J. Phys. Chem. 1983, 87. 3541-3550
relationship between micelle size and shape and surfactant structure focuses on the balance between minimizing hydrocarbon interaction which favors large micelles and minimizing head group repulsion which favors small micelles. The relationship between these quantities is often discussed in terms of the dimensionless ratio v/ (aolc) (6) where V is the volume per hydrocarbon chain, 1, its effective length, and a. the effective area occupied by the surfactant head groups. The prediction in water is that, if V/(aolJis less than 1/3, spherical micelles are obtained. For values larger than this, cylindrical micelles, bilayers, or inverted structures are predicted. For aqueous surfactant solution Vis calculated from eq 4,Zc 0.81, from eq 5, and a. is typically 60-70 A2 for ionic surfactants in the absence of added salt. For fused salt surfactant solution it is difficult to test the generality of eq 6. The quantities given in Table I give ratios of V/(l,ao) of 0.30 as indeed they must, given our assumption of spherical micelles. The free energy of transferring a -CH2- group from EAN to the micelle is -400 cal mol-’ compared to -680 cal mol-l for water. Preliminary measurements on oil-fused salt interfacial tension give a value of 20 dyn/cm compared to a value of
-
3541
50 dyn/cm for the oil-water interfa~e.’~Both of these results suggest that EAN is a more hospitable environment for hydrophobic moieties than is water. Thus small micelles with appreciable portions of the surfactant hydrocarbon chain exposed to the fused salt are not unrealistic. The idnic nature and large size of the solvent will result in very different electrostatic interactions, and consequently different values of ao, than those seen in water. For the mixed micelles containing surfactant and ethylammonium ion as cosurfactant, the assignment of effective head group areas becomes even more nebulous. Neutron scattering experiments could provide information on the distribution of surfactants, terminal methyl groups, and ethylammonium ions within the micelles in this fused salt system.
Acknowledgment. This research was supported in part by NSF Grant CPE-8014567and U.S.Army Contract DAA G29-81-K-0099to D.F.E., and by NSF Grants PCM 7806777 and PCM 81-18107 to V.A.B. Registry No. Hexadecylpyridinium bromide, 140-72-7; tetradecylpyridinium bromide, 1155-74-4; ethylammonium nitrate, 22113-86-6. (14)S. Mukerjee and D. F. Evans, to be published.
Glant Micelles in Ideal Solutlons. Either Rods or Vesicles 0. Portet Laboratolre de Spectrom6trle RayMgh Brlllouln, Unlversitci des Sclences et Technlques du Langueobc, 34060 Montpelller Cedex?France (Received October 18, 1982; In Flnal Form: March 23, 1983)
The phenomenological description of the “sphere-to-rodtransition”in dilute micellar solution,which was proposed by authors such as Mukerjee (1972),Israelachvili et al. (1976),and Mazer et al. (1976),has found extensive experimental support in recent years. However, in its initial form it cannot account for some effects due to the flexibility of giant micelles. We thus reconsider this theory and modify the initial assumptions in the following way: (i) we a priori allow for the coexistence of micelles of different shapes (rods and toroids; disks and vesicles) in solution; (ii) we introduce into the asymptotic expression of the standard free energy of a N-micelle a term which accounts for the configurational entropy associated with the ensemble of bent conformations for the flexible giant micelles. Doing so we are able to explain why the spontaneous formation of closed rings does not inhibit the growth of large rodlike micelles. We also show that in dilute solution the formation of vesicles is the general occurrence when bilayered local structure is involved. We, by the way, point out the connections between the growth of giant micelles and other phenomena such as the reversible polymerization of proteins or the fluid-to-viscous phase transition in liquid sulfur.
Introduction Amphiphilic molecules in aqueous solutions are known to form various types of micelles depending on the particular experimental conditions. U s d y , ionic amphiphiles form small globular micelles in binary solutions at low concentration near the cmc. Deviations from these simple initial conditions sometimes result in a strong modification of the size and shape of the micelles in solution; this is indicated by spectacular changes in some macroscopic properties of the solution such as turbidity and viscosity. t Permanent address, where correspondence should be sent: Laboratoire de Mindralogie, Centre de Dynamique des Phases Condensdes, Universitd des Sciences et Techniques du Languedoc, 34060 Montpellier Cedex, France.
Actually this shape and size “transition” for the micelles can be achieved in various ways such as the following: (i) a large increase of the amphiphile concentration,’ (ii) and/or the use of some organic additives (short-chained alcohols,2benzene, salicylate salts? etc.), (iii) and/or the addition of large amounts of certain mineral salts.”12 The (1) Reiss-Husson, F.; Luzzati, V. J. Phys. Chem. 1964, 68,3904. (2)Staples, E.J.;Tiddy, G. T. J. Chem. SOC.,Faraday Trans. 1 1978, 74,2530.Larsen, J. W.; Magid, L. J.; Payton, V. Tetrahedron Lett. 1973, 29,2663. (3) Ulmius, J.; Wennerstrbm, H.; Johanson, L. B.; Lindblom, G.; Gravsholt, S. J.Phys. Chem. 1979,83, 2232. (4)Anacker, E.W.; Ghose, H. M. J. Phys. Chem. 1963, 67,1713;J. Am. Chem. SOC.1968, 90,3161. Debye, P.;Anacker, E. W. J. Phys. Colloid Chem. 1951, 55, 644.
0 1983 American Chemical Society 0022-3654/83/2087-3541$01.5Q/~
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The Journal of Physical Chemistty, Vol. 87,No. 18, 1983
result is sometimes impressive and aggregation numbers 1 or 2 orders of magnitude larger than those of the initial globular micelles have been estimated on the basis of the experimental r e s ~ l t s . ~ + ~ ~ J ~ J ’ The evidence for such apparently huge micelles, which are fluctuating equilibrium structures, in stable monophasic clear solution is indeed fascinating; and a large number of theoretical as well as experimental studies have been devoted to this subject in the past as well as in recent years. Among the various procedures which are able to induce the shape transition, one is especially convenient: namely, the one where the transition is induced in dilute amphiphile aqueous solutions by the addition of large amounts of some inorganic salts. In such situations, the rare micelles are separated by large mean distances while the long-range electrostatic repulsions are screened out over short distances by the high concentration of the small ions. Neighboring micelles behave independently and the micellar solution can be said to be “ideal”. The study of “ideal solutions” is convenient from both theoretical and experimental points of view. First, simple models exist which relate some macroscopic properties of the solution like its viscosity or its turbidity to the morphological characteristics of the micelles as long as the interactions between micelles can be neglected. Secondly, in ideal solutions, the equations describing the multiple equilibrium between the micelles of various aggregation numbers and the free monomers in solution take the well-known form13
XN = clNe x p N ( p o l - C ~ ~ N ) / ~ B T ] I (1) where pol and F~~ are the standard chemical potentials of the free monomers in solution and of the monomers in the micelles of aggregation number N, c1 is the mole fraction of free monomers, and X , is the mole fraction of N-micelles. These equations together with the relation expressing the conservation of the total amount S (mole fraction in monomeric units) of the amphiphile in the solution
S = c I + CNXN N
(2)
entirely determine the size distribution X , of the micelles once one is given the series of standard chemical potentials VON
Another advantage of inducing the shape transition by addition of mineral salts is that the small inorganic ions remain exclusively soluble in the aqueous medium. Contrasting with what is expected in the case of organic additives, they do not incorporate inside the micelles; in a first approximation, we can consider that continuous variations of their concentration essentially result in continuous variations of the properties of the aqueous solvent. ( 5 ) Ikeda, S.; Ozeki, S.; Tsunoda, M. A. J. Colloid Interface Sci. 1980, 73, 27. Hayashi, S.; Ikeda, S. J . Phys. Chem. 1980, 84, 744. Ikeda, S.; Hayashi, S.; Imae, T. Ibid. 1981, 85, 106. (6) Mazer, N. A.; Carey, M. C.; Benedeck, G. B. J . Phys. Chem. 1976, 80, 1075; Micellization, Solubilization, Microemulsions [Proc. Int. Symp.] 1976, 1977, I , 359.
