Recent Advances in Continuum Modeling of Interfacial and Transport Phenomena in Electric Double...

8/16/2019 Recent Advances in Continuum Modeling of Interfacial and Transport Phenomena in Electric Double Layer Capacitors

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Journal of The Electrochemical Society, 162 (5) A5158-A5178 (2015) A5159

volatility, non-flammability, and the variety of combinations of cationsand anions available.2,4,6 However, their relatively small electricalconductivity typically yields smaller power density than that achievedusing organic electrolytes.2,4,6

To date, optimization of the porous morphology of electrodes andelectrolytes used in EDLCs has been mainly carried out experimen-tally, by trial and error informed by physical intuition. However, thelarge number of variables to be considered andthe different competingphenomena previously discussed make intuitive predictions difficult.In addition, experimental approaches are typically time consumingand costly. By contrast, rigorous physical modeling and accuratenumerical tools could facilitate the design and optimization of theelectrode morphology and the identification of the ideal electrolytein a more systematic and efficient way. Physical modeling of super-capacitors is made difficult by the multiple and intimately coupledinterfacial and transport phenomena simultaneously occurring in thedevice such as (i) charge transport in the electrodes, (ii) ion transportin the electrolyte and through the porous electrode structure drivenby both electric field and concentration gradients, (iii) steric repul-sion among ions, and (iv) local heat generation. Another challengein modeling EDLCs is the multiscale nature of the system and thesometimes complex porous electrode morphology with unsolvated or

solvated ions around 1 nm or less in diameter, pore size ranging fromone to tens of nanometers, electrodes up to hundreds of microns thick,and devices several millimeters in size.

This paper aims to present recent advances in multiphysics andmultiscale continuum modeling of interfacial and transport phenom-ena in EDLCs. The Background section introduces basic conceptsand physical phenomena taking place in EDLCs. The EquilibriumModeling section presents equilibrium models for EDLCs illustratedby detailed simulations of three-dimensional (3D) highly-orderedmesoporous carbon electrodes. This section also offers design rulesfor optimum EDLCs obtained from scaling analysis applied toexperimental data for porous carbon electrodes with pores featuring awide range of shapes and sizes. The Dynamic Modeling section andthe Thermal Modeling section review dynamic models governing thespatiotemporal evolution of electric potential, ion concentrations, and

temperature in EDLCs for different types of electrolytes. They presentnumerical simulations reproducing commonly used experimentalcharacterization techniques such as electrochemical impedancespectroscopy (EIS), cyclic voltammetry (CV), and galvanostaticcycling. Finally, this review ends by discussing limitations of existingcontinuum models and providing recommendations for futureresearch.

Background

Electrical double layer theories.— Figure 1 schematically showsdifferent electrical double layer (EDL) models proposed over time.The concept of EDL was first introduced by Helmholtz 38 who sug-gested that a charged surface immersed in an electrolyte solutionrepels ions of the same charge (positive or negative) but attracts theircounter-ions. The layer of electronic charge at the electrode surface

and the layer of counter-ions in the electrolyte forms what has beentermed the electrical double layer.39,40 The Helmholtz model38 hy-pothesized that counter-ions form a monolayer near the electrodesurface, as illustrated in Figure 1a. This structure is analogous tothat of conventional dielectric capacitors with two planar electrodesseparated by a small distance H , approximated as the radius of anion.38–41

The Helmholtz model was modified by Gouy42 and Chapman43

with the consideration that the ion distribution should be continuous inthe electrolyte solution and given by the Boltzmann distribution. TheGouy-Chapman model accounts for the fact that the ions are mobilein the electrolyte solution under the combined effects of ion diffusiondriven by concentration gradients and electromigration driven by theelectric potential gradient, i.e., the electric field. This results in the

so-called “diffuse layer” illustrated in Figure 1b. However, the Gouy-Chapman model overestimates the electrical double layer capacitancebecause it treats ions as point-charges resulting in unrealistically largeion concentrations at the electrode surfaces.4,39,40

In 1924, Stern proposed a new EDL model accounting for thefinite size of ions. Stern44 combined the Helmholtz model and theGouy-Chapman model to explicitly describe two distinct regions,namely (1) the inner region of thickness H termed the Stern layer and(2) the outer region called the diffuse layer, as shown in Figure 1c.In the diffuse layer, the ions are mobile under the coupled influence of diffusion and electrostatic forces, and the Gouy-Chapman model ap-plies in this layer.36, 39–41 In 1947, Grahame45 improved this model byeliminating a number of deficiencies. The author emphasized that inthe Stern layer, specific (covalent) and non-specific adsorption of ionsat the same electrode surface lead to different closest distances from

the charged surface.4,36,40,41 The inner Helmholtz plane (IHP) consistsof ions adsorbed by specific (covalent) forces and the outer Helmholtzplane (OHP), or Stern layer, of ion adsorbed by non-specific (electro-static) forces. However, continuum models of EDLCs typically con-sider only electrostatic adsorption, corresponding to the Stern model.Note that electroneutrality does notprevail within theelectrical doublelayer.39,46–48

Figure 1. Schematic representation of electrical double layer structures according to (a) the Helmholtz model, (b) the Gouy-Chapman model, and (c) the Gouy-Chapman-Stern model. The double layer distance in the Helmholtz model and the Stern layer thickness are denoted by H while ψs is the potential at the electrode

surface.

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A5160 Journal of The Electrochemical Society, 162 (5) A5158-A5178 (2015)

Electric double layer capacitors.— EDLC devices consist of twoelectrodes supported by their respective current collectors and sepa-rated by a separator impregnated with electrolyte. Each electrode ischaracterized by its capacitance defined per unit surface area, calledthe areal capacitance, and denoted by C s (in F/m

2). Capacitance canalso be defined per unit mass of electrode materials known as gravi-metric capacitance and denoted by C g (in F/g). The total capacitanceC (in F) of an electrode is thus expressed as C

=C s Ai

=C g m where

Ai is the interfacial surface area between the electrode and the elec-trolyte, typically measured by gas adsorption porosimetry, and m isthe mass of the electrode.

Furthermore, one can define the areal differential capacitanceC s,d i f f and the areal integral capacitance C s,in t of an electrode (bothin F/m2) respectively as,36,41,49

C s,d i f f =dqs

dψsand C s,in t =

qs

ψs[1]

where qs and ψs represent the electrode’s surface charge density (inC/m2) and its surface electric potential (in V) relative to the bulk elec-trolyte, respectively. Note that differential and integral capacitancescan also be defined on a gravimetric basis. These definitions are inde-pendent of the experimental method used to measure them. However,

note that capacitances measured using dynamic cycling methods maydependon the cycling periodor frequency. In the limit of slow cycling,the capacitance becomes independent of the cycling rate and its valuecorresponds to the equilibrium capacitance.50,51

In the context of the the Gouy-Chapman-Stern model presented inFigure 1c, the differential, integral, and equilibrium areal capacitanceof one electrode can be computed by considering the Stern and diffuselayer capacitances in series, i.e.,

1

C s= 1

C St s+ 1

C Ds[2]

where C St s and C Ds are respectively the Stern and diffuse layer areal

capacitances on a differential, integral, or equilibrium basis.An EDLC device with electrodes A and B can be treated as two

capacitors in series with integral capacitances C A = C s, A Ai, A andC B = C s, B Ai, B . Thus, the total integral capacitance C T of the deviceis given by

1

C T = 1

C A+ 1

C B. [3]

Note that EDLC devices consisting of identical electrodes, such thatC A = C B , have total capacitance C T = C A/2. Alternatively, forhybrid devices with a pseudocapacitive electrode, such that C B C A,the total integral capacitance equals the smallest capacitance, i.e.,C T = C A.

Finally, the total energy stored in the EDLC with electrodes A andB subjected to potential difference ψ is typically approximated as E = 1

2C in t ,T ψ

2.8,36,52 This suggests that in order to maximize thetotal EDLC energy storage, one needs to maximize the capacitanceC in t ,T and the potential differenceψ between thetwo electrodes. Thetotal capacitance C in t ,T canbe increased by increasing the surface area Ai of the porous electrodes. Note however that the areal capacitanceC s,in t depend on the porous structure of the two electrodes and on theaccessibility of the electrode surface to ions. It may decrease as thetotal surface area of the electrodes increases.

