Langmuir 1993,9,1449-1466 1449

Articles Shape and Size of Micelles in the Sodium Dodecyl

Sulfate-Formamide System A. Ceglie, G. Coldemmina, and M. Della Monica

Dipartimento di Chimica, Universita di Bari, Trav. 200 Re David, I-70126 Bari, Italy

U. Ohson' and B. Jijnsson Division of Physical Chemistry 1, Chemical Center, Lund Univeristy,

P.O. Box 124,S-22100 Lund, Sweden

Received September 28,1992. In Final Form: February 24,1993

We have investigated the surfactant self-mmbly in the sodium dodecyl sulfate (SDS)-formamide system with 2H NMR relaxation on selectively 2H-labeled surfactant. Studiea performed over a large concentration range at 60 O C (slightly above the Krafft temperature) revealed a moderate increase in the micellar size with increasing surfactant concentration. When the relaxation data are analyzed surfactant dynamics in both the aggregated and nonaggregated etatea have to be taken into account. In doing 80 it was possible to evaluate how the fraction of nonaggregated surfactant molecules varies with the concentration in the micellar phase. These results were compared with the predictions from the Poiseon-Boltzmann cell model. The quadrupolar splittinga measured in the hexagonal phase were similar to those measured previously in the SDS-water system indicating similar molecular packing conditions in the cylindrical aggregatea. Below the Kraffttemperature ametastable cubic phaae is formed. Self-diffusion meaeurementa demonstrate that the cubic phase is of the micellar type which is also supported by the relative position of this metastable phase in the phase diagram.

1. Introduction Micelle and microemulsion formation with different

surfactants in nonaqueous media is a topic which has stimulated interest in recent years (see ref 1 for a literature survey), involving studies on phase equilibria1-'0 and microstructure using, for example, NMR methode,t1L16 small angle scattering,l7-19 and fluorescence techniques.20 The interest in these systems arises not only from the

(1) Dee, K. P.; Ceglie, A,; Monduzzi, M.; Merman, 0.; Lindman, B.

(2) Rim, I.; L a t t ~ , A. J. Phys. Chem. 1986,90,6870. (3) Auvray, X.; Petipas, C.; Anthore, R.; Rim, I.; Law, A J. Phys.

(4) Auway, X.; Pepitas, C.; Perche, T.; Anthore, R, Marti, M. J.; Rico,

(6) Auway, X.; Perche, T.; Anthore, R.; Petipas, C.; Rim, 1.; Laths,

(6) WBmheim, T.; Jbuon, A. J. Colloid Interface Sci. 1988,126,627. (7) WWeim,T.; Bokatrbm, J.; William, Y. Colloid Polym. Sci. 1988,

(8) WBmheim, T.; Sjbberg, M. J. Colloid Interface Sci. 1989,191,402. (9) Fribeg, 9. E.; h n g , G. Lcrngmuir 1988,4,796. (10) Fribeg, 9. E.; Liang, Y. C . In Microcmubions: Structure and

D Y M ~ ~ C S ; Friberg, S . E., Bothorel, P., Ells.; CRC P m : Boca Raton, FL, 1987; p 79. (11) Dee, K, P.; Ceglie,A.; Lindman, B. J.Phys. Chem. 1987,91,293& (12) Ceglir, A.; M o n d d , M.; S(lderman, 0. J. Colloid Interface Sci.

1991,142,129. (13) Belmydoub, A; El Bayed, K.; Brondeau, J.; Cmet, D.; Rim, I.;

La-, A. J. hys. Chem. 1988,92,3589. (14) Belmqjdoub, A.; Boubel, J. C.; Cmet, D. J. Phys. Chem. 1989,99,

4844. (16) JonnWmer, M.; Sjaberg, M.; WBmheim, T. J. Phys. Chem. 1990,

94,7649. (16) Sjaberg, M.; Henrilreeon, U.; WArnheim, T. Longmuir 1990,6,

1206. (17) Auvray, X.; Pepitas, C.; Anthore, R, Rico, I.; Lam, A.; Samii,

A. A-Z.; Savignac, A. d. Colloid Polym. Sci. 1987,286,925932. (18) Per&, T.; Auvray, X.; Petipee, C.; Anthore, R; Rim, I.; Lath ,

A.; Belliaaent, M. C. J. Phys. I 1992,2,923. (19) Schubert, K.; Strey, R. J. Chem. Phys. 1991, M, 8632. (20) Almgren, M.; Swmp, S.; Ldfroth, J. E. J. Phys. Chem. 191,89,

4621.

