J . Phys. Chem. 1993,97, 2745-2154
2745
Shape Investigation of Mixed Micelles by Small Angle Neutron Scattering H. Pilsl,'qt H. Hoffmann,t S. Hofmann,* J. Kalus,+A. W. Kencono,i P. Lindner,l and W. Ulbrich6 Experimentalphysik I , Universitat Bayreuth, P.O. Box 10 12 51. W-8580 Bayreuth, Germany, Physikalische Chemie I, Universitat Bayreuth, P.O. Box 10 12 51, W-8580 Bayreuth. Germany, Institut hue-Langevin, BP 156 X,F-38042 Grenoble Cedex, France, and National Atomic Energy Agency, P.O. Box 85 Kly., Jakarta, Indonesia Received: May 28, 1992; In Final Form: November 1 1 , 1992
Several mixtures of the zwitterionic surfactant tetradecyldimethylamine oxide (TDMAO; which normally forms uncharged rodlike micelles) and the anionic sodium dodecyl sulfate (SDS; which normally forms charged globular micelles) were studied by small angle neutron scattering. The aim was to investigate the mixing behavior of these surfactants on one hand and the influence of surfactant charge on the micellar shape on the other hand. A contrast variation experiment on a 60 m M solution with the mixing ratio T D M A 0 : S D S = 8:2 was performed. The fact that the SDS chains were protonated or deuterated, respectively, provided further information about the internal structure of the micelles. Experimental data are in good accordance with a model of elongated micelles consisting of a core of surfactant chains surrounded by a shell of headgroups while the two surfactants mix homogeneously without any significant enrichment of SDS a t the end caps. Taking homogeneous mixing for granted, the shape and surface charge of the micelles were determined for other mixing ratios. A three-axes ellipsoid was chosen as a model for the shape of the micellar core. For the axial ratios a:b:cwe found values of 1:( 1-2):( 1-6) depending on the mixing ratio (1:9 to 9:l). We used the Hayter-Penfold model for globular macroion solutions as well as a semiphenomenological approach according to Farsaci for an attempt at describing the structure factor S(Q) and the micellar interactions.
1. Introduction Molecules of surface active agents can form small aggregates (micelles) above a certain concentration (critical micelle concentration, cmc). These micelles can be, for example, of globular, rodlike or disklikeshape.',2 In mixtures of surfactants the system has several possibilities for organizing itself. Mixed micelles can be formed, for which the composition is in accordance with the overall mixing ratio (ideal miscibility). On the other hand, total inmiscibility would cause the coexistenceof two types of micelles, each type consisting only of one kind of surfactant. The simultaneous presence of two kinds of mixed micelles with different compositions is also possible and represents a transition between the two extreme alternatives mentioned above (microscopic demixing). The latter could be taken as the result of a miscibility gap according to the pseudophase separation model. The miscibility of surfactants in aqueous solutions has been the subject of several theoretical investigations since it became obvious that addition of even small amounts of another surfactant or cosurfactant can vary the properties of a given micellar solution over a wide range.3-* So far there is only limited experimental evidence of whether there is mixing or demixing in a surfactant mixture on a microscopic scale, because there are only a few reliable experimental methods. Haegel and Hoffmann9 showed that two kinds of micelles coexist in tetradecyldimethylamine oxide (TDMAO) and sodium perfluorooctanoate solutions by use of the contrast variation technique in light scattering, where water-glycerol mixtures are used as solvents. This conclusion was supported by selective incorporation of dyes and subsequent separation of the two micellar species by ultracentrifugation. Asakawa et al. were successful in separating two types of micelles by gel filtrationloand ultrafiltration.ll The measurement of selfdiffusion coefficients by FT-NMR also provides an appropriate tool for detecting different kinds of micelles.12 Ottewill et al.I3 presented results of their investigationson the system ammonium
' Experimentalphysik I, Universitat Bayreuth. 1 Physikalische
Chemie I, Universitat Bayreuth. National Atomic Energy Agency. I Institute Laue-Langevin.
decanoate and ammonium perfluorooctanoate which were achieved by the use of the contrast variation technique in small angle neutron scattering. All these publications dealt with the problem of miscibility of hydrocarbon and fluorocarbon surfactants. In an earlier paperI4 we reported data on a surfactant system which consists of sodium dodecyl sulfate (CI2H2$O4Na, SDS) and tetradecyldimethylamine oxide ( C I ~ H ~ ~ ( C H ~ ) ZTDNO, MAO). According to electric- and flow-birefringence measurements, small, unisometric micelles are formed which were assumed to be of elongated, rodlike shape. Their largest dimension does not exceed 30 nm. Mixtures of these systems are able to form shear induced structures (SIS)if a certain critical shear gradient is exceeded. The evolution of these structures proved to be highly dependent on concentration, temperature, mixing ratio, and ionic strength of the solutions. We concluded from the latter observations that the state of the charge of the micelles plays an important role in SIS formation. Therefore we try to find connections between macroscopic properties of the solutions, like SIS formation, and details of microscopic micellar structure, like exact shape of the particles and their surface charge. Before we were able to do this, we had to determine what kind of mixing behavior the zwitterionic/anionicsystem shows. A model of mixed micelles with a composition equal to the overall mixing ratio, but with the SDS molecules aggregating at the end caps of the rodlike micelles (intramicellar demixing), seems plausible if the required surface area per headgroup of SDS is larger than that of the amine oxide,' s ~ 1because 6 of electrostatic repulsion of some charged SDS headgroups. Therefore SDS molecules in one micelle may accumulate at the end caps where the curvature of the surface is larger than in the other regions of the micelle. The strength of the repulsion, and thereby the required surface area of the SDS headgroups, will certainly be diminished if SDS molecules are embedded in a neutral amineoxideenvironment. If the surface area of SDS becomes comparably large or is smallerI7than that of the amine oxide this will support the assumption that SDS is statistically distributed in a mixed micelle. In a recent publication,l8 it was concluded from IR measurements that the latter assumption seems to be true. We used a contrast variation method
0022-3654/93/2091-2145%04.00/0 0 1993 American Chemical Society
2746
Pilsl et al.
The Journal of Physical Chemistry, Vol. 97, No. 11. 1993 TDMAO
7 SDS-D
I
cm-’
6 4 -
2-
homogeneous mixing of TDMAO and SDS
‘d.0
enrichment of SDS a t the micellar endcaps
0.2 0.4 0.6
Q / nm
homogeneous mixing of TDMAO and SDS
W
PSHELL I
p “
1I.O
SDS-D
S(Q)
PSHELL
0.8 -1
1.0
-
0.5
-
*
Ac
enrichment of SDS at the micellar endcaps
Figure 1. (a, top) Schematic longitudinal section through a mixed micelle of TDMA0:SDS = 8:2. (b, bottom) Schematic scattering length density distribution inside the micelles for various models used in this paper.
in small angle neutron scattering to distinguish between these two possibilities (Figure la). Two series of experiments at different contrasts and with a fixed mixing ratio with respect to the number of molecules of 8:2 (TDMAO:SDS), the first one with protonated SDS chains (SDS-H) and the second one with deuterated SDS chains (SDS-D) were performed. A second aim was to achieve more detailed information about the shape of the mixed micelles for different TDMA0:SDS mixing ratios. Furthermore, by an analysis of the structure factors, we tried to get some rough information about the interactionsbetween the micelles and the micellar charge.
