Frumkin-Butler-Volmer theory and mass transfer in electrochemical cells

ISSN 1023�1935, Russian Journal of Electrochemistry, 2012, Vol. 48, No. 6, pp. 570–579. © Pleiades Publishing, Ltd., 2012.Published in Russian in Elektrokhimiya, 2012, Vol. 48, No. 6, pp. 633–644.

570

1. INTRODUCTION

Including effects of the double layer compositionon the charge transfer rate at electrodes was initiatedby Frumkin [1] for one electron proton reduction asan extension to the classical Butler–Volmer equation.Later contributions were e.g. by Parsons [2] whoextended Frumkin’s approach to a multiple electronreaction, Itskovich et al. [3] who assumed a constantconcentration (and thus a constant chemical poten�tial) for the species in the reduced state, and France�schetti and MacDonald [4, 5] who mentionedFrumkin’s approach in the context of electrochemicalimpedance spectroscopy. Calculations for a completeelectrochemical cell were first performed by Bonne�font et al. [6] (in [3–5] only single electrode systemswere used), and extended in later work by Bazant andcoworkers, [7–10] who also introduced the concept ofthe generalized Frumkin–Butler–Volmer (gFBV)equation [9]. In addition, applications of the Frumkinapproach were reported on e.g. corrosion, [11, 12] fuelcells, [13, 14] nano�electrodes, [15] and batteries[16, 17].

To introduce the basic concept of the gFBV theorywe first consider the structure of the double layer,which is a combination of the charge free Stern layerand the diffuse layer (DL), as we will explain next. Weassume that an ion including its solvation (i.e. hydra�tion) shell cannot approach an electrode infinitesimalclose as shown in Fig. 5. As a result, we can identify aplane of closest approach, which coincides with theouter Helmholtz plane. Note that we implicitlyneglect ion adsorption on the inner Helmholtz plane,i.e., we assume that no ions break free from their sol�vation shell. Consequently, we have a charge free layersandwiched between the electrode and the outerHelmholtz plane, which we refer to as the Stern layer.Next, we assume that an electron can tunnel across theStern layer, so that we have electron transfer betweenthe ions at the outer Helmholtz plane and the metal ofthe electrode, and refer to this position as the reactionplane (Fig. 1). Obviously, this view of the reactionplane is very simplistic and in reality many effects caninfluence its position (and thus the thickness of theStern layer), such as the surface roughness of the elec�trode, [18] the presence of an oxide layer, [19] or thedistance across which the electrons can tunnel [20].Therefore we will consider the Stern layer thickness asan adjustable parameter, which we use to study theeffect of the double layer composition.

Frumkin–Butler–Volmer Theory and Mass Transfer in Electrochemical Cells1

M. van Soestbergena, b, z

aMaterials innovation institute, Mekelweg 2, 2628 CD Delft, the NetherlandsbEindhoven University of Technology, Department of applied physics, P.O. Box 513, 5600 MB, Eindhoven, the Netherlands

Received May 31, 2011

Abstract—An accurate mathematical description of the charge transfer rate at electrodes due to an electro�chemical reaction is an indispensable component of any electrochemical model. In the current work we usethe generalized Frumkin�Butler–Volmer (gFBV) equation to describe electrochemical reactions, an equa�tion which, contrary to the classical Butler–Volmer approach, includes the effect of the double layer compo�sition on the charge transfer rate. The gFBV theory is transparently coupled to the Poisson–Nernst–Planckion transport theory to describe mass transfer in an electrochemical cell that consists of two parallel plateelectrodes which sandwich a monovalent electrolyte. Based on this theoretical approach we present analyticalrelations that describe the complete transient response of the cell potential to a current step, from the firstinitial capacitive charging of the bulk electrolyte and the double layers all the way up to the steady�state of thesystem. We show that the transient response is characterized by three distinct time scales, namely; the capac�itive charging of the bulk electrolyte at the fastest Debye time scale, and the formation of the double layersand the subsequent redistribution of ions in the bulk electrolyte at the longer harmonic and diffusion timescales, respectively.

Keywords: generalized Frumkin–Butler–Volmer equation, electrostatic double layer, Poisson–Nernst–Planck transport theory, transient response, electrochemical cell modeling

DOI: 10.1134/S1023193512060110

1 The article is published in the original.z Corresponding author: [email protected]

(M. van Soestbergen).

RUSSIAN JOURNAL OF ELECTROCHEMISTRY Vol. 48 No. 6 2012

FRUMKIN–BUTLER–VOLMER THEORY AND MASS TRANSFER 571

Adjacent to the reaction plane a DL containing anon�zero space charge density is formed. As a resultwe have large electrical field strengths in this layer,while ion concentration gradients counter the forcesacting on the ions due to these electrical fields. Includ�ing the DLs in an electrochemical model goes beyondthe common assumption of electroneutrality, which isubiquitous in the literature of electrochemical model�ing [21–28]. However, including diffuse layers mightbe important as the electrical potential and ion con�centration gradients in these nanoscopic layers arelarge compared to the more gradual changes in theelectroneutral bulk electrolyte. Therefore, the poten�tial drop across the DL might have a significant contri�bution to the overall cell voltage, while ion concentra�tions at the reaction plane near the electrode differfrom their concentration in the bulk region. Conse�quently, the formation of DLs can have a large influ�ence on the charge transfer rate at electrodes, a ratewhich according to the gFBV theory depends on thelocal potential drop and ion concentration at the reac�tion plane, as we will explain in the next section.

