Journal of Colloid and Interface Science 258 (2003) 116–122 www.elsevier.com/locate/jcis
On the determination of the critical micelle concentration by the pyrene 1:3 ratio method J. Aguiar, P. Carpena, J.A. Molina-Bolívar, and C. Carnero Ruiz ∗ Grupo de Fluidos Estructurados y Sistemas Anfifílicos, Departamento de Física Aplicada II, Escuela Universitaria Politécnica, Universidad de Málaga, Campus de El Ejido, E-29013 Málaga, Spain Received 21 December 2001; accepted 16 October 2002
Abstract The aim of this paper is to establish a simple and accurate approach to the treatment of pyrene 1:3 ratio data in the context of critical micelle concentration determination in surfactant solutions. The procedure we propose is based on the assumption that pyrene 1:3 ratio data are properly fitted by a Boltzmann-type sigmoid. From the fitting parameters we consider two singular points as possible candidates to determine the critical micelle concentration of the surfactant, adopting objective criteria for the election of one or the other point. With the purpose of supporting our contention, numerous specific examples are presented and discussed, including single ionic and nonionic surfactants, micellization of surfactants in the presence of additives, and several mixed-surfactant systems. In all the cases examined the experimental data were well fitted by a decreasing sigmoid of the Boltzmann type, and the proposed approach worked in an appropriate way, providing critical micelle concentration values in good agreement with those in the literature obtained using different methods. 2003 Elsevier Science (USA). All rights reserved. Keywords: CMC determination; Fluorescence probe technique; Pyrene 1:3 ratio method
1. Introduction Since Kalyanasundaram and Thomas [1] proved that the characteristic dependence of the fluorescence vibrational fine structure of pyrene could be used to determine the critical micelle concentrations (CMC) of surfactant solutions, the so-called pyrene 1:3 ratio method has become one of the most popular procedures for the determination of this important parameter in micellar systems [2]. This method has been widely used not only in pure micellar solutions, but also in mixed-surfactant systems [3–8], to investigate polymer– surfactant interactions [9–12], and in studies on the effect of additives on the micellar properties of ionic and nonionic surfactants [13–16]. It is well known that the plots of the pyrene 1:3 ratio as a function of the total surfactant concentration show, around the CMC, a typical sigmoidal decrease. Below the CMC the pyrene 1:3 ratio value corresponds to a polar environment; as the surfactant concentration increases the pyrene 1:3 ratio decreases rapidly, indicating that the pyrene is sensing a * Corresponding author.
E-mail address:
[email protected] (C. Carnero Ruiz).
more hydrophobic environment. Above the CMC, the pyrene 1:3 ratio reaches a roughly constant value because of the incorporation of the probe into the hydrophobic region of the micelles. A problem arises from the fact that there is not an objective and unified method to obtain the CMC value from the plots of pyrene 1:3 ratio against surfactant concentration, and different authors seem to take different criteria to choose this point. Zana and co-workers have extensively considered this problem. According to these authors, the CMC values can be obtained using two approaches: (i) from the interception of the rapidly varying part and the nearly horizontal part at high concentration of the pyrene 1:3 ratio plots [17,18]; (ii) from the inflection point of the pyrene 1:3 ratio plots, for surfactants with very low CMCs, typically below 1 mM [6]. This different criterion has been explained in terms of pyrene partition between micelles and bulk phase [19]. In effect, when the volume of the hydrophobic pseudophase is not voluminous enough, pyrene partitions between this phase and the bulk, thereby providing an average value of the pyrene 1:3 ratio index corresponding to a more polar environment, and consequently reflecting an overestimated CMC value. It must be pointed out, however, that the procedure followed by the aforementioned authors to choose between one or the other approach is empirical in
0021-9797/03/$ – see front matter 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0021-9797(02)00082-6
J. Aguiar et al. / Journal of Colloid and Interface Science 258 (2003) 116–122
nature. On the other hand, the problem of a small volume of the hydrophobic phase to solubilize enough pyrene could result in a limitation on the application of the pyrene 1:3 ratio method, so this method could not be applied in the case of surfactants with a very small CMC. Although the scope of the present paper refers to lowmolecular-weight surfactant micelles, it must be pointed out that considerable efforts have been made to implement the pyrene 1:3 ratio method in the field of the block copolymer micelles [20–22]. Since these systems present important structural differences with surfactant micelles of low molecular weight [20], the analysis of the pyrene 1:3 ratio data is more complex. Readers interested in this field can find further interesting discussions in Refs. [20–22]. As the CMC is the most important parameter in investigations concerning the micellization of surfactants, and in order to ease the comparison of the experimental results obtained from different laboratories, the convenience of establishing a standard procedure to obtain the CMC from the pyrene 1:3 ratio versus concentration plots is evident. This is the major aim of this paper, that is, to find an easy analytical method to treat the pyrene 1:3 ratio data, allowing CMC determination on the basis of objective criteria. Throughout this paper, we propose a simple approach to the treatment of pyrene 1:3 ratio data, and we illustrate the application of the procedure to a number of experimental data, which are analyzed and discussed in the context of CMC determination.
