J. Phys. Chem. 1983, 87, 1023-1028
might differ, it also seems that both SERS and extinction would be similarly affected. Each of these speculations will be pursued, the first one experimentally, the second theoretically, in order to determine whether they can contribute to resolving the current dilemma.
Summary 1. A number of colloidal hydrosols with average particle diameters varying from 13 to 38 nm and of varied degree of aggregation have been prepared by modification of the Carey-Lea method. 2. The particles were characterized by electron microscopy and the concentration of adsorbed citrate was determined by a radioactive exchange technique. 3. Raman and absorption spectra were obtained from the near-ultraviolet to the red. Vibrational mode assignmenta were discussed in comparison with dissolved citrate and a copper-coordinated citrate complex. 4. Citrate analyses were used to determine the absolute enhancement of the Raman signals. The measured maximum enhancementa in the range 2 X 105-4 X lo5 agreed approximately with maximum calculated values based upon the electrodynamic theory. 5. The SERS excitation profiles were broadly peaked near 500 nm. Sol preparations containing larger or more aggregated particles showed a shoulder in the absorption spectrum near 500 nm; for such systems the peak in the
1023
SERS excitation profile was shifted to somewhat longer wavelengths. However, the measured SERS excitation profiles, which were very broadly peaked near 500 nm, were in disagreement with the narrow excitation profiles peaking just below 390 nm which were calculated by using the electrodynamic model. These measured results were also inconsistent with the measured narrow absorption band at about 400 nm. 6. Attempts to account for the peaking of the SERS excitation profiles near 500 nm by assuming that the particles were nonspherical or that they comprised a spherical metallic core which was coated by a dielectric shell were unsuccessful because these models would have required that the absorption peak at 400 nm also shift to longer wavelengths. 7. Possibilities for explaining the above discrepancy by a chemical effect or by the effect of particle size and shape on refractive index were noted.
Acknowledgment. This work was supported in part by NSF Grant CHE-801144, by Army Research Office Grant DAAG-29-82-K-0062,and by NIH Grant GM-30904. In addition, part of an IBM grant to the Institute of Colloid and Surface Science was used to upgrade the Raman instrumentation. Also, SERS excitation profiles for sols 1-4 as well as for the dielectric coated silver spheres were computed by Dr. D.-S. Wang. Registry No. Ag, 7440-22-4; citrate, 126-44-3.
Micellar Interactions in Water-in-Oil Microemulsions. 1. Calculated Interaction Potential B. Lemalre,' P. Bothorel, and D. Roux Centre de Recherche Paul Pascal, Domaine Unlversitaire, 33405 Talence Cedex, France (Received:June 9, 1982; In Final Form: October 12, 1982)
A calculation of the mean field intermicellar energy potential is developed for "water-in-oil" (W/O) micelles of microemulsions. Short-rangeattractive interactions are evaluated for penetrable particles formed by a spherical aqueous core and a concentric spherical layer. Hamaker's treatment is generalized to the internal energy of the overlapped region of penetrated micelles. Attractive interactions are calculated through integration of semiempirical interatomic potentials in the overlapping region. Other contributions and particularly entropic terms are considered. The general behavior of the resulting potential is discussed and applicationsare indicated.
