Langmuir 1995,11, 2496-2503
2496
Solution and Adsorption Behavior of the Mixed Surfactant System Sodium Dodecyl Sulfateln-HexaethyleneGlycol Monododecyl Ether J. Penfold Isis Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, U.K.
E . Staples,* L. Thompson, I. Tucker, and J. Hines Unilever Research Laboratory, Quarry Road East, Bebington, Wirral, L69 3JW, U.K.
R. K. Thomas and J. R. Lu Physical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford, U.K. Received November 15, 1994. In Final Form: March 14, 1995@ Small angle neutron scattering and neutron reflectivity have been used to study the adsorption and micellization of the mixed anionic-nonionic surfactant system, sodium dodecyl sulfate (SDS)/n-dodecyl hexaethylene glycol (C12E6) in 0.1 M NaCl . The composition of the surfactant layer adsorbed at the air-solution interface is compared with the micellar compositionover a range of concentrations,extending from 10 to 100 times the critical micellar concentration (cmc). It is found that the results are in close agreement with the predictions of regular solution theory (RST).
Introduction The evolution of micelle compositionin mixed surfactant systems as electrolyte and/or concentration is varied is of much current interest. In domestic and industrial applications surfactants are usually mixtures, either by virtue of impurity or a s a result of a blending process that aims to improve synergy. In a surfactant mixture the micelle that forms a t the critical micellar concentration (cmc)has a compositionbiased toward the component with the lowest free energy, and the micelle composition only tends to that of the bulk composition in the limit of high concentration. As the monomer concentrations must also evolve, it is inevitable that any synergy will vary with concentration. The regular solution theory (RST) can be successfully used to describe the variation of cmc with composition for many systems given only the cmc’s of the pure components and of a mixture. There are, however, many indications that this very pragmatic theory has limitations when its predictions of micellar and hence monomer composition are c0nsidered.l It is an aim of this program of work to establish under what circumstances this simple approach can be successfully applied. In a recent study2we showed that the adsorption behavior of the system SDSln-dodecyltriethylene glycol/O.1M NaCl at the air-water interface is consistent with RST. This system was selected because the areaslmolecule of the constituents were similar and the underlying assumption of RST (ideal entropy of m i ~ i n g would )~ most likely be satisfied. We now extend our study to a system, SDSI in 0.1 M NaC1, in which the aredmolecule of the two components is significantly different and for which surfactant electrode measurements1 are not consistent with the predictions of RST. The choice of the system Abstract published in Advance ACS Abstracts, June 15,1995. (1)Davidson, C.J. Ph.D. Thesis, University ofAberdeen, Aug 1983. (2) Staples, E.;Thompson, L. G.; Tucker, I.; Penfold, J.;Thomas, R. K.; Lu, J. R. Langmuir 1993,9, 1651. (3) Rubingh, D. N. In Solution Chemistry ofSurfactants;Mittal, K. L.,Ed.;Plenum Press: New York, 1979; Vol. 1, p 337. @
also reflects its commercialrelevance, and we have adopted a methodology that minimizes the impact of surfactant impurities. Neutron reflectivity is used to characterize adsorption a t the air-water interface whereas small angle neutron scattering(SANS) is used to obtain the composition of the micelles. By making measurements over a range of supracmc concentrations we can directly compare the measured surface composition and micelle compositions. This approach overcomes the problems associated with surfactant impurity experienced with surface tension studies. In an earlier study2we confirmed the ready incorporation ofthe usual SDS contaminant, dodecanol,into the micelles at concentrations only twice the cmc.
