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Langmuir 2000, 16, 4430-4435
Mixed Micelles of Anionic-Nonionic and Anionic-Zwitterionic Surfactants Analyzed by Pulsed Field Gradient NMR A. M. Misselyn-Bauduin,† A. Thibaut,† J. Grandjean,‡ G. Broze,§ and R. Je´roˆme*,† Center for Education and Research on Macromolecules, and Chimie Fine aux Interfaces, University of Lie` ge, Sart-Tilman, B6, 4000 Lie` ge, Belgium, and 3 Colgate-Palmolive R&D, 4460 Milmort (Herstal), Belgium Received July 28, 1999. In Final Form: February 7, 2000 The micellar composition of two binary surfactant systems, that is, sodium dodecyl sulfate (SDS)/ pentaethylene glycol monodecyl ether (C10E5) and SDS/lauryl amido propyl betaine (LAPB), has been analyzed by pulsed gradient spin-echo NMR (FT-PGSE NMR). The experimental data have been compared to theoretical predictions based on the regular solution theory that takes into account the nonideal mixing of the surfactant pairs. Although good agreement between experiment and theory is observed for the SDS/C10E5 pair, some discrepancy is noted for the LAPB/SDS system, particularly at a high molar fraction of SDS.
Introduction Mixtures of surfactants are widely used in many application fields, such as detergency, painting, coating, cosmetics, oil recovery, and so forth. The thermodynamic analysis of these mixtures has been considered, on the basis of the simplest case of ideal mixing, as reported by Shinoda,1 Lange and Beck,2 and later by Clint.3 In this ideal situation the surfactant headgroups do not interact significantly and mixed micellization is driven by the hydrophobic interaction of the alkyl chains of the surfactants. This approach is successful in handling binary nonionic and binary ionic surfactants, particularly when the surfactant headgroups are of the same composition. When the interactions between the headgroups of the constitutive surfactant cannot be dismissed, a more complex analysis is required, such as the pseudo-phaseseparation model proposed by Holland, Rubingh, and others,4-6 which is based on the approximation of regular solution. The use of this approximation results in a set of relationships between the critical micellar concentration (cmc), composition of mixed micelles, surfactant unimer concentration, and an interaction parameter, βM, which expresses the strength of interaction within the mixed micelles. The regular solution theory has proved successful in accounting for the nonideal behavior of a number of binary surfactant systems, particularly pairs of anionicnonionic surfactants, and is the most largely used theory in the field of mixed micelles. Nevertheless, in some cases, more complete theories are needed.7-12 * To whom correspondence should be addressed. † Center for Education and Research on Macromolecules, University of Lie`ge. ‡ Chimie Fine aux Interfaces, University of Lie ` ge. § Colgate-Palmolive R&D. (1) Shinoda, K. J. Phys. Chem. 1954, 58, 451. (2) Lange, H.; Beck, K. H. Kolloid Z. Z. Polym. 1973, 251, 424. (3) Clint, J. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327. (4) Rubingh, D. N. Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol 1, p 337. (5) Holland, P. M. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 301; American Chemical Society: Washington, DC, 1986; p 102. (6) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (7) Blankschtein, D.; Puvada, S.; Sarmoria, C. Langmuir 1992, 8, 2690.