ing,11-13 ultrasedimentation,'l and electron micro~copy.~"'~ Introduction These microemulsions can be formed by droplets of water Microemulsions are homogeneous mixtures of water, hydrocarbon (oil), surfactant, and in most cases cosur(5) A. Vrij, E. A. Nieuwenhuis, H. M. Fijnaut, and W. G. M. Agterof, factant (which is generally an alcohol).l These systems Faraday Discuss. Chem. SOC., 65, 101 (1978). have been extensively studied in recent years2 in relation (6) A. Graciaa, J. Lachaise, P. Chabrat, L. Letamendia, J. Rouch, C. to their practical applicability, which would be very imVaucamps, M. Bourrel, and C. Chambu, J . Phys. Lett., 38, 253 (1977). (7)A. M. Cazabat and D. Langevin, "Proceedings of the Light Scatportant particularly in the field of improved oil re~overy.~ tering Workshop, Milano, 1979" V. Degiorgio, M. Corti, and M. Giglio, Their structure has been studied with the help of various Eds., Plenum Press, New York. techniques such as light scattering,'*"1° neutron scatter(8) A. M. Cazabat, D. Langevin, and A. Pouchelon, J.Colloid Interface (1) V. K.B a n d and D. 0. Shah in "Micellization,Solubilization, and Microemulsions", Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977; A. A. Calje, W. G. M. Agterof, and A. Vrij, ibid. (2) L. M. Prince, 'Microemulsions", Academic Pres, New York, 1977. (3) R. L. Reed and R. N. Healy in 'Improved Oil Recovery by Surfactant and Polymer Flooding", D. 0. Shah and R. S. Schechter, Eds., Academic Press, New York. (4) W. G.M. Agterof, J. A. J. Van Zomeren, and A. Vrij, Chem. Phys. Lett., 43, 363 (1976). 0022-3654/83/2087-1023$01.50/0
Sci., 73, l(1980). (9) A. M. Cazabat and D. Langevin, J. Chem. Phys. 74, 3148 (1981). (10) A. M.Bellocq and G. Fourche, J. Colloid Interface Sci., 78, 275 (1980). (11) M. Dvolaitzky, M. Guyot, M. Lagues, J. P. Le Pesant, R. Ober, C. Sauterey, and C. Taupin, J. Chem. Phys., 69,3279 (1978). (12) M.Dvolaitzky, M. Lagues, J. P. Le Pesant, R. Ober, C. Sauterey, and C. Taupin, J. Phys. Chem., 84, 1532 (1980). (13) R. Ober and C. Taupin, J.Phys. Chem., 84, 2418 (1980). (14) J. Biais, M. Mercier, P. Lalanne, B. Clin, A. M. Bellocq, and B. Lemanceau, C.R. Hebd. Seances Acad. Sci., 285, 213 (1977).
0 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87,
No. 6, 1983
in oil or oil in water. As pointed out by Vrij,l it is of great interest to apply modern theories of fluids to spherical particles immersed in a liquid medium, even when they are much larger than the molecular size. In this case a mean force interaction potential has to be introduced in the statistical mechanical theories. This potential is largely determined by the structure and composition of the particle surfaces and by the characteristics of the liquid mixture in which the particles are suspended. It is expected that changing the composition of the system induces an almost continuous variation of the potential. Many experimental results have been recently obtained on water-in-oil micelles in microemulsions.'~4-'8 These systems composed of reverse micelles dispersed in an organic solvent provided a very interesting field for investigations on intermicellar interactions. Hamaker and VrijlJghave assumed that the interactions between micelles result essentially from the interactions between water cores. This assumption was supported by the idea according to which the chemical nature of the aliphatic chains of the surfactant and of the alcohol is very similar to that of the organic solvent (oil and dissolved alcohol). Unfortunately the order of magnitude of the attractive interactions so calculated is not satisfying. Several attempts have been made to take into account the contribution of the aliphatic parts20-23including the overlapping effect and mixing entropy. However, up to now no realistic intermicellar potentials have been proposed for water-in-oil micelles in microemulsions. In the last few years simplified forms for this potential have been u s e d ' ~ ~ J *in~ Jlight ~ scattering experiments. Mainly, a hard-sphere potential with an attractive perturbation term and a square well potential were applied to water-in-oil micelles. Significant insights into a realistic potential can be predicted from these first successful investigation~.~ Unfortunately, in these studies, the intermicellar potential was treated as a parameter of the system. In spite of the intrinsic limits of such an approach, the accordance observed shows how the determination of a more realistic potential is of interest in the interpretation of experimental results of more various chemical systems. In this paper we derive such a mean field intermicellar potential including energetic and entropic contributions. Neglecting fluctuation of charges in W/O micelles, one can assume the vanishing of the electrostatic interactions between particles. In the first section we describe the micelle model that we have used for our calculation. In the two following sections, using Hamaker's approach, we derive the calculation of the intermicellar potential resulting from London-van der Waals interatomic attractive interactions. The short-range interactions are mainly determined by the overlapping between micelles which is related to the ability of the continuous phase to penetrate the outside aliphatic layers. The overlapping between micelles is accompanied (15) J. C. Hatfield, Ph.D. University of Minnesota, Minneapolis, MN, 1978. (16) J. Biais, M. Mercier, P. Bothorel, B. Clin, P. Lalanne, and B. Lemanceau, J. Microsc. (Oxford),121, 169 (1981). (17) A. Graciaa, J. Lachaise, A. Martinez, M. Bowel, and C. Chambu, C.R. Hebd. Seances Acad. Sci., 282, 547 (1976). (18) A. Graciaa, J. Lachaise, A. Martinez, and A. Rousset, C.R. Hebd. Seances Acad. Sci., 285, 295 (1977). (19) H. C. Hamaker, Physica IV, 1058 (1937). (20) M. J. Vold, J. Colloid Sci., 16, 1 (1961). (21) D. W. J. Osmond, B. Vincent, and F. A. Waite, J.Colloid Sci., 42, 262 (1973). (22) B. Vincent, J. Colloid Sci., 42, 270 (1973). (23) W. Gerbacia, H. L. Rosano, and J. H. Whittam in "Aerosols, Emulsions and Surfactants". Vol. 11. M. Kerker. Ed.. Academic Press. New York, 1976.
