J. Phys. Chem. 1995,99, 12941 - 12946
12941
AOT Reverse Micelles: Depletion Model G. Cassin?" J. P. Badiali,*s and M. P. F'ileni*~t~* Laboratoire SRSI, Universitk P. et M. Curie, URA 1662, batiment F, 4 place Jussieu, 75005 Paris, France, and CEN Saclay, DRECAM, SCM, 91 I91 Gif sur Yvette, France Received: March 29, 1995; In Final Form: June 6,1995@
The structure of AOT microemulsions in various bulk solvents is investigated by SAXS and conductivity measurements. The intermicellar interactions are analyzed with the sticky hard-sphere model taking into account the discret nature of the solvent. A model is proposed to explain the decrease in the interdroplet attractions with the increase of volume fraction of reverse micelles. It is based on the existence of depletion forces between reverse micelles. It allows the determination of a theoretical relation between the stickiness parameter and the reverse micelle volume fraction which is compared to experimental data. The percolation threshold of reverse micellar systems is then qualitatively determined in three different solvents.
Introduction A microemulsion is a thermodynamically stable mixture of water, oil, and surfactant where the water regions are separated from oil by a monolayer of surfactants. Due to the amphiphilic nature of the surfactant, numerous disordered or partially ordered phases are formed depending on temperature and surfactant concentration.' The ternary phase diagram AOT/water/isooctane shows a large zone where the reverse micellar phase (L2) takes place. In this isotropic liquid phase, the ratio of water over surfactant concentrations (w = [H20]/[AOT]), called the water content, determines the size of the reverse micelles. When w values are lower than 10, the water mobility is greatly reduced (bound water). Above w = 10, the water pool radius, r,, increases linearly with the water content.2 Lower consolute points are observed in nonionic surfactant aqueous solutions, and AOT-reverse micellar solution^.^^^ The existence of a consolution curve is due to a competition between internal energy favoring phase separation and entropy effects promoting miscibility at high temperature. In reverse micellar solutions, the phase separation is obtained by increasing temperature, which implies that the effective intermicellar potential depends on the tem~erature.~ Reverse micellar solutions are also known to present a dynamic percolation phenomenon by increasing temperature or water volume fraction. The percolation threshold corresponds to the maximum in permittivity and to the onset of the conductivity .5 In the present paper, shown is a decrease of intermicellar interactions with the reverse micellar concentration. Due to the discret nature of solvents molecules, this effect is attributed to the appearance of depletion forces between two micelles. A model is proposed from which a theoretical relationship between the stickiness parameter and the volume fraction is deduced and compared with the experimental data.
Experimental Section Materials. Aerosol OT (AOT or sodium bis(2-ethylhexy1)sulfosuccinate) was obtained from Sigma and was used without any further purification. Heptane, isooctane, nonane, and decane
* To whom correspondence'should
be addressed.
UniversitC P. et M. Curie.
* CEN Saclay. @
Abstract published in Advance ACS Absrracts, July 15, 1995.
0022-3654/95/2099-12941$09.00/0
were purchased from Flhka (purity 99.5%). Millipore doubly distilled water was used for all the experiments. Experimental Apparatus. Conductivity measurements were performed using a platinum electrode and a Tacussel CD 810 instrument. All samples were thermostated (T = 22 "C). Small-angle X-ray scattering (SAXS) experiments were performed at LURE (Orsay) on the D22 diffractometer at room temperature.
Definitions of Parameters The water volume fraction, #,
is defined as:
VH~Oand Veil are the water volume and the solvent volume constituting revere micellar solutions. VAOT is the volume occupied by the AOT molecules in reverse micellar solutions. The water volume fraction, #w, can also be described as the following:
where N is the number of micelles. Each micelle is characterized by a water pool radius, r,, and a water volume v,. In (2), the brackets mean an average over the size distribution of the droplets. N and (v,) can be derived from a fit of SAXS experiments. The micellar volume fraction, #,, is equal to
which gives
4 m = 4, + ~ ~ ( ~ A O T V A O T J ~ H ~ O V H , ~ ) where nAOT and nH20 are the number of AOT and water molecules. Taking into account the molecular volume of AOT and water (VAOT = 640 A3 and V H ~ O= 30 A3). The micellar volume fraction, #,, is given by f$m
= 4, (1
+ 21/w)
(3)
The polar volume fraction, #p, includes the volume of water, VH~O, and the volume of the head polar group of AOT which is assumed to be a third of the molecular volume of AOT (VAOT). Similarly, rp and vp,are defined as the polar radius and volume 0 1995 American Chemical Society
12942 J. Phys. Chem., Vol. 99, No. 34, 1995
Cassin et al. structure factor, respectively. The scattering wave vector is given by
respectively. The polar volume fraction is expressed as
4p= @,(I + 7/w)
(4) q = (4dL)sin 8/2
The hard sphere volume fraction, 4d
&Jdr
is defined as
= (N/vO>
(5)
