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Electrochimica Acta 56 (2011) 5707–5716

Contents lists available at ScienceDirect

Electrochimica Acta

journa l homepage: www.e lsev ier .com/ locate /e lec tac ta

lectrochemical digital simulation with highly expanding grid four pointiscretization: Can Crank–Nicolson uncouple diffusion and homogeneoushemical reactions?

rancisco Martínez-Ortiz ∗, Angela Molina, Eduardo Labordaepartamento de Química Física, Universidad de Murcia, Espinardo 30100, Murcia, Spain

r t i c l e i n f o

rticle history:eceived 20 January 2011eceived in revised form 11 April 2011ccepted 13 April 2011vailable online 27 April 2011

eywords:

a b s t r a c t

The digital simulation of electrochemical experiments where homogeneous chemical reactions are cou-pled to the heterogeneous charge transfer is considered by means of the sequential treatment of chemicaland diffusional effects on concentrations. For the first time, this strategy is employed in combination withimplicit time-integration schemes, in particular with a modification of the Crank–Nicolson method (CN),and with highly expanding grid four-point discretization (HEGFPD). CN leads to significant improvementswith respect to explicit integration schemes previously considered in the literature whereas HEGFPD

igital simulationoupled homogeneous chemical reactionsequential methodrank–Nicolsonighly expanding grid four-pointiscretization

gives rise to very fast computations.The errors committed by this method are evaluated for different reaction mechanisms (with first-

and second-order coupled homogeneous chemical reactions) at planar and spherical electrodes. It isdemonstrated that, after selecting the appropriate conditions, this is a very efficient procedure, giving riseto very accurate results. Thus, simple criteria are established for the selection of the adequate simulationparameters.

© 2011 Elsevier Ltd. All rights reserved.

. Introduction

In a recent paper [1] we have shown that the use of anxponentially expanding grid with the asymmetric four-pointpproximation for the derivatives in combination with high expan-ion factors (which will be refereed as Highly Expanding Gridour-Point Discretization, HEGFPD) and high order implicit tem-oral integration schemes is very efficient in digital simulationf electrode processes. So, for example, we have shown that forsimple charge transfer process a grid containing 8 points is

ood enough to obtain four-figure accuracy in planar electrodesnd 26 points in the simulations of spherical microelectrodesor pulse and multipulse techniques. The HEGFPD has the greatdditional advantage that a Thomas-like algorithm [2] and the–v procedure [2,3] for implicit calculation of surface concen-rations can be simultaneously used, allowing the application ofomplex surface conditions with very small modifications in theethod.

It is very well known that using usual three-point spatial dis-

retizations the profile concentration of the species involved inhe chemical reaction are coupled when homogeneous chemi-

∗ Corresponding author. Tel.: +34 868 88 7419; fax: +34 868 88 4148.E-mail address: fmortiz@um.es (F. Martínez-Ortiz).

013-4686/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2011.04.043

cal reactions are present in a given simulation and an implicittime-integration algorithm is used, preventing the application ofthe simple Thomas algorithm and making necessary the use ofmore complex block-tridiagonal matrixes in integration schemas,as shown by Newman [4] and Rudolph [5]. The same cou-pling appears when HEGFPD is used under these conditions,making impossible the use of the scalar-type Thomas algo-rithm.

The sequential method is an alternative to overcome thiscoupling, consisting in calculating sequentially diffusional andchemical changes in concentration in each time step [2]. We haveproceeded calculating first the changes due to the chemical reac-tion, modifying the concentrations by these amounts and thenapplying diffusion operators to these modified values.

As far as we know, the sequential method has only been appliedto explicit time-integration procedures. It was used in the origi-nal monograph by Feldberg [6], who employed known analyticalexpressions to evaluate chemical changes, and later by Flanaganand Marcoux [7] and Nielsen et al. [8] who used Runge–Kutta inte-gration to evaluate chemical changes. Ruzic and Britz have found aproof of the consistence of the method under some specific condi-

tions [9].

In this paper, the sequential method with HEGFPD is usedtogether with some implicit time integration schemes: the Back-ward Implicit method (implicit Euler time integration), the

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708 F. Martínez-Ortiz et al. / Electr

xtrapolation algorithms and the Crank–Nicolson method (trapez-um rule time integration). The value of the extrapolation methodsn the case of an E mechanism with HEGFPD was demonstratedn Ref. [1]. In Appendix A of this paper, the application andsefulness of HEGFPD with Crank–Nicolson and an E mechanismre shown.

The methodology stands out because of its simplicity such thatt can be used by any electrochemist. Thus, all the expressionsequired are available in an explicit form, which contrasts withhe more complex matrix methods. In addition, due to that the

ethod involves independent scalar algorithms for each species,ne can take advantage of parallel computation methods availablen modern cpu multiprocessors.

The rest of the paper is organized as follows:

In the first section, some considerations about the sequentialmethod and the calculation of the changes in concentration causedby the chemical reactions are given, indicating the advanta-geous use of numerical integration methods, such as fourth-orderRunge–Kutta. Next, the value of the sequential procedure isassessed for different reaction mechanisms for which exactsolutions are reported in the literature. The selection of an effi-cient time-integration method is tackled, by comparing differentimplicit schemes. Crank–Nicolson proves to be the most efficientand it enables us to obtain very accurate results with uncouplingof the effects of chemical reactions and diffusion, which greatlysimplifies the resolution of the problem.In Section 3.2, the influence of the simulation parameters (timeand spatial intervals) on the error of the simulation is evaluated,establishing the optimum conditions to ensure accurate resultswith the sequential approach. Most of the simulations in this paperhave been performed with planar electrodes, where the kineticeffects are more apparent. However, the influence of the elec-trode sphericity is also considered in this section. Finally, we havealso studied the voltammetric response of systems with coupledsecond-order homogeneous chemical reactions.

