Determination of the Composition of Mixed Hydrogenated and

solvent H2O/D2O composition in order to determine the micellar neutron ... in the past, had been particularly difficult and the object of considerable...
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J. Phys. Chem. B 1997, 101, 9525-9531

9525

Determination of the Composition of Mixed Hydrogenated and Fluorinated Micelles by Small Angle Neutron Scattering L. Pedone, D. Chillura Martino, E. Caponetti,* M. A. Floriano, and R. Triolo Department of Physical Chemistry, UniVersity of Palermo, Via Archirafi 26, 90123 Palermo, Italy ReceiVed: July 10, 1997X

The compositions of mixed micelles formed in aqueous solutions of sodium dodecanoate and sodium perfluorooctanoate at different total surfactant concentrations and at three (0.33, 0.53, and 0.73) sodium perfluorooctanoate mole fractions were determined by the small angle neutron scattering technique coupled with the external contrast method. At each concentration, measurements were performed as functions of the solvent H2O/D2O composition in order to determine the micellar neutron scattering densities. The method described has allowed, at least for the cases under study, a direct determination of the micellar compositions, which, in the past, had been particularly difficult and the object of considerable debate. At all concentrations considered and within the experimental error, one kind of mixed micelle was always observed; these micelles were always richer in the component present in solution in greater proportion. An overall qualitative agreement between the present results and literature predictions based on regular solution theory was found, although a significant difference was noticeable at large sodium perfluorooctanoate concentrations, suggesting that further tests of the theory are required in order to include subtle interaction effects due to differences in the chemical nature of the surfactants.

Introduction The study of solutions containing different kinds of surfactants has relevant technological implications since nearly all practical applications employ mixed surfactant systems. There are several reasons for this; commercial surfactants are very seldom immune from impurities deriving from the starting materials and from the variability in reaction products during the manufacturing process; the surfactants obtained in this way are less expensive to produce than isomerically pure surfactants and often provide better performances; in addition, mixtures of different surfactant types are often deliberately formulated in order to exploit synergetic behaviors in mixed systems or to provide qualitatively different types of performance in a single formulation (e.g., cleaning plus fabric softening). Finally, practical formulations often require the addition of surfactant additives designed to control the physical properties of the products or to improve their stability. This has led to both theoretical and practical interest in developing a quantitative understanding of the behavior of mixed surfactant systems that can be exploited in applications such as detergency,1-3 enhanced oil recovery,4 and mineral flotation.5 The first problem that arises when dealing even with the simple case of a solution containing just two different surfactants is to determine whether one obtains a mixture of two kinds of micelles, each constituted solely by one of the two surfactants, or mixed micelles, containing both surfactants. In the latter case, one additionally needs to know what is the relative proportion of the two surfactants in the micelle, i.e., the mixed micelle composition, and how this is related to the bulk composition and to changes of the total concentration. For surfactant mixtures, as for solutions of just one surfactant, a critical micellar concentration (cmc) can be defined as the concentration at which an appreciable change in the physical properties of the system is observed and both surfactants selfassemble to form micelles. As an example, the surface tension * Corresponding author. Email: [email protected]. X Abstract published in AdVance ACS Abstracts, October 1, 1997.

S1089-5647(97)02246-3 CCC: $14.00

for ionic-nonionic mixtures that form mixed micelles shows experimental trends similar to those of pure surfactants. Several attempts7-12 have been made in the past to interpret the thermodynamics of micellization of mixed surfactants; this has been done almost exclusively by studying how the cmc values of the mixture changed by varying the stoichiometric composition. Both positive and negative departures from the values expected from ideal mixing have been obtained for the mixture. Negative deviations are always observed for mixtures in which mixed micelles are formed; this is the case, for example, of solutions containing two different hydrogenated surfactants.13 Positive deviations from the values expected from ideal mixing are displayed by solutions containing hydrogenated and fluorinated surfactants, which, depending on their molecular properties, in principle can form micelles containing either both surfactants or just one of them. Rubingh’s9 approach to the thermodynamics of micellization in mixed systems has proven itself capable of reproducing the vast majority of the available experimental cmc trends. This treatment, for a solution containing two different surfactants, 1 and 2, uses regular solution theory (RST) and the model of pseudo-phase separation for micellization;14 a parameter β, depending on C1, C2, and C*, the cmc values for surfactant 1 and 2 and for the mixture respectively, x1 and R1, the micellar and the stoichiometric mole fractions of surfactant 1, respectively, is introduced such that