Compared to the huge effort devoted to the theoretical prediction of micellar and monomeric compositions of nonideal surfactant mixtures, only a few direct measurements are available in the scientific literature. For instance, neutron reflectivity and small-angle neutron scattering were used to study the adsorption at the airwater interface and the micellar composition for a mixture of sodium dodecyl sulfate (SDS) and n-hexaethylene glycol monododecyl ether (C12E6).13 The authors found good agreement between the experimental observations and the predictions based on the regular solution theory. Measurements of surface tension and neutron reflectivity were also used to determine the surface composition of a mixture of anionic (SDS) and sugar-based nonionic surfactants and of a binary system of the same nonionic surfactant and a zwitterionic one.14 The ultracentrifugation technique was used by Huang and Somasundaran15 to monitor changes in the original unimer concentration for mixtures of a cationic surfactant and a nonionic one. The measurement of the surfactant self-diffusion coefficients by Fourier transform pulsed gradient spin-echo NMR spectroscopy (FT-PGSE NMR)16-20 is another promising approach to the analysis of the mixed micellar compositions. In this work the binary surfactant systems, that is, SDS/C10E5 (anionic/nonionic surfactants) and SDS/ LAPB (lauryl amido propyl betaine) (anionic/zwitterionic surfactants), will be studied over a wide concentration and composition range by FT-PGSE 1H NMR spectroscopy. (8) Hoffmann, H.; Po¨ssnecker, G. Langmuir 1994, 10, 381. (9) Huang, L.; Somasundaran, P. Langmuir 1997, 13, 6683. (10) Reif, I.; Somasundaran, P. Langmuir 1999, 15, 3411. (11) Haque, Md. E.; Das, A. R.; Rakshit, A. K.; Moulik, S. P. Langmuir 1996, 12, 4084. (12) Georgiev, G. S. Colloid Polym. Sci. 1996, 274, 49. (13) Penfold, J.; Staples, E.; Thompson, L.; Tucker, I.; Hines, J.; Thomas, R. K.; Lu, J. R. Langmuir 1995, 11, 2496. (14) Hines, J. D.; Thomas, R. K.; Garrett, P. R.; Rennie, G. K.; Penfold, J. J. Phys. Chem. B 1997, 101, 9215. (15) Huang, L.; Somasundaran, P. Langmuir 1996, 12, 5790. (16) Calfors, J.; Stilbs, P. J. Phys. Chem. 1984, 88, 4410. (17) Nilsson, P. G.; Lindman, B. J. Phys. Chem. 1984, 88, 5391. (18) Asakawa, T.; Imae, T.; Ikeda, S.; Miyagishi, S.; Nishida, M. Langmuir 1991, 7, 262. (19) Griffiths, P. C.; Stilbs, P.; Paulsen, K.; Howe A. M.; Pitt, A. R. J. Phys. Chem. B 1997, 101, 915. (20) Ciccarelli, D.; Costantino, L.; D’Errico, G.; Paduano, L.; Vitagliano, V. Langmuir 1998, 14, 7130.
10.1021/la991020l CCC: $19.00 © 2000 American Chemical Society Published on Web 04/21/2000
Pulsed Field Gradient NMR Analysis of Mixed Micelles
Although a similar surfactant pair (SDS/C12E5) was studied by the same technique,17 attention was mainly paid to intermicellar interactions rather than to micellar composition. Recently, the micellar composition of nearly ideal mixtures of two surfactants, SDS and a sugar-based nonionic surfactant (dodecylmalonobis(N-methylglucamide))19 and C6E5/C6SNa (sodium hexanesulfonate),20 was analyzed by FT-PGSE 1H NMR. The surfactants considered in this work (SDS, C10E5, and LAPB) have been selected for their relevance in industry and in basic research. The length of their hydrophobic chain is however such that the cmc of these surfactants and their mixed micelles are low, which explains that the self-diffusion coefficients and thus the micellar compositions will be analyzed at concentrations in the vicinity or higher than the cmc. It must also be pointed out that the mixing behavior of the SDS/C10E5 and SDS/LAPB pairs deviate from ideality. Moderate interactions occur between SDS and C10E5, whereas the anionic and the zwitterionic surfactants interact strongly. Therefore, these systems are quite appropriate to estimate to which extent the mixed micellar compositions analyzed by FT-PGSE NMR fit the predictions of the regular solution theory, thus to estimate the limits of applicability of these predictions in relation to the strength of the mutual interactions of the surfactants. Experimental Section Materials. Sodium dodecyl sulfate (MW 288) from BDH Laboratories was used as received. The cmc of SDS was 8.0 mM, as determined from the surface tension data. No minimum in the concentration dependence of the surface tension was observed in the vicinity of the cmc, proving the high purity of the