Physical Interpretation of Cyclic Voltammetry for Hybrid

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pubs.acs.org/JPCC

Physical Interpretation of Cyclic Voltammetry for Hybrid Pseudocapacitors Henri-Louis Girard, Hainan Wang, Anna d’Entremont, and Laurent Pilon* Henry Samueli School of Engineering and Applied Science, Mechanical and Aerospace Engineering Department, University of California Los Angeles, 420 Westwood Plaza, Los Angeles, California 90095, United States

*S Supporting Information

ABSTRACT: This study aims to elucidate the respective contributions of faradaic reactions and electric double layer formation to charge storage in hybrid pseudocapacitors. It also aims to provide physical interpretation of experimental cyclic voltammetry (CV) measurements. First, a physicochemical transport model was derived from first-principles for simulating coupled interfacial, transport, and electrochemical phenomena in hybrid pseudocapacitors. The model simultaneously accounted for (i) charge transport in both electrodes and electrolyte, (ii) the dynamics of the electric double layer, (iii) steric repulsion due to finite ion sizes, (iv) redox reactions, and (v) intercalation. Then, CV curves were simulated for different electrode thicknesses and Li diffusion coefficients in the planar pseudocapacitive electrode. Particular attention was paid to the so-called b-value characterizing the power law evolution of the total current with respect to scan rate for a given potential. Overall, trends observed in numerically generated CV curves showed good agreement with experimental measurements. In addition, the results indicated that a b-value of unity across the potential window can be associated with purely faradaic charge storage with fast ion intercalation in the thin-film pseudocapacitive electrode. The study also demonstrates that under diffusion- limited conditions of Li intercalation in the pseudocapacitive electrode, the CV curves exhibited two distinct regimes: a faradaic regime dominated by faradaic reactions and a capacitive regime dominated by electric double layer formation. The b-value was near 1.0 in both regimes. However, a dip in the b-value, often observed experimentally, was also obtained and attributed to the transition between the capacitive and the faradaic regimes.

■ INTRODUCTION electrode made of a porous transition metal oxide (e.g., RuO2, MnO2,Nb2O5, MoO3) and an EDLC-type electrode typically Electrochemical capacitors, also known as supercapacitors, have 2,8−12 ff attracted significant attention in recent years due to their made of porous carbon. These di erent devices hold great promise for meeting energy storage needs in current and promise as electrical energy storage devices for high energy and − − − 1 3,5 12 power applications.1 3 They are typically classified as either emerging applications. electric double layer capacitors (EDLCs) or pseudocapacitors In situ measurement of ion concentrations in the electrolyte and of intercalated species concentration profiles in the depending on the energy storage mechanism. EDLCs store ffi energy physically in the electric double layers (EDLs) forming electrode during operation is very di cult, if not impossible. − near the electrode/electrolyte interfaces.1 3 Thus, the charge In addition, it is challenging to discriminate between the storage is highly reversible and the cycle life of EDLCs contributions of the redox reactions and of the EDL formation represents more than 100,000 cycles.1 EDLCs are attractive for to the total current measured experimentally. This information high power applications, such as in the regenerative braking can be obtained through physical modeling and simulations to system of hybrid electric vehicles as well as storing energy and improve the understanding of experimental measurements and balancing the load from intermittent renewable energy sources, to guide the optimization of the electrode and of the − due to their fast charging/discharging rates and long life.1 4 electrolyte. However, the energy density of EDLCs remains low compared This study aims to develop a physicochemical transport with that of batteries. Pseudocapacitors store energy via the model for simulating hybrid pseudocapacitors under cyclic electric double layers as well as via reversible oxidation− voltammetry by accurately accounting for interfacial and reduction (redox) reactions with or without insertion or transport phenomena occurring in the electrolyte and for − intercalation.1,3,5 7 Pseudocapacitors tend to feature larger redox reactions and intercalation of the reaction product in the capacitances and energy densities than EDLCs because the amount of charge they can store is not limited by the surface Received: January 21, 2015 − area of the electrode/electrolyte interface.3,5 7 Finally, hybrid Revised: April 16, 2015 pseudocapacitors are devices combining a pseudocapacitive Published: May 4, 2015

