Eduardo Laborda, Jose´ Manuel Olmos and A´ngela Molina*

10158 | Phys. Chem. Chem. Phys., 2016, 18, 10158--10172 This journal is© the Owner Societies 2016

Cite this:Phys.Chem.Chem.Phys.,

2016, 18, 10158

Transfer of complexed and dissociated ionicspecies at soft interfaces: a voltammetric studyof chemical kinetic and diffusional effects

Eduardo Laborda, Jose Manuel Olmos and Angela Molina*

A new transfer mechanism is considered in which two different ionic species of the same charge can be

transferred across a soft interface while they interconvert with each other in the original phase through

a homogeneous chemical reaction: the aqueous complexation–dissociation coupled to transfer (ACDT)

mechanism. This can correspond to a free ion in aqueous solution in the presence of a neutral ligand

that complexes it leading to a species that can be more or less lipophilic than the free ion. As a result,

the transfer to the organic phase can be facilitated or hindered by the aqueous-phase chemical

reaction. Rigorous and approximate explicit analytical solutions are derived for the study of the above

mechanism via normal pulse voltammetry, derivative voltammetry and chronoamperometry at

macrointerfaces. The solutions enable us to examine the process whatever the species’ lipophilicity and

diffusivity in each medium and the kinetics and thermodynamics of the chemical reaction in solution.

Moreover, when the chemical reaction is at equilibrium, explicit expressions for cyclic voltammetry

and square wave voltammetry are obtained. With this set of equations, the influence of different

physicochemical phenomena on the voltammetric response is studied as well as the most suitable

strategies to characterize them.

1. Introduction

The study of ion transfer processes across the interface betweentwo immiscible electrolyte solutions (ITIES) has attracted greatinterest and has an impact on many fields, such as liquid/liquidelectrochemistry and liquid/liquid extraction, with importantimplications in the development of electrochemical sensorsand the design of drugs.1,2 In addition, the interface betweenthe two phases (usually an aqueous and an organic solutions)can be used as a simple model for ion transport in biologicalmembranes, constituting a biomimetic medium suitable forstudying processes fundamental to bio-catalysis or cellularrespiration.3 Thus, one of the parameters that can be extractedby electrochemical techniques is the standard Gibbs energy ofthe ion transfer, which is directly related to the ion’s lipophilicity.1

The diffusivity of the ion in each phase as well as its concentrationcan also be extracted in a fast, inexpensive and simple way.4–11

These properties determine the distribution of the ionic speciesin biological systems and their potential application as a drugor a drug carrier.

According to the above discussion, electrochemical methodsare very powerful in the study of the transfer of ionic speciesacross soft interfaces. Given the dynamic nature of real systemsand of voltammetric techniques, the rigorous modelling ofsuch processes requires taking into account the influence ofthe species’ chemical reactivity and diffusivity in each phase.Thus, the heterogenous processes of ion transfer can becoupled with homogeneous chemical reactions (as occurredwith the analogous electron transfer processes12–14), such as inthe so-called facilitated or assisted ion transfer mechanisms.For example, in the TOC mechanism the ion transfer ispromoted by a ligand present in the organic solution; i.e. thereis a chemical reaction following the charge transfer that stabilizesthe ion15 (equivalent to the EC mechanism in electron transferprocesses). Other possible transfer routes are the ACT mechanism(a chemical reaction preceding the ion transfer, equivalent to theso-called CE mechanism) and the TIC mechanism (ion transfer byan interfacial chemical reaction).16

Electrochemical techniques can also be employed to char-acterize the thermodynamics (equilibrium constant) and thekinetics (rate constants) of the chemical reaction(s) (exceptfor the TIC mechanism, where the current response is onlya function of thermodynamic parameters17) and to elucidatethe reaction mechanism.18–21 In previous studies, we havereported analytical expressions for the voltammetric response

Departamento de Quımica Fısica, Facultad de Quımica, Regional Campus of

International Excellence ‘‘Campus Mare Nostrum’’, Universidad de Murcia,

30100 Murcia, Spain. E-mail: [email protected]; Fax: +34 868 88 4148;

Tel: +34 868 88 7524

Received 3rd February 2016,Accepted 8th March 2016

DOI: 10.1039/c6cp00780e

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of the TOC and TIC mechanisms at macrointerfaces, as well asfor the case of any number of chemical reactions at equilibriumpreceding and following the ion transfer.22–25 These theoreticaldevelopments have been employed to characterize the transferof several protonated amines facilitated by the ionophore dibenzo-18-crown-6 from water to a solvent polymeric membrane.11,26

In this work, a new transfer mechanism is considered inwhich two different ionic species taking part in a chemicalreaction with a neutral species can be transferred: the aqueouscomplexation–dissociation coupled to transfer (ACDT) mechanism(see Scheme 1). This can correspond, for example, to the transferof a free metal and its complex, provided that the ligand isneutral. From this mechanism two general scenarios can beenvisaged: the situation where the complex is more lipophilic(and so the complexation ‘‘facilitates’’ the ion transfer) and thatwhere the complex is less lipophilic (and then the complexation‘‘hinders’’ the transfer). In the limit when only one of thespecies can be transferred within the electrochemical window,the above two situations would correspond to the so-called ACTmechanism (aqueous complexation followed by transfer18) andto the ADT mechanism (aqueous dissociation followed bytransfer27). Therefore, the theory reported here (which can alsobe applied to the study of equivalent electron transfer reactionschemes) can assist in getting further physicochemical insightinto ion transfer mechanisms in real systems, with potentialimpact on speciation studies, such as the determination ofthe availability of metal complexes and their correspondingbeneficial or toxic properties.28–31 Many experimental andmodelling approaches to speciation and bioavailability/toxicityare based on equilibrium principles.27 In contrast, the considerationof mass transport and chemical kinetic effects in the presenttheory for the ACDT mechanism enables us to analyze thedynamic aspects of speciation in a general and rigorous way.This is particularly interesting given that natural aquatic systemsare generally subjected to changing conditions and are practicallynever at chemical equilibrium.27,32–35

Rigorous and approximate explicit analytical equations forthe chronoamperometric, normal pulse voltammetry (NPV) andderivative voltammetry (dNPV) responses of the ACDT mechanismare deduced under non-equilibrium conditions. A general expressionis also reported for any voltammetric technique (such as cyclicvoltammetry and square wave voltammetry) under total equilibriumconditions. In all cases, the influence of the difference betweenthe formal transfer potentials of the ion and the complex on

the shape and position of the voltammograms is analyzed.Moreover, different criteria are proposed for qualitative andquantitative kinetic and thermodynamic studies based on thevariation of the electrochemical response with time and withligand concentration.