Experimental Section The micelle and surfacecompositions for 5050 and 70:30 SDSI C&6 (mol %) in 0.1 M NaCl have been determined for a series of surfactant concentrationsbetween 1 x and 6.25 x M in the case ofthe micelle and between 1x and 3.1 x M for the surface. The protonated surfactants were obtained from Nikkol (h-ClzE6) and BDH (SDS). The deuterated SDS (d-SDS) was obtained from MSD Isotopes, and alkyl-chain deuterated C12E6 was synthesized and purified by methods described previ~usly.~ The chemical purity of the surfactants was assessed by surface tension measurements and TLC. The surfactants for the neutron measurements were used without further purification. Deuterium oxide (DzO)was supplied by Sigma, and high-purity water (Elga Ultrapure) was used throughout. The glassware and PTFE troughs used for the neutron measurements were cleaned using alkaline detergent (Decon 90) followed by copious washing in high-purity water. Surface Tension Measurements The SDS used in the derivation of the interaction parameter (8)was purified using a foam extraction method for the removal of any dodecanol impurity. The SDS was recovered by freeze drying. Subsequent SDS solutions reached surface equilibrium almost immediately and did not produce a minimum a t the cmc. Surface tensions a t (4) Lu, J. R.; Li, Z. X.; Su ,T. J.;Thomas, R. K.; Penfold, J. Langmuir 1993,9, 2408.
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equilibrium were measured at 298 K using the plate method of W i l h e l m ~ . ~ All glassware and the platinum plate were cleanedprior to use by soaking in chromic acid followed by rinsing with copious quantities of distilled water. The platinum plate was flamed and allowed to cool before use. A Kruss K12 maximum pull tensiometer was used for all surface tension measurements. The solution surfaces were cleaned immediately before each measurement by suction, and the solution concentrations of subsequent dilutions were determined gravimetrically.
It is straightforward to extend this method to the determination of the surface composition of a binary mixture (as is the case here). By selective deuteration of each component in turn the surface excess of each component can be evaluated. Equation 1then becomes”
of
Neutron Scattering Measurements The neutron reflection measurements were made on the reflectometer CRISP,6and the SANS measurements were made on the LOQ,7 difiactometer, both on the ISIS pulsed neutron source at the Rutherford Appleton Laboratory (U.K.),using a broad range of neutron wavelengths and the time of flight method. Neutron Reflection The reflection measurements were made using a single detector at a fixed an le of 1.5”and neutron wavelengths in the range 0.5-6.5 . The absolute reflectivities were calibrated with respect to D20 as described previously.8 The flat background was determined from the reflection signal obtained in the limit of high values of momentum transfer K ( K = (4dA) sin(@), where 8 is the angle of incidence and A the neutron wavelength) and subtracted from the full K range. This has been shown to be a valid procedure provided there is no small-angle scatter from the bulk solution.2 This was verified by making (offspecular) measurements on either side of the specular reflection. For a deuterated surfactant in null reflecting water, nrw (92 mol % H20/8 mol % D20 has a scattering length of 0, that is, a refractive index identical to that of air), the reflected signal arises only from the adsorbed surfactant layer at the interface. The most direct procedure for determining the surface concentration of the surfactant is to fit the measured reflectivity profile by comparing it with a profile calculated using the optical matrix methodg for a simple structural model. Typically, in the determination of the surface concentration, it is sufficient to assume that the surfactant is in the form of a single layer of homogeneous composition. The parameters obtained from such a fit are the scattering length density (e),and thickness (z), of the layer. The aredmolecule in the adsorbed layer is then
e = bJA1t) + bJ(A2t)
(2)
where bi and Ai are the scattering lengths and areas/ molecule of each component. Measurement errors can be reduced by adopting a labeling scheme that maximizes the signal. We have therefore performed measurements with both surfactants deuterated (d-SDS ( b = 2.763 x A) and d-C&e ( b = 2.735 x A)) and with one component hydrogenous (h-ClzE6 (b = 1.32 x A)). The observed change in surface scattering cross section (et)results from the change in b (C12Ee) and, by using eq 2, the aredmolecule is obtained. SANS The scattering from a mixed surfactant system can be described as follows:12
1
where b is the scattering length of the adsorbed molecule and the adsorbed amount (or surface excess, r, expressed in units of mol cm-2) is given by r = 1/(NaA), where Nu is Avagadro’s number. A detailed assessment of the errors in such a prpcedure is given e1sewhere;lOthe errors are typically f 2 A2 at an aredmolecule of 50 Az. ( 5 ) Wilhelmy, L. Ann. Phys. (Leipzig) 1883, 119, 177. (6) Penfold, J.;Ward, R. C.; Williams, W. G.; J.Phys. E: Sci. Instrum. 1987,20, 1411.