SDS sample, particularly by the absence of dodecanol. Pentaethylene glycol monodecyl ether (C10E5, MW 378, cmc ) 0.69 mM) was used as received from Fluka BioChemika. Lauryl amino propyl betaine (LAPB, MW 400, cmc ) 0.17 mM) was provided by Mackam Lmb. Surfactant solutions for surface tension measurements were prepared with HPLC grade water (Altech). All the aqueous solutions used for FT-PGSE experiments were prepared with twice distilled water containing 10% D2O (Aldrich). Surface Tension Measurements. Surface tension was measured at 22 °C to determine critical mixed micelle concentrations (cmc*) for both the SDS/C10E5 and the SDS/ LAPB systems, at different SDS molar ratios, R. An automatic Kru¨ss K12 tensiometer, equipped with a platinum Wilhemy plate, was used for this purpose. Glassware was carefully cleaned by dipping for several hours in 10% H2SO4 solution and then plentifully rinsing with deionized water. The cleanliness of the glassware was checked by surface tension measurement of HPLC grade water, which was also used for the preparation of all the surfactant aqueous solutions. Each surface tension measurement was repeated 3 times and accepted if the results did not differ one from another by more than 0.25 mN/m. The cmc* was determined at a constant surfactant composition from the surface tension measured for ≈1214 solutions of different concentrations. These solutions were prepared by dilution of 40 mM stock solution of the surfactants mixture of the selected composition. The experimental surface tensions were then plotted against the total surfactant concentration, so allowing the graphical determination of the cmc*. NMR Experiments. Self-diffusion coefficients, Ds, were
Langmuir, Vol. 16, No. 10, 2000 4431
measured by the pulsed field gradient NMR technique21,22 using a Bruker AM 300WB spectrometer operating at the proton Larmor frequency of 300 MHz. The basic sequence was used with pulsed field duration, δ, of 6 ms and a time interval, ∆, between the two gradient pulses of 22 ms. The echo attenuation, A, was recorded as a function of the gradient amplitude, g, and calibrated with octanol on the assumption that Ds ) 1.9 × 10-10 m2 s-1 at 20 °C.23 The signal intensity was found to obey the Stejskal-Tanner relation, as predicted by the theory,
A ) A0 exp(-γ2δ2g2Ds(∆ - δ/3))
(1)
where γ is the proton gyromagnetic ratio. In the case of SDS, A was measured as a function of g for the signal of the highest intensity at 1.3 ppm (side chain protons). The other proton signals, particularly, the 4.05 ppm peak (protons in R position of the SO4 group), were less convenient because of poor signal-to-noise ratio under the experimental conditions of this study. In the case of C10E5, the oxyethylenic protons (δ ) 3.7 ppm) were very well suited to the determination of Ds. For LAPB, the -CH2protons in the R position of the carbonyl group at 3.3 ppm was selected. Self-diffusion coefficients were calculated by fitting 13 experimental datum by eq 1. All reported values were the average of three independent measurements. Self-diffusion Coefficients for a Single Surfactant. Ds of a single surfactant in water progressively decreases with concentration, consistently with a two-state mobility model. The observed Ds (Dexp s ) is expressed as the population weighted average of the high self-diffusion coefficient of the unimers (Duni s ) and the lower self-diffusion coefficient of the micelles (Dmic s ). mic ) pDuni Dexp s s + qDs
(2)
with p + q ) 1 Dmic is determined at a surfactant concentration high s enough for the unimer concentration to be negligible with is directly respect to the population of micelles. Duni s measured below the cmc, provided that this concentration to be accurately determined. is high enough for Dexp s is calculated from Dexp When the cmc is too low, Duni s s according to eq 3:
) Dmic + Dexp s s
cmc uni (Ds - Dmic s ) ct
(3)
Mixed Micellization: Pseudo-Phase-Separation Model Based on the Regular Solution Theory. In this model, the critical mixed micelle concentration (cmc*) of nonideal mixtures of two surfactants is expressed as the weight average of the cmc’s of the individual surfactants 1 and 2 (eq 4):
(1 - R) R 1 ) + cmc* cmc1 cmc2
(4)
where R is the molar fraction of surfactant 1 in the surfactant mixture in solution. (21) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288. (22) Stilbs, P. Prog. NMR Spectrosc. 1987, 19, 1.
