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Implementation of a chemical equilibrium constraint in themultivariate curve resolution of voltammograms from systemswith successive metal complexes

José Manuel Díaz-Cruz,* Josep Agulló, M. Silvia Díaz-Cruz, Cristina Ariño, MiquelEsteban and Romà Tauler

Departament de Química Analítica, Universitat de Barcelona, Av. Diagonal 647, E-08028Barcelona, Spain. E-mail: jmdiaz@apolo.qui.ub.es; Fax: +34 93 402 12 33;Tel: +34 93 402 90 83

Received 1st November 2000, Accepted 11th January 2001First published as an Advance Article on the web 20th February 2001

A multivariate curve resolution (MCR) method, using a constrained alternating least squares (ALS) procedure witha new chemical equilibrium constraint, was applied to differential-pulse polarograms of successive metalcomplexes. This new restriction imposes the fulfilment of a chemical model defined by a set of stability constantsthat are optimised along the iterative ALS procedure. The reliability of the method was tested with simulated dataand with polarograms measured for the systems Zn(II) + glutathione and Cd(II) + 1,10-phenanthroline. Thesesystems respectively yield two and three successive and electroactive complexes, which are inert from theelectrochemical point of view, that is, the complexes virtually do not dissociate during the measurement. Althoughthe presence of electrode adsorption could induce overestimation of some concentrations and losses of linearitybetween concentrations and signals, the results showed that the proposed method can yield satisfactory estimationsof the stability constants in this kind of system. The performance of the new method is compared with theperformances obtained using MCR-ALS without the equilibrium constraint and using traditional curve fittingleast-squares approaches.

Aim of investigation

The use of chemometric techniques for the analysis ofvoltammetric data has been limited to date, especially whencompared with the widespread application of chemometrics inspectrophotometric measurements.1 This is mainly due to thelack of linearity between current and concentration in manyelectrochemical processes, and also to the complex relationshipbetween concentrations around the electrode (diffusion layer)and in the bulk of the solution.

A method of multivariate curve resolution by alternatingleast-squares (MCR-ALS), originally developed for spec-trophotometric data,2 was used for the analysis of voltammetricmeasurements of several metal–macromolecule complexes.3The study showed that, although in these systems the signals arenot totally linear with respect to the bulk concentration of thespecies, the method produces results close to those obtained bymeans of the rigorous electrochemical model. Additional workcarried out on the Cd(II)-nitrilotriacetate (NTA) system4

suggests that MCR-ALS can be especially appropriate foranalysing electrochemically inert complexes (i.e., those thatvirtually do not dissociate during the timescale of the measure-ment), where the extension of non-linearity is small.

A further study on the Zn(II)-glutathione system5 showedthat, for electrochemically inert complexes, a standard least-squares curve fitting approach using a program such as SQUAD(Stability QUotients from Absorbance Data)6 can be success-fully applied to voltammetric data to obtain reliable stabilityconstants. However, the use of SQUAD prevents the applicationof restrictions such as selectivity, unimodality or specific shapeof the pure signals, which have been shown to be important inthe resolution of overlapping voltammograms. Moreover, allthe species yielding signals must be included in the complexa-tion model for running SQUAD, which is not always possible,

especially when they are poorly defined or their concentrationsare unknown.

The present work tries to integrate both MCR-ALS andSQUAD approaches with the implementation of a new chemicalequilibrium constraint in the MCR-ALS procedure that at everyiteration of the ALS optimisation forces the concentrationprofiles of some of the species to fulfil a previously assumedmodel of complexation. In this way, stability constants can beoptimised iteratively for the full data set (as SQUAD does) butwith the possibility of imposing additional constraints typicallyused in MCR-ALS and with the option of excluding some of thespecies from the complexation model.

To test the convenience of using this new constraint, theproposed method was applied to the analysis of different datasets: first, numerically simulated voltammograms; second,experimental data from the Zn(II)–glutathione system, whichyields two successive inert complexes and has been previouslyanalysed by both MCR-ALS and SQUAD;5 and third, experi-mental voltammograms measured for the Cd(II)-1,10-phenan-throline system, which gives three successive inert com-plexes.