2. Theory2.1 Influence of the chemical kinetics: application of aconstant potential

Let us consider the reversible transfer of cationic species froman aqueous solution (w) to an organic phase (o) in the form oftwo chemical species Mz+ and MLz+ (ACDT mechanism).

Scheme 1 can correspond to a free metal ion Mz+ andits complex MLz+ in solution. In Scheme 1, k1

0 (M�1 s�1) andk2 (s�1) are the forward and backward rate constants of the

chemical reaction and Dwof

00Mzþ and Dw

of00MLzþ are the formal

ion transfer potentials of species Mz+ and MLz+, respectively.Moreover, the ligand L is assumed to be a neutral speciespresent in a large excess in the aqueous solution, such that thekinetics of the chemical reaction can be considered of (pseudo)-first order (k1 = k1

0cL*(s�1)). Under these conditions, the conditionalor apparent equilibrium constant of complexation can bedefined as follows:

K ¼ cML�

cM�¼ k1

0cL�

k2¼ KccL

� (1)

with ci* (M) being the initial equilibrium concentration ofspecies i (i = M, ML, L) in water and Kc (M�1) the equilibriumconstant based on concentrations.

When a constant potential difference E is applied betweenboth phases with length t, supposing that diffusion is the onlyeffective mass transport mechanism, the variations of theconcentrations of the different species with the distance tothe interface x and with the electrolysis time t (0 r t r t) aredescribed by the following differential equation system

@cwMðx; tÞ@t

¼ DwM

@2cwMðx; tÞ@x2

� k1cwMðx; tÞ þ k2c

wMLðx; tÞ

@cwMLðx; tÞ@t

¼ DwML

@2cwMLðx; tÞ@x2

þ k1cwMðx; tÞ � k2c

wMLðx; tÞ

@coMðx; tÞ@t

¼ DoM

@2coMðx; tÞ@x2

@coMLðx; tÞ@t

¼ DoML

@2coMLðx; tÞ@x2

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(2)

where Dai is the diffusion coefficient of species i in phase a

(a = w, o). The boundary conditions associated with system (2)are given by

x � 0; t ¼ 0

x! �1; t � 0

)cwM ¼ cM

�; cwML ¼ cML� (3)

Scheme 1

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x � 0; t ¼ 0

x!1; t � 0

)coM ¼ coML ¼ 0 (4)

x = 0, t 4 0}

DwM

@cwM@x

� �x¼0¼ Do

M

@coM@x

� �x¼0

(5)

DwML

@cwML

@x

� �x¼0¼ Do

ML

@coML

@x

� �x¼0

(6)

coM(0) = cw

M(0)eZ1 (7)

coML(0) = cw

ML(0)eZ2 (8)

with

Z1 ¼zF

RTE � Dw

of00Mzþ

� �(9)

Z2 ¼zF

RTE � Dw

of00MLzþ

� �(10)

where F, R and T have their usual meanings. Moreover, thecurrent measured can be calculated by means of the followingexpression:

I ¼ �zFA DwM

@cwM@x

� �x¼0þ Dw

ML

@cwML

@x

� �x¼0

� �(11)

where A is the interfacial area.Now, we assume that the diffusion coefficients in a given

phase are equal (DwM = Dw

ML = Dw and DoM = Do

ML = Do) butdifferent for the aqueous and organic solutions (Dw a Do).Indeed, important differences are frequently observed in thediffusivity of ionic species in transfer processes between twoimmiscible electrolyte solutions, such as water and RTILsor solvent polymeric membranes.11,36–38 By defining two newvariables14

x = cwM + cw

ML (12)

f = (KcwM � cw

ML)e(k1+k2)t (13)

with f referring to the perturbation of the chemical equilibriumand x to the total concentration in the aqueous solution,eqn (2)–(8) are transformed as follows:

@xðx; tÞ@t

¼ Dw@2xðx; tÞ@x2

@fðx; tÞ@t

¼ Dw@2fðx; tÞ@x2

@coMðx; tÞ@t

¼ Do@2coMðx; tÞ

@x2

@coMLðx; tÞ@t

¼ Do@2coMLðx; tÞ

@x2

9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;

(14)

x � 0; t ¼ 0

x! �1; t � 0

)x ¼ c� ¼ cM

� þ cML�;f ¼ 0 (15)

x � 0; t ¼ 0

x!1; t � 0

)coM ¼ coML ¼ 0 (16)

x = 0, t 4 0}

@x@x

� �x¼0þe�kt @f

@x

� �x¼0¼ 1þ Kð ÞDo

Dw

@coM@x

� �x¼0

(17)

K@x@x

� �x¼0�e�kt @f

@x

� �x¼0¼ 1þ Kð ÞDo

Dw

@coML

@x

� �x¼0

(18)

coMð0Þ ¼xð0Þ þ e�ktfð0Þ

1þ KeZ1 (19)

coMLð0Þ ¼Kxð0Þ � e�ktfð0Þ

1þ KeZ2 (20)

with

k = k1 + k2 (21)

Moreover, the current response is given by

I ¼ �zFADw@x@x

� �x¼0

(22)

By following a mathematical procedure analogous to thatemployed in ref. 39, based on Koutecky’s dimensionless para-meters method40,41 (see Appendix B), the following rigorousexpression for the current is obtained:

Irig

Id

¼g eZ1 þKeZ2 þ geZ1eZ2ð1þKÞþ Kg eZ2 � eZ1ð Þ2

ð1þKÞ 1þ geZ1ð Þ 1þ geZ2ð ÞSðwÞ" #

ð1þKÞ 1þ geZ1ð Þ 1þ geZ2ð Þ(23)

where

g ¼ffiffiffiffiffiffiffiDo

Dw

r(24)

Id is the diffusion-limited current of a simple ion transferprocess at macrointerfaces12

Id ¼ zFAc�ffiffiffiffiffiffiffiDw

pt

r(25)

and S(w) is given by

SðwÞ ¼X1j¼1

ljw j (26)

where

l1 ¼p2

p0� 1 (27)

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lj ¼1

j!

p2j

p0� 1

� �� g

1þ Kð Þ 1þ geZ1ð Þ 1þ geZ2ð Þ

�Xj�1i¼1

lij � ið Þ! KeZ1 þ eZ2 þ geZ1eZ2 1þ Kð Þ

�

þ p2j

p2i

1þ K þ g eZ1 þ KeZ2½ �g

� ��j4 1

(28)

w = kt (29)

with

p0 ¼2ffiffiffipp

pi4 0 ¼2i

pi�1

8>>><>>>:

(30)

2.1.1 Kinetic steady state aproximation (kss). When thekinetics of the coupled chemical reaction is fast enough, it isadequate to suppose that equilibrium perturbation is independentof electrolysis time42

