Lability of complexes in steady-state finite planar diffusion

Journal of

www.elsevier.com/locate/jelechem

Journal of Electroanalytical Chemistry 588 (2006) 303–313

ElectroanalyticalChemistry

Lability of complexes in steady-state finite planar diffusion

Jose Salvador a,*, Jaume Puy a, Joan Cecılia b, Josep Galceran a

a Departament de Quımica, Universitat de Lleida, Rovira Roure 191, 25198 Lleida, Spainb Departament de Matematica, Universitat de Lleida, Rovira Roure 191, 25198 Lleida, Spain

Received 26 May 2005; received in revised form 20 December 2005; accepted 9 January 2006Available online 17 February 2006

Abstract

The analytical solution of the reaction-diffusion problem of a species forming a complex (with any association and dissociation rateconstants) in solution and disappearing at an active planar surface is presented for a finite diffusion region where the system reachessteady state under diffusion limited conditions. This problem arises in a number of fields ranging from electroanalytical techniques orin situ trace metal sensors to biouptake by organisms. The analysis of the solution allows the introduction of the degree of lability,n, aimed at quantifying rigorously the contribution of the complexes to the metal flux. The differences between the lability degree, n,and the lability parameter, L, used in the statement of lability criteria are shown. A particular expression for the reaction layer thicknessand the lability criterion for the set of conditions of this work are also reported. Finally, when the diffusion layer thickness (such as thegel thickness in some DGT set-ups) can be changed, the lability degree enables the tuning of the relevance of the kinetic contribution ofthe complex to the flux.� 2006 Elsevier B.V. All rights reserved.

Keywords: Speciation; Trace metal analysis; Bioavailability; Lability

1. Introduction

The understanding of the environmental impact of metalcompounds is a subject that has received strong attentionin recent years [1–3]. As it has been pointed out, the seriouspollution hazard of heavy metals demands reliable analyt-ical techniques able to measure the flux of metal thatreaches micro-organisms, algae, plants, and living organ-isms present in the media [4]. The analysis of this flux iscalled dynamic speciation since it depends on the time scaleand on the kinetic parameters of the undergoing processes,as well as on the spatial scale and geometry of the sensor ormicro-organism.

A concept that plays a key role in this issue is lability. Itis used to quantify the ability of the complexes to contrib-ute to the metal flux. In fact, in a natural sample, a greatnumber of ligands such as particles, colloids, polysaccha-

0022-0728/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jelechem.2006.01.005

* Corresponding author. Tel.: +34 973 702828; fax: +34 973 238264.E-mail address: [email protected] (J. Salvador).

rides, proteins, humic and fulvic acids are present. Theinteraction of the metal with these ligands can reduce themetal flux received by a living organism via reduction ofthe mobility of the metal or via a kinetic control of the dis-sociation processes [5,6].

The importance of the lability criteria has been recogni-sed since long time ago. Accordingly, a great effort hasbeen devoted to provide these criteria for different condi-tions of interest in terms of the parameters that characterisethe system [7–15]. In this pursuit, the so-called reactionlayer approximation has become a very useful tool. Intro-duced by Brdicka and Wiesner in the context of the influ-ence of the complex electroinactive species on theelectrodic reduction of a free metal ion [16,17], the reactionlayer approximation is based on the assumption of steady-state conditions and a constant complex concentration inthe system. In this way, the metal transport equationbecomes uncoupled from the transport equation for thecomplex. In the simplest application of the reaction layerapproximation, the complex concentration is assumed tobe the bulk complex concentration. This approximation

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304 J. Salvador et al. / Journal of Electroanalytical Chemistry 588 (2006) 303–313

allows to assess Jkin, a characteristic parameter of the sys-tem defined as the hypothetical ability of the dissociationprocess in contributing to the metal flux in absence of dif-fusion limitation for the complex. The lability criterion isjust based on the comparison of this ability of the dissoci-ation process with the ability of maximum supply ofcomplex to the reaction layer, which in steady-state condi-tions is the maximum diffusional flux of the complex. WhenJkin is greater than the maximum supply of complex, diffu-sion is rate limiting and we say that the system is labile;while if Jkin is less than the diffusional supply we are in con-ditions of kinetic control and we say that the system is par-tially labile or non-labile.

The lability criterion is thus an inequality used to predictif a system is non-labile or labile, but this criterion cannotquantitatively describe the lability degree of the system forpartially labile cases.

This paper focuses on systems with planar diffusion ina finite domain. This is the case of relevant techniquesdeployed ‘‘in situ’’ such as diffusion gradient in thin films(DGT) [18–23] and permeation liquid membrane (PLM)[24–28] which have emerged in the last years as powerfultechniques for the dynamic speciation of metals in differ-ent compartments of the natural media. As the exposuretime of these sensors is usually longer than the effectivetime of the transient regime, steady-state conditions pre-vail. Further consideration of excess of ligand conditionsallows us to provide analytical solutions for the concen-tration profiles and the metal flux. It is the aim of thiswork: (i) to develop analytical expressions for quantify-ing the contribution of the complexes to the metal fluxunder the particular conditions used in this work, (ii)to work out a parameter which, on the basis of the rig-orous solution, quantifies the degree of lability, (iii) todevelop suitable expressions for the reaction layer thick-ness in order to report the lability criterion for the pres-ent case, (iv) to analyse the effect of the change of thethickness of the diffusion layer and provide an analyticalexpression for the thickness that allows measuring a pre-fixed percentage of the contribution of the complex tothe metal flux.