RC circuit and homogeneous models.— Numerous equivalent RCcircuit models53–57 have been proposed and used to numerically in-vestigate and predict the performance of EDLCs. They include butare not limited to (i) the simple series RC circuit, (ii) the so-calledclassical equivalent circuit58 consisting of a resistor in parallel with acapacitor with an additional resistance in series, (iii) the three branchmodel with one voltage-dependent capacitance,59 and (iv) the trans-

mission line network model with an arbitrary number of equivalentRC circuit stages.55,60,61 However, all these models suffer from sev-eral drawbacks.62 First, they require prior knowledge of the resistance

and capacitance of the actual device which are typically determinedexperimentally. 58 Thus, this approach can be used for control but itcannot be used for designing andoptimizing the electrodearchitectureand electrolyte of novel EDLCs. In fact, the above models succes-sively add variables to better fit the experimental measurements. Inaddition, the concept of RC circuit models is inadequate for EDLCssince this approach inherently neglects ion diffusion and the time-dependent and non-uniform ion concentrations in the electrolyte.63–65

Some intuitive arguments have been proposed to justify these RC cir-cuit models. However, no rigorous justifications and validation havebeen provided. For example, the parallel resistance in the classicalequivalent model was attributed to leakage current while the seriesresistance corresponded to the resistance of the electrolyte, the elec-trodes, and current collectors.58 The transmission line network modelwas justified as a way to represent individual pores of the electrodes asa RC circuit connected in series.60 However, the number of RC stageswas typically small and not related to the electrode morphology. 55

Moreover, two different RC circuit models may produce the sameimpedance response. This suggests that fitted values of the resistancesand capacitances in the RC circuit models provides “little or no di-rect information about the physical meaning of the elements for suchmodels”.62 Finally, the fitted capacitance values based on complex

RC circuits were also reported to underpredict those measured usingother techniques.66–70

Alternatively, homogeneous models were also developed to inves-tigate the charging / discharging dynamics of EDLCs. These modelstreat the heterogeneous microstructure of the electrodes as homoge-neous with some effective macroscopic properties determined fromeffective medium approximations and depending on porosity and spe-cific area.71–82 However, these models imposed the specific area ca-pacitance or the volumetric capacitance instead of predicting them. Inaddition, they cannot account for the detailed mesoporous electrodearchitecture.

Thermal considerations.— In applications such as energy stor-age onboard automobiles, EDLCs should be able to operate over awide range of temperatures corresponding to various climates and

seasons. Moreover, during EDLC charging and discharging, heat isgenerated internally, leading to temperature rise in the device. Theheat generation rate depends on the cell’s design, the electrode andelectrolyte materials, and the operating conditions.83 It has been es-tablished that increasing the EDLC temperature within a reasonablerange significantly increases the EDLC performance.84,85 This wasattributed to the increase in electrolyte conductivity with increasingtemperature.84–87 However, EDLC operation at elevated temperaturesmay also result in (i) accelerated aging,83,88–91 (ii) increased self-discharge rates,88–90,92 (iii) increased cell pressure, and possibly (iv)electrolyte evaporation.88 For example, a 10 K temperature rise ora 100 mV increase in cell voltage approximately doubles the agingrate.91,92 EDLC aging results in permanentdecreasein capacitance andincrease in internal resistance leading, in turn, to larger heat genera-tion rate and larger cell voltage.55 In addition, temperature differencesbetween cells in a series-connected EDLC module can cause voltageimbalances and destructive over-voltages of individual cells.83,90 Toavoid these harmful effects, temperature changes in EDLCs should becontrolled.

This paper reviews recent efforts in physical modeling of interfa-cial and transport phenomena in EDLCs under either equilibrium ordynamic cycling conditions. Its aims to provide the interested readerwith all the necessary information to perform multiscale and multi-physics simulations of EDLCs under realistic conditions. It system-atically presents the governing equations, boundary conditions, andconstitutive relationships to predict the local electric potential, ionconcentrations, and/or temperatures based on continuum theory. Allmodels discussed adopt the Gouy-Chapman-Stern model of EDLspresented in Figure 1c. Simulations based on actual experimental

measurements were used to provide physical interpretations of exper-imental observations and design rules to achieve maximum energystorage performance. Similarly, several thermal models with different




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levels of refinement are discussed as ways to predict operating tem-peratures and develop thermal management strategies for existing ornovel EDLC designs. Comparison with experimental data reported inthe literature is presented whenever possible.

Equilibrium Modeling

Governing equations: modified Poisson-Boltzmann models.— Binary and symmetric electrolytes.— For ideal electrolyte solutionswith non-interacting ions treated as point charges, the electric poten-tial is governed by the Poisson-Boltzmann (PB) equation.93–99 Theconcentration profile of each ion species is given as a function of the local electric potential by the Boltzmann distribution. However,numerous studies have established that accounting for finite ion sizeis essential in order to accurately simulate electrical double layers forlarge electric potentials and/or large electrolyte concentrations typicalof EDLCs.93–107 Among the models accounting for the finite size of ions, the modified Poisson-Boltzmann (MPB) models are based onthe local-density and mean-field approximations and are relativelyconvenient both mathematically and numerically.

Binary and symmetric electrolytes consist of two ion species (i

=1 or 2) with identical (i) effective ion diameter, i.e., a1

=a2

=a,

(ii) opposite valency, i.e., z1 = − z2 = z where z is a positive in-teger, and (iii) diffusion coefficient, i.e., D1 = D2 = D. Then, byvirtue of electroneutrality, the bulk concentrations (in mol/L) of bothion species are identical, i.e., c1,∞ = c2,∞ = c∞. Among the differ-ent MPB models, Bikerman’s model is the simplest for binary andsymmetric electrolytes and is expressed as95–104

∇ · (0r ∇ ψ)

=

0 in the Stern layer

2 ze N Ac∞ sinh

zeψ

k B T

1 + 4 N Aa3c∞ sinh2

zeψ

2k B T

in the diffuse layer[4a]

[4b]

where 0 and r are the free space permittivity (0

= 8.854

× 10−12 F/m) and the relative permittivity of the electrolyte solu-tion, respectively. The Boltzmann constant is denoted by k B = 1.38× 10−23 J/K, e = 1.602 × 10−19 C is the elementary charge, N A = 6.022 × 1023 mol−1 is the Avogadro number, while T is thetemperature (in K). Note that under equilibrium conditions, there is nopotential drop across the electrodes since there is no electric current.Therefore, the electric potential at the electrode/electrolyte interfaceis equal to that imposed at the current collectors. The local ion con-centration c i (r) of ion species “i” at location r depends on the localpotential ψ(r), solution of Equation 4, and is expressed as95,99

ci (r) =c∞ exp

− zi eψ(r)

k B T

1 + 4 N Aa3c∞ sinh2

zi eψ(r)

2k B T for i = 1 and 2. [5]

Note that the concentration ci of ions, treated as hard spheres of effective diameter a in the Bikerman model, cannot exceed the max-imum concentration c max = 1/ N Aa3 corresponding to simple cubicion packing. This represents to a maximum ion volume fraction of π/6 ≈ 52%. Asymmetric electrolytes.— Many electrolytes are asymmetric in na-ture due to the difference in (i) diffusion coefficient, (ii) size, and/or(iii) valency between their anions and cations, such as aqueousH2SO4 and Na2SO4. The recent literature reported MPB models validfor asymmetric electrolytes. For example, Borukhov et al.108,109 andSilalahi et al.110 developed MPB models valid for binary electrolyteswith asymmetric valency but identical ion diameters. Their modelwas later extended to binary asymmetric electrolytes with unequalion diameters.111,112 Biesheuvel and co-workers103,104,113 and Alijó

et al.114

developed more general MPB models valid for asymmet-ric electrolytes and/or multiple ion species with different ion sizesand valencies. This was accomplished by incorporating an excess

chemical potential term based on the Boublik-Mansoori-Carnahan-Starling-Leland equation-of-state. It directly relates the excess chem-ical potential to the local ion concentrations, ions’ effective diam-eters, and their exclusion volumes.103,104,113,114 In addition, Li andco-workers107,115,116 developed a model for asymmetric electrolytesbased on the variational principle while accounting for the finite sizesof both ions and solvent molecules. Alternatively, Horno and co-workers102,117–122 developed a MPB model for asymmetric electrolytesby directly applying a Langmuir-type correction to the equilibrium ionconcentration given by the Boltzmann distribution. By contrast, sev-eral studies have considered asymmetric electrolytes with differenteffective ion diameters by defining multiple Stern layers with differ-ent thicknesses near the electrode surface.114,121–124 Only ion speciesof intermediate sizes existed in the intermediate Stern layer(s). How-ever, the associated ion concentrations do not satisfy the overall elec-troneutrality condition across the electrolyte domain.125 The differentstudies104,105,107–109,115,116,123 have demonstrated that asymmetry in ionsize and/or ion valency significantly affects the ion concentration andelectric potential profiles near the electrode surface. However, to thebest of our knowledge, these MPB models developed for asymmetricelectrolytes have not been used to predict the capacitance of EDLCs.

Boundary conditions.— Planar electrodes.— Equations 4a and 4bare second order partial differential equations in terms of ψ(r). Eachrequires two boundary conditions for each dimension of space in theStern and diffuse layers, respectively. Figure 2 shows the schematicand the coordinate system for the reference case of an EDLC withtwo parallel planar electrodes at electric potentials ψs and −ψs sepa-rated by a distance 2 L , with their respective Stern layers of thickness H . This case may seem trivial as it considers very simple geome-try compared with actual porous electrode morphologies. However,such geometry facilitates the numerical simulations and enables oneto understand the contributions and interactions among the differentphysical phenomena taking place simultaneously, as illustrated in thefollowing sections.