@fss

@t¼ 0 (31)

with fss being defined as

fss = KcwM � cw

ML (32)

By taking into account eqn (31) and (32), system (2) leads to thefollowing differential equation for the variable fss

@2fssðxÞ@x2

� kDw

fssðxÞ ¼ 0 (33)

the solution of which is given by

fssðxÞ ¼ fssð0Þ exdr (34)

where x r 0 in the aqueous phase and dr is the thickness ofthe linear reaction layer where the chemical equilibrium isdisturbed

dr ¼ffiffiffiffiffiffiffiDw

k

r(35)

Thus, by applying Koutecky’s dimensionless parameter methodand considering eqn (34), the expression for the current underthe kss approach is obtained

Ikss

Id¼ g

KeZ1 þ eZ2 þ geZ1eZ2 1þ Kð Þ

� ð1þ KÞeZ1eZ2 þ K eZ1 � eZ2ð Þ2F wkssð Þ1þ K þ g eZ1 þ KeZ2ð Þ

" # (36)

where

F wkssð Þ ¼ffiffiffipp

2wksse

wkss2

2erfc

wkss2

� �(37)

wkss ¼2ffiffiffiwp

1þ K þ g eZ1 þ KeZ2ð Þ½ �g KeZ1 þ eZ2 þ geZ1eZ2ð1þ KÞ½ � (38)

and w is defined by eqn (29). The validity of the approximatesolution (36) has been analyzed by comparison with the rigorousone (eqn (23)), finding that the error decreases as w increasesand |Df00| decreases (see eqn (46)). In regard to the influence ofK, the error increases as the proportion of the most lipophilicspecies decreases (i.e. it increases with K if Df00 4 0 andit decreases if Df00 o 0). For example, for 0.2 o K o 5 and|Df00| o 200 mV, both solutions coincide for w Z 3.2 with adifference smaller than 5% and for w Z 14 with a differencesmaller than 1%, independently of the value of g.

2.1.2 Particular case: only the Mz+ ion (or the complexMLz+) is transferred. Under these conditions, when only theion is transferred (aqueous dissociation followed by transfer(ADT), see Scheme 2), the rigorous expression for NPV can

be easily deduced from eqn (23) by making Dwof

00MLþ ! þ1

(i.e. eZ2-0)

IADTrig

Id¼ geZ1

ð1þ KÞ 1þ geZ1ð Þ 1þ KgeZ1

1þ Kð Þ 1þ geZ1ð ÞX1j¼1

lADTj w j

" #

(39)

with

lADT1 ¼ p2

p0� 1 (40)

Scheme 2

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lADTj ¼ 1

j!

p2j

p0� 1

� �� g

1þ Kð Þ 1þ geZ1ð ÞXj�1i¼1

lij � ið Þ! KeZ1½

þ p2j

p2i

1þ K þ geZ1

g

� ��; j4 1

(41)

and pz being defined in eqn (30). In addition, from (36) oneobtains the kss approximate solution for the current

IADTkss

Id¼

geZ1F wMzþ

kss

� �1þ K þ geZ1

(42)

where F wMzþ

kss

� �is given by eqn (37), with

wMzþ

kss ¼2ffiffiffiwp

1þ K þ geZ1ð ÞKgeZ1

(43)

eqn (39)–(43) coincide with those reported in previousstudies for the equivalent CE-mechanism in electron transferprocesses.43,44 Analogous expressions for the case where MLz+

is the only species transferred (aqueous complexation followedby transfer (ACT), see Scheme 2) are also deduced by making

Dwof

00Mþ ! þ1 (i.e. eZ1 - 0) in eqn (23) and (36). Such

expressions are equivalent to (39)–(43) by replacing Z1 byZ2 and K by 1/K.

2.2 Total equilibrium conditions (te): application of anyvoltammetric technique

When the chemical kinetics in the aqueous phase is very fast incomparison with diffusion, we can assume that the chemicalequilibrium is maintained at any time and any point in theaqueous solution. Under these conditions, the expression forthe current greatly simplifies by making w-N(F(wkss)-1) ineqn (36), in a such way that it is obtained that

Ite

Id¼ g eZ1 þ KeZ2ð Þ

1þ K þ g eZ1 þ KeZ2ð Þ

� �(44)

This equation corresponds to a single wave, the half-wavepotential of which is

Ete1=2 ¼ Dw

ofo0MLzþ þ

RT

zFln

1

g

� �þ RT

zFlnð1þ KÞ

� RT

zFln K þ e

zFRT

Df00� � (45)

with

Df00 ¼ Dwof

00MLzþ � Dw

of00Mzþ (46)

Furthermore, the assumption of total equilibrium conditionsenables us to derive an analytical expression for the currentresponse when any sequence of p potential steps E1, E2,. . .Ep isapplied. Thus, it can be demonstrated that the interfacialconcentrations associated with each potential pulse are onlydependent on the value of the applied potential and areindependent of the previous potential pulses. Therefore,the superposition principle can be applied45 in a such way that

the current corresponding to the p-th potential step can bewritten as

Ite; p ¼ zFA

ffiffiffiffiffiffiffiDw

p

r Xpi¼1

cw;i�1T ð0Þ � cw;iT ð0Þffiffiffiffiffitipp (47)

where

cw;0T ð0Þ ¼ c�

cw;iT ð0Þ ¼c�ð1þ KÞ

1þ K þ g eZðiÞ1 þ KeZ

ðiÞ2

� �; i4 0

9>>>=>>>;

(48)

ti;p ¼Xp�1m¼i

tm þ tp (49)

with

ZðiÞ1 ¼zF

RTEi � Dw

of00Mzþ

� �; i4 0 (50)

ZðiÞ2 ¼zF

RTEi � Dw

of00MLzþ

� �; i4 0 (51)

Note that the general solution (47) can be applied to anyvoltammetric technique. For example, the normalized expressionsfor staircase cyclic voltammetry (SCV) and square wave voltam-metry (SWV) are as follows:

cte;SCV ¼ISCV

zFAc�ffiffiffiffiffiffiffiffiffiaDw

p

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiRT

pFDE

rð1þ KÞ

Xpi¼1

G Zi�1ð Þ � G Zið Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip� i þ 1p

(52)

cte;SWV ¼DISWV

ffiffiffitp

zFAc�ffiffiffiffiffiffiffiDw

p

¼ 1þ Kffiffiffipp

X2p�1i¼1

G Zi�1ð Þ � G Zið Þð Þ�"

� 1ffiffiffiffiffiffiffiffiffiffiffiffiffi2p� ip � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2p� i þ 1p

� ��

þG Z2p� �

� G Z2p�1� �#

(53)

where DE is the pulse step in SCV (with DE o 0.01 mV the cyclicvoltammogram (CV) is obtained46) and t is the duration of eachapplied potential in SWV (the frequency is defined as f = 1/2t).Moreover, G(Zi) and a are given by

G Z0ð Þ ¼1

1þ K

G Zið Þ ¼1

1þ K þ g eZðiÞ1 þ KeZ

ðiÞ2

� �; i4 0

9>>>>=>>>>;

(54)

a ¼ FnRT

(55)

with n being the scan rate in SCV and CV.