2. Mathematical formulation

Let us consider, in solution, the complexation of a metalM with a ligand L according to the scheme

Mþ L �ka

kd

ML ð1Þ

and let us assume that the ligand is present in the system ina great excess with respect to the metal so that cL ¼ c�L(from now on, c�i labels the concentration value of speciesi at the bulk solution). The corresponding equilibrium con-ditions then read

K 0 ¼ k0akd

¼ c�ML

c�Mð2Þ

where K 0 ¼ Kc�L, k0a ¼ kac�L and K, ka and kd are respectivelythe equilibrium constant, and the association and dissocia-tion kinetic constants of the complexation process.

When diffusion towards a stationary planar surface isthe only relevant transport mechanism, for steady-stateconditions we can write

DM

d2cM

dx2þ kdcML � k0acM ¼ 0 ð3Þ

DML

d2cML

dx2þ k0acM � kdcML ¼ 0 ð4Þ

Notice that in ligand excess conditions, the kinetics ofinterconversion of M and ML are pseudo first-order andthe system (3) and (4) becomes linear.

Denoting

e ¼ DML=DM ð5Þand introducing a new variable

z ¼ xffiffiffiffiffiffiffiDM

p ð6Þ

the system of Eqs. (3) and (4) becomes

d2cM

dz2¼ k0acM � kdcML ð7Þ

d2cML

dz2¼ kd

ecML �

k0ae

cM ð8Þ

As usual, when the metal is the only one species consumedat the limiting surface, the boundary value problem in dif-fusion limited conditions is given by

z ¼ 0 cM ¼ 0;dcML

dz

� �z¼0

¼ 0 ð9Þ

z ¼ z� ¼ gffiffiffiffiffiffiffiDM

p cM ¼ c�M; cML ¼ c�ML ð10Þ

where g is the thickness of the diffusion domain. For in-stance, in the case of DGT, it would be the thickness ofthe gel layer, if one can assume that some kind of stirringrestores the bulk concentration at the diffusion region edge(x = g). As it has been pointed out [13], when g is of the or-der of the concentration polarisation thickness in the exter-nal solution phase, the boundary condition (10) is notsuitable.

3. Rigorous solution

The rigorous solution of the system (7)–(10) is outlinedin Appendix A. The resulting concentration profiles canbe written as:

cMðzÞc�M¼

zþ eK 0ffiffinp tanh½ ffiffiffinp z�� þ sinh½

ffiffinpðz�z�Þ�

cosh½ffiffinp

z��

h iz� þ eK 0ffiffi

np tanh½ ffiffiffinp z��

ð11Þ

cMLðzÞc�ML

¼z� 1ffiffi

np sinh½

ffiffinpðz�z�Þ�

cosh½ffiffinp

z�� eK 0 tanh½ffiffiffinp

z��h i

z� þ eK 0ffiffinp tanh½ ffiffiffinp z��

ð12Þ

where

J. Salvador et al. / Journal of Electroanalytical Chemistry 588 (2006) 303–313 305

n ¼ kd

eþ k0a ¼

kd 1þ eK 0ð Þe

ð13Þ

Eqs. (11) and (12) indicate that the steady-state metal andcomplex concentration profiles are not linear for the gen-eral kinetic case under excess ligand conditions.

The metal flux at x = 0 can simply be calculated fromthe gradient of the free metal concentration profile, givenin (11), as

J M ¼ DM

dcM

dx

� �x¼0

¼ffiffiffiffiffiffiffiDM

p dcM

dz

� �z¼0

¼ffiffiffiffiffiffiffiDM

pc�Mð1þ eK 0Þ

z� þ eK 0ffiffinp tanh½ ffiffiffinp z��

ð14Þ

3.1. Fully labile system

The fully labile case merits some specific comments.When kd !1 ðk0a ¼ K 0kdÞ, we have n!1, as Eq. (13)shows. Taking into account that tanh tends to a constantfor an increasing argument, cMðzÞ=c�M and cMLðzÞ=c�ML

given in (11) and (12) become

cMðxÞc�M

¼ cMLðxÞc�ML

¼ xg

ð15Þ

which means that equilibrium conditions at any spatialpoint are reached and that the steady-state metal and com-plex concentration profiles of a labile system in excess ofligand conditions are linear.

Likewise, applying the same limit, kd !1 ðk0a ¼ K 0kdÞ,the metal flux given by (14) reduces to

J M ¼ J labile ¼DMc�Mð1þ eK 0Þ

g¼ DM

c�Mgþ DML

c�ML

gð16Þ

Eq. (16) indicates that in a labile system, the metal flux isonly limited by diffusion and that, under steady-state con-ditions, the diffusion of the total metal towards the limitingsurface is just the addition of the independent maximumdiffusion of the free metal and of the complex. Recallingthe expressions for the metal and complex concentrationprofiles given in (15), Eq. (16) can be rewritten as

J M ¼ J labile ¼ DM

dcM

dx

� �x¼0

þ DML

dcML

dx

� �x¼0

ð17Þ

Notice that (16) and (17) have been obtained as the limitkd!1 of (14), which yields the metal flux in the generalkinetic formulation of the problem. It is worth highlighting

that although the general problem states dcMLðxÞdx

� �x¼0¼ 0, as

indicated in (9), after the limit kd!1, the gradient of cML

at x = 0 is not null, dcMLðxÞdx

� �x¼06¼ 0 (see (15)), and JM defi-

ned as J M ¼ dcMðxÞdx

� �x¼0

for the general kinetic case becomes

J M ¼ DMdcMðxÞ

dx

� �x¼0þ DML

dcMLðxÞdx

� �x¼0

, with the second term

differing from zero. This analytical result here obtainedsupports the use of (17) for the metal flux in direct mathe-matical formulations of fully labile systems [29–32].