© 2015 American Chemical Society 11349 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article pseudocapacitive electrode. This model was derived from first- current”14,25 or described more cautiously as “capacitive principles in that it relies on the fundamental equations behavior”,22,31,32 as “pseudocapacitive behavior”,20,21 or as governing electrodynamics. It was used to elucidate the physical “non-diffusion controlled charge storage”.28 In addition, a dip phenomena contributing to charge storage in pseudocapacitors is often observed in the b-value plotted as a function of ψ 18,25,28 and to provide physical interpretation of experimental measure- s. This has been attributed to the presence of redox ments. peaks from the faradaic reaction in the CV curves.18,25,28 To the best of our knowledge, despite its extensive usage, this ■ BACKGROUND semiempirical analysis and its physical interpretation have not Electric Double Layer Structure. According to the Stern been rigorously demonstrated, particularly for simultaneous model, the electrolyte is divided into two domains: the Stern electric double layer formation, redox reactions, and ion layer and the diffuse layer.13 The Stern layer is a compact layer insertion in the pseudocapacitive electrode. of charges adjacent to the electrode surface characterized by the Models of Pseudocapacitors. A variety of models for absence of free charges. Within the diffuse layer, ions are free to pseudocapacitors have been proposed. Equivalent RC circuit move under the influence of diffusion, electromigration, and and transmission line models consist of modeling a physical steric repulsion.13 The thickness of the EDL can be device by a succession of ideal capacitors and resistors 33 approximated as the Debye length λ and varies with the integrated in a circuit. They have been used to study the D 33−35 diameters, valencies, and concentrations of ions as well as with charging/discharging dynamics of pseudocapacitors. How- temperature.13 ever, these models have to be fitted with experimental data to Empirical Characterization of Pseudocapacitors. A retrieve the values of the different resistances and capacitan- 35 semiempirical approach for analyzing cyclic voltammetry (CV) ces. They can be used to control the device operation, but measurements has been developed and used extensively to they cannot be used to design the electrodes or electrolyte of characterize capacitive effects in the total current. Specifically, novel pseudocapacitors. Moreover, the classical RC circuit ff the measured total current density jT at a given surface potential models neglect ion di usion and nonuniform ion concentration ψ 36 s was assumed to relate to the scan rate v according to the in the electrolyte. As faradaic reactions depend strongly on semiempirical relationship14 the ion concentrations at the reaction plane, these models are therefore not suitable for modeling pseudocapacitors with j ()ψ T s 1/2 simultaneous faradaic and capacitive currents. =+kv1()ψψ k2 () v1/2 ss (1) Continuum models, numerically solving the Poisson fi ψ ψ equation for electric eld and the mass conservation equations where k1( s)andk2( s) are semiempirical functions for ion concentrations, have also been developed to analyze ψ − independent of v but dependent only on the cell potential s pseudocapacitors.37 45 Different approaches investigated the ψ − imposed between the current collectors. The functions k1( s) ff 46 37 39,42 ψ e ects of porosity, redox-active particle size, solid and k2( s) respectively correspond to the slope and intercept in ff 42 1/2 1/2 ψ 14 phase di usion limitation in the case of ion intercalation, and the plot of jT/v versus v for a given potential s. This moving reaction fronts.40 However, these models imposed the approach was supported by the facts that (i) the capacitive double layer areal capacitance (in F/m2)orvolumetric current jC, associated with EDL formation or dissolution, is capacitance (in F/m3) as constant properties independent of − known to vary linearly with the scan rate v in the absence of the electric potential instead of predicting them.37 46 redox reactions13,15 and that (ii) the theoretical faradaic current fi Unfortunately, the double layer capacitance is known to vary jF due to surface redox reactions in a semi-in nite electrolyte 1/2 nonlinearly with electric potential under large electric potentials domain is proportional to v when the presence of the EDL 47 13 and electrolyte concentrations. Furthermore, some models and the associated capacitive current are ignored. This assumed the concentration of ions at the reaction front in the method has been used for full-cell hybrid pseudocapacitors with 37,39,40,44,46 16 electrolyte was the bulk concentration although the a chemically inert counter electrode as well as for electric field and the presence of EDLs near the electrodes pseudocapacitors with Li or Na metal as the counter electrode affect the concentrations of the ions involved in the faradaic to measure specifically the performance of the metal oxide 17−24 reaction. electrode. Furthermore, three-electrode experiments also Finally, first-principles molecular dynamics (MD) simula- used this approach to characterize pseudocapacitive electro- − tions have been performed to explore the fundamental des.7,14,25 31 electrochemical behaviors of pseudocapacitive materials includ- The presence of redox peaks in many pseudocapacitor CV 48,49 50 51 52 ing RuO , TiO , MnO , and MoO . However, the curves shows that different processes can be dominant at 2 2 2 3 computational complexity and cost limit MD simulations to different potentials. This led to another approach for analyzing extremely small time and length scales on the order of 10 ns experimental CV measurements. Assuming that the current at 48,53 and 10 nm, respectively. These are significantly smaller any potential is generally dominated by either a capacitive or a than those encountered in actual characterization methods such faradaic process, eq 1 can replaced by as cyclic voltammetry or galvanostatic cycling, which are on the ψ 7,8 b()s order of several micrometers and seconds. For example, He j ()ψψ= kv () (2) T ss et al.54 simulated cyclic voltammetry for two cylindrical pores ψ where the exponent b( s) is the so-called b-value expected to with subnanometer width at unrealistically large scan rates of 1 vary between b = 1/2 in the case of purely faradaic current and to 10 MV/s to study the local dynamics of charging and b = 1 for purely capacitive current.14,25 The value of b is discharging. Because of the small time and length scales, such fi ψ ff typically evaluated by least-squares tting of jT( s) versus v for studies cannot simulate relatively slow and long-range di usion ψ 14,25 given values of s. A b-value of 1 corresponds to a greater processes across an electrode. Therefore, MD simulations seem rate capability and is therefore desirable for all potentials in the inadequate for reproducing experimental cycling with realistic operating window. It has been attributed to “capacitive time scales for electrodes or devices with realistic dimensions.

11350 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article

Figure 1. Schematic of the simulated one-dimensional hybrid pseudocapacitor cell consisting of a redox-active pseudocapacitive electrode and a carbon electrode with LiClO4 electrolyte in PC. The dashed line encloses the computational domain simulated.

This study aims to develop a continuum model to accurately they approach the electrode. Therefore, the solvated ion simulate coupled interfacial, transport, and electrochemical diameter may depend on the location considered.58 However, phenomena in hybrid pseudocapacitors subjected to cyclic to the best of our knowledge, no quantitative model exists voltammetry. The model simultaneously accounted for (i) relating the partially solvated ion diameter to the local electric charge transport in both electrodes and electrolyte, (ii) the field and position with respect to the electrode. (3) The dynamics of the electric double layer, (iii) steric repulsion due diffusion coefficients of all ion species in the electrolyte and that to finite ion sizes, (iv) redox reactions, and (v) ion intercalation of Li in the redox-active electrode were independent of their in the pseudocapacitive electrode. The goal of this study is also respective concentrations. Their values corresponded to those to rigorously examine the respective contributions of capacitive of dilute solutions reported in the literature. Note that in reality and faradaic current densities to the total current density and the ion diffusion coefficients vary with ion concentration. provide physical interpretations of experimentally obtained CV Umino and Newman59 reported that the diffusion coefficient of curves. Finally, it aims to assess whether b = 1 in eq 2 can be + 2− H and SO4 for H2SO4 in water varied nonlinearly with salt associated with faradaic reactions and why a dip is often 18,25,28 concentration. However, it did not exceed 40% of its observed in the b-value versus potential plot. asymptotic value corresponding to dilute solutions for concentrations ranging from 0.3 to 7.5 M. In fact, it was ■ ANALYSIS verified numerically that accounting for concentration-depend- Schematics and Assumptions. Figure 1 shows the ent diffusion coefficient had a negligible impact on the schematic of the one-dimensional hybrid pseudocapacitor predicted current at the scan rates considered in this study simulated as well as the associated computational domain and when ion diffusion in the electrolyte was not limiting (see coordinate system. The cell consisted of a planar thin-film Supporting Information). (4) Isothermal conditions prevailed pseudocapacitive electrode made of transition metal oxide throughout the device. (5) Advection in the electrolyte was MpOq of thickness LP and a planar carbon electrode of thickness negligible. (6) The reversible redox reaction given by eq 3 was LC. They were separated by an electrolyte of thickness 2L made heterogeneous and occurred at the electrode/electrolyte of LiClO4 salt in propylene carbonate (PC) solvent. The interface near the pseudocapacitive electrode at electrolyte domain was divided in three regions corresponding x = −L + H.13 This assumption is commonly used in modeling to a Stern layer of thickness H near each electrode surface and a battery and supercapacitor,13 as free electrons are assumed to ff fi di use layer beyond. This simpli ed geometry was chosen in be confined in the electrode and free Li+ ions are confined in order to assess the contribution of multiple and closely coupled the electrolyte. (7) The intercalation and deintercalation of Li physical phenomena in detail without considering the multi- atoms into and out of the pseudocapacitive electrode were dimensional nature of actual electrodes. Note also that the treated as diffusion processes. (8) The redox-active material did ff electrode curvature was found to have negligible e ect on the not undergo any phase transition. This assumption is consistent areal capacitance of an EDL on a sphere of radii larger than 40 fi 55 with observations on Nb2O5 thin lms during lithium nm. The following reversible redox reaction was assumed to 60 σ insertion. (9) The electrical conductivity P and the Li take place at the pseudocapacitive electrode ff ffi di usion coe cient D1,P of the pseudocapacitive electrode were +− constant. In practice, they may change with Li insertion. mLi++⇌ Mpq Om e Li m M pq O (3) However, to the best of our knowledge, no quantitative model To make the problem mathematically tractable, the following or experimental measurements exist capturing these effects in a ff + − fi fi assumptions were made: (1) electrodi usion of Li and ClO4 thin lm of metal oxide. (10) The speci c ion adsorption due ions in the electrolyte and insertion (or intercalation) of Li to nonelectrostatic forces was assumed to be negligible. Note atoms in the electrode were one-dimensional. (2) The Stern that previous simulations of EDLCs based on this assumption layer thickness H was approximated as the radius of the largest agreed well with experimental data.61,62 (11) The potential − 13,56,57 solvated ion species, namely, ClO4 , so that H = a2/2. In drop across the highly conducting current collectors was practice, the ions may lose part or all of their solvation shell as negligible so that only the electrodes and electrolyte domains

