Supercapacitor Modeling

Supercapacitor modeling enables us to predict their behavior in different apphcations, on the basis of a representation of the main physical phenomena occurring in the coirqxment. There are many different models for snpeicapacitors (two-branch model, model based on a transmission line, single-pore model, multi-pore model, etc.) [BEL 01 HAM 06]. These models are in the form of equivalent electrical circuits. Using them, we can describe the supercapacitor s behavior quite accurately. 

The impedance of a porous electrode has been studied by de Levie PEL 67], Frequency dependency is primarily caused by the dynamics of the ions in the electrolyte. At high fiequency, the ions do not have the time to reach the difficult-to-access surfaces, located deep in the pores. This results in a decrease of the capacitance and the series resistance with increasing frequency. 

K is the conductivity of the electrolyte, C the low-frequency capacitance of a pore, n the number of pores, h their height and r their radius. This model does not take account of the variation of the capacitance with the voltage. 

The capacitance Q is made up of a constant capacitance and a capacitance which is proportional to the voltage  

The two-branch model of the supercapacitor may be presented with a third branch, made up of an indnctance and a parallel resistance which represents the leakage current. 

Classic equivalent model of supercapacitor circuit. ESR = equivalent serial resistor. EPR equivalent parallel resistor. 

A simplified portrait of this technique describes the use of a small amplitude sinusoidal voltage (often 10 mV) applied to produce a resulting sinusoidal current. Measurement of this current permits the calculation of impedance and phase angle through which the double-layer capacitance can then be assessed. Further characterization of capacitance can be achieved through sweep analysis of the capacitance at various voltages and temperatures to gain practical assessment of device performance. 

Nyquist impedance plot comparing ideal vertical impedance of capacitor (thin line) and that of supercapacitor (thick line). Equivalent series resistance (ESR) is derived from the intercept of the real impedance axis followed by the equivalent distributed resistance (EDR) of a porous electrode. Source Kotz, R. 2000. Electrochimica Acta, 45,2483-2498. With permission.) 

Advanced equivalent circuits describing supercapacitor. L is an inductor, R, is internal resistance, and Zp is a complex pore impedance element. (Source Duller, S., Member, E. Karden et al. 2002. IEEE Transactions on Power Electronics, 38,1622-1626. With permission.) 

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Devillers, N., S. Jemei, M. C. Pera, D. Bienaime, and F. Gustin. 2014. Review of characterization methods for supercapacitor modelling. Journal of Power Sources 246 596-608. 

The field of supercapacitor modeling has been extremely active in recent years with many papers appearing each month. Here we will... 

DFT calculaticHi [8] and closer to the experimental data. This implies that adding the polarizability of electrode (even the imis) could help increase the accuracy of supercapacitor modeling. 

In the sixth paper of this chapter, Kierzek et al., mainly focus on modeling of pore formation vs surface area growth phenomena upon activation of coal and pitch-derived carbon precursors. These authors briefly touch on other precursor carbons as well. The properties of newly synthesized materials are being looked at from the point of view of their application as active materials in the supercapacitor electrodes. Editors thought this work by the Institute of Chemistiy and Technology of Petroleum and Coal in Poland, could be of genuine interest to the practical developers of carbon materials for the supercapacitor industry. 

The contribution by Rouzaud et al. teaches to apply a modified version of high resolution Transmission Electron Microscopy (TEM) as an efficient technique of quantitative investigation of the mechanism of irreversible capacity loss in various carbon candidates for application in lithium-ion batteries. The authors introduce the Corridor model , which is interesting and is likely to stimulate active discussion within the lithium-ion battery community. Besides carbon fibers coated with polycarbon (a candidate anode material for lithium-ion technology), authors study carbon aerogels, a known material for supercapacitor application. Besides the capability to form an efficient double electric layer in these aerogels, authors... 

Verbrugge M, Liu P. Microstructural analysis and mathematical modeling of electric double-layer supercapacitors. Journal of the Electrochemical Society 2005 152(5) D79-D87. 

Rafik F, Gualous H, Gallay R, Crausaz A, Berthon A. Frequency, thermal and voltage supercapacitor characterization and modeling. Journal of Power Sources 2007 165 928-934. 

