J. Phys. Chem. 1995,99, 14819-14823
14819
Shape Fluctuations of Microemulsion Droplets F. Sicoli and D. Langevin" Laboratoire de Physique Statistique de I'ENS,' 24 rue Lhomond, 75231 Paris cedex 05, France Received: May 24, 1995@
The surfactant film bending elasticity can be described by a spontaneous curvature COand two elastic constants K and E, associated with the mean curvature and the Gaussian curvature, respectively. These parameters are very important in the determination of the structure of the dispersions stabilized by the surfactant (droplets or sponge-like structures). We have studied ternary mixtures of oil, water, and nonionic surfactants of different chain lengths. Depending on the temperature, the microemulsions are in equilibrium with excess oil (oil in water structure), excess water (water in oil), or both excess oil and water (sponge-like structure). The interfacial tensions between the microemulsions and the excess phases are ultralow. We present neutron scattering experiments for droplet microemulsions, from which we determine the droplet radius and polydispersity. (Experiments were performed in Laboratoire Ldon Brillouin, CE Saclay, 91 191 GifTYvette, France, Laboratoire commun CEA-CNRS.) These results, in combination with interfacial tension determinations, are used to estimate K and E.
I. Introduction Microemulsions' are dispersions of oil and water stabilized by surfactant molecules. They are frequently made of droplets (oil in water (olw) or water in oil (wlo) microemulsions) surrounded by a surfactant monolayer and dispersed in a continuous phase (water or oil, respectively). When the composition of the medium is known, the droplet radius can be predicted quite accurately by using the following relation
can change its topology: positive k favors saddle-splay structures as in bicontinuous cubic or sponge phases, while negative K favors lamellar or spherical structures. In many droplet microemulsions, the magnitude of CO determines the maximum droplet size R, (maximum solubilization power). This size can be calculated by minimization of the total free energy:4
R = - 34 cs
=
where 4 is the dispersed volume fraction, cs the number of surfactant molecules per unit volume incorporated in the droplets, and C the area per surfactant molecule. This relation expresses the fact that these surfactant molecules sit at the oilwater interface and that each of them occupies a well-defined area, independent of the composition. This is because in order for the microemulsion to be thermodynamically stable, the surfactant monolayer must reduce the oil-water interfacial tension to about zero: its surface pressure must balance the tension of the bare interface, thus fixing the value of 2. In the following, we will assume, as in recent microemulsion models, that is constant. The free energy of the system is then the sum of the dispersion entropy and the higher order surface energy term, the curvature energy.2 The curvature energy here is the surfactant film bending energy, the expression for which has been given by H e l f r i ~ h : ~ 1
F = -K ( C , 2
+ C2 - 2C0)2+ KC,C2
(2)
where Cl and C2 are the two principal curvatures of th_e surfactant layer, COis its spontaneous curvature, and K and K are the mean and Gaussian bending elastic constants. The type of microstructure is closely related to the sign of the spontaneous curvature of the surfactant layer CO:by convention CO> 0 for aqueous dispersions and CO < 0 for reverse systems. The second term of eq 2 takes into account the fact that the system
@
Associated with CNRS, Universities Paris 6 and 7. Abstract published in Advance ACS Abstracts, September 1, 1995.