2. Experimental The surfactants used in this study, which are listed in Table 1, were acquired from Sigma, except sodium dodecyl sulfate and octaethylene glycol monododecyl ether, which were purchased from Fluka. In all cases these surfactants were used as received. The fluorescence probe pyrene, from Sigma, was also used without pretreatment. All other chemicals used were of analytical grade, water was doubly distilled (Millipore), and all experiments were carried out with freshly prepared solutions. Fluorescence measurements were performed in a FluoroMax-2 (Spex) spectrofluorometer. Fluorescence emission spectra of a number of surfactant solutions containing 1–2 µM of pyrene were recorded using an excitation wavelength of 335 nm, and the intensities I1 and I3 were mea-
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sured at the wavelengths corresponding to the first and third vibronic bands located near 373 and 384 nm. The ratio I1 /I3 is the so-called pyrene 1:3 ratio. All fluorescence measurements were carried out at 25.0 ± 0.1 ◦ C. The conductivity measurements were performed with a Metrohm 712 digital conductometer using a dip-type cell of cell constant 0.88 cm−1 . All measurements were done in a jacketed vessel, which was maintained at 25.0 ◦ C (±0.1 ◦ C) with a circulating water thermostat bath.
3. Treatment of pyrene 1:3 ratio data As stated above, Zana and co-workers [6,17,18] suggest that the CMC can be alternatively obtained from two singular points in the pyrene 1:3 ratio plots. In this section we propose a straightforward procedure to determine these points. Our approach is based on the assumption that the pyrene 1:3 ratio plots can be adequately described by a decreasing sigmoid of the Boltzmann type, which is given by A1 − A2 + A2 , (1) 1 + e(x−x0 )/ x where the variable y corresponds to the pyrene 1:3 ratio value, the independent variable (x) is the total concentration of surfactant, A1 and A2 are the upper and lower limits of the sigmoid, respectively, x0 is the center of the sigmoid, and x is directly related to the independent variable range where the abrupt change of the dependent variable occurs. Figure 1 illustrates the meaning of these parameters. In this figure we have noticed two singular points, labeled as (xCMC )1 and (xCMC )2 . These points correspond to those mentioned above. That is to say, (xCMC )1 should give the CMC values when the surfactant has a very low CMC value, and (xCMC )2 the corresponding values in the remaining cases. That is clear that the (xCMC )1 value is the center of the sigmoid, x0 , and it is given as a fit parameter of the experimental data to Eq. (1), but the value of (xCMC )2 must be analytically determined. From Fig. 1, it is observed that (xCMC )2 can be obtained from the intersection of the straight lines, y2 = A2 and the tangent to the sigmoid passing through its center, that is, the line y3 = f (x). Let us obtain, first, the equation of y3 . The slope of the tangent line at the sigmoid center is dy A2 − A1 . (2) = dx x=x0 4 x y=
Table 1 Studied surfactants together with the CMC literature values at 25 ◦ C Surfactant Anionic Cationic
Nonionic
Name
Abbreviation
CMC (mM)
References
Sodium dodecylsulfate Dodecyltrimethylammonium bromide Tetradecyltrimethylammonium bromide Cetyltrimethylammonium bromide Triton X-100 Octaethylene glycol monododecyl ether Tetraethylene glycol monododecyl ether
SDS DTAB TTAB CTAB TX-100 C12 E8 C12 E4
8.20 14.6–16.0 3.60–3.72 0.92–1.00 0.24–0.27 0.11 0.064
[1,15,23] [23–25] [2,24,25] [2,14,23] [1,2,18] [6,23] [23]
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this question than analyzing experimental results by the proposed procedure.