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In the case of nonideal mixing, this equation is modified by an activity coefficient (f1 and f2) for each surfactant:
(1 - R) R 1 ) + cmc* f1 cmc1 f2 cmc2
(5)
According to Rubingh,4 these activity coefficients depend on an interaction parameter, βM, defined as the change in enthalpy (H) when the surfactants 1 and 2 are mixed together (eq 8),
ln f1 ) βM(1 - x1)2
(6)
Table 1. cmc* and βM Values at 22 °C for SDS/C10E5 and SDS/LAPB Surfactant Mixtures at Different SDS Molar Ratios (r) SDS/C10E5 cmc* (mM)
βM
R
cmc* (mM)
βM
0 0.1 0.3 0.5 0.7 0.9 1
0.69 0.75 0.8 0.84 1.0 1.8 8.0
-2 -2.2 -2.9 (-4)a -3.3 (-4.3)a -3.1 (-3.8)a
0 0.1 0.3 0.5 0.7 0.9 1.0
0.17 0.11 0.09 0.1 0.13 0.36 8.0
-7.8 -8.6 -8.5 -8.3 -6.3
a
2
ln f2 ) βMx1
(7)
βM ) (H11 + H22 - 2H12)/RT
(8)
SDS/LAPB
R
Optimized values.
where x1 is the mole fraction of surfactant 1 in the mixed micelles. So when the mixture of surfactants is ideal, βM ) 0, and deviation from ideality results in negative βM. In very few cases, βM may be positive, so indicating that the two surfactants are incompatible and that micellar demixing occurs. The composition of the mixed micelles, (x1), at the cmc and the interaction parameter, βM, can be calculated by iterative resolution of the following equations:4
[
][
]
(9)
][ ]
(10)
R cmc* 1 cmc1 x1 (1 - x )2 1
βM ) ln and
[
βM ) ln
(1 - R)cmc* 1 cmc2(1 - x1) x12
It must be stressed that these calculations are only valid at the cmc*. At total surfactant concentrations higher than the cmc*, Clint3 proposed an analytical solution that gives the concentration of the unimeric species (c1,uni and c2,uni), and hence the micellar composition (x1 and x2), as a function of the total surfactant concentration (C) in the case of an ideal mixing behavior (eq 11),
x1 )
-(C - ∆) + ((C - ∆)2 + 4RC∆)1/2 2∆
(11)
where ∆ ) cmc2 - cmc1. In the case of nonideal mixing, ∆ ) f2 cmc2 - f1 cmc1, and x1 is calculated by iteratively solving eq 11. Thus, f1 and f2 are defined by eqs 6 and 7 in which βM is calculated by solving eqs 9 and 10, which rely on the experimental cmc*, measured at different bulk compositions (R). Results and Discussion Surface Tensions. From the surface tension data, the critical mixed micelle concentrations (cmc*) have been determined for both the SDS/C10E5 and the SDS/LAPB systems, at different SDS molar ratios (R), as explained in the Experimental Section (Table 1). In the case of the C10E5/SDS system, the experimental values of cmc* have been plotted as a function of R in Figure 1 and compared to the dependence of the cmc* predicted in the ideal case of mixing (eq 4). Consistently with data reported for similar binary anionic/nonionic systems, the behavior of the C10E5/SDS system is nonideal. Deviation from ideality is also observed for the SDS/LAPB
Figure 1. cmc* for mixtures of C10E5 and SDS at different molar ratios (R). The open circles are experimental values. The full curve is calculated in the case of ideal mixing behavior.
system. To account for this nonideality, the interaction parameter, βM, has been calculated by iterative resolution of eqs 9 and 10 for each R composition (Table 1). βM has never been found to be zero, so assessing that the mixing of the surfactants is nonideal. Actually, all the βM values are negative, which is the signature of interaction between the surfactants within the mixed micelles. The absolute values of βM are moderately high for the SDS/C10E5 mixture and roughly constant over the entire range of composition, which is in line with the regular solution theory as developed by Rubingh, as will be discussed later in this article. These values are also comparable to data published for similar systems.14,24 The significantly more negative βM values for the anionic/zwitterionic mixtures suggest that the two types of headgroups strongly interact, which is not surprising for the combination of the negative charge of SDS and the strong dipole of the zwitterionic surfactant. Self-diffusion Coefficients. Single Surfactants. Selfdiffusion coefficients have been measured by FT-PGSE 1 H NMR spectroscopy for aqueous solutions of SDS, C10E5, and LAPB, respectively, at various concentrations. The self-diffusion coefficients of SDS, C10E5, and LAPB micelles ) 4.8 × 10-11, 4.6 × 10-11, and 9.3 × 10-11 m2/s, (Dmic s respectively) have been determined at concentrations well above their respective cmc values. On the other hand, Duni for SDS (48 × 10-11 m2/s) has been directly deters mined at a concentration below the cmc, which was (23) Herden, H.; Ka¨rger, J.; Pfeifer, H.; Kube, C.; Scho¨llner, R. J. Colloid Interface Sci. 1992, 152, 281. (24) Rosen, M. J.; Hua, X. Y. J. Am. Oil. Chem. Soc. 1982, 59, 582.