Belhachemi F, Rael S, Davat B. A physical based model of power electric double-layer supercapacitors, IEEE-IAS 00, Rome, 2000. 

Since the appearance of the redox [ii, iii] and conducting [iv] polymer-modified electrodes much effort has been made concerning the development and characterization of electrodes modified with electroactive polymeric materials, as well as their application in various fields such as -> sensors, actuators, ion exchangers, -> batteries, -> supercapacitors, -> photovoltaic devices, -> corrosion protection, -> electrocatalysis, -> elec-trochromic devices, electroluminescent devices (- electroluminescence) [i, v-viii]. See also -> electrochemically stimulated conformational relaxation (ESCR) model, and -> surface-modified electrodes. 

The porous electrode theory was developed by several authors for dc conditions [185-188], bnt the theory is usually applied in the ac regime [92,100,101,189-199], where mainly small signal frequency-resolved techniques are used, the best example of which are ac theory and impedance spectra representation, introdnced in the previons section. The porous theory was first described by de Levi [92], who assumed that the interfacial impedance is independent of the distance within the pores to obtain an analytical solution. Becanse the dc potential decreases as a fnnction of depth, this corresponds to the assnmption that the faradaic impedance is independent of potential or that the porons model may only be applied in the absence of dc cnrrent. In snch a context, the effect of the transport and reaction phenomena and the capacitance effects on the pores of nanostructured electrodes are equally important, i.e., the effects associated with the capacitance of the ionic donble layer at the electrode/electrolyte-solntion interface. For instance, with regard to energy storage devices, the desirable specifications for energy density and power density, etc., are related to capacitance effects. It is a known fact that energy density decreases as the power density increases. This is true for EDLC or supercapacitors as well as for secondary batteries and fnel cells, particnlarly due to the distributed nature of the pores... 

FIGURE 1.4 Double-layer models (a) Helmholtz model, (b) Gouy-Chapman model, (c) Stem model, and (d) Grahame model. (With kind permission from Springer Science+Business Media Electrochemical Supercapacitors Scientific Fundamentals and Technological Applications, 1999, Conway, B.E. Originally published by Kluwer Academic/ Plenum Pubhshers, New York in 1999.)... 

Huang, J. S., B. G. Sumpter, and V. Meunier. 2008. A universal model for nanoporous carbon supercapacitors applicable to diverse pore regimes, carbon materials, and electrolytes. Chemistry—A European Journal 14 6614-6626. 

A supercapacitor stores energy in electrical double layers at electrode/electrolyte interfaces. In molecular modeling of supercapacitors, the... 

The electrodes of supercapacitors, in particular carbon-based electrodes, have very complex microstructure. Consequently, many types of EDLs can exist in a supercapacitor. For example, some EDLs are formed near open surfaces that can be planar (e.g., flat graphene sheet), cylindrical (e.g., CNT), or spherical (e.g., OLC) some EDLs appear inside narrow pores (e.g., sUt-shaped and cylindrical (Mies). By taking each EDL as a separate capacitor, a supercapacitor can be considered as many capacitiM s connected in series and/or in parallel. Thus, it is reasonable to study some of these capacitors (i.e., EDLs) to gain insights into the charge storage mechanism in a real supercapacitor. Many different techniques can be used to model the EDLs, and here we focus on molecular simulations and density functional theory (DFT) calculations. 

Molecular dynamics (MD) and Monte Carlo (MC) simulations are popular molecular simulation techniques. These methods are well suited for modeling supercapacitor because EDLs in supercapacitors are essentially molecular phenomena, e.g., the thickness of EDLs in supercapacitors is typically less than a few nanometers. The uni(]ue advantage of these methods is that they provide direct information on both the microstructure (e.g., ion density distribution across EDL, which is difficult to measure experimentally) and the macroscopic properties (e.g., its capacitance) of EDLs. This allows one to establish the microscopic origins of the capacitance of supercapacitors and thus helps guide the design and selection of electrode/electrolyte materials for supercapacitors. MD simulation, as... 

Once the ion density distribution inside a modeled supercapacitor (or EDL) is determined using molecular simulations or DFT calculations, the macroscopic properties of supercapacitors can be computed. The most important property... 

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