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2.2.1 Particular case: only the Mz+ ion (or the complexMLz+) is transferred. Under these conditions, by makingeZ2-0 in eqn (44) and (45), the NPV current and the half wavepotential when only the free ion is transferred (ADT mechanism)are given by

IADTte

Id¼ geZ1

1þ K þ geZ1(56)

Ete;ADT1=2 ¼ Dw

ofo0Mzþ þ

RT

zFln

1

g

� �þ RT

zFlnð1þ KÞ (57)

Moreover, the general solution (47) for an arbitrary sequence ofpotentials can also be used in this particular case, by making eZ2-

0 in the expressions of surface concentrations (eqn (48)). Hence,the CV and SWV responses are also given by eqn (52) and (53) with

GADT Z0ð Þ ¼1

1þ K

GADT Zið Þ ¼1

1þ K þ geZðiÞ1

; i4 0

9>>>>=>>>>;

(58)

Analogous expressions for the case where MLz+ is the only speciestransferred (ACT mechanism) are also deduced by making

Dwof

o0Mþ ! þ1 (i.e. eZ1-0) in eqn (44), (45), (48) and (54). Such

expressions are equivalent to (52), (53) and (56)–(58) by replacing

Dwof

o0Mzþ by Dw

ofo0MLzþ (Z1 by Z2) and K by 1/K.

3. Results and discussion3.1. Chemical kinetics: normal pulse voltammetry (NPV),differential voltammetry (dNPV) and chronoamperometry

The effect of the chemical kinetics on the NPV voltammogramsis parameterized here by the variable w, defined as w = kt(eqn (29)). The increase of the chemical rate constant k(k = k1 + k2) or the duration t of the potential pulse (0 r t r t)results in higher values of w. Therefore, the ‘‘effective’’ chemicalkinetics can be varied experimentally by changing the duration ofthe applied potentials. Note that this is qualitatively equivalent tochanging the scan rate in cyclic voltammetry and the frequency insquare wave voltammetry. Thus, the decrease of the scan rate inCV or the frequency in SWV is analogous to the use of longerpulse times in NPV. Furthermore, the value of k can also bemodified by means of the initial concentration of the ligand(cL*), since k1 = k1

0cL*.The influence of w is analysed in Fig. 1 and 2, where NPV-

voltammograms are plotted for a ratio of diffusion coefficientsDo/Dw = 0.001. Note that this large difference between thediffusivity of the ionic species in phases w and o is not uncommonin the present context given that 2- and even 3-order-of-magnitudedifferences have been reported between the D-values in con-ventional solvents and solvent polymeric membranes or roomtemperature ionic liquids.11,36–38 An apparent equilibrium con-stant K = 2 is considered (that is, cM* = c*/3 and cML* = 2c*/3)and several values of the difference between the formal ion

transfer potentials Df00 ¼ Dwof

00MLzþ � Dw

of00Mzþ

� �, taking into

account that the transfer of the complex can be thermodynamicallyless favourable (Df00 4 0, Fig. 1) or more favourable (Df00 o 0,Fig. 2) than the transfer of the free species.

As can be seen in Fig. 1A and 2A, when the difference Df00 islarge (|Df00| = 200 mV) and two waves can be distinguished, thefirst signal is related to the transfer of the most lipophilicspecies, i.e. the species with less positive formal transferpotential (Mz+ in Fig. 1A and MLz+ in Fig. 2A). Hence, the

Fig. 1 Effect of the chemical kinetics on the NPV-voltammograms(eqn (23), (36) and (44)) when the free ion is more lipophilic than thecomplex (Df00 4 0). K = 2, g2 = Do/Dw = 0.001, t = 1s, z = 1, T = 25 1C.

Id ¼ zFAc�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDw=pt

p.

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limiting current of the first wave (I1lim) is larger in Fig. 2A than in

Fig. 1A since initially MLz+ is present in a higher concentrationaccording to the chemical equilibrium constant considered(cML* = 2cM*). As the w-value is higher, the magnitude of thefirst wave increases. This can be understood by the fasterinterconversion between Mz+ and MLz+. As discussed in Section2.1.2, under the Df00-values in Fig. 1A and 2A, the limiting current of

the first wave (I1lim) and their half-wave potential (E1

1/2) are identical tothose obtained for the equivalent CE-mechanism (eqn (39)–(43)) withthe species transferred being Mz+ in Fig. 1A and MLz+ in Fig. 2A.

The voltammograms also undergo a transition from a dou-ble wave to a single one as the w-value is increased. In the limitof very fast kinetics, they coincide with the response predictedunder total equilibrium conditions (grey line, eqn (44)), thelimiting current of the wave being proportional to the sum ofthe initial concentrations of species Mz+ and MLz+ in water (c*).On the other hand, when the value of w is small (w r 0.01),the voltammograms show two waves corresponding to the‘‘independent’’ transfer of each species, which are centred onthe corresponding half-wave potential given by14

E1=2; j ¼ Dwof

00j þ

RT

zFln

1

g

� �j ¼Mzþ;MLzþð Þ (59)

Note that the variation of the signal shape with pulse durationis a simple criterion to discriminate between the mechanismdepicted in Scheme 1 and the situation where there is nointerconversion between species Mz+ and MLz+, that is, thechemical process does not take place (at least within thevoltammetric time-scale). Also note that the above conclusionsare qualitatively applicable to other voltammetric techniquesand the corresponding current–potential curves show an analogousbehaviour as the time-scale of the experiment is changed.