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Table 2. Composition (xSDS) of Mixed Micelles for the SDS/C10E5 System in Relation to the Total Surfactant Concentration and the Global SDS Molar Fraction (r) as Calculated from Dexp Measured from the Signals at 3.7 and 1.3 s ppm total concn. (mM)
11 m2/s) Dexp s (×10 3.7 ppm
Dexp (×1011 m2/s) s 1.3 ppm
DSDS (×1011 m2/s) s
qC10EO5
qSDS
xSDS
0.1
40 20 5 2.5
6.9 8.6 11.4 13.5
6.9 8.7 11 13.4
7.4 9.4 7.4 12.3
0.97 0.95 0.90 0.87
0.94 0.90 0.94 0.83
0.10 0.10 0.10 0.10
0.3
40 20 10 5 2.5
7.3 8.2 8.8 11.8 16.2
7.9 9.5 10.9 14.8 19.1
9.4 12.4 15.8 21.7 26
0.96 0.95 0.94 0.90 0.83
0.90 0.83 0.75 0.61 0.51
0.29 0.27 0.25 0.23 0.21
0.5
40 20 10 5 2.5
5.6 7.4 8.1 10.4 15.3
6.6 9.9 11.8 14.5 19.8
7.6 12.3 15.5 18.6 24.3
0.99 0.96 0.95 0.92 0.84
0.94 0.83 0.76 0.68 0.55
0.49 0.46 0.44 0.43 0.40
0.7
40 20 10 3
7.0 7.9 10.7 17.0
9.7 12.3 18.7 27.0
10.8 14.2 22.1 31.3
0.97 0.96 0.91 0.82
0.86 0.79 0.60 0.39
0.67 0.66 0.61 0.52
0.9
40 30 20 10
7.0 8.6 9.5 10.1
8.5 14.9 19.9 28
8.7 15.6 21 30
0.97 0.95 0.93 0.92
0.91 0.75 0.63 0.42
0.89 0.88 0.86 0.80
R
impossible for C10E5 and LAPB because of their exceedingly -11 m2/s for C E and 165 × 10-11 small cmc. Duni 10 5 s (71 × 10 m2/s for LAPB) has accordingly been determined from the on 1/Ct (eq 3). linear dependence of Dexp s SDS/C10E5 Mixed Micelles. In the case of binary surfactant mixtures, the Ds value measured for each surfactant reflects its partition between the unimeric and the micellar species. Therefore,
DSDS ) pSDSDuniSDS + pSDSDMM s s s
(12)
10E5 + DMM DCs 10E5 ) qC10E5DuniC s s
where pSDS stands for the molar fraction of SDS unimers, pC10E5 is the molar fraction of C10E5 unimers, qSDS and qC10E5 are the molar fractions of SDS and C10E5 in the mixed is Ds for the mixed micelles, micelles, respectively, DMM s 10E5 and DuniC are Ds for the SDS and C10E5 and DuniSDS s s unimers, respectively. and DsC10E5 at The experimental determination of DSDS s different R compositions and different total surfactant concentrations allows qSDS and qC10E5 to be determined and the micellar composition within the mixed micelles to be calculated (xSDS and xC10E5). As explained in the Experimental Section, the determination of DCs 10E5 is straightforward by using the signal at 3.7 ppm. In contrast, the is more problematic because the determination of DSDS s signal to be used at 1.3 ppm results from the superposition of proton resonances typical of both the nonionic and the anionic side chains. Progress in the processing of FT-PGSE NMR data has recently been reported, namely, the CORE (component-resolved) data analysis, which is well suited to very complex systems.25 In this study, the signal intensity versus field gradient followed a monoexponentiel and DsC10E5 decay at 1.3 ppm, indicating that both DSDS s exp values were comparable. Therefore, Ds has been ap(25) Stilbs, P. In Polymer-Surfactant Systems; Kwak, J. C. T., Ed.; Surfactant Science Series; Marcel Dekker Inc.: New York, 1998; Vol. 77, chapter 6, p 239.