2. GENERALIZED FRUMKIN–BUTLER–VOLMER EQUATION

Contrary to the classical Butler–Volmer equation[29], where the bulk ion concentration and the poten�tial drop from the metal of the electrode to the bulkelectrolyte are the stimuli that drive the reaction, weuse the local ion concentration and electrical fieldstrength at the reaction plane in the gFBV equation asthese stimuli instead. At the reaction plane we assume anelectrochemical reaction of the form, O + ne– R,where we have ideal thermodynamics for the species inthe reduced and oxidized states, and use their concen�

tration, ci, instead of their activities, ai, to obtain aFaradaic current according to,

(1)

where Ki are rate constants, αO and αR are the transfercoefficients (αO + αR = 1), f is equals F/RT, with F isFaraday’s constant, R is the gas constant and T is tem�perature, ΔVS is the potential drop across the Sternlayer, subscript rp indicates that the concentration isevaluated at the reaction plane, and subscript O and Rdenote the oxidized state (or oxidation reaction) andreduced state (or reduction reaction), respectively.

For the potential drop across the Stern layer werequire an additional boundary condition, which canbe obtained from the electrical field strength at thereaction plane and the properties of the Stern layer asindependently proposed by Itskovich et al. [3],Franceschetti and Macdonald [5], and Bonnefontet al. [6]. Namely, the potential drop across the Sternlayer follows from the electrical field strength, E = –�V,at the reaction plane according to,

(2)

where λS is the Stern layer thickness, n is the outwardnormal vector on the reaction plane, thus pointing outof the electrolyte phase, and subscript rp indicates thatthe gradient in electrical potential is evaluated at thereaction plane. Substitution of Eq. (2) into Eq. (1)results in the physically intuitive expression [9],

(3)

for the Faradaic charge transfer relation, which onlydepends on local properties at the reaction plane.Consequently, Eq. (3) clearly shows the difference

JF KRcO rp,αRnfΔVS–( )exp=

– KOcR rp,αOnfΔVS( ),exp

ΔVS λS∇Vrp

n,=

JF KRcO rp,αRnλSf∇Vn–( )exp=

– KOcR rp,αOnλSf∇Vn( ),exp

Potential Reaction of outer Helmholtz plane

Met

alPlane

Inner Helmholtz

Specific adsorption

Solvation or

Ion

Stern layer Electrolyte Distance Electron transfer

Met

al

of zero charge

hydration shell

plane

Vgr

Vbulk

Vm

ΔV

DL

ΔV

S

(а) (b)

Fig. 1. Schematics of the double layer structure; (a) the potential drop across the double layer, and (b) the plane of closestapproach.

572


van SOESTBERGEN

between the gFBV equation and the classical Butler–Volmer approach, since for the latter case the potentialin the exponential terms is an “over potential” relativeto some reference value and condition, while the con�centration of species are determined at the bulk�dif�fuse layer interface. As a result, the classical Butler–Volmer approach cannot be applied to systems wherewe have charging of the bulk region due to currentsabove the diffusion limitation [7, 8, 30–32] or theoverlap of DLs, such as found in micro� andnanochannels [33–35]. Additionally, the dependenceof the electrical potential on the charge transfer rate inthe gFBV equation is easier�to�grasp than the “overpotential” in the classical Butler–Volmer approach,while it can be transparently incorporated in theboundary conditions of a system without the require�ment of a reference or equilibrium condition.

To show the effect of the double layer compositionon the charge transfer rate we consider the limit ofinfinitesimal thin DLs, i.e. the thin�DL limit. Thethin�DL limit is a very useful approximation for elec�trochemical cells with a relatively large electrode spac�ing compared to the characteristic length scale for thethickness of the DL, i.e. the Debye length,

λD = [7, 8], where ε is the permittiv�ity, and c

∞ the ionic strength of the electrolyte. In the

thin�DL limit we can use the equilibrium DL proper�ties to describe the ion concentration at the reactionplane [6–8, 30, 36], which result from the Poisson�Boltzmann (PB) ion distribution [36–38],

(4)

where zi is the valence of ionic species i, ΔVDL is thepotential drop across the DL (“zeta potential”), andsubscript bulk refers to the bulk region (“plane of zerocharge”). Beyond the assumption of ions as pointcharges and thus the PB distribution, steric effects canplay an important role on the structure of the DL athigh electrode potentials, so that Eq. (4) changes sig�nificantly, as e.g. discussed in [18, 19, 39]. However,these effects are not considered in the current work.