4. Results and discussion 4.1. Single-surfactant systems
Fig. 1. Decreasing sigmoid of the Boltzmann type showing its characteristic parameters, as well as two singular points labeled as (xCMC )1 , center of the sigmoid, and (xCMC )2 , which corresponds to the intersection of the straight lines y2 = A2 and y3 = f (x).
The equation of y3 = f (x) can be written in general as y3 = a + bx,
(3)
where the slope b is given for Eq. (2). As this line passes through the center of the sigmoid, (x0 , y0 ), it is possible to determine the parameter a, so that, after reorganization, Eq. (3) becomes A1 + A2 A1 − A2 A1 − A2 + x0 − x. (4) 2 4 x 4 x We can now determine the interception between the lines y2 = A2 and y3 = f (x). In effect, y3 =
A1 + A2 A1 − A2 A1 − A2 + x0 − (xCMC )2 = A2 , (5) 2 4 x 4 x from which, after simplifying some terms, we finally obtain (xCMC )2 = x0 + 2 x.
(6)
Therefore, fitting the pyrene 1:3 ratio data versus the surfactant total concentration to Eq. (1), we can obtain A1 , A2 , x0 , and x as fit parameters and, subsequently, (xCMC )1 or (xCMC )2 . However, at this moment we have two problems: First, to demonstrate that the pyrene 1:3 ratio data are appropriately described by a Boltzmann-type sigmoid, and second, to establish a criterion to choose between (xCMC )1 or (xCMC )2 in an objective way. There is no other way of solving
To test the above treatment we have selected some surfactants (see Table 1), frequently considered as model systems, which have highly reliable CMC data. In addition, the CMC values of the ionic surfactants studied in this investigation have been analyzed by the common conductance method, as this technique is available in our laboratory. In Table 2 we present the most significant fit parameters for all singlesurfactant systems studied, including the regression-square (r 2 ) and the chi-square (χ 2 ) coefficients, together with the (xCMC )1 and (xCMC )2 values, and the CMCs as obtained from the conductance method for the ionic surfactants. In the same table we have included the ratio x0 / x with the purpose of analyzing whether this provides some interesting information. Representative results for SDS and DTAB, from both pyrene 1:3 ratio and conductance methods, are presented in Fig. 2. Figure 3 shows the pyrene 1:3 ratio plot for TX-100 and C12 E8 . There is a clear difference between the cases in Fig. 2 and those in Fig. 3. Note that whereas the micellar formation of ionic surfactants (Fig. 2) is accompanied by an abrupt decrease in the pyrene 1:3 ratio values, in the case of the nonionic ones that process occurs in a more gradual way. This behavior can be explained by two different ways: (i) pyrene is associated with some kind of premicellar aggregates, which at higher surfactant concentrations are converted into micelles; and (ii) pyrene and the surfactant may self-associate into small aggregates that do not exist in the absence of pyrene, pyrene being responsible for their formation. From Figs. 2 and 3 and Table 2 it seems obvious that the pyrene 1:3 ratio data for the studied surfactants are well fitted by a Boltzmann-type sigmoid. In addition, it is observed that, in the case of ionic surfactants, the found values for (xCMC )2 are very close to the CMC obtained by the conductance method and also to the literature values (see Table 1). However, for nonionic surfactants (xCMC )1 stands better for the CMC value. From data in Table 2 additional
Table 2 Fitting parameters of pyrene 1:3 ratio data to Eq. (1) for the investigated surfactants: n is the number of points used in the fit, x0 is the center of the sigmoid,
x is the time constant, r 2 and χ 2 have their usual meaning, and CMC is the critical micelle concentration as obtained by the conductance method Surfactant
n
x0
x
r2
χ2
(xCMC )1 (mM)
(xCMC )2 (mM)
CMC (mM)
x0 / x
2.8 × 10−4 6.8 × 10−5 1.7 × 10−4 1.4 × 10−4
7.62 13.6 3.40 0.897
8.18 15.14 3.82 1.011
8.22 15.2 3.75 0.98
27.21 17.66 16.19 15.74
1.9 × 10−5 2.8 × 10−5 7.0 × 10−5
0.245 0.110 0.059
0.345 0.148 0.095
– – –
4.90 5.09 3.28
SDS DTAB TTAB CTAB
15 11 14 11
7.62 13.6 3.40 0.897
0.28 0.77 0.21 0.057
0.9970 0.9985 0.9975 0.9978
TX-100 C12 E8 C12 E4
12 15 15
0.245 0.110 0.059
0.050 0.019 0.018
0.9995 0.9958 0.9983
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Fig. 4. Plots of pyrene 1:3 ratio versus concentration of TX-100 in ethylene glycol–water mixtures of different composition: (•) 10 wt%, (◦) 20 wt%, () 30 wt%, and ( ) 40 wt%.