11351 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article were simulated. The goal of this study is to reproduce trends in second terms of eq 8 represent the ion flux due to diffusion and CV curves and to provide physical interpretation of behaviors electromigration, respectively, while the last term represents a observed experimentally. The assumptions considered have correction accounting for finite ion size.56 Finally, the presence been widely used in the literature to make possible simulations of the Stern layers near each electrode was accounted for via of pseudocapacitors. boundary conditions,56,62 and no governing equations for ψ ψ Governing Equations. The local electric potential (x, t) (x,t) and ci(x,t) needed to be formulated or solved within the − − ≤ ≤− ≤ in the pseudocapacitive ( LP L x L) and carbon (L Stern layer. ≤ x L + LC) electrodes is governed by the one-dimensional Boundary and Initial Conditions. The one-dimensional Poisson equation expressed as56,63 governing eqs 4 to 8 are second-order partial differential equations in space and first-order in time. Each equation ∂ ⎛ ∂ψ ⎞ ⎜⎟ requires two boundary conditions and one initial condition in ⎝σP ⎠ = 0 in the pseudocapacitive electrode ∂xx∂ (4) each region it is solved. First, the initial electric potential was assumed to be uniform ⎛ ⎞ ∂ ⎜⎟∂ψ across the device and given by σC = 0 in the carbon electrode ∂xx⎝ ∂ ⎠ (5) ψ(,0)x = 0V (9) σ σ where P and C are the electrical conductivities of the + − Initially, the Li and ClO4 ion concentrations in the electrolyte pseudocapacitive and carbon electrode, respectively. − ≤ ≤ − The local molar concentration of intercalated Li atoms in the ( L + H x L H) were taken as uniform and equal to their bulk concentrations satisfying electroneutrality according pseudocapacitive electrode, denoted by c1,P(x,t), is governed by the mass diffusion equation given by63,64 to ∂c ⎛ ∂c ⎞ cx11,2(,0)==− c∞∞ andc(,0) x zc 11,2 / z (10) 1,P ∂ ⎜ 1,P ⎟ = D1,P in the pseudocapacitive electrode ∂tx∂ ⎝ ∂x ⎠ Similarly, the initial Li concentration in the pseudocapacitive − − ≤ ≤− (6) electrode ( L LP x L) was uniform and equal to c1,P,0, ff ffi i.e. where D1,P is the di usion coe cient of intercalated lithium atoms in the pseudocapacitive electrode. cx1,PP(,0)= c 1, ,0 (11) Moreover, the potential and ion concentrations in the diffuse layer of the electrolyte solution (−L + H ≤ x ≤ L − H) are The potential at the current collector/pseudocapacitive fi − − ψ − − ψ governed by the generalized modi ed Poisson Nernst Planck electrode interface was imposed as ( L LP,t)= s(t). 56 ψ (GMPNP) model derived by Wang et al. The GMPNP During cyclic voltammetry measurements, s(t) varied linearly model was developed for asymmetric electrolytes with multiple with time according to56 fi ions species of nite size. For binary and asymmetric ⎧ ff 56 ψτ−−−vt[( n 1)] electrolytes the GMPNP in the di use layer is expressed as ⎪ max cCV ⎪ ⎧ 2 ⎪ −≤<−ττ ∂ ⎛ ∂ψ ⎞ for (ntncCVc 1) ( 1/2) CV ⎪ ⎜⎟ ψ = ⎨ ϵϵ =−Fzc∑ (7a) s()t ⎪ ∂xx⎝ 0 r ∂ ⎠ ii ⎪ψτ+−−vt[( n 1/2)] ⎨ i=1 ⎪ min cCV ⎪ ⎪ ∂ ∂ ⎩ for (ntn−≤< 1/2)ττ ⎪ ciiN cCVcCV (12) ⎩ =− fori = 1, 2 (7b) ∂t ∂x τ where nc is the cycle number and CV is the cycle period, while ψ ψ where ci(x,t) is the local molar concentration of ion species i in max and min are the maximum and minimum values of the cell + ψ the electrolyte solution at time t. Here, i = 1 refers to Li and potential s(t), respectively. In addition, the carbon electrode − −1 ψ i = 2 refers to ClO4 . Moreover, F = 96 485 C mol is the surface was electrically grounded so that (L + LC,t)=0V. ϵ × −12 −1 Faraday constant, 0 = 8.854 10 Fm is the free space Note that the choice of electrical ground is arbitrary and was ϵ ff permittivity, and r is the relative permittivity of the electrolyte made for convenience. It did not a ect the predicted CV curves fl solution. The local mass ux of ion species i, denoted by Ni(x,t) or the associated physical interpretations. in mol m−2 s−1,isdefined for a binary and asymmetric The electric potential varied linearly across the Stern layers electrolyte as56 so that the electric fields at the pseudocapacitive and carbon electrodes satisfied56,62 ∂ciiiiDFzc ∂ψ Nii(,)xt=− D − ∂ψ 1 ∂x RTu ∂ x − (,)−+LHt =[(ψψ −Lt ,) − ( −+ L Ht ,)] ∂x H ⎛ 2 ⎞ Dcii ∂ ⎜ ⎟ (13a) − 2 ⎜∑ ccjjmax/ , ⎟ 1/− ∑ ccjjmax, ∂x ⎝ j=1 ⎠ ∂ψ 1 j=1 (8) (,)LHt−=[(,)ψψLt −− ( L Ht ,)] ff ffi ff ∂x H (13b) where Di and ai are the di usion coe cient and the e ective ion diameter of ion species i in the electrolyte solution, These boundary conditions accounted for the presence of the respectively. The temperature is denoted by T (in K), while Stern layers at both electrodes without explicitly simulating × 23 −1 −1 −1 62 NA = 6.022 10 mol and Ru = 8.314 J K mol are the them in the computational domain. Avogadro constant and the universal gas constant, respectively. The current density at the pseudocapacitive electrode/ 3 − The maximum concentration cj,max = 1/(NA aj ) corresponds to electrolyte interface, located at x = L was equal to the sum of fi simple cubic ion packing at the electrode surface. The rst and the capacitive jC(x,t) and faradaic jF(t) current densities (in