Theory

General overview of MCR-ALS

MCR-ALS requires that the voltammograms recorded atdifferent ligand concentrations are arranged in a matrix ofcurrents I with as many rows as the number of recordedvoltammograms and as many columns as potentials scannedduring the current measurements. Then, singular value decom-position (SVD) is applied to estimate the number of electro-

This journal is © The Royal Society of Chemistry 2001

DOI: 10.1039/b008802l Analyst, 2001, 126, 371–377 371

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active species linearly contributing to the current and, finally,the matrix I is decomposed as a product of a matrix C(containing the concentration of every electroactive species)and a matrix VT (containing the corresponding pure voltammo-grams) plus an error matrix X:2,3,7

I = C VT + X (1)

This procedure is carried out in an iterative way from an initialestimation of concentration profiles or pure signals obtainedfrom the data matrix by evolving factor analysis (EFA).2,3Whenno constraints are applied, every iteration by alternating least-squares (ALS) involves two steps:

C = I (VT)+ (2a)

VT = C+ I (2b)

where ‘T’ and ‘+’ denote transposed and pseudo-inversematrices, respectively. However, the ALS procedure allowsseveral restrictions (constraints) to be imposed such as selectiv-ity (only one species is present in some parts of the matrix), non-negativity (of concentrations and/or signals), unimodality(single peak shape of concentration profiles and/or unitvoltammograms), closure (application of mass balance) andsignal shape (fitting of the signals to an empirical equationwhich properly describes them).7

The relative error of the matrix decomposition is expressed asa percentage of lack of fit (lof), according to the equation:

lof =

( )

,

,

i j

ij ij

ij

i j

I I

I

ÂÂ

-

◊

ˆ 2

2100 (3)

where Iij are the elements of the experimental matrix I, and Îij

the corresponding calculated elements of the matrix Î as theproduct of C and V estimated by ALS.

Stability constants in MCR-ALS (without equilibrium con-straint) are calculated from the average of the log K valuescomputed point by point from the concentration profiles in theregions of predominance of the species involved in eachequilibrium. The standard deviations of these mean log K valueswere computed in the usual way.

Implementation of the equilibrium constraint inMCR-ALS

In the present work the implementation of an additionalchemical equilibrium constraint was tested for the case ofsuccessive and mononuclear metal complexes. The approachconsists of fitting a set of complexation constants to theconcentration profiles obtained at every ALS iteration. Thisrequires the assumption of a complexation model, i.e., thenumber and stoichiometry of the metal complexes and someinitial estimations of the stability constants.

For this purpose, it can be useful to do a preliminary MCR-ALS study of the matrix with less restrictive constraints(selectivity, non-negativity and/or signal shape) in order to havea qualitative view of the system (which helps in the selection ofthe complexation model). It is also practical to use the resultingconcentration profiles as an initial estimation for a secondMCR-ALS run including the equilibrium constraint. Suchpreliminary profiles can be of further use in order to estimate theinitial values of the stability constants to be employed in theequilibrium-constrained MCR-ALS analysis.

The equilibrium constraint is based on the resolution of apolynomial equation which results from the combination of the

overall formation constants and the mass balance for the metaland the ligand.

For instance, a system including a metal ion M and a ligandL that form two successive and mononuclear complexes (MLand ML2) would be described by the equations:

b1 = [ML]/([M][L]), b2 = [ML2]/([M][L]2) (4)

cTM = [M] + [ML] + [ML2] (5)

cTL = [L] + [ML] + 2 [ML2] (6)

where cTM and cTL are the total concentrations of the metal andthe ligand, respectively, and b1, b2 are the overall formationconstants of the complexes ML and ML2. The combination ofeqn. (4)–(6) as a function of the free ligand concentration [L]yields the expression:

[L]3 b2 + [L]2 {b2 (2cTM 2 cTL) + b1}+ [L] {b1 (cTM 2 cTL) + 1} 2 cTL = 0 (7)