The value of the difference Df00 also has an effect on theNPV-voltammograms, as shown in Fig. 1B, C, 2B and C. Thus,smaller differences between the formal transfer potentials giverise to the overlapping of the two waves. Hence, the effect ofchemical kinetics is less apparent and so more difficult todetect experimentally. The response finally turns into a singlesignal when Df00-0 (see Fig. 1C and 2C), which is insensitiveto the w-value since under these conditions it is fulfilled thatZ1 = Z2 = Z (see eqn (9) and (10)) in a such way that eqn (23) leadsto the following expression for the current–potential response:

IDf00 ¼0Id

¼ geZ

1þ geZ(60)

which corresponds to a single sigmoidal curve (Fig. 1C and 2C).As can be inferred from eqn (60), the current is independentof the chemical reaction since both species have identicallipophilicity and they are transferred at the same potential

Dwof

00Mzþ ¼ Dw

of00MLzþ

� �. In fact, the above expression coincides

with that obtained for an ion transfer reaction without anykinetic constraint in the aqueous phase.14

The effect of the equilibrium constant is shown in Fig. 3, forDf00 = 200 mV and Do/Dw = 0.001 (note that the conclusions arequalitatively analogous for any other values of Df00 and Do/Dw).As expected, when K c 1 or K { 1 the curves tend to thosecorresponding to the simple transfer of the complex or the freespecies, respectively, given that under such conditions they arepractically the only species present in the aqueous medium. Onthe other hand, two waves are obtained for non-extreme valuesof K when total equilibrium conditions are not achieved(Fig. 3A and B). As can be seen, lower K-values result in a

Fig. 2 Effect of the chemical kinetics on the NPV-voltammograms (eqn (23),(36) and (44)) when the free ion is less lipophilic than the complex (Df00o 0).K = 2, g2 = Do/Dw = 0.001, t = 1s, z = 1, T = 25 1C. Id ¼ zFAc�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDw=pt

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higher limiting current of the first signal (I1lim), due to the

increase of the initial concentration of the most lipophilicspecies Mz+. The variation of the limiting current with K isless notorious for large k-values, since the chemical reactionprovides a higher amount of species Mz+. Note that even if theinitial concentration of the most lipophilic species is not very

large in solution (e.g. K = 10 in Fig. 3B), its correspondingelectrochemical signal can be significant if the interconversionMLz+ $ Mz+ is fast. This point is important for the correctinterpretation of electrochemical signals in speciation studies.

For large w-values such that total chemical equilibriumconditions are attained (Fig. 3C), a single curve is obtainedindependently of the values of K and Df00 (see also Fig. 1 and 2)according to eqn (44), the position of which is given by eqn (45).Thus, the voltammogram shifts towards more positive potentials asthe equilibrium constant takes larger values, since the proportion of

Fig. 3 Influence of the equilibrium constant (K = cML*/cM*) on theNPV-voltammograms (eqn (23), (36) and (44)) for different chemicalkinetics. Df00 = 200 mV, g2 = Do/Dw = 0.001, t = 1s, z = 1, T = 25 1C.


p.

Fig. 4 Effect of the ligand concentration (cL*) on the NPV-voltammograms((A) eqn (23) and (36)), Ehalf (B) and I1lim (C). Df00 = 200 mV, g2 = Do/Dw =0.001, t = 1s, z = 1, T = 25 1C, c* = 10 mM. Kc =k1

0/k2 = 10 M�1,


p. E1/2,Mz+ is given by eqn (59).

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the less lipophilic species (i.e., the complexed species in this figure)increases with K (see eqn (1)).

The effect of the variation of the concentration of species L(cL*) is shown in Fig. 4. The value of cL* modifies the value ofboth the apparent equilibrium constant (K = KccL*) and the‘‘effective’’ rate constant of complexation (k1 = k1

0cL*), in such away that both K and k1 increase with cL*. Taking the above twoeffects into account, NPV-curves have been plotted in Fig. 4A fordifferent concentrations of the ligand (assuming that cL* is ina large excess with respect to c* in all the cases) and Kc = 10 M�1

(Kc = k10/k2). As can be seen, given that Df00 4 0

(i.e., Dwof

00MLzþ 4Dw

of00Mzþ ), higher concentrations of the ligand

result in a lower limiting current of the first wave (I1lim) and the

shift of the voltammogram towards more positive potentials.This is a consequence of the increase of the concentration ofspecies MLz+, which has been considered to be less lipophilicthan Mz+ (the opposite behaviour would be observed if MLz+

were the most lipophilic species). Obviously, the voltammo-grams would be insensitive to the value of cL* in the absence ofthe homogeneous chemical reaction (i.e., w { 1).

The study of the variation of the position of the signal and ofthe value of I1

lim (provided that the difference of formal transferpotentials is large enough) can be used to determine the rateand equilibrium constants. In Fig. 4B and C, the variation ofI1lim and the position of the signal (parameterized by means of

the potential Ehalf at which the normalized current takes thevalue I/Id = 0.5) with cL* have been plotted for different values ofk10 and k2, maintaining the ratio k1

0/k2(=Kc) constant. The shapeof the curves is very similar for any Kc value, the influenceof which is the shift of the curve towards smaller cL*-values as Kc

increases (not shown). The rate constants and the real equilibriumconstant can be extracted by fitting the variation of I1

lim and Ehalf

with cL* within the adequate range of ligand concentrations (KccL*approximately between 1 and 10). The two limit values of thecurves in Fig. 4B correspond to the half-wave potentials of speciesMz+ (very small ligand concentration) and MLz+ (very high ligandconcentration), which are given by eqn (59). Therefore, when therange of cL* is not limited by the nature of the ligand and/or by itssolubility, the half-wave potentials of the ion transferred can bedetermined directly from these two limit values.

The theory developed for the ACDT mechanism depicted inScheme 1 also allows us to study the current–potential responsein derivative voltammetry (dNPV), obtained by differentiationof the experimental NPV curve. A notable advantage of thistechnique is the higher resolution that it offers when thesignals are overlapped. In Fig. 5A, the influence of w on thedNPV-voltammograms is shown for Df00 = 200 mV. As discussedfrom the NPV curves, the increase of w leads to the transitionof the signal from two peaks centred at the correspondinghalf-wave potentials to a single peak. The latter coincides withthe response obtained under total equilibrium conditions(grey line) with Epeak being given by eqn (45). Furthermore,the peak current of the first signal increases with w whereas thesecond peak undergoes the opposite behaviour. Note that thetwo peak potentials take more positive values as the chemicalinterconversion is faster.

The influence of the equilibrium constant on the dNPVresponse is examined in Fig. 5B, for Df00 = 200 mV and w = 1.In this case the effect is also analogous to that observed in NPV(see Fig. 3A). Thus, the voltammograms show two peaks when Ktakes intermediate values, the magnitude of the first one(transfer of Mz+) decreasing with K whereas the second signalincreases as K is larger. In addition, the two peaks converge into asingle one when K-N (simple transfer of species MLz+) or K-0(simple transfer of species Mz+). This figure illustrates the ideamentioned above that the conclusions reached for the NPV signalare qualitatively applicable to other voltammetric techniques.

