Application of multivariate curve resolution to the voltammetric study of the complexation of fulvic acids with cadmium(II) ion

Analytica Chimica Acta 459 (2002) 291–304

Application of multivariate curve resolution to the voltammetric study of the complexation of fulvic acids with cadmium(II) ion M.C. Antunes a , J.E. Simão b , A.C. Duarte b , M. Esteban c , R. Tauler c,∗ a

c

Departamento de Qu´ımica, Universidade de Trás-os-Montes e Alto Douro, 5000-911 Vila Real, Portugal b Departamento de Qu´ımica, Universidade de Aveiro, 3810-312 Aveiro, Portugal Departament de Qu´ımica Anal´ıtica, Facultat de Qu´ımica, Universitat de Barcelona, Avinguda Diagonal 647, 08028 Barcelona, Spain Received 10 October 2001; received in revised form 8 February 2002; accepted 20 February 2002

Abstract The interaction of fulvic acids with Cd(II) ions has been studied by differential pulse anodic stripping voltammetry (ASV), at pH 7.0 and 0.1 M KNO3 . Voltammetric data obtained during the titration of mixtures of Cd(II) and fulvic acids at different concentrations have been analyzed using multivariate curve resolution (MCR). The application of this method allowed the resolution of the major contributions involved in the titration experiments. Apart from free Cd(II), two more contributions related with the complexation process were detected and resolved, and their corresponding pure voltammograms and concentration profiles were estimated. Simultaneous analysis of independent voltammetric titrations using the proposed MCR methods is shown extremely recommended because it overcame some of the limitations observed in the analysis of individual titrations. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Fulvic acids; Multivariate curve resolution; Voltammetry; Cd(II) ion; Metal complexation

1. Introduction Fulvic acids (HF) present in humic substances are known to influence the role of metal ions in aquatic environments and soils because they drastically affect mobility and toxicity of metal ions. Therefore, the study of interaction of metal ions with fulvic acids represents an important area of research in environmental analytical chemistry. Due to the chemical heterogeneity and the complexity of humic substances, the mechanisms of complexation with metal ions are very complex phenomena that depend on many factors, such as, pH, ionic strength of the medium, and ∗ Corresponding author. Tel.: +34-93-402-1276; fax: +34-93-402-1233. E-mail address: [email protected] (R. Tauler).

HF concentration [1]. Furthermore, HF show polyfunctional, polyelectrolic and conformational effects that contribute to further complicate the study of the formation and stability of the HF metal ion complexes. Voltammetric methods have become an important tool in metal specialization studies, owing to their high sensitivity and rich signal information content. Voltammetric techniques have been systematically used in the study of the interactions between HF and heavy metals [2–6]. However, the applicability in these studies of anodic stripping voltammetry (ASV), one of the most interesting techniques from the analytical point of view, is hampered by many factors. Thus, the correct interpretation of experimental data is significantly more difficult in ASV than in direct voltammetric techniques such as pulse polarography. This is due to the fact that ASV data depend both

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on the complex formation in the stripping step, and on the dissociation of the complex occurring in the deposition step. Metal-to-ligand concentration ratios at the electrode surface during the stripping step (which are not necessarily the same as in the bulk) and adsorptive effects on the electrode play a key role [7,8]. Accordingly, the interpretation of voltammetric data for these systems is extremely difficult, and it has been usually based on theoretical models (hard modelling) assuming either homogeneity [9,10] or heterogeneity [11,12] or other assumptions. In these theoretical studies, the diffusion coefficient of the free metal ion is considered to be much larger than that of the bound metal species and the kinetics of association–dissociation is widely discussed [9,10]. In the recent years, new multivariate data analysis methodologies (soft modelling) have been introduced for the study of the interactions between metal ions and ligands, in particular for macromolecular ligands [13,14]. Among these methods, multivariate curve resolution with alternating least squares (MCR–ALS) [15,16], has been proposed as a powerful tool for the investigation of voltammetric data from studies of metal ion complexation with different ligands [17–19]. The aim of the present paper is to extend the use of MCR–ALS to the investigation of voltammetric data from Cd/HF systems, as a complementary tool to the hard modelling approach. Although, some metal fulvic acid complexation systems have been already systematically studied by a variety of voltammetric techniques, to the best of our knowledge, a soft modelling approach as the one here proposed has been not previously attempted.