(7) Young, C. Y.; Missel, P. J.; Mazer, N. A.; Benedeck, G. B. J . Phys. Chem. 1978,82, 1375. (8) Missel, P. J.; Mazer, N. A.; Benedeck, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980,84, 1044. (9) Porte, G.; Appell, J.; Poggi, Y. J. Phys. Chem. 1980, 84, 3105. (10) Appell, J.; Porte, G. J . Colloid Interface Sci. 1981, 81, 85. (11) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (12) Appell, J.; Porte, G.; Poggi, Y. J . Colloid Interface Sci. 1982,87, 492. (13) Tanford, C. “The Hydrophobic Effect”; Wiley: New York, 1973.
These are the reasons that most of the literature about the shape and size transition in micellar solutions deals with ideal s i t ~ a t i o n s . ~ - l ~ Nevertheless, our knowledge remains unclear on important points. First, it appears that the determination of the shape of unknown microscopic objects of probably various sizes, randomly dispersed in a diluted isotropic solution, is not so obvious. The usual procedure is to measure independently several physical properties of the micellar solutions (intensity, angular dissymmetry, and spectral distribution of the scattered light, viscosity, etc.) and to compare their correlative variations with the predictions of different geometrical models for the shape of the micelles. Generally, the geometrical models which are considered as possible candidates to describe the micellar shape are spheroids (sphere, oblate, or prolate ellipsoids). In fact, rather good fits are often obtained with prolate m~dels.~-’J~ And at the present time it is generally admitted that large micelles of simple ionic amphiphiles are rodlike: the shape “transition” is often referred to as the ”sphere-to-rod tran~ition”.’~ However, the above-mentioned fitting procedures necessarily imply that the different geometrical candidates are a priori quantitatively specified in order to produce quantitative predictions to be compared to the experimental results. This sometimes leads to embarrassing ambiguities: one easily finds in the literature examples where a given set of experimental data can be fitted with equal success by models which are differently specified (the light scattering data of Young et al.’ on SDS + NaCl fitted with a monodisperse rigid-rod model’ or a flexible polydisperse rod model; l5 the viscosity data of Kushner et d.16 on CTABr NaBr treated with the flexible-pearl-necklace model by Stigterl’ or with the rigid-rod model by Nagarajan18). These few examples illustrate how difficult it is, at the present time, to obtain an unambiguous description of the micellar shape on the basis of the interpretation of the experimental results. Moreover, there is no a priori reason to preclude the possible coexistence of micelles of various shapes in solution. This renders the analysis of experimental results even more ambiguous. These difficulties have made even more imperative the necessity of a satisfying theory for the shape and size transition. At the present time one can distinguish in the literature between two different approaches. In the first of these, one tries to estimate separately the contribution of each component of the delicate balance of opposing forces which is known to rule the micellization process. This procedure had some success in predicting the order of magnitude of the cmc of various amphiphiles under various conditions. It was extended with the aim of predicting which shape of lower free energy a micelle should adopt once it is too large to accommodate the spherical shape.lg Such theory often predicts that the shape of large micelles should be oblate (disklike) rather than prolate (rodlike),a conclusion
+
(14) Nagarajan, R.; Khalid, S. M.; Hammond, S. Colloids Surf. 1982, 4, 147-62.
(15) Mazer, N. A. In “Dynamic Laser Scattering: Applications of Photon Correlation Spectroscopy”; Pecora, R., Ed.; Plenum Press: New York, 1981. (16) Kushner, L. M.; Hubbard, W. D.; Parker, R. A. J . Res. Natl. Bur. Std. (US.) 1957, 59, 113. (17) Stigter, D. J . Phys. Chem. 1966, 70, 1323. (18) Nagarajana, R. “Are Large Micelles Rigid or Flexible? A Reinterpretation of Viscosity Data for Micellar Solutions”, to be submitted for-publication. (19) Tanford, C. J. Phys. Chem. 1974, 78, 2469. See also: Leibner, J. E.; Jacobus, J. J . Phys. Chem. 1977,81, 130.
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
Giant Micelles In Ideal Solutions
which is generally in disagreement with the experimental results when very large aggregates are present in solution. The other approach is much more phenomenological: it is based on the simple intuitive idea that, in the limit of large aggregation numbers N, the standard free energy of the micelle necessarily takes a simple asymptotic form. The first one to propose this type of approach was Mukerjee in 1972 the basic idea is that the contribution, to the standard free energy of a N-micelle, of an extra added monomer becomes independent of N when N is sufficiently large. In this frame, the standard free energy GoN of the N-micelle becomes necessarily linear in N at large N a n d can be expressed according to the convenient notation of Israelachvili et alez1in the form G o N = NpoN = Np, kTa (3)
+
The consequences of this form for GoN are well-known: the micelles grow to large sizes when the excess term (kTa) in eq 3 is large enough; the mean aggregation number (N) increases linearly with the square root of the soap concentration; and the polydispersity is large when (N) is large. These predictions actually found extensive experimental support during the past 10 y e a r ~ . ~ . ~ J l J ~ It should be noted at this point that the asymptotic form for GoN in eq 3 has a wide range of generality; it may be stated each time that one deals with aggregates with unidimensional growth driven by short-range forces. One can think of the equilibrium polymerization of sulfur in the pure liquid phase (Tobolsky and Eisenberg, 19591%or in binary solution (Scott, 1965Iz3and also the reversible polymerization of proteins (Oosawa and Asakura, 1975)" as illustrative examples. A more rigorous approach to these phenomena has been recently developed in a series of papers by Wheeler, Kennedy, and PfeutyZ5and Wheeler and Pfeuty: 26 the equilibrium polymerization is described by the n 0 limit of the n-vector model of magnetism in a small magnetic field. This point of view emphasizes the analogy between unidimensional equilibrium polymerization and critical phenomena. The connection of relation 3 with unidimensional growth was however missed in the paper of Mukerjee.20 In fact, the excess kTa term accounts for end effects: it represents the excess standard free energy in the ends of the micelle where the environment of one constituting monomer is different from what it is everywhere else in the micelle. This point was recognized by Israelachvili et al. in 1976.21 They underlined that this excess term is independent of N only in the case of unidimensional cylindrical micelles where the number of monomers standing in the ends is independent of N. If the micelles are rigid disks, the number of monomers which are at the border of the disk indeed increases linearly with the square root of the total aggregation number N,in this case G o N would rather take the form
-
CON = N p O N = Npoa + MI2kTa
(4)
(20) Mukerjee, P. J. Phys. Chem. 1972, 79, 565. (21) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Trans. Faraday SOC.1976, 72, 1525. 1959,81, 780; (22) Tobolsky, A. V.;Eisenberg, A. J.Am. Chem. SOC. J. Am. Chem. SOC. 1960, 82, 289; J. Colloid Sci. 1962, 17, 49. (23) Scott, R. L. J. Phys. Chem. 1965, 69, 261. Scott, R. L. In 'Elemental Sulfur";Meyer, B., Ed.; Wiley: New York, 1965; Chapter 17, p 237. (24) Oosawa, F.;Aeakura, S. In 'Molecular Biology, an International Series of Monographs and Textbooks"; Academic Pres: New York, 1975. (25) Wheeler, J. C.; Kennedy, S. J.; Pfeuty, P. Phys. Rev. Lett. 1980, 49, 1748.