Prog. Colloid Polym. Sci. 1987, 73, 167.

Chem. l989,sS, 7468.

I.; Law, A. J. Phys. Chem. 1990,94,8804.

A Lcrngmuir 1991,7,2386.

266,562.

possible pratical applications but also from the goal of a deeper understanding of surfactant self-association in general. Qualitatively, the general behavior with non- aqueous polar solvents is similar to the corresponding aqueous systems. Micelle formation has been demon- strated in a number of nonaqueous polar solvents as well as the formation of liquid crystalline mesophases at higher surfactant concentrations. Quantitative differencies, how- ever, are the much higher Krafft temperatures and critical micelle concentration (cmc) values with nonaqueous polar solvents compared to the aqueous systems. Since the hydrophobic effect2' is important for micelle formation in an aqueous medium, this indicates that the analogous solvophobic effect in solvents like formamide is signifi- cantly weaker.

Also in microemulsions, water may be replaced by alternative polar solvents. However, by gradually replac- ing water by formamide, there is a gradual transition from a structured to nonstructured solutions,11 in certain cases passing a disorder line and a wetting-nonwetting tran- sition.19

For a particular system, the binary sodium dodecyl sulfate (SDS)-formamide system, the Krafft temperature has been determined to 57 OC, significantly higher than the corresponding value in water (20 "C), and a cmc value at 60 OC, based on surface tension measurements, of 5.6 wt 96 has been reported.2 Furthermore, a sequence of liquid crystalline phases (hexagonal-cubielamellar) have been identified at higher surfactant concentrations and elevated temperatures.6 At room temperature, well below the Krafft point, the surfactant has a limited solubility of about 7-8 w t 96 in the solvent. Here, it has been demonstrated by several experimental techniques that

(21) Tanford, C. The Hydrophobic Effect; Wiley New York, 1980.

0743-7463/93/2409-1449$04.00/0 Ca 1993 American Chemical Society

1460 Langmuir, Vol, 9, No. 6, 1993 Ceglie et 41.

micelle formation does not occur within the solubility

In this paper we investigate the surfactant self-assembly in the SDS-formamide system above the Krafft temper- ature, using 2H NMR relaxation on selectively deuterium labeled SDS. This well-established experimental tech- nique has been shown to be very useful for obtaining structural as well as dynamical information on aggregated surfactant systems.22* We focus our study on the micellar phase of the system where the experimental data allow us to determine the variation of the micellar size as well as the fraction of monomerically solubilized surfactant ions. The latter results are compared with the results of a Poisson-Boltzmann calculation within the cell model.24 In addition, we also report on quadrupolar splitting8 measured in the hexagonal phase, which are compared with those obtained previouslys in the corresponding aqueous system, and the formation of a metastable cubic phase below the Krafft temperature.

2. Experimental Section Materials. ~,CY-~H&DS, sodium dodecyl sulfata, was pre-

pared following the procedure previously deacribed.s The purity of the product (mp 196-198 "C) was checked by NMR spectroscopy. Formamide (FM) purchased from Fluka P99 % 1 was purified following the suggestions of Notley and SpirOn and then stored in a drybox on molecular sieves, activated at T > 360 "C, from at least 24 h before use.

Methods. 2H NMR relaxation measurements were p e r f o d at 4.7 T on a Varian XL-200 spectrometer. Longitudinal relaxation rates, R1, were measured using the standard inversion recovery sequence while transverse relaxation rates, Rz, were measured using the Carr-Purcell-Meibolm-Gill sequence. The temperature was controlled with an accuracy of h0.5 OC. Self- diffusion experimenta were performed on a modified JEOL FX- 60 spectrometer.

3. Theoretical Considerations 2H relaxation of selectively deuterium labeled surfac-

tants is a well-established experimental technique to study surfactant dynamics and solution microstructure in self- aggregated surfactant s y s t e m ~ . ~ ~ > ~ 3 For aliphatic deuter- ons, the relaxation is governed by the quadrupolar interaction with an approximately cylindrically symmetric electric field tensor. The longitudinal (R1) and transverse (R2) relaxation rates are given by2*

(1)

limit.1JlJ2,20

3a2 R, = 40 x2[2j(w0, + 8 j ' , ( 2 ~ o ) l

where x is the quadrupole coupling constant, j (w> is a reduced spectral density function of the relevant molecular dynamics, and 00 is the resonance frequency.