4.2. Micellization of surfactants in the presence of additives
Fig. 2. Plots of pyrene 1:3 ratio (•) and conductivity (◦) versus surfactant concentration for (a) SDS and (b) DTAB. Arrows indicate the CMC values of the respective surfactants.
Fig. 3. Plots of pyrene 1:3 ratio versus concentration of surfactant for TX-100 (•) and C12 E8 (◦). Arrows indicate the CMC values of both surfactants.
interesting conclusions can be extracted. Note that the x parameter value does not reflect satisfactorily the different behavior of ionic and nonionic surfactants commented on above. Nevertheless, if we analyze the behavior of the ratio x0 / x, it is observed that in the case of ionic surfactants this ratio is higher (typically > 10) than for the nonionic ones. Therefore, this coefficient, x0 / x, can be useful to establish an objective approach when selecting a singular point able to provide a reliable CMC value in the pyrene 1:3 ratio plots.
With the purpose of studying whether the trend observed in the preceding section is followed by more complex systems we have decided to examine two different cases: the micellization of TX-100 in ethylene glycol (EG)–water mixtures and of CTAB in the presence of increasing urea concentrations. Figure 4 shows the pyrene 1:3 ratio plots for TX100 in solvent systems with increasing EG content, and Table 3 summarizes the fitting parameters of the curves in Fig. 4 together with the same parameters discussed in the previous section. As can be seen from data in Fig. 4, the addition of EG produces two effects. First, it inhibits the micellar formation of TX-100, as revealed by an increase in the CMC. Second, the reduction in the slope of the pyrene 1:3 ratio plots, around the CMC, indicates that the micellization process results to be a less cooperative process as the EG content increases in the solvent system. These effects, which have recently been investigated [26], are a consequence of the ability of EG to act as a structurebreaking agent and of the interactions of the cosolvent with the oxyethylene groups of the surfactant. From both Fig. 4 and data in Table 3, it can be seen that the experimental data are also well fitted by a sigmoid of the Boltzmann type. In Table 3 we have included, for comparison, CMC data obtained for the same system from both surface tension measurements [26] and a spectroscopic procedure [27]. By comparing the results presented in Table 3, it is evident that the CMC values obtained from (xCMC )1 are very similar, saving the inherent differences to the utilization of different experimental techniques, to those previously reported. In addition, the found values of x0 / x for this system are significantly smaller than 10 and comparable to those of nonionic surfactants. Therefore, according with the observations in the preceding section, the election of (xCMC )1 as the CMC value of TX-100 in solvent systems formed by EG–water mixtures would be justified.
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Table 3 Fitting parameters of pyrene 1:3 ratio to Eq. (1) for TX-100 in EG–water mixtures (symbols as in Table 2) n
EG (wt%) 0 10 20 30 40
12 13 18 13 14
x0 0.245 0.256 0.394 0.636 1.067
x 0.050 0.055 0.084 0.144 0.288
r2
χ2
(xCMC )1 (mM)
(xCMC )2 (mM)
CMCa (mM)
CMCb (mM)
x0 / x
0.9995 0.9977 0.9985 0.9963 0.9994
1.9 × 10−4
0.245 0.256 0.394 0.636 1.067
0.345 0.366 0.562 0.924 1.643
0.238 0.280 0.347 0.560 0.740
0.240 0.292 0.389 – 0.991
4.90 4.65 4.69 4.41 3.70
8.8 × 10−5 5.0 × 10−5 8.3 × 10−5 1.7 × 10−5
a Data obtained from surface tension measurements (Ref. [26]). b Data obtained using a spectroscopic method with EG content expressed in vol% (Ref. [27]).