This deduction can be repeated for different values of n, wheren indicates the maximum number of successive mononuclearcomplexes formed (from ML to MLn). In this case, eqn. (4)–(6)have the more general form:

b1 = [ML]/([M][L]), … , bn = [MLn]/([M][L]n) (8)

cTM = [M] + [ML] + … + [MLn] (9)

cTL = [L] + [ML] + … + n[MLn] (10)

When comparing eqn. (7), obtained for n = 2, with theanalogous expressions deduced from eqn. (8)–(10) at different nvalues, we find that all of them fulfil the general expression:

[L]n+1bn + [L]n {bn (n cTM 2 cTL) + bn21} + [L]n21 3

{bn21 ((n 2 1) cTM 2 cTL) + bn22} + … 2 cTL = 0 (11)

where b0 · 1. If the values of n, cTM, cTL and b1…bn are known,it is possible to obtain the free ligand concentration [L] from theroots of the associated polynomial. This can be done numer-ically by the command roots of MATLAB.8 It then becomes aquestion of selecting the proper value of [L] from among thenon-complex and non-negative roots (usually it is the minimumone). Once [L] is known, the equilibrium concentrations of theremaining species can be easily computed from the combinationof eqn. (8) and (9), which yields:

[M] = 1/(1 + b1 [L] + b2 [L]2 + … + bn [L]n) (12)···

[MLn] = bn [L]n/(1 + b1 [L] + b2 [L]2 + … + bn [L]n)

Using this strategy, a new MATLAB function (betacalc) waswritten which is able to compute the concentrations of all metalspecies (from M to MLn) from the values of n, cTM, cTL andb1…bn.

In order to perform the opposite operation, a new functionbetafit was developed which is based on the MATLABoptimisation function lsqcurvefit. Thus, from a set of concentra-tions of the different metal species plus the corresponding cTM

and cTL values, betafit optimises a set of log b values by least-squares. The program also computes numerically the Jacobianmatrix J, which contains the derivatives of the optimisedfunction with respect to the fitted parameters. As discussedlater, J can be used to estimate the standard deviation of thefitted parameters.

The implementation of the above-mentioned functions in theMCR-ALS program is summarised in Fig. 1. After determina-tion of the number of species by principal components analysis(PCA), the process is started with an initial estimation of theconcentration profiles or the unit voltammograms of thesespecies (which can be obtained by EFA). Then, a series of ALSiterations is started. At each of these steps, different constraintsare applied which modify the concentration matrix C. Thebetafit function is applied to these modified concentrations andfrom the known values of n, cTL, cTM and initial guesses of logb1 … log bn a set of stability constants is optimised. Then,

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betacalc computes new values for the concentrations in Caccording to the obtained stability constants. After that, the purevoltammogram matrix VT is obtained from the resulting matrixC and modified according to the corresponding constraints. Ifconvergence is not achieved, a new matrix C is obtained fromVT and a new iteration is started, keeping the last set ofoptimised log b values as an initial guess in this newiteration.

It must be noted that the log b fitting is carried out with theconcentration profiles of all metal species normalised by theinitial total metal concentration. The program also has an inputto assign every complex to one of the species assumed.Although in some cases the free ligand is able to produce anoticeable current, for the moment it is better not to include thiscurrent in the fitting. This is because the currents produced bymany of the ligands previously studied by us are not linear withrespect to the free ligand concentration (they have importantcontributions from capacitive currents or anodic oxidation ofthe mercury drop).

According to the usual methodology in non-linear least-squares fitting, the standard deviations of the fitted log b valueswere estimated from the diagonal elements of the variance–covariance matrix P:9

si

2 = r rii

T

Pdf

(13)

where si is the standard deviation of the fitted log b parameteri, r is the matrix of residuals (difference between theconcentration matrix and the matrix predicted from the fittedlog b values), df are the degrees of freedom of the system andthe variance–covariance matrix P can be computed from theJacobian matrix as:9

P = (JT J)21 (14)

In this case, the Jacobian matrix contains the partial derivativesof the concentration of the different metal species with respectto log b1 … log bn and it is numerically computed by thelsqcurvefit function. All computations assume the same weight

Fig. 1 Implementation of the chemical equilibrium constraint in the MCR-ALS procedure.