Let us move now to the study of the chronoamperometricresponse; i.e. the current–time curves (I vs. t) obtained underthe application of a constant potential. In Fig. 6, the influenceof the chemical kinetics and of the equilibrium constant on theI–t response is analyzed for Df00 = 200 mV, upon the application

of a potential E � Dwof

00Mzþ ¼ 200 mV that corresponds to the

limiting current of the first wave of the NPV curve (see Fig. 1A).With regard to the effect of k (Fig. 6A), one can observe that the

increase of this variable results in larger values of the current,reaching the limit of total equilibrium for very fast kinetics (seeinset in Fig. 6A). Thus, the plot of I1

lim/Id versus w is a sigmoid withtwo limit values corresponding to the simple transfer of species

Fig. 5 Influence of the chemical kinetics (A) and the equilibrium constant(B) on the dNPV-voltammograms, obtained by differentiation of thecorresponding NPV curves (eqn (23), (36) and (44)). Df00 = 200 mV,g2 = Do/Dw = 0.001, t = 1s, z = 1, T = 25 1C. Id ¼ zFAc�


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Mz+ (lower limit, slow chemical kinetics) and the ion transfer undertotal equilibrium conditions (upper limit, fast chemical kinetics).The position of this sigmoid is a function of the K-value, since itdetermines the initial concentrations of each species and so thevalue of current obtained (the smaller the apparent equilibriumconstant, the higher the current measured).

The effect of K is shown in Fig. 6B and C, where the currentdensity-time curves are plotted for two different values of k andseveral apparent equilibrium constants (indicated on thegraphs). As can be observed, the decrease of K leads to highervalues of the current density, according to the increase of theinitial concentration of the free species. Moreover, for a fixedvalue of K, the current density takes larger values as k increases,since the chemical reaction is able to provide a larger amountof species Mz+. Note that the plots I1

lim/Id vs. t are constantswhen K-0 or K-N, since under these conditions there ispractically a single species present in the aqueous mediumand the chronoamperogram follows a Cottrelian behaviour.12

Thus, I1lim/Id E 1 for very small K-values (cM* = c*) and

I1lim/Id E 0 for extremely high K-values (cM* = 0). On the other

hand, deviations from the Cottrell equation are observed forintermediate values of K (Fig. 6B and C) and k(Fig. 6A).

3.2. Total equilibrium conditions: cyclic voltammetry (CV)and square wave voltammetry (SWV)

As discussed in Section 2.2, under total equilibrium conditions(very fast kinetics), an analytical equation can be obtained for anyvoltammetric technique (eqn (47)), such as cyclic voltammetry (CV)and square wave voltammetry (SWV). In Fig. 7A and B, theinfluence of the scan rate (n) in CV and of the frequency ( f ) inSWV is shown. Unlike the cases in Fig. 1 and 2 (under theinfluence of the chemical kinetics), the time scale of the experi-ment has no influence on the shape of the voltammograms. Theonly effect of n and f is the increase of the peak current whereas theposition and shape of the signal are not altered. Note that thiseffect is also observed when there are no chemical reactionscoupled to the ion transfer processes,12,13,47 such that, as expected,the variation of the response with the experimental time-scalecannot be used to detect the occurrence of the chemical reaction.

The presence of the coupled chemical process can be probedby varying the bulk concentration of ligand cL*. In Fig. 7C andD, one can observe that the only influence of the cL*-valueon the CV and SWV-voltammograms under total equilibriumconditions is the shift of the voltammograms. For Df004 0 (i.e.the complex is less lipophilic than the free species) the shifttakes place towards more positive potentials as cL* is increased,whereas the shape of the response is not affected (compared withthe results shown in Fig. 4A). This is a consequence of the very fastkinetics of the interconversion between Mz+ and MLz+, such thatthe presence of ligand only influences the value of the apparentequilibrium constant K. Obviously, the shift of the voltammogramswould take place towards less positive potentials if MLz+ were themost lipophilic ionic species (Df00 o 0, not shown) and theposition of the signal would remain unaffected if there were nocoupled chemical reactions. Therefore, the variation of the signalwith cL* is a direct and straightforward way to detect the presenceof the chemical process as well as to distinguish the degree oflipophilicity of the two ionic species.

Variable cL*-experiments can also be performed to quantifythe value of the real equilibrium constant Kc. The shift ofthe voltammograms can be quantified through the values ofthe peak potential of the forward peak in CV (ECV

p,forward) and thepeak potential in SWV (ESWV

peak). The latter coincides with the half-wave potential under equilibrium conditions48 (eqn (45)),whereas the forward peak potential in CV differs from this byE28.5 mV at 25 1C for z = 1:12

ESWVpeak �E1=2;Mzþ ¼Df00 �RT

zFln

KccL�þe

zFRT

Df00

1þKccL�

0@

1A

ECVp;forward�E1=2;Mzþ ¼Df00 �RT

zFln

KccL�þe

zFRT

Df00

1þKccL�

0@

1Aþ1:109RT

zF

9>>>>>>>=>>>>>>>;

(61)

Fig. 6 Effect of the chemical kinetics and the equilibrium constant on thechronoamperometric response (eqn (23), (36) and (44)). Df00 = 200 mV,g2 = 0.001, E � Dw

of00Mzþ ¼ 200 mV, z = 1, T = 25 1C. Id ¼ zFAc�


p.

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with E1/2,Mz+ being defined by eqn (59). By fitting the variationsof the peak potentials with the ligand concentration (see Fig. 7Eand F) one can determine Kc and the difference between thehalf-wave potentials (which coincides in this case with thedifference between the formal transfer potentials). Note thataccording to eqn (61) the influence of the coupled chemicalreaction on the position of the signal depends on the charge ofthe free and complexed ions (z).

Finally, the influence of the charge of the ionic species isanalyzed in Fig. 8. Fig. 8A shows that the forward peak currentin CV is affected by z, such that ICV

p,forward is proportional to |z|3/2,as described for single charge transfer processes (the Randles–Sevckic equation12). Thus, the ratio between the slopes ofthe two plots depicted in Fig. 8A is 23/2 (=2.8). In addition,the peak-to-peak separation for z = 2 (E29 mV) is half of thatcorresponding to z = 1 (E58 mV), as it is also well-known forsimple charge transfer processes.12

Regarding the effect in SWV, one can observe that the peakcurrent increases with the ion charge z (Fig. 8B). Indeed, thepeak current is a function of the product |z|Esw (with Esw beingthe square wave amplitude), in a such way that for a given|z|Esw-value the same value of |ISWV

peak/z| is obtained47 (see greylines in Fig. 8B). The half-peak width (W1/2) of the SWV-curveis also a function of the charge of the ions (Fig. 8C). TheseW1/2-values have been calculated from the following equationreported for a simple ion transfer49 that, as expected fromeqn (53), also describes the ACDT mechanism under totalequilibrium conditions:

W1=2 ¼RT

jzjF ln1þ e2ZSW þ 4eZSW þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2ZSW þ 4eZSWð Þ2�4e2ZSW

q1þ e2ZSW þ 4eZSW �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ e2ZSW þ 4eZSWð Þ2�4e2ZSW

q264

375

(62)

Fig. 7 Influence of the scan rate (A), frequency (B) and ligand concentration (C–F) on the CV and SWV curves (eqn (52) and (53)), forward peak potentialin CV and peak potential in SWV under total equilibrium conditions. A = 12.5 cm2, Df00 = 200 mV, g2 = Do/Dw = 0.001, z = 1, T = 25 1C, DE = 0.01 mV,Esw = 25 mV, Es = 2 mV. (A and B) c* = 1 mM, Kc = 10 M�1, cL* = 0.2 M. (C and D) c* = 0.1 mM, Kc = 10 M�1, n = 20 mV s�1, f = 10 Hz. (E and F) c* = 0.1 mM,n = 20 mV s�1, f = 10 Hz. E1/2,Mz+ is given by eqn (59).