2. Experimental Details 2.1. Materials. The zwitterionic surfactant TDMAO was a gift of the Hoechst Co., Gendorf, Germany. It was delivered as a 25% solution and purified by freeze-drying and twice recrystallizing from acetone p.a. The anionic SDS was bought from Serva (Heidelberg, Germany) as “SDS cryst. reinst”. The water for preparation of the solutions was demineralized and distilled twice. Thedeuterated SDS-D (C12D25S04Na;99.4 at.% D) was bought from IC Chemicals GmbH, Munchen, Germany, and used without further treatment. The deuterium oxide (D20) was delivered by Carl-Roth GmbH & Co., Karlsruhe, Germany, with 99.7% purity. 2.2. Data Collection. The small angle neutron scattering experiments were mainly performed at the D 11- and D 17 instrument^^^ at the Institut Laue-Langevin in Grenoble, France. The sample-to-detector distances were chosen to be 1.1 and 2.6 m at the D 11 camera and 0.8 and 2.8 m at the D 17 camera, respectively. A momentum transfer of Q = 0.015-2.5 nm-1 was covered. Qisdefined as (47~/X)sin(e/2). Odenotes thescattering angle and X the wavelength of the neutrons. The wavelength was X = 1.O nm in both cases. Additional measurements were carried out at the SANS-2 facility of the GKSS Forschungszentrum Geesthacht, Germany. The sample-to-detector distances were chosen to be 1.3 and 2.6 m; the wavelength, 0.525 and 1.0 nm, respectively. 2.3. Data Treatment. The two dimensional scattering intensities I ( Q ) were divided by the isotropic scattering intensity of
0.0
~
’
1
I
1
.
Figure 2. (a, top) Comparisonof scattering data of the mixtures TDMAO: SDS = 8:2 with protonated (SDS-H) and deuterated (SDS-D) SDS dissolved in DzO. The solid lines are due to a fit assuming a shell structure for the micelles. (b, bottom) Comparison of the two structure factors used to fit the scattering data of the mixtures with SDS-H and SDS-D (Hayter-Penfold model).
light water in order to normalize the intensities to the solid angle and to the sensitivity of each detector pixel. Normalization with respect to the transmission of each sample and subtraction of the incoherent background followed in the usual way. The absolute value of the differential cross section dZ/dfl was calculated using the scattering of light water for calibration.20
3. Results 3.1. Mixing Behavior, Contrast Variation. Our first aim was to distinguish whether the two surfactants mix homogeneously in one micelle or whether SDS is enriched or depleted at the micellar end caps. To study the internal structure of the micelles, we used the contrast variation technique. By the use of solvents of different D20:H20 ratios regions with different scattering length density inside the particles become visible. As we want to distinguish thescattering intensity fromTDMAO and SDS, we have to provide a significant difference in the scattering length densities of both materials. Therefore we performed two series of experiments: (A) TDMAO mixed with SDS both with normal protonated hydrocarbon chains (and thereforevery similar scattering length densities of the surfactant chains) and (B) TDMAO and perdeuterated SDS (different scattering length densities of the surfactant chains). We chose a mixing ratio TDMA0:SDS = 8:2 and a 60 mM total surfactant concentration for both mixtures. In series A the contrast was changed by solving the surfactants in various H20-D20 mixtures with different concentrations x of D2O ( x = 100,60, 25, 15,8, and 0%). In series B solutions with x = 100, 60, 50,40, 30, 25, 15, 8, and 0% were prepared. The first step in the evaluation of the scattering data is a rough determination of the shape of the particles. We assumed that the micelles are homogeneous, showing constant scattering length density. Using a least-squares routine, the data with the highest scattering intensity (x = 100%; the solvent is pure D20; see Figures 2 and 3) were analyzed. The best fits were obtained assuming a three-axes-ellipsoidal shape (see Figure 4) for the micelles. Using form factors squared (IF(Q)12)of simpler shapes like rods or ellipsoids of revolution, we in fact could not fit the measured intensities satisfactorily at Q-values higher than 1.6 nm-I. An
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2747
Shape Investigation of Mixed Micelles
In (
4.0
I
(Q)) 2.0
I
SDS-H
0.0
100% D,O 407. D,O
-2.0
0% D,O
-4.0
-6.0
1
-8.01
$.Fit-region.?
'
-
0.0
.
'
-
'
0.2
.
'
0.6
0.4
0.8
I
'
157. D2O
-I
1.0
Q2/ n m 2 0
1
2
3
4
5
6
Q2
4.0
In (%(Q)) 2.0
Figure 3. Scattering intensity and fit result (solid line) of the mixture TDMA0:SDS-H = 8:2 dissolved in D2O in a logarithmic plot as an example for the fit quality in the high Q-region (the micelles are assumed to show shell structure).
0.0
shell: thickness
2.0
-
4.0
-
6.0
-
8.0
1OOX D,O
60% D,O O X D,O
8 X D,O 307. D,O
0.0
0.2
-
Figure 4. Model for the micellar shape. Three-axes ellipsoid with axes a = a; t , b = bj t , and c = Cj t and shell structure.
+
0.8
1.0
Figure 5. Scattering data of the mixture TDMA0:SDS = 8:2 with SDS at various D2O concentrations. Solid lines are fit results of Guinier plots for obtaining (dZ/dQ)(O): (a, top) with protonated SDS; (b, bottom) with deuterated SDS. 31,
+
0.6
0.4
Q2/ nmb2
core axes a i. b i tci a
-
assumed polydispersity for the more simple shapes could solve the problem too. But the distribution function for the values in question turned out to be very broad. Therefore the model of an ellipsoid with three different axes was the simplest one in accordance with the measurements. (For the calculations of the scattering intensity and details about the fit procedure see Appendix). We expected that deuteration of SDS does not change the shape of the micelles. But when we compared the scattering curves of the samples with SDS-H and SDS-D in more detail, we observed some differences (Figure 2). The position of the maximum of intensity was shifted to a higher Q-value for deuterated SDS that might give a hint for the existence of smaller particles in this sample. (The effect that the exchange of H by D in the SDS chain leads to smaller micelles, and hence smaller aggregation numbers, was also observed in electric birefringence experiments.2') The fitted values we found (we used the model of Hayter and Penf0ld22.2~for a calculation of the structure factor S(Q) and assumed micelles with constant scattering length density; see Appendix) for the axes a and bof the three-axes ellipsoid are 1.85 f 0.05 and 3.15 f 0.08 nm for the SDS-D mixture, and a = 1.85 f 0.05 and b = 2.77 f 0.08 nm for SDS-H, respectively. The longest micellar axis c is -4.9 nm for the deuterated sample and -6.1 nm in the mixture with SDS-H. Evidently there are differences in the micellar dimensions induced by deuteration, but at this stage of interpretation we cannot find any conclusions that the miscibility of the surfactants has changed by the deuteration of the SDS tails. In order to investigate the internal structure of the micelles, we analyzed scattering curves taken from solutions of different solvent mixtures. If the shape of these curves is always the same, the particles have constant scattering length density throughout the micellar volume. In this case the square root of the measured intensity should depend linearly on the D2O concentration for any Q,because (IF(Q)12>S(Q)is the same for any contrast p = P m - p s (see eq 1; Appendix). The experimental error for a proper evaluation of the measured intensity can be reduced by a least-
I
c
cm -
+
I
1
'-1 l ,i p
,;z
=
= * SDS-D
I E
22.4k.3 % D,O
E'
-1
I
r
SDS-H 1 - 7 . l f . 4 % D,O 0
1
-'
I
I
I
I
I
-
Figure 6. Square roots of the extrapolated intensities at Q 0 of Figure 5 as a function of the D2O concentration. As expected a straight line is observed for both the SDS-H and SDS-D micelles.