We now substitute the PB ion distribution in Eq. (1)and follow [40], namely; we express the potential dropacross the DL as function of the double layer capaci�tance. We write the double layer capacitance, Cd, as thecapacitance of the DL and the Stern layer in series,i.e., 1/Cd = 1/CDL + 1/CS, where CDL and CS refer tothe capacitance of the DL and the Stern layer, respec�tively [19]. The capacitance of two parallel planes is bydefinition given as C = Δq/ΔV, where Δq and ΔV arethe difference in charge stored and the electricalpotential difference between both planes, respectively.Since the Stern layer remains charge neutral we canwrite ΔVDL = (1 + CDL/CS)–1(Vm – Vbulk), whichrelates the potential drop across the DL to the poten�tial drop across the complete double layer, and ΔVS =

εrε0RT/2F 2c∞

ci gr,ci bulk,

zifΔVDL–( ),exp=

Cd/CS(Vm – Vbulk) for the potential drop across theStern layer. Substitution in Eq. (1) yields

(5)

for the charge transfer rate, where α* = αR ×

Equation (5)

has the same form as the classical Butler–Volmerequation, however, the effective transfer coefficient,α*, differs from its true value depending on the DL toStern layer capacitance ratio. The capacitance of theDL and Stern layer per unit area is given as CDL =⎯qDL/ΔVDL, and CS = ε/λS, where qDL is the chargestored in the DL. According to the Gouy–Chapmantheory the charge stored in the DL equals qD =

± [38], which

results in

(6)

for the capacitance ratio of a binary electrolyte, i.e.zi = ±1, where δ is the Stern layer thickness relative tothe Debye length, δ = λS/λD. Next, we can identify twolimits for the effective transfer coefficient. The firstlimit is the condition of a zero Stern layer thickness toDebye length ratio, i.e., δ 0, where we have α* =zO/n. We will refer to this limit as the Gouy–Chapman(GC) limit [7, 9], since in this limit the DL, which isat equilibrium described by the GC theory, dominatesthe charge transfer rate at the electrode. The secondlimit is the opposite condition where the Stern layerthickness to Debye length ratio is infinite, i.e., δ ∞,where we thus have α* αR, and we retain the clas�sical Butler–Volmer equation. We will refer to thislimit as the Helmholtz (H) limit [7, 9], since in thislimit the potential drop across the double layer exactlyequals the potential drop from the metal phase to thereaction, or outer Helmholtz, plane.

Finally, we show results for the effective transfercoefficient as function of the potential drop across thedouble layer for various values of the Stern layer toDebye length ratios (Fig. 2). To determine the potentialdrop across the double layer (ΔVDL + ΔVS = Vm–Vbulk)we use ΔVS = λDδqDL/ε [38], where we assume a con�stant bulk ion concentration, which is valid in the lowcurrent regime. Furthermore, we consider an electro�chemical reaction involving the transfer of one elec�

tron (αR = αO = see [7]) with the valence of the ions

in the oxidized state equal to one. The results pre�sented in Fig. 2 show that for δ = 0, i.e. the GC�limit,we have an effective transfer coefficient of 1, while forδ = ∞, (H�limit) we have α* = αR. For small values of

JF KRcO bulk,α*nf Vm Vbulk–( )–{ }exp=

– KOcR bulk,1 α*–( )nf Vm Vbulk–( ){ }exp

CS/CDL 1+( ) 1– zO

nαR

�� CDL/CS 1+( ) 1–+⎩ ⎭⎨ ⎬⎧ ⎫

.

2RTε ci bulk ,zifΔVDL–( )exp 1–{ }

i∑

CDL

CS

�� δfΔVDL

��fΔVDL

2��⎝ ⎠

⎛ ⎞sinh=

12��,



δ and Vm – Vbulk we find clear deviations from theH�limit, and thus the classical Butler–Volmer equa�tion, while for increasing values of Vm–Vbulk all linesconverge towards the H�limit as the capacitance ratioCDL/CS blows up for high values of the potential dropacross the DL (note that f = ~40 1/V at T = 273 K).Consequently, deviations from the classical Butler–Volmer approach are likely to occur at small electrodepotentials in combination with small Stern layer thick�ness to Debye length ratios. However, note that the DLcapacitance cannot increase unbounded if stericeffects for ion crowding in the DL are taken intoaccount, which result in a maximum capacitance forcell voltages that are not much larger than the zero�voltage [19], and might thus lead to deviations fromthe H�limit at higher voltages as well.