2. Experimental 2.1. Reagents and solutions A commercial humic acid (HA) from Fluka was used after sample pre-treatment as described by van den Hoop et al. [20]. The stock HA solution was prepared by dissolving 2.5 g of commercial HA in 1 l of decarbonated water and pH adjusted to 9.0 with a 0.1 M NaOH solution. This solution was stirred under nitrogen atmosphere for 24 h, after which 1 M HNO3 solution was added to reduced the pH to 3.0, and was left for 24 h with continuous stirring under a nitrogen

atmosphere. After this was centrifuged at 7600 rpm during 60 min, the pH of the supernatant was increased to seven, treated with an ion exchange to convert the material into the acid form and dialysed. The resulting solution was stored at 4 ◦ C in the dark. Stock solutions of Cd(II) were prepared by dilution of a Cd(II) standard solution for atomic absorption (1000 ␮g l−1 in 1% HNO3 ), supplied by Merck. Solutions of potassium nitrate, nitric acid and potassium hydroxide were prepared from analytical-reagent grade reagents (Merck). All solutions were prepared and diluted with ultrapure-water obtained from a Millipore MilliQ-Plus purification system. Purified nitrogen was used to remove dissolved oxygen from solutions prior to voltammetric analysis. 2.2. Apparatus and software An Autolab System (Ecochemie) attached to a Metrohm 663VA stand and a personal computer using GPES3 packages (Ecochemie) was used to perform the voltammetric measurements. A system with three electrodes was used: a hanging mercury drop electrode (HMDE) as working electrode, an Ag/AgCl reference electrode, and a glassy carbon auxiliary electrode. A Orion 720A pH meter was used for pH measurements. The calculations associated to the MCR–ALS method [15,16] were performed using several programs [21] implemented in MATLAB [22]. 2.3. Experimental procedure ASV titrations were performed using two different methodologies: (i) addition of Cd(II) to a solution containing fulvic acids, (ii) addition of HF to a solution containing Cd(II), as shown in Table 1. In all cases, the titrations were performed in the presence of 0.1 M KNO3 and pH 7.0. At this pH, most of the carboxylate groups binding sites of HF were deprotonated. In a typical measurement ASV experiment 20 ml of solution with a fixed KNO3 concentration was transferred to the cell and the pH was adjusted with HNO3 or KOH solutions. Nitrogen was passed through the solution for 10 min. The voltammograms were recorded from −900 to −500 mV under the following conditions: deposition potential, −900 mV; deposition time, 20 or 60 s; equilibration time, 30 s; pulse duration,

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Table 1 Experimental conditions of voltammetric titrations Titration A B C D

CHA (mol l−1 )a 10−7

CCd (mol l−1 ) 10−5

4× to 2.2 × 1 × 10−6 to 2×10−5 2.5 × 10−4 5 × 10−6

10−6

1× 5 × 10−7 4 × 10−7 to 1.5 × 10−5 1.25 × 10−7 to 3 × 10−6

tdep (s)b

Matrix

20 60 60 60

I1 I2 I3 I4

a This concentration corresponds to the concentration of the deprotonated groups at pH 7.0, determined by potentiometric titrations with KOH. b Deposition times for pre-electrolysis.

20 ms; pulse height, 50 mV; and scan rate, 5 mV s−1 . Successive additions of titrant were made and successive voltammograms were recorded after each titrand addition. During the measuring step a continuous flow of nitrogen was passed over the surface of solutions. 2.4. Data treatment

VT

and find matrix C which gives



min
Iˆ PCA − CV T
, V T constant C

For each titration, all the voltammograms measured after each addition of titrant, were arranged in a data matrix I . This matrix I had m rows (number of recorded voltammograms) and n columns (number of different potentials, where the intensities were measured). Assuming that every electroactive species contributes linearly to the whole measured signal (shown later), the bilinear decomposition of the data matrix I can be described by the following equation: I = CV T + R

derived from Eq. (1) and described by Eqs. (2) and (3) find matrix V T which gives



min
Iˆ PCA − CV T
, C constant (2)

(1)

where C is the matrix describing the concentration changes of every detected component (chemical species), V T the matrix describing the individual ‘pure’ voltammograms of these components, and R the residual matrix describing the variance not explained by CV T . The linear behavior assumption postulated by Eq. (1) is strictly valid for electrochemically inert systems [18,23] and it has been shown that it is also a good approximation in the study of different macromolecular systems using voltammetry [19,24]. The main task of the MCR–ALS method [15,16,25], is the estimation of C and V T matrices from the original I matrix without any previous assumption about the nature and/or composition of the system. This is achieved by solving iteratively two ALS problems