β)

ln(C*R1/C1x1) (1 - x1)2

(1)

and

1)

x21 ln(C*R1/C1x1) (1 - x1)2 ln[C*(1 - R1)/(1 - x1)C2]

(2)

It is then possible, from experimental cmc determinations, to solve eq 2 by iteration to obtain x1, which, by substitution in eq 1, yields a value for β. © 1997 American Chemical Society

9526 J. Phys. Chem. B, Vol. 101, No. 46, 1997

Pedone et al.

The above parameter has been interpreted as

β)

N(W11 + W22 - 2W12) RT

(3)

where N is Avogadro’s number, Wii is the potential energy for a pair of two surfactant molecules, R is the universal gas constant, and T is the absolute temperature. As a consequence, it has been associated to intermolecular interactions in the mixed micelle9 that depend on the balance between attractions and repulsions involving all regions of the surfactant molecule.15,16 A negative value of β indicates dominant attractive interactions in the mixed system and, thus, the formation of mixed micelles, whereas a positive β indicates that W12 < (W11 + W22)/2. Positive β values are usually displayed by hydrocarbon/ fluorocarbon surfactant mixtures. As a consequence of a positive ∆S of mixing, in order to obtain a mixture of micelles, each containing only one of the two surfactants, a positive value of β is not sufficient. RST, on the basis of calculated values of ∆G of mixing, predicts that phase separation occurs when β g 2; in the case of surfactant mixtures this implies that mixed micelles cannot form if β > 2, and one obtains aggregates containing only one of the two surfactants. Shinoda et al.,17 once again by using RST, have shown that in a mixture of sodium perfluorooctanoate and sodium decyl sulfate, for which β ) 1.8, mixed micelles are formed, whereas in a mixture of ammonium perfluorononanoate and ammonium dodecyl sulfate, for which β ) 2.2, a mixture of micelles is obtained. De Lisi et al.,18 on the basis of measured heat capacities, enthalpies, osmotic coefficients, and cmc values, have calculated the excess free energy for mixtures of surfactants as functions of stoichiometric mole fractions. They predicted that, up to β ≈ 1.6, a mixture of surfactants will form mixed micelles, while a value of β > 1.6 implies the coexistence of two pseudophases (mixture of micelles at different composition). In this context, aqueous solutions of sodium dodecanoate (SD) and sodium perfluorooctanoate (SPFO) are particularly interesting because the two surfactants contain the same head group, the same counterion and hydrophobic chains by roughly the same volume (364 Å3 for the hydrogenated surfactant and 384 Å3 for the perfluorinated one); in addition, the cmc values of the two surfactants (28 mM for the hydrogenated surfactant and 30 mM for the perfluorinated one)18 indicate similar hydrophobic characteristics. The experimental18 trend of the cmc of solutions containing mixtures of SD and SPFO shows positive deviations from that expected from ideal mixing. Rubingh’s theory reproduces the experimental cmc vs bulk composition trends with a value of β ) 1.00 ( 0.05, thus indicating prevailing repulsive interactions as the result of different H-H, F-F, and H-F interactions. Shinoda’s approach is able to reproduce the experimental cmc trend with β ) 1.6 ( 0.1.18 According to both studies, the formation of mixed micelles is expected, but in the past, by observing positive deviations from the ideal behavior in the cmc vs bulk composition trends, other authors19-21 have suggested that two kinds of micelles, each rich in one of the two components, might coexist. Although the last hypothesis cannot be ruled out in principle, there is no unambiguous experimental evidence; in fact, opposite conclusions17,19 have been drawn on the same system by using different approaches in interpreting the same experimental cmc trends of mixtures. In a previous study,22 small angle neutron scattering (SANS) measurements, coupled with the external contrast technique, have allowed us to determine that in a solution containing SD and SPFO at a total surfactant concentration 0.49 M and an SPFO mole fraction of 0.67, mixed micelles were formed. In