For binary and symmetric electrolytes, the equilibrium potentialψ( x ) is antisymmetric. Thus, only half of the one-dimensional EDLC

illustrated in Figure 2 may be simulated from the surface of one of theelectrodes to the cell centerplane. Then, four boundary conditions areneeded to solve Equation 4. They are given at the electrode/electrolyteinterface ( x = L), at the Stern/diffuse layer interface ( x = L − H ),and at the centerplane ( x = 0) by93ψ( L) = ψs and ψ( L − H +) = ψ( L − H −) in the Stern layer

[6a]

∂ψ

∂ x ( L − H +) = ∂ψ

∂ x ( L− H −) and ψ(0) = 0 V in the diffuse layer

[6b]

where ψs is the potential imposed at the electrode surface and H is

the Stern layer thickness corresponding to half of the effective iondiameter a, i.e., H = a/2. It is important to note that the condi-tion ψ( x = 0) = 0 V at the centerplane is not valid for asymmetricelectrolytes.125 Then, the entire electrolyte domain between the twoelectrodes needs to be simulated. Finally, extension of these bound-ary conditions to two-dimensional (2D) and three-dimensional (3D)geometries is straightforward.

Equation 4a indicates that the potential is linear in the Stern layer.Thus, the presence of the Stern layer can be accounted for via a differ-ent boundary condition to Equation 4b imposed at the Stern/diffuselayer interface located at x = ( L − H ) and substituting Equations 6aand 6b by95,99,126,127

∂ψ

∂ x ( L− H ) = ψs −ψ D

H and ψ(0) = 0 V in the diffuse layer

[7]where ψ D = ψ( L − H ) is the potential at the Stern/diffuse layerinterface computed numerically by solving Equation 4b. Then, one




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Figure 2. Schematic and coordinate system of an EDLC consisting of Stern and diffuse layers between two planar electrodes. Here, the ion diameters of largeand small ion species were denoted by a1 and a2, respectively.

does not need to solve the MPB model and the associated boundaryconditions in the Stern layer [Equations 4a and 6a].

Cylindrical or spherical electrode surface.— Wang and Pilon128

extended Equation 7 to cylindrical and spherical electrodes orpores. For planar, cylindrical, and spherical electrodes illustrated inFigure 3a, the boundary conditions at the Stern/diffuse layer interfaceaccounting for the presence of the Stern layer can be expressed as 128

− 0r ( E H )∇ ψ ·

r H

r H

= C St s

R0

R0 + H

p[ψs −ψ(r H )] . [8]

Here, r H is the local position vector of the Stern/diffuse layer inter-face located at r H = R0 + H near the electrode surfaces. Here, theparameter p is such that (i) p = 0 for planar electrodes, (ii) p = 1for cylindrical electrodes, and (iii) p = 2 for spherical electrodes of radius R

0.128 In addition,

r ( E

H ) is the electrolyte relative permittivity

evaluated at the Stern/diffuse layer interfacebased on the local electricfield E H = E (r H ) as discussed in the next section. Finally, C St s is theStern layer capacitance predicted by the Helmholtz model and givenby93,128,129

C St s =

0r ( E H )

H for planar electrodes,

0r ( E H )

R0 ln

1 + H R0

for cylindrical electrodes of radius R0,0r ( E H )

H

1 + H

R0

for spherical electrodes of radius R0.

[9a]

[9b]

[9c]

Moreover, for cylindrical and spherical pores of radius R0 illus-trated in Figure 3b, the boundary condition at the Stern/diffuse layerinterface located at r H = R0 − H can be written as128

− 0r ( E H )∇ ψ ·

r H

r H

= C St s

R0

R0 − H

p[ψs −ψ(r H )] . [10]

Here also, the Stern layer capacitance C St s for cylindrical ( p = 1) orspherical ( p = 2) pores is given by the Helmholtz model expressedas129

C St s =

0r ( E H )

R0 ln

R0 R0− H

for cylindrical pores of radius R 0,

0r ( E H ) H

R0 − H

R0

for spherical pores of radius R0.

[11a]

[11b]

Note that when R0 H , Equations 8 and 10 reduce toEquation 7 for planar electrodes.These alternative boundary conditions for planar, concave, and

convex electrode surfaces were shown to accurately account for theStern layer without explicitly simulating it in the numerical domain.128

Figure 3. Schematic of the EDL structure forming near (a) a cylindrical or

spherical electrode particle and (b) a cylindrical or spherical pore with radius R0 and Stern layer thickness H and illustrating the arrangement of solvatedanions and cations as well as the Stern layer and the diffuse layer.




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Table I. Summary of Booth model parameters for commonly usedaqueous and organic electrolyte solvents.

Solvent r (0) n β (m/V) Ref.

Water 78.5 1.33 1.41 × 10−8 144, 145PC 64.4 1.42 1.314 × 10−8 146, 147AN 35.97 1.34 3.015

×10−8 146, 148, 149

This presents the advantages of reducing the number of mesh ele-ments used by the numerical method and thus the computational time.Such advantages become more significant with increasing electrode orpore radius and geometric complexity of the electrode architecture ingeneral.

Constitutive relationships.— In order to solve the MPB model andthe associated boundary conditions for the potential ψ(r) in EDLCsunder equilibrium conditions, several properties of the electrolyte arenecessary including (i) the electrolyte relative permittivity r , (ii) theions’ effective diameter a , and (iii) the ions’ valency z. The relativepermittivity r of polar electrolytes has often been treated as constantand equal to that of the solvent at zero field.94,97–99 However, the latter

significantly decreases as the electric field increases to the large valuestypically encountered near the electrode surface of EDLCs.130–132 Infact, the solvent dipole moments become highly oriented under largeelectric fields. Therefore, further orientation of the dipole momentscan hardly provide more polarization leading to saturation and a de-crease in the relative permittivity.130–134 The Booth model accountsfor the dependence of electrolyte solvent relative permittivity on thelocal electric field at large electric field.130–132 It is expressed as

r ( E )

=

n2 + [r (0) − n2] 3β E

coth(β E ) − 1β E

for E ≥ 107 V m−1

r (0) for E


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the Stern/diffuse layer interface. Second, the diffuse layer areal equi-librium capacitance C Ds,eq and the total areal capacitance C s,eq can be

respectively estimated as146,164

C Ds,eq =Qs

ψ D Ad = 1

ψ D Ad

As

0r ( E )E · n d A and C s,eq =Qs

ψs As.

[14]

Here also, ψ D denotes the potential at the Stern layer/diffuse layerinterface. Note that C St s,eq , C

Ds,eq , and C s,eq satisfy Equation 2.

In the case of planar electrodes with binary and symmetric elec-trolytes with relative permittivity r treated as constant, analyticalexpressions for the Stern and diffuse layers capacitances C St s and C

Ds

can be obtained from the MPB model. They are given by 93,99,146

C St s,eq, pla nar =0r

H and

[15]

C Ds,eq, pla nar =2 ze N Ac∞λ D

ψ D

2

ν pln

1 + 2ν p sinh2

zeψ D

2k B T

where λ D is the Debye length for binary and symmetric electrolytes

defined as λ D = 0r k B T /2e2 z2 N Ac∞ and corresponding to an es-timate of the EDL thickness.39 The packing parameter ν p is defined asν p = 2a3 N Ac∞.95,99 It represents the ratio of the total bulk ionconcen-tration 2c∞ to themaximum ionconcentration cmax = 1/ N Aa3 assum-ing a simple cubic ion packing and is always less than unity.93,95,99,146

Then, combining Equations 2 and 15, the equilibrium areal capaci-tance for planar electrodes C s,eq, pla nar can be written as

146

1

C s,eq, pla nar = a

20r + ψ D

2 ze N Ac∞λ D

×

2

ν pln

1 + 2ν p sinh2

zeψ D

2k B T

−1/2. [16]

Despite the convenient expression for C Ds,eq, pla nar given by

Equation 15, the potential ψ D remains unknown and needs to bedetermined by solving numerically the MPB model. Scaling analysispresented in Section Scaling Analysis will demonstrate that ψ D isconveniently related to ψs by a power law.

Simulations of EDLCs with ordered porous electrodes.— Mostcontinuum simulations of EDLCs have been limited to one-dimensional (1D) or quasi-two-dimensional electrode structuresand/or have relied on fitting parameters.146 To the best of our knowl-edge, Wang et al.146 presented the first simulations of EDLCs withthree-dimensional (3D) mesoporous electrodes consisting of highly-ordered and monodisperse mesoporous carbon spheres arranged inclose packed structures. These simulations also accounted for thefinite ion size as well as for the dependence of the electrolyte rela-tive permittivity on the local electric field given by the Booth model

[Equation 12] and were obtained using only properties reported inthe literature. Due to the complex electrode morphology, the meso-porous structure was simplified by considering the areal capacitancesof closely-packed non-porous carbon spheres and of a single meso-pore inside a carbon matrix separately. Their Stern layer capacitanceC St s,eq was estimated from the Helmholtz model while the diffuse layer

capacitance C Ds,eq was computed numerically by assuming ψ D = ψs ,i.e., by ignoring the Stern layer in the computational domain. The nu-merical predictions were in good agreement with the results reportedby Liu et al.165 for closely packed monodisperse mesoporous carbonspheres 250 nm in diameter with 10 nm mesopores for 1 M TEABF 4in PC as the electrolyte. The results also indicated that the Stern layercapacitance and the field-dependent permittivity r ( E ) need to be ac-counted in orderto properly estimate the porous electrode capacitance.