3. POISSON–NERNST–PLANCK TRANSPORT THEORY

Ion transport in a dilute electrolyte is generallydescribed by the Nernst–Planck (NP) equation,where it is assumed that ions behave as ideal pointcharges [21]. We can substitute the NP flux equation ina mass balance to obtain,

(7)

where is the time�derivative of the ion concentra�tion, Ji is the flux, Di the diffusion coefficient, zi thevalence of the ions, and V is the local electrostaticpotential in volts. In Eq. (7) we do not consider advec�tion of the solvent or chemical reactions within thebulk of the electrolyte. The former is important whene.g. flow cells are considered [14, 35], while the lattercannot be omitted when describing transport of pro�tons and hydroxyl ions due to the self�dissociation ofwater [13, 41, 42]. The electrical potential in the local�density mean�field approximation is related to thecharge density by Poisson’s equation

(8)

If we use Eqs. (7) and (8) and combine them withappropriate boundary conditions we obtain a self�con�sistent set of equations for the description of masstransfer in the bulk electrolyte of an electrochemicalcell according to the full Poisson–Nernst–Planck(PNP) transport theory.

The boundary conditions for an applied electrodepotential are straightforward and given by

(9)

for the potential at the reaction (outer Hemholtz)plane, while the fluxes of the reacting species are cou�pled to the charge transfer according to Jin = JF/F andthe fluxes of the inert species are zero. The boundaryconditions for a system where a constant current isapplied are more elaborate and can be derived by con�

c· i ∇ Ji⋅– ∇ Di ∇ci zicif∇V+( )[ ],⋅= =

c· i

∇ ε∇V( )⋅ F zici.i∑–=

Vrp

V m λS∇Vn–=

sidering that in an electrolyte the electrical current isequal to the sum of the ionic conduction current andthe Maxwell displacement current, i.e. [6, 11, 43, 44],

(10)

where the first term represent the conduction currentand the second term the Maxwell current. At the elec�trodes the conduction term equals the Faradaic cur�rent, and we obtain [11, 43]

(11)

for the potential gradient at the reaction plane. Thecontribution of the Faradaic current in Eq. (11) istime�dependent until the steady�state is reached,where it exactly equals the applied current, and theMaxwell current vanish completely.

The assumption of electroneutrality to describemass transfer is ubiquitous in the literature of electro�chemistry [21–28]. Considering Poisson’s equation,i.e. Eq. (8), goes beyond this assumption since it cou�ples the electrical potential to the space charge den�sity. Interesting in this equation is the factor betweenthe second derivative of the electrical potential andthe space charge density, which is F/ε and equals1.4 × 1014 V m/mol in aqueous electrolytes. The largemagnitude of this term results in an enormous electri�cal field for even a small excess of space charge, which

I F ziJi( )i∑– ε d

dt��∇V,–=

∇Vrp

1ε�� JF t '( )n I–[ ]dt '

0

t

∫=

1.00

0.75

0.50

10001 10010f(Vm – Vbulk)

α*

δ = ∞

δ = 0

δ = 10–3

δ = 0.2

δ = 1

δ = 5

Fig. 2. The effective transfer coefficient for a typical oneelectron transfer reaction for various values of the Sternlayer thickness to Debye length ratios; zO = 1, n = 1, andαR = 1/2.

574


van SOESTBERGEN

will counteract the separation of charges [21]. Also,this term implies that very small deviations from elec�troneutrality can results in a non�linear electricalpotential distribution [21]. Therefore, simply setting thespace charge term in Eq. (8) equal to zero and solvingthe resulting Laplace’s equation will result in a flawedmodel. The proper way of dealing with electroneutralityis to replace Poisson’s equation by ci = 0. As a

result we reduce the number of field variables in ourmodel and make our system of equations more tracta�ble for analytical or numerical solutions. Obviously,electroneutrality is violated near the electrodes wherewe have the formation of DLs, as here large gradientsin electrical potential develop. Consequently, we canonly use the electroneutrality assumption in the bulkelectrolyte when the DLs are negligibly small com�pared to feature size of the electrochemical cell, char�acterized by ε = λD/L � 1, where L is the electrodespacing [6–10, 18, 19, 30, 36, 45–47]. While in addi�tion we need to apply appropriate boundary condi�tions for the bulk electrolyte, which account for thepotential and concentration gradients at interfaces dueto the formation of DLs. In general these gradients arewell described by the equilibrium properties of theDLs, see [6–10, 18, 19, 30, 36, 45–47].

Note that in many electrochemical cells models theassumption of electroneutrality is accompanied byLaplace’s equation for the electrical potential. How�ever, electroneutrality does not imply a Laplaciandirectly from Poisson’s equation, as we will discussbelow. Let us consider a system consisting of two par�allel electrodes with infinity large area spaced by afinite distance, L, that sandwich a monovalent andbinary electrolyte, with the anion and cation concen�tration denoted as ca and cc, respectively. Due to theinfinite electrode area we can describe this systemusing a one�dimensional model with spatial coordi�nate X, which according to Eq. (7) results in

(12)

for the mass balance of the dimensionless salt concen�tration, c = (cc + ca)/2c

∞, while for the dimensionless

charge density, ρ = (cc – ca)/2c∞

, we have

(13)

which approaches zero for ε � 1. From Eq. (13) wecan obtain an expression for the migration term (i.e.the second term between square brackets), which wesubstitute in Eq. (12) to obtain [7, 8, 21, 39, 48–52],