squares fit procedure of a theoretically derived curve to substantial parts of the scattering curve in a Q-region where S(Q) = 1. As long as the particles are not too large and not too unisometric, it is possible to fit the scattering data in a Guinier plot (Figure 5) of ln(I(Q)) versus Q2 by a straight line in a certain Q-range. Such Guinier plots are shown in Figure 5 as solid lines. (The shape of the scattering curve in a Guinier approximation is given by I ( Q ) exp(-Q2Rg2). R, is the radius of gyration.) The square root of the extrapolated intensity to Q = 0 is shown in Figure 6 and depends linearly on the D20 concentration of the solvent. The concentration XM of D20 where the extrapolated intensity becomes zero is called the matching point. For inhomogeneous micelles (surfactant tail- and headgroups in general have different scattering length densities; see Al.2 in the Appendix) a matching point exists too, defined by dZ/dQ = 0 in eq 3 for Q = 0 with ps = pmatch. (Notice that Fi(Q) = Fa(Q) = 1 for Q = 0). It may happen that the usually used Guinier plot with extrapolation to Q 0 is not always a good route for an evaluation of the intensity at Q+ 0. In such a case the scattering curve of an inhomogeneous particle can show large deviations from this simple exponential law at relatively small Q-values. Nevertheless: in reality a definite value of the intensity at Q 0 can be founxin most cases. If the shape of the scattering curves changes with D2O concentration x , this indicates that regions of different scattering length densities exist inside the particles. Those differences are
-
-
-
2148
Pilsl et al.
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
-
4;::El!; .0r1 0.01
0.0
1.0
2.0
I
I
3.0
0.0
Q /nm-'
x d (1 Q )
cm-'
2.0
1.0
2.0
3.0
0.0
Q /nm-'
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
Q /nm-'
Q /nm-'
1.0
2.0
3.0
Q /nm-'
0.5
0.0
0.0 0.0
I
0.0 1.0
2.0
Q /nm-'
3.0
0.0
1.0
2.0
3.0
0.0
Q /nm-'
1.0
2.0
Q /nm-'
3.0
0.0
1.0
2.0
Q /nm-'
3.0
0.0
1.0
2.0
3.0
Q /nm-'
Figure 7. Scattering data and fit results for the contrast variation experiment on the mixture of TDMAOSDS = 8:2: (a, top) for protonated SDS; (b, bottom) for deuterated SDS. The solid lines are due to a fit where homogeneously mixed micelles with a shell structure were assumed. caused by the shell structure of the micelles. In the case of the mixture with deuterated SDS, some enrichment of the SDS-D may cause regions of different scattering length densities, too. Some of the measured curves are shown in Figure 5 as plots of ln(l) versus Q2. The shape of the curves changes with varying D20concentration for the protonated and the deuterated mixture (the differences are most significant in the region around the maximum of the scattering intensity), and we found different slopes of the straight lines fitted in a Guinier plot (Figure 5b). As the shape of the curves varies, we conclude that we have to replace the simple model of homogeneous particles by a shell model. Thesquare roots of the intensities at Q-0, extrapolated from a Guinier approximation, depend linearly on the D 2 0 concentration for mixtures with both protonated and deuterated SDS, respectively (Figure 6). For a solution of two or more kinds of particles with different distributions of scattering length densities, this plot would show deviation from linearity near the matching point. As we do not find such deviations within the errors of the experiment, we can conclude that in both samples scattering arises from nearly identical particles with the same internal structure. From the values of the D20 concentration for the matching points XM = 7.1 f 0.4 and 22.4 f 0.3% for SDS-H and SDS-D,respectively, we got interrelations (eq 3) for micellar parameters that confirmed our fit results for the shape. In order to show whether the surfactants mix homogeneously inside the micelle, we tried to fit the whole scattering data of both contrast variation series using standard techniques that have been applied in a large number of works on ionicand mixed micelles.*"26 To find a model for the mixing behavior, we have to analyze the shapeof the scattering curve of the mixture with deuterated SDS under conditions where this curve is most sensitive to mixing or demixing. This happens near the matching point where unfortunatelythescattering intensity isvery low. In thecaseofcomplete mixing, the scattering intensity of the mixture with SDS-D at the matching point (by definition then the intensity at Q = 0 is zero) will be almost zero for the whole Q-range because no significant difference in scattering length density exists between micellar core and shell or between shell and solvent, respectively. If SDS is gathered at the end caps, the intensity for Q # 0 becomes larger than that for the former case because the local contrast is larger now, giving rise to more intensity at finite Q-values. The low intensity measured for the mixture with SDS-D with D20 concentration x near the matching point XM for all Q-values leads to the speculation of complete mixing, which was tested in a first fit attempt (see Appendix). The micelles
were assumed to consist of a hydrocarbon core, the shape of which was a three-axes ellipsoid (this shape fitted the scattering data best) with axes a,, b,, and c,, and an outer shell of thickness t . The whole micelle again had the shape of a three-axes ellipsoid with axes a, + t , b, + t , and c, + t (see Figure 4). It turned out that the errors of the fit parameter t were rather large. Therefore we finally fixed t to a value of 0.47 nm, which is an averaged headgroupdiameter. The interparticle structure factorS(Q) was calculated according to Hayter and P e n f ~ l d(see ~ ~section , ~ ~ A3.1) and was assumed to be the same for any solvent. Notice that there might be changes in the shape of the micelles with changing H2O:DtO ratio in the solvent. Such changes were not considered in the framework of these fits. We performed one common fit for each contrast variation series, fitting all data points at various D20 concentrations x with one set of parameters simultaneously, provided the scattering intensity was sufficiently high. (The data of the mixtures with SDS-H where x = 15, 8, and 0% and for SDS-D with x = 30, 25, 15, and 8% D2O were not used in the fit.) From the fitted values of the ellipsoidal axes a,, b,, and c,, we calculated the number density n of the particles according to eq 15. Toobtain the structurefactorS(Q) in the Hayter-Penfold approach, which is valid for spherical particles, an effective macroion diameter u and the electric charge z , of one particle were determined by the fit procedure. In order to takeinto account the absolute scattering intensity, we also used a scaling factor in eq 3 as a fit parameter. One scaling factor was fitted for each contrast variation series. The values were slightly different but in both cases near 