Figure 1. Difference between the micellar mole fraction, xSPFO, and the stoichiometric mole fraction, RSPFO, of the same component vs stoichiometric mole fraction at the following total surfactant concentrations (M): 0.07, solid line; 0.1, dotted line; 0.2, dashed line; 0.5, dasheddotted line.

this paper we show that the same is true also for other compositions and that, more important, the above technique can be used to determine the micellar composition and test Rubingh’s theory, the only one that is able to predict the micellar composition. In Figure 1 is shown, as a function of the stoichiometric SPFO mole fraction, RSPFO, the difference between the predicted micellar SPFO mole fractions, xSPFO, calculated according to eq 2, and RSPFO, at four different total surfactant concentrations; it can be seen that, in all cases, the micelles are enriched, with respect to ideal behavior (the micelles are always richer in the component having the lower cmc), in the bulk majority component; in addition, the magnitude of the deviation increases on decreasing the total concentration and depends on RSPFO, being largest at RSPFO ≈ 0.3 and 0.7 and negligible at RSPFO ≈ 0.5. For this reason the above compositions were chosen for the present investigation. Experimental Section D2O was an Aldrich (99.8 D at. %) product; doubly distilled H2O was used. SPFO was prepared by neutralizing perfluorooctanoic acid (Fluka) with NaOH (Merck). SD was a Sigma product. Before use, SPFO and SD surfactants were dehydrated under vacuum at 60 °C for 72 h. SPFO and SD 0.35 M stock solutions in H2O and D2O were prepared. By mixing proper volumes of the above solutions, SPFO-SD solutions in H2O and D2O were prepared at the same total surfactant concentration and three different SPFO mole fractions: 0.33, 0.53, and 0.73. For contrast studies, appropriate volumes of the stock H2O and D2O solutions at the three mentioned SPFO mole fractions were mixed to provide the desired solvent compositions. For measurements as functions of the total surfactant concentration, the 0.35 M solutions were diluted with the appropriate solvent mixture. SANS measurements were performed with the 30m SANS Camera of the W.C. Kohler Center for Scattering Research at Oak Ridge National Laboratory, TN. A 64 × 64 cm bidimensional detector was placed 150 cm from the sample. The selected neutron wavelength was λ ) 4.75 Å with ∆λ/λ ) 6%. With the above geometry, the accessible range of momentum transfer Q was 0.048-0.35 Å-1. Solutions were contained in quartz cells with a thickness of 1 or 2 mm depending on the H2O content; an external thermostat maintained the temperature

Compositions of Mixed Micelles Determined by SANS

J. Phys. Chem. B, Vol. 101, No. 46, 1997 9527

Figure 3. Experimental SANS scattering data, d∑(Q)/dΩ, for SPFOSD mixtures at an SPFO mole fraction RSPFO ) 0.33. Symbols refer to solvent compositions (% D2O) as follows: b 100, O 80, 2 60, 4 40, 9 20, 0 0.

Theory and Data Analysis The scattering cross section for a two-phase system constituted by a continuous phase and a scatterer can be written as23,24

d∑(Q)/dΩ ) Np〈F(Q)〉2S(Q) + Np∆(Q) + C

(4)

where Q ) 4π sin(θ)/λ, 2θ being the scattering angle, is the modulus of the momentum transfer vector, Np is the particle number density, F(Q), is the scattering amplitude, S(Q) is the static structure factor, and C is a constant including terms such as instrumental background and incoherent scattering. The term ∆(Q) ) 〈F(Q)2〉 - 〈F(Q)〉2 describes deviations from spherical symmetry and size polydispersion. For monodispersed spheres 〈F(Q)2〉 ) 〈F(Q)〉2 ) P(Q) and eq 4 becomes Figure 2. Experimental SANS scattering data, d∑(Q)/dΩ, for SPFOSD mixtures in D2O at the three considered SPFO mole fractions, RSPFO: top 0.33; middle 0.53; bottom 0.73. In all cases symbols refer to total surfactant concentrations (M) as follows: b 0.07, O 0.11, 2 0.17, 4 0.28, 9 0.35.

at 25.0 ( 0.2 °C. SANS data from the detector were radially averaged after corrections for background, detector sensitivity, empty cell scattering, and sample transmission. Differential scattering cross sections, d∑(Q)/dΩ, were obtained after calibration with water. Although we are convinced that, in principle, all experimental data should be available to the reader, because of the large number of runs, we chose to report only some typical data. The interested reader can obtain the full set upon request. Experimental SANS data for SPFO-SD mixtures in D2O as functions of the total surfactant concentration and at the three considered SPFO mole fractions are reported in Figure 2. An interaction peak is observed in all cases; on decreasing the concentration, the scattering intensity decreases and the interaction peak shifts toward smaller Q values. In Figure 3 are shown SANS data for a total surfactant concentration 0.35 M and SPFO mole fraction 0.33 at different solvent compositions. The intensity is higher at 99% D2O, it decreases, going through a minimum, and it increases again on lowering the D2O content. The peak position is not appreciably affected by changes in the D2O content, indicating the absence of a detectable isotope effect on particle dimensions and interactions. Similar trends have been observed for the other concentrations at the three compositions examined in this paper.

d∑(Q)/dΩ ) Np P(Q) S(Q) + C

(5)

In the simplest case of spherical aggregates of radius R, having a uniform scattering length density, the scattering amplitude is defined as25

F(Q) ) ∆FVΦ(QR)

(6)

where the contrast ∆F is the difference between the scattering length density of the particles (Fp) and of the solvent (Fs), V is the total volume of aggregates, and

Φ(QR) ) 3

sin(QR) - QR cos(QR) (QR)3

(7)

In most cases, deviations from a monodisperse spherical shape or nonhomogeneities in the scattering density have to be taken into account.26-28 The SANS technique is useful for determining structural parameters of the systems under analysis;26-28 in this paper we have used the SANS technique with the purpose of deriving the mixed micelles composition. Previously,29 the SANS technique had been applied to suggest that, in an aqueous solution containing ammonium decanoate and ammonium perfluorooctanoate, mixed micelles are formed. Recently, Penfold30 applied this technique, coupled with internal contrast variation (variation of the surfactant isotopic composition), in order to obtain the micelle composition for a mixture of hydrogenated surfactants. In our case, because SD and SPFO have different scattering length densities, the internal contrast

9528 J. Phys. Chem. B, Vol. 101, No. 46, 1997

Pedone et al.

TABLE 1: Adjustable and Derived Parameters from Fitting of SANS Data for Water-SD-SPFO Solutions at SPFO Molar Fraction rSPFO ) 0.33 as a Function of D2O Content (%v/v) and Total Surfactant Concentration C (mol/L)a C [D2O]

0.350(2)

0.280(2)

0.170(3)

0.103(2)

0.068(2)

TABLE 2: Adjustable and Derived Parameters from Fitting of SANS Data for Water-SD-SPFO Solutions at SPFO Molar Fraction rSPFO ) 0.53 as a Function of D2O Content (%v/v) and Total Surfactant Concentration C (mol/L) C [D2O]

0.350(2)

0.278(2)

Z 99.8 79.6(2) 59.7(3) 39.9(1) 0

13.8(6) 13.1(7) 13.0(7) 11(3) 13.3(9)

99.8 79.6(2) 59.7(3) 39.9(1) 0

57.7(6) 58.5(7) 58.9(6) 63(2) 57.4(9)

99.8 79.6(2) 59.7(3) 39.9(1) 0

24.0 24.1 24.3 24.9 23.9

13.8(5) 14.2(3) 13.5(5) 13.0(3) 12.0(9)

99.8 79.6(2) 59.7(3) 39.9(1) 0

9.03 7.22 3.55 1.14 2.12

ν 55.8(5) 57.4(3) 58.1(5) 55(2) 53(1) R2 23.7 23.9 24.0 23.5 23.4 χ 6.35 2.55 2.10 1.38 2.33

99.8 79.6(2) 59.7(3) 39.9(1) 0

3.660 1.916 0.709 0.091 0.642

2.740 1.449 0.535 0.059 0.474

12.9(5) 12.3(6) 13.6(0.8) 12.8b 13(2)

11.5(6) 12.9(8) 14(2) 13(1) 10(5)

99.8 79.7(2) 58.9(2) 20.2(1) 0

12.7(3) 11.3(7) 8(1) 14(1) 13.9(5)

52.4(4) 52.5(5) 54.3(6) 52.0b 50(2)

51.7(4) 57.1(5) 59(1) 43(6) 36(3)

99.8 79.7(2) 58.9(2) 20.2(1) 0

55.5(3) 54.1(6) 59(1) 58(1) 52.7(4)

23.2 23.2 23.5 23.5 22.7

23.2 23.9 24.2 21.4 20.4

99.8 79.7(2) 58.9(2) 20.2(1) 0

23.7 23.6 24.5 24.0 23.2

4.76 2.97 2.15 2.09 P(0) 1.462 0.762 0.295 0.037b 0.229

3.11 2.32 2.22 1.67 2.43 0.736 0.425 0.167 0.017 0.082

0.388 0.202 0.076 0.010

a

Numbers in parentheses represent uncertainties in the last digit. b For these mixtures, the intensities were very small and it was not possible to perform the fits. In these cases ν and Z were kept constant and equal to the averages obtained from solutions with the same chemical and different isotopic compositions.

can be varied very easily by varying the molar fraction of the two surfactants; however, Penfold’s approach cannot be applied because the structure of SD and SPFO pure micelles are very different;22,23 in what follows we show the alternative procedure used. The thermodynamic limit (Q ) 0) of the form factor P(0) is equal to

P(0) ) [(Fp - Fs)V]2

(8)

This makes possible the evaluation of the particles scattering density; in fact, the scattering length densities of D2O (6.325 × 10-10 cm-2) and H2O (-0.56 × 10-10 cm-2) cover the Fp range of most surfactants, and therefore, by using solvent H2O/ D2O mixtures with the appropriate composition, the particles scattering density can be matched. Under these conditions, ∆Fs ) 0 (match point, m.p.) and therefore P(0) ) 0. In practice the square root of I(0) is plotted vs Fs or the solvent composition, and the m.p. is obtained either graphically or analytically. For multicomponent aggregates, the particle scattering length density, Fp, is given by

Fp )

0.172(2)

0.103(2)

12.8(3) 12.7(5) 10(1) 14(3) 13(1)

11.9(4) 9.4(6)

0.068(2)

Z

∑iFiXi

(9)

where Xi is the mole fraction in the aggregate of component i of scattering length densities Fi. Clearly, for a two-component particle, the knowledge of Fp allows the evaluation of the composition.

13.2(3) 12.6(4) 9(1) 14(2) 14.6(6)

99.8 79.7(2) 58.9(2) 20.2(1) 0

3.40 4.05 1.99 1.67 2.26

ν 51.6(2) 52.0(4) 55.6(8) 56(2) 50.4(5) R2 23.1 23.2 23.9 23.8 22.7 χ 2.96 1.65 1.41 2.04 2.12

99.8 79.7(2) 58.9(2) 20.2(1) 0

2.482 1.066 0.277 0.327 1.149

1.791 0.795 0.204 0.244 0.841

13(9) 13(3)

48.1(2) 48.9(3) 50(1) 51(2) 46.0(8)

46.7(2) 47.9(4)

22.5 22.6 23.1 22.9 22.1

22.3 22.7

1.65 1.10 1.28 1.70 2.09 P(0) 0.943 0.425 0.108 0.118 0.418

52(6) 44(2)

22.1 21.8 1.66 1.21 1.82 2.40 0.453 0.208

0.240 0.108

0.055 0.190

0.104

a

Numbers in parentheses represent uncertainties in the reported figure.

Micellar aggregates are usually considered as being formed by two regions with different scattering length densities: a core containing the hydrocarbon chains and a shell containing the ionic or polar heads, the counterions, if present, and solvent molecules. Within this model, eq 6 becomes

F(Q) ) (Fcore - Fshell)VcoreΦ(QRcore) + (Fshell - Fs)VtotalΦ(QRtotal) (10) where V is the volume corresponding to the radius R. As a consequence eq 8 can be rewritten as

P(0) ) [Vcore (Fcore - Fshell) + Vtotal (Fshell - Fs)]2 (11) In the case of interest to the present work, i.e., aqueous solutions of SPFO-SD mixtures, the two surfactants have the same head group and hence Fcore is the only parameter depending on micellar composition. At the match point, eq 11 can be solved for Fcore, yielding the following equation

Fcore ) [Vcore Fshell - Vtotal (Fshell - Fs)]/Vcore

(12)

where the only unknown is the solvent scattering density at the match point, Fs, since all other quantities are related to this quantity and to known molecular properties of the two surfactants. In conclusion, once P(0) values as functions of the solvent D2O content are known, one can determine the micellar composition. Incidentally, if two different kinds of micelles were present at the same time, there would never be a solvent composition such that P(0) ) 0. One method to extract the value of P(0) from the SANS scattering intensity uses Guinier’s law,31 which is strictly valid

Compositions of Mixed Micelles Determined by SANS TABLE 3: Adjustable and Derived Parameters from Fitting of SANS Data for Water-SD-SPFO Solutions at SPFO Molar Fraction rSPFO ) 0.73 as a Function of D2O Content (%v/v) and Total Surfactant Concentration C (mol/L) C [D2O] 99.8 80.3(2) 59.4(2) 40.7(2) 19.4(1) 0

0.350(1) 11(1) 12.1(9) 13.5b 13.5b 14.4(5) 13.8(6)

0.281(2) 12.6(7) 11.8(7) 13.3b 13.3b 15(1) 14(1)

99.8 80.3(2) 59.4(2) 40.7(2) 19.4(1) 0

46(1) 48.2(7) 47.6b 47.6b 48.9(4) 46.9(5)

44.0(6) 46.9(6) 45.7b 45.7b 45.2(8) 46.6(8)

99.8 80.3(2) 59.4(2) 40.7(2) 19.4(1) 0

22.2 22.6 22.4 22.4 22.6 22.2

21.8 22.4 22.1 22.1 21.9 22.2

99.8 80.3(2) 59.4(2) 40.7(2) 19.4(1) 0

7.99 2.26

4.79 1.74

0.98 0.95

1.64 3.52

99.8 80.3(2) 59.4(2) 40.7(2) 19.4(1) 0

1.377 0.483 0.037b 0.090b 0.675 1.704

1.003 0.359 0.026b 0.073b 0.489 1.325

0.174(2) Z 13.3(7) 11.9(9) 14.6b 14.6b 17(2) 16(1) ν 39.4(4) 42.5(6) 41.2b 41.2b 42.6(7) 40.3(4) R2 20.9 21.6 21.1 21.1 21.2 20.8 χ 2.43 1.05 0.87 1.23 P(0) 0.490 0.177 0.013b 0.036b 0.252 0.624

0.112(2)

0.069(1)

J. Phys. Chem. B, Vol. 101, No. 46, 1997 9529 TABLE 4: Total Surfactant Concentration (C), Solvent Composition at the Match Point (D2O), Micellar Core Scattering Density (GCORE), and Micellar SPFO Mole Fraction (XSPFO) for Aqueous SPFO-SD Mixtures at the Three Stoichiometric Compositions C, mol/L

D2O, %v/v

FCORE, 10-10 cm-2

XSPFO

RSPFO ) 0.33 11.9(9) 12(2) 13.4b 13.4b 18(4) 11(1) 34.7(4) 39.1(8) 37b 37b 40(1) 33.7(6) 20.0 20.9 20.3 20.3 20.8 19.9

11(3) 9(6) 13.5b 13.5b 18(17) 16(3) 32.3(6) 40(2) 37.2b 37.2b 40(3) 36.2(6) 19.5 21.5 20.3 20.3 20.6 20.6

1.69 0.84

2.37 1.84

0.80 0.98

1.07 1.21

0.234 0.088 0.006b 0.018b 0.132 0.289

0.079 0.032 0.001b 0.012b 0.066 0.148

a

Numbers in parentheses represent uncertainties in the reported figure. b For these mixtures, the intensities were very small and it was not possible to perform the fits. In these cases ν and Z were kept constant and equal to the averages obtained from solutions with the same chemical and different isotopic compositions.

for noninteracting systems [S(Q) ) 1]. Previously this method has been applied to demonstrate the existence of just one kind of mixed micelles in one particular SPFO-SD mixture.22 In the present work, in addition to verifying the existence of mixed micelles at bulk surfactant compositions different from that previously considered, the objective is to determine the micellar composition, and therefore, in order to find P(0) values, rather than using Guinier’s law, it was decided to adopt the same fitting procedure of the whole scattering curve that has been successfully used24,26-28 to derive structural information from micellar systems. The procedure has been fully described elsewhere,24 and no further details will be given here. Suffice it to say that eq 4 was fitted to the experimental d∑(Q)/dΩ vs Q curves by using a model of monodispersed spherical aggregates constituted by a hydrophobic core containing the hydrocarbon and fluorocarbon chains and a shell containing the ionic groups, a fraction of counterions, and solvation water molecules (11 for each counterion and 13 for each head group32). An iterative procedure was used to calculate the unknown micellar composition required in the fitting procedure: as a starting value, the bulk composition was used and P(0) values were obtained from the fit; from the P(0) values vs D2O solvent content, the m.p. composition was determined, and from this, a new value for the micellar composition was obtained, which was then used to repeat the fitting procedure; iterations were

0.35 0.28 0.17 0.11 0.07

29.70 29.36 28.47 28.01 26.76

0.35 0.28 0.17 0.11 0.07

40.84 40.78 40.31 39.70 39.04

0.35 0.28 0.17 0.11 0.07

51.95 51.75 52.46 52.73 55.36

1.485 1.461 1.400 1.368 1.282

0.324 0.324 0.308 0.295 0.296

2.252 2.248 2.215 2.173 2.128

0.537 0.538 0.527 0.518 0.517

3.571 3.003 3.052 3.070 3.251

0.753 0.759 0.758 0.761 0.825

RSPFO ) 0.53

RSPFO ) 0.73

performed until the calculated composition agreed with the previous one to (1%. The adjustable parameters used in the calculations were the micellar total apparent charge Z and the aggregation number ν. The solid lines in Figures 2 and 3 represent the results of the fit for the cases shown; they can be considered typical of the whole set of results. In Tables 1, 2, and 3 are reported ν and Z fitting parameters values, the total radius R, the P(0) values, and derived parameters for the three SPFO mole fractions considered. In addition, the values of the standard deviation of the fit, χ, are also reported:

χ)

x

∑[Wi(Ic,i - Io,i)]2

no. of points - no. of parameters + 1

(13)

expressing the agreement between calculated, Ic,i, and experimental, Io,i, intensities. In each row are shown the parameter values at constant solvent composition and the corresponding ones at various surfactant concentration values, while in each column are shown the same data at constant surfactant concentration and various solvent composition. Due to the low intensities obtained for the samples whose composition is close to that corresponding to the m.p., the values noted refer to calculations of the experimental intensity performed using ν and Z values equal to the averages obtained from solutions with the same chemical and different isotopic composition, while the P(0) values of Tables 2 and 3 related to the lower surfactant concentration, C ) 0.07 M, have been obtained using ν and Z values extrapolated from the trends obtained by plotting the corresponding values at various surfactant concentration at the same isotopic composition (ν ) 51, Z ) 10.2 and ν ) 45.6, Z ) 10.8, respectively). As can be seen (Figures 2 and 3 and χ values in Tables 2-4), the agreement between experimental and calculated intensity is generally good. Only in a few cases, at high concentrations, large χ values, reflecting poorer agreement between calculated and experimental results, are obtained. This is due to the choice made about the micellar form: we use a spherical model, but, on increasing the concentration, deviations from sphericity are expected; an ellipsoidal model would give better results at higher concentration. Since, in the present work, the focus is on micellar composition rather than structure, in order to determine

9530 J. Phys. Chem. B, Vol. 101, No. 46, 1997

Figure 4. P(0)1/2 (see text) vs D2O solvent content for SPFO-SD mixtures at a mole fraction RSPFO ) 0.33. Symbols refer to concentrations (M) as follows: b 0.07, O 0.11, 2 0.17, 4 0.28, 9 0.35.

Pedone et al.

Figure 6. P(0)1/2 (see text) vs D2O solvent content for SPFO-SD mixtures at a mole fraction RSPFO ) 0.73. Symbols refer to concentrations (M) as follows: b 0.07, O 0.11, 2 0.17, 4 0.28, 9 0.35.

Figure 5. P(0)1/2 (see text) vs D2O solvent content for SPFO-SD mixtures at a mole fraction RSPFO ) 0.53. Symbols refer to concentrations (M) as follows: b 0.07, O 0.11, 2 0.17, 4 0.28, 9 0.35.

P(0) values in a consistent manner, we chose to use in all cases the spherical model. Discussion Some general observations on the structural parameters can be derived from inspection of the tables. For example, it can be observed that, at a given SPFO mole fraction, the aggregation number ν increases with increasing total surfactant concentration. Pure SD22 micelles, in agreement with the lower cmc, are bigger than those formed by SPFO.33 At a fixed total surfactant concentration, ν of the mixtures decreases with increasing SPFO mole fraction, and it is always intermediate between that of aqueous solutions of pure SD and that of pure SPFO. In Figures 4-6 are shown plots of P(0)1/2 vs the D2O content of the solvent mixtures. Within the experimental uncertainties, a single match point is observed in all cases; this leads us to suggest that, at least at the examined compositions, a single kind of mixed micelle is always formed. It can be noticed that at RSPFO ) 0.33, on decreasing the total surfactant concentration, the match point shifts toward lower scattering densities (lower D2O content), indicating that the micelles become enriched in hydrogenated surfactant, the majority component. The same is true at RSPFO ) 0.53, although the absolute variation is less pronounced. At RSPFO ) 0.73 the match point shifts toward

Figure 7. Comparison between the micellar compositions obtained from SANS data (symbols) vs total surfactant concentration, C, and those predicted by the ideal (dashed line) and regular (solid line) solution theories. Base lines correspond to the stoichiometric mole fractions.

higher scattering densities, indicating that the micelles become enriched in fluorinated surfactant, the majority component. From the match point solvent compositions, obtained from linear least-squares fits of the plots in Figures 4-6, the solvent scattering densities and, from eq 12, the scattering densities of the micellar core were calculated. The micellar SPFO mole fractions, xSPFO, were thus readily obtainable. In Table 4 the above results are summarized. Finally, in Figure 7, the above obtained micellar compositions are compared with those predicted by the ideal and regular

Compositions of Mixed Micelles Determined by SANS solution theories; the general observed trend is that the mixed micelles are always richer in the component present in solution in greater proportion; there is an overall qualitative agreement between the present results and those predicted by Rubingh’s approach, although a significant difference is noticeable at large SPFO concentrations, suggesting that further tests of the theory are required in order to include subtle interaction effects due to differences in the chemical nature of the surfactants. In conclusion, it has been shown that, when dealing with mixed hydrogenated-fluorinated surfactant solutions, the SANS technique can be effectively used to determine (a) whether a single kind of mixed micelle is formed and (b) the composition of the mixed micelles. Hall et al.34 have recently used ionselective electrodes to derive the composition of mixed micelles by performing electromotive force measurements; this technique cannot be applied to the systems under study in the present work since the surfactants here considered have the same ionic group and counterions. More generally, since SANS relies on contrast differences, the technique is easily applicable to surfactants possessing sufficiently different scattering densities, although, if this were not the case, deuteration of one of the surfactants (internal contrast) might be employed. Acknowledgment. The authors are grateful to the Oak Ridge National Laboratory (ORNL) for the opportunity to perform the measurements and to the CNR (Consiglio Nazionale delle Ricerche) and the Ministero dell’Universita` e della Ricerca Scientifica e Tecnologica (MURST) for financial support. References and Notes (1) Jost, F.; Leiter, H.; Schwuger, M. J. Colloid Polym. Sci. 1988, 266, 554. (2) Rubingh, D. N.; Jones, T. Ind. Eng. Chem. Prod. Res. DeV. 1982, 21, 176. (3) Kurzendorfer, C. P.; Schwuger, M. J.; Lange, H. Ber. Bunsen-Ges. Phys. Chem. 1978, 82, 962. (4) ImproVed Oil RecoVery by Surfactant and Polymer Flooding; Shah, D. O., Schechter, R. S. Eds.; Academic Press: New York, 1977. (5) Dobia`s, B. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986; pp 216-224. (6) Ogino, K.; Abe, M. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical

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