(14)

where D* = 2DcDa/(Dc + Da), which is the “ambipolardiffusivity” of the salt. Eq. (14) now resembles Fick’s

zii∑

c 12�� ∂

X∂�� Dc Da+( ) c∂

X∂�� Dc Da–( )cf V∂

X∂��+⎝ ⎠

⎛ ⎞=

ρ 12�� ∂

X∂�� Dc Da–( ) c∂

X∂�� Dc Da+( )cf V∂

X∂��+⎝ ⎠

⎛ ⎞ ,=

c D* ∂2c

X2∂��,=

second law for the diffusion of neutral species [50, 53],indicating that the salt behaves like one species due tothe high forces required to separate the cat� and anionsin solution. From Eq. (13) we obtain a replacement forPoisson’s equation according to

(15)

which, by neglecting any redistribution of species andafter the Maxwell current died out, follows Ohm’s lawwhere the electrical current is given by I = κ∂XV, withthe electrolyte conductivity, κ = fFc

∞(Dc + Da).

Though Eq. (15) is derived for our particular system ofa binary monovalent electrolyte, similar “modifiedPoisson’s equations” can always be deduced by assum�ing electroneutrality. In systems where we have a largeexcess of inert supporting electrolyte the potential isscreened across the DLs and a relation for the electri�cal field in the bulk is not required [46]. Note that inmany electrochemical cell models ion redistributiondue to mass transfer is neglected, and Eq. (15) con�denses to a simple Laplacian equation. However, thisis only appropriate under stringent conditions anddoes not follow from neglecting the space charge den�sity in Eq. (8) directly.

Let us continue our discussion on mass transfer inthe system described above by considering the casewhere we impose a constant electrical current onto theelectrodes, while one species is inert and the otherelectrochemically active at the electrodes. Conse�quently, we have conservation of the inert species, i.e.∫cinertdx = c

∞L, so that from Eq. (14) we find that at

steady�state we have a linear ion concentration profileof the inert species with a maximum slope equal to±2c

∞/L. When we substitute this concentration profile

into the flux equation for the inert species (whichequals zero at steady�state) we find that ±2c

∞/L =

⎯zici/∂XV, which after substitution into the flux equa�tion for the electrochemically active species leads tothe diffusion limiting current, Ilim = 4FDc

∞/L, where

D is the diffusion coefficient of the reactive species.Note that we cannot apply a current above this limita�tion, unless we break the electroneutrality conditionand have an expansion of the DLs or introduceanother electrode reaction [7, 8, 21, 30, 31, 46].

Next, we consider the diffusion time scale of thesystem, i.e. τ = tL2/D. At this time scale the Maxwellcurrent is negligible compared to the conduction cur�rent for systems where ε2 � 1 [47]. As a result theapplied current to the system equals the conductioncurrent, a situation for which Eq. (14) has an exactsolution in terms of an infinite series [50, 51]

(16)

∂X∂

�� cf V∂X∂

��⎝ ⎠⎛ ⎞ Dc Da–( )

Dc Da+( )�� ∂2c

X2∂��– ,=

c x τ,( ) 1 2iapp12�� x– fn

2NxL

��⎝ ⎠⎛ ⎞cos⎝ ⎠

⎛ ⎞

n 1=

n = ∞

∑–⎩ ⎭⎨ ⎬⎧ ⎫

,+=



where fn = exp(–4N2τ)/N2, N = π(2n – 1), x = X/L

and we have scaled the applied current, Iapp, accordingto iapp = Iapp/Ilim. The exponential term in Eq. (16) willvanish for times that are large compared to the diffu�sion time scale leaving c(x) = 1 + iapp(1 – 2x), which isexactly the classical steady�state solution for planarelectrochemical cells [6–9, 26, 45].

Though Eq. (16) will provide an exact solution forthe ion redistribution across the cell as function oftime for an infinite number of terms, its use for deriv�ing analytical current�voltage relations is limited evenfor n = 1. We will demonstrate this by considering thepotential drop across the electroneutral bulk region ofthe electrolyte. Previously we found that the Maxwellcurrent vanishes at the diffusion time scale [47], sothat the conduction current equals the applied electri�cal current. Converted to the dimensionless parameterintroduced above, and writing ϕ = fV, Eq. (10)becomes 2iapp = c∂xϕ, which results in

(17)

for the potential drop across the bulk region, wheresuperscript 0 and 1 indicate the concentration at x = 0and x = 1, respectively. Note that the potential dropacross the bulk is positive when positive charge flowfrom the left to the right through the cell. The integralin Eq. (17) has no exact solution when c is given by

12��

Δϕbulk ϕ0 ϕ1–iapp

c��dx

x = 0

x = 1

∫= =

Eq. (16). However, it can be solved rather easily byassuming a linear distribution of species,