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for all experimental data. Although this procedure mayunderestimate the standard deviations of the fitted constants (itonly takes into account the error in the fitting and not the errorin the matrix decomposition), it can prove useful to compareresults obtained with the same methodology.

Experimental procedures

All reagents used were of analytical-reagent grade (Merck).Cd(II) stock solutions were prepared by dissolvingCd(NO3)2·4H2O in water and were standardised complex-ometrically.10 Stock solutions of 0.01 mol L21 1,10-phenan-throline were prepared by dissolving an accurate weight of thereagent in water. Acetic acid–acetate solutions containing 0.01mol L21 acetate were used for pH regulation at different pHvalues, and also as a supporting electrolyte. Ultrapure water(Milli-Q plus 185 system, Millipore) was employed in allexperiments. Nitrogen SEO N50 was used for de-aeration of thesolutions to be measured voltammetrically (20 min prior to theset of measurements and 2 min after every addition of reagent tothe solution).

Voltammetric measurements by differential-pulse, normal-pulse and reverse-pulse polarography (DPP, NPP and RPP,respectively) were carried out with an Autolab System PSTAT10 (Eco Chemie) attached to a Metrohm 663 VA stand and to apersonal computer by means of the software package GPES3(Eco Chemie). The system was also connected to a Metrohm665 Dosimat, for the addition of solutions, and to an Orion SA720 pH-meter for monitoring the pH value during theexperiments. In all cases the reference electrode, to which allpotentials are referred, was Ag/AgCl, KCl (3 mol L21), and thecounter electrode was a glassy carbon electrode. A staticmercury drop electrode (SMDE) was used as the workingelectrode. Drop times of 0.5 s, pulse times of 40 ms and scanrates of 10 mV s21 were used in all polarographic measure-ments, unless otherwise indicated. For DPP, pulse amplitudes of50 mV were applied. Glass cells were used in all themeasurements, which were carried out at 20 ± 1 °C.

Details regarding Zn(II)–glutathione system measurementsare provided elsewhere.5

The voltammograms were smoothed and converted into datamatrices by means of GPES3 software and several home-madeprograms implemented in MATLAB.8 MCR-ALS analysis ofthe data was carried out through several programs alsoimplemented in MATLAB (MATLAB programs for MCR-ALS are available at the web site http://www.ub.es/gesq/mcr/mcr.htm).

The conditional stability constants of Cd(II)–1,10-phenan-throline complexes determined by MCR-ALS were correctedfor side reactions in the usual way11 by using the literaturevalues of log K = 1.56 for the stability constant of the Cd(II)–acetate complex (at 25 °C and ionic strength 0.1 mol L21) andlog K1 = 4.93, log K2 = 1.9 for the protonation constants of1,10-phenanthroline (at 25 °C and ionic strength 0.1 molL21).12

The simulation of differential-pulse polarograms of succes-sive metal complexes was carried out in the same way asexplained in ref. 7.

Results and discussion

Analysis of simulated data

Fig. 2 shows a set of voltammograms simulated for a systemthat yields three successive inert and electroactive complexes byusing the values log b1 = 4.5, log b2 = 8.5 and log b3 = 12.0.In order to test the influence of the experimental error, 0.5%

Gaussian noise was added to the matrix values. Fig. 3 comparesthe concentration profiles and the pure signals used in thesimulation with those obtained by SQUAD and MCR-ALS withand without equilibrium constraint. Table 1 summarises thestability constant computed in every case.

Comparison of these results shows that the use of theequilibrium constraint in MCR-ALS improves the accuracy ofthe stability constants with a very small increase in the lack offit (from 1.70 to 1.74%). In comparison with both MCR-ALS

Fig. 2 Differential-pulse polarograms simulated for three successive inertand electroactive complexes with stability constants log b1 = 4.5, log b2 =8.5, log b3 = 12.0, total metal concentration 1 3 1025 mol L21 and totalligand concentration ranging from 0 to 2 3 1023 mol L21. The thick linedenotes the signal measured in the absence of ligand. The matrix valuescontain 0.5% Gaussian noise.