(3)

where Iˆ PCA is the principal component analysis (PCA) reproduced data matrix using a particular number of components. See [26] for more details about PCA and other factor analysis methods, which are not explained here for brevity; the singular value decomposition (SVD) [27] is also a common procedure to perform a similar matrix decomposition. Usually, the visual inspection either of the plot of the magnitude of eigen values or of the singular values of matrix I, already provides a first estimation of the number of the major linearly independent components. This number is a first indication of the number of components contributing significantly to the observed data variance and is confirmed afterwards using evolving factor analysis (EFA) [28,29] and the ALS optimization procedure described later. The unconstrained solutions of Eqs. (2) and (3) by least squares are: V T = C + Iˆ PCA

(4)

C = Iˆ PCA (V T )+

(5)

where (V T )+ and C + are the pseudo-inverses of V T and C matrices [27], respectively. However, in order to have physically meaningful solutions, a set of constraints are applied at each iteration, such as non-negativity [30,31] for V T and C matrices, unimodality [32,33] and selectivity [15,34]. Especially

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relevant for a correct resolution of species profiles in V T and C without rotational ambiguities are the selectivity and local rank constraints [15,34,35]. Constrained solutions finally obtained by MCR–ALS are then readily interpretable from a chemical point of view. In titrations A and B, closure constraints for the mass-balance of total metal concentration can also be applied. However, at very high concentrations of the ligand, adsorption phenomena of the complex at the electrode surface may hinder the fulfillment of the closure condition. The ALS iterative process described earlier by Eqs. (2)–(5) needs a starting with an initial estimation either of matrix C or V T . To obtain these initial estimations EFA was used [28,29]. As ASV voltammograms have a peak shape, EFA may be used in both measurement directions, following the evolution of the titration to obtain an initial estimation of concentration profiles or following the evolution of voltammograms (intensity at different potentials) to obtain an initial estimation of the pure individual voltammograms of the detected components. In this work, the second approach was preferred. Under closure constraints, once C matrix is obtained by the ALS iterative optimization, relative quantitative information about the resolved contributions is readily available. In particular, estimations of equilibrium concentration quotients between the different resolved contributions are easily evaluated from the columns of matrix C. In this way, approximate estimations of complexation constants are possible. On the other hand, qualitative interpretation of the resolved voltammetric contributions is also easily available from the rows of matrix V T . The main advantage of the proposed chemometric method is that it allows an easy interpretation of the obtained solutions C and V T , even though these solutions were obtained only under soft constraints like non-negativity, unimodality, closure or selectivity constraints. None of these constraints impose a pre-considered equilibrium hard model and therefore, the proposed approach should be considered a ‘self soft modelling’ approach in contrast to the so called ‘hard modelling’ approaches, often used in electrochemistry [1–12,36,37]. In previous works [15,16,34,38,39], it has been demonstrated that the simultaneous analysis of several experiments giving data matrices obtained for the same chemical system under different experimen-

tal conditions improve considerably the resolution power of the MCR–ALS method. In this work, the data matrices obtained in four independent titration experiments (data matrices I 1 , I 2 , I 3 and I 4 shown in Table 1) were arranged in an augmented column wise data matrix. This augmented data matrix has a number of rows equal to the total number of acquired voltammograms in the different titrations and it has a number of columns equal to the number of potentials in the measured voltammograms (the same potentials were measured in the four data experiments). This column wise data matrix augmentation assumes that common vectors span the column vector spaces of the different individual data matrices, i.e. that common voltammetric contributions are present in the four data sets. The linear model given by Eq. (1) can be easily extended to the simultaneous analysis of the four data matrices as follows:       I1 C1 R1  I 2   C 2  T  R2       I aug =   I 3  =  C 3  V +  R3  I4 C4 R4 = C aug V T + R aug

(6)

with the unconstrained ALS solutions of C aug and V T :     C1 I1  C2   I 2  T + T +    C aug =  (7)  C 3  =  I 3  (V ) = I aug (V ) C4 I4 