. Sequential method: computation of the effect of coupledhemical reactions

The sequential method is based on forcing the decoupling of theffects of chemical reactions and diffusion on the concentrationsf the species participating in the process, by making sequentialalculations of the changes in concentrations due to each process.ence, first the variations produced by chemical reactions are eval-ated for the time interval ��, and next we deal with the problems if this was purely diffusive.

To show the strategy followed in the sequential method, we willonsider the case of the catalytic mechanism:

C + e− � B

C + Qk1�k2

B + P(1)

or the sake of simplicity we will consider that species P and Q are inarge excess such that the chemical reactions are pseudo-first order.hus, the dimensionless differential equations describing the con-entration changes of the different species at a spherical diffusioneld are given by:

∂CC ∂2CC 2 ∂CC 1st 1st

⎫⎪

∂�

=∂x2

+R0 + x ∂x

− K1 CC + K2 CB

∂CB

∂�= DB

DC

(∂2CB

∂x2+ 2

R0 + x

∂CB

∂x

)+ K1st

1 CC − K1st2 CB

⎪⎬⎪⎪⎭ (2)

ica Acta 56 (2011) 5707–5716

where:

Ck = ck

c∗B + c∗

C

, k ≡ B, C, P, Q

� = t

tr

x = r − r0√DCtr

R0 = r0√DCtr

K1st1 = k1c∗

Q tr

K1st2 = k2c∗

Ptr

(3)

ck is the concentration of species k, c∗k

is the bulk concentrationof species k, Dk is the diffusion coefficient of species k, t is time,tr is an appropriate reference time (for example, the duration ofthe experiment for chronoamperometric experiments or the valueRT/nFv for cyclic voltammetry where v is the scan rate), r is thedistance from the centre of the electrode and r0 is the radius of theelectrode.

By examining the right-hand term in Eq. (2), it can be inferredthat the variation of concentrations with time are due to simultane-ous diffusional and chemical changes. According to the sequentialmethod, this can be treated by first taking into account the changesin the initial concentrations (denoted as CB and CC) due to thecoupled chemical reaction for the time interval ı�, which leadsto intermediate concentration values (denoted as C̃B and C̃C ), andthen calculating the changes caused by diffusion starting from theintermediate chemically-changed concentration profiles.

Regarding the calculation of the effects of the coupled chemicalreactions, this can be carried out in different ways, for example, byusing explicit expressions obtained from direct integration of thekinetic equations, or by numerical integration of the differentialequations referred, exclusively, to the chemical kinetics.

The first method can be implemented very easily. For exam-ple, in the case of the chemical reaction included in Eq. (1), theconcentration variations in an interval time �� are given by:

C̃B = CB +[

K1st1

K1st1 + K1st

2

(CB + CC ) − CB

]

× {1 − exp[−(K1st1 + K1st

2 )ı�]}

C̃C = CB + CC − C̃B

(4)

where C̃B and C̃C are the concentration values modified by thechemical reaction, and CB and CC are the known initial concentra-tion values.

In spite of the simplicity of this method, it requires a particularexpression for each kinetic scheme. So, numerical integration is amore general alternative if employed adequately, such as, for exam-ple, the Runge–Kutta method (RKI). Thus, the expression equivalentto Eq. (4) with fourth-order RKI is:

C̃B = CB + 16

(p1 + 2p2 + 2p3 + p4)

C̃B = CC − 16

(p1 + 2p2 + 2p3 + p4)(5)

where:

p1 = K1st1 ı�CC − K1st

2 ı�CB

p2 = K1st1 ı�

(CC − p1

2

)− K1st

2 ı�(

CB + p1

2

)

p3 = K1st

1 ı�(

CC − p2

2

)− K1st

2 ı�(

CB + p2

2

)p4 = K1st

1 ı�(CC − p3) − K1st2 ı�(CB + p3)

(6)

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F. Martínez-Ortiz et al. / Electr

Despite the more complex form of Eq. (5) with respect to Eq. (4),hey offer easier systematic implementation when several chemicaleactions are simultaneously taking place. Moreover, with not veryigh values of the product (K1st

1 + K1st2 )ı�, the results obtained from

4) and (5) are equivalent. Thus, depending on the kinetic scheme,n accuracy of 6–7 figures in the concentrations is ensured forK1st

1 + K1st2 )ı� < 0.15 and 4–5 figures for (K1st

1 + K1st2 )ı� < 0.40,

hich does not limit the accuracy of the simulations in any case.his approximation is not adequate for (K1st

1 + K1st2 )ı� > 1.5 since

he error in the calculated concentrations is greater than 1%. All theesults presented in this paper are obtained by using this fourth-rder Runge–Kutta approximation.Once the effect of the chemicaleactions on the concentration profiles is calculated, the problems treated as if the chemical processes did not exist. That is, thehanges due to diffusion transport on the concentrations C̃B and

˜C are calculated, the boundary value problem applied, and theagnitudes of electrochemical interest extracted. For digital simu-

ation, the corresponding diffusion equations must be written iniscrete form; considering an exponentially expanding grid theeneral form of the approximate first and second derivatives atoint xi is given by [1]:

′i(p, q) = 1

h�ie

i+p−q∑l=i+1−q

ˇl−iul (7)

′′i (p, q) = 1

h2�2ie

i+p−q∑l=i+1−q

˛l−iul (8)

here h is the amplitude of the first interval, �e is the expansionactor of the spatial expanding grid, p represents the number ofoints considered to evaluate the derivative, q the relative position

n this set of points at which the derivative is evaluated, and ˇ and ˛re the coefficients for the first and second spatial derivatives. Thexpressions for these coefficients depend on the number of pointsaken to the left and right of the point considered and they are givenn Ref. [1] for all the possible forms taking three, four and five-pointpproximations.