11352 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article

A/m2) at the Stern/diffuse layer interface, located at jt() 56,65 F x = −L + H, so that N1(,)−+LHt = and zF1 ∂ψ −−21 −σ (,)(−=−++Lt j L Ht ,)() j t NL(,)0molms−= Ht (21) P ∂x CF(14) 1 Both the carbon and pseudocapacitive electrodes were However, near the carbon electrode at x =+L, only the − capacitive current contributed to the total current density so impermeable to ClO4 ions (i = 2) so that that −−21 N22(,)(,)0molms−+LHtNLHt = − = (22) ∂ψ −σ (,)Lt=− j ( L Ht ,) C ∂x C (15) Finally, the present model can be extended to pseudocapacitors with two redox-active electrodes by adding the ff In both cases, jC(x,t) is the displacement current density due to corresponding faradaic current jF at the Stern/di use layer the electric double layer formation at the pseudocapacitive and interface near the counter electrode (x = L − H) and solving 66 carbon electrodes and defined as the mass conservation equation for intercalated Li in both ∂2ψ electrodes. However, current symmetric pseudocapacitors have ±− =−ϵϵ ±− been shown to have numerous limitations, including lower jC ((LHt ),)0 r ((LHt ),) ∂∂xt (16) gravimetric energy and power densities68 and smaller potential In addition, the faradaic current density jF(t) is typically window due to irreversible redox reactions occurring at one of described by the generalized Frumkin−Butler−Volmer model the electrodes.68 Hybrid pseudocapacitors enable better control evaluated at the pseudocapacitive electrode/electrolyte inter- of the intercalation reaction in the pseudocapacitive electrode face and expressed as13 and feature better overall performance.69 ⎧ ⎡ ⎤ ⎡ ⎤⎫ Constitutive Relationships. A total of 23 input parameters ⎪⎪(1− αη )zF − αηzF were needed to solve eqs 4 to 22 including (i) the electrolyte j ()tjt= ()⎨ exp⎢ 11⎥ − exp⎢ ⎥⎬ FF,0 ⎪⎪⎣ ⎦ ⎣ ⎦ properties ϵ (0), β, n, a , a , z , z , D , D , and c ∞, (ii) the ⎩ RTuuRT ⎭ r 1 2 1 2 1 2 1, pseudocapacitive electrode properties σ , D , c , k , and (17) P 1,P 1,P,0 0 Δψ , (iii) the carbon electrode conductivity σ , (iv) the cell η Δψ − Δψ Δψ eq C where (t) = H(t) eq is the overpotential and H(t) = dimensions L, LP, and LC, and (v) the operating conditions T, ψ(−L,t) − ψ(−L+H,t) is the electrical potential drop across the ψ , ψ , and v. Typical values were collected from the 13 max min Stern layer near the pseudocapacitive electrode. For electro- literature. des made of transition metal oxides, the equilibrium potential The electrolyte considered in the present study was LiClO ff Δψ fi 4 di erence eq is typically tted experimentally as a function of in the organic solvent propylene carbonate (PC) at T = 298 K. ϵ the state-of-charge c1,P/c1,P,max, where c1,P,max is the maximum The relative permittivity of the electrolyte r is known to concentration of intercalated lithium atoms in the pseudoca- decrease significantly at large electric fields due to the 42,45,67 pacitive electrode. It can be obtained by fitting saturation of the polarization of the solvent molecules. The 63,65 experimental data for the open-circuit potential. The so- Booth Model has been developed to account for this effect and 63,65 70,71 called exchange current density jF,0(t) can be written as is expressed as j ()tFzkcLHtc=−+ [ ( , )]1−α [ −− c ( Lt , )]α ⎡ ⎤ F,0 10 1 1,Pmax , 1, P 223 ⎢ 1 ⎥ ϵ=+ϵ−rr()En ( (0) n ) coth(βE ) − α βE ⎣ βE ⎦ (23) [(cLt1,P − ,)] (18) fi ϵ where k0 is the reaction rate constant expressed in where E is the norm of the local electric eld, r(0) is the m1+3α mol−α s−1. Here, the transfer coefficient α was assumed relative permittivity at zero electric field, n is the index of to be 0.5, corresponding to identical energy barriers for forward refraction of the electrolyte at zero electric field, and β is an and backward redox reactions.13 In this case, the faradaic empirically defined fitting parameter. For propylene carbonate, ϵ β current j (t) has the same sign as the overpotential η(t). these parameters are as follows r(0) = 64.4, n = 1.42, and = F −8 61 + Moreover, the current collector was impermeable to the 1.314 × 10 m/V. The effective solvated ion diameters of Li − − lithium atoms intercalated in the pseudocapacitive electrode so (z1 = +1) and ClO4 (z2 = 1) in PC were taken as 72 that the mass flux of Li vanished at the pseudocapacitive a1 = 0.67 nm and a2 = 1.0 nm, respectively. The Stern layer electrode/current collector interface, i.e. thickness H, corresponding to the distance of closest approach to the electrode plane, was equal to the largest solvated ion ∂c1,P 56 ff ffi −D (,)0molms−−LLt = −−21 radius, i.e., H = a2/2 = 0.5 nm. The ion di usion coe cients 1,P P + − × −10 2 ∂x (19) of Li and ClO4 in PC were taken as D1 = 2.6 10 m /s × −10 2 72 The mass flux of Li exiting through the pseudocapacitive and D2 = 3.3 10 m /s, respectively. The initial bulk ion concentrations were imposed as c1,∞ = c2,∞ = 1 mol/L, typical electrode/electrolyte interface was related to the faradaic 18,20,73 current density j (t) based on stoichiometry as of electrochemical capacitors. F The pseudocapacitive electrode properties were inspired by ∂c1,P jt() those of Nb2O5 whenever available since Nb2O5 was shown to F 74 −D1,P (,)−=Lt be a good material for intercalation pseudocapacitance. The ∂x zF (20) 1 electrical conductivity of metal oxides may increase with the The mass flux of Li+ ions (i = 1) across the Stern/diffuse layer amount of intercalated lithium.75 Here, however, a constant σ −5 interface near the pseudocapacitive electrode was related to value P =10 S/m was selected based on the range of lithium 76 jF(t) in the same way, while it vanished near the carbon concentration observed. The electrical conductivity of the σ electrode such that carbon electrode was taken as C = 5 S/m corresponding to

11353 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article

ψ Figure 2. Top: Capacitive jC, faradaic jF, and total jT current densities versus cell potential s for a scan rate v = 1 V/s for (a) Case A and (c) Case B. Bottom: Intercalated lithium concentration normalized by the maximum concentration at the pseudocapacitive electrode surface (x = −L) and at the − − current collector (x = L LP) for (b) Case A and (d) Case B at scan rate v = 1 V/s. The red circle and blue square are guides corresponding to the same instant in the cycle in ﬁgures panels c and d.