0022-365419512099-14819$09.00/0
where RO = C0-1.4 Equation 1 shows that the droplet radius increases when the dispersed phase volume fraction 4 increases: when R exceeds R,, the system separates into two phases: a microemulsion with droplet size R, and an excess phase (emulsification f a i l ~ r e ) . ~ In microemulsion systems, the elastic constants K and E are typically of order kTe6-' This means that the surfactant film at the surface of the droplet will be undulated due to thermal energy. As a result, the shape of a given droplet constantly fluctuates around an average spherical form with a mean radius given by eq 1. Because this droplet can exchange material with the other surrounding droplets, the shape fluctuations are in average fluctuations at constant total surface area and constant total internal volume, and their amplitude is related to both K and ks-I0These shape fluctuations are equivalent to an average polydispersity, the nature of which is obviously very different from the polydispersity in more conventional colloidal systems such as dispersions of solid particles. It has been observed indeed that the polydispersity of microemulsion droplets seems to depend on the particular experiment used for the determination of this quantity: it is much smaller when measured with dynamic light scattering than with elastic light or neutron scattering." This is simply because in dynainic measurements, averages are performed over times less than the time scale of the experiment: in light scattering experiments, time scales are typically microseconds, Le., comparable or longer than the time for material exchange between the droplets and the surrounding medium.I2 In static experiments, one measures an average over all instantaneous droplets shapes, and the measurement probes the complete shape distribution. 0 1995 American Chemical Society
Sicoli and Langevin
14820 J. Phys. Chem., Vol. 99, No. 40, 1995
In the following, we will present small angle neutron scattering investigations of the size and the polydispersity of microemulsions made with nonionic surfactants. In a previous study, we have reported a first series of results on surfactantoil-D2O mixtures.13 In the present study, we have used D20 and deuterated oils to achieve the so-called "shell contrast". This allows us to determine the polydispersity with a better accuracy. We have enlarged the range of experimental conditions, to investigate how the polydispersity depends on temperature, oil, and surfactant chain length. Let us recall that the bending elastic constants depend strongly on the surfactant chain length 1, they are predicted to scale as l3.I4 As a consequence, the polydispersity should strongly depend on 1, as already suggested by our previous study. The interfacial tension y between the microemulsion and the excess phase also depends on K and K. We will compare the values found for the elastic constant using small angle neutron scattering for the size and polydispersity determinations, surface light scattering for the interfacial tension determinations, and ellipsometry for the determination of the bending constant K.
11. Theoretical Background Thermal fluctuations in droplet microemulsions have been analyzed theoretically by describing the droplet deformation with an expansion of spherical harmonics Ylm8-Io by Safran and coworkers : (4) /m
where 6' and 4 are polar angles. These authors have shown that the main contribution comes from the two first harmonics, 1 = 0 and 1 = 2. At the emulsification failure limit, their mean square amplitude is given by
(1.212>
=
4(4K -
kT kT - -[ln
(5b)
4 - 11
plays a role here, because when changing the droplet radius or shape at constant C, one changes the total number of droplets and thus the topology. The type of deformation described by Yo0 is a mere change in radius and is favored by negative values of K. The ones described by Ylm are peanut-like deformaticns, favored by positive values of K. This is why the sign of K is opposite in eqs 5a and 5b. The droplet size fluctuations described by (IU/mI2) are equivalent to what is usually called droplet polydispersity:
It is also possible to obtain information on Kif the interfacial tension y between the microemulsion and the excess phase is measured. Indeed, one can show that6 2K+K y = 2 ++In 4nR 4 - 1) (7) Rm In the above description, the entropy of mixing of the droplets has been taken to be equal to the value for lattice theories.
TABLE 1: Upper (Tu) and Lower (TI)Temperatures of Two-Phase-Three-phase Boundaries for Systems Containing Equal Amounts of Oil and Water, at 1% ClfES, 2% c10E.1, and 4% CgE3, Respectively surfactant oil TI("0 Tu("C) CizEs hexane 26.5 29 heptane 28 32 octane
a
C I oE4
decane" octane decane"
C8E3
decane
30 36 23 21 16
35 41 28 33 28.5
Data from ref 17.
Different expressions have been derived by other authors in which In 4 is replaced by In a@ A discussion can be found in ref 15.
111. Experimental Procedures
1. Phase Behavior. We have studied temary oil-waternonionic surfactant mixtures, on which previous determinations of the modulus K have been performed with ellipsometry.' The surfactants are alkyl polyethylene glycol ether surfactants with alkyl chains of n carbon atoms and polar parts of m ethoxy groups: C12E5, CloE4, and C&. The two first surfactants were purchased from Nikko Chemicals, the last one was purchased from Bachem, and all were used as received. The microemulsions were prepared with deuterated water and alkanes. The sample composition is such that at a given temperature, a dilute microemulsion phase (4 2%)is in equilibrium with an excess phase. o/w systems were obtained at low temperature, T < T, w/o systems at high temperature, T > Tu. The values of and Tu are given in Table 1. Details of the phase diagrams can be found in refs 7 and 16. The temperatures and Tu are given for mixtures of hydrogenated components. Deuteration slightly changes the phase boundaries (they are 2 "C lower for D20-octane-ClzEs and change much less with alkane de~teration).'~ 2. Neutron Scattering Experiments. We have measured the droplet polydispersity by static small angle neutron scattering experiments in Saclay (PAXE spectrometer). The droplet volume fraction is about 2%, such that the interactions between droplets are negligible. Figure 1 shows three typical spectra, together with the fit with a Gaussian distribution of ~he1ls.l~ Water which scatters isotropically was used to normalize the intensities. An altemative procedure making use of graphite was also used and found to give equivalent results. Correction factors for inelastic scattering were taken to be 0.855 for 2 = 6 A, 0.901 for 2 = 7 A, and 0.952 for 2 = 8 A, the wavelengths used. The incoherent background was assumed to correspond to the high q intensities and has been subtracted. The scattering length densities were taken to be 6.40 for D20, 6.20, 6.30,6.40, and 6.70 for deuterated hexane, heptane, octane, and decane, respectively, -0.58, -0.54, and -0.49 for hydrogenated hexane, octane, and decane, respectively, and 0.1 1 for the surfactants, in units of The spectra have been fitted with a Gaussian distribution of spheres with a polydispersity papp.The instrumental function contains contributions from the spread of neutron wavelengths and collimation effects. It can be shown that it can be approximated by a Gaussian function.I8 On the spectrometer used, the width of the instrumental function introduces the equivalent of a supplementary polydispersity plnst of lO%.I9 According to the well-known properties of the convolution of Gaussian functions, the actual polydispersity p is simply such as papp2 = p 2 plnst2.The data for the different samples are summarized in Table 2 .
-
+
Shape Fluctuations of Microemulsion Droplets
J. Phys. Chem., Vol. 99,No. 40, 1995 14821 TABLE 3: Scattering Cross Sections, Internal Radii, Shell Thickness, and Polydispersity for C&,d-0ctane-D20 Microemulsions for Different Ratio of Deuterated to Hydrogenated Oil X I(q=O) R (A) d (4 P (%)
I A (log units)
0 0.2 0.3 0.35 0.4 0.45 0.5 0.7
1 average
L
I
o
002
1
aoc
I
I
I
I
0.1
012
OM
I
0.m
ma
P
3
Figure 1. Neutron scattered intensity I versus wave vector q (in k') for three different o/w microemulsions in the shell contrast, T = 14 "C. The lines are fits: CIZES,R, = 100 A, p = 0.24; c&4. R, = 96 A, p = 0.29; CsE3 R, = 73 A, p = 0.4.
TABLE 2: Radii and Polydispersities for the Studied Microemulsions. For the Sphere Contrast and for o/w Systems, R = R,. For the Shell Contrast and for w/o Systems in All Cases, R is the Internal Radius surfactant
R,
(A)
oil
T("C)
contrast
hexane
14
sphere shell shell sphere shell shell sphere shell sphere shell sphere shell sphere shell sphere shell sphere sphere
76 68 56 76 48 35 132 120 100 108 101 74 43 46 34 40 63 27
sphere shell shell sphere shell shell sphere sphere sphere shell sphere shell shell
65 53 92 104 94 65 102 149 39 51 35 22 57
heptane octane decane hexane
20
heptane octane hexane heptane hexane
30 34
octane octane
40
decane
8
octane octane
10 14
decane octane
16 20 30 34
decane decane
8 10 14 30 30.4 34
sphere shell sphere shell shell shell shell
d(A) 32 32 28 19 26 18 30
14
13 8 12 21
D (%)
21 26 23 17 21 20 16 28 18 18 20 25 36 21 28 25 35 23 23 22 22 26 29 23 30
12
39 32
11 11
42 36
47
44 50
10
55 29 17 12
9.5 6 9.5 10
42 32 37 28
44 49 25
The results show that the polydispersity increases rapidly when the surfactant chain length decreases, as already expected from the measured variation of K.' For a given system, the
43.3 8
1.35 1.71 1.19 1.3 2.32 14.2 63.7
49.5 50.9 47.5 53.4 54.3 31.7 43 29 45
13.3 11.4 9.2 10.8 12.4 11.8 9 10
32 30 38 30 54 74 37 88 39
45 f 9
11 f 2
47 f 21
polydispersity increases somewhat with increasing temperature. The role of oil chain length is not very marked. We do not observe the large decrease of the bending elastic constants with increasing oil chain length reported for AOT systems and attributed to an easier oil penetration into the surfactant layer for the small chain lengths.20 The theoretical spectrum is in fact not exactly the same as the one of polydisperse spheres, because part of the polydispersity arises from the u2 modes which correspond to distorted spheres. The exact spectrum has been calculated by Farago et al.'" Instead of two adjustable parameters for polydisperse spheres (R and p ) , this spectrum depends on three parameters, the mean radius R and the mean square amplitudes (w2) and ( ~ 2 ~ ) In . view of the limited experimental accuracy and of the very small differences between the two kinds of spectra, we did not attempt to perform the three-parameter fits with our data. However, it can be shown that the sensitivity of the shape of the spectrum to u2 is limited and that instead of eq 6, the polydispersity deduced from the fit is ratherz1
+
In our earlier paper,I3 we used p2 = ((w2) 5 ( ~ 2 ~ ) ) / 4(see n eq 6), and this led to difficulties for the interpretation of the data. In order to c o n f i i that t h i s procedure for the determination of the polydispersity is correct, we have carried out two series of experiments in which the contrast of the solvent was varied around the point of zero contrast for the average sphere, one series for o/w droplets, the second for w/o droplets. In the first series of experiments, reported in ref 13, we used C I ~ E ~ , deuterated heptane, and mixtures of H20 and D20 at 20 "C. In the second series, we used C I O E ~D20, , and mixtures of deuterated and protonated octane at 30 "C. This type of contrast variation minimizes the change in droplet radius due to the shift in transition temperatures induced by changing the deuterated solvent into an hydrogenated one. The surfactant being protonated, for a given ratio x = (H-solvent)/(D-solvent), the contrast between the spheres and the solvent is minimal. If the spheres were monodisperse, the contrast should be identically zero at this point. But because the droplets are polydisperse, the residual intensity is a direct measure of the polydispersity. This procedure has also been used recently in light scattering experiments on AOT microemulsions.22 The spectra were fitted with a dispersion of polydisperse shells, and the results are given in Table 3. A theoretical treatment by T e ~ b n e allows r ~ ~ one to relate the value of the intensity at q = 0 to the polydispersity. If the
Sicoli and Langevin
14822 J. Phys. Chem., Vol. 99, No. 40, 1995
TABLE 4: Relation between Interfacial Tensions: Radii? and Polydispersity Values surfactant
oil
T
hexane 14 20 34 octane 14 20 34 40 CIoE4 octane 10 14 16 20 30 34 CgE3 decane 8 10 14 30 30.5 34
C&
-1001 0
I
0.2
I
I
0.4
0.6
1
1
0.8
1
-
X
Figure 2. Square root of the scattered intensity versus x = (D-octane/ H-octane) for C10E4 water in oil microemulsions; T = 30 "C. The
lines show the deviation due to polydispersity. shell thickness d is sufficiently small compared to the radius R
where I is the scattered intensity, e is the scattering length density, the subscripts ext, int, s, denote the continuous phase, the droplet core, and the surfactant layer respectively; Q is the average value for the droplet. When (I(q=O))In is plotted versus x and symmetrized (Figure 2), one obtains, except close to the minimum, a straight line passing through the horizontal I = 0 at x 0.38. Using the measured I values, one finds with eq 9, p = 0.29, a value which is in good agreement with those of Table 2. The polydispersity deduced from this type of experiments is a measure of volume changes independent of shape changes and has contributions from both uo and u2 (eq 6). Because the different determinations of polydispersity (contrast variation and spectral shape) are comparable, this means that u2 is small. Taking into account the experimental uncertainty, we expect (uo2) 1 and ( ~ 2 ~ < )0.17 for the sample studied (this is compatible with 2K = 1.3kT (Table 3) and K = 0.5kT (ref 5) which leads to ( ~ 2 = ~ 0.11 ) with eq 5b). 3. Correlation with Ellipsometry and Interfacial Tension Measurements. Ellipsometry allows one to measure the mean square amplitude of the thermal deformations at the macroscopic (flat) oil-water interface. Combined with interfacial tension measurements, it leads to the determination of K . We have previously found that for CloE4-octane microemulsions, K is OSlkT, for CsE3-decane microemulsions, K = 0.31kT, K being temperature independent in both cases. For C12Es-hexane microemulsions K is larger: K 1-2kT; in the narrow threephase range of this system, the oil-water interfaces were difficult to stabilize in the ellipsometric cell and it- was only possible to roughly evaluate K. In an earlier paper,13we tried to use these determinations of K and found difficulties in the analysis of the polydispersity (partly because we used eq 6 to fit the polydispersity data). The accuracy and the number of studied systems was too limited, and it was not possible to determine the temperature variation of the polydispersity. The temperature range investigated here is wider, and for a given system, there is a slight increase of the polydispersity when the temperature increases. This suggests that and eventually K might be temperature dependent. We have chosen here to use the interfacial tensions and the droplet radius determinations to estimate 2K K from eq 7:
-
-
+
-
+
(10)
The corresponding values are reported in Table 4. When using
a
Value measured at T
yRm2/kT1/4np2 y (mN/m) (In C#J - 1)/4n (In C#J - 1)/4n 1.2 x lo-' 4 x 10-2 8.6 x 2.5 x IO-1 1.3 x lo-' 2.7 x 1 x lo-'
3.5 2.5 1 4.1 4 0.9 0.8
2 1.8 2.1 2.6 2.1
2.5 x 10-I 7.8 x low2 6.5 x 1.7 x 8x 1 x 10-1
4.7 3.1 2.3 1.4 1.3 0.8
2.4 1.7
1.4 x lo-'
1.5 1.2 1.5 1.5 1.1 1.1
1.6
1.2 x 10-1 1 x 10-I 3 x lo-' 3 x lo-' 5 x 10-1
1.6 1.2
1.8 1.2 1.1 2.1
+ 2 "C, data taken in part from ref 6.
the experimental polydispersities of Table 2 and eqs 5a and 6, we find similar values of 2K f E (Table 4). It must be stressed that the accuracy of these values is extremely limited: the polydispersity and the droplet radius are not accurately known, and the shift in temperature introduced to account for the differences between H20 and D20 samples introduces a supplementary uncertainty. The values of Table 4 can therefore only suggest general trends for the temperature behavior of 2K
+ E.
+
The values of 2K E show a slight decrease when temperature increases, an effect not evidenced in ellipsometry for K alone. In this method, K is obtained from the variation of the ellipsometric coefficient with the interfacial tension and the temperature range where the ellipsom tric ccefficient varies the most is limited. We also find &at K is larger than 2K as determined by ellipssmetry. This suggests that is positive. Let us recall that K is more likely to be negative in droplet systems (although po$ive, but small positive, values are not excluded). Positive K values have also been reported recently in a different system.24 4. Role of the Droplet Volume Fraction. One can see from eqs 5a and 8 that the polydispersity values should depend on the droplet volume fraction 4 and that these variations should be larger for the surfactants of smaller chain lengths. We have performed systematic measurements for different series of samples in which the droplet volume fraction has been varied between 0.05% and 3%. Unfortunately, even within a series, the polydispersity values for different volume fractions were scattered at random, and no systematic evolution has been noted. Previous experiments on AOT microemulsions have also shown that the droplet polydispersity depends little upon the droplet volume f r a ~ t i o n . ~ ~ . * ~ We have also attempted to measure the interfacial tension variation with volume fraction for the CloE4-octane-HnO system at 32 "C. The results are shown in Figure 3, together with the fit with eqs 3 and 7. Again, the experimental accuracy is not good enough and the range of accessible 4 values too limited to enable us to draw significant conclusions. Measurements of R , and y for a similar range of volume fractions (1% < 4 < 50%) have been performed by Fletcher for AOT micro emulsion^.^^ However, the accuracy of the calculated yRm2/kT(25%) is also not sufficient to support a linear variation versus In 4.
s+
Shape Fluctuations of Microemulsion Droplets
J. Phys. Chem., Vol. 99, No. 40, 1995 14823 Acknowledgment. We are grateful to L. T. Lee and J. Teixera for their help during the neutron scattering experiments. We also thank S. A. Safran and R. Strey for very useful discussions and sharing their unpublished results. This work has been partially supported by CEC, contract CT 92.0019, of the program Human Capital and Mobility.
..
References and Notes
t 01
1
1
I
I
I
0
2
4
6
8
10
@$A)
Figure 3. Interfacial tension variation versus droplet volume fraction for the system CloE4-octane-water at 32 "C. The line is the fit with eqs 3 and 7.
Let us recall that the exact form of the entropy term in 4 1 in eqs 3 and 7 is not even clearly established. In their treatment, Overbeek et al. propose that instead of In 4 - 1, one should use In 4 - 6 where
3 2
d=-ln-
-
-
16R3 vw
-
where vw is the water molecular volume (v, 30 A3). Typically, for R 100 A, 6 20. In a recent paper, Borkovec finds a different expression for 6, but again 6 20.28 The role of the entropy of mixing remains to be clarified so that the values of the elastic bending constants could be properly determined.