In a previous paper [1], we showed the value of the four-oint approximation (4,2) (i.e., p = 4 and q = 2) in combinationith an exponentially expanding grid. Using the (4,2) form for

he spatial discretization and the Backward Implicit (BI) and therank–Nicolson (CN) schemes for the time-integration, the discretexpressions of the differential diffusion equations will be given by:

Backward Implicit

C ′C,i − C̃C,i

ı�= 1

h2�2ie

i+2∑l=i−1

˛l−iC′C,l + 2

R0 + xi

1

h�ie

×i+2∑

l=i−1

ˇl−iC′C,l

C ′B,i

− C̃B,i

ı�= DB

DC

(1

h2�2i

i+2∑˛l−iC

′B,l + 2

R0 + xi

1

h�i

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪

(9)

e l=i−1 e

×i+2∑

l=i−1

ˇl−iC′B,l

) ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

ica Acta 56 (2011) 5707–5716 5709

Crank–Nicolson

C ′C,i

− C̃C,i

ı�= 1

2h2�2ie

i+2∑l=i−1

˛l−i(C′C,l + C̃C,l) + 1

R0 + xi

1

h�ie

×i+2∑

l=i−1

ˇl−i(C′C,l + C̃ ′

C,l)

C ′B,i

− C̃B,i

ı�= DB

DC

(1

2h2�2ie

i+2∑l=i−1

˛l−i(C′B,l + C̃B,l) + 1

R0 + xi

× 1

h�ie

i+2∑l=i−1

ˇl−i(C′B,l + C̃B,l)

)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(10)

where C̃k,i is the known value of the concentration of species k atpoint xi (0 ≤ i ≤ iMAX − 2) after allowing for the chemical change C ′

k,i

is the unknown value of the concentration at time � + ı� at point xi,and the expressions for the ˛ and ˇ coefficients are [1]:

˛−1 = 2�3e (2 + �e)

(1 + �e)(1 + �e + �2e )

˛0 = 21 − 2�e − �2

e

1 + �e

˛1 = −21 − �e − �2

e

�e(1 + �e)

˛2 = 21 − �e

�e(1 + �e)(1 + �e + �2e )

(11)

ˇ−1 = − �3e

1 + �e + �2e

ˇ0 = −2 − �2e

1 + �e

ˇ1 = 1�e

ˇ2 = − 1

�e(1 + �e)(1 + �e + �2e )

(12)

From Eqs. (9) and (10), and after applying the Thomas algorithm[2], it is obtained for each species k the following recursive relation-ship relating the values of concentration at two adjacent points ofthe grid:

C ′k,i+1 = ek,i + ak,iC

′k,i (13)

where the concrete expressions for the coefficients ek,i and ak,idepend on the time-integration method and on the form used toapproximate the derivative, and they are given in Appendix B forthe (4,2) approximation.

By means of the u–v method [3], the surface gradients of theparticipating species can be expressed as linear functions of thecorresponding surface concentrations, which in combination withthe approximate expression for the first derivative (Eq. (7)) leadsto:(

∂Ck

∂x

)x=0

= w̄k + ıkC ′k,0 (14)

where the expressions for the coefficients w̄k and ık depend on thenumber of points (N) taken to evaluate the derivative:

w̄k =N−1∑l=0

l−1∑m=0

ˇlek,m

l−1∏n=m+1

ak,n (15)

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710 F. Martínez-Ortiz et al. / Electr

k =N−1∑l=0

ˇl

l−1∏n=0

ak,n (16)

This simple relationship between the surface gradient and theurface concentration enables us to easily apply the boundary valueroblem as well as to calculate the current. In order to minimizeny contribution in the global error, we have considered the five-oint approximation (5,1) (i.e., p = 5, q = 1) for the calculation of theurface derivative. Thus, the expressions for the w̄ and ı coefficientsre given by Eqs. (15) and (16) with N = 5 and:

ˇ0 = −4 + 5�e + 7�2e + 5�3

e + 3�4e + �5

e

(1 + �e)(1 + �2e )(1 + �e + �2

e )

ˇ1 = (1 + �e)(1 + �2e )

�3e

ˇ2 = − (1 + �2e )(1 + �e + �2

e )

�5e (1 + �e)

ˇ3 = (1 + �e)(1 + �2e )

�6e (1 + �e + �2

e )

ˇ4 = 1

�6e (1 + �e)(1 + �2

e )

(17)

. Test of the sequential method

For the determination of the error committed with the sequen-ial method when calculating the electrochemical response, two

echanisms have been considered for which exact analyticalolutions are available in potential step chronoamperometry atpherical electrodes: first-order catalytic mechanism [10] and first-rder CE mechanism [11].

The scheme for the catalytic mechanism is given by Eq. (1) andor the CE mechanism by:

C + Qk1�k2

B + P

C + e− � D

(18)

nd considering that species P and Q are in large excess we canefine the equilibrium constant as:

1st =k1c∗

Q

k2c∗P

= c∗B

c∗C

(19)

nd the dimensionless kinetic parameter:

1stR = K1st

1 + K1st2 (20)

In all the cases, the sequential method has been used, firstlyalculating the influence of the chemical reactions. In the extrapo-ation methods this is done prior to the application of each BI stepsee below). For the spatial discretization, we have employed anxponentially expanding grid with asymmetric four-point approx-mation for the derivatives and high expansion factor (�e). Thee-values used have always been selected according to the opti-um values reported in our previous work [1]. Moreover, given

hat there are chemical interactions between the different species,unique diffusive field is used for all of them.