To avoid this simplification and faithfully simulate the porouselectrode morphology, Wang and Pilon128 derived and used the jumpboundary conditions given by Equations 8 and 10 along with the MPB

model [Equation 4] to simulate 3D ordered bimodal mesoporous car-bon electrodes. Figures 5a and 5b show SEM images of the bimodalporous carbon electrode synthesized by Woo et al.166 with (a) highlyordered macropores and (b) mesoporous walls separating the largepores. Figures 5c and 5d show schematics and dimensions of themesoporous structure simulated. The dimensions of the simulatedelectrode structure were identical to those of the bimodal mesoporouscarbon CP204-S15 electrodes synthesized.166 Figure 6 shows themea-sured gravimetric capacitance C g (in F/g) of different bimodal carbonfilms166 as a function of the specific surface area, ranging from 910 to1030 m2 /g, obtained using 1 M TEABF4 in PC.Figure 6 also shows thenumerically predicted gravimetric capacitance as a function of specificsurface area. The latter was varied by changing the macropore radius R0 from 50 to 150 nm while other geometric parameters such as thecarbon wall thickness and mesopore radius respectively remained 2and 7 nm and identical to those of CP204-S15 mesoporous carbon.166

The numerical results were obtained using either the solvated or thenon-solvated effective ion diameters reported in the literature as a =1.40 nm159,160 or a = 0.68 nm,158 respectively. It is evident that thepredicted and experimentally measured gravimetric capacitances C gincreased linearly with increasing specific surface area. The predictedslope of C g versus specific surface area corresponded to constant areal

equilibrium capacitances of C s = 7.4 or 10.2 µF/cm2

when using sol-vated or non-solvated ion diameter, respectively.128 This trend agreedvery well with the experimental data reporting an areal capacitance of 9.4 F/cm2.166 These results also indicate that the choice of solvatedversus non-solvated effective ion diameters makes a significant dif-ference in the simulation results. Unfortunately, their values are notalways reliably known.

Scaling analysis.— The previous section discussed equilibriumsimulations accounting for the 3D porous structure of carbon elec-trodes. However, this type of simulations is limited to highly-orderedporous structures for which symmetry in the geometry and antisym-metry in the potential can be invoked to simplify the geometry andboundary conditions of the simulations. Unfortunately, many elec-trodes feature disordered, non-spherical, and/or polydisperse pores.At this time, such complex structures cannot be faithfully simulatednumerically, even under equilibrium conditions, due to the computa-tional cost required to simulate a representative elementary volume of the electrode. Alternatively, scaling analysis has been used by physi-cists andengineers to identify similarity behavior in complex systems.This section present how scaling analysis could be used to reduce thelarge number of design parameters of EDLCs to a few meaningfuldimensionless numbers accounting for the dominant physical phe-nomena governing the behavior of the device. Then, engineering cor-relations can be derived relating the different dimensionless numbersidentified using semi-empirical coefficients determined from a widerange of experimental data.

Planar electrodes.— Dimensional analysis of the MPB model for bi-nary and symmetric electrolytes given by Equations 4a and 4b can be

performed by scaling the spatial coordinate x by the Debye length λ Dand the local potential ψ by the thermal voltage k B T /ez. The latterrepresents the voltage that would induce an electrical potential energyequivalent to thethermal energy of an ionof charge z. Then, thedimen-sionless variables are such that x ∗ = x /λ D and ψ∗ = ψ/(k B T /ez).167Four dimensionless similarity parameters arise from the scaling anal-ysis of the MPB model given by Equations 4 to 11, namely167

ψ∗s =ψs

(k B T / ze), ν p =

c∞1/2 N Aa3

, a∗ = aλ D

, and L ∗ = Lλ D

[17]

where ψ∗s is the dimensionless surface potential, ν p is the packingparameter, a∗ is the dimensionless ion diameter, and L ∗ is the dimen-sionless electrolyte layer thickness for planar electrodes. For cylin-drical or spherical particles or pores, the spatial coordinate is r in

cylindrical or spherical coordinates and the characteristic length scaleis the average pore radius R 0. Then, x

∗ and L ∗ should be replaced byr ∗ = r /λ D and R∗0 = R0/λ D, respectively.




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Figure 5. (a) FE-SEM images of 3D ordered bimodal porous carbon CP204-S15 and (b) TEM image of the mesoporous walls separating the large pores.Reproduced from Ref. 166 with permission of The Royal Society of Chemistry. (c-d) computational domain, coordinate system, and dimensions of the orderedbimodal mesoporous carbon simulated corresponding to CP204-S15.

Analytical expressions for the potential at the Stern/diffuse layerinterface ψ D, used in Equations 7 and 16, or ψ

∗ D do not exist

when accounting for the finite ion size, i.e., when ν p = 0. How-ever, they can be determined numerically by solving the equilibriumMPB model with the Stern layer. For planar electrodes, a correla-

Figure 6. Predicted128 and experimentally measured166 gravimetric capac-itance C g for bimodal carbons as a function of their specific surface area.

Numerical results were obtained by solving Equation 4 using boundary condi-tions given by Equations 8 and 10 with solvated or non-solvated ion diameters

(a = 1.40 or 0.68 nm), c∞ = 1 mol/L,ψs = 2 V, and theelectrolyte permittivityr ( E ) given by Equation 12. The specific surface area was adjusted by varyingthe inner macropore radius R0 was varied from 50 to 150 nm.

tion was found167 relating the dimensionless potentials ψ∗ D and ψ∗s

according to ψ∗ D = 0.37ψ∗1.16s . This correlation was valid for a widerange of parameters, namely 0.01 ≤ ψ∗s ≤ 20, 16 ≤ L∗ ≤ 329,0.052 ≤ ν p ≤ 0.94, and 1.15 ≤ a∗ ≤ 3.03. It was also shown toapply to cylindrical or spherical pores with acceptable accuracy. 167

Then, a dimensionless equilibrium areal capacitance C ∗s,eq, pla nar can be defined as C ∗s,eq, plan ar = C s,eq, plan ar /(20r /a). Based onEquation 16, C ∗s,eq, pla nar can be expressed in terms of the three di-mensionless numbers identified previously ψ∗s , ν p, and a

∗ as167

1

C ∗s,eq, plan ar

= 1 + 0.74ψ∗1.16sa∗

2

ν pln

1 + 2ν p sinh2

0.185ψ∗1.16s−1/2

[18]

This analytical expression can be conveniently used to predict theequilibrium areal capacitance of planar electrodes, assuming r to beconstant, without solving the MPB model.

Actual porous carbon electrodes.— In order to assess the applicabilityof theabove scaling analysis to actual carbon-basedporous electrodes,a wide range of experimental data was collected from the literature.They were selected to ensure that the electrolytes were binary andsymmetric and the reported capacitance corresponded to the equilib-rium areal capacitance.167 Table III summarizes the experimental datareported in the literature for EDLCs with various electrolytes, elec-trode average pore radii R 0, and potential windows ψ = 2ψs , alongwith the corresponding range of the experimentally measured equilib-rium areal capacitance, denoted by C s,eq,ex p (µF/cm

2). The electrodesconsisted of various micro and mesoporous carbons with average pore




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Table III. Summary of experimental data reported in the literature for various carbon electrodes, binary and symmetric electrolytes, potentialwindow ψ = 2ψ s , and average pore radius R0 along with their integral areal capacitance C s,eq,exp (in µF/cm

2).

Ref. Electrode R0 (nm) Electrolyte ψmax − ψmi n (V) C s,eq,ex p (µF/cm2)158 TiC-CDC 0.68-1.09 1 M TEABF4 in AN 2.3 6.0-13.6

168 OMC-M 2.15-4.25 6.88 M KOH 0.8 16.8-27.5

168 OMC-K 1.95-4.7 6.88 M KOH 0.8 12.0-22.5

169 GNS/CB 0.364-0.37 6 M KOH 1 28.3-46.6170 HOMC 0.37-0.41 6 M KOH 1 8.2-11.2

171 FSMC 2.15 6 M KOH 0.6 19.4

172 OMC 1.35-3.0 6 M KOH 1 5.8-11.8

172 OMC 1.35-3.0 1 M TEABF4 in AN 2 5.2-6.7

173 OMC 2.7-3.25 6.88 M KOH 0.89 11.9-15.0

174 CMK-8 2.39 2 M KOH 1 13.3

174 H-CMK-8 2.33 2 M KOH 1 20.2

175 OMC 2.25 6 M KOH 0.8 18.8

176 MC spheres 1.34 2 M KOH 1 11.1

177 MC 0.625-0.69 1 M TEABF4 in AN 2 10.3-11.6

178 OMC 0.395-0.555 1 M TEABF4 in PC 2 5.5-6.7

179 Carbon foam 1.9 6 M KOH 1 12.5

180 OMC 0.6 6 M KOH 0.8 14.1-19.6

181 GAC 0.245-0.26 6 M KOH 1 13.4-17.7

182 C-CS 1.95 6 M KOH 0.9 10.6-16

Note: solvent for KOH was water.

radii ranging from 0.36 to 3.25 nm. Three different electrolytes wereconsideredand treated as binary and symmetric, namely aqueous KOHand TEABF4 in PC or AN at concentrations c∞ ranging between 1and 6.88 mol/L. The effective diameters of anions and cations wereassumed to be identical and equal to that of the smallest non-solvatedion.