(18)

where the concentration at x = 0 and x = 1 follow fromEq. (16) according to

(19)

where the ± sign refers to the positive sign at x = 0 andthe negative sign at x = 1, which from τ ~ 0.1 onwardcan be approximated by the first term of the summa�tion only [47], so that

(20)

where g(τ) = 1 – 8/π2exp(–π2τ). In Fig. 3a we presentresults for c according to Eq. (16) relative to Eq. (20).We observe that Eq. (20) is indeed a good approxima�tion for τ > 0.1. In Fig. 3b we show that the corre�sponding profiles of c are almost linear at τ ~ 0.1.Therefore we will substitute Eq. (20) into Eq. (18) toobtain

(21)

for the distribution of ions between the two electrodesat times that are comparable to the diffusion timescale. Finally, substituting Eq. (21) into Eq. (17) gives,

(22)

c x τ,( ) c0 c1 c0–( )x,+=

c τ( ) 1 iapp 1 2 fn τ( )n 1=

n = ∞

∑–⎩ ⎭⎨ ⎬⎧ ⎫

,±=

c τ( ) 1 g τ( )iapp,±=

c x τ,( ) iappg τ( ) 1 2x–( ) 1+=

Δϕbulk2

g τ( )�� g τ( )iapp[ ]1–

tanh=

1.00

0.95

0.900.10.010.001τ

c (Eq. (16)/c (Eq. (20)

iapp = 1/3

iapp = 2/3

iapp = 1

1.8

1.0

0.21.00.50x

c

iapp = 1/3

iapp = 2/3

iapp = 1

(а) (b)

Fig. 3. Comparison between the salt profiles across the cell according to the exact solution and its first order approximation;(a) the relative error between the dimensionless salt concentration at x = 0 according to Eq. (16) with n = 10 and its first orderapproximation for n = 1, i.e. Eq. (21) as function of dimensionless time τ for iapp equal to 1/3, 2/3 and 1; (b) the concentrationprofiles corresponding to (a) for τ = 0.1 according to Eq. (16) (full lines) and their linear approximation (dashed lines).

576


van SOESTBERGEN

for the potential drop across the electroneutral bulk.Combining this potential drop with the potentialdrops across the DL and the Stern layer will enable usto construct the transient behavior of the cell potentialof an electrochemical cell under an imposed current aswe will discuss in the next section.

4. DEBYE, HARMONIC, AND DIFFUSION TIME SCALE

In this section we derive analytical relations for thetransient response of the cell potential to a step inapplied current. These relations include the initialperiod of capacitive charging of the system as well asthe redistribution of ions across the bulk region. As wewill show, combining the relations derived here willenable us to construct the complete transient behaviorof the cell potential for an electrochemical cell con�taining a monovalent electrolyte as discussed in theprevious section. We consider the case where the cat�ion is the reactive ion, which is formed from the metalof the electrode at x = 0 and incorporated again in themetal of the electrode at x = 1. Furthermore, weassume that the metal atom concentration is large andnot affected by the electrochemical process. Conse�quently, we can write KOcR = JO for the oxidation rate.

First, we derive an expression for the capacitivecharging of the bulk region of the electrochemical cell.In the absence of ion redistribution, i.e. c(x, τ) = 1 andρ(x, τ) = 0, the potential drop across the bulk region islinear and equals Δϕbulk = –∂xϕ, so that from Eq. (10)converted to dimensionless parameters we obtain thedifferential equation, dτ = ε2/(2Iapp + Δϕbulk)dΔϕbulk,which results in

(23)

for the initial charging of the bulk region. Note thatτ/ε2 in Eq. (23) represents the Debye time scale, whichis a material property of the electrolyte and has a char�acteristic value for aqueous solutions in the range of nsto μs [36].

Next, we derive an expression for the potential dropacross the double layers. To do this we consider thecurrent at the reaction plane, i.e., Eq. (11), fromwhich we obtain,

(24)

for the time derivative of the potential drop across theStern layer, where the ±�sign refers to the positivevalue at position x = 1 and the negative value at posi�tion x = 0, and jF = JF/Ilim, which is the Faradaic cur�rent scaled to the diffusion limiting current. Weassume that the DLs are in quasi�equilibrium (validfor ε � 1) and are thus described by the PB distributionaccording to Eq. (4), while again no redistribution of

Δϕbulk 2iapp 1 τ/ε2–( )exp–[ ]=

dΔϕS

dτ��

2δ jF iapp–( )ε

��±=

ions in the bulk region occurs, so that the Faradaiccurrent equals,

(25)

where ΔϕDL = 2sinh–1 [38], and the reaction

rate constants are scaled to the diffusion limiting cur�rent, kR = KRc

∞/Ilim and jO = JO/Ilim. Note that in

Eq. (24) the dynamics is characterized by the timescale τ/ε, which is the harmonic time scale of the sys�tem [36]. Consequently, we have the charging of thebulk region at the Debye time scale, while the chargingof the DLs is at the longer harmonic time scale, fol�lowed by the redistribution of ions at the longest diffu�sion time scale.