4. Lability of the complex

Let us obtain the contribution of the complex to themetal flux, which we label Jcomplex [15]. It can be obtainedby just subtracting from the metal flux (JM) the term thatcorresponds to the diffusional transport of the free M pres-ent in the system, Jfree, totally uncoupled from complexa-tion. Thus,

J complex ¼ J M � J free ð18Þwhere for the particular conditions of the present system(diffusion in a finite domain),

J free ¼ DM

c�Mg

ð19Þ

The maximum contribution of the complex arises when thesystem is labile. Thus, in order to quantify the contributionof the complex to the metal flux, we can define a lability de-gree, n, as the fraction of the present contribution with re-spect to the maximum contribution of the complex to themetal flux [5,12,33,34]:

n ¼ J M � J free

J labile � J free

ð20Þ

Recalling the expression of Jlabile given in Eq. (16), n re-writes as

n ¼J M � DM

c�M

g

DMLc�

ML

g

¼ J complex

J dif;ML

ð21Þ

The denominator of Eq. (21) indicates that the maximumcontribution of the complexes to the metal flux is indepen-dent of the kinetics of complexation, but is limited by dif-fusion. This is just the situation in labile systems.

By means of Eq. (14), Eq. (21) rewrites:

n ¼ z� � tanh½ ffiffiffinp z��= ffiffiffinp

z� þ eK 0 tanh½ ffiffiffinp z��= ffiffiffinp ð22Þ

where n has been defined in (13). In terms of the degree oflability, the system tends to be labile as n becomes close to1. Notice that Eq. (22) predicts rigorously the percentagewith which the complex contributes to the metal flux (withrespect to its maximum contribution) once the kinetic con-stants and the spatial dimensions of the sensor are known.However, the evaluation of the metal flux requires both thelability degree and the mobility of the system embedded inJfree and Jdif,ML

J M ¼ J free þ nJ dif ;ML ð23ÞAs some in situ speciation techniques like PLM or DGT al-low working with sensors of different thickness of the sens-ing layer, Eq. (21) can also be used to determine thethickness of the diffusion layer of the sensor for a given per-centage of the kinetic contribution of the complex. In thisway we can exclude the contribution of the complex or in-clude it in the sensor answer (see Section 9 below).

The degree of lability can also be easily related with thecharacteristics of the complex concentration profile. Eq.


(A-8), which corresponds to the integration of the additionof Eq. (7) and Eq. (8) multiplied by e, can be written as

cMðzÞ þ ecMLðzÞ ¼ ec0ML þ ½c�M þ eðc�ML � c0

MLÞ�zz�

ð24Þ

where c0ML labels the complex concentration at the elec-

trode surface (where ‘‘electrode’’ denotes, in general, theactive surface where M disappears). Recalling that

dcML

dz

� �z¼0¼ 0, the derivative of (24) becomes

d

dz½cMðzÞ þ ecMLðzÞ�

z¼0

¼ dcM

dz

� �z¼0

¼ c�M þ eðc�ML � c0MLÞ

z�

ð25Þ

and the metal flux,

J M ¼DMc�Mð1þ eK 0nÞ

g¼DM

c�MgþDML

c�ML

g1� c0

ML

c�ML

� �ð26Þ

The first term of the r.h.s. of Eq. (26) can be identified withthe metal flux when only metal, at a concentration given byc�M, is present in the system. It is denoted as Jfree. The sec-ond term indicates the increase of the flux due to the pres-ence of complex species which is denoted as Jcomplex. Using(26) in (21),

n � 1� c0ML

c�ML

� �ð27Þ

a result previously obtained in the context of the sphericaldiffusion [12].

5. Limiting cases

5.1. The limiting case eK 0 � 1 and g� l1

A very intuitive description of the diffusion-reactionprocess relies upon the concept of reaction layer that willbe developed in the next section. For planar or sphericalsemi-infinite diffusion, the thickness of the reaction layer(which we label l1 in this work) can be shown to be[12,35]:

l1 ¼ffiffiffiffiffiffiffiDM

k0a

sð28Þ

In most configurations, g will be much larger than l1 (forinstance, in Ref. [20], the thickness of the gel layer of theDGT device was varied from 0.16 mm to 2 mm while a typ-ical l1 is of the order of lm).

Furthermore, we are mainly interested in cases wherethe contribution of the complex is large, this implyingeK 0 � 1. Otherwise the metal flux would not appreciablydiffer from Jfree, as can be deduced from Eq. (26). So,one can approximate n � k0a and

ffiffiffinp �

ffiffiffiffiffiDMp

l1 . Using bothapproximations, tanh½ ffiffiffinp z�� 1. Thus, from Eq. (14), weobtain an approximation for the flux:

J M �DMc�Mð1þ eK 0Þ

g þ eK 0l1ð29Þ

Substitution of this expression into (21), yields a very sim-ple expression for the lability degree

n � gg þ eK 0l1

ð30Þ

With the additional condition g� eK0l1, Eq. (30) becomes

n � kdg

ek01=2a D1=2

M

ð31Þ

which reverts to the well known Davison condition [7,36]whenever e = 1. Recall that the set of conditions for a rig-orous use of the Davison condition in the present case are:eK 0 � 1, g� eK 0l1 (more restrictive) and g� l1.