(7) Heenan, R. K.;Osborn, R.; Stanley, H. B.; Mildner, D. F. R.; Furusaka, M. J. Submitted to J. Appl. Crystallogr. (8)Lee, E. M.; Thomas, R. K.; Penfold, J.;Ward, R. C. J.Phys. Chem. 1989, 93, 381. (9) Penfold, J. In Neutron, X-ray and Light Scattering; Lindner, P., Zemb, T., Eds.; Elsevier: New York, 1991; p 223. (10)Simister, E. A.; Thomas, R. K.; Penfold, J.;Aveyard, R.; Binks, B. P.; Cooper, P.; Fletcher, P. D. I.; Lu, J. R.; Sokolowski, A. J. Phys. Chem. 1992,96,1383.
where N = micelle number density, V = micelle “dry” volume, eip= micelle scattering length density, e, = solvent scattering length density, K = scatteringvector ( K = (4nJA) sin(8/2), 8 = scattering angle),P ( K = ) particle form factor, S(K)= structure factor, and the summation is over all micelle compositions. In the limit of small scattering vector P ( K )= 1and in the presence of electrolyte (0.1 M NaC1) S(K)is close to unity for the relatively dilute solutions studied here. For SANS measurements, instrumental systematic variations such as detector response and calibration errors introduce errors in the determination of absolute scattering cross sections which are typically 10%. However, in this instance we are interested only in the composition of the micelles, and if we assume that the composition distribution is narrow, we can ratio the (background subtracted) intensities obtained from the system with both surfactants hydrogenous and where one component is deuterated and reduce many of these uncertainties. Specifically for the combination of h-SDS/h-C12E6 and h-SDS/d-ClzEs in DzO, the volume and mole fractions of the micelles are determined from
v, =
(&
- ‘)(@h-SDS
- @D20)
(@h-C12E6 - @h-SDS) - &(@d-C12E6
(4)
- @h-SDS)
Where VF = volume fraction of C12E6 , RI = Ih-sDsm-clzEd = scattering length density of component i, and Ii = scattered intensity ( K =. 0)for system i,
Ih-SDSld-Cl2E6, ei
where hf~= mole fraction of 6 and voli = molecular volume of component i. Mixed micelles can of course have large composition fluctuations, and from eq 3, it can be seen that different isotopic labeling will reflect different weightings of the micelle composition distribution. This results in a small (11)Penfold, J.;Thomas, R. K.; Simister, E. A.; Lee, E. M.; Rennie, A. R. J. Phys.: Condens. Matter 1990,2, SA 411. (12)Cabane, B. In Surfactant Science Series Vol.22; Zana, R., Ed.; M. Dekker: New York, 1987; p 57.
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error in the micelle compositionsmeasured using the above procedure. The error can, however, be determined because in the limit of high concentration the true micelle composition must approach that of the total solution. Consequently, high accuracy can be obtained as the approach does not involve modeling of micelle interactions, geometry, composition distribution, or micelle concentration and obviates the need for accurate absolute intensities. In the presence of appreciable intermicellar interactions, isotopic substitution of the different components will produce different “effective”structure factors,lSreflecting the additional polydispersity due to composition fluctuations. This offers the possibility that, in future studies, the width of the micelle composition may be measured. As we required the evaluation of micelle composition at very low micelle concentrations, the SANS measurements were made over a limited wavelength range. LOQ normally uses a chopper (25 Hz)to select alternate neutron pulses from the source, allowing use of long-wavelength neutrons. By using the source frequency of 50 Hz the flux a t the sample is doubled at the expense of information a t small scattering vectors but without compromising the extrapolation procedure implied into eq 3. When DzOis used as the solvent, the need for expensive deuterated chemicals is minimized and, importantly, the lower incoherent scattering results in higher transmissions and allows long path length cells to be used. The required sensitivity was hence obtained by using a large (12 mm) beam and 5 mm path length cells.