(26) Wheeler, J. C.; Pfeuty, P. Phys. Rev. Lett. 1981, 46, 1409; Phys. Reu. A 1981,24, 1050.
3543
Introducing eq 4 into the classical multiple equilibrium equations 1 and 2 (which stand in the case of ideal solutions) produces very different conclusions compared to what was predicted with eq 3 for the prolate shape: an abrupt transition driving to the formation of an infinitely large aggregate in equilibrium with small globular micelles should occur when the soap concentration exceeds a critical value which depends on a. The same analysis of the bidimensional growth leading to the prediction of an abrupt transition was previously derived by Oosawa (1975) dealing with the polymerization of proteins.24 The predictions are therefore very different depending on whether the large micelles grow along one dimension (prolate shapes) or two dimensions (oblate shapes). And one may think of the variations of the micellar mean size with the soap concentration as a sensitive criterion in order to determine the shape of the large micelles. However, one should wonder about the stability of such very large aggregates with respect to thermally induced fluctuations in curvature. Israelachvili et al.21J7and Mitchell and Ninham28discussed the bidimensional case to some extent. They showed that the bilayered structure sometimes opposes only weak resistance to local bending (spontaneous curvature can even be expected in some cases). In such a situation they arguez7that, since in a closed bilayer the energetically unfavorable rim regions are eliminated at a finite, rather than infinite, aggregation number, which is entropically favored, spontaneous formation of spherical vesicles may occur. This is indeed in agreement with much experimental evidence. However such an argument, if it is completely self-consistent, should apply with equal success to the unidimensional case. Let us start with a large rodlike micelle and bend it until a closed ring with same aggregation number N is formed. This process indeed eliminates the unfavorable ends; the corresponding energetic gain per micelle is typically 20-30kF1920while the elastic energy spent to bend the cylinder decreases roughly as 1/N. As a consequence the formation of closed rings should inhibit the growth of long unidimensional micelles at some early stage. Actually, this prediction is in definite contradiction with the experimental evidence reported a b ~ v e . ~ * ~And J l Jthe ~ problem is serious. Actually from the crudest common-sense point of view, complete stiffness for giant micelles with liquidlike cores is hardly conceivable. As mentioned above, many experimental s t ~ d i e s ~ ~actually ~ J ~ J ~suggest J ~ flexibility. The recent experimental work in ref 9 is especially convincing: the evidence of flexibility is provided via a qualitative comparison of magnetic birefringence and quasi-elastic light scattering measurements which does not require the specification of detailed geometrical models. Moreover, it could have been derived without making the a priori assumption for the rodlike shape for the micelles; qualitatively similar variations of the magnetic birefringence together with those of the hydrodynamic radius would also unambiguously indicate flexibility in the case of bidimensional giant micelles.30 Since, clearly the "zeroth-order approximationn21is incapable of providing a correct explanation for the apparently indefinite growth of very large rodlike micelles once some flexibility is allowed, there is an instant need for a more rigorous approach. (27) Israelachvili, J. N.; Marcelja, S.;Horn, R. G. Q. Reu. Biophys. 1980, 13, 121.
(28) Mitchell, D. J.; Minham, B. W. J. Chem. SOC.,Faraday Trans. 2 1981, 77,601.
(30) De Gennes, P.G.; Taupin, C. J. Phys. Chem. 1982,86, 2294.
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The Journal of Physical Chemistry, Vol. 87,No. 18, 1983
In our opinion, such an approach should fulfill the following requirements: (i) no a priori selection should be made between the different shapes which are possible candidates for giant micelles (micelles of different shape may coexist in solution); (ii) as far as possible the calculations should not involve specific features (such as optimum surface per polar head, detailed balance between opposing forces) in order to preserve some predicting power; (iii) the configurational entropy associated with the variously bent conformations for the giant flexible micelles should be accounted for in the model (the recent Talmon and Pragger31model for microemulsions has demonstrated the importance of such a contribution when very large or even infinite objects are involved). We therefore proceed as follows. The possible shapes for giant micelles are not so many. They can be classified according to their dimensionality (one or two at most, since empty regions in micelles are prohibited) and connectivity (opened or closed shapes); in simpler words we may have long cylinders, closed rings, large opened bilayers, and closed vesicles. These are the only shapes where the large majority of the constituting monomers experience a homogeneous environment. This general classificationforgets about many unimportant specific variations: the cross section of unidimensional micelles may be more or less ellipsoidal; the local structure may be more or less flexible; etc. For each type of shape we classically introduce an asymptotic form of Go$ G o N = N p o= ~ N / . L o b d k + G o b ( N ) - kT In (z(N)) (5) where p o b d k is determined by the homogeneous environment experienced by the large majority of the monomers. /.Lobulk only depends on the structure (bilayer or cylinder) of the homogeneous part of the micelle; it does not depend on N. Gob(N) is the excess standard free energy at the border or in the ends of the micelles. And Z ( N ) is the partition function associated with the ensemble of bent conformations for the flexible micelle. Such partitioning of the micelle into a favorable homogeneous “bulk” and unfavorable “border” or “ends” becomes valid when the size of the micelle is larger than the range of the interaction responsible for the aggregation process (comprising the forces arising from geometrical packing constraints). The problem in the following sections will thus be to derive the variations of Gob(N) and of Z ( N ) with N for each of the four possible shapes for the giant micelles. The second step will then be to express the equilibrium of the micelles of all shapes and sizes according to relations 1and 2. The following remark will help to simplify a priori this second step: p o b u l k is the same for opened- and closedshape micelles of the same dimensionality. Their coexistence in the same solution cannot be a priori excluded. But except under extremely exceptional circumstances, the bulk standard chemical potentials /.Lobulk of the monomers in cylindrical or in bilayered local structures differ somehow. Because of the increasing predominant (Npob,&) bulk contribution to GON, the type of giant micelle which corresponds to the smaller /.Lobulk will be alone present in measurable proportions in one given solution. We can, therefore, consider separately the cases of unidimensional and bidimensional giant micelles in what follows. Doing so in the following sections, we actually are able to elucidate the above-discussed apparent contradiction. We show that the reason that the formation of closed rings does not inhibit immediately the growth of giant unidi(31) Talmon, Y.; Prager, S. J. Chen. Phys. 1978,69, 2984.