77 that of activated carbon. The reaction rate constant k0 for electrode may reach zero resulting in the absence of faradaic transition metal oxides has been reported to range between current at any subsequent time. However, it is important to −11 −8 2.5 −0.5 −1 63,65,78 ffi 10 and 10 m mol s . Its value was taken as note that the choice of a su ciently large value of c1,P,0 had no −8 2.5 −0.5 −1 ff k0 =10 m mol s , i.e., the most favorable redox reaction e ect on the oscillatory steady state solution. Note also that ff Δψ condition. In addition, the equilibrium potential di erence eq some metal oxide electrode synthesis processes result in was assumed to be constant and equal to zero for the sake of electrodes preloaded with Li atoms.80 simplicity.79 Note that, in reality, the equilibrium potential Moreover, the electrolyte thickness was taken as 2L =2μm, ff di erence varies linearly with the state of charge c1,P/c1,P,max of while the electrode thicknesses LP and LC were identical and the electrode.34 We verified that in all cases considered here, equal to either 20 or 500 nm. The potential window was Δψ ff the variation of eq did not a ect the current magnitude or selected to be large enough to show all relevant phenomena the b-value (see Supporting Information). The maximum occurring during charging and discharging. Consequently, the intercalated lithium concentration in the redox active electrode pseudocapacitive electrode potential was cycled between −0.8 ρ ρ was estimated as c1,P,max = m /M where and M are the density and +0.8 V, while the carbon electrode was grounded. Finally, and molar mass of the fully intercalated metal oxide. For the scan rate v was arbitrarily varied from 10−2 to 10 V/s. The ρ ρ ≈ 3 Li2Nb2O5, and M were reported as 4.6 g/cm and M = scan rate values were large compared with those used 42 ≈ ff ff 279.7 g/mol yielding c1,P,max 32.9 mol/L. The di usion experimentally due to the fact that di usion limitation occurs ffi coe cient D1,P of intercalated lithium atoms in transition metal at a much higher scan rate for the simulated planar electrodes oxides typically ranges from 10−16 to 10−10 m2/s.78 Here, it will than for actual porous electrodes. The scan rate was chosen to be treated as a variable parameter to explore asymptotic explore these effects. In addition, simulations with relatively behaviors of hybrid pseudocapacitors. Finally, the initial large scan rate took less time and reached an oscillatory steady concentration of Li in the electrode c1,P,0 was adjusted on a state faster. case by case basis. First, imposing c1,P,0 = 0 mol/L initially Method of Solution and Data Processing. The would prevent any faradaic current jF(t) [eq 18] and ion governing eqs 4 to 8 were solved along with the boundary insertion at any later time. Similarly, if c1,P,0 is too low during and initial conditions given by eqs 9 to 22 using the commercial the first deintercalation step, the Li concentration in the finite element solver COMSOL 4.3b running in parallel on 8

11354 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article cores with 64 GB of RAM. The numerical convergence Figure 2a shows the predicted capacitive jC, faradaic jF, and fi ff criterion was de ned such that the maximum relative di erence total jT current densities for Case A under oscillatory steady ff ψ in the predicted current densities jC and jF at the Stern/di use state as functions of the imposed potential s(t) for scan rate ψ layer interface near the pseudocapacitive electrode v = 1 V/s. The CV curve jT vs s featured a peak in both − ψ ψ (x = L + H) was less than 1% when dividing both the charging, for decreasing s, and discharging, for increasing s, ψ mesh size and the time step by a factor of two. This currents around s = 0 V. Figure 2a also shows that the faradaic Δ ≈ τ corresponded to imposing a time step of t CV/1000 = current dominated during the entire cycle and the capacitive ψ − ψ ( max min)/1000v. The mesh size was the smallest at the current was negligible. This can be attributed to fast redox Stern/diffuse layer interface due to the large potential gradient reactions and intercalation of Li atoms in the thin electrode. and gradually increased away from the electrode. The mesh size Indeed, the large reaction rate guaranteed that Li+ ions at the was specified to be less than 0.025 nm at the Stern/diffuse layer Stern/diffuse layer interface were consumed by the redox ff interface, while it was less than LP/25 and LC/25 in the reaction during charging. In addition, the large di usion electrode domains and less than L/50 in the diffuse layer. The coefficient of Li+ ions in the electrolyte ensured sufficient total number of finite elements was about 500 for all cases supply of Li+ during charging. Similarly, the large diffusion simulated in the present study. The criterion for reaching coefficient of Li in the pseudocapacitive electrode ensured oscillatory steady state was similar when running an additional sufficient supply of Li atoms to be oxidized at the electrode/ 10 cycles. This corresponded to running 10 cycles for each case electrolyte interface during discharging. Then, no significant studied. capacitive current resulting from EDL formation was observed. The instantaneous total current density jT(t)atthe Figure 2b shows the intercalated lithium saturation in the ψ fi pseudocapacitive electrode/electrolyte interface was computed redox-active electrode as a function of s(t)de ned (i) at the − − as the sum jT(t)=jC(t)+jF(t) where the capacitive jC(t) and electrode/current collector interface as c1,P( L LP,t)/c1,P,max faradaic jF(t) current densities were given by eqs 16 and 17, and (ii) at the electrode/electrolyte interface as fi − respectively. CV curves at a speci c scan rate were obtained by c1,P( L,t)/c1,P,max. First, the two curves were superimposed ψ plotting jT(t), jC(t), and jF(t) as functions of s(t). Finally, the indicating that the concentration of intercalated Li was uniform 2 areal integral capacitance Cs,int (in F/cm ) was computed from throughout the pseudocapacitive electrode. This was verified by ψ the total current jT(t) and the imposed potential s(t) plotting the concentration profile in the electrode. Second, according to81 neither curve shows any hysteresis during the entire cycle. This can be attributed to the fast intercalation and deintercalation 1 j ()ψ and to the small electrode thickness, which did not limit the C = T s dψ sint, ∮ s redox reactions. ψψmax− min 2v (24) Figure 2c shows the current densities jC, jF, and jT predicted for Case B under oscillatory steady state as functions of ψ − potential s(t) ranging between 0.8 and 0.8 V for a scan rate v ■ RESULTS AND DISCUSSION of 1 V/s. In contrast to Case A (Figure 2a), Figure 2c shows General CV Curve Behavior. To study the effects of the that, for slow Li diffusion in a thick electrode, the faradaic pseudocapacitive electrode properties on the performance of a current jF dominated only in the lower end of the potential hybrid pseudocapacitor, two extreme cases were defined. First, window, while the capacitive current jC dominated in the higher Case A corresponded to a thin-film pseudocapacitive electrode end during both charging and discharging. In addition, the total −10 with fast Li intercalation such that LP = 20 nm and D1,P =10 current jT and the integral capacitance were smaller in Case B m2/s. Case B corresponded to a thick pseudocapacitive than in Case A for the same v = 1 V/s. In Case B, the CV curve ψ electrode with slow Li intercalation so that LP = 500 nm and featured three peaks in the total current: at s = 0 V in both the −14 2 ψ − D1,P =10 m /s. These electrode thicknesses are charging and the discharging phase, and at s = 0.15 V in the representative of actual pseudocapacitive nanoparticles on a charging phase. Figure 2d shows the intercalated lithium 25 82 ψ conductive substrate and thick film coating or microspheres, saturation in the redox active electrode as a function of s(t) (i) respectively. Additionally, the values of the diffusion coefficient at the electrode/current collector interface ff − − D1,P was selected such that in Case A, the di usion length LD c1,P( L LP,t)/c1,P,max and (ii) at the electrode/electrolyte fi τ 1/2 μ fi − de ned as LD =(D1,P CV/2) = 12.6 m was signi cantly interface c1,P( L,t)/c1,P,max. At the current collector/electrode − − larger than the electrode thickness, whereas in Case B, the interface, the saturation c1,P( L LP,t)/c1,P,max remained ff ff di usion length LD = 126 nm was on the order of the electrode constant throughout the entire cycle. Indeed, the low di usion ffi thickness. Thus, these parameters were chosen as extreme coe cient D1,P and the thick electrode prevented the values to explore both the lithium diffusion limited and the intercalated lithium from reaching the current collector before diffusion-independent regimes in the pseudocapacitive elec- the current was reversed. Note, however, that the value of ff ffi − − ff trode. The di usion coe cient of Li in a metal oxide electrode c1,P( L LP,t) was di erent from its initial value c1,P,0. During − would typically increase with the concentration of intercalated charging, the saturation c1,P( L,t)/c1,P,max at the electrode/ ψ ≈− Li. However, to the best of our knowledge, no model capturing electrolyte interface increased sharply at potential s 0.15 V this effect is available. Therefore, the values were selected corresponding to the sudden increase in the faradaic current arbitrarily within a reasonable range to explore asymptotic reaching a peak at −0.15 V, as observed in Figure 2c (red behaviors. Other input parameters were identical for both cases circle). During discharging, the saturation at the pseudocapa- ψ and given in Constitutive Relationships. In both Cases A and B, citive electrode/electrolyte interface reached zero around s = charging corresponded to reduction of Li+ ions into Li atoms, −0.3 V corresponding to the steep drop observed in the − which intercalated into the redox-active electrode. Discharging faradaic current jF in Figure 2c (blue square). Then, c1,P( L,t)/ corresponded to deintercalation and oxidation of Li atoms into c1,P,max was very small, and the faradaic current was limited by Li+ ions that diffused in the electrolyte. the diffusion of Li out of the pseudocapacitive electrode. Note