All the computer programs were written in C++ and compiledith mingw32 3.80.0 under Windows XP in a personal computer

ith AMD Phenom 9850 2.5 GHz processor with 4 Gb RAM, and

n the same system under LINUX Suse 11.2 using the gcc compiler,ersion 4.2.0. The “long double” type was always used in arithmeticalculations.

ica Acta 56 (2011) 5707–5716

3.1. Time-integration method

First, the influence of the time-integration method on the errorof the simulation will be studied in order to find the most effi-cient scheme for the sequential approximation with HEGFPD. Withthis aim, a chronoamperometric experiment under limiting currentconditions is considered, comparing the results for the dimension-less flux obtained in the simulation (G = (∂CC/∂x)x=0) with thoseobtained from the exact solution given in references [10] and [11](Gexact):

Error = G − Gexact

Gexact(21)

such that the relative error in percentage, Error (%), is given by thevalue obtained from the above equation multiplied by 100.

We have used implicit time-integration schemes: the ImplicitEuler method (commonly known in electrochemistry simulation asthe Laasonen method [12] or Backward Implicit, BI), the extrapo-lated implicit Euler method (the so-called extrapolation algorithms[13]) and the trapezium rule method (the so-called Crank–Nicolsonmethod (CN) [2]). In extrapolation methods, each time intervalis divided into subintervals and the BI method is successivelyapplied by following an adequate strategy. Depending on thenumber of subintervals and the way of applying the BI method,the different extrapolation schemes arise: EXTRAP2 (second-orderextrapolation), EXTRAP3 (third-order extrapolation) and EXTRAP4(fourth-order extrapolation). When using the sequential method incombination with extrapolation schemes the splitting of diffusionand chemical changes is carried out for each time subinterval.

Regarding the Crank–Nicolson algorithm, this is a second-ordermethod in time where the average of the known (C̃k,i) and unknown(C ′

k,i) concentration values is considered. The value of CN with

HEGFPD is demonstrated in Appendix A, where we show the con-venience of using four previous steps of BI or EXTRAP2, which isdenoted as CN(4BI) and CN(4EXTRAP2), respectively (see AppendixA).

Although the results obtained are somewhat dependent on thekinetic parameters, the grid parameters and the time intervals, theyare shown in a representative way by Fig. 1. This figure has beenobtained for a first-order CE mechanism with K1st = 3, a time inter-val ı� = 0.01 and a spatial interval h = 0.001. For the abscissa axiswe have taken the product �1st

R ı� since, as will be seen below, thisproduct removes the dependence with the particular values of therate constants. We can see that the results obtained are quite dif-ferent from those corresponding to an E mechanism (see AppendixA). In that case, the extrapolation methods are very favourable,whereas in this case they lead to significant errors, the higherthe extrapolation order, the greater the error. Thus, we concludethat the most accurate results are obtained with the CN-basedmethods, followed by the EXTRAP2 method. Among the meth-ods based on CN, that including initial second-order extrapolationsteps, CN(4EXTRAP2), is the most efficient, and so we have selectedthis time-integration scheme for the subsequent studies.

3.2. Optimization of simulation parameters

Next, the influence of the different simulation parameters willbe evaluated in order to establish the optimum conditions for theseto be adjusted prior to calculations. We found that the stability ofthe method here proposed is in accordance with that reported byBieniasz for Crank–Nicolson in presence of homogeneous chemicalreactions [14], that is, it is stable for any � = ı�/h2 value. However,

in the sequential approach if the fourth-order Runge–Kutta methodis used for the evaluation of the chemical contribution, instabilityis observed for �1st

R ı� > 3. This is a consequence of the error intro-duced by the Runge–Kutta method for such values of �1st

R ı� but it

F. Martínez-Ortiz et al. / Electrochimica Acta 56 (2011) 5707–5716 5711

Fig. 1. Relative error (in percentage) for different time-integration schemes in achronoamperometric experiment of limiting current at � = 1 in function of the prod-uct �1stı� at a planar electrode for a first-order CE mechanism. �e = 1.55, ı� = 0.01,hp

iu

ct

spaswTwwGmatlfedcswatctea0Ttaor

Fig. 2. (A) Influence of ı� on the relative error (in percentage) in the simulation ofthe limiting current at � = 1 for a first-order CE mechanism at a planar electrode,for different values of the equilibrium constant, indicated in each case on the corre-sponding curve. For comparison, the E mechanism is also included. �e = 1.55, h = 0.01,�1st

R= 20. CN(4EXTRAP2). (B) Influence of �1st

Rı� on the relative error (in percent-

age) in the simulation of the limiting current at � = 1 for a first-order CE mechanismat a planar electrode for two values of the equilibrium constant (indicated on the

R= 0.001. K1st = 3. The �1st

Rvalues are adjusted in each case to have the desired

roduct �1stR

ı�.

s not a practical limitation because the accuracy recommends these of �1st

R ı� ≤ 1 as stated below.The results included below are obtained under limiting current

onditions, although equivalent results are found for other poten-ial values.

Firstly, the influence of the equilibrium constant will be con-idered. The results obtained for the first-order CE mechanism arelotted in Fig. 2. As can be seen in Fig. 2A, the error committedlways increases with the time interval and the equilibrium con-tant. This is logical since the sequential approach is expected toork worse when the kinetic contribution to the current increases.

hus, for an equilibrium constant K1st = 1 the kinetic componentould contribute to the total current with a maximum of 50%hereas the kinetic contribution can exceed 95% for K1st = 100.iven that the kinetic component also depends on �1st