Figure 7a plots a total of 56 experimental data points for the equi-librium areal capacitance C s,eq,ex p as a function of the average poreradius R0 for the experimental data summarized in Table III. The equi-librium areal capacitance varied between 5.5and 47µF/cm2 due to thewide range of electrolytes, electrodemorphologies, pore sizes, andpo-

tential windows considered. No obvious trend was apparent. Figure 7bshows the same data plotted in terms of the ratio C s,eq,ex p/C s,eq, plan ar as a function of ( R∗0 −a∗/2) where C s,eq, pla nar was predicted by Equa-tion 18. The dimensionless numbers associated with the experimentaldata were such that 23 ≤ ψ∗ = 2ψ∗s ≤ 90, 0.1 ≤ ν p ≤ 0.57,1.6 ≤ a∗ ≤ 4.05, and 1.4 ≤ R∗0 ≤ 40.2. Figure 7b indicates thatC s,eq,ex p/C s,eq, pla nar decreased from 0.5 to about 0.1 as ( R

∗0 − a∗/2)

increased from 0 to 40. First, it is remarkable that the experimen-tally measured equilibrium areal capacitances C s,eq,ex p of such differ-ent mesoporous carbon electrodes had the same order of magnitudeas the theoretical areal capacitance for planar electrodes C s,eq, pla nar .In addition, plotting the data in terms of C s,eq,ex p/C s,eq, pla nar ver-sus ( R∗0 − a∗/2) significantly reduced the scatter compared withFigure 7a. In fact, the dimensionless plot described a consistent trend.The capacitance ratio C s,eq,ex p/C s,eq, pla nar increased as the dimen-sionless pore radius R∗0 decreased and approached the dimensionlessnon-solvated ion radius a ∗/2. On the other hand, the capacitance ra-tio C s,eq,ex p/C s,eq, plan ar reached a plateau around 0.08 as R

∗0 − a∗/2

exceeded ∼10. The scatter in the experimental data and the fact thatC s,eq,ex p differs from C s,eq, plan ar for large values of R0 can be at-tributed to the following main reasons:167 (i) The electrodes’ poresfeature a polydisperse size distribution whereas the scaling analysisused the average pore radius. (ii) The relative permittivity r wasassumed to be constant in the predictions of C s,eq, pla nar . However,r decreases significantly under the high electric fields encounteredin actual EDLCs.93,99,146 (iii) The electrolytes were assumed to besymmetric while actual anions and cations may have different ion di-ameters. And (iv) the assumption of simple cubic packing associatedwith the packing parameter ν p used in the MPB model may be overly

simplistic.Overall, scaling analysis indicates that the equilibrium areal capac-itance of mesoporous carbon electrodes with binary and symmetric

Figure 7. (a) Experimental data of equilibrium capacitance C s,eq,ex p as afunction of average pore radius R0 and (b) the ratio C s,eq,ex p/C s,eq, pla nar as

a function of R∗0 − a∗/2 for EDLCs with various mesoporous carbon elec-trodes and binary symmetric electrolytes. Reproduced with permission fromRef. 167.




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electrolytes can be expressed as the product of the equilibrium arealcapacitance of a planar electrode C s,eq, plan ar and a geometric function f ( R∗0 − a∗/2) correcting for the fact that pore/electrolyte interfacesare not planar, i.e.,167

C s,eq, pred = C s,eq, plan ar f ( R∗0 − a∗/2). [19]The geometric function f ( R∗0 − a∗/2) increased sharply as the poreradius approached the ion radius, i.e., R0

≈(a/2)+ or ( R∗

0 −a∗/2)

→0+. On the other hand, it converged to a constant in the limiting casewhen pores were large compared with the ion radius, i.e., R0 a/2or ( R∗0 −a∗/2) 1. Then, the effect of the pore curvature on the arealcapacitance was negligible. Note also that the function f ( R∗0 − a∗/2)reached this constant value very rapidly for ( R∗0 − a∗/2) larger than2-3. This suggests that electrode’s pores should be as monodisperseas possible and their diameter should match those of the ions. Bothof these criteria can be achieved with carbide-derived carbons whichfeatured large capacitances.158,183

Dynamic Modeling

Modified Poisson-Nernst-Planck model.— Governing equations.—

The modified Poisson Nernst-Planck (MPNP) model governs the

spatiotemporal evolutions of the electric potential ψ(r, t ) and ionconcentrations ci (r, t ) in binary and symmetric electrolytes (a1 = a2= a, z 1 = − z2 = z > 0, and D1 = D2 = D ) according to64,95,184

∇ · (0r ∇ ψ) =


− ze N A(c1 − c2) in the diffuse layer[20a]

[20b]

with∂ci

∂t = − ∇ · Ni in the diffuse layer, for i = 1, 2. [20c]

Here, Ni (r, t ) isthe ion massflux vectorof ion species “i” (in mol/m2s)

at location r and time t defined as

Ni (r, t ) = − D∇ ci − z F Dci

Ru T ∇ ψ

− D N Aa3ci

1 − N Aa3(c1 + c2)∇ (c1 + c2) for i = 1, 2 [21]

where F = e N A is the Faraday constant and D is the diffusioncoefficient of both ion species. This model accounts for finite ionsize and is applicable to cases with large electric potential and/orelectrolyte concentrations. On the right-hand side of Equation 21, thethree terms contributing to the ion mass flux Ni correspond to iondiffusion, electrostatic migration, and a correction due to the finite ionsize, respectively.95,99

Finally, the electric potential within the electrodes is governed bythe continuity equation combined with Ohm’s law to give,185,186

∇ · (σs∇ ψ) = 0 [22]where σs is the electrical conductivity of the electrode material. Note

that Equations 20 to 22 are coupled and thus must be solved simulta-neously subject to proper initial and boundary conditions.

Initial and boundary conditions.— Zero electric potential and uniformion concentrations equal to the bulk concentrations ci,∞ are typicallyused as initial conditions for solving the MPNP model in the elec-trolyte, i.e.,

ψ( x , t = 0) = 0 V and ci ( x , t = 0) = ci,∞. [23]The boundary condition imposed at the current collector/electrode

interface depends on the experiments simulated such as impedancespectroscopy (EIS), cyclic voltammetry (CV), or galvanostatic charg-ing and discharging of EDLCs. For EIS measurements, the imposedelectric potential can be expressed, in complex notation, as 41,187–189

ψs (t )=ψdc

+ψ0e

i2π f t . [24]

This harmonic potential consists of two components: (i) a time-independent “DC potential” ψdc and (ii) a periodically oscillating

potential with a small amplitude ψ0 around the DC component.189–191

The imaginary unit is denoted by i (i.e., i 2 = −1) while f is thefrequency expressed in Hz.

In CV measurements, the potential at the current collector/ electrode interface ψs varies periodically and linearly with time t according to,125

ψs (t )

=ψmax − v[t − 2(m − 1)τCV ] for 2(m − 1)τCV ≤ t


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Electrochemical impedance spectroscopy (EIS) consists of impos-ing the potential ψs (t ) given by Equation 24. The resulting electriccurrent js (t ) is measured and decomposed as

js (t ) = jdc + j0ei(2π f t −φ) [31]where jdc and j0 are the DC current density and amplitude of thecurrent density around its DC component, respectively; while φ( f )

is the frequency-dependent phase angle between the harmonic poten-tial ψs (t ) and the current density js (t ). The complex electrochemicalimpedance Z is defined as41,187-189

Z = ψ0 j0

eiφ = Z + i Z [32]

where Z and Z (expressed in m2) are the real and imaginary partsof the impedance, respectively. Based on the equivalent series RCcircuit, the resistance and differential capacitance per unit surfacearea are given by41,87–89,96

R = Z and C s,d i f f ( f ) = −1

2π f Z [33]

where Z is the out-of-phase component of the measuredimpedance.41,197 Note that EIS cannot measure the integral capaci-tance C s,in t .

198

Alternatively, cyclic voltammetry (CV) consists of imposing atriangular potential ψs (t ) as a function of time [Equation 26] andmeasuring the resulting current density js (t ). The results are typicallyplotted in terms of current or current density js (in A/m

2) versuselectrode potential ψs , referred to as “CV curves”. The surface chargedensity qs accumulated at the electrode surface during one cycle canbe estimated by computing the area enclosed by the CV curves. Then,the areal differential C s,d i f f and integral C s,in t capacitances of oneelectrode are obtained as functions of the measured current density jsat the scan rate v = |dψs/dt | (in V/s) according to,198

C s,d i f f (v) =| js|v

and

[34]

C s,in t (v) = qsψmax −ψmi n

= 1ψmax −ψmi n

js

2vd ψs .