1. Furthermore an additional flat background having its origin in incoherent scattering was fitted too. Some of the scattering curves of the two experimental series as well as the fit results are shown in the Figure 7 (for those D20 concentrations x not used for the fit procedure, the solid lines in Figure 7 are calculated intensities using the fitted parameters). The values of the fit parameters are given in Table I. For both the protonated and the deuterated mixture we got a good agreement between the measurements and our calculations in all cases. The remaining small discrepancies between theory and experiment are of similar quality for both mixtures. The remaining discrepancies in the case of deuterated SDS cannot be attributed to inhomogeneous mixing of the surfactants at this stage of interpretation. To strengthen the assumption of homogeneous mixing, we calculated the scattering intensity for the case of inhomogeneous mixing and compared the result to experimental data. For these calculations, both the overall shape of the micelles and the
Shape Investigation of Mixed Micelles
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2149
TABLE I: Fit Results for Calculations According to the Shell Model with Two Models for the Structure Factors for Various Mixtures of TDMAO and SDS Fitted Values for the Micellar Core Axes ah bh and ci (Thickness of the Headgroup Shell Fixed at 0.47 nm in AH Cases), Aggregation Number Y, Fit Parameters for S(Q) in the Hayter-Penfold Model, Macro-Ion Charge zm, and Diameter 'u at various TDMAOSDS 8:2 8:2 5:5 6:4 7:3 SDS-H SDS-D 9: 1 1.6 1.6 f 0.05 1.63 f 0.05 1.59 f 0.05 1.63 f 0.5 1.62 f 0.05 1.66 f 0.05 1.6 1.90 f 0.01 1.6 1.90 f 0.01 2.08 += 0.02 2.08 f 0.03 2.18 f 0.06 3.18 f 0.06 3.37 f 0.08 3.29 f 0.08 2.52 f 0.08 2.69 f 0.08 2.44 f 0.08 5.5fO.l 5.6f0.1 5.6fO.I 5.9h 3.6f0.1 4.8h 1.90f0.01 2.34f0.08 2 . 8 f 0 . 1 9.4h 7.1, 14.5, 83 93 111 145 315 340 324 257 223 406 10.3 10.5 11.5 11.9 13.9 15.4 16.2 17.4 22.7 32.9 4.60 5 .OO 5.3 1 5.80 6.95 7.28 7.90 9.04 7.44 11.7 0.200 0.191 0.183 0.163 0.1 19 0.121 0.127 0.148 0.181 0.162 0,0243 0.0242 0.0238 0.0232 0.0217 0.0217 0.0222 0.0231 0.0234 0.0228 8.95 9.41 10.41 11.5 14.96 15.96 16.38 16.65 15.98 21.5 6.03 6.44 6.76 7.34 9.09 9.39 9.48 11.30 8.58 13.2 VI 26 30 40 55 122 I48 159 168 148 359 instrument [ DI 1 ILL,Grenoble Geesthacht ] [ D17 ILL j[Geesthacht 1 :9
0:lO
2:8
3:7
1o:o 1.6 f 0.05 1.6 f 0.05 >150
Il\.IL,'
IC
0
The inverse screening length
K
DII
and the volume fraction 7 are calculated from fit parameters; the values for the 'preferred interparticle distance"
L and the root mean square deviation U L , which measures the degree of disorder in the liquid, are fitted using S(Q) according to Farsaci. From L we
calculated Y F , , ~ , . , ~assuming , that the micelle number density is equal to k3.* Obtained withS(Q) model of Hayter-Penfold. of Farsaci. 4,
0.10
1
A
40% D,O cm 0.3
1 i\
Obtained withS(Q) model
-1
I 00
20
10
30
0.0
1 .0
2.0
3.0
I 0.0 006 I
(9) cm
1
0.4
0.6
0.6
1.0
1
ic'\
25% D,O
1
XSDS
Figure 9. Calculated reduced x2as a function of the molar fraction XSDS of SDS-D in the micellar end cap regions for the solution with 30% D20 (Figure 8b (top, right).
-I
I
00
0.2
10
Q/
20
nm
-1
30
0.0
1 .0
Q / nm
2.0
3.0
-1
Figure 8. Comparison of model calculations for an inhomogeneous mixing of SDS-D with measured intensities for different contrasts. Solid lines are calculations that are in accordance with experimental data for the S D S molar fraction in the end cap regions XSDS = 0.2 (condition for homogeneous mixing) and XSDS = 0.33. Dashed lines are calculations that definitely are not in accordance with experimental results (XSDS = 0.375,0.5, and I). The arrows show the direction of increasing values of XSDS. The D2O concentrations were 40% (a, top, left), 30% (b, top, right), 25% (c, bottom, left), and 0% (d, bottom, right).
structure factor were assumed to be the same, as those for the above mentioned model. The core was subdivided into two parts, both of ellipsoidal shape again. The difference in the length of the largest axes of these ellipsoids Ac in this model (see section A4.2 and Figure lb) was fixed at 0.85 nm. In this case the volume of the end cap regions is equal to the total volume of the SDS tails in the micellar core. The molar fraction of the SDS tails in the end cap regions XSDS was varied to study how scattering intensity is changed by SDS-D enrichment. (Notice that changes of the scattering length distribution in the shell due to SDS enrichment are not included in this simple model. The incorporation of this effect would increase the deviations of calculations with and without enrichment.) In Figure 8 thecalculated intensity is compared to the scattering data for XSDS = 0.2, 0.33, 0.375, 0.5, and 1 (XSOS = 0.2 is the condition for homogeneous mixing; XSDS = 1 means that only SDS tails are in the end cap regions). The best agreement between experimental data and theory is
given for homogeneous mixing. With rising enrichment of SDS the discrepancies increase. In the case of a concentration of 40% D 2 0of the solvent (Figure 8a), the theoretical curve with regard to SDS-D enrichment is higher; at 0% D20 (Figure 8d) it is lower than the curve calculated for the case of complete mixing. This shows that small errors in the determination of the absolute intensity cannot be responsible for the observed differences between calculations and experimental results. The solid lines in Figure 8 (calculations for SDS-D molar fractions in the end cap regions of XSDS = 0.2 and 0.33) are in agreement with experimental data whereas the deviations from scattering data observed for the dashed lines (XSDS = 0.375, OS, and 1.) show that those values are not suited for modeling the micellar system. For a more quantitative description of the goodness of our calculations for different degrees of SDS enrichment, we performed x2tests for several values of XSDS.~' Here we compared only the shape of measured and calculated curves regardless of absolute intensity in the Q-range of 0.1-1.5 nm-I. Therefore a prefactor for the absolute intensity dZ/dQ (eq 3) was determined for any value of XSDS by an individual fit. In Figure 9 the reduced x2 is given as a function of the molar fraction of SDS in the end cap regions for the solution with 30% D20. (Notice that in the previous fits we used one common scaling factor for all measured curves within a contrast variation series; this is the reason why in Figure 8b the fit quality does not seem to be satisfactory whereas now in Figure 9 the values of the reduced x2 are near 1.) We observed a minimum of the reduced x2for XSDS around 0.2 and a steep increase for larger xsDs-values. In the same way we extracted the minima of the reduced x2 for the other solvent mixtures. The values of these minima varied from 0 to 0.32 with a mean of 0.17 and a standard deviation of approximately 0.1.