11355 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article

η Figure 3. Temporal evolution of (a) the faradaic jF(t) and capacitive jC(t) current densities, (b) the overpotential (t), (c) the exchange current − + − − ff density jF,0(t), and (d) the ion concentrations c1( L + H,t)ofLi and c2( L + H,t) of ClO4 at the Stern/di use layer interface near the τ pseudocapacitive electrode as functions of dimensionless time t/ CV for Case B at scan rate v = 1 V/s. Here, the vertical dashed lines indicate τ ψ − τ ψ − transitions at t/ CV = 0.3 ( s = 0.15 V) and t/ CV = 0.65 ( s = 0.3 V). The red circle and blue square correspond to the identical indicators in Figure 2. that this behavior illustrates the need to account simultaneously these regimes, it is worth considering the temporal variation of for redox reactions and EDL formation and to determine the other important variables. concentration of each species in the electrodes and in the Figure 3 shows the temporal evolution of (a) the faradaic jF η electrolyte during the entire cycle. Indeed, previous mod- and capacitive jC current densities, (b) the overpotential (t), 37,39,40,44,46 els assumed constant and uniform ion concen- (c) the exchange current density jF,0(t), and (d) the ion + − − − trations in the electrodes and electrolyte when simulating concentrations of Li c1( L+H,t) and of ClO4 c2( L + H,t)at ff faradaic reactions. The present results indicate that such the Stern/di use layer interface as functions of dimensionless τ assumptions may be valid for parts of the cycle but not for time t/ CV for Case B at scan rate v = 1 V/s. Figure 3a clearly others. Then, diffusion limitations or reactant starvation cannot shows the two prevailing regimes. A dominant capacitive current was observed at the beginning of the charging step for be accounted for while they play an important role in the τ τ t/ CV < 0.3 and during part of the discharging step for t/ CV > energy storage mechanisms. In addition, in both Cases A and B, ψ − 0.65. This corresponded to 0.8 V > s > 0.3 V encompassing the state of charge or the Li saturation c1,P/c1,P,max was very fi charging and discharging. A dominant faradaic current prevailed small. This justi es the assumption of perfect faradaic behavior for t/τ between 0.3 and 0.65 across the charging/discharging Δψ CV characterized by eq =0V. − ψ − transition corresponding to 0.8 V < s < 0.3 V. Overall, Case B featured two regimes: a faradaic regime and a Figure 3b shows that the overpotential η(t) varied linearly capacitive regime. Note that the faradaic regime is desirable with a large slope during the capacitive regime and a smaller because it results in larger total current and capacitance. Indeed, slope around η = 0 V during the faradaic regime. Indeed, in the μ 2 Δψ η η Δψ ≈ the integral areal capacitance Cs,int was 47.9 F/cm in Case A case of eq =0V, (t) is such that (t)/H = H/H at 1 V/s when the current was exclusively faradaic. This was ∂ψ/∂x(−L + H,t). Then, combining this expression with the μ 2 ∂η ∂ ≈ ϵ ϵ almost double the capacitance of 24.8 F/cm achieved by the expression of jC(t) given by eq 16 yields / t jC(t)H/ 0 r. device simulated in Case B, also at 1 V/s. To further elucidate Thus, the time rate of change of η(t) depends on the value of