Lemaire et ai.
Figure 1. Schematic picture for reverse micelle.
by oil removing. In the fourth section the usually entropic contributions due to chain mixing and volume restriction are treated. In the last section we analyze the numerical behavior of the total potential for actual parameters. Finally, we briefly indicate immediate applications and developments of this potential in the experimental and theoretical field of microemulsions. Micelle Model A schematic representation of a reverse spherical water-in-oil micelle is given in Figure 1. Micelles are composed of two parts. The external aliphatic layer formed by end surfactant chains is assumed to be penetrated by the continuous phase. The internal part, limited by the ends of the alcohol chains, is assumed to be an impenetrable core. So for micellar overlapping this region can be considered as a hard sphere defined by its radius RHS. The larger limit of the micelle defined by radius R T is determined by the position of the ends of the surfactant chains in their all-extended conformation. In this model the radii R H S and R T can be calculated from the real chemical composition of the micelle and from the difference (1) between positions of the ends of surfactant and alcohol chains in an all-extended conformation ( I = 1.26 A). n is the difference between the number of carbons of the alcohol and surfactant. Experimental determination of the chemical composition of micelles can be deduced from dilution procedure17J8or by neutron ~cattering."-'~ The amount of continuous phase penetrating the outside aliphatic layer plays an important role since its ability to penetrate is related to the extent of overlap between micelles. experiment^^^,^^ on direct micelles show that the partial molar volume of oil molecules which penetrate the micelle is larger than in the pure liquid phase. The effect in the case of reverse micelles is expected to be similar but probably smaller as in direct micelles. It is obvious that this partial molar volume is directly related to the specific volume of CH2and CH, groups in the aliphatic layers. We define Y R C H as the reference volume of a CH2 group in pure alkane liquids which is known to be equal to 27 A3. (24) E. Vikingstad and H. Huiland, J.Colloid Interface Sci., 64, 510 (1978). (25) E. Vikingstad, J. Colloid Interface Sci., 68, 287 (1979).
Micellar Interactions in W/O Microemuisions
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
TABLE I: Energetic and Geometric Parameters of Lennard-Jones Potentialsa C,C H,H 0,O S,S Na,Nab rii, A E=,
a
cal mol-’
rij =
4.12 38
2.92
31
(riirjj)I’* ; ei’ = (eiiejj)”*;
Not given in ref 23.
3.24 94
3.78 43
oil .removing
Oil
/-
_ - --
2.92
74
i, j = C, H, 0, S, Na. hard sphere
The modified value in the aliphatic layer will be noted ‘VLcHz and is expected to be different from 27 A3. Intermicellar Interactions After Vrij we have chosen to separate the total interaction potential due to dispersion forces into two terms U = UHS + UA,where UHs is a hard-sphere potential due to repulsion forces and UA the attractive part. It is recalled that in our model the hard-sphere radius is defined by the end of the alcohol chains. The evaluation of UA as a function of the chemical components of the system and of the intermicellar distance is required. This evaluation takes into account the variation of the interaction of each particle with its surroundings. This potential can be written as UA = UA(192) + uA(o,o) - uA(1,o) - UA(2,o) (1) U(0,O)is the interaction energy between the two droplets if they are composed of solvent, U(1,O) is the interaction energy between particle 1and a particle identical with 2 but composed of solvent, and U(2,O) has a similar definition. A general formula for the internal energy UAcan be used, for example:
..--I- isolated
g(r) = 1
c uWi + a d for r > aWi + uWj for r
(34 (3b)
uWi being the van der Waals radius of group i.