R , Fig. 2B isore representative for the study of the influence of K1st, where the

bscissa is the product �1stR ı�. This removes the effect of �1st

R , sincehe results obtained for different �1st

R values but the same equi-ibrium constant are identical. The most unfavourable conditionsor the simulation are those with a large kinetic component. Nev-rtheless, under conditions near the kinetic steady state, the erroroes not increase with the equilibrium constant any more. Thisan be observed in Fig. 3, which shows the error in very differenttrong kinetic conditions: catalytic mechanism and CE mechanismith equilibrium constants 100, 200 and 1000. All the results are

lmost superimposable and, given that they are obtained underhe worst conditions from a kinetic point of view, we can con-lude that this is an upper limit for the error committed withhe sequential method and CN time-integration preceded by fourxtrapolation steps with HEGFPD. Thus, we find that we can ensuren error smaller than 0.5% by taking �1st

R ı� < 0.35, smaller than.1% with �1st

R ı� < 0.15 and smaller than 0.01% with �1stR ı� < 0.05.

hese can be considered as very good results taking into account1st

hat a value �R ı� = 0.35 involves that, approximately, the vari-

tion in concentrations due to the chemical kinetics can be evenf 30% in a single time step. Under these conditions, besides accu-ate results the method provides a very high efficiency even for fast

curves) and different �1stR

values: 20 (circle points), 50 (square points), 100 (trianglepoints) and 200 (diamond points). The ı� values are adjusted in each case to havethe desired product �1st

Rı�. �e = 1.55, h = 0.001. CN(4EXTRAP2).

chemical kinetics. As an example, with the desktop computers pre-viously described cyclic voltammograms or chronoamperogramswith 105 time steps are obtained for a first-order CE mechanism inless than 1 s, including its representation through graphical inter-face. This enables fast access for the simulation of kinetic systemswith k1 + k2(s−1) ≈ 5 × 104 × v(V/s) ≈ 5 × 104/tr(s), with v being the

scan rate and tr the duration of the potential pulse. The simu-lation runtime slightly exceeds 1.5 s with an economical laptopcomputer (Celeron E1200 1.6 GHz, 2 Gb RAM). Nevertheless, forvery fast chemical reactions it would be preferable to resort to

5712 F. Martínez-Ortiz et al. / Electrochimica Acta 56 (2011) 5707–5716

Fig. 3. Influence of �1stR

ı� on the relative error (in percentage) in the simulation ofthe limiting current at � = 1 for first-order CE and catalytic mechanisms at planarelectrodes. For the CE mechanism three values of the equilibrium constant are con-sidered: K1st = 100 (red points), 200 (green points) and 1000 (maroon points); for thecatalytic mechanism K1st = 0 (blue points). The �1st

Rvalues are: 20 (circle points), 50

(Ct

al(ro

ttisFtute

dmbtitblv

gttfIitmap

Fig. 4. (A) Influence of h on the relative error (in percentage) obtained in the sim-ulation of the limiting current at � = 1 for a first-order CE mechanism and different�1st

Rvalues (indicated on the curves) at a planar electrode. K1st = 3. The ı� values are

adjusted in each case such that �1stR

ı� = 0.01. For comparison, the E mechanism isalso included. �e = 1.55. CN(4EXTRAP2). (B) Influence of h on the relative error (inpercentage) obtained in the simulation of the limiting current at � = 1 for a first-ordercatalytic mechanism and different �1st

Rvalues (indicated on the curves) at a planar

square points), 100 (triangle points) and 200 (diamond points). �e = 1.55, h = 0.001.N(4EXTRAP2). (For interpretation of the references to color in this figure legend,he reader is referred to the web version of the article.)

pproximate steady state approaches like those described in theiterature [15–17]. These approximations behave very well fork1 + k2)tr > 15. Hence our method (with simulation times in theange of 0.2 ms to 1 s) and kinetic steady state approximationsverlap in more than three orders of magnitude of rate constants.

We have also studied the influence of the spatial interval h forhe first-order CE and catalytic mechanisms. Thus, we have fixedhe value of the equilibrium constant and we have studied the hnfluence for several �1st

R values, keeping the product �1stR ı� con-

tant. The results are shown in Fig. 4A for a CE mechanism and inig. 4B for a catalytic mechanism. In both cases it is found that ifhe h value is small enough, it has no effect on the error of the sim-lation, and from a given h value, the error increases with h suchhat we can select the appropriate h value (hlim) to ensure a givenrror.

In Fig. 5 we can see that for a given error the 1/hlim value linearlyepends on

√�1st

R and the dependence is almost the same for bothechanisms. This behaviour points out that the space interval must

e sufficiently small in comparison with the linear reaction layerhickness (see Eq. (22)) for there to be a sufficient number of pointsnside this region for an accurate description. Thus, for the condi-ions here considered, it is found that the error is smaller than 0.1%y including 4 points of the spatial grid inside the linear reaction

ayer. This is a simple criterion to easily determine the adequate halue, once the kinetic constants are given.

For the above studies a planar electrode has been considerediven that the kinetic effects are more noticeable. The application ofhe sequential method with HEGFPD and CN has also been extendedo spherical electrodes and in Fig. 6 the results obtained are shownor a wide range of electrode sizes, including ultramicroelectrodes.t can be seen that, in general terms, the error committed dimin-shes with the electrode radius since the chemical contribution to

he current is smaller due to the enhancement of the diffusion

ass transport. So, the simulation strategy here presented is fullypplicable to spherical electrodes, with even better results than forlanar electrodes.

electrode. K1st = 1. The ı� values are adjusted in each case such that �1stR

ı� = 0.01.For comparison, the E mechanism is also included. �e = 1.55. CN(4EXTRAP2).