Finally, we can solve Eqs. (24) and (25) numericallyand combine the results for ΔϕS and ΔϕDL with theresult for Δϕbulk of Eq. (23) according to

(26)

to obtain the initial cell potential as function of time.Next we discuss a simplified solution of Eq. (24) in thelimit of small potentials drops across the double layer.For small potential drops across the Stern layer, whiche.g. occur in case of fast electrode kinetics, we can lin�earize the Faradaic current, such that

(27)

where i* = kR – jO, which after substitution intoEq. (24) and integration results in,

(28)

for the potential drop across the Stern layer, where the±�sign again refers to the positive value at positionx = 1 and the negative value at position x = 0. Forsmall values of ΔϕS we can derive the potential dropacross the DL from Eq. (28) according toΔϕDL = ΔϕS/δ, which after substitution of Eqs. (23)and (28) in Eq. (26) results in an analytical descriptionof the initial capacitive charging of the system for lowpotential drops across the double layer. However, notethat for a large unbalance in the gFBV equation, i.e.,when |kR – jO| � 0, the assumption of a small potentialdrop across the double layer is not satisfied, even forfast electrode kinetics, due to the relatively high opencell potential. Next, we assume that kR = jO = k at both

jF± kR12��ΔϕS– ΔϕDL–⎝ ⎠

⎛ ⎞exp jO12��ΔϕS⎝ ⎠

⎛ ⎞ ,exp–=

ΔϕS

2δ��⎝ ⎠

⎛ ⎞

ϕcell ΔϕS ΔϕDL+[ ]x 0= Δϕbulk+=

= ΔϕS ΔϕDL+[ ]x 1=

jF±ΔϕS

2�� kR

δ 2+δ

��⎝ ⎠⎛ ⎞ jO+– i*,+=

ΔϕS2δ iapp i*–( )

kR δ 2+( ) jOδ+��±=

× kR δ 2+( ) jOδ+( )–( )τε��⎝ ⎠

⎛ ⎞exp 1–⎩ ⎭⎨ ⎬⎧ ⎫



reaction planes, so that an unbalance in the gFBV equa�tion is absent. As result we obtain from Eqs. (23), (26)and (28),

(29)

for the cell potential as function of time up to the har�monic time scale.

Solutions for the cell potential at the diffusion timescale were already presented in [47] for both the GC�and H�limit as previously discussed in the currentwork. Here we restate the results presented in [47] andbriefly describe their derivation. First we consider thegFBV equation for the electrochemical charge trans�fer, which equals the applied current at the diffusiontime scale, and where we again use the PB relation, i.e.Eq. (4), to obtain,

(30)

where cm is the concentration at the very edge of thebulk region and subscript m either denotes the anodeside (A) at x = 0 or the cathode side (C) at x = 1. Equa�tion (30) differs from Eq. (25) in not assuming that theion concentration in the bulk remains constants butcan vary in time. Note that in many models a chargetransfer relation is used with some fixed pre�term forthe reaction rate constant that does not account forion redistribution due to mass transfer. Consequently,the applicability of these models is limited to the rela�tively small field where ion redistribution is negligible.Returning to our limiting cases, we can derive thepotential drop across the double layer from Eq. (30) asfunction of the applied potential and ion concentra�tion at the very edge of the bulk region. In theGC�limit the Stern layer does not sustain any voltagedrop, so that from Eq. (30) we obtain,

(31)

while for the H�limit, where the potential across theDL equals zero, we obtain,

(32)

ϕcell 2iapp=

× 1 τ/ε2–( )exp–[ ] 1k�� 2k δ 1+( )τ

ε��–⎝ ⎠

⎛ ⎞exp 1––⎩ ⎭⎨ ⎬⎧ ⎫

jF± iapp± kRcm12��ΔϕS– ΔϕDL–⎝ ⎠

⎛ ⎞exp= =

– jO12��ΔϕS⎝ ⎠

⎛ ⎞ ,exp

ΔϕDL GC,

kR m, cm

jO m, iapp+−��⎝ ⎠

⎛ ⎞ ,ln=

ΔϕS H, 2iapp± iapp

2 4jO m, kR m, cm++2jO m,

��⎝ ⎠⎜ ⎟⎛ ⎞

.ln=

Finally, we can combine Eqs. (20), (22), and (26)with either Eqs. (31) or (32) to obtain

(33)

for the cell potential in case of the GC�limit and,

(34)

for the H�limit, where βm = 4jO,mkR,m andϕ0 = ln(jO,CkR,A/jO,AkR,C), which is the open cellpotential. Equations (33) and (34) become accurateafter τ ~ 0.1 when the ion concentration and thepotential drop across the bulk region are well predictedby Eqs. (20) and (22), respectively. For long times, i.e.τ ∞, we obtain the steady�state solutions reportedin [9] except for a sign reversal of all terms due to thereversed definition of the cell potential in [9]. It isinteresting to see that the difference between Eqs. (33)and (34) vanishes when the kinetic rate constantsbecome very high. Namely, the second term inEq. (32) becomes negligible when jO,m � i, and thesame holds for the second and third term of Eq. (34)for βm � i2/(1 ± g(τ)i). In addition, it was observedthat the solutions for increasing thickness of the Sternlayer converges rapidly to the H�limit [47]. Therefore,for higher values of the Stern layer thickness the differ�ence between the classical BV and the gFBV equationis indistinguishable. An effect which is similar to thatpresented in Fig. 2.