Fig. 3 Concentration profiles (a) and pure voltammograms (b) obtained byanalysis of the data matrix in Fig. 2 according to different methods: MCR-ALS with closure and non-negativity (2), MCR-ALS with closure, non-negativity and equilibrium constraint (5) and SQUAD analysis ofsmoothed signals (+). Solid lines show the ‘true’ curves used in thesimulation. The free metal ion and the successive complexes are denoted asM, ML, ML2 and ML3, respectively.

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methods, SQUAD is more sensitive to random noise, since noreliable values for log b were obtained from the simulatedmatrix until the noise was minimised by Savitzky–Golaysmoothing. This is not unexpected considering that MCR-ALSoptimisation is carried out on a PCA-reproduced data matrix(Fig. 1) which has been noise-filtered for the particular numberof selected components.

Finally, the analyses of the standard deviations in Table 1show them to be much smaller when using the equilibriumconstraint. This seems to confirm that, as stated above at the endof the Theory section, the calculation of standard deviationsfrom the Jacobian matrix may underestimate them, because itdoes not take into consideration the error in the matrixdecomposition of eqn. (1).

Analysis of DPP signals of the Zn(II)–glutathione system

The next step in testing the convenience of employing theequilibrium constraint was the study of its application to a dataset previously analysed by conventional MCR-ALS andSQUAD. This set of differential-pulse polarograms is shown inFig. 4, and it corresponds to a recently published study on theZn(II)–glutathione system.5

The analysis in Fig. 5 and Table 2 indicates that the resultsobtained by MCR-ALS with the new constraint are similar tothose from conventional MCR-ALS and SQUAD. Althoughthere are no significant differences among the log b valuesobtained by all three methods (they are in the range ±0.1), theconcentration profiles and the pure signals from the equilib-rium-constrained MCR-ALS approach more closely resemblethose computed with SQUAD than those coming from MCR-ALS without equilibrium constraint.

Table 2 also shows that the standard deviations obtainedusing the equilibrium constraint are higher than in the simulatedcase (Table 1). This is not surprising because the deviationsfrom the assumed complexation model should be moreimportant for experimental data than for data simulated to fulfilperfectly a given complexation model.

Analysis of the Cd(II)–1,10-phenanthroline system

A study of the more complex system Cd(II)–1,10-phenanthro-line was the final step in the work described here. In this systemthree successive complexes are expected instead of two. Alsoexpected was an important complication that can to somedegree affect the linear relationship between signals and speciesconcentrations: the strong adsorption of the complexes onto theelectrode surface.

In the preliminary part of this study, NPP, RPP and DPPmeasurements were performed on solutions containing differentproportions of Cd(II) ion and 1,10-phenanthroline. As a generalpattern, NPP signals presented important maxima whichconfirm the adsorption of the complexes on the electrodesurface.13 However, this is hardly surprising since 1,10-phenan-throline has a great affinity for mercury. In fact, this reagent hasbeen proposed for the determination of Cu(II) by adsorptivestripping voltammetry through the formation of adsorbedcomplexes.14,15 The use of RPP decreases the interferences

from adsorption and produces normal-shaped sigmoidal polaro-grams. However, the RPP signals measured in this system areaffected by a very poor signal-to-noise ratio. As for differential-pulse polarograms, they could be affected by adsorption. In anycase, they present a regular peak shape and a good signal-to-noise ratio, which suggests that this is the best technique for thestudy of the Cd(II)–1,10-phenanthroline system.

DPP titrations of Cd(II) with 1,10-phenanthroline at differentpH values showed that the best conditions for the estimation ofstability constants by MCR-ALS are attained at pH 5.0. Atlower pH values the dissociation kinetics of the complexesbecomes too fast, and a single signal is observed for thesimultaneous reduction of all species, which progressivelymoves to more negative potentials as the ligand concentrationincreases. This strongly labile behaviour of the complexationprocess can seriously affect the linearity between the measuredsignals and both the species concentrations and pure signals(bilinearity). On the other hand, at higher pH values the 1+1complex is virtually absent, and this hinders the calculation ofb1.