proximated to the weighted average of DSDS and DCs 10E5, s respectively (eq 13):
Dexp ) RDSDS + (1 - R)DsC10E5 s s
(13)
is obtained from Dexp at At a given R composition, DSDS s s C10E5 separately measured at 3.7 ppm. From 1.3 ppm and Ds DSDS and DCs 10E5 data, qSDS and qC10E5 have been calculated s is constant whatever the on the assumption that DMM s has been measured in reference to the composition. DMM s signal of the -CH2-O- protons of the nonionic surfactant (3.7 ppm) at R ) 0.1 and total surfactant concentration of 100 mM, thus far above the cmc* (0.75 mM), so that the concentration of unimers is negligible. Table 2 shows the Dexp values measured at various compositions (R) for s different total surfactant concentrations. From these data qSDS and qC10E5 have been calculated and, hence, the molar fraction of SDS in the mixed micelle (xSDS). All the Dexp s values have been determined three times, each providing an independent xSDS through application of eqs 12 and 13 and for which the standard deviation has been calculated. The micellar composition (xSDS) has been plotted against the total concentration of the surfactant mixtures whose composition values of R are 0.1, 0.3, 0.5, 0.7, and 0.9, respectively (Figure 2). This plot also emphasizes the relationship between the micellar composition (xSDS) and the composition of the surfactant mixtures (R) at constant total surfactant concentration. The standard deviations of the micellar compositions are small but expectedly increase when the total surfactant concentration is decreased (Figure 2) because the signal/noise ratio of the NMR signals decreases in parallel and results in less accurate self-diffusion coefficient determinations. The experimental data have been compared to the micellar compositions (xSDS) predicted by the regular solution theory (eq 11) at different compositions (R) and total surfactant concentrations (solid lines). The interaction parameters, βM, used in these calculations for each value of R are listed in Table 1. Figure 2 shows that, at low R, there is a good agreement between the experimental data and the theoretical predictions, and that the micellar composition
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Figure 2. Relationship between the micellar composition (xSDS) and the total surfactant concentration for the C10E5/SDS system at global composition R ) 0.1 (0), 0.3 (b), 0.5 (4), 0.7 (2), and 0.9 (-). The curves are data calculated by eq 11, by using βM values listed in Table 1.
Figure 3. Experimental data (4) compared to theoretical ones (βM ) 0, dashed curve; βM ) -4.0, full curve) for the dependence of xSDS on the total C10E5/SDS concentration at R ) 0.5.
is the same as the original surfactant composition at a total concentration higher than ≈10 mM. Nevertheless, slight deviations between experiment and theory are observed for R ) 0.5, 0.7, and 0.9. In these composition ranges, the theory predicts that the mixed micelles contain slightly more C10E5 than experimentally observed. Thus, alternative βM values have been used in eq 11 to estimate to what extent the interaction parameter must be changed for the theoretical micellar composition and the experimental data including standard deviations to be fitted properly at R ) 0.5, 0.7, and 0.9. The optimized βM values are -4.0, -4.3, and -3.8 at R ) 0.5, 0.7, and 0.9, respectively, compared to -2.9, -3.3, and -3.1 as reported in Table 1. The quality of the fit by the optimized βM is illustrated in Figure 3 for R ) 0.5 and compared to the theoretical prediction in the case of ideal mixing behavior (βM ) 0). The experimental errors on the micellar composition, particularly at low total surfactant concentration, explain only partial deviations from predictions by the regular solution theory. These deviations are better
Misselyn-Bauduin et al.