5.2. Immobile complex, DML = 0

The case of an immobile complex, DML = 0, alsodeserves specific mention, because it could be a goodapproach when macromolecular ligands are present or incases where a layer with complexing sites in its structureis covering the limiting surface: sensors coated with a gellayer, biofilms, etc. In steady-state conditions, when ametal is complexed to a immobile ligand, we have e = 0,n!1, and JM, cMðzÞ=c�M and cMLðzÞ=c�ML given respec-tively by (14), (11) and (12) reduce to (15) and to

J M ¼DMc�M

g¼ J labile ¼ J free ð32Þ

Thus, we are facing a case where the profiles and the metalflux indicate that we are in labile conditions, independentlyof the kinetic constants value since: (i) the concentrationprofiles of M and ML are related everywhere by theequilibrium constant K 0, as deduced from (15) or alterna-tively by cancelling the diffusion term in (4); (ii) cM andcML drop to zero at the electrode surface; and (iii) JM

reduces to Jlabile. However, this JM-value also coincideswith Jfree, in agreement with the fact that the maximum dif-fusive flux of ML, which equals the maximum kinetic con-tribution, drops to zero for immobile complexes.

The result presented above indicates that in steady-stateconditions, the presence of complexing sites in the gel layerdoes not modify the flux respect to the absence of this com-plexing sites. The situation would be different in the tran-sient regime since then the presence of complexing sites inthe gel layer: (i) increases the metal flux since these sitesact as a reservoir of M which is released as the metal con-centration profile goes into the gel layer, (ii) increases thetime to reach steady-state conditions.

6. The reaction layer approximation and the lability criterion

The reaction layer approximation was historically intro-duced to compute the kinetic currents in systems with excessof ligand. When the complex is the relevant metal species(K 0 > 1), metal consumption at a limiting surface is suppliedby chemical dissociation in a quasi-steady state situation(after a time t > 1

k0a) while the complex concentration

remains almost unchanged. Mathematically, the reaction


layer approximation relies on assuming steady-state condi-tions and a flat complex concentration profile. In this way,the diffusion equation for the metal becomes uncoupledfrom the transport equation of ML and can be easily solvedgiving rise to an approximation for the metal flux.

The reaction layer approximation has been used to for-mulate lability criteria, which provide a simple way of deter-mining which is the process limiting the metal flux: either thedissociation of the complex or the supplying of complexfrom the bulk solution. The maximum ability of the kineticprocess to produce metal flux, i.e., the hypothetical maxi-mum kinetic flux in absence of diffusion limitation for thecomplex, is evaluated by using the reaction layer approxima-tion and the condition cMLðxÞ ¼ c�ML. The resulting metalflux has been denoted as Jkin in the literature. The maximumsupply of complex is the maximum diffusional flux, to which,for transient cases, an extra contribution coming from thedecrease of the complex concentration at each volumeelement of the system has to be added. If Jkin� Jdif, thetransport of complex limits the metal flux available at theactive surface and we say that the system is labile. Inthe opposite case, when Jkin� Jdif, we have a metal fluxkinetically limited and we say that the system is partiallylabile or non-labile. This comparison is classically assessedby means of the evaluation of the lability parameter L. Then,the application of the lability criteria corresponds to L� 1for a labile complex and L� 1 for a non-labile one.

A key parameter in the evaluation of Jkin is the reactionlayer thickness, l, whose expression for one complex inplanar semi-infinite diffusion was derived by Kouteckyand Brdicka [37]. It can be seen that, in planar geometry

and semi-infinite diffusion, the expression l1 ¼ffiffiffiffiffiffiffiDM

kac�L

qstands for: (i) an operational parameter to evaluate Jkin

by means of J kin ¼ kdc�MLl1 [35], (ii) the effective thicknessof the disequilibration layer [38] since J M ¼ DM

c�M

l1 and (iii)the average distance traveled by the metal ion before reas-sociation [39]. When changing the system under consider-ation (e.g. the reaction scheme, the geometry, etc.) orunder other conditions, these three concepts do not alwayscoincide in just one mathematical expression and we haveto decide which one of the 3 concepts (see page 346 inRef. [40]) is the one to be retained as ‘‘reaction layer’’.We adhere here to the operational definition [12,35], asexplained in (i), so that the reaction layer thickness is anoperational parameter (different for different scenarios)that yields the hypothetical maximum kinetic flux, i.e. with-out any limitation due to the diffusion of the complex.

We aim now at providing this parameter and the corre-sponding lability criterion for one complex that diffuses ina finite domain.

6.1. Expression for the thickness of the reaction layer under

finite planar diffusion

Let us consider

cMLðxÞ � c�ML; 8x ð33Þ

despite the fact that the metal concentration drops to zeroclose to the limiting surface. Using Eq. (33), we are evalu-ating the hypothetical metal flux in absence of diffusionlimitation of the complex. Eq. (3) becomes

d2cM

dx2¼ cM � c�Mðl1Þ2

ð34Þ

The integration of (34) with the boundary condition givenin (9) leads to

dcM

dx

� �x¼0

¼ c�Ml1

cothg

l1

� �ð35Þ

The operational definition [12,35] of the thickness of thereaction layer, l, is based on splitting the overall flux intotwo components,

J M ¼ J kin þ J free ¼ J kin þDMc�M

gð36Þ

whereDMc�

M

g is the steady-state metal flux in absence of com-plex. With this splitting, Jkin reflects the contribution of thecomplex to the metal flux. Notice also that this splitting isnot necessary in planar semi-infinite diffusion since the stea-dy-state metal flux of the free metal is zero. Writing Jkin as

J kin ¼ kdc�MLl ð37Þ

the combination of Eqs. (36), (37) and (35), leads to the fol-lowing expression for l

l ¼ l1 cothg

l1

� �� l1

g

ð38Þ

As stated above, we are adhering to an operational defini-tion of l which allows reproducing the metal flux as indi-cated in (36). Each change of the geometry of the sensor,boundary conditions, presence of other ligands, formationof complexes with other stoichiometric metal to ligand ra-tios, etc., that modifies the dependencies of JM on the char-acteristic parameters will also modify the expression for l.