Fig. 6 shows the DPP signals obtained at the optimum pH of5.0. The addition of the ligand causes a drop in the peak of freeCd(II) reduction (which stays at the same peak potential) and anincrease in a series of overlapping signals at more negativepotentials corresponding to the reduction of successive Cd(II)complexes. These facts confirm the (electrochemically) inertbehaviour and the bilinearity of the system, which was one ofthe conditions of the present study.

Fig. 7 shows the singular value decomposition of the data setin Fig. 6. It suggests that the number of components of the datacould be four (the PCA-reproduced data matrix using three, fourand five principal components is 7.1, 2.5 and 2.0%, re-spectively). These four components can be attributed to the freeCd(II) ion and three successive complexes with 1,10-phenan-throline.

The application of MCR-ALS considering four species, andwith the constraints of non-negativity (for both speciesconcentrations and signals) and closure, yields the concentra-tion profiles and pure voltammograms shown in Fig. 8, with alack of fit of 5.9%. From these concentration profiles, thestability constants were estimated as the average of the valuescomputed point by point in the regions where the involvedspecies predominate. Results are given in Table 3 and are inagreement with those found in the literature.12

When, in addition to the constraints of non-negativity andclosure, the restriction of chemical equilibrium is applied, thelack of fit increases considerably (13.4%), but still producesreasonable concentration profiles and pure voltammograms(Fig. 8), and the values for the stability constants are close tothose found in the literature (Table 3).

From a qualitative point of view, the shapes and the regionsof predominance of signals and concentration profiles obtainedby both methods very closely coincide. However, from aquantitative point of view, the formation of ML and ML2

species is higher in the concentration profiles obtained byMCR-ALS without the equilibrium constraint and, as aconsequence, the corresponding pure voltammetric signals arelower. This discrepancy may be due to MCR-ALS solutionswithout the equilibrium constraint being more affected by theadsorption of some of the complexes on the electrode,

Table 1 Comparison among the overall stability constants used in the simulation of the voltammograms of Fig. 2 and the values obtained by fitting thevoltammograms according to different methods. Standard deviations are given in parentheses (see Theory section)

Method lof (%) log b1 log b2 log b3

Values used for simulation — 4.5 8.5 12MCR-ALS (non-negativity + closure) 1.70 4.49 (0.07) 8.61 (0.09) 11.9 (0.3)MCR-ALS (non-negativity + closure + equilibrium) 1.74 4.496 (0.001) 8.509 (0.002) 12.015 (0.002)SQUAD (after smoothing) 2.58 4.535 (0.002) 8.530 (0.003) 12.002 (0.004)

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increasing the apparent concentration of such complexes.Nevertheless, these differences do not seem to be veryimportant, since the values for the stability constants obtained inboth cases are highly coincident.

Finally, it is notable that in this case the SQUAD programwas unable to estimate any set of stability constants from the

Fig. 4 Differential-pulse polarograms measured during the titration of a 13 1025 mol L21 Zn2+ solution with glutathione at pH 8.5 with 0.15 mol L21

borate buffer. (Data taken from ref. 5.)

Fig. 5 Concentration profiles (a) and pure voltammograms (b) obtained byanalysis of the data matrix in Fig. 4 by different methods: MCR-ALS withclosure and non-negativity (2), MCR-ALS with closure, non-negativityand equilibrium constraint (—) and SQUAD (---) . The free metal ion andthe successive complexes are denoted as M, ML and ML2, respectively.

Table 2 Values obtained for the overall stability constants of the Zn(II)–glutathione complexes by fitting the voltammograms according to differentmethods. Standard deviations are given in parentheses (see Theorysection)

Method lof (%) log b1 log b2

MCR-ALS (non-negativity +closure) (ref. 5) 3.69 4.87 (0.04) 8.40 (0.20)

SQUAD (ref. 5) 3.70 4.69 (0.01) 8.56 (0.02)MCR-ALS (non-negativity +

closure + equilibrium) 3.72 4.78 (0.05) 8.66 (0.09)

Fig. 6 Differential-pulse polarograms measured for a solution containing1 3 1025 mol L21 Cd(II) and 0.01 mol L21 acetate buffer at pH 5.0 aftersuccessive additions of 1,10-phenanthroline (up to 7.3 3 1025 mol L21).The thick line denotes the signal measured in the absence of ligand.