11356 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article

Figure 4. CV curves for multiple scan rates for (a) Case A and (b) Case B. Total current density jT as a function of scan rate v in log scale for a ψ ψ selection of imposed potentials s(t) for (c) Case A and (d) Case B. Computed b-value as a function of imposed potential s(t) for (e) Case A and (f) Case B using the charging current for all scan rates shown in panels a and b, respectively. The regression coefficient R2 obtained in the fitting of ψ the b-value was larger than 0.95 for all values of s. fi − τ the capacitive current jC(t). The latter is signi cant during the c1,P( L,t) reached zero at dimensionless time t/ CV = 0.65 ψ − capacitive regime and very small during the faradaic regime. corresponding to s = 0.3 V (blue square). This corresponded ff η This explains the large di erences in slopes of (t) in the two to (i) a vanishing exchange current density jF,0, (ii) the start of regimes shown in Figure 3b. the sharp decrease in the faradaic current jF(t)around ψ − Figure 3c shows the exchange current density jF,0(t) s = 0.3 V, (iii) the onset of the EDL formation (Figure computed from the concentrations of Li+ at the Stern/diffuse 3d), and (iv) the associated capacitive current (Figure 3a). This − fi layer interface c1( L + H,t) and of Li at the pseudocapacitive con rms that the lack of lithium atoms at the electrode surface − electrode/electrolyte interface c1,P( L,t), according to eq 18. was the cause of the transition from the faradaic to the During the capacitive regime, the exchange current density was capacitive regime corresponding to a switch in the charge − − negligibly small. This was due to the fact that c1,P( L,t) was storage mechanism observed in the CV curve around 0.3 V nearly zero during this regime (Figure 2d). It explains why the during discharging shown in Figure 2c (blue square). Then, the ff faradaic current jF(t) was small despite the large overpotential faradaic current was limited by out-di usion (or deintercala- η (t) (Figure 3b). Similarly, during the faradaic regime, jF,0(t) tion) of intercalated Li in the electrode. was large resulting in large faradaic current jF(t) despite the Similarly, Figure 3d also displays the two distinct regimes in η + − ff small overpotential (t) . Notably, during the discharging step the Li and ClO4 concentrations at the Stern/di use layer

11357 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article interface. During the faradaic regime, both ion concentrations 4c shows that, for Case A, the total current jT increased linearly ψ − remained constant and nearly equal to the bulk concentration, with scan rate v for s = 0.4 V and +0.4 V. The same − − ≈ i.e., c1( L + H,t)=c2( L + H,t) c1,∞ = c2,∞ = 1 mol/L. This observation was made for all potentials across the potential indicates that the redox reaction was fast enough to consume window including the current at the charging peak, also known any excess Li+ at the interface, while ion diffusion in the as cathodic peak. However, Figure 4d indicates that, for Case B, ffi electrolyte was su ciently fast to replenish the ions consumed the evolution of the total current density jT(t) with the scan rate by the reaction and available in large amounts in the relatively v depended on the potential considered. The b-value was equal ψ − large electrolyte domain. Therefore, during this regime, the to (i) 1 for s = 0.5 V, when the current was exclusively ψ contribution of the capacitive current jC to the total current jT faradaic, (ii) 0.96 for s = 0.5 V when the total current was was negligible. By contrast, during the capacitive regime, the ψ − exclusively capacitive, and (iii) 0.91 for s = 0 V when the total anion ClO4 formed an EDL near the pseudocapacitive current was a combination of faradaic and capacitive electrode surface resulting in a significant capacitive current contributions. In addition, the current at the charging peak (Figure 3a). Indeed, in the capacitive regime, the overpotential evolved as j ∝ v0.64. This decrease in the b-value around the η fi T,pa (t) was positive so that the electric eld attracted the peak associated with the faradaic process was consistent with negatively charged anions electrostatically to the electrode experimental observations for both two- and three-electrode − 7,18,22,25−27,29 surface. In fact, c2( L + H,t) quickly reached its maximum experiments. As discussed earlier, the charging 3 value c2,max =1/NA a2 . The saturation of the electrode surface − peak, corresponding to the lowest b-value, was located at the with ClO4 ions caused the capacitive current to reach a transition between the faradaic and capacitive regimes, as τ maximum around t/ CV = 0.8 (Figure 3a) and was responsible shown in Figure 2c. ψ for the hump observed in the CV curve at s = 0 V in the ψ Figure 4e,f shows the b-value as a function of the potential s discharging step (Figure 2b). Note that similar observations computed by the least-squares method from the total current were made in CV curves of EDLCs.81 density jT vs scan rate v shown in Figure 4c,d for Cases A and B, Finally, Figure 3a shows that the faradaic current remained respectively. Figure 4e indicates that for Case A the b-value was positive, corresponding to deintercalation of Li, during most of independent of ψ and close to unity across the potential the charging step. The late reversal of the faradaic current was s η window. Note that the faradaic current with fast intercalation due to the fact that jF(t) always has the same sign as (t) [eq η dominated for all potentials (Figure 2a). Thus, these results 17]. Therefore, (t) had to become negative for the faradaic indicate that a b-value of unity can be associated with fast current to change sign. However, Figure 3b shows that, in the reversible faradaic reaction in thin electrodes. Then, the faradaic capacitive regime during the discharging step, the overpotential − current j was proportional to the scan rate v, as also observed increased due to the formation of the EDL by the anion ClO F 4 in EDLCs. Thus, this behavior can be referred to as “capacitive at the electrode surface. This EDL had to dissolve before the behavior”. However, the charge storage mechanism is truly overpotential became negative and the faradaic current reversed faradaic and not capacitive. By contrast, for Case B, Figure 4f sign. Figure 3d confirms that the onset of the faradaic regime − indicates that the b-value featured a drop from 1 to about 0.6, as coincided with the complete dissolution of the ClO EDL, 4 often observed in experimental studies of pseudocapaci- while the concentration of Li+ c (−L + H,t) at the Stern/diffuse 1 tors.14,25,83 In fact, it is interesting to note that Figure 4f is layer interface increased from zero to c1,∞. As a consequence, − analogous to Figure 10c of ref 22 obtained experimentally with the formation of a thick EDL of ClO4 ions in the capacitive regime during the discharging step causes the late onset of the a two-electrode cell with a working electrode made of anatase faradaic current during the charging step. Therefore, preventing TiO2 and a counter electrode made of lithium metal with 1 mol/L LiPF6 in a 1:1 ethylene carbonate and dimethyl or delaying the capacitive regime during discharging would also 22 increase the length of the faradaic regime during charging. carbonate mixture as electrolyte. Similar results were also Analysis of the b-Value. Figure 4a,b shows the numeri- reported in Figure 5 of ref 25 obtained experimentally in a cally predicted CV curves for scan rates v ranging from 0.1 to three-electrode experiment with a working electrode made of 10 V/s for Cases A and B, respectively. Figure 4a indicates that, nanocrystalline anatase TiO2 nanoparticles deposited on glassy in the case of a thin electrode with fast intercalation (Case A), carbon with 1 mol/L LiClO4 salt in PC as electrolyte and the CV curves were similar in shape with the total current lithium metal for both the counter and the reference ψ electrodes.25 density jT( s) increasing with scan rate. This suggests that the cell was not limited by diffusion of any species for the range of The dip in the b-value has often been attributed to the ∝ 1/2 scan rates considered. Figure 4b shows similar CV curves for intercalation process on the basis that jT v for reversible fi ff different scan rates in the case of a thick electrode with slow redox reactions with semi-in nite di usion of the reaction intercalation corresponding to Case B. The peaks and humps products and in the absence of electric double layers, as ψ 13 apparent during charging shifted to lower potential s as the previously discussed. However, the dip in the b-value scan rate v increased. Note that this was consistent with predicted numerically did not occur during the faradaic regime + − ψ experimental observations of Li insertion in nanocrystalline but for 0.2 V < s < 0 V corresponding to the transition fi 7 14 lms of orthorhombic Nb2O5 or in TiO2 anatase. Here also, between the capacitive and faradaic regimes (Figure 2c). Previous models accounting separately for redox reactions and the total current jT achieved in Case B was much smaller than − that observed in Case A, for any given scan rate v and potential for electric double layer37 46 would not be able to predict the ψ s. In other words, the capacitance of the thin electrode with dip in the b-value caused by the interplay between the two fast intercalation (Case A) was much larger than that of the phenomena requiring the prediction of the concentrations of all thick electrode with slow intercalation (Case B). species in the entire domain to account for reactant starvation. ψ Figure 4c,d shows the total current density jT(t) during For larger potentials, s > 0 V, the b-value increased again to a charging as a function of scan rate v in log scale for different value close to unity. In this region, the capacitive current ψ values of potential s(t) for Cases A and B, respectively. Figure dominated, and the b-value was consistent with the expression