Two limiting cases can also be considered. Wheng� l1, Eq. (38) yields

limg=l1!1

l ¼ l1 ¼ffiffiffiffiffiffiffiDM

k0a

sð39Þ

which is the well-known Koryta expression (28) for semi-infinite diffusion.

In the opposite limiting case of thin enough diffusionlayers, g� l1, using the asymptotic expansion of thehyperbolic cotangent for small arguments, one obtains:

limg=l1!0

l ¼ 0 ð40Þ

and further substitution into (37) leads to

limg=l1!0

J kin ¼ 0 ð41Þ

which means that all systems tend to be inert for thin en-ough diffusion layers.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

-5 -3 -1 1 3 5

108 J

/ (m

ol m

-2s-1

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

)1d,1log( / sk

0ML*ML

c

c

kinJ

MJ

kinJ f

0ML*ML

c

c

Fig. 1. Metal flux at the electrode surface, JM (continuous line), thekinetic flux, Jkin (short dashed line) and J/

kin (long dashed line), referredto the left ordinate axis and c0

ML=c�MLðDÞ, referred to the right ordinateaxis as functions of the dissociation kinetic constant log(kd/s�1).Parameters: c�T;M ¼ 0:1 mol m�3, c�T;L ¼ 3:0 mol m�3, K = 100 m3 mol�1,g = 8 · 10�4 m, DM = 10�9 m2 s�1, e = 0.1. In each point ka takes thevalue required to maintain a fixed K. The horizontal bullets on theright side indicate the metal flux for the labile case.


Using (38), the lability parameter reads

L ¼ J kin

J dif

¼ kdlDML

g ¼ 1

eK 0g

k01=2a

D1=2M

coth gk01=2

a

D1=2M

!� 1

" #ð42Þ

and the lability criterion becomes

L� 1 ð43Þfor a labile system.

For the usual case of g� l1, Eq. (42) reduces to

L ¼ geK 0l1

¼ k01=2a g

eK 0D1=2M

ð44Þ

which coincides with the lability parameter reported by Jan-sen et al. [36]. The derivation of the lability parameter byJansen et al. was obtained via substitution of t with g2=D(where D means the average diffusion coefficient of M andML) in the lability parameter obtained for planar semi-infi-nite diffusion and, as found in Eq. (44), this is a goodapproximation for the present case whenever g� l1.

Notice that Jkin, the hypothetical maximum contribu-tion of the complex to the metal flux in absence of diffusionlimitation for the complex, differs from Jcomplex, the actualcontribution of the complex to the metal flux. Only whenthere is no diffusion limitation, i.e., when the metal flux islimited by the kinetics of the dissociation Jkin will coincidewith Jcomplex. Otherwise, c0

ML < c�ML and Jkin� Jcomplex.This is seen in Fig. 1 which shows the metal flux JM com-puted from (14) vs. the kinetic dissociation constant of thecomplex, kd. For the data of the figure, Jfree is negligible, sothat JM and Jcomplex converge. Thus, the sigmoidal shapeexhibited by the metal flux, depicted in continuous line, isdue to the contribution, by dissociation, of the complex,as indicated by the decrease in the ML concentration atthe surface (also plotted in Fig. 1) and the concomitantincrease in JM.

Fig. 1 also depicts Jkin obtained applying (37). As can beseen, Jkin reproduces accurately JM for low kd-values (Jfree

is negligible for neK 0 � 1). However, Jkin� JM when thecML-profile diverges from the flat c�ML-value, i.e., for thepartially labile regime, as expected. The development ofthe complex concentration profile is just an indication ofthe incapacity of the diffusion process to keep the bulkcomplex concentration unaltered at the limiting surface.For this reason, the hypothetical metal flux in absence ofdiffusion limitation, Jkin, exceeds the actual metal flux.However, with the appropriate changes, we are now goingto see that the reaction layer approximation could still beused to approximate the contribution of the complex tothe metal flux.

6.2. The partially labile case

For the partially labile case, there is a non-flat concen-tration profile of ML, cMLðxÞ 6¼ c�ML, "x, and Eqs. (33)and (34) are not realistic to approach the contribution ofthe complex to the metal flux. Previous work [14,15] has

shown that the reaction layer approximation can still pro-vide accurate values for the metal flux by using ML con-centration at x = 0. Labelling this value as c0

ML, we canintegrate the differential Eq. (34) in the range where cML(x)is constant and approximately equal to c0

ML. This rangeapproximately spans the reaction layer. An implicit finalexpression for l is thus obtained, but simple algebra indi-cates that it reduces to (38) whenever l < g. The complexcontribution to the metal flux can then be computed,within the reaction layer approximation, by means of J/

kin

which is formally identical to Jkin, but computed by usingconcentrations at the reaction layer, c0

ML, instead of thebulk concentrations:

J/kin ¼ kdc0

MLl ð45ÞJ/

kin is expected to be an accurate approximation of Jcomplex

for all the kd range. This is seen in Fig. 1, which shows thatJ/

kin reproduces accurately JM (or Jcomplex as bothconverge).