In the present system, the surfactant aggregation is weak and dynamics in both the aggregated and nonaggregated (monomeric) state contribute to the relaxation. The lifetime in a given state is expected to be short relative to the relaxation time in which case the spectral density can

(22) Lindman, B.; Sbderman, 0.; Wennemtr(lm, H. In Surfactant Solutions. New Methods oflnvestigation.; Zana, R., Ed.; Marcel Dekker, New York, 1987.

(23) Chachaty, C. Rog. Nucl. Magn. Spectrosc. 1987,19,183. (24) Gunnarseon, G.; J(lneeon, B.; Wennemtr(lm, H. J. Phys. Chem.

1980,84, 3114. (26) Sbderman, 0.; Carletram, G.; Olsson, U.; Wong, T. C. J. Chem.

SOC., Faraday Trans. 1 1988,84,4475-4486. (26) Monduzzi, M.; Ceglie, A.; Lindman, B.; Sbderman, 0. J. Colloid

Interface Sci. 1990, 136, 113. (27) Notley, J. M.; Spiro, M. J. Chem. SOC. B 1966, 362. (28) Abragam, A. The Principles of Nuclear Magnetism; Clnrendon

Press: Oxford, 1961.

be written as a weighted sum:

j<w> = pj,,(w) + (1 - P)lw(d (3) Here, P is the fraction of surfactant molecules in the monomeric state and jmon(w) and jcrlle(w) are spectral density functions describing the dynamics in the monomeric and aggregated stat&, respectively.

In the aggregated state, it is well established that there is a clear timescale separation offast localmotions (internal modes) and slow motions associated with the aggre-

The fast local motions are slightly anisotropic, due to the preferred orientation of the surfactant molecules leaving a residual anisotropy which is averaged to zero by the isotropic slow motion. For spherical micelles the local motions are approximately cylindrically symmetric around the preferred orientation (aggregate normal) and the residual anisotropy is described by a single-order param- eter S, given by

(4)

where 8 is the angle between the C-D bond vector and the aggregate normal. The spectral density can be written as a weighted sum

(5) where jfcw) and j&) describe the fast (local) and slow (global) dynamics, respectively.

In the case of spherical aggregates, j8(w) is Lorenzian

1 2 s = - ( 3 cos2 e - 1 )f

i,(w) = (1 - S2) j f (w) + S2jj,<w)

where T~ is a correlation time due to the combined motions of aggregate tumbling and surfactant lateral diffusion. Statistical independence of these two motions imply

(7) The correlation time for the rotational tumbling of a sphere in a viscous medium is given by

TS-' = T i 1 + T i 1

where q is the solvent viscosity, R is the radius of the sphere, and ~ B T the thermal energy. The diffusion correlation time can be associated with the diffusion on a sphere of radius R

where Dht is the lateral diffusion constant. Note that Dht is defiied in relation to the radius, R, where it is evaluated (in practice, eq 9 can be considered as a definition of Dht). From the strong radius dependence of T# it is clear that spin relaxation is a technique capable of monitoring small changes in aggregate size and shape.

For small deviations from spherical shape, we may consider prolate or oblate ellipsoids. This case was recently treated by Halle who developed numerical schemes to evaluate the joint spectral density function of lateral diffusion and aggregate tumbling.33 In this case, the spectral density function has to be evaluated numerically.

(29) Wenneretriim, H.; Lindman, B.; Sdderman, 0.; Drakenberg, T.; (30) Hnlle, B.; Wennemt&m, H. J. Chem. Phys. 1981, 76,1928. (31) Sbdennan,O.; Walderhaug,H.;Henrikeson,U.;Stilbe,P. J.Phys.

(32) Sbderman, 0.; Henrikseon, U.; Ohon, U . J. Phys. Chem. 1987,

Roeenholm, J. B. J. Am. Chem. Soc. 1879,101,6880.

Chem. 1986,89,3693.

91,116. (33) Halle, B. J. Chem. Phys. 1991,94,3150.

Surfactant Self-Assembly Langmuir, Vol. 9, No. 6, 1993 1451

LV 0 1

Weight fraction SDS

Figure 1. A schematic phase diagram of the SDS-formamide system based on the data of refs 2 and 5 and the present study. The shaddowed area indicates a metastable cubic phase.

The order parameter, or rather the product 1x81, can be measured independently in anisotropic mesophases when the structure is known. In the present system a hexagonal mesophase is formed at higher surfactant concentration (see Figure 1). Assuming a structure of infinite cylinders, the residual quadrupolar coupling, A, is given by

(10)

It has been demonstrated for several surfactant systems that the order parameters measured in hexagonal phases and in solutions of micelles are very similar.

3 A = (1 - P)gXlSI

4. Phase Behavior In formamide, and other nonaqueous polar solvents,

such as methylformamide and ethylene glycol, Krafft temperatures, TK, and the concentrations for the onset of micellar formation are significantly higher than in the corresponding aqueous systems. In the present system the Krafft temperature has been determined to 57 "C and onset of micelle formation, as deduced from surface tension measurements, to 5.6 wt %,2 which can be compared to 16 "C and 0.2 wt % , respectively, in water. Further studies of the phase behavior in the SDS-formamide system were recently presented by Auvray et aL5 who identified hexagonal, bicontinuous cubic, and lamellar phases at higher surfactant concentrations. A schematic phase diagram of the SDS-FA system, based on the data of refs 2 and 5 and observations in the present study, is shown in Figure 1.

An interesting observation is that when cooling samples in the liquid phase, within a narrow concentration range of about 40-45 wt % , below 45-50 "C the samples become very viscous while still being optically isotropic. A comparison with phase diagrams of other surfactant systems, as for example the dodecyltrimethylammonium chloride (D0TAC)-water indicates that the viscous isotropic samples correspond to the formation of a metastable cubic phase, where the micelles crystallize within a cubic lattice. Such a cubic phase is commonly found at concentrations between the liquid micellar and hexagonal phases. We have indicated the approximate extension of this metastable cubic region in Figure 1.

Below the Krafft temperature the thermodynamically stable state is crystalline surfactant, possibly ~olvated,3~

(34) Balmbra, R. R.; Clunie, J. S.; Goodman, J. F. Nature 1969,222,

(35) Laughlin, €2. G.; Munyon, R. L.; Fu, Y.-C.; Fehl, A. J. J. Phys. 1159.

Chem. 1990,94,2546.

in equilibrium with a relatively dilute solution. Samples, cooled from the liquid phase to room temperature can remain as a homogeneous liquid for several days or longer before crystallization of the surfactant takes place. Crys- tallization may however easily be induced by adding a trace amount of crystalline surfactant or scratching the glass walls of the sample tubes with a hard object, thereby creating nucleation centers for the crystallization.

5. The Metastable Cubic Phase A sample containing 43 wt % SDS was investigated by

small angle X-ray diffraction, using a Kiessig camera with pinhole collimation, in the metastable cubic region. The diffraction pattern was recorded at room temperature, after equilibration at higher temperatures in the liquid phase. As commonly observed with cubic phases,36 the film showed a number of spots, rather than complete diffraction rings, due to the incomplete powder averaging. This is a characteristic property of cubic phases, which, compared to other lyotrophic liquid crystalline phases, from relatively large microcrystallites. Hence it is difficult to obtain a complete powder in the small sample sizes (=1 mm) used in small angle X-ray scattering. With some uncertainty, due to the "spottiness", five reflections could be identified, at the spacings: 42.8, 38.6, 34.3, 30.8, and 26.8 A, respectively. Cubic phases appearing between the liquid micellar and hexagonal liquid crystalline phases generally belong to the space group Pm3n.36 The spacings observed here could also be indexed to this space group, where the spacings correspond to m = (h2 + it2 + P) = 8, 10, 12, 16, and 20, respectively. With this indexing, the observed spacings are consistent with a lattice parameter a = 120 A.

Fontell et al.37 have proposed a structure of this cubic phase with eight micelles per unit cell, which are slightly elongated. With a = 120 A and assuming the micelle volume fraction to be 0.4, we obtain with eight micelles per unit cell a micellar volume of 86 nm3. Assuming a prolate shape, and the length of the minor semiaxis to be equal to 20 A, this micellar volume is consistent with an axial ratio of 2.6. A similar axial ratio is found in the liquid phase at higher temperatures, vide infra. A smaller axial ratio (=1.4, evaluated as a prolate), however, was obtained in the micellar cubic phase of the DOTAC-water system38 from a fluorescence quenching study.

Another sample, containing also 43 w t % SDS, was prepared in the metastable cubic region at 37.7 "C by lowering the temperature from the liquid phase. The self- diffusion constants of SDS and FA were measured and for SDS it was found to be very low DSDS = 5.6 X m2 s-l, while for FA it was high DFA = 1.2 X 10-lo m2 s-l. These results show (i) that the structure of the metastable cubic phase is of the micellar type and (ii) that the concentration of monomeric SDS is low. A quantitative estimate can be obtained by assuming that the diffusion constant can be written as a weighted sum D = PAD,, + (1 - P)D,, where A = 0.8 is an obstruction factoS9 and Pis the fraction of monomers. In the cubic lattice we expect D, to be very low and if we neglect this contribution we have P - D/(ADmon). mt 25 "C Dmon has been measured to 2 X 1 0 - l O m2 s-l.ll At 38 "C we expect a somewhat higher value, say 4 X 1 0 - l O m2 s-l, which gives P = 0.02. We will return to

(36) Fontell, K. Colloid Polym. Sci. 1990,268,264. Fontell, K. Adv.

(37) Fontell, K.; Fox, K. K.; Hansson, E. Mol. Cryst. Liquid Cryst.

(38) Johansson, L. B.-A.; Siiderman, 0. J. Phys. Chem. 1987,91,5275. (39) Jtinsson, B.; Wennerstrtim, H.; Nilsson, P.-G.; Linse, P. Colloid

Colloid Interface Sei. 1992,41, 127.

Lett. 1985, 1, 9.

Polym. Sci. 1986, 264, 77.

1452 Langmuir, Vol. 9, No. 6, 1993

Table I. tH Quadrupolar Splitting#, A, Becorded at Various Temperatures in the Hexagonal Pham and in

Metartable Sampler below the Krafft Temperatun, in the Present System. and for a Sample in the SDS-Water

Systemb system T/'C A/kHz system T/OC A/lrHz

Ceglie et a1.

larger than R1. R2 continues to increase, almost linearily, with the concentration, while R1 levels of at higher concentrations to a value of about 32 s-l.

As discussed in section 3 we need to consider three types of motions: the dynamics in the monomeric state and the fast and slow motions in the aggregated state. These three motions are characterized by the spectral densities jmon, jf, and in, respectively. At lower concentrations we expect the surfactant to be mainly dissolved as monomers, and from those concentrations we can extract information on

Monomer Dynamics. As evident from R1= R2 at high dilution, the monomer dynamics are in extreme narrowing, i.e. jmon(0) = jmon(w) = jmon(2~), at the present resonance frequency. R1 = R2 = 7 s-l corresponds to an effective correlation time Tmon,& = jmon(0)/2 = 14 pa. This value is the same as found in a spin relaxation study of an aqueous SDS solution below the cmc at 20 O C . 4 0 Similar values are to be expected considering the lower viscosity but also the lower temperature in the aqueous system.

Micellar Growth. In the time scale separation of fast and slow motions in the aggregated state it is in practice implicitly aesumed that also the fast motions are in extreme narrowing, i.e. jd0) = j d w ) = jd2w). Such conditions, at the present resonance frequency, have been verified in aqueous solutions of spherical SDS micelles% as well for a number of other surfactant system~.~~93~ Moreover, jdO) values have been found to be roughly independent of micellar shape."

If the fast motions in the aggregated state and the monomer dynamics both are in extreme narrowing, it then follows that these two motions contribute as a constant offset to R1 and RZ and the difference AR = R2- R1 depends only on the slow dynamics in the aggregated state

AR = (1 - P) - (xS)~L~,(O) + j 8 (wd - 2j,(2wo)1 (11)

In Figure 2 we also show variation of AR with the surfactant concentration. Below WSDS = 0.15 AR is almost zero, while it increases almost linearly with WsDs above this concen- tration.

The question we first need to answer is if AR increases due to an increase in micellar size or if it is only an effect of a decrease in P with increasing concentration. For spherical micelles AR depends on R, q, and Dht. Assuming R = 20 A and using q = 1.63 cP, the value AR = 30 8-1 observed at higher concentrations corresponds to Dht I 2 X lo-" m2 s-l. This value is much smaller than the value 1 X m2 s-l obtained in aqueous micellar solutions26 at a much lower temperature: 25 "C. From this we can conclude that the micelles in the present system indeed grow in size with increasing concentration.

Indications of micellar growth can also be found when comparing the variations of R1 and R2 with the concen- tration. In accordance with eq 3 we may write

R1= PRmon + (1 - P)RI, (12a)

Imon.

9T2 40

a 26 11.4 a 76 9.7 a 40 10.9 b 26 11.6 a 60 10.3

0 60 wt % SDS in fonnamide. b 46.5 wt % SDS in water. Data from ref 26.

60 I . , , , , , . 3 5 0 Lm /

/ /

i .A

0 0.1 0.2 0.3 0.4 0.5

WsDs Figure 2. Variation of the relaxation rates, RI (0) and Rn (01, and their difference AR = R1 - R1 (A) with the Surfactant concentration in the micellar phase at 60 "C.

the question of the surfactant monomer concentration in the solvent below when discussing the NMR relaxation data from the micellar phase.

6. The Hexagonal Phase The quadrupolar splitting was measured for one sample

(50 wt 5% SDS) at several temperatures in the hexagonal phase, both in the stable phase above the Krafft tem- perature and in the metastable state at lower temperatures. The results are shown in Table I. For comparison we also give the value previously measured in the corresponding aqueous system at 25 OC.= Similar values are obtained with the two solvents which indicates that the local structure and the surfactant packing conditions in the aggregates are similar.

7. The Liquid Phase Spin relaxation rates, R1 and R2, were measured in a

wide concentration range and at several temperatures both in the liquid phase above the Krafft temperature and on metastable liquid samples below the Krafft temperature. The most extended measurements were performed at 60 OC, just above the Krafft temperature, and we will start by discussing the results obtained at this temperature.

7.1. Micelle Formation at 60 "C. R1 and& measured at 60 "C in the micellar phase are plotted as a function of the weight fraction SDS, Wsm, in Figure 2. A t lower concentrations R1 = R2, a condition which is usually referred to as extreme narrowing. At high dilution, the relaxation rates level off at about 6 s-l. At low concen- trations, the relaxation rates increase slightly with in- creasing concentration. From 1 to 7 wt % R1 and R2 stay almost constant; only a minor increase from 6.5 to 8.5 s-1 is observed. With the concentration increased further, the relaxation rates increase more strongly with the concentration. Above 15 wt % R2 becomes significantly

where Rmon = Rlwon R2won. In the case of constant micellar size, the ratios

should be constant. This is not the case, as is seen in

(40) Jonetrdh", M. Thesis, Lund University, Lund, 1990. (41) Quist, P.-0.; Halle, B.; Fur6, I. J. Chem. Phys. 1991, 95, 6946.

Surfactant Self-Assembly

2.5 0 Langmdr, Vol. 9, No. 6, 1993 1453

1 n 1 2

M- +E 1.5 I

t p1 5 0 .5

0 0 0 . 1 0 . 2 0.3 0.4 0 . 5

WSDS

Figure 3. Variation of the ratio (R1- R")/(RI - Rmon), using R,, = 7 s-l with the surfactant concentration at 60 "C. The fact that thie ratio increases with increasing concentration, taking values >1, signifies an increasing micellar size with increasing concentration.

h

v)

" v

v1

M-

1 5

1 0

5

0 1 1.5 2 2.5 3 3 . 5 4

r Figure 4. Calculateda variations of R1, with the axial ratio of prolate ellipsoidal micelles, according to the theory of Halle." In the calculations the length of the minor semiaxis is kept constant to 20 A and the four different curves correspond to four different valuea of Dht (0.5, 1 2, and 5 x 10-10 m2 8-1, respectively).

Figure 3 where we have plotted (R2 - Rmon)/(RI- Rmon), using Rmon = 7 s-l. Instead, the ratio increases from being close to unity, consistent with the extreme narrowing, at 1 O w t % toavalueof2.5at40wt%.

Variation of the Monomer Concentration. Having established a micellar growth, we now turn to evaluate how P varies with the Concentration. We note that R1 levels off at higher concentrations, indicating that R1 is insensitive to the micellar growth. We may investigate this further by calculating how j l ( w ) varies with the axial ratio when spherical micelles growth into a prolates. In particular, we investigate how RI , = (3r2/40) (xS)~~~ .~ (OO) + 8jB(2wo)l varies with the axial ratio, r. This variation is shown in Figure 4 for four different values of Dkt between 0.5 and 5 X W0 m2 s-l. The curves were calculated numericallyt2 according to the theory of Halleess and in the calculations we have assumed a constant length of the minor axis, b = 20 A. (Keeping instead the interfacial area per enclosed volume constant, which corresponds to a constant area per headgroup, gives veri similar results.) As is seen in Figure 4, R1, values decrease by about 25 % when r increases from 1 to 4. Since RlJR1 = 0.3-0.5, as is seen by comparing R1, with the observz R1 values,

(42) The lab frame spectral densitiea referring to lateral m i o n and partide tumblingwem calculated numerically with a program written by Bedl Halle, using the direct method.=

.- Be 0 . 3 n I I

0 0 .1 0.2 0.3 0.4 0 .5

WsDs Figure 6. Varations of W- and Wmie with the total surfactant concentration, Wsm, at 60 O C . The experimentally determined values of W,, (0) and W, (0) are obtained from the P values calculated according to eq 12a, using R1- = 7 8-1 and R1, = 32 s-l. The dashed and the solid lines correspond to the W, and W-, respectively, calculated from the Poisson-Boltzmann cell model. In the model calculations we have used a dielectric constant of c = 96,e a constant aggregation number N- = 60, andacmc=lOwt%.

and the contribution from the fast motions, R ~ J , is expected to be constant, R1, is expected to decrease by only 10% with r increases in the range 1 I r I 4 (as will be shown below, r does not exceed 3). Neglecting this descrease, we may assume that R1 approaches a constant value R I , at higher concentrations whenPapproaches zero. Assuming Rl,, and to be constants, we can find P * (Rim - R I ) / @ I , - &on) using eq 12a. From the P values, we may then calculate the concentration of monomeric and micellizsd surfactant. The results, obtained from aeauming R,,,, = 7 s-l and R E , = 32 s-l, are shown in Figure 5 where we have plotted the monomer weight fraction, Wmm, and micellar weight fraction, W , = W s ~ s - Wmon, versus the total surfactant concentration, Wsm. From these data we see that aggregation starts to occur at WSDS 10 wt %, and above this concentration Wmon values decrease when increasing the total concentration. The latter effect is a common property of ionic amphiphiles. It is due to the increasing electrostatic screening in the system which lowers the electrostatic potential on the micellar surface and, hence, lowers the chemical potential of the surfactant ion in the micellar state. This effect can be calculated quantitatively within the Poisson-Boltzmann cell modelU which has been shown to work well for monovalent surfactants.'s

In Figure 5 we have also compared the variation of Wm, and WF with the concentration, as obtained from the relaxation data, with those predicted by the Poisson- Boltzmann cell model. The principles of the calculations have been described previously.U*u The results are given in Figure 5 as a solid line (W& and a dashed line (Wmon), respectively. The calculations were performed with spherical geometry for a constant aggregation number N- = 60, which is the number found in a large concentration range of the corresponding aqueous system, and a cmc value of 0.4 M (= 10 wt 9% ) and a dielectric constant of the solvent, c = 96.4 In spite of simplifications involved, a good agreement between the experimentally determined and the theoretically predicted variations of Wmon and

(43) J ~ ~ n , B.; Wennerstr6m, H.; Halle, B. J. Phys. Chem. 1980,84, 2179. -. (44) Jtlneson, B. Thesis, Lund University, Lund, 1981. (46) Baee, S. J.; Nathan, W. 1.; Meighan, R. M.; Cole, R. H. 3. Phys.

Chem. LS64,68,509.

1464 Langmuir, VoI. 9, No. 6,1993

3 5 ) . I , , . I , , . I

Ceglie et al.

i% 15 1 10

5 c n 0 4

0 0 . 1 0.2 0.3 0 .4 0.5

WsDs Figure 6. Variation of the relaxation rate difference in the aggregatadstate,AR ,calculatedfromAR,= AR/(l-P),where P is calculated accogmg to eq 12a, with the surfactant concen- tration.

Wdc is obtained. From the relaxation data we know that the aggregation number is not constant but increases with the concentration and slightly nonspherical micelles are formed at higher concentration. On the other hand, the variation in micellar size is not very strong and, moreover, W,, does not depend strongly on the aggregation number within the cell model (choosing N, = 40 gives a similar curve). Our conclusion from the comparison is that the variation of WmOn with the total concentration can be understood in terms of current theories of monovalent ionic amphiphiles.

The cmc value 10 wt % indicated by the present experimental data is higher than the value 5.6 wt % reported from surface tension measuremenh2 This difference may be due to different water contents in the formamide, which is known to be hygroscopic. Here, particular care was taken to minimize the water content, while no such attempts were made in the work of ref 2. It is reasonable to assume that the cmc varies monotonically with the water to formamide ratio and that a lower cmc corresponds to a higher water content. We also note that in the surface tension plot of ref 2 there are no data points in the concentration range between 4.1 and 13 wt % ; hence it is difficult to estimate the accuracy of their reported cmc value. Also the solubility limit of SDS at room temperature (far below the Krafft temperature) has been found to be higher than the cmc value of ref 2 which supporta the fiiding of the present work of a higher cmc value. On the other hand, it is most likely that the surfactant association at low concentrations occurs grad- ually over a larger concentration interval as has been reported for the cetyltrimethylammonium bromide (CTAB)-formamide system.& In this case, a critical micellization concentration is not well def ied and a comparison of cmc values is of less value.

Relaxation in the Aggregated State-Micellar Size. From knowing P we may calculate AR, = AR/(1- P, which we have plotted versus WSDS in Figure 6. At lower concentrations, close to the onset of micelle formation, hR, is small, =2 a-1. With increasing concentration AR, increases and at WSDS = 0.4AR, = 30 s-l. From AR, we can determine the micellar size, by interpreting the data in terms of spherical or prolate ellipsoidal particles, aa discussed in section 3. The only unknown parameter is the lateral diffusion constant Dht. Considering spherical and prolate shapes we have calculated numerically,3' using the theory of Halle,a the variation of AR, with the axial ratio, r, keeping the length of the minor semiaxis constant

n

v1 W

3

M M

erla d

140

120

100

80

60

4 0

2 0

I I I I I I 1 1.5 2 2.5 3 3.5 4

r Figure 7. Numerical calculations" of AR,,, as a function of the axial ratio of prolate ellipsoidal particles according to the theory of Halle." The length of the minor semiaxis ie kept constant at 20 A and the four different curves correspond to four different values of Dk (0.5, 1 2, and 5 X W0 m2 s-l, respectively).

5 0

4 0

30

2 0

A AR L I

-I A

I/ A

0 " 2 ' A I ' I ' 0 0 . 1 0 . 2 0.3 0.4 0 . 5

WsDs Figure 8. Variation of the relaxation rates, R l ( 0 ) and R2 (o), and their difference AR = R2 - R1 (A) with the surface concentration in the micellar phase at 75 OC.

to 20 A. This length corresponds roughtly to the extended length of the surfactant molecule. The results of the calculations, done for different values of Dht are presented in Figure 7.

A value of Dht can be obtained by assuming that the micelles formed at lower concentrations, close to the onset of micell formation, are spherical with a radius of 20 A. In this case, AR, = 2 s-l is consistent with Dht = 4 X 10-10 m2 s-l. This value of Dht is reasonable if we compare with the value 1 X 1&l0 m2 s-l obtained for spherical SDS micelles in water at room temperature.%

On the basis of the assumption of spherical micelles at lower concentrations, having a radius of 20 A, we may obtain the variation of a prolate axial ratio from the variation of AR, with the surfactant concentration. By comparing the ART values presented in Figure 6 with the calculated curves in Figure 7, we see that the data are consistent with a moderate micellar growth from spheres at lower concentration to an axial ratio of roughly 2.5 at higher concentrations, where ARw = 30 a-1. The value r = 2.5 should be considered as an upper limit. Intermicellar interactions are expected to slow down the particle tumbling at higher concentrations if the particles overlap which, since the analysis is based on the tumbling of an isolated particle, leads to an overestimation of the axial ratio. It is also possible that the micelles formed at lower concentrations are smaller and less compact compared to spherical micelles in water. This has been indicated in a

Surfactant Self- Assembly

similar system.17@ In this case our Dht value of 4 X 1 0 - l O m2 s-l may be an overestimation which in turn leads to an overestimation of the particle axial ratios at higher concentrations. With a frequency-dependent NMR re- laxation study or a small angle scattering study, one may possibly resolve this issue. To summarize we find that from the variation of AR it is clear that the micelles grow with increasing concentration although the growth is only moderate and the actual numbers are model dependent.

7%. Temperature Dependence. Additional relaxation studies performed at 75 "C gave results which were very similar to those obtained at 60 OC. In Figure 8 we have plotted R1, R2 and AR as a function of the SDS concen- tration. From comparing the data in Figure 8 with those obtained at 60 "C (Figure 2), we can conclude that the ~~

(46) Olofsaon, G. J. Chem. SOC., Faraday Tram. 1991,87,3037-3042.

Langmuir, Vol. 9, No. 6, 1993 1455

association behavior and the variation of micellar size are similar for the two temperatures.

Some additional measurements were also performed in the metastable region below the Krafft temperature a t 25 and 40 "C (data not shown). Also here, the results were similar to those obtained at higher temperatures, i.e. a weak micellar growth with increasing surfactant concen- tration.

Acknowledgment. This work was supported by Con- siglio Nazionale delle Ricerche-Progetto F i n h t o Chim- ica Fine I1 and the Swedish Natural Science Research Council (NFR). We are grateful to Bertil Halle, for sharing with us the software for spectral density calculations in prolate and oblate symmetry, and to Krieter Fontell for performing the small angle X-ray measurements.