For �1stR ı� < 0.1 the errors obtained with spherical electrodes

of medium size (R0 ≈ 1) are slightly greater than those for the pla-nar case. Anyway, whatever the electrode size the error committedis warranted to be smaller than 0.1% by including 7 points in thelinear reaction layer (ıR, Eq. (22)). In case of medium and fast chem-ical reactions (�1st

R � > 1), the thickness of this layer can be easilyestimated from the following expression [16]:

ıR =(

1R0

+√

�1stR

)−1

(22)

F. Martínez-Ortiz et al. / Electrochim

Fig. 5. Influence of√

�1stR

on 1/hlim (see text) for two values of the relative error

(indicated on the graph) for a first-order CE mechanism with K1st = 100 (trianglepeC

s

ı

c

Flv√

C

oints) and for a first-order catalytic mechanism with K1st = 1 (circle points). Planarlectrode. The ı� values are adjusted in each case such that �1st

Rı� = 0.01. �e = 1.55.

N(4EXTRAP2).

For slow chemical kinetics, the linear reaction and linear diffu-ion layers almost coincide, and so they can be calculated from:

≈ ı =(

1 + 1√)−1

(23)

R D R0 ��

It is worth highlighting that this can be done with a great effi-iency with HEGFPD even under strong kinetic conditions (i.e., large

ig. 6. Influence of the electrode sphericity (R0) on the error in the simulation of theimiting current at � = 1 for a first-order catalytic mechanism and different �1st

Rı�

alues (indicated on the curves). The �e-value is adjusted in the range of 1.55 −2 depending on R0 according to the results obtained in Ref. [1]. K1st = 0, h = 10−5.N(4EXTRAP2).

ica Acta 56 (2011) 5707–5716 5713

�1stR values and narrow reaction layers). Thus, for the very high

value �1stR = 5 × 104 only 26 points are needed for the simulation of

the complete concentration profile at planar electrodes and spher-ical microelectrodes. However, for the most practical situations12–14 point grids give rise to excellent results.

According to our previous results [1], four figure accuracy iswarranted for the simulation of an E mechanism with HEGFPD bytaking 5 points inside the linear diffusion layer whatever the elec-trode size. Therefore, we can infer that, in relation to the spatialdiscretization, the accuracy of the simulation seems to be mainlydetermined by the resolution of the grid in the linear reaction ordiffusion layer such that, with 7 spatial points inside this region,errors smaller than 0.1% are always warranted.

4. Higher-order chemical reactions: other electrochemicaltechniques and mechanisms

In the above sections, only first-order mechanisms have beenconsidered for which exact solutions were available for the valida-tion of the methodology proposed.

Let us now consider the situation where species P and Q arenot in excess and so the chemical reactions are second-order. Inthis case, the differential equations of the species involved in thechemical reaction are given by:

∂CC

∂�= ∂2CC

∂x2+ 2

R0 + x

∂CC

∂x− K2nd

1 CCCQ + K2nd2 CBCP

∂CB

∂�= DB

DC

(∂2CB

∂x2+ 2

R0 + x

∂CB

∂x

)+ K2nd

1 CCCQ − K2nd2 CBCP

∂CP

∂�= DP

DC

(∂2CP

∂x2+ 2

R0 + x

∂CP

∂x

)+ K2nd

1 CCCQ − K2nd2 CBCP

∂CQ

∂�= DQ

DC

(∂2CQ

∂x2+ 2

R0 + x

∂CQ

∂x

)− K2nd

1 CCCQ + K2nd2 CBCP

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(24)

where:

K2nd1 = k1(c∗

B + c∗C )tr

K2nd2 = k2(c∗

B + c∗C )tr

(25)

and the equilibrium constant is defined as:

K2nd = k1

k2= c∗

Bc∗P

c∗Cc∗

Q

(26)

To show the effect of second-order kinetics we have consideredthe case of cyclic voltammetry. In Fig. 7 the cyclic voltammo-grams obtained for the second-order CE mechanism (with k1c∗

Q +k2c∗

P(s−1) = 500 × v(V/s) for a single-electron transfer at 298 K) andthe second-order catalytic mechanism (with k1c∗

Q + k2c∗P(s−1) =

62.5 × v(V/s) under the same conditions) with a reversible chargetransfer process are plotted. Different bulk concentrations ofspecies P and Q are considered, showing the transition from second-to pseudo first-order kinetics.

Although there are no analytical expressions available for com-parison, there are some other evidences of the validity of thesequential method for these second-order mechanisms. Firstly, theresults obtained for both mechanisms stabilize as ı� and h decreaseand the difference between a given experiment and the stable valuefollows the rules given for these parameters (ı� and h) in Section3.2, being now �2nd

R = C∗Q K2nd

1 + C∗PK2nd

2 . Therefore, the conditionspreviously established to warrant accurate results also apply tosecond-order mechanisms.

Moreover, for large enough concentrations of the electroinactivespecies P and Q the responses coincide with that corresponding tofirst-order mechanisms with K1st

1 = K2nd1 C∗

Q and K1st2 = K2nd

2 C∗P (see

Fig. 7).

5714 F. Martínez-Ortiz et al. / Electrochim

Fig. 7. (A) Cyclic voltammograms of a second-order CE mechanism for different con-centrations of the electroactive species (P and Q) at a planar electrode: C∗

P= C∗

Q=

1(a), 2(b), 5(c), 10(d) and 500(e). The curve corresponding to a first-order CE mech-anism is also plotted (superimposed to curve e). �1st/2nd

Rı� = 0.25, K1st/2nd = 5, 2000

time steps, h = 0.03 (14-point grid), �e = 1.55, CN(4EXTRAP2). (B) Cyclic voltammo-grams of a second-order catalytic mechanism for different concentrations of theelectroactive species (P and Q) at a planar electrode: C∗

P= C∗

Q= 1(a), 2(b), 5(c), 10(d),

25(e) and 500(f). The curve corresponding to a first-order catalytic mechanism isah

srKaEaht

lso plotted (superimposed to curve f). �1st/2ndR

ı� = 0.25, K1st/2nd = 0, 250 time steps,= 0.03 (14-point grid), �e = 1.55, CN(4EXTRAP2).