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where

Zsw ¼jzjFRT

Esw (63)

Therefore, the half-peak width is a function of |z| and |z|Esw

such that equal |z|Esw values lead to the same value of |z|W1/247

(see grey lines in Fig. 8C). As can be observed in Fig. 8C,W1/2(z = 1)/W1/2(z = 2) E 2 when the value of the amplitude issmall but this ratio tends to unity as Esw takes higher values.

4. Conclusions

Analytical explicit equations have been reported for the responsein different voltammetric techniques of two ionic species thatinterconvert through a chemical reaction and both can betransferred from water to an organic solution, a solvent polymericmembrane or ionic liquid (the ACDT mechanism). Rigorous andapproximate explicit solutions are presented for normal pulsevoltammetry (NPV), derivative voltammetry (dNPV) and chrono-amperometry including the effects of the chemical kinetics andalso of the likely different diffusion coefficients in each phase.Moreover, a general expression has also been obtained for anyvoltammetric technique when the chemical kinetics is very fast(chemical equilibrium conditions).

The above equations enable us to study the variation of thefeatures of the electrochemical response with the time scale ofthe experiment and with the concentration of the ligand for anyvalue of the difference between the formal transfer potentials ofthe free and complexed species. Different criteria and procedureshave been proposed for the detection of the chemical processcoupled to the ion transfers, as well as for the determination ofthe equilibrium and rate constants and the lipophilic characterand the charge of both ionic species. Thus, the theory reportedhere enables the rigorous analysis of the dynamic aspects ofchemical speciation in voltammetric studies as well as a deeperunderstanding of the voltammetric response and the differentphysicochemical processes behind it (heterogeneous chargetransfer, mass transport and chemical interconversion).

Appendix ASymbols

A Interfacial areac* Initial total concentration in the aqueous solution

(c* = cM* +cML*)cai Concentration profile of species i (i = M, ML, L) in

the phase a (a = w, o) under the application of aconstant potential

ci* Bulk concentration of species i (i = M, ML, L)cw, j

T (0) Total surface concentration in the aqueous solutionassociated in the j-th potential pulse applied in multi-pulse techniques, under the assumption of totalequilibrium conditions (cw, j

T (0) = cw, jM (0) + cw, j

ML(0))Da

i Diffusion coefficient of species i (i = M, ML) in phasea (a = w, o)

Da Diffusion coefficient in phase a (a = w, o)E Applied potential in normal pulse voltammetry

(NPV) and chronoamperometry

Fig. 8 (A) Influence of the charge z of the ionic species on the forwardpeak current in CV and (B and C) the peak current and half-peak widthin SWV under total equilibrium conditions. A = 12.5 cm2, Df00 = 200 mV,g2 = Do/Dw = 0.001, T = 25 1C, c* = 1 mM, K = 2, DE = 0.01 mV, Es = 2 mV,f = 10 Hz.

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Ej Applied potential in the j-th potential pulse inmultipulse techniques

E11/2 Half-wave potential of the first wave in NPV for

|Df00| 4 120 mVE1/2,i Half-wave potential of the simple ion transfer of

species i (i = M, ML)Ete

1/2 Half-wave potential under the assumption of totalequilibrium conditions

Ete,ADT1/2 Half-wave potential under the assumption of total

equilibrium conditions when Mz+ is the only speciestransferred

Ehalf Potential at which the NPV-current takes half of itsmaximum value (I/Id = 0.5)

ECVp,forward Forward peak potential in cyclic voltammetry (CV)

ESWVpeak Peak potential in square wave voltammetry (SWV)

Es Potential step of the staircase in SWVEsw Square wave amplitude in SWVf SW frequencyF Faraday constantI Current response in NPVISCV Current response in SCVId Diffusion-limited current of a single transfer process

of an ion at concentration c* with charge zIkss Solution for the current response in NPV under the

kss approachIADTkss Solution for the current response in NPV under the kss

approach when Mz+ is the only species transferredIDf00=0 Solution for the NPV response when Mz+ and MLz+

have the same lipophilicityI1lim Limiting current of the first wave obtained in NPV

for |Df00| 4 120 mVICVp,forward Forward peak current in CV

ISWVpeak Peak current in SWV

Irig Rigorous solution for the current response in NPVI ADT

rig Rigorous solution for the current response in NPVwhen Mz+ is the only species transferred

Ite Solution for the current response in NPV under theassumption of total equilibrium conditions

I ADTte Solution for the current response in NPV under the

assumption of total equilibrium conditions whenMz+ is the only species transferred

Ite, j Current associated with the j-th potential pulseapplied in multipulse techniques, under theassumption of total equilibrium conditions

k10 Forward rate constant of the complexation reaction

(M�1 s�1)k1 Effective forward rate constant of the complexation

reaction (s�1) (k1 = k10cL*)

k2 Backward rate constant of the complexation reaction(s�1)

Kc Equilibrium constant based on concentrations

(M�1) Kc ¼cML

�

cM�cL�¼ k01

k2

� �K Apparent equilibrium constant of complexation

K ¼ cML�

cM�¼ k01cL

�

k2

� �

R Molar gas constantt Electrolysis timeT Absolute temperature (K)n Scan rate in CV and SCVW1/2 Half-peak width of the square wave voltammogramx Distance to the interfacez Electrical charge of species Mz+ and

MLz+ dr thickness of the linear reaction layerDE Pulse step in NPV and CV and SCVDISWV Current response in SWV

Dwof

00i Formal ion transfer potential of species i (i = Mz+, MLz+)

Df00 ¼ Dwof

00MLzþ � Dw

of00Mzþ

� �Difference between the formal transfer potentials ofspecies Mz+ and MLz+

k Effective chemical rate constant (k = k1 + k2)x Sum of the concentrations of the free and complexed

ions in the aqueous solution as a function of timeand distance to the interface (x = cw