The reaction layer can be visualised in Fig. 2. Despitethe operational definition of l, we can see that for the dataof the figure, the reaction layer is a region close to the lim-iting surface characterised by the absence of equilibriumbetween metal and complex. This is easily recognised bythe divergence of the normalised concentration profiles,cML=c�ML and cM=c�M. Notice then that for the partiallylabile regime, the system can be spatially split into a non-labile and a labile region, separated approximately by theboundary of the reaction layer. This division has been sche-matically depicted in [41] noting that the kinetic flux due to

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20

x /μ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

108 J

M( x

) / (

mo

l m-2

s-1

)

*i

i

c

c

MJ (x)

M*M

c (x)

c

ML*ML

c (x)

c

Fig. 2. Normalised concentrations profiles, ci=c�i , of M (·) and ML (D)referred to the left ordinate axis and the free metal flux (continuousline) referred to the right ordinate axis as a function of the distance tothe electrode surface measured as x/l. Parameters: kd = 0.1 s�1, ka =10 m3 mol�1 s�1. The rest of parameters as in Fig. 1.

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

(c L/mol m-3)-1/2

or

L

*

(30)

(22)

(44)L

Fig. 3. Plot of the lability degree, n vs. c��1=2L . Continuous line: n

calculated by means of (22). Marker full triangle, approximated n-valuecalculated by means of (30). The approximate lability parameter L,calculated by means of the simple expression (44) that renders L

proportional to c��1=2L is also depicted in dotted line in the figure.

Parameters: as in Fig. 1. except kd = 10�2 s�1 and ka = 1 m3 mol�1 s�1,DM = 5 · 10�10 m2 s�1, e = 0.8.


the dissociation equals the diffusive flux of ML towards thereaction layer.

For data of Fig. 2, l = 5.83 · 10�6 m. As g = 8 · 10�4 m,the approach g� l1 applies and l and l1 converge(l1 = 5.87 · 10�6 m).

7. Concentration profiles and metal flux

Let us now analyse the concentration profiles and thetransport of metal along the diffusion space. Fig. 2 plotsM and ML concentration profiles, together with the metalflux J MðxÞ ¼ DM

dcM

dx

� �x

at different distances from the elec-trode surface (i.e. the flux crossing a plane with constantabscissa). As can be seen, neither the metal nor the complexconcentration profiles are linear close to the electrode sur-face. But, in the region x� l, equilibrium is practicallyreached and the normalised metal and complex concentra-tion profiles tend to be linear. Accordingly, the metal fluxtends to be constant in agreement with the linear metalconcentration profile.

As we are in steady state, JM(x) + JML(x), whereJ MLðxÞ ¼ DML

dcML

dx

� �x, should be constant across planar sur-

faces defined by constant x. (This is easily seen by additionand integration of M and ML diffusion equations). Since atx = 0 there is no flux of ML, as prescribed by the boundarycondition, the decrease in JM(x) when x increases is com-pensated by an increase in JML(x).

Notice that far enough from the electrode surface(x� l), JM(x) and JML(x) are related by

J MLðxÞ ¼ eK 0J MðxÞ x� l ð46Þindicating the fulfilment of local equilibrium conditions be-tween M and ML at this distance. Moreover, in this region(x� l), there is no net dissociation since there is equilib-rium and JM(x) is constant, in agreement with the almost

linear metal concentration profile. The same is also truefor ML, indicating that both species diffuse independentlyas in the fully labile case. On the other hand, for x < l,there is net dissociation and the incoming JM(x + dx) toany volume element is smaller than the outgoing JM(x)flux, the difference between both being just the net dissoci-ation of ML to keep constant the M concentration in theelement as required by the steady-state conditions.

8. Dependence of the degree of lability on c�L and g

The degree of lability decreases as the ligand concentra-tion increases. In excess of ligand, the complexation pro-cess is lineal with an effective association rate constant,k0a, dependent on c�L. Additionally, most metal in the systemis in the complex form whenever K 0 � 1. In steady state, anincrease of the ligand concentration modifies the balancebetween the association and dissociation rates, becausethe former is increased. As a consequence, Jcomplex

decreases with increasing ligand concentration and thelability degree, which is depicted in Fig. 3, follows the sametrend. Notice that the decrease of n follows the dependencec��1=2

L for low n-values. This dependence is also exhibited bythe approximate expression (44) reported for the labilityparameter L, which is close to n for low n-values, as canbe seen in the figure.

The applicability of the simple expression (30), whichyields results quite close to the rigorous n-values obtainedfrom (22), is also remarkable. At high enough c�L-values,g� eK 0l1 and n reduces to Eq. (31) and depends onc��1=2

L as indicated above.Fig. 4 plots the rigorous analytical expression for the

metal flux at the electrode surface, Eq. (14), together withJcomplex (computed as Jcomplex = JM � Jfree) and the labilitydegree n given by (21) in terms of logg. In this figure, wecan see the monotonous decrease of n as the diffusion

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

-9 -8 -7 -6 -5 -4 -3 -2

log (g /m)

106 J

/ (m

ol m

-2s-1

)

0

10

20

30

40

50

60

70

80

90

100

ξ(%

)

ξ

MJ

complexJ kinJ

labileJ

freeJ

Fig. 4. Metal flux at the electrode surface JM (continuous line), thecomplex contribution to the metal flux Jcomplex (marker +), the kinetic fluxJkin (short dashed line), Jlabile (short dashed line with marker ·) and Jfree

(short dashed line with marker h), referred to the left ordinate axis and thelability degree n of the system (s) (in %) referred to the right ordinate axisas function of log (g/m). Parameters: kd = 100 s�1, ka = 104 m3 mol�1 s�1.The rest of parameters as in Fig. 1.


region decreases while, parallel to this decrease, JM

diverges from Jlabile to approach Jfree. Thus any system islabile for a large enough diffusion domain thickness, whileall systems tend to be inert when g� l. This result agreeswith the behaviour obtained for the short and long timeregimes in planar semi-infinite diffusion [10] according tothe fact that changing the thickness of the diffusion domainimplies a change in the time required to reach steady state.