Regular Solution Theory For a mixed surfactant system the variation of mixed cmc, C*, with composition (mole fraction), a , is given by3
l-a +c* flC2 f2C2
1- a
C*a
1 p=-ln(1- x)2
CIX
and
The supra-cmc micelle compositionx is derived by iterative solution of X=
-(C
+ ((C - D)’+ 4aCD)”
- D)
20
(9)
where D = fzCz - f1CI and C is the total surfactant concentration. Hol1andl4introduced a surface interaction parameter, ,Bs, to describe the excess free energy of mixing for surfactant a t the air-water interface and explicitly included the change in monomer activity associated with surface pressure. An expression was derived that provides a direct link between the activity coefficient and mole fraction in the micellar and adsorbed “phases”. Assuming that the surfactant areas/molecule are invariant, pSand the surface composition can be obtained by iterative solution of
RT
fi.i
Ai
fsixsi
+ ximaX
n = -In -
(10)
where Ai is the area/molecule of surfactant i, is the surface pressure of surfactant i above the cmc, n is the surface pressure of the mixed surfactant, and f s i = exp(Pgsi‘).
(6)
where C1 and CZare the cmc’s of the pure components and thefi are activity coefficients. Within the regular solution approximation the activity coefficients can be expressed in terms of the micelle composition x and a single interaction parameter p,
where Wlland WZZare the self-interaction energies and W12 is the distinct interaction energy within the mixed micelle. The assumption that a single interaction parameter is appropriate for all mole fractions is synonymous with ideal entropy of mixing. The ,B parameter can be identified with simple pairwise monomer interactions and consequently does not impose any explicit structure on the monomer-monomer distribution within the micelle. Any treatment that involves a functional (mole fraction) form for the interaction parameter introduces a nonideal entropy of mixing, that is, the /3 parameter changes with the monomer self and distinct “radial” distribution functions within the micelle and implicitly includes multibody interactions. The micelle composition ( x ) a t the cmc and the interaction parameter ,B are obtained by iterative solution of the following: (13) Hayter, J.B.;Penfold, J. Colloid Polym. Sci. 1988,261, 1072.
To complete a description of the system we also need to know the monomer concentrations that are in equilibrium with the measured micelle composition. There is insufficient accuracy in the determination of the absolute intensities, especially where micelle compositions vary with Concentration, to make a direct determination. However, from the micelle composition obtained it is possible, using the principles of mass conservation, to predict the possible monomer concentrations that may coexist with the micelle. At one extreme the monomer could be entirely that of the component (relatively) deplete in the micelle while higher monomer concentrations tend to, and in the limit of negligible micelle concentration reach, the bulk composition. This situation is described by the expression
(Mt, - Mm,) ~ m o *= (Mmo, -
MmJ ct
(11)
where Mti = mole fraction of component i in total, Mmi = mole fraction of component i within the micelle, Mmoi = mole fraction of component i as monomer, C ,,, = monomer concentration, and Ct = total surfactant concentration. The true monomer concentration is obtained from the point a t which the curve of possible monomer concentrations crosses the surface tension derived cmc vs composition curve. Thus we can obtain the monomer concentrations that are in equilibrium with the measured micelle composition,and this provides the basis for a direct comparison with the predictions of RST. Furthermore, if we assume that intermicellar interactions have a negligible effect on the micelle compositions, (14) Holland, P. M. Colloids Surf: 1986, 19, 171.
Solution and Adsorption Behavior of a Mixed Surfactant I
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Scattering Vector, K (in A " ) Figure 1. Ratio of the SANS scattering intensities from two isotopic combinations(~-SDS:~-C~~E~)/(~-SDS:~-CI~E~) for the system 70 mol % SDS/30 mol % ClzE$O.l M NaCl at a surfactant concentrationof 0.01 M. Inset shows scattering intensities for 2.0 x M 70:30 mol % SDS/&& in DzO: (0,top curve) h-SDS:h-C&; (0,bottom curve) d-SDS:h-C&. Solid lines are model fits as described in the text.