+   C1 I1     C I 2  2 + VT =   C 3   I 3  = (C aug ) I aug C4 I4

(8)

where I aug and C aug are the corresponding column wise augmented data and concentration matrices. The augmented concentration matrix, C aug contains the concentration profiles resolved for each data matrix I i , one on the top of each other, and the single matrix V T contains the pure individual voltammograms of the detected contributions in the four voltammetric experiments. Matrix augmentation (Eq. (6)) also helps to overcome possible rank deficiency and poor resolution problems [38,39]. In order to evaluate the fitting error in the reproduction of the original matrix using the solutions either found by PCA or by MCR–ALS,

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a lack of fit (lof) value was calculated using the following equation:   i,j (I i,j − Iˆ i,j )2 × 100 (9) lof(%) =

 2 i,j I i,j where I i,j and Iˆ i,j represents the current intensity at voltammogram i and potential j in the experimental and calculated matrices, respectively. This magnitude allows an easy comparison between different models and methods in the explanation of the same data set. It is preferred to other fitting values based on variance error, since it is directly interpretable and it is in the same units as the data are (in current intensities relative units). 3. Results and discussion Fig. 1 shows the set of voltammetric curves obtained in titration A (Table 1). As it can be seen, along that titration (ligand addition), a progressive

295

shift of the peak maxima towards negative potential is observed as well as a decrease in the peak height. The same sort of results were obtained for titration B, whereas for titrations C and D, corresponding to the addition of Cd(II) to a HF solution, the observed behavior was the opposite: a systematic shift of the peak maxima to more positive potentials and an increase of the peak height. This behavior can be attributed to the complexation of Cd(II) by HF, and it is characteristic of labile systems with diffusion coefficients of metal complexes (DML ) lower than the diffusion coefficient of free metal ion (DM ). Moreover, in these systems, peak current becomes constant when the concentration of the metal complex is equal to the total concentration of metal [40]. Thus, the systematic shift observed for the peak potential is a consequence of the labile complexation process. If EpM and EpM+L are the peak potentials of Cd(II) in the absence and presence of fulvic respectively, the magnitude of the potential shift should be related to the stability constants of the formed complexes. At the beginning of the titration, when the

Fig. 1. Voltammetric curves obtained in titration A (Table 1). From the beginning of the titration (right side) to the end of the titration (left side), the ligand (fulvic acid)-to-metal concentration ratio increases and Cd(II) ion is more complexed.

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ligand-to-metal concentration ratio was low, the potential shifts were also small. On the contrary, at the end of the titration, when there was a larger excess of the ligand concentration over the metal concentration in the bulk (see Table 1), and the metal was more strongly complexed, the potential shifts were larger. In the same way, if ipM and ipM+L are the peak currents in the absence and presence of ligand, the ratio ipM+L /ipM decreased when the ligand-to-metal concentration ratios increased and the complexation was stronger. This fact is a result, essentially, to the much lower diffusion coefficient of the metal complex when compared to that of the free metal ion (D ML < D M ) [41]. Besides that, it must also be mentioned that the half-peak widths for the complex species are higher than for the free metal ion. This could be due to two facts: (i) the existence of overlapping (non-resolved) signals (see further), and/or (ii) the absence of an

excess of ligand near the electrode surface during the oxidation of Cd(Hg) in the stripping step [7]. Calculations according to Mota et al. [7], considering the metal ion concentration present on the electrode surface during the stripping step, showed that the excess of ligand condition is not attained in this case for most of the sample solutions during the titration. This is a common situation in titration experiments where large metal-to-ligand ratios are covered in detail. Most of the hard modelling approaches however, assume excess of ligand conditions, since then theoretical solutions of the problem are not so complex. It is for this reason that the application of MCR–ALS to this sort of titration data is still more interesting and it has an added value. Fig. 2 shows the normalized singular values for each individual data matrix. Larger singular values were obtained for matrices I 1 and I 2 than for matrices I 3 and I 4 . Third singular value was only clearly

Fig. 2. Singular value decomposition (SVD) of the different individual data matrices I 1 (titration A), I 2 (titration B), I 3 (titration C), and I 4 (titration D, see Tables 1 and 2) and of the augmented data matrix [I 1 ; I 2 ; I 3 ; I 4 ]; simultaneous analysis of the four titrations A, B, C and D.