Notice that in Fig. 4, JM increases monotonously as g

decreases. This behaviour is independent of any set ofparameters of the system, since dJM

dg is always negative ascan be analytically seen by inspection of the derivative of(14).

Interestingly, Jcomplex = JM � Jfree shows a maximum inFig. 4. A qualitative understanding of this behaviour canbe provided: when the diffusion layer is large enough (forinstance, g = 10�3 m in Fig. 4), the complex behaves aslabile and JM, Jcomplex and Jlabile coincide. Decreasing g,all the fluxes JM, Jcomplex and Jlabile increase due to thereduction of the distance from the electrode to the bulk

Table 1Comparison of the thickness of the diffusion domain that renders Jcomplex maxsolution of the derivative of (47)

e K 0 kd log (gmax) from (48)

0.1 290.03 1 · 10�5 �1.430.1 290.03 1 · 10�2 �3.930.1 140.07 1 · 106 �7.900.01 70.07 2 · 10�7 �1.900.01 70.07 2 · 10�1 �4.900.01 70.07 2 · 106 �8.40

We also compare the degree of lability at this gmax, calculated with (22) withThe rest of parameters as in Fig. 1.

concentrations. However, the increase of both JM andJcomplex is not as large as that of Jlabile indicating that theloss of lability of the system has started. A further decreaseof g not only reduces the deviation from bulk conditions,but also reduces the lability of the system. The result is thatJcomplex decreases (see points for log (g) < �6 in the figure)while JM still shows an increase due to the increase of Jfree

also depicted in the figure.Let us obtain the thickness of the diffusion layer corre-

sponding to this maximum. With JM given by (14) and Jfree

defined in (19), Jcomplex can be written as:

J complex ¼effiffiffiffiffiffiffiDM

pc�MLð

ffiffiffinp

z� � tanh½ ffiffiffinp z��Þz�ð ffiffiffinp z� þ eK 0 tanh½ ffiffiffinp z��Þ ð47Þ

Although it can be solved numerically, the derivative of(47) with respect to z* equalled to zero has no explicit ana-lytical solution. However, if

ffiffiffinp

z� � 1 (valid whenevereK 0 � 1 and g� l1), then tanh½ ffiffiffinp z�� ffi 1 andðsech½ ffiffiffinp z��Þ2 ffi 0. Under these conditions, the thicknessof the diffusion layer that renders Jcomplex maximum, de-noted as gmax, becomes

gmax ¼ffiffiffiffiffiffiffiDM

pz�max;Ap ¼

ffiffiffiffiffiffiffiffiffiDML

p 1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ eK 0p

ffiffiffiffiffikd

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ eK 0p ð48Þ

Replacing gmax in (22), the degree of lability of the complexis:

n ffi 1

1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ eK 0p ð49Þ

a value dependent on e and K 0. Notice that gmax corre-sponds to the thickness of the diffusion layer more sensitiveto the presence of the complex, since the complex contribu-tion to the metal flux in this configuration is maximum.Notice that the degree of lability of the complex is not veryhigh for this thickness of the sensor (recall eK 0 > 1) (seeTable 1). Although a higher lability degree would be ob-tained at a greater g, the absolute complex contributionwould be lower, a consequence of the variation of bothJM and Jfree. Table 1 checks the accuracy of the approxi-mate Eq. (48) with the rigorous numerical solution of thederivative of (47).

Together with the peaked behaviour of Jcomplex, thetransition from the labile to the non-labile regime of thecomplex is evidenced in Fig. 4 by a shoulder in JM which

imum, gmax, calculated via (48), with the exact one, obtained by numerical

log(gmax) num. solution n (%) as (49) n (%) as (22)

�1.43 15.4 15.4�3.93 15.4 15.4�7.90 20.5 20.5�1.96 43.4 39.1�4.96 43.4 39.1�8.46 43.4 39.1

the approximate degree of lability calculated with (49).


moves JM from practically being Jlabile to values close toJfree. The difference in the abscissa for a fixed ordinatevalue between Jlabile and Jfree can be found from (16) and(19) as

Dðlog gÞ ¼ logð1þ eK 0Þ ð50Þwhich estimates the thickness of the JM-shoulder.

9. Conclusions

The contribution of the complex to the metal flux underdiffusion limited conditions in a planar finite domain canbe fully analysed for any value of the association/dissocia-tion rate constants. The explicit analytical expression forthe metal flux (14) gives rise to a rigorous quantificationof the lability degree of the system (Eq. 22). We highlightthe simple expression (29) for a good approximation ofthe flux under the usual relevant conditions (eK 0 � 1 andg� l1).

Complementary, the specific lability criterion (42) andthe reaction layer thickness (38) for the present system havebeen worked out within the reaction layer approximation.

The lability degree, n, describes the percentage of thekinetic contribution with respect to the maximum kineticcontribution of the complex to the metal flux, once the con-centrations, kinetic constants, diffusion coefficients andspatial dimensions of the sensor are known.

Alternatively, the lability parameter, L, compares thekinetic flux, Jkin, a measure of the hypothetical ability ofthe complexation process to produce metal flux in absenceof diffusion limitation of the complex, with the maximumdiffusional supply of ML, in order to determine which isthe process limiting the metal flux in the system. If the dif-fusional supply is limiting we say that the system is labilewhile the system is partially labile or non-labile when thekinetics of dissociation is the limiting process.

The analytical expression reported for the lability degreecan also be used to determine the thickness of the diffusionlayer of sensors like DGT or PLM, so that we measure aprefixed percentage of the maximum kinetic contributionof the complex. In this way we can select the contributionof the complex in the sensor answer.