accounted for by the errors on the experimental cmc* values (determined by surface tension) as explained by Hoffmann and Po¨ssnecker.8 These errors directly affect βM and, thus, the predicted micellar compositions. In this respect, the cmc* has been calculated from the optimized βM values and found to be 0.69, 0.8, and 1.5 mM at R ) 0.5, 0.7, and 0.9, respectively. These values are ≈15-20% lower than the data in Table 1, consistently with the wellknown lack of accuracy in the graphical estimation of cmc from surface tension data. Indeed, errors of 10-20% on the cmc estimated by this method are not rare. It thus appears that the apparent variation of βM with R does not indicate that the regular solution fails to account for the mixing behavior of SDS and C10E5, but it merely emphasizes the difficulty in collecting cmc* data accurate enough for finding a single βM that supports the applicability of the theoretical approach of Rubingh. SDS and n-dodecylβ-D-maltoside mixtures were previously studied,14 which are quite comparable to the SDS/C10E5 pair, the cmc’s of C10E5 and n-dodecyl-β-D-maltoside being close to each other. The βM values calculated for these two systems of moderately interacting surfactants by the same method are actually in the same range, that is, from -2.0 to -5.0. Their mixing behavior, which deviates from ideality, can be adequately described by the pseudo-phase-separation model based on the regular solution theory, at least within the limits of accuracy of the experimental data used. SDS/LAPB Mixed Micelles. The same experimental methodology has been used in the case of the LAPB/SDS system as that for the SDS/C10E5 pair. As mentioned in has been determined the Experimental Section, DLAPB s from the proton signal at 3.7 ppm, which is well separated from the other ones in the NMR spectrum of the surfactant mixture. The resonance of the alkyl side chain protons is observed at 1.3 ppm for both the SDS and LAPB surfactants. Therefore, eq 13 has been used to extract DSDS from the signal at 1.3 ppm. Results are listed in s Table 3. The LAPB/SDS micellar composition (xSDS) has been plotted versus the total surfactant concentration, for different R values, and compared to the theoretical dependence calculated by eq 11 (Figure 4) and the βM values listed in Table 1. Theoretical predictions and experimental data are quite consistent for R ) 0.1 and R ) 0.3. In the two cases, the micellar composition and the original surfactant composition are the same, whatever the total surfactant concentration. The predictions based on the regular solution theory with βM ) -7.8 (at R ) 0.1) and βM ) -8.6 (at R ) 0.3) are consistent with the experimental data. At higher R values (R ) 0.5, 0.7, and 0.9), the experimental micelle compositions deviate from the bulk composition, particularly when the total surfactant concentration is smaller than 10 mM. According to the experimental data, the surfactants interact more strongly in the micelles than theoretically predicted. The major discrepancy is observed for R ) 0.9. The observed difference could not be accounted for merely by experimental errors in the cmc* determination, as proposed in the case of the C10E5/SDS system. Effort has been made to match better the experimental and theoretical data, by using alternative βM values in eq 11. At R ) 0.7, βM should be changed from -8.3 to -12. The situation is still worse at R ) 0.9 because βM should be decreased to -40 for the theoretical predictions to fit the experimental data! Such highly negative values have no physical meaning and merely indicate that the regular solution theory is no longer able to predict the composition of mixed micelles in the LAPB/SDS system. Thus,
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Langmuir, Vol. 16, No. 10, 2000 4435
Table 3. Composition (xSDS) of Mixed Micelles for the SDS/LAPB System in Relation to the Total Surfactant Concentration and the SDS Molar Fraction (r) as Calculated from Dexp Measured from the Signals at 3.3 and 1.3 ppm s R