Note that at low scan rates, the areal integral capacitance is inde-pendent of scan rate and equal to the equilibrium capacitance of theelectrode, i.e., C s,in t (v → 0) = C s,eq . Note that Equation 34 givesthe expressions for the capacitances C s,dif f and C s,in t of one electrodeand not those of the entire EDLC device.

The galvanostatic cycling method consists of charging and dis-charging the EDLC at constant current ± js [Equation 27] whilemeasuring the cell or device potential ψs (t ). It can be usedto measure both differential and integral capacitances accordingto,198

C s,d i f f ( js )

= js

|dψs/dt |and C s,in t ( js )

= jsτG/2

ψmax −ψmi n[35]

where τG/2 is the time required to vary the electric potential from ψmi nto ψmax or vice versa under imposed current ± js . The integral capac-itance is more commonly reported than the differential capacitancefor EDLCs tested using galvanostatic cycling method.50,199–204 Notethat in galvanostatic measurements, the areal capacitances C s,d i f f andC s,in t calculated using Equations 35 are identical only when the mea-sured electric potential varies strictly linearly with time. In fact, undervery low electric potential ψs (for CV measurements) and low current js (for galvanostatic measurements), the surface charge density varieslinearly with potential and C s,d i f f = C s,in t .99Sample simulations: physical interpretation of CV curves.— Wang andPilon195 performed 1D simulations of the two-electrode system illus-trated in Figure 2 to provide physical interpretations of experimentally

observed CV curves. Figure 8a shows the current density js versussurface potential ψs curves predicted from CV simulations for aque-ous KCl electrolyte, treated as binary and symmetric, for three values

0.0 0.1 0.2 0.3 0.4 0.5

-1.0

-0.5

0.0

0.5

1.0

D=2×10-9 m

2/s, c∞=1 mol/L

C u r r e n t d e n s i t y ,

j ( 1 0 A / m )

s

7

2

0.0 0.1 0.2 0.3 0.4 0.50.0

1.0

2.0

3.0

4.0

5.0

6.0

D=2×10-9 m

2/s, c∞=1 mol/L

cmax=1/NAa3

Surface potential, ψ s

(V)

Surface potential, ψ s (V)

S u r f a c e i o n c o n c e n t r a t i o n , c ( m

o l / L )

2

(a)

(b)

Figure 8. Predicted (a) js versus ψs and (b) c2( x = 0) versus ψs curvesdetermined from CV simulations for three values of potential window, i.e.,ψmax − ψmi n = 0.3, 0.4, and0.5 V. Results were obtained by numerically solv-ingthe MPNP model with a Stern layerwithoutaccounting fortheelectrode( L s= 0 nm) for v = 107 V/s, D = 2 × 10−9 m2 /s, c∞ = 1 mol/L, and L = 80 nm.195

of potential window ψmax −ψmi n = 0.3, 0.4, and 0.5 V. Results wereobtained by solving the MPNP model with a Stern layer without ac-counting for the potential drop across the electrode, i.e., σs → ∞.Other parameters were identical for all three cases and equal to v =107 V/s, a

=0.66 nm, D

=2

×10−9 m2 /s, c

∞ =1 mol/L, and L

=80 nm. Note that the scan rate was very large compared with exper-imental measurements because the simulations were performed forplanar electrodes separated by a relatively thin electrolyte domainfeaturing small ionic resistivity.

First, it is worth noting that the numerically generated CV curvesare very similar to those obtained experimentally with EDLCs fordifferent potential windows.205,206 Second, the three simulated CVcurves reached a maximum current density at about ψs = 0.2 Vduring discharging while js decreased at larger surface potential.This hump is typical of experimental CV curves.1,205,207–209 Differ-ent interpretations have been proposed in the literature to explain thishump including (i) electrolyte starvation due to limited amount of ions at low concentrations,158,210 (ii) redox reactions at the electrodesurface,1,207–209,211–217 (iii) different ion mobilities between anions and

cations,215

and (iv) saturation of ions at the electrode surface.205,206

However, there was no clear evidence and definitive explanation tothe commonly observed hump in the CV curves.




12/21


To physically interpret the simulated CV curves, Figure 8b showsthe surface anion Cl− concentration c2 as a function of surface poten-tial ψs for the same cases considered in Figure 8a. It also plots themaximum ion concentration cmax = 1/ N Aa3. It is evident that the sur-face anion concentration c2 increased rapidly with increasing potentialup to ψs = 0.2 V. This regime corresponded to the increase in currentdensity js shown in Figure 8a due to the formation of an EDL of Cl

−

anions before their concentration reached its maximum at ψs=

0.2 V.For ψs > 0.2 V, the anion concentration asymptotically approachedits maximum value cmax . In this regime, the ion accumulation near theelectrode surface became slower as the electric potential increaseddue to steric repulsion resulting in decreasing current density js[Figure 8a]. Overall, these results demonstrate that the hump observedexperimentally in CV curves for EDLCs can be attributed to the satu-ration of the electrode surface with closely packed ions as proposed inRef. 205 and 206 based on experimental results. The hump does notappear to be caused by electrolytestarvation, redoxreactions, or differ-ent ion mobility since these phenomena were not accounted for in thesimulations.

Sample simulations: Galvanostatic cycling.— Simulations of galvano-static cycling for binary and symmetric192 or asymmetric218 elec-trolytes and of EIS198 can be found in the literature. These studies

were also able to qualitatively reproduce experimental measurementsand provide physical interpretation based on interfacial and trans-port phenomena. For example, Figure 9a shows the electric potentialψ( x = − L , t ) computed at the electrode surface located at x = − L(Figure 2 as a function of time t .192 The simulations were performedfor electrolyte of thickness L = 50 µm at 298 K consisting of 1 MTEABF4 in PC and treated as binary and symmetric with z = 1, a= 0.68 nm, D = 1.7 ×10−7 m2 /s, c∞ = 1 M, and r = 66.1.192 TheEDLC was cycled at a constant current js = 14 mAcm−2 with a cy-cle period t c = 10 ms. This current density was within the range of typical experimental values. The combination of js and t c was cho-sen to yield a realistic maximum potential of 2.5 V for commercialEDLCs using organic electrolytes. However, the cycle period t c wasvery small because the planar EDLC simulated charged very rapidlycompared with actual porous electrodes. Figure 9a shows that the sur-

face potential varied almost linearly between the minimum potentialof 0 V and the maximum potential of 2.5 V and resembles typicalgalvanostatic cycling measurements on EDLCs. Figure 9b shows thecorresponding concentration c 2( x , t ) of BF

−4 anions as a function of

location x at several times during a charging step near the same elec-trode. It indicates that surface ion concentration increased over timefrom initial concentration c∞ = 1 M to cmax = 1/ N Aa3 = 5.3 M asthe EDL formed. As charging proceeded, the closely packed ion layerat the electrode surface became thicker and the EDL region propa-gated inside the electrolyte domain. It is interesting to note that the ionconcentration decreased from c max to c∞ over a very narrow spatialregion featuring steep concentration gradients. This will result in sig-nificant local heat generation, as discussed in the Thermal Modelingsection.

Generalized modified Poisson-Nernst-Planck model.— Governingequations.— The above MPNP model is valid only for binary andsymmetric electrolytes. To overcome this limitation, Wang et al.125

derived the so-called generalized MPNP (GMPNP) model from firstprinciples based on excess chemical potential and Langmuir-type ac-tivity coefficient. It is valid for asymmetric electrolytes and/or in thepresence of an arbitrary number N of ion species. The GMPNP modelis expressed as125

∇ · (0r ∇ ψ) =


−eN A N

i=1 zi ci in the diffuse layer

[36a]

[36b]

∂ci

∂t = − ∇ · Ni in the diffuse layer. [36c]

Figure 9. (a) Electric potential ψ(− L , t ) at the electrode surface at theStern/diffuse layer interface as a function of time and (b) BF−4 anion concen-tration c2( x , t ) as a function of location x at various times during galvanostaticcycles of period t c = 10 ms and current density js = 14 mAcm−2.192

Here, the general expression for mass flux vector Ni of ion species “i”is defined as

Ni = − Di ∇ ci − Di ci zi F

Ru T ∇ ψ− Di ci

N A N

i=1a3i ∇ ci

1 − N A N

i=1a3i ci

. [37]

For vanishing ion diameter (ai = 0), the GMPNP model reducesto the classical PNP model.95 In addition, for binary and symmetricelectrolytes such that N = 2, a1 = a2 = a, z1 = − z2 = z, and D1 = D2 = D, the GMPNP model reduces to the MPNP model99presented in the Dynamic modeling section.

Here also, uniform ion concentrations and zero electric potentialgiven by Equation 23 can be used as initial conditions. Note that elec-

troneutrality in the electrolyte requires that N

i=1 zi ci,∞ = 0. Unlike in

binary and symmetric electrolytes, c i,∞ may differ from one speciesto another. The boundary conditions used in the MPNP model to




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simulate EIS [Equation 24], CV [Equation 25], and galvanostatic cy-cling [Equation 27] and accounting for the Stern layer remain validat all times for asymmetric electrolytes. The ion mass fluxes Ni stillvanish at the Stern/diffuse layer interface [Equation 29]. Finally, con-stitutive relationships and properties used in the MPNP model canalso be used for the GMPNP model. Note that the continuum modelstreat ion diffusion coefficient Di , ion diameter a i , and valency zi asindependent variables. In reality, ions with large effective diameterand/or valency have typically smaller diffusion coefficient,153,154,156

as illustrated in Table II.