2750 The Journal of Physical Chemistry, Vol. 97, No. 11 1993 I
A
1
Pilsl et al.
by Hayter and Penfold. The fit parameters were the core axes a,, b,, and c,, a scaling factor for the absolute differential cross section, an additional background, and for S(Q) the macroion charge zm and the macroion diameter u. The thickness of the headgroup shell again was fixed at 0.47 nm in all cases. Pure SDS forms globular micelles, where we determined the hydrocarbon core radius to be 1.90 f 0.05 nm. The results (see Table I) of the fit for the pure SDS sample are in fair agreement with those found in a former investigation25that used thesame micellar model. Another model for the SDS micelle that also led to comparable results has been presented in ref 32 where the globular particles were described by a two-shell structure. TDMAO forms rodlike uncharged micelles. The radius of the rod core was determined to be 1.60 f 0.05 nm, and the length turned out to be greater than 30 nm. In this case it was not possible to fit the length of the particle because the information about this dimension is hidden in the Q-region in which the Figure 10. Scattering data of the TDMAOSDS mixing series. The deviation of the structure factor from unity is large and none of solid lines are due to a fit using the shell model for the micellar structure. the models for S(Q), developed for spherical particles, could be used in this case. The values for the radius and the length of the In the framework of our model calculations, the scattering data TDMAO micelles are comparable to those found in electric of the contrast variation experiment can be described with XSDSbirefringence and light scattering experiments.33 values between 0.09 5 XSDS 5 0.25 with a confidence level of To fit the scattering data of the mixturesof the two surfactants, 90%. Comparing this to XSDS = 0.2 for homogeneous mixing, we we used the model of a three-axes ellipsoid (see Figure 4). As can conclude that a mixture of the two surfactants in the first already mentioned simpler models like rods or ellipsoids of approximation forms homogeneously mixed micelles. revolution were not suited for fitting the scattering data in the The conclusion of homogeneous mixing is also supported by high Q-range. By an inspection of Figure 10 one has the investigations of the pH-value of C12-DMAO and C14-DMAO impression that the quality of the fit is good. But for higher solutions as a function of SDS addition.'8,21As soon as even small Q-values the scattering intensity becomes low. Then a logarithmic amounts of SDS are added to pure (212-or C14-DMAO solutions plot of ln[dZ/dQ Q] versus Q2 is a better indication of the fit (pH = 7) the pH increases sharply to a much higher value (e.g. quality in this Q-region. This is shown in Figure 3 (in this plot pH = 1 1 in the case of a 100 mM solution of Cl4-DMAO). This the scattering of rodlike particles would result in a linear is only possible if the pK,-value of the amine oxide headgroups relationship in a certain Q-range; for a test with a plot dZ/dQ is changed by neighboring SDS molecules and requires a smooth Q" versus Q,2Sour largest Q-values were too small). The values internal mixing of both types of surfactants. Effects of synergism we found for the axes of the hydrocarbon core of each surfactant which requires close interactions of both types of molecules in mixture are given in Table I. amine oxide4DS mixtures are also reported in BetainSDS mixed At higher TDMAO contents we always found the smallest micelles.28 core axis to be 1.60f 0.05 nm; therefore we assumed this to be On the other hand mixed surfactant-cosurfactant systems (e.g. potassium laureate and decanol) show internal s e g r e g a t i ~ n . ~ ~ +valid ~ ~ too in the mixtures of TDMA0:SDS = 3:7, 2:8, and 1:9 where the Q-range measured was too small to give full information But as these molecules do not show distinctive attractive or about all characteristic dimensions of the particles. For the repulsive interactions, these results cannot be compared to those mixture 1:9even a model of a spherical particle would give a good in the case of our mixed zwitterionic-ionic surfactant system. fit result but with an unusually large radius of 2.0 nm. Such a 3.2. Investigations on the Micellar Shape. We have shown large radius is not in accordance with the length of a monomer. that a 8:2 mixture of TDMAO and SDS forms micelles, whose This fact again favors the ellipsoidal model. The radius of the structure can be described by a shell model where the scattering spherical micelles of pure SDS (TDMA0:SDS = 0: 10)of 1.9nm length densities in the core and in the shell are homogeneously is unusually large too compared to the length of a stretched C12H25 distributed within the accuracy given by our experiment. We chain.34 The Q-range measured is toosmall for deciding whether assumed this to be valid for all mixtures of the two surfactants SDS forms globular micelles indeed or also micelles with ellipsoidal and investigated the shape and the interaction of the micelles a t shape, as both models fit our scattering data. (The fit results for the mixing ratios TDMA0:SDS = 10:0,9:1, ...,5:5,3:7, ...,0:lO the axes of the ellipsoid using a fixed axis a, = 1.6 nm (as found for a constant total concentration of 60 mM. for a, for the mixtures) were 6, = c, = 2.0 nm.) In the cases of We tried to fit the scattering intensity of these mixtures with the mixtures 9:l and 8:2 the largest axes c, could only be fitted S(Q) = 1 (which can be done provided Q is sufficiently large), exploiting a model for the structure factor S(Q). The two models with the structure factor S(Q) of Hayter-Penfold theory22(section for S(Q), H a y t e r - P e n f ~ l d ~ and ~ . ~ ~Farsaci,-" showed quite A3.1) and with the structure factor according to Farsaci3' (section different results. Therefore we are not able to give unambiguous A3.2). In nearly all cases these three techniques revealed the data about the particle size, and the two values given in Table same results for the micellar shape and dimensions. In a few I are therefore reported without the indication of an error. (A cases it was not possible to find unambiguous conclusions about comparison of the two values of c, with the interparticle distance shape and dimensions. The differences between the shape of the L (obtained in the fit with the structure factor according to Farsaci) particles determined with a model of a constant scattering length seems to favor the larger values because then the length of the density and those using the shell model described before turned particles is comparable to L. This might explain the results of out to be small. (Notice that for these measurements the solvent electric birefringence measurements2' on these mixtures where, always was D20, the tails of the surfactants always were e.g., the shape of the birefringencesignal and the relaxation times protonated, and therefore the contrast is large. Small differences show that indeed the overlap concentration is exceeded in the of thecontrast ofthecoreand theouter shell areof minor influence case of the TDMA0:SDS = 9:l mixture.) on the scattering intensity.) The scattering data of the series of TDMAOSDS mixtures are reported in Figure 10 as well as the The results show that the smallest particle axis a, has the same calculations using the shell model and the structure factor proposed value for all mixing ratios, whereas the axes b, and c, grow for
Shape Investigation of Mixed Micelles 100 1 ,
I
tension 'nte*aci41
O.l
,
. .
.