we can extend this approach and, from one determination of micelle composition, deduce the total solution concentration C,' a t which solutions of a different total composition M,'will contain that measured micelle composition:
c,'
(Mt, - Mmi)
= (Mt,' - Mm,)',
(12)
Mt, = mole fraction of component in in total Mm, = mole fraction of component i within the micelle (measured)
C, = total surfactant concentration Results and Discussion Figure 1 shows the ratio of the SANS scattering intensities obtained from two isotopic combinations (h-SDS:h-Cl2E6)/(d-SDS:h-ClzE6)forthe system 70 mol % SDS/30 mol % C1zE$O.l M NaCl at a concentration of 0.01 M. The data was typical of the measurements made here, and the collection time was approximately 40 min. In this case the intercept was determined to be 6.4f 0.2, which translates, using eq 4,to a variation in composition of only f0.5%. The observed decrease in the intensity ratio with increasing K results from the different particle form factors for the two different isotopically labeled surfactant combinations. The ethylene oxide headgroup has a significant scattering cross section with respect to D20, and when Cl2E6 dominates the scattering, a n effectively larger micelle radius is observed, indicated by the more rapid decay of intensity with K . This of course
Table 1. Parameters of Micelle Geometry from Model Fits to SANS Data for 70:SO mol % SDS/CnEa concn (M) aggregation no. total radius (A) 10-2 10-3 10-3
5 2
100 95 86
24.8 24.3 24.0
does not affect the volume term used in eq 3, which can be identified with the total number of scattering molecules. The SANS data was also modeled to establish changes in micelle geometry with composition. Adequate results could be obtained if the micelles were assumed to be polydisperse (Schultz distribution) inhomogeneous spheres. Following previous work13the micelle was modeled as a central core containing only hydrocarbon chains and a n outer layer containing solvent, headgroup, and the remainder of the alkyl chains and interparticle interactions are specificallyin~1uded.l~ The scattered intensities from a surfactant DzO solution with a concentration of 2.0 x M and composition 70 mol % SDS/30 mol % C&6 are shown, along with model fits, in the inset in Figure 1. The micelle composition was assumed to be the uncorrected value obtained from eq 3. The results of the analyses (the key model parameters for the fits to the data) are shown in Table 1and indicate that the micelle radius vanes very little with composition. In Figure 2 the predictions of RST for the system SDS/ CI2E$O.l M NaCl are shown. The analysis is based on surface tension data for the pure components and on a 70 mol % SDS/30 mol % C&6 mixture. The curves depict the concentration dependence of the micelle composition (15)Hayter, J. B.;Penfold, J. Mol. Phys. 1981, 42, 109.
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2500 Langmuir, Vol. 11, No. 7, 1995 10
-’
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-*
1
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.4
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I
,
.8
1
.-c0
+ m u
C
U
0
Mole Fraction SOS
Figure2. PredictionsofRST (interactionparameterp= -2.7) for SDSIC12EdO.lM NaC1. The SANS determined micelle compositions for 5050 ( x ) and 70:30 (0) mol % SDS/C12Es are shown. for solution compositions indicated by the high concentration limits of the individual curves. Also shown in Figure 2 is the composition of the micelle, as measured by SANS, a t a series of concentrations and a t the solution compositions of 5050 and 70:30 SDS/ClzE,j (mol %). It can be seen that the high concentration points are not consistent with the known bulk composition, but the general curve suggested by the data does closely follow the predicted contour. The accuracy obtainable in the SANS measurements suggests that the observed discrepancy cannot be attributed to errors in measurement but can be identified with two assumptions implicit in eq 4. Firstly we have assumed that the molecular volumes within the mixed micelle are identical to those within the single-component micelle, and we have assumed that the distribution of micelle compositions is narrow. A more rigorous analysis of the scattering from mixed micellar systems, especially where isotopic substitution is used, must include the distributions of both composition and size. In the absence of any robust theoretical treatment we have adopted a simplified approach. By performing the analysis where the micelle composition is known, i.e., in the limit of high concentration, a correction factor for the measured value of micelle composition can be determined. By repeating such measurements at a range of compositions, in the limit of high concentration, a n appropriate correction factor for a measured composition a t any concentration can be obtained. In Figure 3 the RST prediction and the corrected data are compared. The magnitude of the correction is too large (e.g., a result of 66 mol % was obtained for a composition of 70 mol % SDS) to be attributed to changes in molecular volume alone. Agreement between measured and known composition can, however, be obtained if a skewed distribution of micelle compositions is used in eq 3. That is a distribution in which there is a very low probability of compositions rich in SDS. Such a distribution is consistent with the variation of the cmc, and of the micelle composition a t the cmc, with solution composition (see Figure 2). In Figure 4 the above data and the RST predictions for ? -2.5 and -4.0 are presented and alternative ,lvalues indicate the sensitivity of predicted micelle composition