Scaling laws in CV measurements.— Dimensional analysis of theGMPNP model along with the boundary conditions reveal that CVsimulations with binary and asymmetric electrolytes with planar elec-trodes were governed by twelve dimensionless similarity parametersexpressed as125

v∗ = λ2 D/ D1

( Ru T / z1 F )/v,ψ∗max =

ψmax

Ru T / z1 F ,ψ∗mi n =

ψmi n

Ru T / z1 F , L ∗ = L

λ D,

a∗1 =a1

λ D, a∗2 =

a2

λ D, ν p,1 =

c1,∞1/2a31 N A

, ν p,2 =c2,∞

1/2a32 N A,

D∗2 = D2 D1, z∗2 = z2 z1

, L∗s = Lsλ D

, and σ∗s = σs (ψmax −ψmi n )/ L s z1 F c1,∞ D1/ L.

[38]

Here, v∗ is the dimensionless scan rate. It can be interpreted as the ra-tio of the ion diffusion time scale (τ D1 = λ2 D/ D1) to the characteristictime τth = ( Ru T / z1 F )/v for reaching the thermal potential Ru T / z1 F at scan rate v .125 Here, the Debye length for asymmetric electrolytes

with N ion species is expressed asλ D = (0r Ru T /F 2 N

i=1 z2i ci,∞)

1/2.39

Moreover, ψ∗max and ψ∗mi n are the dimensionless maximum and mini-

mum surface potentials, respectively, scaled by the thermal potential Ru T / z1 F . They can also be interpreted as the ratio of characteristictimes to reach ψmax or ψmi n and the characteristic time τth at scan

rate v.125

Here, ν p,1 represents the packing parameter due to finiteion size of ion species “1”. Note that v∗, ψ∗max , ψ

∗mi n , L

∗, a∗1 , andν p,1 were identical to or direct combinations of the similarity param-eters identified for the CV simulations of EDLCs with binary andsymmetric electrolytes.195 When considering binary and asymmetricelectrolytes, three additional dimensionless numbers appear, namelyν p,2, D

∗2 , and z

∗2 associated with ion species 2 in addition to those

obtained for binary and symmetric electrolytes. Accounting for theresistive losses through the electrode of thickness L s and electricalconductivity σs results in dimensionless thickness L

∗s = L s/λ D and

electrical conductivity σ∗s representing the ratio of the characteristiccurrent density in the electrode [σs (ψmax − ψmi n )/ L s ] to that in theelectrolyte F z1c1∞ D1/ L .

Moreover, a dimensionless areal integral capacitance can be de-fined as

C ∗s,in t =C s,in t

z1eN A D1c1,∞/λ Dv= 1

ψ∗max −ψ∗mi n

j∗s2v∗

d ψ∗s [39]

where C s,in t is definedby Equation 34. Graphically, C ∗s,in t corresponds

to the area enclosed by the j∗s versus ψ∗s curve. The dimensionless

capacitance C ∗s,in t , for binary asymmetric electrolytes, depends onthe twelve dimensionless numbers defined in Equation 38. However,finding a correlation such as that found for C ∗s,eq, pla nar could be verychallenging and time consuming. Instead, Wang et al.125 identified ascaling law for EDLC integral capacitance in CV measurements withbinary and asymmetric electrolytes between two planar electrodes.Figure 10a shows the double layer areal integral capacitance C s,in t predicted from CV simulations as a function of scan rate v rangingfrom 10−2 to108 V/s for caseswith different dimensionless parameters

L∗, z ∗2, ψ∗max , ψ∗mi n , D∗2 , a ∗1 , a ∗2 , ν p1, ν p2, σ∗s , and L∗s .125

It is evidentthat the C s,in t versus v curves differed significantly from one caseto another due to the broad range of parameters considered. Figure

Figure 10. Predicted (a) capacitance C s,in t as a function of scan rate v and(b) ratio C s,in t /C s,eq as a function of (τ RC /τCV )(1 + 80/σ∗s ) obtained fromCV simulations for different cases with various dimensionless parameters L ∗, z∗2 , ψ

∗max , ψ

∗mi n , D

∗2 , a

∗1 , ν p1, ν p2, σ

∗s , and L

∗s .

125

10b shows the same data as those shown in Figure 10a but plottedin terms of C s,in t /C s,eq as a function of (τ RC /τCV )(1 + 80/σ∗s ) whereC s,eq is the equilibrium capacitance corresponding to the maximumintegral capacitance and the plateau observed in C s,in t at low scanrates. Here, τ RC is the “RC time scale” for binary and asymmetricelectrolytes corresponding to the characteristic time of ions’ electro-diffusion63 and τCV is the half cycle period of CV measurements.They are expressed as,125

τ RC =λ D L

( D1 + D2)/2= √ τ Dτ L and τCV =

ψmax −ψmi nv

[40]

where τ D = 2λ2 D/( D1 + D2) and τ L = 2 L

2

/( D1 + D2) represent thecharacteristic times for ions to diffuse across the EDL and from oneelectrode to the other, respectively. The ratio τ RC /τCV can also be




14/21


expressed in terms of the dimensionless numbers125

τ RC

τCV = 2v

∗ L∗

(1 + D∗2 )(ψ∗max −ψ∗mi n )[41]

First, it is remarkable that all the curves in Figure 10b collapsedon a master curve irrespective of the different values of the abovedimensionless similarity parameters. Second, Figure 10b indicates

that two regimes can be clearly identified:125 (i) τ RC τCV corre-sponds to the quasi-equilibrium or ion diffusion-independent regimeand (ii) τ RC τCV corresponds to the ion diffusion-limited regime.In the diffusion-independent regime, ion and charge transports are fastenough to follow the change in the surface potential ψs (t ), unlike inthe diffusion-limited regime. Note that when σ∗s is very large (e.g.,σ∗s 80), the charge transport in the electrode is much faster thanthat in the electrolyte. Then, the potential drop across the electrode isnegligible andit suffices to simulate theelectrolyteto numerically gen-erate CV curves.125 This result also suggest that the electrode materialshould be such that σ∗s 80 for potential drop through the electrodeto be negligible. Finally, note that this analysis is also valid in thelimiting case of binary and symmetric electrolytes when z∗2 = −1,a∗2 = a∗1 , and D∗2 = 1, as considered in Refs. 63, 184, 195, and 219.

Thermal Modeling

Introduction.— Experimental studies of EDLCs have shown thatgalvanostatic cycling under current ± I s (in A) caused an overall tem-perature rise from cycle to cycle as well as superimposed temper-ature oscillations with the same frequency as the charge-dischargecycles.83,88,89,92,220–222 The overall temperature rise corresponded toirreversible Joule heating, while temperature oscillations were at-tributed to reversible heating.92,220,221 Temperature measurements atthe surface of a 5000 F commercial EDLC during galvanostatic cy-cling showed that the overall temperature rise in a thermally insulatedEDLC was approximately linear and proportional to I 2s .

92 The re-versible heat generation rate within an EDLC was empirically foundto be exothermic during charging, endothermic during discharging,

and proportional to I s .92,221Thermal modeling of EDLCs aims to predict the temperature of

EDLCs under specific operating conditions in order to avoid the neg-ative effects associated with high temperature operation discussedearlier. In addition, detailed thermal models can be used to identifythe different local interfacial and physical phenomena responsiblefor heat generation and to determine their respective contributions.Several fundamental questions are of particular interest: while Jouleheating is well understood, what physical processes are responsiblefor reversible heat generation resulting in the experimentally-observedtemperature oscillations? Also, does the reversible heat generationratevary significantly with time and location within the EDLCs? Thesequestions are addressed in the following sections.

Volume-averaged thermal modeling of EDLCs.— Volume-averagedthermal models of EDLCs assumethat the temperature in thedevice is uniform and dependent only on time. Then, thermal energybalance performed on the entire EDLC yields the following governingequation for the volume-averaged device temperature T̄ (t )92,223,224

C thd T̄

dt = Q̇(t ) − (T̄ − T ∞)

Rth[42]

where C th is the heat capacity of the device (in J/K) and Q̇(t ) is theinternal heat generation rate (in W). The term (T̄ −T ∞)/ Rth representsthe heat transfer rate (in W) from the device to its surroundings attemperature T ∞. The associated thermal resistance Rth (in K/W) canbe expressed as Rth = 1/h̄ A, where h̄ is the average convective heattransfer coefficient (in W/m2K) and A is the external surface area of

the device (in m2

). The total heat generation rate Q̇(t ) consists of an irreversible and a reversible contribution, i.e., Q̇(t ) = Q̇ ir r (t ) +Q̇rev(t ).