Limiting expressions for the fully labile case areobtained from the general kinetic ones. These expressions(see, for instance, Eq. (17)) indicate that the gradient ofthe complex at the limiting surface is not zero in fully labileconditions in agreement with previous mathematical for-mulation of the fully labile case in the literature.

The limiting case of immobile complex has also beenstudied. In steady-state conditions, the concentration pro-files of M and ML are in equilibrium everywhere, regard-less of the value of the kinetic constants. However, JM

coincides with Jfree this indicating that there is no contri-bution of the complex to the metal flux in steady-stateconditions, as expected (in steady-state conditions, themaximum complex contribution is just the maximum dif-fusional flux of the complex which is null in this case).

However, immobile complexes contribute to the metalflux in the transient regime and increase the time requiredto reach steady state.

Special attention has been devoted to the behaviour ofthe system when the thickness of the diffusion domainchanges. It is shown that, decreasing the thickness of thediffusion domain, the free metal flux always increases, whilethe degree of lability of the system decreases, i.e., JM

approaches Jlabile for large enough thickness of the diffu-sion domain while JM approaches Jfree for low enoughthickness of the diffusion domain. In this transition, JM

develops a shoulder, while Jcomplex shows a peaked behav-iour. The thickness of the diffusion domain that rendersJcomplex maximum can be obtained from Eq. (22).

Appendix A. Rigorous solution of the system (7)–(10)

Let us combine Eqs. (7) and (8) to rewrite this system interms of new unknown functions, y0 and y1, so that eachnew equation depends only on one unknown function[33,42]. Following d’Alembert methodology, a general lin-ear combination of (7) and (8) can be written as

d2ðcMðzÞ þ wcMLðzÞÞdz2

¼ k0a 1� we

� �cMðzÞ þ kd

we� 1

� �cMLðzÞ

ðA-1Þ

where w is the weight of cML in the combination. Imposingthat

kd

we� 1

� �¼ wk0a 1� w

e

� �and solving for w, we have

y0 ¼ cMðzÞ þ ecMLðzÞ

y1 ¼ �cMðzÞ þcMLðzÞ

K 0ðA-2Þ

In terms of y0 and y1, Eqs. (7) and (8) become

d2y0ðzÞdz2

¼ 0 ðA-3Þ

d2y1ðzÞdz2

¼ ny1ðzÞ ðA-4Þ

with

n ¼ kd

eþ k0a ¼

kdð1þ eK 0Þe

ðA-5Þ

The general solutions of (A-3) and (A-4) are

y0ðzÞ ¼ A1;0zþ A2;0

y1ðzÞ ¼ A1;1 expðffiffiffinp

zÞ þ A2;1 expð�ffiffiffinp

zÞðA-6Þ

Rewriting the boundary conditions at the bulk solution interms of y0 and y1 we have

y�0 ¼ c�Mð1þ eK 0Þy�1 ¼ 0

ðA-7Þ

and their application to (A-6) lead to


y0ðzÞ ¼ c�Mð1þ eK 0Þ þ T 0ðz� z�Þ ðA-8Þy1ðzÞ ¼ T 1 sinh½

ffiffiffinpðz� z�Þ� ðA-9Þ

where T0 and T1 denote the remaining integration con-stants. These constants can be obtained by imposing theboundary conditions at z = 0. Then,

T 0 ¼c�Mð1þ eK 0Þ ffiffiffinpffiffiffi

np

z� þ eK 0 tanh½ ffiffiffinp z�� ðA-10Þ

and

T 1 ¼�c�Mð1þ eK 0Þffiffiffi

np

z� cosh½ ffiffiffinp z�� þ eK 0 sinh½ ffiffiffinp z�� ðA-11Þ

Replacing these values in (A-8) and (A-9), we have thesought solution (similar to that in Ref. [42]), which in termsof the original unknowns can be written as indicated inEqs. (11) and (12).

Appendix B. Table of the most relevant symbols

Symbol Meaning Equation

c0ML
Complex
concentration at theelectrode surface

(24)

c�i
Bulk concentration ofspecies i
(2), (10)

Di
Diffusion coefficient ofspecies i
(3), (4)

g
Thickness of thediffusion domain
(10)

Jcomplex
Contribution of thecomplexes to the metalflux
(18)

Jdif
Maximum diffusiveflux due to complexML
(44)

Jfree
Flux if the complexeswere inert (i.e. due tofree M)
(19), (36)

Jkin
Hypotheticalmaximum kineticcontribution if therewas no limitation fromthe diffusion of thecomplexes
(36), (37), Fig. 1, (44)

J/kin
Approximation to the
contribution ofcomplex ML to themetal flux usingreaction layerconcentrations

Fig. 1, (45)

Jlabile
Flux if the complexeswere labile
(16), (17)

JM
Total metal flux atx = 0, i.e., JM(x = 0)
(14)

JM(x)
Metal flux at a given x JMðxÞ ¼ DMdcM
dx

� �x

Symbol Meaning Equation

JML(x)
Flux of complex at agiven x
JMLðxÞ ¼ DMLdcML

dx

� �x

ka, kd
Association anddissociation rateconstants
(2)

L
Lability parametercomparing Jkin and themaximum diffusionalflux of the complexes
(42)

n
Uncoupling parameter (13), (A-5) z New spatial variable (6) z* Bulk position
expressed in thevariable z

(10)

e
Dimensionlessdiffusion coefficient forthe complex ML
(5)

l
Diffusion layerthickness in finitediffusion
(38)

l1
Diffusion layerthickness in semi-infinite diffusion
(28)

n
Degree of lability (20)
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Documents

Lability of complexes in steady-state finite planar diffusion