Table 4 Fitting parameters of pyrene 1:3 ratio data to Eq. (1) for CTAB in different aqueous solutions of urea (symbols as in Table 2) Urea (M)
n
x0
x
r2
χ2
(xCMC )1 (mM)
(xCMC )2 (mM)
CMCa (mM)
CMCb (mM)
x0 / x
0 2 4 6
11 12 12 14
0.897 1.000 1.277 2.489
0.057 0.088 0.136 0.158
0.9978 0.9983 0.9971 0.9961
1.4 × 10−4 1.0 × 10−4 1.1 × 10−4 2.0 × 10−4
0.897 1.000 1.277 2.489
1.011 1.177 1.549 2.805
0.98 1.20 1.73 3.10
1.0 1.1 1.6 –
15.74 11.30 9.39 15.75
a Data obtained from conductance measurements (Ref. [14]). b Data obtained from fluorescence measurements using pyrene-3-carboxylaldehyde as a probe (Ref. [14]).
We have also analyzed micellization data of CTAB in the presence of high urea concentrations, previously obtained in our laboratory [14]. Table 4 shows the corresponding fitting parameters for this system. As can be seen in this case, the (xCMC )2 values compare well with the CMC values obtained from conductance and fluorescence probe techniques (using pyrene-3-carboxylaldehyde as a probe). In contrast to the previous case, the x0 / x values are, in general, greater than 10 and similar in magnitude to those of ionic surfactants. Consequently, it seems that the behavior of the ratio x0 / x for more complex systems, such as those presented in this section, also follows a tendency similar to that which is seen in the case of the single-surfactant systems. Although we have not presented the corresponding plots for this system, from data in Table 4 it is possible to infer that, besides the inhibition of micellar formation, the micellization process occurs in a more gradual way as the urea concentration increases, as indicated by the increasing values of x. Recently, a study on the effect of KCl on micellar properties of TX-100 has been carried out in our laboratory [16]. We have examined the corresponding pyrene 1:3 ratio data under the present approach, finding that the above condition, in relation to the ratio x0 / x, is also fulfilled. In this case, the ratio x0 / x is smaller than 10, which requires the election of (xCMC )1 as the CMC value. 4.3. Mixed-surfactant systems In the case of surfactant mixtures, the determination of the CMC has an added interest. This is because some properties of mixed-surfactant systems, such as the composition of the micellar phase, cannot be measured directly, but must be estimated using proper mixing thermodynamic models, and these models are based on the knowledge of the CMC over the whole range of composition. Moreover, it is of interest
Fig. 5. Plots of pyrene 1:3 ratio versus total surfactant concentration for the TX-100/SDS system at different mole fractions of cosurfactant in the solution (αSDS ): () 0.1, () 0.3, () 0.5, ( ) 0.7, (•) 0.8.
and of considerable practical importance to know the CMC and the composition of the micellar pseudophase because, unlike single-component surfactant systems, these two quantities exhibit substantial variations as the total surfactant concentration is increased [28]. As mentioned above, the pyrene 1:3 ratio method has been used to determine the CMC in mixed surfactant systems. In fact, in recent contributions from our laboratory we have employed this method to study the micellar formation of TX-100 with SDS, CTAB, and C12 E8 [7] and to examine the effect of alkyl chain length of n-alkyltrimethylammonium bromides on the formation of mixed micelles with TX-100 [8]. Figure 5 shows representative pyrene 1:3 ratio vs concentration plots for the mixed system formed by TX-100 and SDS. In the mentioned studies [7,8], all the CMC values were obtained from the interception of the rapidly varying part and the nearly horizontal part at high concentration of the pyrene 1:3 ratio plots, that is, from
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Fig. 6. Variation of the x0 / x ratio as a function of the mole fraction of cosurfactant in the solution (α2 ) for different mixed surfactant systems formed by TX-100 with SDS (•), C12 E8 (◦), DTAB (), TTAB ( ), and CTAB ( ). The dashed line indicates the value of the x0 / x ratio below which the point (xCMC )1 must be taken as the CMC.