The Journal of Physical Chemistry, Vol. 97, No. 11, 1993 2751
, . . . . ,
,
t 0.0
2oot/
0.8
0.8
1.0
0.0
0.2 0.4
%MA0
Figure 11. Surface tension against decane of different T D M A O S D S mixtures for a total surfactant concentration of 100 mM. XTDMAO is the T D M A O molar fraction of the surfactant mixture. XTDMAO= nTDMAO/ (nTDMAO
,I
0
0.2 0.4
+ nSDS).
the TDMA0:SDS mixing ratios from 0:lO to 7:3. In all these cases the ratio of the axes c,:blis smaller than 2 and the shape of the particle is more disk- than rodlike. For the mixtures 8:2 and 9:l the axis b, decreases while c, is still increasing, giving a ratio of cl:blgreater than 2. These micelles show rodlike shape. At the mixture 10:0 the axes a, and b, are equal. Here the experimental data are fitted in the high Q-regime where S(Q) = 1 with the form factor appropriate to long rods. These results may also be supported by interfacial and surface tension measurements as a function of the mixing ratio. Former investigations35revealed that surfactants which show an interfacial tension against decane of 1 mN/m or more usually form spherical micelles. Surface active agents with interfacial tensions from -0.2 to 1 mN/m tend to form rodlike aggregates while substances with still lower values prefer to aggregate in micelles of disklike shape or in lamellar structures. Our system starts to form threeaxial spheroids when the contents of TDMAO are raised above 20% with a steady growth of the second axis b (see Table I) with an increasing share of TDMAO. The system thus has the tendency to form a disklikestructure. The interfacial tension of this mixed system is shown in Figure 1 1. The curves show a minimum of 0.3-0.4 mN/m between themixing ratios 8:2 and 5 : s ofTDMAO: SDS. This is very close to the value of 0.2 mN/m mentioned above. If one considers furthermore that these values are not strict limits but may vary to some extent depending on the special surfactant, it may well be possible that the system TDMAOSDS is very close to a sphere-to-disk or -to-lamellar transition. Therefore the results of SANS are supported by interfacial tension measurements, and it seems plausible to consider three-axial spheroids as a model structure of TDMAO-SDS mixed micelles above a certain concentration of SDS. 3.3. Analysis of S( Q).As there is no simple analytical model available for the structure factor for charged nonspherical particles, we tried to extend the use of the Hayter-Penfold model22323 (section A3.1) to our problem. As long as the deviation of the micellar shape from a sphere is sufficiently small, this might be reasonable. The model fitted the experimental data quite well with the exception of the mixtures of 9:l and 8:2, where the shape of the particles is rodlike and thus surely not covered by the model. For the calculation of S(Q) in this model four parameters are required: The macroion diameter Q and the charge of the ions zm were determined by fit. The volume fraction t) is calculated by eq 9 to take into account the solvent association in the micelles. The values for particle number density and particle volume are given from the fit of the micellar core dimensions. The inverse screening length K is calculated from the ionic strength of the solution, and therefore related to the total number of charges on all the macroions. (Notice that the ionic strength in this calculation is only caused by the dissociated counterions. Sur-
0.8
0.8
1.0
%MA0
Figure 12. Comparison of micellar volume (calculated from fitted micellar dimensions) and the effective volume of the spherical macroion entering +ns~s). Hayter-Penfold theory for S(Q). XTDMAO = ~ T D M A O / ( ~ T D M A O 401,
. .
,
.
, . .
.
.
=m
30 I
0.0 0.2 0.4 0.6 0.8 1.0 XTDMAO
1-1u
0.1 0.0
0.0 0.2 0.4
0.6
0.8 1.0
XTDMAO
Figure 13. (a, top) Micellar charge z, for different T D M A O S D S mixtures. (b, bottom) Surface charge per nm2 surface area of the ellipsoids.
factant monomers in the solution (cmc) are not taken intoaccount (section A3.1.)) The fit results for the macroion diameter gave us a hint as to whether or not the use of Hayter-Penfold theory was a reasonable approximation. The volumeof the spherical macroions at different TDMA0:SDS ratios is almost the same as the volume calculated from the fitted micellar dimensions. This is shown in Figure 12. Only the mixtures of TDMA0:SDS of 8:2 and 9:l show differences due to reasons discwed earlier. The fitted results of these mixtures are reported too (Table I), even when the results are more than doubtful in these,cases. We found an increase of the macroion charge with increasing TDMAO contents, whereas the values of charge per unit surface'area of the ellipsoids show only small variations throughout the mixing series (see Figure 13). One has to keep in mind that together with micellar charge the screening length of the solutions changes too. There is a decrease of the inverse screening length K with increasingTDMA0 for low TDMAO concentrations (Figure 14a). The increase of K that begins at TDMA0:SDS = 6:4 might also be a hint for discrepancies between model assumptions and reality. The degree of dissociation a (defined as the number of charges on a micelle divided by the total number of SDS molecules in a micelle) (Figure
2152
Pilsl et al.
The Journal of Physical Chemistry, Vol. 97, No. 1 1 , 1993
“.-‘I 1
0.31,
nm
-1
0.0
. .
. ,
.
. .
,
500 Aggregation number 400
300 200 100
0.0 0.2 0.4
0.6 0.8 1.0
I 0
Ceesthacht
M
0 0.0 0.2 0.4
‘TDUO
0.6
0.8
1.0
%MA0
o.6 0.6
Figure 16. Comparison of the aggregation numbers obtained from the interparticle distance L (number density of the micelles n = L?; open symbols) and calculated from the micellar core dimensions (filled symbols). XTDMAO = nTDMAO/(nTDMAO + nSDS).
i
micelles, the addition of SDS to TDMAO solutions might be a useful way of varying the surface charge of rodlike micelles. The mixing behavior of the two surfactants was investigated by two series of contrast variation experiments of constant ratio TDMAO: SDS = 8:2. In the first series SDS with a normal protonated hydrocarbon chain was used. It was replaced by perdeuterated SDS in the second series to provide a significant difference in scattering length density between the two surfactants. The scattering data of these series revealed no signs of enrichment of SDS in some parts of the micelle. We investigated a series of different mixtures of the surfactants with mixing ratios from TDMA0:SDS = 10:0 to 0:lO in order to determine shape and charge of the mixed micelles. The shape of the particles changes from globular micelles (TDMA0:SDS = 0:lO) to more disklike and finally switching to rodlike aggregates and rods at TDMAO: SDS = 1O:O. Basically an increase of particle volume with increasing TDMAO contents was observed. The effective surface charge also increases linearly with a rising share of TDMAO.
Q /nm
-1
Figure 15. Comparison of the structure factors S(Q) used to fit the scattering data of the TDMA0:SDS = 6:4 mixture according to Farsaci and Hayter-Penfold. The latter curve is shifted upward about a value of 0.1 for a better representation.