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mensional micelles lies in the tremendous reduction of the number of possible bent conformations when a long cylinder is closed as to form a ring. We also show in the same consistent way that, in the case of bilayered structure, the abrupt transition toward the formation of an infinite aggregate will never occur in ideal solutions; the spontaneous formation of closed vesicles will certainly inhibit the growth of such huge bilayers however large the intricic rigidity of the structure. But prior to the detailed investigations, two remarks are required. First, the present analysis is for a pure onecomponent surfactant system. In the case of multicomponent surfactant systems the local composition inside the micelle may change from place to place, especially in the ends or at the border. Very different behavior is therefore expected in the case of mixed micelles. Second, the present approach is phenomenological. It cannot explain why a particular surfactant molecule eventually prefers to assemble in cylindrical or bilayered local structure rather than in spherical structure. More specific d e s c r i p t i o n ~ ~involving ~ J ~ 3 packing constraints and the balance of opposing forces are needed to answer such questions. But once the local structure is prescribed by the specific constraints, the analysis aims to select in a consistent way which shape compatible with such local structure will predominate in the solution. Thus, phenomenological approaches and specific descriptions usefully complement each other when giant micelles are involved.
Case of Unidimensional Micelles Asymptotic Form for G O N . In the case of more or less flexible rodlike opened micelles, the number of monomers which stand in the ends does not depend on the aggregation number; neither does it depend on the various conformations of the micelle. Gb(N) is constant at fixed temperature and added salt concentration: Gb(N)
=
Gb
(6)
In order to evaluate the bending partition function Z(N), one must quantify in some way the flexibility of the micelle. This is conveniently achieved by introducing the persistence length 1,: 9,32 the local orientation of the micellar axis at a point of contour coordinate l being defined by a unit vector t(l) tangent to it, 1, is defined by the relation
(Z(l).Z(l+lp)) = l / e
(7)
where e is the natural logarithm basis and the brackets stand for thermal average. The completely stiff limit indeed corresponds to 1, m and the opposite completely flexible limit corresponds to 1, 0. In the particular case where the micellar rigidity arises from elastic resistance to local bending, it can be shown that32
- -
1, = a / k B T
(8)
where a is the elastic bending modulus and T the absolute temperature. In a first approximation, we neglect the excluded volume effects inside one given micelle: the fluctuations in the curvature of two adjacent parts AB and BC of the AC micelle will remain statistically independent and we have for the corresponding bending partition functions the relation Z(AB)-Z(BC) = Z(AC) (9) (32) Lifshitz, E. M.; Pitaevskii, L. P. “Landau and Lifshitz Course of Theoretical Physics”, 3rd ed.; Vol. 5 , part I, pp 396-400. Kratky, 0.; Porod, G.R e d . Trau. Chin. Pays-Bus 1949, 68, 1106.
Giant Micelles in Ideal Solutions
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983 3545
The asymptotic expression for Z(N) then necessarily takes the form
Z(N) =
(aN
(10) where the quantity Z is independent of N but depends on the micellar flexibility and thus on 1,. If we call Q(N)the number of bent configurations of the N micelle, we indeed also have Q (AB)*Q(BC) = Q( AC) (11)
Q(N,O)zz N-5J2n;312zN
(20)
In order to derive Z(N) from Q(N,O)we use the general relation between free energy G, internal energy E , and entropy S
G=E-TS
(21)
which simply gives
and thus Q(N)also has the form Q(N) =
WN
(12)
It can be otherwise shown that the ratio z / Z stands in the range 1 < z/Z < e. (In fact, this ratio does not directly depend on 1, but rather on some smallest cutoff wavelength for bending sinusoidal modes for the micelle). Introducing eq 6 and 10 into the general asymptotic expression 5 and putting pow = pobulk- kT In Z (13) Gb
= kTa
(14)
we obtain GoN
= NpON = Npow
+ kTa
Q ( N , x ) / Q ( N= ) x3/R,3 (15) As long as the excluded volume interaction can be neglected, Re depends on the micellar size by the simple relation32 (16)
We introduce n,, the number of monomers in a length of micelle equal to 21,: 21, = n g . In order to obtain a closed loop the end-to-end distance x must first be driven down to an order of magnitude comparable to the length occupied by a monomer on a micelle: x zz L / N
(17)
We get
Q ( N , x = L / N ) = N-3/2n;3/2Q(N)
Z ( N ) = A(l,)n;3/2N-5/2ZN
(18)
and finally
Q ( N , x = L / N ) r N-3/2n;3/2(z)N (19) Now, once the loop is closed, it does not keep any trace of the particular place where the ends have welded together; the number Q(N,x=O)of configurations for the closed toroid is thus some N times smaller than Q(N,L/N). We therefore admit
(23)
=
with A(1,) 1. (A(1) slightly depends on 1, as long as (l/P)(d(n,(l)3/2)/dp1is not zero because 1, typically varies like l / K J Z in relation 23 for closed rings has the same value as for the opened micelles in eq 10. We express now GoN(rings). Putting
(3)
an expression which is formally identical with what was obtained by Israelachvili et al.21neglecting all about bent conformations. Let us now consider the case of a closed-loop-like unidimensional micelle. Gob(N) is indeed zero. The problem remains to evaluate Z(N). One can easily imagine that closing an initially opened long flexible cylinder enormously reduces the number of accessible bent conformations; one of the ends which was initially randomly located in a volume of the order of 4uR,3/3 (where R,2 is the quadratic average of the end-to-end distance) must be driven into a small volume 4ax3/3 in the immediate vicinity (at a small distance x ) of the other end. In a first approximation, one may admit that the number of configurations Q(N,x)with the end-to-end distance smaller than x and the total number of configurations Q(N)are related by
Re = (21&)'/2
where 0 = l/(kBT). Careful attention paid to the differential relation 22 shows that Z ( N ) has a form analogous to that of Q(N,O):
= p0bu& - kT In (2)
PO,.