11358 DOI: 10.1021/acs.jpcc.5b00641 J. Phys. Chem. C 2015, 119, 11349−11361 The Journal of Physical Chemistry C Article of capacitive current evolution with the scan rate given by H Stern layer thickness (nm) 13,15 2 jC = Csv where Cs is the areal capacitance. jC predicted capacitive current density (A/m ) 2 jF predicted faradaic current density (A/m ) ■ CONCLUSION j predicted total current density (A/m2) T 2.5 −0.5 −1 This article presented a new physicochemical model, derived k0 reaction rate constant, eq 18 (m mol s ) from first-principles, for hybrid supercapacitors featuring a L half of interelectrode distance (nm) pseudocapacitive electrode and a carbon electrode subjected to LC thickness of the carbon electrode (nm) cyclic voltammetry. The model simultaneously accounted for LP thickness of the pseudocapacitive electrode (nm) (i) charge transport in both electrodes and electrolytes, (ii) the m stoichiometric number of intercalated Li, LimMpOq dynamics of the electric double layer, (iii) steric repulsion due M molecular weight of the fully intercalated active to finite ion sizes, (iv) redox reactions, and (v) intercalation. electrode material (g/mol) The results showed that for fast intercalation in a thin electrode n index of refraction of the electrolyte the current was always due to the faradaic reaction and that the nc cycle number × 23 −1 b-value was unity across the potential window. For relatively NA Avogadro constant, NA = 6.022 10 mol fl −2 −1 thick electrodes and slow intercalation, the CV curves exhibited Ni ion ux of species i (mol m s ) two distinct regimes. First, a faradaic regime prevailed for the p,q stoichiometric numbers of the metal and oxygen in the more negative potentials and was dominated by the faradaic metal oxide MpOq −1 −1 reaction and limited mainly by the diffusion of Li in the Ru universal gas constant, Ru = 8.314 J mol K pseudocapacitive electrode during discharge. Second, a T local temperature (K) capacitive regime dominated at larger potentials with current t time (s) due to the formation of an EDL at the electrode surface. A b- v scan rate of the cyclic voltammetry (V/s) μ value of unity was associated with both regimes. The dip in b- x location in one-dimensional space ( m) value, often observed experimentally,18,25,28 was shown to occur z ion valency during the transition between these two regimes. ■ GREEK SYMBOLS ■ ASSOCIATED CONTENT α transfer coefficient, eqs 17 and 18 *S Supporting Information β fi ff ffi tting parameter for the Booth Model, eq 23 Results for concentration-dependent ion di usion coe cients ϵ ϵ × −12 −1 0 vacuum permittivity, 0 = 8.854 10 Fm and effect of variable equilibrium potential difference on the ϵ r relative permittivity of electrolyte current magnitude and the b-value. The Supporting Informa- Δψ ff eq equilibrium potential di erence (V) tion is available free of charge on the ACS Publications website Δψ H potential drop across the Stern layer (V) at DOI: 10.1021/acs.jpcc.5b00641. λ D Debye length (m) η η Δψ − Δψ ■ AUTHOR INFORMATION overpotential, = H eq (V) ρ density of the fully intercalated pseudocapacitive Corresponding Author electrode material (kg/m3) * Phone: +1 (310) 206-5598. Fax: +1 (310) 206-2302. E-mail: σ electrical conductivity of electrode (S/m) [email protected]. τ CV cycle period (s) Notes ψ electric potential (V) fi ψ ψ The authors declare no competing nancial interest. min, max minimum and maximum of the potential window (V) ■ ACKNOWLEDGMENTS ψ imposed cell potential (V) s This material is based upon work supported as part of the Molecularly Engineered Energy Materials, an Energy Frontier ■ SUBSCRIPTS Research Center funded by the U.S. Department of Energy, ∞ Office of Science, Office of Basic Energy Sciences under Award refers to bulk electrolyte Number DE-SC0001342. i refers to ion species i C refers to the carbon electrode ■ NOMENCLATURE P refers to the pseudocapacitive electrode a effective ion diameter (nm) b ■ REFERENCES bb-value, power constant in jt = av c ion concentration (mol/L) (1) US Department of Energy. Basic Research Needs for Electrical 3 cmax maximum ion concentration, cmax =1/NAa (mol/L) Energy Storage, 2007. http://www.osti.gov/accomplishments/ 2 Cs,int integral areal capacitance (F/m ) documents/fullText/ACC0330.pdf. c initial concentration of intercalated Li in the (2) Simon, P.; Gogotsi, Y. Materials for Electrochemical Capacitors. 1,P,0 − pseudocapacitive electrode (mol/L) Nat. Mater. 2008, 7, 845 854. (3) Zhang, Y.; Feng, H.; Wu, X.; Wang, L.; Zhang, A.; Xia, T.; Dong, c1,P,max maximum concentration of intercalated Li, c1,P,max = mρ/M (mol/L) H.; Li, X.; Zhang, L. Progress of Electrochemical Capacitor Electrode Materials: a Review. Int. J. Hydrogen Energy 2009, 34, 4889−4899. D diffusion coefficient of ions in electrolyte (m2/s) ff ffi (4) Abbey, C.; Joos, G. Supercapacitor Energy Storage for Wind D1,P di usion coe cient of intercalated Li in the pseudoca- Energy Applications. IEEE Trans. Ind. Appl. 2007, 43, 769−776. pacitive electrode (m2/s) − (5) Wu, Z.-S.; Zhou, G.; Yin, L.-C.; Ren, W.; Li, F.; Cheng, H.-M. e elementary charge, e = 1.602 × 10 19 C Graphene/Metal Oxide Composite Electrode Materials for Energy × 4 −1 − F Faraday constant, F =eNA = 9.648 10 C mol Storage. Nano Energy 2012, 1, 107 131.

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