Usually, semiempirical interatomic potentials are used in the evaluation of Eij.2sIn the usual cases where Lennard-Jones potentials are used the interaction energy Eij between two atoms i and j as a function of their distance rij depends only on two parameters cij and r*ij
Eij = ~ ~ ~ [ ( r *- 2(r*-/r-)6] ~ ~ U / rU ~ ~ ) ~ ~ (4) Only the attractive part of this potential concerns the rij* term. Comparison between eq 2 and 4 leads to Eij= We have adopted for these parameters values given by Scheraga et aL2‘ (see Table I). In fact, parameters cij and r*ij are only determined for pairs of atoms of the same nature (i = j). Parameters corresponding to mixed pairs (i # j) are evaluated according to the arithmetic or geometric means of the preceding ones; consequent results are (26) H. A. Scheraga, Adu. Phys. Org. Chem., 6, 103 (1968). (27) F. A. Momany, L. M. Carruthers, R. F. MacGhire and H. A. Scheraga, J. Phys. Chem., 78, 1595 (1974).
Jl
penetrated
Flgure 2. Schematic representation of the concentration change in the overlapping region.
J
where r is the distance between the interacting molecules i and j, E, is the interaction potential between particules i and j, and qi, qj are the number densities of these molecules in volumes Q1 and Q p Furthermore, g(r) is the radial correlation function between species i and j. This formula is valid even for overlapping volumes. For long-range interactions g(r) = 1and eq 2 reduces to the equation used by Hamaker.lg Taking into account the excluded distance due to van der Waals radii of species i and j, one can obtain a simplified form of eq 2 by taking
g(r) = 0
1025
isolated
II - penetrated
Figure 3. Volume partition of the overlapping regions for separated (I) or interpenetrated (11) micelles.
not sensitive to the type of mean used. In the following calculations we have adopted, for convenience, the geometric mean according to Berthelot’s principle.28
Internal Energies of Overlapping Volumes In our model we only consider interpenetration of the outside layers of micelles. In Figure 2 we have schematically indicated the variation of composition and structure during overlap of these layers, which can be predicted. Before penetration, as already indicated, the outside aliphatic layers are composed of the ends of surfactant chains with a large amount of continuous phase. During penetration, in the overlap region it is possible to distinguish essentially two situations. In the first, where overlapping is not too large, the ends of surfactant chains of one micelle have not reached the hard surface defined by the ends of the alcohol chains of the second micelle. There is only an exchange of solvent molecules dissolved in the interface by the chain segments of surfactant molecules. In the second situation, where there is a large overlap, surfactant chains are confiied in a smaller volume due to the presence of the two hard alcohol layers. This process induces removing of continuous-phase molecules leading to an excess volume and a large increase of CH2 and CH3 surfactant group concentration. A t this stage of our calculation we are concerned with the variation of the internal energy of this overlapped region before and after penetration, according to Ha(28) D. Berthelot, c. R. Hebd. Seances Acad. Sci., 126, 1703 (1898).
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The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
Lemalre et ai.
t
maker’s scheme. In Figure 3 we have given a schematic picture of the various volumes Viand V: with their internal energies Eiand E: which occur in the change of internal energy between situations I and 11. In situation I micelles are situated at infinity and the total energy of volumes Vl Vz V3 for the two micelles is VI = 2(E1 + E2 + E3) (6)
9
+ +
In situation I1 the totalenergy of the overlapping volume added to the internal energy of the removed continuous phase is VI, = E$ E’1 E‘, E $ (7) It must be noticed that volumes V,, Vrl, and V\ belong to the alcohol hard layer, so El and E’, = E\. Consequently, we obtain U = VI1 - VI = E’, + E\ - 2(E, + E3) (8) where
+
+
+
Figure 4. Illustration of the volume restriction effect. I I
I
I
I
A
l,,’^‘\2
i
I
I
X
I I I
‘d Figure 5. Extension of an aliphatic chain.
V b = (V,
+ 2V,)Y,O/Y,
(12)
Y, is the molar partial volume of the continuous phase in the aliphatic layers and V,“its value out of the vesicle. Hb, H;, and H’, expresa the interaction of all atoms in each molecule and of all the molecules in the volume according to their number densities, so, for example
Hi= CpkCnkjpj k
for i = 0,1, 2
J
(13)
where k describes all the molecules in volume Vi whose number densities are pk. The index j describes all the different atoms of molecules k whose numbers are n i and the right-hand-side term is pj = (cir*i!)1/2. g(r) mainly characterizes the radial correlation function of CHz groups in the various volumes. Hb,H;, and H’, are calculated according to the actual densities of surfactant and continuous phase in the corresponding volumes and take into account the large variation of surfactant concentration due to overlapping. Integrals of eq 9-11 have not been exactly solved. Neglecting boundary effects we have chosen to derive, for these various volumes, a mean field calculation according to the formula
S= = =
-jvJT
dr‘
-v -v
4?rrZg(r)
dr
r6 ((3/4*)v)1/3
4a
- dr r4
= -V(4r/3)[l/a3 - 47~/3Vl
(14)
where a is the excluded radius of a CH2 group. This parameter a has been chosen equal to 4 A, twice the van der Waals radius of a CHz group. We have verified that the introduction of a more precise function g(r) = e-U(r)/kT for CH, groupsz9does not significantly change results. (29)W.L. Jorgensen, J. Am. Chem. SOC.,103, 4721 (1981).
As it will be shown later, this part of interactions in the overlapping region which is attractive is found to be the most important and results from increase of surfactant concentration in this volume. Nevertheless, entropic contributions which are repulsive are induced by the constraints applied to surfactant chains during large overlapping. Calculation of the interaction energies of nonoverlapping volumes leads to negligible values relative to the overlap contribution. Entropic Contributions These entropic contributions arise from intermicellar penetration and are mainly due to change in configuration and conformation entropy of molecules during overlap of the aliphatic interfaces. The first contribution, called the “volume restriction effect”, has been extensively studied in relation to adsorption of polymers on colloidal partic l e ~ .This ~ ~ effect results from the change of conformational entropy of surfactant chains of one micelle when constrained by the hard alcohol layer of the second micelle (Figure 4). When the distance between the two alcohol layers is small, extended conformations of surfactant chains are forbidden. Unfortunately usual treatments concerning long polymer chains are inadequate in this situation. In order to obtain the order of magnitude of this contribution, we have derived an analytical whose treatment results were confirmed by a Monte Carlo simulation using a method previously d e ~ c r i b e d . ~ ~ One of the most unambiguous results obtained from Monte Carlo simulation of dense aliphatic monolayer is that the conformational equilibrium of chains can be described by two classes of particular conformations called “kinks” and “helices”.32 It is possible to relate the extension perpendicular to the interface of such conformations and their partition function. Consequently, we have obtained an analytical expression of the change of the chain entropy as a function of its length and of the space available. Then this expression is averaged over the position of the chain in the overlapped region. We consider n successive bonds of a chain following two virtual bonds 1 and 2 whose plane is taken as a reference (Figure 5). The extension d of the n bonds is measured along the x axis defined by the two reference bonds. Let (30)F.T.Hesselink, A. Vrij, and J. T. H. Overbeek, J. Phys. Chem., 75, 2094 (1971).
(31)B. Lemaire and P. Bothorel, MacromolecuZes, 13, 311 (1980). (32)J. Belle and P. Bothorel, Nouu. J . Chim., 1, 265 (1977).
Micellar Interactions in W/O Microemulsions
The Journal of Physlcal Chemistry, Vol. 87,No. 6, 1983
1027
TABLE 11: First-Order Terms for the Recurrence Equations on the Partition Function of Constrained Chains Ptn (d ) d,A
n = l n=2
0
0
1.26 2.52 3.78 5.04
1
Pk ( d ) n=4
n=3
0 2w 1
0 2wz 4w
1
0 2w2 2w + 4 w ' 4w 2wz
+
+ 2w3
n = l
n=2
2w
2w2 2w 0
0
1
n=3 2wQ 2 w t 2wz 2w 0
n=4
+ 2w3
2w 3 2(wz + 3w3) 2(w + w 2 + w 3 t w') 2w * 0
us define Ptn(d) and P , ( d ) as the partition function of the n bonds, the first of which being trans (t)or gauche (g) and whose extension is d. We note w as the statistical weight of the gauche isomer relative to the trans one, i.e. w = exp(-AU/RT) (15) where AU is the intramolecular energy difference between these two isomers. It can be shown that the following recurrence equations approximately hold for kinks and helices:
P,(d) = P,-l(d
-
1.26) + P,-l(d - 1.26)
(16)
+
P , ( d ) = w2[2P,-3(d- 2.52) + Pn-3(d - 2.52)] w 3 [ P n 4 ( d - 2.52) + P,-d(d - 2.52)] (17) where 1.26 A is the extension of a bond in the direction of bonds 1 or 2. The total partition function of the n bonds of extension d is Zn(4
p n ( 4
+Pn(4
(18)
The first terms needed in the evaluation of eq 16 and 17 can be determined by direct enumeration and lead to values shown in Table 11. Equation 17 can be completed by the low-order approximate equations: p,(1.26) P,(1.26)
-
2~"-~
(19)
2d-l
(20)
In addition, we have calculated an average value of d for surfactant chains in the overlapping region according to a uniform distribution on the polar surface; we obtain
with
+RHS- D COS 00 = X / ( R T + R H s ) X = RT
Finally, we have taken the mean partition function per surfactant in the overlapping region as Z,(& giving rise to the volume restriction effect (free energy): Us = -RTN In [Z,(d)/(Zn(1.26n)}] (22) where N is the number of surfactants in this region. The second entropic term which is usually taken into account in the theories of the stability of colloidal particles in the effect due to mixing e n t r ~ p y Polymer . ~ ~ ~ ~adsorbed on the colloidal particles is solvated by the solvent of the continuous phase. During interpenetration of the adsorbed layers of two micelles there is a large exchange of solvent molecules and of polymer segments corresponding to a real change of mixing. Such a description is not appropriate for reversed micelles of microemulsions because of the few aliphatic bonds concerned with overlapping. It must be pointed out that taking into account of short alcohol chains
Figure 0. Interaction potentials for various YLCH, vs. the intermicellar distance.
there is generally a large space available for surfactant end chains in the outside layers. Therefore, after overlapping, position configurations of surfactants on their respective surfaces are only constrained by mutual exclusion of their atomic clouds. Considering the flexibility of these molecules and the space available around them, this exclusion probably induces a little change in conformational entropy. We have calculated this contribution; however, according to its low order of magnitude and its imprecision, we have not kept this term. Furthermore, taking into account the equilibria of the continuous phase between outside layers and the bulk of the microemulsion, we have not introduced any entropy variation for solvent molecules. From eq 8-12 it can be easily shown that, for distances corresponding to V'l = VI = V3= V i = 0, that is, for which the repulsive energy of entropic nature is null, the attractive energy is approximately proportional to the overlapping volume V 2or V',. This situation gives the most important contribution to the attractive interaction potential.
Discussion Almost all micellar parameters required in the evaluation of the total intermicellar free energy potential according to the model just defined can be determined from light scattering and dilution procedure."J8 Only one parameter of physical meaning remains unknown and will be discussed now. This parameter that we have already introduced is the variation of molar partial volumes of the components of the aliphatic layers (in both situations I and 11, Figure 2) relative to their reference values. The reference phase for the oil molecules is the pure liquid, and that for the surfactant molecules is the dilute solution. The relative variation K can be mainly related to the variation of the specific volume of CH2 (and CH,) groups in the form K = VLCH,/27. We have given in Figure 6 several curves of the intermicellar free energy potential U = U, + Us for various valus of YLCH, in the case of a microemulsion experimentally studied and whose components are given in
1028
J. Phys. Chem. 1983, 8 7 , 1028-1034
TABLE I11 Droplet and Continous-Phase Compositions of Experimentally Studied Microemulsiona composition, cm3 droplet 7.967
water
continuous phase 0.193 71.56
dodecane
SDS hexanol a
4.66 6.10
9.52
See part 2. R T = 54 A .