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Table 2 Percentage of lack of fit values between the experimental data matrix and the reproduced data matrix by PCA and ALS Matrix I1 I2 I3 I4 [I 1 ; I 2 ; I 3 ; I 4 ]f

PCA (2)a (% lof)b 9.98 4.07 4.33 4.88 10.46

PCA (3) (% lof)

ALS (2) (% lof)

ALS (3) (% lof)

2.80 1.18 3.30 3.71 4.32

11.14c

4.47c 1.21d –e –e 5.75

4.14d 4.60d 5.35d 11.00

a

Number of components considered in the calculation are given in parenthesis. Lack of fit values calculated by following the expression given in Eq. (9). c Lack of fit values obtained using the following constraints: non-negativity (concentrations and voltammograms), unimodality (concentrations and voltammograms), closure (concentrations, i.e. mass balance) and selectivity (at the beginning of the titration only free metal is present). d Lack of fit values obtained using the following constraints: non-negativity (concentrations and voltammograms), unimodality (only to voltammograms). Selectivity could only be applied to data matrix I 2 (like in I 1 ). e Optimization with three components fail in this case, giving lof values larger than with two components and with totally unreasonable profiles. f Simultaneous analysis of the four experimental titrations (augmented data matrix [I ; I ; I ; I ]) using Eq. (6). 1 2 3 4 b

distinguishable for I 1 matrix. The more important conclusion from these plots is that the assumption of experimental data to behave in a bilinear way according to Eq. (1), is fully supported from pure mathematical analysis of experimental data. Only a reduced set of linear components are necessary to describe satisfactorily well the experimental data at constant pH using a bilinear model. Therefore, no strong deviations from linearity appear to be present in the experimental data. In previous works [19] with well defined macromolecular ligands, this was also observed, even for those cases where the theoretical electrochemical approach assumes that the complexation behavior is labile and therefore, with the presence of non-linear contributions. However, in practice, such non-linearities resulted to be low, compared with those from the linear part of the signal, and the data analysis and interpretation was still possible using the proposed MCR–ALS approach. To further confirm this reasoning, in Table 2, PCA [26] considering two or three components in the individual analysis of the different data matrices are given in the first two columns of the Table 2. These results showed that three components already explained satisfactorily the data variation of matrices I 1 and I 2 , with percentages of PCA lof of only 2.80 and 1.18%, respectively. Table 2 also shows that the change in unexplained data variance for matrices I 3 and I 4 when either two or three components were considered was rather small, around only a 1% of change, which

clearly indicates that the resolution of a third component for these two titrations should be very difficult. SVD of the augmented data matrix including the simultaneous analysis of the four voltammetric titrations ([I 1 , I 2 , I 3 , I 4 ], see Eq. (6)) was given also in Fig. 2. As for the individual analysis of data matrices I 1 and I 2 , three major components were detected showing that whatever is the titration approach (adding metal-to-ligand or ligand-to-metal) the four data sets can be well explained assuming the contribution of the same three (common) major components, with a percentage of total unexplained variance (lof) of only 4.32%. 3.1. MCR–ALS individual analysis of the different voltammetric titrations Experimental data matrices obtained in the course of voltammetric titrations were first analyzed using EFA [28,29]. Due to the known expected peak shape of the voltammograms, EFA was applied in the direction of the change of potential (i.e. columns of data matrix, D), and not in the direction of concentration changes (i.e. rows of data matrix, D), as it is traditionally performed in spectroscopic titrations [13,14] or in chromatography. In this way, initial estimates provided by EFA gave good initial estimations of the voltammetric contributions. These EFA estimations were used to initiate the ALS optimization process. The constraints imposed during the ALS optimization procedure were

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Fig. 3. Resolved voltammograms for Cd(II) ion (curve I) and for Cd(II)–HF complexes (curves II and III) in the MCR–ALS analysis of data matrix I 1 (titration A).

non-negativity and unimodality, both for concentration and voltammetric profiles. Additionally, for matrices I 1 and I 2 , selectivity and closure constraints [15] can be applied too. However, the closure constraint is difficult to be totally fulfilled if, as often is the case, a certain unknown amount of metal ion is loosed by adsorption on the electrode surface. Selectivity, on the other hand, can be applied at the beginning of the titrations A and B, because in these two titrations it is known that only the free metal ion species was present at the beginning of the titration. Fig. 3 shows the voltammetric contributions for the three considered components, resolved by MCR–ALS of matrix I 1 . Voltammogram I in Fig. 3 can be easily assigned to the reoxidation of Cd(Hg) to free Cd2+ with a characteristic peak potential around −0.68 V. Voltammograms II and III, appeared at more negative potentials than voltammogram I, and they can be associated to the formation of two different type of complexes between Cd(II) and HF. The peak potentials of these two species appeared approximately at −0.71 and −0.75 V, respectively. Since HF have a large number of different types of complexing sites, voltammograms II and III in Fig. 3 should not be directly interpreted as voltammetric sig-