M +cwML)

t Duration of each potential applied in NPV and inSWV

tj Duration of the j-th potential pulse applied inmultipulse techniques

f Perturbation of the chemical equilibrium (f =(Kcw

M �cwML)e(k1+k2)t )

fss Perturbation of the chemical equilibrium underkinetic steady state treatment (fss = Kcw

M � cwML)

cCV Normalized SCV signal cSCV ¼ISCV

zFAc�ffiffiffiffiffiffiffiffiffiaDw

p� �

cSWV Normalized SWV signal cSWV ¼DISWV

ffiffiffitp

zFAc�ffiffiffiffiffiffiffiDw

p� �

w Effective chemical kinetics (w = kt)

Appendix B

In order to solve the equation system (14), the followingchanges of variables are made

sa ¼x

2ffiffiffiffiffiffiffiffiDatp ; a ¼ w; o

w ¼ kt

8<: (B1)

with xo0 for the aqueous phase (w) and x 4 0 for the organicsolution (o). Thus, the system (14) and the conditions(15)–(20) become

@2f@sw2

þ 2sw@f@sw� 4w

@f@w¼ 0

@2x@sw2

þ 2sw@x@sw� 4w

@x@w¼ 0

@2coM@so2

þ 2so@coM@so� 4w

@coM@w¼ 0

@2coML

@so2þ 2so

@coML

@so� 4w

@coML

@w¼ 0

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(B2)

sw - �N} f = 0; x = c* (B3)

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so - N} coM = co

ML = 0 (B4)

sw = so = 0}

@x@sw

� �sw¼0þe�w @f

@sw

� �sw¼0¼ ð1þ KÞg @coM

@so

� �so¼0

(B5)

K@x@sw

� �sw¼0�e�w @f

@sw

� �sw¼0¼ ð1þ KÞg @coML

@so

� �so¼0

(B6)

coMð0Þ ¼xð0Þ þ e�wfð0Þ

1þ KeZ1 (B7)

coMLð0Þ ¼Kxð0Þ � e�wfð0Þ

1þ KeZ2 (B8)

According to Koutecky’s method,40,41 the solutions f(x,t), x(x,t),co

M(x,t) and coML(x,t) are given by

fðx; tÞ ¼ f sw; wð Þ ¼P1j¼0

aj swð Þw j

xðx; tÞ ¼ x sw; wð Þ ¼P1j¼0

bj swð Þw j

coMðx; tÞ ¼ coM so; wð Þ ¼P1j¼0

sj soð Þw j

coMLðx; tÞ ¼ coML so; wð Þ ¼P1j¼0

dj soð Þw j

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(B9)

that introduced in (B2) lead to the following homogeneousdifferential equation system in a single variable sa (a = w, o):

aj00swð Þ þ 2swaj

0swð Þ � 4jaj swð Þ ¼ 0

bj00swð Þ þ 2swbj

0swð Þ � 4jbj swð Þ ¼ 0

sj00soð Þ þ 2sosj

0soð Þ � 4jsj soð Þ ¼ 0

dj00soð Þ þ 2sodj

0soð Þ � 4jdj soð Þ ¼ 0

8>>>>>><>>>>>>:

(B10)

The functions which are solutions of the system (B10) have thefollowing form:

aj swð Þ ¼ ajc2j swð Þ þaj sw ! �1ð Þ

limsw!�1

L2jL2j

bj swð Þ ¼ bjc2j swð Þ þbj sw ! �1ð Þ

limsw!�1

L2jL2j

sj soð Þ ¼ cjc2j soð Þ þsj so !1ð Þ

limso!1

L2jL2j

dj soð Þ ¼ djc2j soð Þ þdj so !1ð Þ

limso!1

L2jL2j

9>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>;

8j (B11)

where aj, bj, cj and dj are constants that are determined byapplication of the boundary value problem, L2j are numericseries of s potencies and ci are Koutecky’s functions40,41,50

The initial and bulk conditions (eqn (B3) and (B4))establish that

b0ð�1Þ ¼ c�; bj4 0ð�1Þ ¼ 0;

ajð�1Þ ¼ 0

sjð1Þ ¼ 0

djð1Þ ¼ 0

9>>>=>>>;8j (B12)

in a such way that by taking into account that the function ew

can be expressed as

ew ¼X1k¼0

wk

k!(B13)

the surface conditions (eqn (B5)–(B8)) are transformed asfollows:

b0 ¼ a01þ K 1þ geZ1ð Þ þ geZ2

Kg eZ2 � eZ1ð Þ ; a0 ¼Kc�g eZ2 � eZ1ð Þ

ð1þ KÞð1þ geZ1Þ 1þ geZ2ð Þ

for j4 0

aj ¼p0a0

j!p2j� KeZ1 þ eZ2 þ geZ1eZ2ð1þ KÞ

eZ2 � eZ1

Pji¼1

p2ibi

ð j � iÞ!p2j

aj ¼a0

j!þ 1þ geZ1 þ K 1þ geZ2ð Þ

g eZ2 � eZ1ð ÞPji¼1

bi

ð j � iÞ!

8>>>><>>>>:

(B14)

According to eqn (22), for the final expression of the current weare only interested in coefficients bj included in functions bj

(eqn (B11)) corresponding to the variable x (eqn (B9)), which areobtained by equating the last two equations in (B14):

for j4 0:

bj ¼ �g eZ2 � eZ1ð Þ

1þ geZ1ð Þð1þ KÞ 1þ geZ2ð Þa0

j!1� p0

p2j

� ��

þXj�1i¼1

bi

ð j � iÞ!1þ geZ1 þ K 1þ geZ2ð Þ

g eZ2 � eZ1ð Þ

�

þ KeZ1 þ eZ2 þ geZ1eZ2ð1þ KÞeZ2 � eZ1

p2i

p2j

��(B15)

By introducing the following definition for convenience:

lj ¼ �bjp2jð1þ KÞ 1þ geZ1ð Þ 1þ geZ2ð Þ

a0g eZ2 � eZ1ð Þp0; j4 0 (B16)

with pz being defined in eqn (30), the expression for current (23)is finally obtained.

Acknowledgements

The authors greatly appreciate financial support provided bythe Fundacion Seneca de la Region de Murcia under theProjects 19887/GERM/15 and 18968/JLI/13. EL also thanks theMinisterio de Economia y Competitividad for the fellowshipawarded ‘‘Juan de la Cierva – Incorporacion 2015’’ and JMO forthe grant received under the Project CTQ2012-36700, co-fundedby the European Regional Development Fund.

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Eduardo Laborda, Jose´ Manuel Olmos and A´ngela Molina*