(24)
-kT In KO= -kT In ( A ( l , ) n ~ ~ / ~ ) (25) it comes out that GoN
= Npom- kT In
(KO) + Y2kTIn (N)
(26)
In this expression we have neglected the elastic energy necessary to bend the rod so as to obtain the ring. In the case of a very rigid structure it may be of some importance. However, it decreases as 1 / N . Thus, its effect will be essentially to prescribe some minimum size for a closed ring. Size and Shape Distribution of Unidimensional Micelles. We now introduce these expressions for G o N into the multiple equilibrium classical relation 1 in order to determine the composition of an actual solution. Since pow is the same for opened cylinders as well as for closed rings, we must allow for the coexistence of both shapes in solution for any values of N . We obtain for the respective populations XN(opened) and XN(closed): Xdopened) = (cl exp((pol- p o m ) / k T ) ) N exp(-a)
(27)
XN(Cl0Sed) =
(28)
(C1
eXp((pOl- p 0 m ) / k T ) ) N & N - 5 / 2
The relation expressing the conservation of the total mass of amphiphile S is written m
m
s = C1 -t n=no NXN(opened) + n=nd NXN(C10Sed)
(29)
where no and n,,' are the aggregation number of minimum-size micelles of each type. no is thus the aggregation number of small spherical micelles (typically no = lo2) while we admit that the length of the minimum-size torus is about 2 or 3 times the persistence length (n,,' = few n J . If
Y = c1 eXp((pol - P o m ) / k T )
(30)
it comes out that m
S - c1 = exp(-a) N=no
NYN + KO C NW3I2YN
(31)
N=%,'
The size and shape distribution for the micelles is thus entirely determined by the values of the total amphiphile
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TABLE I : Mole Fraction (Monomeric Units) of Opened Unidimensional Micelles (A), Closed Toroids ( B ) ,and Free M o n o m e r s (C,) as a F u n c t i o n of Sa
S
a.
1.62 1.75 1.75 1.77 1.78 1.79 1.79 1.8
1.62X 2.16 X 3.71 X 6.6 X 1.6 x 10-5 3.6 x 10-5 2.2 x 1 0 - 4 2 x 10-3
Y
B
A
106C,
01
-0 4.1 x 10-7 1.9 x 10-6 4.8 x 1.44 x 10-5 3.41 x 10-5 2.13 x 10-4 2 x 10-3
(N)opened
= 20
-0 10-2, rr 0 10-16 1 0 10-14
lo-'*
2
0 0 0
10-10 1 9.2 x 10-9 1.4x 10-7
0.9 0.97 0.98 0.985 0.99 0.993 0.997 0.999
130 150 165 200 240 430 1100
b. c y = 30' 1.6 x 10-9 8 x 1O-l1 0.993 240 9.9 x 10-9 9.2x 10-9 0.997 430 9.3 x 1.4 x 10-7 0.999 1100 1.04 X 5.0 x 10-7 0.9997 3300 2.33 x 6.31 X lo-' 0.9998 5000 9.3 X 10-6 8.44 x 10-7 0,9999 10000 (cmc = a e x p ( ( p o m- p n ,)/kT) = 1.8 X M). n o is t a k e n equal t o 100,nc to 1200,a n d no' to 1000. C o m p u t a In a tion IS according to relation 31. (Abopenedis t h e m e a n aggregation n u m b e r of t h e large unidimensional micelles. small range f o r S close to t h e c m c (S = 1.813 X a n d 2.03 X t h e mole fraction of t o r o i d s is of t h e s a m e order of magnitude a s t h e m o l e fraction of o p e n e d micelles. A b e c o m e s again largely d o m i n a n t when S increases m o r e than typically 10(c m c ) .
1.789x 1.813 X 2.03 X 3.33 x 4.76 X 1.2 x 10-5
1.787 1.795 1.799 1.799 1.799 1.799
mole fraction S and of the end excess a (ais expected to depend sensitively on the temperature and on the nature and mole fraction of the added saltsJ1). c1 is the mole fraction of free monomers: it remains of the order of magnitude of the cmc; and the case of interest corresponds to situations with S >> cmc e cl. As long as a remains small (typically S exp(a) > n:), the convergence properties of both series in eq 31 will discriminate between which type of micelle will predominate in the solution depending on the value of S. The second series, which corresponds to the toroids, converges to a finite value sf for Y = 1. On the other hand, the fist series corresponding to the opened cylinders converges for Y < 1 but diverges for Y = 1. S being finite, Y remains strictly smaller than 1; therefore, the mole fraction S, of amphiphile present in the form of rings remains strictly bounded by -
(35) Taking reasonably n, z 1200 (which corresponds to the results on cylindrical micelles of cetylpyridinium bromide which have been studied in ref 9 and 12) and n,,' z 1000 1.5 X lo4 in mole fraction, that is one evaluates to say, M. The partial concentration of toroids remains smaller than the cmc (typically 10-3-10-4 M for
-
cetylpyridinium bromide micelles with added NaBr). In practice, where the total amphiphile concentration typically lies in the range 10-3-10-1 M, one therefore expects then that toroids will remain unobserved. As an example, we have used relation 31 to compute in Table Ia the mole fraction of free monomers, opened-shaped micelles, and rings as functions of S for a = 20 and for a cmc of M (1.8 X lo* mole fraction). These values typically correspond to situations where cetylpyridinium bromide micelles grow in 0.2 M NaBr aqueous solution, at 30 OC.ll Actually in this situation the mole fraction of toroids is very small compared to that of opened micelles for all values of S. This is not always true, especially when a grows much larger; Table Ib is computed for the same value of the cmc but a is set equal to 30. Considering the experimental results in ref 11and 33, perhaps such a high value for a can be experimentally reached with a high concentration of added salt (NaBr) >> 0.8 M, (NaC103)>> 0.2 M) in a micellar solution of cetylpyridinium bromide. In such a plausible situation, there is a small range for S around the cmc, where toroids are predominant. This predominance of closed rings in the immediate vicinity of the cmc increases when a increases. At the present time values for (Y very much larger than 30 seem difficult to obtain. But the reversible polymerization of sulfur is an equivalent example of unidimensional growth. Keeping the present notation, at the transition temperature between the fluid and the viscous phase, sulfur realizes a very high value of a. And actually the detailed behavior of the specific heat at the transition temperature (equivalent to the cmc in a certain way) is indeed strongly affected by the presence of a broad distribution of closed rings.26 Any way the predominance of opened cylinders will quickly be restored as soon as S becomes significantly larger than the cmc. Although our estimation of Sf," is very crude, we conclude that, except in the above-discussed special circumstances, large unidimensional micelles essentially exist in the opened shape and that their size sensitively grows with S. The reason for the predominance of opened cylinders is essentially statistical: opened shapes have a much (33) Porte, G.; Appell, J., presented at the International Symposium on Surfactants in Solution, Lund, Sweden, 1982.