to relatively small changes in mixed cmc. Figure 4 may also indicate a systematic trend in measured micelle composition not consistent with a single p; however, the trend is within the error implied by the surface tension data. The value of the cmc obtained from the surface tension data was found to vary significantly with equilibration time. In the short time limit (minutes)thepvalues obtained with 30:70, 5050, 70:30, and 9O:lO SDS/C12Es were -2.6 f0.2, whereas at equilibrium (40 min) p values as large as -4.0 were obtained. The origin of this phenomenon is unclear as the surface tension data indicate no minimum and the measured areas/molecule are consistent with the absence of impurity. We have ignored the inevitable composition fluctuations because in this simplified model such contributions are second order, and it is unlikely that a more rigorous analysis would alter the conclusion. As stated earlier, even with a refined model, estimates of the cmc based on this scattering approach would be subject to large error due to experimental limitations. We can, however, identify the measured micelle composition with a monomer concentration using the mass conservation approach (equation 6). In Figure 3 the possible monomer concentrations consistent with the measured micelle composition at 2.5 x M 50 mol % SDS/BO mol % ClzEs obtained using eq 6 are plotted (dashed line). It can be seen that this line intersects the point representing the surface tension derived cmc for a 90 mol % SDS/10 mol % Cl&& thereby identifying the micelle composition with an SDSdominated monomer composition of 9O:lO. This result is in close agreement with the predictions of RST. It is also possible to analyze the variation of micelle composition with concentration in terms of a “local” ,8 parameter using the RST formalism. In this instance no information on mixed cmc is required, but taking the cmc’s of the individual surfactants as fixed parameters, a least squares fit to the micelle composition data yields p parameters of -2.8 f 0.5 (50:50)and -1.8 f 0.5 (70:30). Hoffmann and Poesnecker16have evaluated the error in the p parameter associated with a 1%error in the (16)HofFmann, H.; Poesnecker, G . Langmuir 1994, 10, 381.
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10 -l
10-2
c
.-
:
4-
c
c v
C U
10-4
0
1
I
I
.2
.4
.6
1
.0
Mole Fraction SOS
Figure3. As Figure 2, with SANS data correctedt o coincide with theory in the limit ofhigh concentration.The monomer composition M) is shown as a dotted line. The consistent with point “a” (the measured micelle composition at a concentration of 2.5 x concentrations obtained for surface tension data are shown (0). 10 -2
IO -3
IO - 4
IO
-s 0
I
I
I
I
.2
.4
.6
.a
1
Mole Fraction SDS
Figure 4. Comparison of SANS with predictions of RST for /3 values in the range -2.5 t o -4.0, data is shown as ( x ) .
individual surface tension derived cmc values using a geometrical method. To evaluate the errors associated with the surface tension derived p parameters obtained in this study, we have used a method based on Bayesian error propagation (seeAppendix). The evaluation of errors is difficult for a set of nonlinear equations that must be solved iteratively, and the Bayesian error propagation approach provides a convenient and direct method. Despite its apparent complexity it is relatively straightforward to apply. The results of such a n analysis for a 1% and 10% error in the individual surface tension derived cmc values are shown in Figure 5. It can be seen that the /3 parameters obtained from the micelle composition data
(-) -2.5
and (- - -) -4.0. The SANS
at the different bulk compositions are consistent with errors in the cmc values of only a few percent. The areas/molecule of the individual surfactants were measured ;sing neutron reflection (C12E6) and surface were obtained tension; 38 A2/moleculeand 54 ~21molecule for SDS and CI2E6,respectively. From these values and the supra-cmc surface tension values was obtained a surface interaction parameter, /Is, of -2.5 f 0.3. The surface concentrations obtained by neutron reflection are shown in Figure 6a,b. The associated surface compositions are shown in Figure 7 together with the predictions of RST for the surface. The same compositions and concentrations a s in the SANS experiment were used. It can
2502 Langmuir, Vol. 11, No. 7, 1995
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Interaction Parameter, p Figure 5. Probability distribution of interaction parameter p (from Bayesian error propagation analysis)due to errors (0)in cmc values, u = 1%(dotted line) and u = 10% (solid line).