nals of a single specific complex but as an average of electrochemical responses related to the formation of two major different types of complexes between Cd(II) and HF. According to previous studies [1–12,36,37], HF present in humic substances contains a complex mixture of potential binding sites. Due to their heterogeneity and to electrostatic and conformational effects, it is not possible to assign unambiguously macroscopic stability constants for the species at equilibrium, since they can be only valid at microscopic level. However, it is usually accepted [1–12], that most of the binding sites of these heterogeneous ligands are rather weak (like carboxylate and hydroxylate groups), but that they also contain a lower but significant number of stronger binding sites (like o-dicarboxylate and o-hydroxylcarboxylate groups). Thus, despite of fulvic acids having large number of functional groups with different affinities, the voltammetric experimental data variance can be easily interpreted considering only two major contributions. One of these two contributions (component II) is the metal ion more weakly bound to HF and the other contribution (component III) is the metal ion more strongly bound to HF. The corresponding resolved concentration profiles of these three components are shown in Fig. 4. As

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Fig. 4. Resolved concentration profiles for Cd(II) ion (curve I) and for Cd(II)–HF complexes (curves II and III) in the MCR–ALS analysis of data matrix I 1 (titration A).

expected, the contribution of the component I (free Cd(II)) to the global electrochemical signal decreased when the HF concentration increased. At the same time, the concentration of component II increased first, followed by its decrease and the simultaneous increase of component III concentration. At the beginning of the titration, at low CL /CM bulk values, the percentage of available ligand weak binding sites will be large compared to the fewer strong binding sites [36,37]. Therefore, at low CL /CM bulk values, weak binding sites will be pre-dominantly occupied and the MCR–ALS resolved component II in Figs. 3 and 4 describe the concentration profile of the metal ion bounded to the weakest (but major) binding sites of HF. With the increase of HF concentration and CL /CM ratio, the number of stronger binding sites available

for complexation will also increase considerably, and the stronger complexing sites will be preferentially occupied. Accordingly, the contribution of component II (metal interaction with weak binding ligand sites) will decrease and the contribution of the metal bound to the stronger binding ligand sites (component III in Fig. 4) becomes more important. Summarizing, the reasons why formation of complex II is considered to be weaker than formation of complex III are: (i) its formation constant deduced from concentration profiles is lower (see later), (ii) peak shifts observed for complex II are less negative than for complex III, and (iii) our previous external chemical knowledge about the much higher abundance of weak binding sites than strong binding sites in fulvic acids (shown earlier and in [1–7]).

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From the resolved concentration profiles in Fig. 4, a rough estimation of the apparent [36] complexation constants or equilibrium concentration quotients was attempted. The following equations were used for the evaluation of the two constants corresponding to species II and III:

[ML] log K1 = log (10) [M][L∗ ] and log K2 = log



[ML ] [M][L∗ ]

(11)

where [M] corresponds to MCR–ALS resolved concentration profile for the free metal ion (profile I) in Fig. 4, [ML] corresponds to MCR–ALS resolved concentration profile for the first metal complex (profile II) in Fig. 4 and [ML ] corresponds to MCR–ALS resolved concentration profile for the second metal complex (profile III) in Fig. 4. [L∗ ] corresponds to the

concentration of free ligand binding sites evaluated from the amount of ligand added during the titration (considering that all of them are not protonated and available for complexation at pH 7.0) and subtracting the fraction of them that were already bound to the metal ion (ML and ML complexation). In Figs. 5 and 6, the log Kapp values obtained by Eqs. (10) and (11) are represented as a function of the increase ligand concentration. The average log Kapp values were 5.5 and 6.4, respectively. These values are of the same order than those previously reported for the complexation of similar samples of fulvic acids with log K values between 5 and 7.5 [11]. For matrix I 2 , the results followed the same trends as for matrix I 1 , with the formation of three components, two major metal complexes and the free metal ion. However, for matrices I 3 and I 4 , as it was already shown in SVD plots (Fig. 2) and confirmed by MCR–ALS, two components could only be resolved. In these two titrations (C and D, where a fixed amount

Fig. 5. Rough estimation of first apparent complexation constant (log K1 in Eq. (10) from MCR–ALS resolved profiles corresponding to components I and II from Fig. 4 (titration A). The average log K1 value was 5.5 with a standard deviation of 0.1 log units.