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
Giant Micelles in Ideal Solutions
larger number of bent conformations than closed rings. This statistical average is so large that it can overcome an energetic disadvantage of some 2Ck30kT per micelle!!! This is indeed a very surprising result. When S >> Sf,", we can approximately write m
S - c1 - Sf," r exp(-cu) C NYN N=n
(36)
The giant opened micelles grow when LY and/or S increases and their mean aggregation number ( N ) asymptotically scales according to the classical law predicted by Mukerjee and others:
( N ) z ((8- C1- Sflmm)e x p ( 4 V 2
(37)
This simple treatment of the unidimensional case rests on the same assumptions as the calculations of Harris (1970) for the molecular composition of liquid sulfur." We in particular neglect the intramicellar excluded volume interactions. In fact, the correction associated with this effect cannot be introduced simply in the standard free energy G o N of the individual N-micelle; it becomes significant for such large micelles that intermicellar interactions also become important even at very low concentrations. In other words, one cannot distinguish between the intramicellar and intermicellar contributions of the excluded volume effect to the total free energy of the solution. In this situation the only consistent starting point is the partition functions of the whole solution!! And the problem is much more difficult. Perhaps the recent methods developed by Wheeler and Pfeuty26will yield a consistent treatment with excluded volume interaction. Case of Bidimensional Micelles A similar study of the case of bidimensional micelles is actually much more difficult. A first problem arises when one tries to evaluate the number of monomers which stand at the border of the N-micelle; this number is not a univocal function of the only aggregation number N . Even if the micelle is everywhere plane, it may have a circular or more or less ellipsoidal (or even more exotic) contour and the length of its contour as well as the number of monomers which stands in it will fluctuate greatly. If, on the contrary, the border of the micelle is set circular but if the micelle is allowed to present a homogeneous curvature more and more pronounced, its border will decrease continuously until a closed spherical vesicle is formed. Therefore the excess standard free energy at the border of a bidimensional N-micelle depends both on the shape of the border and also on the various bent conformations for the micelle. The second problem arises when one intends to count the ensemble of bent configurations of a bidimensional object whose size is much larger than its persistence length the number of configurations will be thoroughly affected by the impossibility of the micelle cutting through itself. Within these conditions, the evaluation of Q(N)and therefore of Z ( N ) seems extremely difficult and to our knowledge it has not been done at the present time. Nevertheless, in order to derive some firm conclusions, one can proceed as follows: (1)Let us first consider the particular (somewhat academic) case where the elastic resistance to bending of the micelle is large enough so that it remains stiff even for very large sizes (despite the thermal fluctuations of curvature). (This is the rigid-plate limit; the persistence length is very large.) (34) Harris, R.E.J. Phys. Chem. 1970, 74,3102.
3547
(2) We then show that, although this situation is the most unfavorable to the formation of closed vesicles, the bidimensional opened micelles become unstable compared to the vesicles. Actually the closing process takes place much before the micelle reaches such sizes that it becomes unstable with respect to the thermal fluctuations of curvature. (3) The conclusion that bidimensional micelles appear in ideal solution, essentially as closed vesicles, is then even more valid when the elastic resistance to curvature is weak. We recall, in the Conclusion, within what quite restrictive conditions this statement actually holds. We thus start with the assumption that the bidimensional structure, consisting of the double layer of amphiphilic molecules, presents a resistance to bending such as is considered by Helfrich in ref 35; the local density of curvature energy ilF/dS depends on the radius of curvature R according to
dF = (K/2R2) dS where dS is the area element and K is the elastic modulus. (We do not consider here the case where spontaneous curvature occurs.) De Gennes and Taupin have recently3" estimated the quadratic average ( d 2 ( l ) ) of the misorientation thermally induced between the normal to the surface at two points separated by a distance 1: (d2(l)) =
( W " / 4 In U/a)
(38)
where a is some cutoff wavelength of the order of the size of one constituting monomer. Therefore, the misorientation ( d 2 ( l ) ) , in the case of a bidimensional object, increases logarithmically with the distance 1; this is extraordinarily slow compared to what is expected in the unidimensional case (where ( d 2 ( l ) ) increases linearly with 1; see,.for instance, ref 23). Correlatively the persistence length tP which is deduced from the definition relation (cos d(S,)) = l / e
(39)
depends very sensitively (exponentially) on K
lp= 8 ~ X P ( ~ T K / ~ B T )
(40)
To illustrate this point, let us suppose that K increases in the range 10-14-10-13erg; 5, increases from a few 10' to lo6 A. Therefore, 1order of magnitude variation of K around a physically reasonable value erg typically corresponds to the splay elastic constant for one monomolecular layer of a thermotropic smectic liquid crystal) completely transforms the situation; we start with a completely flexible micelle (4smaller than the thickness of the bilayer) and reach a situation where the structure remains stiff over quasi-macroscopic distances. This evaluation indicates that, in the bidimensional case, the rigid limit does not necessarily correspond to physically absurd values of K. Within this limit, the bidimensional micelle is submitted to thermally induced bending fluctuations which remain small; the planelike shape of the micelle remains always recognizable. We are interested in situations where the boundary excess free energy is large enough to promote the growing size of the micelle; we may thus admit that this excess thoroughly favors a circular border of minimum (35) Helfrich, W.Phys. Lett. A 1973, 43,409; 2.Naturforsch. 1973, 28, 693.
(36) Mazer, N.A.;Krasnick, R. F.; Carey, M. C.; Benedek, J. B. In 'Micellization, Solubilization and Microemulsions"; Mittal, K., Ed.; Plenum Press: New York, 1977.