Fig. 7 Singular value decomposition of the data matrix shown in Fig. 6.

Fig. 8 Concentration profiles (a) and pure voltammograms (b) obtained byanalysis of the data matrix in Fig. 6 according to different methods: MCR-ALS with closure and non-negativity (2) and MCR-ALS with closure, non-negativity and equilibrium constraint (—). The free metal ion and thesuccessive complexes are denoted as M, ML, ML2 and ML3, re-spectively.

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data matrix due to high divergence during the iterativeoptimisation procedure. This fact is not unexpected, taking intoaccount the high lack of fit yielded by the equilibrium-constrained MCR-ALS. This also suggests than the MCR-ALSmethod implemented with equilibrium constraints is a morerobust tool than SQUAD for obtaining rough estimations ofstability constants even in the presence of secondary phenom-ena such as electrode adsorption, noise or decrease in datalinearity.

Conclusions

The results reported here show that the implementation of a newconstraint in the MCR-ALS procedure to take chemicalequilibrium into account provides a useful tool for thevoltammetric analysis of systems producing successive inertelectroactive complexes.

The presence of undesired phenomena such as electrodeadsorption can overestimate the concentration of some speciesor affect the linearity between signals and concentrations. Thus,the estimation of the constants is, in general, less accurate thanwith techniques such as spectrophotometry or potentiometry.However, the combination of MCR-ALS with voltammetricmeasurements proves very useful at concentration levels so lowas to be inaccessible to these techniques, as is the case, forinstance, in the study of the metal complexes of metallo-thioneins and related molecules.

In such cases, the equilibrium-constrained MCR-ALS seemsto be more convenient than the traditional equilibrium hard-modelling methods such as SQUAD, since the former is morerobust (i.e., less affected by experimental noise and deviationsfrom linearity) and versatile (the equilibrium constraint does nothave to be imposed on all electroactive species and it can beused simultaneously with other constraints).

As a general rule for applying the equilibrium constraint, it isadvisable to start the MCR-ALS analysis with less restrictiveconstraints (e.g. selectivity and signal shape) from the estima-tion of concentration profiles provided by EFA. If a qualitativeinspection of the results allows both the formulation of acomplexation model and the identification of the involvedspecies, a second MCR-ALS run can be performed with theequilibrium constraint to refine the results and to obtain thevalues of the stability constants. In this second run it is advisableto use the preliminary MCR-ALS results as initial values for theiterations. In any case, if the complexation model is well-known

and reasonable estimates of the stability constants are available,the calculations can start directly with the application of theequilibrium constraint.

Acknowledgements

The authors gratefully acknowledge M. Maeder, ProfesorVisitante IBERDROLA de Ciencia y Tecnología, on leave fromthe University of Newcastle (Australia), as well as A. de Juan(Universitat de Barcelona) for very helpful discussions. Finan-cial support from the Spanish Ministries of Education andCulture and Science and Technology (Projects PB96-0379-CO3and BQU2000-0642-C03-01) and from the Generalitat deCatalunya (1999SGR-00048) is gratefully acknowledged.

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Table 3 Values obtained for the overall stability constants of the Cd(II)–1,10-phenanthroline complexes by means of MCR-ALS using different constraints.The values from ref. 12 are also given. Standard deviations are given in parentheses (see Theory section)

Method lof (%) log b1 log b2 log b3

MCR-ALS (non-negativity + closure) 5.9 5.5 (0.2) 10.5 (0.3) 15.2 (0.4)MCR-ALS (non-negativity + closure + equilibrium) 13.4 5.64 (0.05) 10.95 (0.06) 15.95 (0.07)Ref. 12 (25 °C and ionic strength 0.1 mol L21) — 5.8 (0.1) 10.6 (0.1) 14.6 (0.3)

Analyst, 2001, 126, 371–377 377

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