Table 111. The entropic term is calculated with AU = 600 cal mol-'. All these curves of free energy show the same behavior as a function of the intermicellar distance D. Each of these curves is the sum of two contributions. The first is a repulsion term due to the volume restriction effect which arises for distances smaller than R T RHS and which increases up to infinite hard-sphere repulsion. The second term is due to attractive interactions. Its most important vlaues appear in the overlapping region corresponding to 2Rm I D I 2RT and results of short interatomic attractive interactions. The total energy potential exhibits a minimum in the overlapping region. Figure 6 shows that a slight increase of YLCH2, the specific volume of CH2groups in the aliphatic micellar layers, induces a sharp variation of the attractive part of the potential in the overlapping region. The interaction potential becomes more attractive
+
when increasing the partial molar volume of oil, surfactant, and cosurfactant. It seems difficult to obtain an exact value of Y L C H z in an independent way but it can be predicted that this value only depends on the chemical nature of oil and surfactant. So, this parameter must be the same for all series with the same oil and surfactant. It is observed that the absolute value of the attractive energy between particles is mainly determined by this parameter. For a given oil and surfactant this parameter can be considered as an energetic parameter which should be constant in view of the variation of the micellar size and of the alcohol chain length. Such accordance is examined in the experimental part (part 2) of this study. One point of interest of this calculated potential is to allow the prediction of the strength of the micellar attraction according to size and composition of the continuous phase. It is possible to check the influence of various molecular parameters and of the chemical nature of components. Such a comparison is developed in the experimental study presented in part 2 through the measurement of the second virial coefficient of microemulsions. A more precise analysis of the thermodynamical behavior of micellar solution in a large concentration range can be envisaged by application of modern theories of liquids. Evaluation of the radial correlation function from this potential and of the osmotic pressure would lead to a direct evaluation of the light scattering intensity curves and more generally to many structural and thermodynamical data.
Micellar Interactions in Water-in-Oil Microemulsions. 2. Light Scattering Determination of the Second Viriai Coefficient S. Brunettl, D. ROUX,A. M. Bellocq, G. Fourche, and P. Bothorel' Centre de Recherche Paul Pascal, Domine Universltaire, 33405 Talence Cedex, France (Received: June 9, 1982; In Final Form: October 12, 1982)
We have studied both the intensity and the autocorrelationfunction of the light scattered by water-in-oil (W/O) microemulsions. The investigated microemulsions were formed from water, dodecane, sodium dodecyl sulfate, and pentanol or hexanol or heptanol. Micellar size and virial coefficients were extracted from experimental data. Our data evidence that interactions in these systems mainly depend on alcohol chain length. They are strongly attractive in microemulsions containing pentanol and much less in hexanol and heptanol systems. The intermicellar potential presented in paper 1allows one to interpret the scattering results and provides an approach for the understanding of the variations of the B values; the agreement between theory and experiment is very good.
I. Introduction The scattering properties of microemulsions were first investigated by Hoar and Schulman.' Since that time, these systems have attracted considerable attention. In a large region of the phase diagram their structure can be pictured as a dispersion of water droplets surrounded by a film of surfactant and alcohol molecules in a continuous medium mainly made of oil. Light scattering techniques have been extensively applied to these systems, since these methods provide information on micellar sizes and interactions between droplet^.^-^ (1)J. H.Shulman and T. P. Hoar, Nature (London), 152,102(1943). (2)A. Graciaa, et al., C. R. Hebd. Seances Acad. Sci., Ser. B , 282,547 (1976).
0022-3654/83/2087-1028$0 1.50/0
Previous results have shown that in water-in-oil (W/O) microemulsions a large range of interaction forces can be obtained by varying chemical composition. Modern theories of fluids were used by several authors to explain the experimental result^.^^^ But, up to now, difficulty in evaluating the interaction potential for inverted micelles did not permit a complete analysis of the data. A more precise knowledge of this intermicellar potential allows one to foresee a more complete interpretati~n.~ In the preceding paper we have presented an intermicellar potential; (3)A. A. Calje, W. G . M. Agterof, and A. Vrij, Micellization, Solubilization, Microemulsions, [Proc. Int. Symp.], 1976,2, 779 (1977). (4)(a) A. M. Cazabat and D. Langevin, J. Chem. Phys., 74, 3148 (1981);(b) M.Corti and V. Degiorgio, J.Phys. Chem., 85,711-7(1981). (5) D.Roux, Surf. Chem., Proc. Scand. Symp., 7th, 141 (1981).
0 1983 American Chemical Society
