final equivalent circuit (6e), which is known as a transmission line, the distributed resistances represent the

ESR intrinsic to each pore as the ions from the electrolyte diffuse towards the electrode. The distributed capacitances represent the non-Faradaic double-layer capacitance of each pore [40, 43]. It is important to note, as well, that this equivalent circuit could be modified to model a porous pseudocapacitor electrode by incorporating the Faradaic pore equivalent circuit (6d) [23-24].

4.2. Empirical Relationships

There has also been considerable research on the empirical relationships between pore size, surface area, and capacitance [10, 44-47]. As discussed in section 3.1.1, despite the proportional relationship between surface area and capacitance found in theory, early evidence from physical experiments suggested that surface area and capacitance were uncorrelated [8, 10, 46]. Two competing mathematical models have been developed to explain this discrepancy between theory and experiment. The first

C

_{a. b.}

C R_s

^CDL

c. d.

R_L

C_P

^R_L’

^CDL

S

R_L

_{Electrolyte - Electrode Interface}

Po e

Po e

Po e

Po e

_{Pore Pore Pore Pore}

_e.

Figure 6

Hierarchy of equivalent circuits for porous electrodes: (a) capacitor; (b) capacitor with series resistance; (c) simple double-layer pore circuit: capacitor and leakage resistance in parallel, with series resistance; (d) simple pseudocapacitor pore circuit that builds on (c) by adding a parallel circuit consisting of capacitor in parallel with leakage resistor; (e) transmission line model for a porous electrode that consists of a line of circuits like (c) in parallel. Note that, alternatively, circuits like (d) could be used to model a pseudocapacitor.

model proposes that, because of unique electrosorption behavior found in micropores, the capacitance per micropore surface area and capacitance per external surface area must be calculated separately [10]. The second model, which is now widely accepted, suggests that electrolyte ions cannot diffuse into pores beneath a size threshold and therefore the surface area of those pores cannot contribute to the capacitance [2, 44-46].

In considering the second model, there have been efforts to determine the optimal pore size and size distribution needed to maximize ion accessibility [45-48]. As a corollary result, an inverse relationship between pore size and ESR has also been demonstrated.

4.3. Theoretical Limits

Quantitative modeling also has been used to estimate the theoretical limits of the energy and power densities for supercapacitors. Additionally, by determining the limiting factors that prevent supercapacitors from reaching their theoretical limit, this research has generated new insights on methods to optimize supercapacitor design. While there has been consistent interest in developing improved electrode materials to increase energy densities, theoretical models suggest that it is the ion concentration and breakdown voltage of the electrolyte that often limit the energy densities of supercapacitors [49-51]. Furthermore, additional research suggests that the power densities of supercapacitors can be limited, as well, by the electrolyte [52-53]. Thus, the research results emphasize that the optimization of the electrolyte is as important as the optimization of the electrode for achieving energy and power densities closer to the theoretical limits of supercapacitors.

5. Prospectus on the Future of Supercapacitor R&D

Over the last several years, supercapacitor R&D has focused upon efforts to increase the capacitance of electrode materials and to develop improved quantitative models. However, recent research trends suggest that new areas may be rising to the forefront of supercapacitor R&D. In particular, R&D efforts concerning hybrid capacitors, equivalent series resistance, electrolyte optimization, and self-discharge are likely to expand and enable major performance advances in supercapacitors.

5.1. Hybrid Capacitors

Hybrid capacitors have been demonstrated to exhibit a combination of performance characteristics that formerly was unattainable. They combine the best features of EDLCs and pseudocapacitors together into a unified supercapacitor. (See Section 3.3 for a discussion of hybrid supercapacitors.) Although hybrid capacitors have been explored less than EDLCs or pseudocapacitors, the research that is available suggests that they may be able to outperform comparable EDLCs and pseudocapacitors [7, 12, 26]. As a result, R&D efforts concerning the fabrication of

improved hybrid capacitors and the development of more accurate quantitative models of hybrid capacitors have continued to expand [31-38, 44, 48]. Along with the increasing interest in developing high cycle life, high-energy supercapacitors, the tremendous flexibility in tuning the design and performance of hybrid capacitors is leading them to surpass EDLCs as the most promising class of supercapacitors.

5.2. Equivalent Series Resistance

Their ESR prevents supercapacitors from achieving power densities closer to the theoretical limits. Thus, determining how to lower the ESR of supercapacitors is becoming an important area of R&D. Several methods for reducing the ESR already have been developed, including polishing the surface of the current collector, chemically bonding the electrode to the current collector, and using colloidal thin film

suspensions [15, 18, 20]. In addition, there has been research in defining the relationship between pore size and ESR in electrode materials and determining the intrinsic ESR of various electrolytes [14, 47]. As these R&D efforts progress, they should allow supercapacitors to achieve power densities closer to their theoretical limits.

5.3. Electrolyte Optimization

In the scientific literature, electrolyte optimization has been emphasized consistently as the critical step towards improving supercapacitors [2, 49-53]. While the resistance of an electrolyte can limit power density, its ion concentration and operating voltage can limit the energy density of a supercapacitor. Despite the impact of electrolyte properties on supercapacitor performance, R&D efforts towards improving electrolytes have yet to become as rigorous or to be as fruitful as the comparable R&D efforts

towards improving electrodes. However, the authors believe that, due to the importance of electrolyte optimization and the emphasis upon that in the literature, it is necessary to encourage more R&D efforts to refine electrolytes and improve the synergy between electrolyte and electrode.

5.4. Ameliorating Self-Discharge

Another step that needs to be taken for supercapacitors to fulfill their promise is to ameliorate their tendency to self-discharge. Because charged supercapacitors are in a higher state of potential energy than discharged supercapacitors, there is thermodynamic pressure for a supercapacitor to discharge. This pressure sometimes manifests itself in

the undesirable phenomenon known as self-discharge, which occurs when a capacitor discharges internally on an open circuit. Self-discharge is intrinsic to all electrochemical energy storage systems including batteries, as well as capacitors. However, it occurs at a higher rate for supercapacitors. Hence, self-discharge tends to be more detrimental for them. This is because there is not an intrinsic barrier to the supercapacitor operating in reverse as there is in the case of systems based upon chemical reactions, in which the reverse process often is retarded by thermodynamic or kinetic barriers in the absence of an external connection between the electrodes. Also, in supercapacitors the potential difference between the electrodes often is very large and the distance is very small. As a

result of these several factors, the potential difference within an EDLC can be much more difficult to maintain than that within a battery.

There are a number of different mechanisms for self-discharge, but they commonly result from uncontrollable Faradaic reactions, such as the reduction and oxidation of impurities in the electrode material. Thus, improving material purity has been identified as one way to decrease the rate of self-discharge in supercapacitors.