3548
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
length (with indeed some small thermal fluctuations around this form). Broadly speaking, the micelle has the shape of a circular rigid disk. In agreement with Israelachvili et a1.21we then write Gob(RT) z
kTC"l2
(41)
In the stiff limit, the entropy associated with bending strains remains small. We neglect thus its contribution to GoN,which takes the very simple form G'N
+
= NpoN = Npom k T a N j 2
(4)
Israelachvili et have investigated the case where micelles of this type alone would be present in the ideal solution. Putting again
Y = ~1 exp((wo1- p o m ) / k n
(30)
we write the equilibrium relations
XN = YN exp(-aW*)
(42)
The mass conservation relation becomes m
S
- c1
= C NYN exp(-aN1/2) N=no
(43)
The series in eq 43 converges for Y = 1 up to a maximum value = C N exp(-aN/2) N=no
(44)
The series in eq 43 and 44 have no simple analytical form. One can, however, make some estimations using a computer. Giving to n,a typical value, 100, we see that, as long as a remains smaller than a number of the order of 1, s, is larger than 1; there is no physically reasonable upper limit for S - c1 in this case. We then also see that for any reasonable value for S - c1 (S - c1 < l), ( N ) remains in the range 100 C N C 125. Thereare no large bidimensional micelles in this case. When a is notably larger than 1, S - c1 decreases very fast with a as indicated in Table 11. The maximum values for ( N ) , (N),=, defined by
(N)"
sibility that the formation of a closed vesicle becomes a preferable alternative much before the size of the disk becomes so large. Such a closed vesicle has no border. But its size being much smaller than .$, its standard free energy contains a term related to its curvature which cannot be neglected: 35
G o b ( N = (1/2)KS(1/R2) d S = 27~K S
= C N exp(-aN1/2)/Cexp(-aN1/2) (45)
remains, here again, of the order of no. Thus, as long as S - c1 increases in the range 0 C S - c1 C S,, ( N ) increases slowly in the narrow range no C ( N ) C (N),=; there are still no large bidimensional micelles in this case. When the total quantity of amphiphile reaches a value such that S - c1 is equal to s,, we then have pol k T In c1 = pom (46)
+
We have equilibrium between the free monomers and a giant micelle of infinite size. On the basis of these considerations, Israelachvili et al.21 suggested that the growth of bidimensional micelles should yield the separation of an infinite aggregate. However, we must keep in mind that, although large, E, remains finite; before it becomes macroscopic, the size of the giant aggregate will overpass 6,. The thermal bending fluctuations will then make the initial disk unrecognizable, and the asymptotic form of G o N will be completely transformed. Very probably the infinite growth of the giant aggregate and thus the occurrence of a true transition will be inhibited at this point. However, such thermal fluctuations have a significant influence only when the size of the disk becomes of the order of E,. In fact, we cannot a priori exclude the pos-
(47)
We neglect again (rigid limit) the configurational entropy and we get for G o N GoN(vesicle)= NpoN = Npom+ k T y
(48)
where = 2rK/kBT
(49)
Giving to y the value 12 (which corresponds to K r erg) and assuming that a is of the order of unity, we see, comparing GoN(vesicle)(eq 48) with GoN(disk)(eq 4) that the closed vesicle is very quickly prefereable (as soon as N is of the order of 102-103),thus, long before the size of the micelles becomes of the order of E,. Therefore, even in the rigid limit, one must consider the eventual coexistence of opened- and closed-shaped bidimensional micelles in the solution. The form of GoNfor the vesicles implies XM(vesicle) = YN exp(-y)
m
, ,s
Porte
(50)
The mass conservation relation becomes
s - c1 = N=no 5 NYN exp(-aN/2) + exp(-y) N=n,' 5
NYN (51)
The fiist series in eq 51, as we have seen before, converges for Y = 1 up .to , ,s The second series converges only if Y is strictly smaller than 1 and increases infinitely when Y increases close to 1. Equation 51 has thus the same structure as eq 31 and can be discussed in the same manner: (1) if a is smaller than a number of the order of 1,the only present micelles me the small spherical micelles. (2) If a is large, we consider two extreme cases: (a) if the total amount of amphiphile is small (S - c1 ,,,s all the excess amphiphile forms closed vesicles. Therefore, large disklike micelles, as well as the transition to an infinite bilayer expected t o occur in some cases in ref 27, will never be observed. Both these shapes are unstable in ideal solutions; however large is the rigidity of the bidimensional structure, vesicles will spontaneously form when both the tendency of the monomers to aggregate into a bilayered local structure (a)and the overall concentration S are large enough. As it is well analyzed by Israelachvili et al. in ref 27, the size of the closed vesicles may depend on the excess elastic energy y due to curvature. We can approximately write m
( S - c1 - S,=) exp(y) = C NYN N=n,'
(52)
It is then easy to show that, as long as ( S - c1 - S ), exp(y) C nd2, the mean aggregation number of the vesicles remains of the order of nd. If, on the contrary, y increases to large values, one expects to observe large vesicles; in the limit of very large y values their mean aggregation number increases sensitively with the total amount of amphiphile in solution according to
(N)vesieles (6' - c1 - s m d exp(y)Il/*
(53)
Giant Mlcelles in
Ideal Solutions
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
TABLE 11: smm,Defined in Eq 44, and,,,CN, Defined in Eq 45, as Functions of 'a
a
w)max
Smax b
ff
0.8 1.2 1.8 2.2 2.8 no = 100.
1.26 (>1) 1.36 X l o - * 2.08 x 10-5 3.05 X lo-' 5.86 X Mole fraction.
131 119 111 109 107
Assuming that the minimum aggregation number for closed vesicles is typically 5000 (-100-A radius), the transition value for (S - c1 - smm) exp(y) is
y
+ In (S- c1 - smm)
N
2 In n,,'
N
17
(54)
If we set S in the typical mole fraction range 104-10-3, this gives for K a value of the order of
K = (ykBr)/27r
2
(1.7-2)
X
erg
(55)
Such a value very probably is larger than what can be expected for usual ionic amphiphiles. But it is not physically unrealistic; one can imagine some more exotic amphiphilic molecules involving a rigid part similar to the rigid part involved in smectic mesogenic molecules. It would probably be interesting to synthesize such amphiphilic molecules in order to verify this conjecture. At the end of this discussion of the limiting case of rigid bidimensional micelles, it appears that an elastic resistance to bending which is large enough to make negligible the confiiational entropy is actually incapable of preventing the closing of the disks and the formation of vesicles. In the opposite case (very flexible micelles) we thus do not expect that the configurational entropy is important enough to modify the predominance of closed vesicles. We can therefore conclude that the formation of vesicles is the general occurrence when bidimensional micelles grow in ideal solution.
Conclusion The two important starting conditions for the validity of our discussion are the following: the solution is ideal; the mean aggregation number is large. This second condition underlies the homogeneity assumption. In this frame p"N = G"N/N is a strictly decreasing function of N when N increases toward a. This last statement however fails in the case of mixed micelles or when the elongation is induced by organic additives; such additives can be incorporated into the micelle and can remain preferentially either in the bulk or at the boundary. Such inhomogeneous distribution of the additive in different parta of the micelle can relax the excess energy at the boundary; large disks may remain stable under these conditions (see, for instance, the case of mixed micelles of bile salts ref 28). The stabilization of the disks with respect to vesicles can also be achieved in the absence of additives as soon as intermicellar interactions become important (nonideality); at high enough amphiphilic concentration a local piling up of disks may counterbalance the boundary effects. Such a process can yield (but not necessarily) some long-range orientational order in the solution (lyotropic nematic or lamellar mesophase). Also, in ideal solution, an oblatelike shape, but of small size, might be stable if it corresponds to an absolute minimum for p o N The conditions for the validity of our discussion are thus quite restrictive. They however exactly correspond to the
3549
experimental classical situation where the shape transition is induced in a dilute solution of one given ionic amphiphde through the addition of large amounts (a few lo-' M) of some salts. When they are fulfilled, the obtained conclusions are summed up simply: (1)the possible shapes for giant micelles can be classified according to their dimensionality (one or two) and their either opened or closed character; (2) giant micelles of different dimensionality do not coexist in the same solution; (3) giant micelles, if they are unidimensional, are at usual concentrations M) predominantly of the opened type. Their size sensitively increases with the soap concentration; (4) if they are bidimensional they essentially exist in the closed vesicle shape. The growth of their size much over their minimum aggregation number (typically 5 X 103-104)will be observed only if the elasticity bending modulus K is larger than 2 X erg (a condition which is not expected to be fulfiied with usual amphiphiles). At the end of the present discussion, we have thus obtained the natural explanation for our embarrassing starting problem: the formation of energetically favored closed rings does not inhibit the growth of giant energetically disfavored rods because their statistical probability of formation is much lower than that of the rods. They can accommodate a too small number of discernible bent conformations. Such decisive importance of a statistical argument in the field of surfactants is not so common. And the present one deserves to be noticed. The conclusion that vesicles will systematically be favored in ideal solution when bidimensional local structure is involved is also an important result with respect to the current literature. The important point is that a phase whose dimensionality is smaller than that of the space where it grows (two-dimensional micelle in three-dimensional space) cannot lead to the formation of an infinite aggregate. The case of giant micelles is a particular situation where an intermediate approach between the oversimplistic pseudophase approximation and the rigorous statistical treatment (!!) is available. This is a mean field approximation. But it can be developed in a consistent way with no introduction of specific "ad hoc" assumptions, so that the present conclusions really have predictive power. They can thus help to elaborate experimental protocols in order to determine the shape of giant micelles in a given set of solutions. As an example, let us reconsider the experimental results on cetylpyridinium bromide with NaBr as added salt which are given in ref 9. First, quasi-elastic light scattering measurements indicated low diffusion coefficients (a few cm-l at about 40 "C) for the micelles in diluted (a few M) CPBr solutions as soon as the NaBr concentration was high enough (0.2-0.8 M). The diffusion coefficient markedly decreases with increasing CPBr concentration." Such results can have two explanations: (i) critical slowing down around a second kind of phase transition; 29 (ii) very large size for the micelles. The absence of any discontinuous effects when all pertinent parameters (temperature, salt concentration, amphiphile concentration) are widely varied precludes the critical explanation. Micelles are thus very large; the corresponding hydrodynamic radius being in the range 200-650 A according to our asymptotic discussion, they necessarily are either spaghetti-like very long cylinders or rigid spherical vesicles. Helfrich has shown that the magnetic birefringence of elastic vesicles, if it is not zero, necessarily increases with the vesicle size.35 The measured magnetic birefringence of the CPBr + NaBr solution being constant over the range where the hydrodynamic radius
3550
J. Phys. Chem. 1983, 87,3550-3557
of the micelles increases considerably, they certainly are not vesicles; therefore, they are unidimensional and flexible. The main features of the giant CPBr micelles are so unambiguously derived with no requirements for sophisticated fitting procedures.
Acknowledgment. I thank J. Appell and P. Pfeuty for interesting discussions. The paper benefitted from the criticisms of 0. Parodi. Registry No. Sulfur, 7704-34-9.
Photoprocesses on Colloldal Clay Systems. 2. Quenching Studies and the Effect of Surfactants on the Luminescent Propertles of Pyrene and Pyrene Derivatives Adsorbed on Clay Colloids R. A. DellaGuardla and J. K. Thomas’ Deparhnent of Chemlstry, University of Notre Dame, Notre Dame, Indiana 46556 (Recelved: November 17, 1982; In Final Form: February 8, 1983)
The cationic fluorescent probe [4-(1-pyrenyl)butyl]trimethylammonium bromide (PN+) is adsorbed by the colloidal particles of the clay minerals montmorillonite and kaolin. The emission spectrum, polarization of fluorescence measurements, and transient fluorescence decay characteristicsof PN+ are used to study the nature of its adsorbed state. Quaternary ammonium surfactants of varying hydrocarbon chain length cause a rearrangement of the PN’ molecules on the surface and decrease its interaction with the mineral surface. Quenching studies with nonionic and cationic molecules indicate that diffusion on the surface of montmorillonite is reduced below that observed in aqueous solution while the apparent rates obtained with kaolin particles are increased. Montmorillonite particles with a surfactant bilayer surrounding their surfaces are formed by the addition of an excess amount of surfactant to the colloid. The emission spectrum and steady-state quenching studies yield information on the location of pyrene on these particles, as well as on the nature of the colloidal particles.
Introduction There have been numerous studies over the past several years on the influence of interfacial phenomena on photochemical One group of systems studied, cationic, anionic, and neutral micelles, has received much attention due to their ability to solubilize hydrophobic molecules in their hydrocarbon-like core. Dramatic increases or decreases in photoinduced electron-transfer reactions have been reported and are a result of the micellar properties such as charge, size, and shape. Indeed, the charge on these assemblies and the resulting potential field are responsible for the charge separation of photochemically produced radicals. Charge separation, Le., inhibition of back electron transfer, is essential for any practical solar conversion system. Luminescence quenching techniques have been especially useful in characterizing the nature of many of these systems and explaining their catalytic effects. These ideas and techniques are presently being used in this laboratory to study the photochemistry that occurs on colloidal clay minerals. These colloids have been chosen for study because of their stability in aqueous media and their large cation-exchange capacity (cec). The two minerals used in this work were montmorillonite and kaolin. Both are aluminosilicates containing aluminum in octahedral configuration which shares oxygen atoms with silicon in tetrahedral configuration. Montmorillonite is referred to as a 2:l layered mineral because its aluminum shares oxygen atoms with silicon on either side of it. Then there occurs an expandable layer into which water, organic molecules,
or cations may be intercalated. The aluminosilicate sheet and expandable layer keep repeating throughout the clay structure in a periodic manner. Kaolin, on the other hand, is a nonexpandable 1:l layer mineral. There is a sharing of oxygen atoms between one silica sheet and one aluminum sheet in a continuous network that cannot be easily disrupted for the intercalation of ions or organic molecules. Hence, only the surface of kaolin particles participates in chemical reactions whereas both the surface and the internal layers are available in montmorillonite particles. The cation-exchange capacity of these two mineral appears as a periodic negative charge along their structure. It occurs due to the isomorphous replacement of aluminum for ferrous or magnesium ions in the octahedral layer or by the replacement of the silicon in the tetrahedral layer by either aluminum or ferric ions. The replacement of an atom of higher positive valence for one of lower valence results in a net negative charge. This excess of negative charge is balanced by the adsorption of cations on the layer surfaces. In the presence of water, charge-balancingcations may be exchanged with other cations available in solution. The expandable layer in montmorillonite and cationexchange properties of both minerals have been utilized to adsorb the fluorescent probe [4-(l-pyrenyl)butyl]trimethylammonium bromide (PN+). This has enabled a study of the probe’s interaction with the clay mineral surface, the effect of the addition of adding cationic surfactants to this surface on the PN*-clay binding, and the accessibility of PN+ on these colloids by various fluorescence quenching molecules added to the solution.
(1) Fendler, J. H.; Fendler, E. J. “Catalysis in Micellar and Micromolecular Systems”; Academic Press: New York, 1975. (2) Two, N. J.; Braun, A. M.; Griitzel, M. Angew. Chem. 1980,19,675. (3) Thomas, J. K. Chem. Rev. 1980, 80, 283.
Experimental Section Chemicals. Cetylpyridinium chloride (Sigma), nitrobenzene (Eastman), nitromethane (J. T. Baker), sodium iodide (Baker), and thallium(1) sulfate (Fisher) were used
0022-365418312087-3550$01.50/0 0 1983 American
Chemical Society