It is worth highlighting that the most unfavourable situation forimulations corresponds to large kinetic contributions for the cur-ent (large �1st/2nd

R , catalytic mechanism and CE mechanism with1st/2nd � 1). Both for the CE mechanism with K1st/2nd < 20 (Fig. 2)nd for other common electrochemical mechanisms, such as EC,

C2 (dimerization of the electrode product), ECE,. . ., the conditionsbove given for simulations are excessively conservative. Thus, weave confirmed that, even with very fast kinetics, errors smallerhan 0.1% are obtained with �1st/2nd

R ı� ≈ 1 in these cases.

ica Acta 56 (2011) 5707–5716

5. Conclusions

The digital simulation of electrochemical problems where first-and second-order homogeneous chemical reactions are coupledto the heterogeneous charge transfer has been successfully car-ried out by combining the sequential evaluation of chemical anddiffusion changes, with the Crank–Nicolson method for the time-integration and HEGFPD for the spatial discretization. In order toavoid the oscillatory behaviour with Crank–Nicolson, four initialextrapolation steps have been used.

The above procedure enables to uncouple the effects of thechemical reactions and the diffusion transport, which greatly sim-plifies the resolution of the problem such that it can be easilyimplemented from explicit expressions. Moreover, since each con-centration profile is calculated independently, the use of parallelcomputation methods available in modern cpu multiprocessorsis possible as well as GPU (Graphics Processing Units) currentlyavailable at low cost. Hence, this method has many advantages ofimplicit methods (accuracy, speed and stability) together with thesimplicity and easy parallelization of explicit methods.

Simple criteria are given to select the simulation parameters(time and spatial intervals) such that accurate results are warrantedwith short cpu-times for any electrode radius. Thus, even for veryfast chemical kinetics, errors smaller than 0.1% are obtained with�1st/2nd

R ı� < 0.1 and 7 points of the spatial grid inside the linearreaction or diffusion layer, which involves only 26 points for thecomplete concentration profiles whatever the electrode size, fromplanar electrodes to spherical microelectrodes. Nevertheless, forthe most common situations with 12–14 point grids and values of�1st/2nd

R ı� ≈ 1 excellent results are obtained.

Acknowledgements

The authors greatly appreciate the financial support provided bythe Dirección General de Investigación Científica y Técnica (ProjectNumber CTQ2009-13023), and the Fundación SENECA (ProjectNumber 11989). E. Laborda thanks the Ministerio de Educación yCiencia for the grant received.

Appendix A. The Crank–Nicolson method as an alternativeto the extrapolation methods with HEGFPD

In this Appendix the value of the Crank–Nicolson method (CN)as an alternative to extrapolation methods is demonstrated withHEGFPD.

CN follows an implicit time-integration scheme where theaverage of the second spatial derivative at the time � (known con-centration values, Ci) and at � + ı� (unknown concentration values,C ′

i) is taken [2]. The implementation of the CN method is essentially

identical to that of the BI algorithm, although the coefficients for theapplication of the Thomas algorithm are different (see Appendix B).

The CN method is stable for any �-value (with � = ı�/h2) and itoffers two major advantages with respect to BI: it is a second ordermethod in time and it leads to more precise values of the calcu-lated magnitudes. Nevertheless, CN has the important drawback ofpresenting an oscillatory response for large �-values when there isany kind of time discontinuity in the boundary value problem, asoccurs in pulse techniques.

To overcome the problem with oscillations, several procedureshave been used by different authors. The first method found in the

literature consists in subdividing the first time interval, for exam-ple, by using an exponential expansion similar to that employedin our spatial discretization. This method was firstly suggested byPearson [18] and later by other authors [19,20].

F. Martínez-Ortiz et al. / Electrochimica Acta 56 (2011) 5707–5716 5715

Fig. A.1. (A) log(|Error|) for the simulation of a Cottrell experiment for an E mecha-nism by using the Crank–Nicolson method for time integration preceded by some BIsteps. Planar electrode, �e = 1.55, ı� = 0.025, h = 0.04. The number of BI steps is indi-cated on each curve. (B) log(|Error|) for the simulation of a Cottrell experiment foran E mechanism by using the Crank–Nicolson method for time integration precededbo

t2og

ofidFC

Fig. A.2. Comparison of the efficiency of different time integration methods inthe simulation of a Cottrell experiment for an E mechanism at a planar electrode.�e = 1.55, h = 0.04. The ı� values are adjusted in order that the same � value involvesthe same cpu-time for all the integration schemes: 0.0125 for BI, 0.0149 for CN,0.0385 for EXTRAP2, 0.0625 for EXTRAP3 and 0.1 for EXTRAP4. For the previous

y some EXTRAP2 steps. Planar electrode, �e = 1.55, ı� = 0.025, h = 0.04. The numberf EXTRAP2 steps is indicated on each curve.

It is also possible to make other type of subdivisions, such ashe division in half of the first interval N times (with N betweenand 10) [21]. These methods, conveniently employed, damp CN

scillations but they increase the number of times needed for aiven simulation, and so they may not be efficient.

Another alternative is to use a time-integration algorithm with-ut oscillations for the first steps [22,23]. The BI method is therst candidate since it is very similar in concept to CN and it

oes not present any oscillations, although it is less accurate.ig. A.1A shows the removal of oscillations in the response of theN method when including a number N of BI steps at the begin-

BI or EXTRAP2 steps in CN, the ı� values corresponding to these schemes areused.

ning of the potential pulse (referred to as CN(NBI)) for a simplecharge transfer process (E mechanism) in a Cottrell experiment.However, the error committed also increases with the number ofBI steps, the error approximating to that corresponding to the BImethod.