14b) seems to be nearly constant for all TDMAO concentrations (this is in agreement with former investigations on mixed micelles26),except for the mixtures 8:2 and 9:l that already have been discussed. The use of the structure factor according to Farsaci3’ (section A3.2) isan alternative way offitting thescattering dataaccording to eqs 1 and 3. In general the fit quality was not as satisfactory as in the case of the Hayter-Penfold theory because S(Q) shows less structure at higher Q-values. For a comparison of S(Q) of both models, see Figure 15. We found an increase in the fitted values of the preferred interparticle distance L in the Farsaci model (seeeq 10)with rising TDMAOcontents. Thisisconsistent with the increase of the particle size, as shown in Figure 16, where the fitted values of the akgregation number Y, determined via the volume of the micelles, are compared to those values Y F ~which ~ ~ are ~ estimated ~ , from the assumption that the micelle number density is equal to L-3 (see Table I). The differences between the two Y values show that a determination of the aggregation number from the position of the maximum of the structure factor can only be used as rough estimates. 4. Conclusions As the pure anionic surfactant SDS forms charged globular micelles and the pure zwitterionic TDMAO uncharged rodlike
Acknowledgment. This work was supported by the Bundesministerium fiir Forschung und Technologie under Contract No. 03-KA3 BAY-7. We also thank F. Frisius from the GKSS Forschungszentrum for his assistance a t the SANS-2 experiments. A. W.K. thanks the Bavarian Government for financial support. Appendix A l . Calculation of the Scattering Intensity. A l . 1 . Homogeneous Particles. For a system of monodisperse particles with a constant scattering length density, the absolute scattering cross section per unit volume can be expressed by36
where n denotes the number density of the particles, p, and ps are the scattering length densities of the micelle aJd the solvent, respectively, and V, is the particle volume. (IF(Q)I2) and S(Q) are the orientationally averaged particle form factor squared and the interparticle structure factor. S(Q) is given by
g(r) is the pair correlation function or the probability of finding a micelle at a distance r from a micelle at r = 0. S(Q) has to be calculated in an appropriate model described below in the text. Equation 2 is an approximation. For a more detailed description see ref 37. A1.2. Shell Model. As surfactant tail- and headgroups have different scattering length densities, we use a shell model. The micelles consist of a core with scattering length density pi and
Shape Investigation of Mixed Micelles
The Journal of Physical Chemistry, Vol. 97, No. I I , 1993 2153
volume vi, surrounded by a shell with density pa. The total volume of the particle is V,. The intensity related to this model is calculated by
g(O)
n([(P, -pJVmFm(O)
+ (Pi-Pa)vi'i(0)12)s(Q)
(3) Fi and Fmare the form factors of the micellar core and the whole micelle, respectively. Notice, that for random orientation of the unisometric micelles in space an adequate averaging has to be performed. A2. Determination of the ParticleShape. In order to examine the shape of the micellar aggregates we fitted the experimental scattering data according to eqs 1 and 3 by minimizing the sum of the squared differences between measured and calculated intensities weighted by the inverse of the squared absolute error (weighted least-squares fit). For convenience the form factors squared of some assumed shapes are listed.3840 For a spherical homogeneous particle of radius R the form factor squared is given by (IF(Q)12) =
L is the length, R the radius of the cylinder, and J I the Bessel function of the first kind and order 1. For an ellipsoid with three different axes (a, b, and c), the form factor is given by
( sin2 0
K2
8ne21N, tk,T
=-
(7)
The ionic strength I is calculated by ( N , is Avogadro's number, e theelementary charge, t thedielectricconstant, k , theBoltzmann constant, and T the temperature)
(3(sin(X) -x3X cos(X))
where X = QR. For cylinder-like particles the form factor squared is calculated by
x=~
There are four basic parameters that enter the calculations of S(Q) in this case. The diameter of the macroions u was used as a fit parameter as we extended the use of the Hayter-Penfold theory to nonspherical particles. u has to be interpreted as an "effective diameter" of the micelles. The strength of the interaction is determined by the number of elementary charges ZM on the surface of the particles. zmalso was obtained by fit. The screening of the interaction can by expressed in terms of the inverse Debye-Hiickel screening length K which was calculated from the ionic strength I of the solution
e~cos' cp + b2 sin2 e sin2 cp + c2 cos2
There are two ways to deal with the problem of the structure factor S(Q). If an appropriate model for S(Q) is available, it is possible to obtain information about the shape by using a leastsquares routine to fit the scattering data with the product S(Q) (IF(Q)l2)according to eqs 1 and 3. In cases where no model for the structure factor is available, any fit about the shape of the micelles is restricted to the high Q-region, where S(Q) has the value 1. Due to the basic behavior of the scattering law, this restriction leads to the loss of information about the larger particle dimensions because they contribute to the scattering intensity predominantly at smaller Q-values. In a first attempt we fitted the experimental data with S(Q) = 1 and tried afterward two models for a structure factor S(Q) # 1 and checked the results for consistency. For these fits we assumed homogeneous micelles with constant scattering length density over all their volumes to get first information about shape and size of the particles. As we realized that this model cannot describe the whole experimental data, in a second attempt we used the shell model for a determination of the micellar shape. A3. Structure Factor for Interacting Particles. A3.1. The Model of Hayter and Penfold. In order to get an analytic expression for the structure factor, it was assumed that the interaction energy of particles is given by a repulsive screened Coulomb pair potential between finite spherical m a c r ~ i o n s . ~ * J ~
As the ions in the solution with charge zi = 1 are dissociated from the macroions, the number density ni of these ions is the number density of the macroions nmacrotimes the number of elementary charges of the macroions z,. The last parameter entering the calculations for S(Q) in this model is the volume fraction q of the solution calculated by where nmacrois the number density of the particles. V, is the particle volume. ( q was calculated that way to take into account the solvent association in the micelles.) It is not possible to calculate a structure factor for charged nonspherical particles in a simple closed form. As long as the micellar shape does not differ too much from spheres, we can assume that calculations according to the model of Hayter and Penfold can fit the experimental data quite well, because the center-to-centerinteraction potential for large distancesr between the micelles depends only weakly on the actual orientation of the nonspherical particles. Of course in the case of rather long particles an agreement with this theory cannot be expected. A3.2. The Model of Farsaci. A semiphenomenologicalmodel to describe particle interactions has been developed by F a r ~ a c i . ~ ' He gives an analytic expression for the structure factor S(Q) based on calculations where the next neighbor distances between the particles are randomly distributed around a "preferred distance"L with a root mean square deviation u ~ f r o mthisdistance.