(xCMC )2 . In order to examine the influence of the election of (xCMC )1 instead of (xCMC )2 on the interaction between the two surfactants in the mixed micelle, we have analyzed our pyrene 1:3 ratio data in some representative cases under the present approach. We have applied the described treatment to our pyrene 1:3 ratio data for binary mixtures of TX-100 with five different surfactants (which we will call henceforth the cosurfactant). We have paid special attention to the value of the ratio x0 / x. Figure 6 shows the value of this parameter as a function of the mole fraction of cosurfactant in the solution. As can be seen, for all the mixed systems and for pure C12 E8 the value of x0 / x is less than 10, suggesting that for these systems the point (xCMC )1 must be selected as the CMC value. The CMC data of the mixed surfactant systems are usually analyzed by using a regular solution theory [29,30], which considers an interaction parameter (β12 ) to characterize the interactions between the two surfactants in the mixed micelle. This parameter is defined by NA (W11 + W22 − 2W12 ) , (7) RT where NA is the Avogadro number and Wij are the pairwise interaction energies between monomeric species in the micelle. Note that β12 not only is an indication of the degree of interaction between the surfactants but also accounts for the deviation from ideality. A negative value of β12 implies an attractive interaction; the more negative the β12 value the greater the attraction. According to the regular solution approach, the activity coefficients in the mixed micelles are expressed as f1 = exp β12 (1 − x1 )2 , f2 = exp β12 x12 , (8) β12 =
where x1 is the mole fraction of surfactant 1 in the mixed micelle. The interaction parameter, β12 , can be determined when the CMC for the mixed system (C ∗ ) is known. From
Fig. 7. Variation of (a) CMC and (b) micellar composition, xCTAB , with the mole fraction of cosurfactant (αCTAB ) for the TX-100/CTAB system. Filled circles represent CMC and micellar composition results as obtained from our pyrene 1:3 ratio data, whereas open circles are the corresponding values from Ref. [31]. The dashed lines represent the ideal behavior and the solid lines are for a regular solution (β12 = −1.2).
this treatment, the following equation can be derived, x12 ln(α1 C ∗ /x1 C1 ) 2 (1 − x1 ) ln((1 − α1 )C ∗ /(1 − x1 )C2 )
= 1,
(9)
which relates the mole fraction of surfactant 1 to the CMC of the binary system, to the mole fraction of surfactant 1 in the solution (α1 ), and to the CMCs of pure surfactants (C1 and C2 ). Equation (9) can be solved iteratively for x1 and then the interaction parameter β12 can be determined from the equation β12 =
ln(α1 C ∗ /x1 C1 ) . (1 − x1 )2
(10)
With the aim of evaluating the effect of the determination of the CMC value by one, (xCMC )1 , or the other point, (xCMC )2 , we have selected the TX-100/CTAB system, as we can compare our results with those recently reported by Gharibi et al. [31], using surface tension measurements. Figure 7 shows the CMC values obtained for the TX100/CTAB system, as well as the micellar composition, as functions of the mole fraction of cosurfactant in the solution (αCTAB ). In the same figure, we have included, for comparison, the CMC data reported by Gharibi et al. [31] together with the corresponding values for the micellar composition calculated by us. First of all, it can be seen that our CMC data compare well with those determined
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using surface tension measurements, in particular at low participation of cosurfactant in the solution. The agreement between these data is better when we examine the results for the micellar composition versus solution composition (Fig. 7b). This figure shows that for 0 αCTAB 0.6 the mixed micelle predominantly consists of a nonionic component (TX-100), and only for high αCTAB values does the participation of cosurfactant become significant. On the other hand, we have carried out a nonlinear curve fit with the new CMC values, obtaining a value of β12 = −1.2. It must be pointed out that this value is a little smaller than that (β12 = −1.0) previously reported [7].
5. Conclusions This study has allowed us to demonstrate that the plots pyrene 1:3 ratio versus total surfactant concentration can be adequately described by a decreasing sigmoid of the Boltzmann type. In these plots we have characterized two singular points, which can provide the CMC value. Through numerous examples we have shown that the ratio x0 / x can be used to establish an objective approach to choosing between one and the other point. Besides providing a fast, simple, and accurate procedure to determine the CMC value of a surfactant system, the proposed method is of considerable importance because it allows to use a unified procedure to treat the pyrene 1:3 ratio data and, in this way, to compare the values of CMC obtained by different researchers.
Acknowledgment This work has been financially supported by the Spanish Science and Technology Ministry (Project MAT2001-1743).
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