-
IV. Conclusion We have investigated the polydispersity of microemulsion droplet systems and shown that the polydispersity mainly depends on the surfactant chain length. The influence of temperature and oil chain length is less pronounced. We have used independent determinations of the interfacial tension between the microemulsions and the excess phases to obtain information on the surfactant layer bending elastic constants K and K. The accuracy of all these determinations is limited. We found a tendency for 2K K to decrease with increasing temperature. It must be noted that we have used in our analysis the form for the entropy of mixing of microemulsion droplets of the lattice theories. Additional terms in the entropy of mixing derived in other theories would increase the values determined for 2K K. Neutron spin echo experiments, where the roles of the different droplet shape fluctuation modes responsible for the measured polydispersity can be better disentangled, are also projected to improve the accuracy of the determination of the elastic constants.
+
+
(1) de Gennes, P. G . ; Taupin, C. J . Phys. Chem. 1982, 86, 22942304. (2) Safran, S . A.; Roux, D.; Cates, M. E.; Andelman, D. Phys. Rev. Lett. 1986,57,491-493. Cates, M. E.; Andelman, D.; Safran, S. A.; Roux, D. Langmuir 1988, 4, 802-806. (3) Helfrich, W. Z. Naturforsch. 1973, 28, 693. (4) Safran, S. A. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S. H., Huang, J. S . , Tartaglia, P., Eds.; NATO AS1 Series; Kluwer Acad. Pub.: Dordrecht, 1992; Vol. 369. (5) Safran, S. A.; Turkevich, L. E. Phys. Rev. Lett. 1983, 50, 19301933. (6) Binks, B. P.; Meunier, J.; Abillon, 0.;Langevin, D. Langmuir 1989, 5, 415-421. (7) Lee, L. T.; Langevin, D.; Meunier, J.; Wong, K.; Cabane, B. Prog. Colloid Polym. Sci. 1990, 81, 209-214. (8) Safran, S . A. J . Chem. Phys. 1983, 78, 2073. (9) Milner, S . T.; Safran, S. A. Phys. Rev. A 1987, 36, 4371. (10) Farago, B.; Richter, D.; Huang, J. S.; Safran, S. A.; Milner, S . T. Phys. Rev. Lett. 1990, 65, 3348. (11) Compare for instance the neutron scattering data of Ober and Taupin, Ober, R.; Taupin, C. J . Phys. Chem. 1980, 84, 2418, and the light scattering data of Cazabat and Langevin, Cazabat, A. M.; Langevin, D. J . Chem. Phys. 1981, 74, 3148. (12) Zana, R. In Sur$actants in Solution; Mittal, K. L., Bothorel, P., New York, 1986; Vol. 4, p 115. , F.; Langevin, D.; Lee, L. T. J . Chem. Phys. 1993 99,4759. (14) Szleifer, I.; Gamer, D.; Ben Shaul, A.; Roux, D.; Gelbart, W. M. Phys. Rev. Lett. 1988, 60, 1966. (15) Overbeek, J. Th.; Verhoeckx, G. J.; de Bruyn, P. L.; Lekkerkerker, H. N. W. J . Colloid Interface Sci. 1987, 119, 422. (16) Kahlweit, M.; Strey, R.; Firman, P.; Haase, D.; Jen, J.; Schomacker, R. Langmuir 1988, 4, 499. Kahlweit, M.; Strey, R.; Firman, P. J . Phys. Chem. 1986, 90, 671. (17) Strey, R. Private communication. (18) Pedersen, J. S . ; Posselt, D.; Mortensen, K. J . Appl. Crystallogr. 1990, 23, 321. (19) Teixera, J. F'rivate communication. (20) Binks, B. P.; Kellay, H.; Meunier, J. Europhys. Lett. 1991, 16, 53. (21) Gradzielski, M.; Farago, B.; Langevin, D. Submitted for publication. (22) Ricka, J.; Borkovec, M.; Hofmeier, U. J . Chem. Phys. 1991, 94, 8503. (23) Teubner, M. J . Chem. Phys. 1991, 95, 5072. (24) Kegel, W.; Bodnar, I.; Lekkerkerker, H. J . Phys. Chem. 1995, 99, 3272. (25) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79, 2461. (26) Wu, X. L.; Tong, P.; Huang, J. S. Preprint. (27) Flechter, P. D. I. Chem. Phys. Lett. 1987, 141, 35. (28) Borkovec, M. J . Chem. Phys. 1991, 91, 6268.
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