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Fig. 6. Rough estimation of first apparent complexation constant (log K2 in Eq. (11)) from MCR–ALS resolved profiles corresponding to components I and III from Fig. 4 (titration A). The average log K2 value was 6.3 with a standard deviation of 0.1 log units.

Fig. 7. Resolved voltammograms for Cd(II) ion (curve I) and for Cd(II)–HF complexes (curves II and III) in the MCR–ALS analysis of augmented data matrix [I 1 ; I 2 ; I 3 ; I 4 ]; simultaneous analysis of data from titrations A, B, C and D using Eq. (6).

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of ligand was titrated by the metal ion) the heterogeneous character of the ligand is emphasized and only two components could be resolved, one related with the free metal ion and another one related with the complexation of metal ion. This conclusion was confirmed by inspection of the voltammograms and concentration profiles obtained for these two components by MCR–ALS. These results could be interpreted considering that the metal complex component was representative of the whole complexation process, strong and weak, and that it was not possible to differentiate between them under the experimental conditions used for these two individual titrations. It should be noted that titration A was the one with optimal condition for the evaluation of equilibrium quotients given in Figs. 5 and 6.

3.2. MCR–ALS simultaneous analysis of the different voltammetric titrations Figs. 7 and 8 gives respectively the MCR–ALS resolved voltammetric and concentration profiles of the three components obtained in the simultaneous analysis of the four titrations A, B, C and D (see Table 1) using the model described in Eq. (6) (augmented matrix [I 1 ; I 2 ; I 3 ; I 4 ]). All four data matrices were first normalized to the same maximum current intensity (equal to one). Concentration profiles of Fig. 8 were obtained considering only the mass balance of Cd(II) ion for first titration A (the total concentration of Cd(II) in this titration was equal to 1 × 10−6 M). This concentration was used as a closure constraint during the MCR–ALS optimization. As a consequence of that,

Fig. 8. Resolved concentration profiles for Cd(II) ion (curve I) and for Cd(II)–HF complexes (curves II and III) in the MCR–ALS analysis of augmented data matrix [I 1 ; I 2 ; I 3 ; I 4 ]; simultaneous analysis of data from titrations A, B, C and D using Eq. (6).

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the concentration of the different resolved components in the titrations B, C and D in Fig. 8 (y-axis) are in relation to the Cd(II) ion concentration in titration A. As the deposition times for pre-electrolysis were different for the different titrations and also because of the normalization procedure used for the different data matrices (shown earlier), no reliable quantitative relationship was possible between these three experiments. Therefore, for these three titrations, results should be compared only qualitatively and in relation to what was obtained for titration A. For data matrices I 1 and I 2 , good agreement was obtained between the concentration and voltammetric profiles resolved by MCR–ALS using both approaches, the individual (Eq. (1)) and the simultaneous analysis (Eq. (6)). Moreover, in the simultaneous analysis of the four matrices, the resolution of three contributions was also possible in titrations C and D, which was not possible in the individual data analysis of these two matrices (only two contributions were resolved in each case). These results point out the relevance of the simultaneous analysis approach to overcome some limitations observed in the analysis of single titration experiments. In the individual analysis of I 3 and I 4 matrices, the resolution of three contributions was much more difficult, both because of the lower ligand concentrations achieved in the titration) which did not push towards stronger complexation) and also because of the much higher overlap and embedding nature of their respective concentration profiles as it is clearly seen in Fig. 8. Thus, the MCR–ALS resolution power increases when it is applied to the simultaneous analysis of different voltammetric titrations containing complementary information.

4. Conclusions A new chemometrics methodology is proposed for the analysis of voltammetric data in the study of metal complexation by fulvic acids. The proposed MCR method allowed the detection and resolution of a reduced number of contributions, with their corresponding changes in concentration and voltammetric responses. These results also indicate that methods like PCA, EFA and MCR can be very useful and powerful to investigate voltammetric data obtained during metal complexation studies of fulvic acids.

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Therefore, the results obtained in this study are intended to contribute to the description, characterization, and macroscopic modelling of interaction processes between fulvic acids and metal ions using voltammetric techniques. Moreover, this soft modelling approach can be very useful complementary tool to the hard modelling approaches developed until now.

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