S(Q) = 2
1 - exp(-aL2Q2/4) cos QL 1 - 2 exp(-a2Q2/4) cos QL
+ exp(-o;Q2/2)
-1
(10) The structure factor is characterized by peaks at the positions QL = 2nu, n = 1,2,3, .... For U L 0 the shape of the peaks behave like &functions (perfect order). S(Q) becomes 1 for UL 00, which means perfect disorder. Intermediate states can be modeled by finite U L -values. A4. Model Calculations for Mixed Micelles. A4.1. Homogeneous Mixing. The scattering length densitiespi of the micellar cores can be approximately calculated by
-
pi = O.~P(CI,H~,)+ 0*2~(Ci2H25 or C12D25)
with
-
(11)
2754 The Journal of Physical Chemistry, Vol. 97, No. 11, 1993
p(C12H25) = -0.387 X lo-" cm/nm3; V(Cl2H25) = 0.354 nm3 p(CI2DZ5)= 6.97 X lo-" cm/nm3; V(C12D2,)= 0.354 nm3
Pilsl et al. of the SDS tailgroups XSDS in the end cap region between the a r e ellipsoids, the influence of various degrees of enrichment on the scattering intensities was studied. XSDS = 0.2 corresponds to normal homogeneous mixing of the surfactants. Any enrichment of SDS is described by larger xsDs-values. The number of tailgroups in the end cap region is calculated by
From these known valuesZSwe estimated p(CI4Hz9)= -0.376 X lo-" cm/nm3; V(CI4H2,) = 0.408 nm3 assuming that the volume of C14H29 is the sum of the volumes of C12H25and ( C H Z ) ~ . The scattering length density pa of the micellar shell is the same in both mixtures. pa is calculated from the scattering length 6 of the surfactant headgroups and associated solvent molecules. The number nsolvof these H2O or DzO molecules associated to one headgroup can be calculated from the difference of the total volume of the outer shell and the volume Vhead occupied by the headgroup molecules. The averaged volume of the headgroups Vhedd is 'head = (o*8Vhead(TDMAO)+ 0*2Vhead(SDS))V (12) Vhcad(TDMAO)and Vhead(SDS)are the VOlUmeS Of the TDMAO and SDS headgroups, respectively. Vhead(SDS)is 0.068 nm3,25and we estimated for Vhead(TDMA0) a value of 0.135 311113. v is the aggregation number. Then the volume of the shell is given by
'shell 'head + ynsolv'solv (13) The volume VsOlvof a water molecule is 0.03 nm3.25 The total scattering length of a surfactant headgroup, for exampleTDMA0, is the sum of the scattering length b, of the atoms i of the TDMAO headgroup: bhead(TDMA0) = Zb, The averaged total scattering length of a headgroup in a mixed micelle is bhead = O.Bbhead(TDM~0) + 0*2bhcad(SDS)Therefore pa is given by
(14) pa = IVbhead + Ynsolvbsolvl / 'shell To calculate the scattering intensity according to eq 3, we need the micellar number density n given by
n = nmacro = (c-c&/v
(15) where the aggregation number Y was calculated by v = Vcore/ Vta,,. c is the number concentration of the monomers and co the cmc of the solution (CO was assumed to be small compared to the total surfactant concentration and was fixed to zero for all solutions). V,,, is the micellar core volume which was determined from the fit results of the core dimensions. V,,,I is the averaged tailgroup volume of a monomer calculated in the same way as the averaged headgroup volume in eq 12. Using one of the theories for S(Q) and for the shape, respectively, we fitted all the contrast variation data simultaneously. As already mentioned an ellipsoidal shape turned out to give the best agreement with the scattering curves. A4.2. Inhomogeneous Mixing. In the model we used to calculate scattering intensity in the case of enrichment of SDS molecules a t the micellar end caps, the a r e was subdivided into two three-axes ellipsoids differing only in the length of the largest axis c (see Figure 1b). The length of this difference Ac was fixed at 0.85 nm. In this case the volume of the end cap regions A V ( A V is the difference between the volumes of the two ellipsoids that model the core) is equal to the total volume of all SDS tailgroups in one micelle. By the variation of the molar fraction
Changes in the scattering length density in the headgroup shell due to SDS enrichment are neglected for simplicity.
References and Notes ( I ) Zana, R., Ed. Surfactant Solutions-New Methods of Investigations; Surfactant Science Series; Dekker: New York, Basel, 1987; Vol. 22. (2) Hoffmann, H.; Ebert, G. Angew. Chem., Int. Ed. Engl. 1988, 27, 902. (3) Asakiawa, T.; Johten, K.; Miyagishi, S.; Nishida, N. Langmuir 1985, 1. 347. (4) Mukerjee, P.; Mysels, K. J. ACS Symp. Ser. 1975, 9, 239. (5) Mukerjee, P.; Young, A. J. S. J . Phys. Chem. 1976, 80, 1388. (6) Kamrath, R. F.; Franses, E. J. Ind. Eng. Chem. Fundam. 1983, 22, 230. (7) Kamrath, R . F.; Franses, E. J. J. Phys. Chem. 1984, 88, 1642. (8) Nagarajan, R. Langmuir 1985, I, 331. (9) Haegel, F. H.; Hoffmann, H. Prog. Colloid Polym. Sei. 1988, 76, 132. (IO) Asakawa, T.; Miyagishi, S.; Nishida, M. Langmuir 1987, 3, 821. ( I I ) Asakawa, T.; Johten, K.; Miyagishi, S.; Nishida, M. Langmuir 1988, 4, 136. (12) Calfors, J.; Stilbs, P. J . Colloid Interface Sei. 1985, 103, 332. (13) Ottewill, R. H.; Burkitt, S. J.; Hayter, J. B.; Ingram, B. T. Colloid Polvm. Sei. 1987. 265. 628. (14) Hofmann, S.;Rauscher, A.; Hoffmann, H. Ber. Bunsen-Ges. Phys. Chem. 1991, 95 (2), 153. (15) Hofmann, S.; Hoffman, H. Unpublished results. (16) Otter, G. Dissertation, Universityof Bayreuth, 1988. Hoffmann, H. Proa. Colloid Polvm. Sei. 1990. 83, 16. 07) Wilson, A.; Epstein, M. B.; Ross, R. J . ColloidSci. 1957, 12, 345. ( I 8) Weers, J. G.; Rathman, J. F.; Scheuing, D. R. Colloid Polym. Sei. 1990, 268, 832. (19) Ibel, K. J. Appl. Crystallogr. 1976, 9, 296. (20) Kalus, J.; Hoffmann, H.; Reizlein, K.; Ulbricht, W . ; Ibel, K. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 37. (21) Hofmann, S.; Hoffmann, H. Unpublished results. (22) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109. (23) Hayter, J. B.; Penfold, J. Mol. Phys. 1982, 46, 651. (24) Sheu, E. Y.; Chen, S. H. J. Phys. Chem. 1988, 92, 4466. (25) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. (Paris) 1985,46,2161, (26) Bucci, S.; Fagotti, C.; Degiorgio, V.; Piazza, R. Lmgmuir 1991, 7, 824. (27) Bevington, P. R. Data Reduction and Error Analysis f o r the Physical Sciences; McGraw-Hill Book Co.: New York, 1969. (28) Iwasaki. T.; Ogawa, M.; Esumi, K.; Meguro, K. LPngmuir 1991, 7, 30. (29) Hendrikx, Y.; Charvolin, J.; Rawiso, M. Phys. Reo. 1985,833,3534. (30) Hendrikx, Y.; Charvolin, J.; Rawiso, M. J . Colloid Interface Sei. 1984, 100, 597. ( 3 1 ) Farsaci, F. Phys. Chem. Liq. 1989, 20, 205. (32) Hayter, J. B.; Penfold, J. J . Chem. SOC.,Faraday Trans I 1981,77, 1851.
(33) Hoffmann, H.; Oetter, G.; Schwandner, B. Prog. Colloid Polym. Sci. 1987, 73, 95.
(34) Tanford, C. The Hydrophobic Efject; John Wiley & Sons: New York 1971 _ _ _._ __ _. ,
(35) Hoffmann, H.; Oetter, G. J. DispersionSci. Technol. 1988,9(5 and 6). 459. (36) Chen, S.H.; Lin, T. L. Methods Exp. Phys. 1987, 23 (e), 489. (37) Kalus. J.; Hoffmann, H. J . Chem. Phys. 1987. 87 ( I ) . (38) Kostorz, G. Treatise Mater. Sei. Technol. 1979, I S , 217. (39) Guinier, A.; F0urnet.G. Small AngleScatteringofX-Rays; Chapman and Hall Ltd.: London, 1955. (40) Mittelbach, P.; Porod, G.Acta Phys. Austriaca 1961, Ed XV/I-2, 122