0
so Concentration (
100
* rmr
I
0
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Concentration I
100
*cmc 1
Figure 6. Surface excess, r ( x mol cm-?, as a function of concentration for (a) 70:30 mol % SDS/C&6 and (b) 50:50 mol % SDS/C&s: ( 0 )SDS, ( 0 )C12E6, and (*) total.
be seen that, a t all but the lowest concentrations, the surface and micellar compositions are similar. This finding is in agreement with our surface tension data, which is largely invariant with concentration above the cmc a t this composition. The deviation near the cmc is again consistent with the likely presence of a very surface active contaminant (dodecanol).lo This level of impurity, although significant a t the air-water interface, has no impact on the micelles when they present large surface areas as in the measurements performed here.
Summary The picture that emerges is that, contrary to previous studies,l this system can still be adequately described by RST, at least a t the compositions studied here and with any discrepancy being accountable by expected experimental errors. As in earlier work with the SDS/C12E3 mixture, the micelle and surface compositions are found to be very similar. In the case of SDS/C12E3 this was expected as the dispersed surfactant was a lamellar phase. However, in this instance, the SANS data is consistent with the existence of spherical micelles, a geometry
expected from geometric packing considerations. The observed geometry-independent composition of the adsorbed layer would appear to vindicate the assumptions ofRST. That is, our finding is consistent with a n enthalpydominated mixing process in which geometry-based entropic contributions are insignificant.
Acknowledgment. We acknowledge the contribution of the Bayesian error analysis of the interaction parameter by D. Sivia, DRAL. Appendix: Evaluation of Error in Interaction Parameter /3 Using Bayesian Error Propagation Given a best estimate of the cmc’s c1, c2, and c* and a measure of their reliabilities, c1 = cl0 f q ,c2 = cz0f a,, and c* = co* f a*
(Al)
to obtain a n estimate of the error in the interaction parameter p we need to evaluate the conditional probability distribution function (pdf): prob(PlI)(where ‘‘I” means “given”I and includes all the relevant background information such as that stated in eq Al). The maximum
Solution and Adsorption Behavior of a Mixed Surfactant 10 -1
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1
10 -2
-E C
z
2 +
10-3
J V 0
10-4
10-5
0
Mole fraction SO5
Figure 7. Comparison of surface and micellar compositions with RST. Solid and dotted lines are the theoretical surface and
micellar compositions. The experimental data for the surface and micellar compositions are plotted as +, + and #, 0 for solution compositions of 70:30 and 5050 mol % SDSK12Es.
of this pdf represents the optimal estimate of ,8, and its width, or spread, gives an indication of the confidence interval (or uncertainty). To calculate the desired pdf, we can use the rules of probability theory to express it as
and the pdf for /3, given
p r o b ( P ) n = ~ ~ ~ p r o b ( ~ , c , , c , , cdc, * I Idc, ) dc* =
Its value is uniquely determined by the (numerical) solution Ac~,cz,c*) of eqs 8 in the main text. Thus the whole procedure can easily be carried out with a short computer program. The desired pdf is obtained by placing a d-function with an amplitude given by eq A3 at a position determined by eq A4 in ,&space for each point on a discrete 3-D grid in c-space and summing over all suchcontributions. Note that not all combinations of cl, CZ, and c* lead to a real solution forb; these can be excluded from the sum as being unphysical (and this exclusion represents an appropriate correction to the simple pdf of eq A3).
~ ~ ~ p r o b ( P l c , , c , , c * xS )prob(c,,c,,c*lI) dc, dc, dc*
(A21 This equation is just a 3-D integral over the product of two pdfs, where the pdf for c1, CZ, c* is a multivariate Gaussian, prob(c,,c2,c*lI) = exp
+
2
+
LA940912C
~ 1 , 1 2 2 ,and
c*, is a d-function:
