J . Phys. Chem. 1988, 92, 6689-6693
6689
Also, most of phenol molecules may exist in the micelles. (They may be mixed micelles.) Altogether, the concentration of the solutes which act as the water structure promoter is expected to be not so high when it is compared with that of the soluble or miscible nonelectrolyte in water. For elucidating precisely the structures of the nonelectrolytes and surfactants, it seems necessary to obtain more thermodynamic or static information for such solutions under the present investigation even if the systems are quite complicated. Studies of other nonelectrolytes and surfactant solutions by ultrasonic methods are also desired in order to clarify quantitatively the relation between the structures of both solutes and the ultrasonic properties. They are under investigation and the results will be reported in due course.
Le., it is a fraction of non-hydrogen-bonded water or less structured water. In pure water, it has been estimated to be about 0.24,24 which is quite larger than that obtained in this study. Then, the solutes (phenol and SDS monomers) may be regarded to act as structure promoters. However, it is not so small when it is compared with that obtained in the aqueous solution of triethylene glycol monobutyl ether with SDS4 As has been pointed out, the surfactant monomers in the aqueous phase may also be expected to promote the water structure. In order to estimate the concentration of the monomers of SDS in phenol solution, a conductivity measurement at 0.8 mol dm-3 phenol has been measured with changing SDS concentration and the break point (cmc) has been found to be 1.8 mmol dm-3, which is quite lower than that of the cmc of SDS alone.25 This result indicates that the concentration of monomeric SDS is smaller when phenol coexists.
Acknowledgment. This work was partly supported by the Naito Foundation. Registry No. SDS, 151-21-3; H20, 7732-18-5; phenol, 108-95-2; methanol, 67-56-1.
(24) Davis, Jr., C. M.; Jarznski, J. Adv. Mol. Relax. Processes 1968, I , 155. (25) Miura, M.; Kodama, M. Bull. Chem. SOC.Jpn. 1972, 45,428.
Evaluation of Micellar Radii and Counterion Binding from Self-Diffusion Coefficients As Applied to Ionic/Zwitterlonic Mixed Micellar Systems Mikael Jansson,**tPer Linse,l and Roger Rymd6nt Institute of Physical Chemistry, University of Uppsala, P.O. Box 532, S-751 21 Uppsala, Sweden, and Physical Chemistry 1, Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden (Received: December 17, 1987; In Final Form: May 24, 1988)
The radii of mixed ionic/zwitterionic micelles have been obtained from micellar self-diffusion data by taking intermicellar interactions into account via a kinetic theory of Brownian motion. Experimental self-diffusiondata of counterions, surfactants, and micellar aggregates were obtained for the complete composition interval for the systems (dodecyldimethy1ammonio)propanesulfonate (DDAPS)/lithium dodecyl sulfate and DDAPS/dodecylammonium acetate. A considerably different dependence of mixed micellar radius on composition was observed in the two systems. Aggregates containing the anionic amphiphile were found to be larger than the corresponding cationic mixed micelles throughout the complete composition interval. As a check for consistency, the micellar radii were used as input values in model calculationsof fractional ion association ratios. A good agreement was observed with experimental ion association data.
1. Introduction
Studies of micellar translational diffusion constitute a main source of information on the size and shape of micellar aggregates (ref 1 and 2 and references given therein). Mixed micelles formed from zwitterionic and nonionic micelles have also been s t ~ d i e d . ~ - ~ The dynamics of micelles in solution also reflects intermicellar interactions which complicates the interpretation of micellar diffusion data in terms of micellar size/shape. This is of particular importance for solutions of ionic micelles where strong electrostatic micelle-micelle interactions prevail. In the present paper we have attempted to extract information on the size of micelles from micellar self-diffusion coefficients by taking electrostatic micelle-micelle interactions into account via a kinetic theory of interacting Brownian particles.6 As a check for consistency in the evaluated aggregate radii we have also determined experimental degrees of ion association from measurements of counterion self-diffusion. These data were compared to an electrostatic theory of ion association based on the nonlinearized Poisson-Boltzmann equation where the aggregate radius enters as a fundamental computational parameter with regard to both surface charge density and micellar volume. To whom correspondence should be addressed. +University of Uppsala. 'University of Lund.
0022-3654/88/2092-6689$01.50/0
The systems studied in this paper entail mixed micelles formed from anionic/zwitterionic and cationic/zwitterionic surfactants with the same hydrocarbon chain length. The size and shape of surfactant aggregates are to a large extent determined by simple geometrical constraints based on the packing of amphiphiles into closed aggregates.' Since the effective size of ionic surfactant headgroups is largely dependent on repulsive electrostatic interactions, it can be anticipated that the mixed micelles will show a dependence of aggregate size on ionic surfactant content. It can also be expected that ionic-zwitterionic interactions may contribute differently to the composition dependence of aggregate size in the two systems.
2. Materials and Methods Lithium dodecyl sulfate (LiDS), CH3(CHZ)l10S03Li', biochemical grade, was obtained from BDH chemicals, and (do(1) Nilsson, P-G.; Wennerstrom, H.; Lindman, B. Chem. Scr. 1985, 25, 61. (2) Brown, W.; Rymden, R. J . Phys. Chem. 1987, 91, 3565. (3) Nilsson, P. G.; Lindman, B. J . Phys. Chem. 1984, 88, 5397. (4) Faucompr;. B.; Lindman, B. J. Phys. Chem. 1987, 91, 383. (5) Jansson, M.; RymdCn, R. J . Colloid Interface Sci. 1987, 119, 185. (6) Ohtsuki, T. Physica A (Amsterdam) 1982, llOA, 606. (7) Mitchell, J. D.; Ninham, B. W. J . Chem. Soc.,Faraday Trans. 2 1981, 77, 601.
0 1988 American Chemical Society
6690 The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 X , , , , EM11
*--
7
Jansson et al. first step and the procedure is repeated until convergence in micellar radius is achieved. A micelle is modeled as a charged hard sphere. All micelles at a given composition are assumed to be identical; thus we neglect polydispersity. We are using the simplest form of an effective micelle-micelle potential, the screened Coulomb potential U ( r ) = $ exp[-X(r - l ) ] / r
t
(1)
where U is the potential between micelles in units of k T and r the intermicellar distance given in units of micellar diameter. The potential parameters $ and X are given by
'
(Xtnagg)
*
1
+
= 4 x ~ , ~ , k T ( l X/2)' 2Rmic
X = K*~R,,
where by
x:is the fraction of ionic amphiphiles in micelles given p i c
=
Xioncamp
ion
t
Xioncamp
t
T-----7f
and
K-'
decyldimethy1ammonio)propanesulfonate (DDAPS), CH3(CH2)11N+(CH3)2CH2CH2CH2S0,-, high-purity sample, from CALBIOCHEM, and dodecylammonium acetate (DDAAc), CH3(CH2)11N+H3 -OOCCH3, was synthesized from dodecylamine and acetic acid. Deuterium oxide (99.8 wt %), which was used as solvent, was purchased from Norsk Hydro, Rjukan, Norway. Micellar, counterion, and amphiphile self-diffusion coefficients were obtained from 'H and 'Li pulsed-gradient spin-echo experiments on a JEOL FX-100 FT N M R spectrometer. The measurement procedures were recently reviewed in detaiL8 The diffusion coefficient of the micellar entity was obtained by monitoring the diffusion of a solubilized probe molecule, hexamethyldisiloxane, known to have a negligible solubility in water. Addition of small amounts of hexamethyldisiloxane to the micellar solution had no effect on the self-diffusion coefficients of counterions or amphiphiles. All measurements were performed at 25 OC.
3. Theoretical Calculation 3.1. Micellar Radii. In the present work we discuss the change in micellar radius as the micelle is gradually changed from a zwitterionic to a ionic one. In this section we describe the underlying theory and computational strategy used. The calculation will give the answer to how the micellar radius changes from the pure zwitterionic, XI,, = 0, to the pure ionic, X,,, = 1, micelle in order to be consistent with experimental self-diffusion data. The micellar radii for the pure zwitterionic and ionic micelles are assumed to be known in advance. Each composition, except x,,, = 0 and XI,, = 1, involves a full calculation consisting of four steps calculated iteratively until self-consistency has been achieved. Figure 1 shows how the four steps are arranged and how the parameters are passing through the steps. Briefly, in the first step only known data are entered except the micellar radius which is guessed. As a result of the last step, where the experimental diffusion data are used, a new micellar radius is obtained. This new radius is now used in the (8) Stilbs, P.Prog. Nucl. Magn. Reson. Spectrosc. 1987, 19, I .
- [AMPI + (1 - X i o n I C a m p
is the screening length given by K~
Figure 1. Flow chart of the procedure of calculating R,,,. The denotation of the variables is given in the text. The variables to the left are constants except V,,, which depends on the ionic amphiphile considered. [AMP] and D are the experimental values for a given composition,X,,,, and R,,, is the result of the calculation.
- [AMPI
= (XionCamp + [AMP])e2/q,e,kT
In the last equation XionCamp should be interpreted as the concentration of the monovalent counterions. The meanings of the other symbols are as follows: nag the aggregation number, eo the dielectric permittivity of vacuum, e, the dielectric constant of water, k Boltzmann's constant, T the absolute temperature, Rmicthe micellar radius, Campthe stoichiometric concentration of amphiphile, [AMP] the free amphiphile concentration, and e the unit charge. The use of more elaborate potentials of the form given by eq 1 but with different interpretations of $ and X has recently been d i s c ~ s s e d . ~ - The ~ ' volume of an aggregated amphiphile is and is denoted by V,, and V, assumed to be independent of for the ionic and zwitterionic amphiphiles, respectively. The free ionic amphiphile concentration depends on Xi,, and is determined from diffusion data (see below). The free zwitterionic amphiphile concentration is, due to lack of experimental data, approximated to be zero. For a pure DDAPS-water system, the critical micellar concentration is -3.0 mM.4 This simplification of the model has no significant effect on our results. In the first step, the micellar number density, p , and the potential parameters, $ and A, are calculated. The amphiphile volumes used are V,,, = 522 A3 (LiDS), V,, = 444 A3 (DDAAc), and V , = 650 A3 (DDAPS). The stoichiometric concentration of amphiphile is the same, Camp= 0.100 M, at all compositions. Finally, T = 298 K and e, = 78.3 are used. The micelle-micelle radial distribution function, g(r), is needed in the third step. There exist a set of different statistical mechanical methods to calculate g(r) given the potential and the density. Monte Carlo simulations and molecular dynamics give an exact answer, although the results are affected by statistical noise. Analytic liquid theories give approximate answers and the degree of accuracy depends on the system and the method used. One of them, the hypernetted chain approximation (HNC), is known to accurately describe charged systems and deviations from simulation results are generally ~ r n a I l . ' ~ - ' W ~ e have used the algorithm of GillanI6 to solve the H N C equation.
e l
(9) Beresford-Smith, B.; Chan, D. Y . ;Mitchell, D. J. J . Colloid Inrerface
Sei 19RS -.. -. - -, -105 - - , -716 - -.
(10) Belloni, L. J . Chem. Phys. 1986, 85, 519. ( 1 1) Woodward, C. E.; Jonsson, B. J. Phys. Chem. 1988, 92, 2000. (12) Rasaiah, J. C.; Friedman, H. L. J . Chem. Phys. 1969, 50, 3965. (13) Hansen, J. P.; McDonald, I. R. Phys. Reu. A 1975, 11, 2111. (14) Linse, P.; Jonsson, B. J . Chem. Phys. 1983, 78, 1983. (15) Linse, P.; Andersen, H. C. J . Chem. Phys. 1986, 85, 3027. (16) Gillan, M. J. Mol. Phys. 1979, 38, 1751.
Ionic/Zwitterionic Mixed Micellar Systems
The Journal of Physical Chemistry, Vol. 92, No. 23, 1988 6691
In the third step the micellar density, the potential, and the radial distribution function are used to calculate the change in the micelle diffusion coefficient due to micellemicelle interactions. The change is given in terms of DIDo, where D and Do are the diffusion coefficients at finite concentration and at infinite dilution, respectively. Two main approaches exist to calculate DIDo for Brownian particles: the memory and the relaxation approach. Recently Lekkerkerker and Dhont17 showed that these approaches give the same leading term in a density expansion of D / D o for hard-sphere systems, viz., D/Do = 1-24, where 4 is the volume fraction of hard spheres. The relaxation approach for strongly interacting particles at low density has been developed by Ohtsuki and Okanol8 and carried out to higher densities by O h t s ~ k i . ~ , ' ~ A semiquantitative agreement of the DIDo results in comparison with Brownian dynamic results by Gaylor20 was obtained for a latex system.6 Following Ohtsuki6 the reduction in the self-diffusion coefficient (at zero frequency) is given by
D/Do = 1
+ ( 2 a p / 3 ) L m U ' ( r ) g(r)f(r)
r2 d r
(2)
where f(r) is the solution of the integro-differential equation
with d F(r,s) = -[r2g(r) A(r,s)] - 2rg(r) B(r,s) dr
until self-consistency is achieved. Normally this is obtained after 2-3 turns. Since the proposed procedure uses both experimental and theoretical results to obtain micellar radii, no direct comparison of experimental and theoretical micellar diffusion results can be made to estimate the accuracy of the procedure. The judgment of the accuracy of the method has to be made on the soundness of the obtained micellar radii (cf. below and section 4) and from other experimental results (counterion association, cf. sections 3.2 and 4). There are three types of sources giving rise to errors in the radii obtained. The first, and probably the most severe, is the very simplified model of the micellar solution (hard sphere, monodispersity, and screened Coulomb interaction). The second is of statistical mechanical nature and lies in the calculation of g(r) (HNC approximation) and in the evaluation of DIDo (neglect of hydrodynamic effects, linear response assumption, and a superposition approximation). The third type, which most likely is less important, is the accuracy of the measured micellar diffusion constant. In principle we could calculate the radii for all compositions, including &, = 0 and Xi, = 1 . However, deficiencies in the model for noncharged micelles could, at least partly, be remedied by using the experimental value of the micellar radii for Xi,, = 0 in order to calculate A (in eq 4), which essentially is a fitting parameter but could be viewed as an increase of the bare micellar radius to give the hydrodynamic radius. The use of Rmic= 24.6 A and 7 = 0.898 X Pa s for water at 298 K gave A = 3.0 A. This reasonable value of A shows the consistency of the procedure for noncharged micelles. The same A was subsequently used in all calculations. Instead of fitting A, we could set A = 0 and fit 7 with almost the same result, as A 0.4. The experimental data also indicate a slightly larger ion association to the pure DDAAc micelles compared to LiDS, pointing to the influence of specific effects on ion association. Effects of ion specificity related to the chemical nature of the counterion has been observed in a number of different micellar systems.24 The fraction of associated counterions calculated on the basis of purely electrostatic interactions is in very good agreement with the trend of the experimental data for the two systems. This gives support to the present calculation procedure for extracting aggregate radii from micellar self-diffusion coefficients. Qualitatively, the almost invariant ion association to LiDS/DDAPS micelles at high molar ratio of LiDS can be understood from the concomitant increase in radius with increasing amounts of added DDAPS. The dilution of the surface charge is thus counteracted by an increase in aggregation number of the mixed micelles. Likewise, the more rapid decrease in /3 observed in the DDAAc/DDAPS system is due to the invariance of the micelle radius with composition resulting in a more composition-dependent surfacce charge density for this system. It is implicitly assumed in the modeling of the electrostatic interactions between ions and charged micelles that a uniformly smeared out surface charge density is adequate to describe the discrete nature of the charged surfactant headgroups. It can be
shown that this approximation is surprisingly good even for dilute surface charges.26 This is confirmed by the good overall agreement between experimental ion-binding data and those predicted by the Poisson-Boltzmann model for the present systems. It thus appears that ion association to mixed ionic/zwitterionic micelles is dominated by ion-ion interactions and that the influence from ion-dipolar interactions, due to the presence of zwitterionic headgroups, is of minor importance for ion association. It was suggested by Tanford in connection with the salt dependence of the cmc’s of betaines that the terminal part of the zwitterion could be incorporated in the diffuse double layer, acting as a covalently bound c o ~ n t e r i o n . ~This ~ view is at variance with our results, however, since in the present case Ac- would be expelled from close contact with the micellar surface whereas an enhanced ion association would result for Li+. This type of mechanism is also anticipated to be more important at low Xi,, where we in fact observe almost identical degrees of ion association in both systems. The general principles of micelle formation also apply to the formation of mixed micelles;28Le., the size and shape of aggregates are largely determined by geometrical constraints based on the volume and length of the hydrocarbon chain while maintaining an optimal size for the hydrophilic headgroup area. The effective size of ionic headgroups mainly reflects repulsive electrostatical interactions acting in the plane of the micellar surface. Although the inclusion of zwitterionic headgroups appears to be of minor importance for ion association, they certainly contribute to interheadgroup interactions. This is indicated by the dependence of the cmc’s of zwitterionic micelles on electrostatic screening by simple salt.27,29 Since large effective headgroup areas favor the formation of small micelles, the observation that DDAAc/DDAPS forms considerably smaller aggregates than those containing LiDS indicates stronger repulsive surface interactions in the former system. This can be rationalized by assuming that the ionic headgroup interacts more strongly with the positive nitrogen of the zwitterion compared to the negative end. A recent study of dynamics and order in zwitterionic (decyldimethy1ammonio)propanesulfonate micelles demonstrates a distinct maximum in both the order parameter and correlation time profiles for the methylene groups at the nitrogen site,30 which is consistent with the nitrogen atom being firmly anchored in the micellar interface. This would indicate a closer packing of headgroups in the case of LiDS/DDAPS due to dominantly attractive ionic-zwitterionic interactions, Le., a smaller headgroup area as compared to the DDAAc/DDAPS system. 5. Summary We have obtained experimental self-diffusion data for mixed ionic/zwitterionic micelles. The two different ionic amphiphiles, one cationic and one anionic, gave rise to different variation of micellar self-diffusion upon a change in ionic-zwitterionic composition. By using a simple micellar model and statistical mechanical approximations, we have inferred that micelles containing the anionic amphiphile are larger than the corresponding cationic mixed micelle at the same composition. A tentative explanation of the different micelle size based on surfactant headgroup interactions has been presented. Registry No. DDAPS, 14933-08-5; LiDS, 2044-56-6; DDAAc, 2016-56-0.
(26) Israelachvili, J. N. Intermolecular and Surface Forces; Academic: New York, 1985. ( 2 7 ) Tanford, C. The Hydrophobic Effect; Wiley-Interscience: New York, I win (28) Szleifer, I.; Ben-Shad, A,; Gelbart, W. M. J. Chern. Phys. 1987, 86, 7094.
(29)Herman, K. W.J . Colloid Interface Sci. 1966, 22, 352. (30) Jansson, M.; Li, P.; Stilbs, P. J . Phys. Chem. 1987, 91, 5279.