A new alternative is to consider more precise methods for theinitial steps. Given that CN is a second order method, an extrapola-tion scheme with similar characteristics is chosen, that is, EXTRAP2.In Fig. A.1B the effect of initiating the process with some EXTRAP2steps is shown. As can be observed, few previous EXTRAP2 stepscompletely remove the CN oscillations and they also increase theaccuracy of results offered by EXTRAP2 (also shown in the figurefor comparison). This is a very interesting result since the cpu-timefor CN is only about 20% longer than BI, and so it is almost threetimes faster than EXTRAP2, providing more accurate results.

According to the above, in Fig. A.2 the efficiency of CN with BI orEXTRAP2 previous steps and other implicit time-integration meth-ods are compared. With this aim, the relative cpu-time taken byeach scheme has been taken into account such that the numberof time steps are selected in order that a given �-value refers tothe same cpu-time. It is clearly observed that the most efficientmethods are EXTRAP4 and CN with four preceding EXTRAP2 steps(that is, CN(4EXTRAP2)), although they offer very different char-acteristics. Thus, for a given cpu-time, with EXTRAP4 much lesscurrent-time points are obtained. So, this method will be moreadequate for pulse techniques where we are only interested inthe current value at the end of the potential pulse. Contrarily,if we want to have the complete chronoamperometric curves,for a given cpu-time they will be much more defined by usingCN(4EXTRAP2). The same applies to techniques where the per-

turbation applied varies continuously with time, such as in linearor triangular sweep voltammetry or in many galvanostatic tech-niques.

5716 F. Martínez-Ortiz et al. / Electrochim

Table B1Coefficients of the Thomas algorithm for the four-point approximation (4,2).

i < iMAX − 3

ek,i =Bk

i+1−Fk

2,i+1ek,i+2−ek,i+1(Fk

1,i+1+ak,i+2Fk

2,i+1)

1+ak,i+1(Fk1,i+1

+ak,i+2Fk2,i+1

)

ak,i = −Fk−1,i+1

1+ak,i+1(Fk1,i+1

+ak,i+2Fk2,i+1

)

i = iMAX − 3

ek,iMAX−3 =Bk

iMAX−2−Fk

2,iMAX−2C∗

k−ek,iMAX−2Fk

1,iMAX−2

1+ak,iMAX−2Fk1,iMAX−2

ak,iMAX−3 = −Fk−1,iMAX−2

1+ak,iMAX−2Fk1,iMAX−2

i = iMAX − 2

AC

adtaFtwl

aocaaa

[[

[[[[

[[[[[

Daimi PB-558, Dept. of Computer Science, Aarhus University, 1998.[21] A. Molina, C. Serna, F. Martínez-Ortiz, E. Laborda, J. Electroanal. Chem. 617

(2008) 14.[22] M. Luskin, R. Rannacher, Appl. Anal. 14 (1982) 117.[23] R. Rannacher, Numer. Math. 43 (1984) 309.

ek,iMAX−2 = BkiMAX−1

− C∗k(Fk

2,iMAX−1+ Fk

1,iMAX−1)

ak,iMAX−2 = −Fk−1,iMAX−1

ppendix B. Coefficients of the Thomas algorithm for therank–Nicolson and Backward Implicit methods

For the resolution of the equation system given by Eqs. (9)nd (10), the Thomas algorithm [2] (with some small adaptationsepending on the case) can be applied when the symmetricalhree-point approximation (3,2) or the asymmetrical four-pointpproximation (4,2) are employed for the spatial derivatives inick’s second law [1]. In the (3,2) form, a total of three points areaken to evaluate the derivatives at the second point of the set,hereas in the (4,2) form, four points are considered for the calcu-

ation of the derivatives at the second point.By applying the Thomas algorithm, the tridiagonal or quadradi-

gonal system is reduced to a didiagonal linear system of equationsf the type (13) that can be solved once the value of the surfaceoncentration is provided. In Table B1 the expressions of ek,i andk,i for the (4,2) form are given, considering semi-infinite diffusionnd both the BI and CN methods where Bk

i= bk

i/f k

0,i, Fk

m,i= f k

m,i/f k

0,ind:

f k−1,i

= Dk

DC

(˛−1 + 2h�i

e

R0 + xiˇ−1

)

f k0,i

= Dk

DC

(˛0 + 2h�i

e

R0 + xiˇ0

)− ε�2i

e

�

f k1,i

= Dk

DC

(˛1 + 2h�i

e

R0 + xiˇ1

)

f k = Dk

(˛ + 2h�i

e ˇ

)(27)

2,i DC2 R0 + xi

2

bki

= −ε�2ie

�Ck,i −

ica Acta 56 (2011) 5707–5716

where Dk is the diffusion coefficient of species k and the ˛ and ˇcoefficients are given by Eqs. (11) and (12). For the BI method � = 1and = 0, and for the CN method ε = 2 and:

= Dk

DC

(i+2∑

l=i−1

˛l−iCk,l + 2h�ie

R0 + xi

i+2∑l=i−1

ˇl−iCk,l

)(28)

References

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Dekker, New York, 1969, p. 199.[7] J.B. Flanagan, L. Marcoux, J. Phys. Chem. 77 (1973) 1051.[8] M.F. Nielsen, K. Almdal, O. Hammerich, V.D. Parker, Acta Chem. Scand. A 41

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1966.16] A. Molina, I. Morales, M. López-Tenés, Electrochem. Commun. 8 (2006) 1062.17] I. Ruzic, J. Electroanal. Chem. 144 (1983) 433.18] C.E. Pearson, Math. Comput. 19 (1965) 570.19] D. Britz, O. Østerby, J. Strutwolf, Comput. Biol. Chem. 27 (2003) 253.20] O. Østerby, Five Ways of Reducing the Crank–Nicolson oscillations. Tech. Rep.