During galvanostatic cycling, the current is a square signal of magnitude I s with cycle period t c. Then, the irreversible heat gen-eration rate is constant and equal to Q̇ir r = I 2s R, where R is theelectrical equivalent series resistance of the EDLC.92,221,225 Note,however, that most thermal models ignored Q̇rev and determined Rexperimentally. 88,91,92,224 Schiffer et al.92 developed an ad hoc modelfor the reversible heat generation rate Q̇rev based on the fact that theentropy of the ions (i) decreased during charging as ions became moreordered due to EDL formation and (ii) increased during dischargingas the ions returned to their disordered state corresponding to uniformion concentrations.92 These processes were respectively exothermicand endothermic to satisfy the second law of thermodynamics.92 Thederivation assumed that the EDL consisted of a monolayer of ions(Helmholtz model) and that the cell capacitance C T was independentof the cell voltage. For a binary and monovalent electrolyte with sym-metric ion size, the total reversible heat generation rate (in W) wasexpressed as92

Q̇rev = −2T̄ k B

eln

V S

V 0

C T

d ψs

dt = −2 T̄ k B

eln

V S

V 0

I s (t ) [43]

where V 0 and V S are the total electrolyte volume and the Stern layervolume, respectively. The expression of Q̇rev given by Equation 43

has been used in various studies.91,222,224 Unfortunately, the volumesV 0 and V S are difficult to evaluate for porous electrodes of commercialEDLCs. Instead, thevalue of ln (V S /V 0) wasusedas a fittingparameterto match the model predictions with experimental data. A similar butsimpler approach is to assume the reversible heat generation rateto be a square wave expressed as Q̇rev(t ) = +α I s during chargingand Q̇rev(t ) = −α I s during discharging where α is a positive semi-empirical parameter specific to each device (in V). 225

Finally, the energy conservation Equation 42 is a first-order linearordinary differential equation requiring only one initial condition oftentaken as room temperature, i.e., T̄ (t = 0) = T̄ 0. The temperature T̄ (t )can be expressed as the sum T̄ (t ) = T̄ ir r (t ) + T̄ rev(t ) where T̄ ir r (t )and T̄ rev(t ) correspond to the contributions of the irreversible andreversible heat generationrates to the temperature rise, respectively.225

They are given by

T̄ ir r (t ) = T̄ 0 + I 2s R Rth + T ∞ − T̄ 0)(1 − e−t / Rth C th

and [44]

T̄ rev(t ) = C −1th e−t / Rth C th t

0

et / Rth C th Q̇rev(t

)dt . [45]

Temperature predictions obtained by solving Equation 42 showedgood agreement with experimental data reported in the literature92,88

for different commercial devices using C th , Rth , R provided bythe manufacturer while the parameter α was estimated as α =2C thT̄ rev/ I s t c from the amplitude of the temperature oscilla-tions T̄ rev measured experimentally under oscillatory steady-stateconditions.225

Overall, the above volume-averaged model provides a rapid and

easy way to predict the temperature of commercial EDLCs. It can beused to design appropriate thermal management strategies of individ-ual or modules of EDLCs and to control charging and discharging formaximizing performance and minimizing aging.91,224 Unfortunately,it cannot predict the thermal behavior of new EDLC designs withoutpreliminary testing. More importantly, it cannot predict the spatiotem-poral evolution of temperature within the device.

Local thermal modeling of EDLCs.— Local thermal models aimto predict the spatiotemporal evolution of the local temperature withinthe electrolyte, electrode, and current collector of an EDLC device.This can be achieved by solving the heat diffusion equation within thecell expressed as223

ρc p∂T

∂t = ∇ ·(k

∇ T )

+ q̇. [46]

Here ρ, c p , and k are the density, specific heat, and thermal conduc-tivity of the current collector, electrode, or electrolyte. In addition, q̇




15/21


is the local volumetric heat generation rate (in W/m3).223 It can betreated as uniform in the current collector and in the electrode andcorresponds to irreversible Joule heating, i.e., q̇ = j 2s /σs . In the elec-trolyte, the heat generation rate can be divided into irreversible andreversible contributions such that q̇ = q̇ir r + q̇rev .

Equation 46 has been solved for two-dimensional89,222 and three-dimensional cases.220,226,227 Here also, most simulations consideredthe local temperature rise due only to irreversible heating but did notaccount for reversible heat generation.89,90,220,226,227 The heat gener-ation rate was prescribed as either (i) uniform throughout the en-tire device,89,220,222,226,227 (ii) uniform in the “active components,”i.e., the electrodes and separator,226,227 or (iii) as having differentvalues in the current collectors, electrodes, and separator.90 Ther-mal models accounting for reversible heating222 used the reversibleheat generation rate in the electrolyte predicted by Schiffer et al.92

assumed to be uniform, i.e., q̇rev = Q̇rev/V where V is the elec-trolyte volume. Moreover, most of the local thermal models usedexperimental measurements to retrieve input parameters necessaryto solve the model.89,90,226,227 Local thermal models with uniformheat generation rates have shown good agreement with experimentaltemperature measurements for experimentally characterized EDLCdevices.89,222 However, they are semi-empirical models relying on

experimentally measured device properties and ignoring the detailedelectrochemical phenomena taking place during cycling. Thus, theyare unable to predict (i) local variations in the heat generation rate,(ii) the performance of novel and untested EDLC designs, or (iii)how different device designs or materials would affect the thermalperformance.

Recently, d’Entremont and Pilon192,218 developed a local thermalmodel of EDLCs, from first principles, to predict the local irreversibleand reversible heat generation rates based on the ion transport andon conservation of energy. The ion transport was modeled using(a) the MPNP model, presented in the Modified Poisson-Nernst-Planck model section, for binary and symmetric electrolytes192 or (b)the GMPNP, discussed in the Generalized modified Poisson-Nernst-Planck model section, for asymmetric electrolytes.218 The derivationof the energy conservation equation accounted for heat conduction

and energy transport by ion mass fluxes.192,218 The authors also ob-tained the well-known heat diffusion equation given by Equation 46and rigorously derived an expression for q̇ given by

q̇ = q̇ E + q̇S = q̇ir r + q̇rev. [47]

Here, q̇ E = j · E is the heat generation associated with ions decreas-ing their electric potential192,228,229 and q̇S is the so-called “heat of mixing”.228,230 First, the irreversible volumetric heat generation rate,corresponding to Joule heating, was derived as one of three contribu-tions to q̇ E and expressed as q̇ir r = j 2/σe, where j is the local ioniccurrent density (in A/m2) and σe is the local electrolyte conductivity(in S/m). For an electrolyte with N ion species “i” of valency z i and

concentration c

i , the current density j

and the electrolyte conductivityσe can be expressed as,48,231

j = N

i=1 zi F Ni and σe =

F 2

Ru T

N i=1

z2i D i ci (r , t ) [48]

Second, the reversible heat generation rate q̇rev arose in the EDLregion due to the concentration and temperature gradients.192,218 Itwas expressed as the sum of four different contributions,

q̇rev = q̇ E ,d + q̇ E ,s + q̇S ,c + q̇S ,T [49]

where q̇ E ,d and q̇ E ,s are contributions from q̇ E arising from iondiffusion and steric repulsion, respectively and such that q̇ E =q̇ir r + q̇ E ,d + q̇ E ,s . For asymmetric electrolytes with N ion species,

they were expressed as218

q̇ E ,d = j

σe·

F

N i=1

Di zi ∇ ci

and

q̇ E ,s = j

σe · F

N

i=1 Di zi ci

N A N

i=1a3i ∇ ci

1 − N A N

i=1a3i ci

. [50]

Note that q̇ E ,d and q̇ E ,s differ from zero only in the presence of anion concentration gradient ∇ ci . On the other hand, the heat of mixingwas such that q̇S = q̇S ,c + q̇S ,T with the terms q̇S ,c and q̇S ,T arisingfrom concentration gradients and from the temperature gradient, re-spectively. In the general case of asymmetric electrolytes, they wereexpressed as218

q̇S ,c =3

32π

eF 2

(0r )3/2

Ru T N

i=1 z2i ci

1/2

N i=1

z2i Ni

·

N i=1

z2i ∇ ci

and q̇S ,T = −3

32π

e F 2

N i=1

z2i ci

1/2

(0r )3/2 R1/2u T 3/2

N i=1

z2i Ni · ∇ T . [51]

Note that q̇S ,T was found to be negligible compared with q̇ E ,d , q̇ E ,s ,and q̇S ,c for all cases simulated.

192,218 However, the latter three termswere of similar magnitude and particularly large near the electrodesurface.

The numerical results indicated that q̇ir r was constant and uniformthroughout the electrolyte for both symmetric192 and asymmetric218

electrolytes, as assumed in existing local models.89,90,220,226,227 It de-creased with increasing valency | zi | or diffusion coefficient D i of oneor both ion species due to the resulting increase in electrical conduc-tivity σe of the electrolyte. By contrast, the reversible heat generation

term q̇rev was highly non-uniform, unlike what was assumed in thedifferent implementations of Schiffer’s model.222 It was exothermicduring charging, endothermic during discharging, and localized to anarrow region mainly in the EDL forming near the electrodes. 192,218

For symmetric electrolytes, q̇rev was proportional to the current den-

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