Results that show the correspondence between thefull PNP�gFBV theory and the simplified modelsderived above are presented in Fig. 4a. The full PNP�gFBV model results are computed by implementingthe dimensionless equivalents of Eqs. (7) and (8) withboundary conditions according to Eq. (11) in thecommercial finite element code COMSOL (a moredetailed description of this procedure can be found in[47]). The results show that even at a relatively lowvalue of the kinetic constant, namely; k = 0.3, the sim�plified model according to Eq. (29) is a good approxi�mation for the transient current, while for k = 0.1 thesolution based on solving the simple differential equa�tion of Eq. (24) gives a good approximation. In addi�tion we can use the relations for the cell potential at thediffusion time scale, i.e. Eqs. (33) and (34), in combi�nation with either Eq. (24) or (29) to construct thewhole start�up behavior from τ = 0 up to the steady�state of the electrochemical cell (Fig. 4b). Here weclearly see that for the model parameters consideredhere (iapp = 0.75, k = 2, and δ = 10) the GC� and

ϕcell GC,ϕ0

1 iapp/jO C,+1 iapp/jO A,–��⎝ ⎠

⎛ ⎞ln+=

+ 21 g+g

�� giapp[ ]1–tanh

ϕcell H, ϕ0 2iapp

βA 1 giapp+( )��⎝ ⎠

⎛ ⎞1–

sinh+=

+ 2iapp

βC 1 giapp–( )��⎝ ⎠

⎛ ⎞1–

sinh 21 g+g

�� giapp[ ]1–tanh+

578


van SOESTBERGEN

H�limit give a lower and upper boundary for the time�dependent cell potential that are sufficiently closetogether to fairly accurately predict the response for anarbitrary value of the Stern layer thickness δ.

To summarize, we first have the capacitive chargingof the bulk region at the Debye time scale, followed bythe formation of the DLs at the harmonic time scale,while finally the cell potential further evolves due tothe redistribution of ions at the diffusion time scale.Consequently, the transient behavior of the cell poten�tial of an electrochemical cell in response to an appliedelectrical current proceeds at three distinct timescales.

CONCLUSIONS

We have used the generalized Frumkin–Butler–Volmer (gFBV) equation to describe the charge trans�fer rate due to an electrochemical reaction at an elec�trode. In the gFBV equation it is assumed that theelectrochemical reaction proceeds at a predefinedplane near the electrode. We have shown that for aninfinitely large spacing between this reaction planeand the metal of the electrode relative to the Debyelength, which is a characteristic length scale for thediffuse layer containing a non�zero space charge den�sity, we obtain the classical Butler–Volmer equation asthe correct limit of the gFBV equation. In the oppositelimit, where the reaction plane coincides with themetal electrode, the charge transfer rate is dominatedby the properties of the diffuse layer. Consequently, inthis case the gFBV deviates significantly from the clas�sical Butler–Volmer equation, since the latter equa�

tion does not consider the non�zero space charge den�sity in the diffuse layer.

In addition we have shown that the gFBV equationmore naturally combines with the Poisson–Nernst–Planck (PNP) transport theory for ionic species thanthe classical Buter–Volmer equation. From the fullPNP�gFBV model we have derived analytical relationsfor the transient behavior of the cell potential of anelectrochemical in response to a step in the appliedcurrent. It followed from these relations that the evo�lution of the cell potential proceeds at three distincttime scales. Namely; (i) the Debye time scale for thecapacitive charging of the electroneutral bulk region,(ii) the harmonic time scale for the formation of thediffuse layer, and (iii) the diffusion time scale for theredistribution of ions across the cell. We showed thatby combining the solutions for all three time scales wecan construct the complete transient behavior of thecell, from the very initial capacitive charging all theway up to the steady state.

ACKNOWLEDGMENTS

This research was carried out under project numberM81.6.10383 in the framework of the StrategicResearch Programme of the Materials innovationinstitute in the Netherlands (www.m2i.nl).

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15

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5

01010–8 10–210–5

Ion redistribution,

GC�limit,

Charging of DLs,

Charging of bulk,

Full model, k = 0.1

Full model, k = 0.3

Full model, k = 100

Eq. (29), k = 0.1

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Documents

Frumkin-Butler-Volmer theory and mass transfer in electrochemical cells