total concn (mM)
(×1011 m2/s) Dexp s 3.3 ppm
Dexp (×1011 m2/s) s 1.3 ppm
DSDS (×1011 m2/s) s
qLAPB
qSDS
xSDS
0.1
40 20 10 3
7.7 10 11.7 17.3
7.2 9.1 11.2 16.1
3.0 1.1 6.8 5.3
0.98 0.97 0.96 0.93
1.00 1.00 0.94 0.97
0.10 0.10 0.10 0.10
0.3
40 30 20 10 5 2.5
4.0 6.7 8.7 9.8 11.9 13.6
3.9 7.0 8.1 9.4 11.7 13.1
3.8 7.8 6.6 8.2 11.4 11.9
0.99 0.99 0.98 0.97 0.96 0.95
1.00 0.91 0.94 0.90 0.83 0.82
0.30 0.28 0.29 0.28 0.27 0.27
0.5
40 20 15 10 5 2.5
7.7 8.1 8.8 8.7 9.9 12.5
8.2 9.0 9.8 10.3 11.9 15.1
8.8 9.9 10.7 11.8 13.8 17.6
0.98 0.98 0.97 0.97 0.97 0.95
0.89 0.87 0.85 0.82 0.78 0.69
0.48 0.47 0.46 0.46 0.44 0.42
0.7
40 30 20 10 5 2.5
7.9 9.5 10.1 11.5 9.5 11.5
9.5 11.8 13.8 14.9 14.2 16.7
10.1 12.8 15.4 16.4 16.2 18.9
0.98 0.97 0.97 0.96 0.97 0.96
0.86 0.80 0.74 0.72 0.72 0.66
0.67 0.66 0.64 0.64 0.63 0.62
0.9
40 20 10 5 2.5
9.5 10.1 11.1 10.0 12.4
12.3 17.0 17.5 22.8 27
12.6 17.8 18.2 24.2 28.6
0.97 0.97 0.96 0.97 0.95
0.80 0.69 0.68 0.54 0.44
0.88 0.86 0.86 0.83 0.80
respect to the micellar composition and that βM is constant over the whole composition range8,14 are not valid in any case. This limitation results from the simplification of the equation proposed to correlate the activity coefficients and the composition of the micellar pseudophase (eq 6), as discussed elsewhere.8-10
Figure 4. Relationship between the micellar composition (xSDS) and the total surfactant concentration for the LAPB/SDS system at global composition R ) 0.1 (0), 0.3 (b), 0.5 (4), 0.7 (2), and 0.9 (-). The curves are data calculated by eq 11, by using βM values listed in Table 1.
compared to the previous system, that is, the SDS/C10E5 pair, for which βM was found in the range from -2 to -4.3 in the whole composition domain, βM for the SDS/LAPB pair has to change dramatically with the composition (from -7 to -40) for fitting the predictions by the regular solution theory. Such a dependence largely exceeds the consequence of the errors on the cmc* calculated from surface tension data. Composition-dependent βM values were reported elsewhere,11,14,26,27 values that support the limitation of the regular solution theory. The underlying assumptions that the excess free energy of mixing is symmetrical with (26) Bakshi, M. S.; Crisantino, R.; De Lisi, R.; Milioto, S. J. Phys. Chem. 1993, 97, 6914. (27) Desai, T. R.; Dixit, S. G. J. Colloid Interface Sci. 1996, 177, 471.
Conclusions This work has reported on the composition of mixed micelles calculated from self-diffusion coefficients measured by FT-PGSE NMR for binary surfactant mixtures. Data collected for different total surfactant concentrations and compositions of the binary mixtures, that is, SDS/ C10E5 and SDS/LAPB, have first confirmed that SDS and C10E5 moderately interact. There is a good agreement between the micellar composition and the theoretical prediction by the regular solution theory as discussed by Rubingh and Clint.3,4 The minor differences (when observed) between experiment and theory have been mainly ascribed to experimental errors in the measurement of the cmc* of mixed micelles. Compared to the moderate strength of interaction between SDS and C10E5, LAPB and SDS interact very strongly, and the regular solution theory fails to account for the experimental data at high SDS molar fractions. The single βM parameter approximation as proposed by Rubingh is no longer valid, and βM has to be changed drastically to an extent which depends on the bulk surfactant composition for the fitting between experiment and theory to be restored. The experimental determination of the micellar composition thus gives valuable information on the interaction between the surfactants in solution and on the limits of applicability of the regular solution theory. Acknowledgment. R.J., A.M.M.B., and A.T. are indebted to Colgate-Palmolive (Lie`ge) and to the “Services Fe´de´raux des Affaires Scientifiques, Techniques et Culturelles” (PAI, 4/11) for support to CERM. LA991020L