where d is the hard-sphere diameter of the micelle. The packing fraction, qd, is usually defined in the liquidstate theory. We have to determine a relationship between the hard-sphere volume fraction, #d, and the packing fraction, qd. The sum of the volume of water, solvent and AOT molecules constituting reverse micellar solutions, VO,differs from the total volume of the solution after mixing of componant, V. From thermodynamics, the ratio (VdV) can be easily calculated as the following:6 ( v ~ v= , 1
+ 4m(ms/(n/6)(d3)>xs
(6)
where tir, and x are the solvent molecular volume and isothermal compressibility, respectively. The ratio VdV can be approximate to unity and the hardsphere volume fraction, $d, can be replaced by the packing fraction, r,?& since the two following conditions are verified: (i) The solvent molecular volume, m,,is very small compared to the micellar volume. (ii) The isothermal compressibility xs of pure solvent is highly smaller than 1. The average value of the hard-sphere diameter, dhs, is assumed, to be proportional to (rp)for the various &, values. From the results given in ref 7, we assume that dhs = 2[(rp) 41 8, and from which we get
+
qd
= 4 d = @p( 1
+ 12/(rp))= d,( 1 + 7/w)(l f 12/(rp))= &,,(I + 12/(rp))(1+ 7/w)/(l + 21/w) (7)
It is assumed that (r,,") = (rp)"for n = 2 or 3. The solvent volume fraction, &, is defined as
4 s = voil/(vH20 + veil + vAC),) = voi,/vo $J~ is related to the solvent packing fraction qs as the following:
where eS and vs are the density of pure solvent and the geometrical volume of one solvent molecule, respectively. As above, we must determine a relationship between the micellar volume fraction, $m, and the solvent packing fraction, qS. For small @m and the definition of &, we get C#J~ = 1 - d m . So in (8) we can approximate VdV to 1 and expand & in terms of &,, with
+
with B(qri) = 3
sin qri cos qri
The polydisperse form factor, P,(q), is obtained from P ( q ) by assuming a gaussian distribution of sizes as:
where CT is the root-mean-square deviation from the mean polar radius (rp). Structure Factor. In the Chen et al. approximation,8instead of calculating a sum of partial structure factors an effective monodisperse structure factor S(q) is introduced, it is related to Sm(q) through the relation
where p(q) is a corrective term taking in account the size polydispersity in the form factor. To calculate S(q), the McMillan-Mayer formalismIO is used. That is to say, it is assumed that the solvent structure appears implicitely via the potential of mean force Umm(r) in which r is the distance between two micelles. It is generally assumed that Umm(r) contains a repulsive part at short distance and an attractive part having a short range compared to the micellar radius. It is possible to mimic the physical ingredients which determine Um( r ) by using a mathematical convenient form which is the socalled sticky hard-sphere potential VAs(r) defined as' ' . I 2
where
where q- is the packing fraction in the pure solvent.
A(r) =
Treatment of the SAXS Experiments The analysis of scattered X rays intensity Z(q) requires to take into account the existence of an inherent polydispersity in droplet size. According to Chen et al.8-9an approximate relation for I(q) is
I(q) = ("P,(q)Sm(q)
where 8 and L are the scattering angle and the wavelength of the X-ray beam. Form Factor. The form factor, P(q), depends on water pool The via r,, v,, and the electronic density ew= 0.334 eform factor also depends on the electron density of sulfonate groups of AOT molecules which is assumed uniformly distributed with the density ep = 0.55 ein a spherical shell located between the radius r, and rp = r, 4 A. The solvent electronic density, ees,is estimated to 0.25 e- A-3. P ( q ) can be written as
(10)
where q, P,(q), and Sm(q) are the scattering wave vector, the average polydisperse form factor, and the average intermicellar
+
m
for r < dhs
= [-ln[dhs/12z(6 - dhs)]
=O
for dhs < r < 6
forr'd
(15)
here dhs and z-' are the average value of the micellar hardsphere diameter and the stickiness parameter, respectively. The real pair potential Umm(r)and its limiting form VAS(T) are related via the second virial coefficient; we have l/z = J[exp - PUmm(l,2)- 13 dr
(16)
AOT Reverse Micelles: Depletion Model
6.0
2.0
5
cd
0.8
.
c G 0.4
.
-8- Isooctane
3
n
w
OK-
J. Phys. Chem., Vol. 99, No. 34, 1995 12943
0 .
4.8
-
0
0.04
0.08
0.12
Figure 1. Scattered intensity versus wave vector for AOT-waternonane microemulsions at w = 40 varying water volume fraction, &. (A) & = 0.04; (B) r # ~ = ~ 0.06: (C) @w = 0.12. (0)experimental scattering intensity: (-) polydisperse sticky hard-sphere simulation.
From VAS@)and the PY approximation, an analytical expression of S(q) can be obtained. It is in good agreement with the available Monte Carlo simulation although some weaknesses exist. In this approximation the structure factor can be written s(q*rJ,lddhs). By using dhs = 2[(r,)
+ 41 8,for various q$,, values, from (7)
it is possible to relate, we can relate q d to &, or &,. Hence, for a given & or #J,, value, the fits of SAXS experiments can be obtained by varying three parameters: the mean polar radius (rp), the size polydispersity u,and the stickiness parameter T-I. From these, we get the others parameters such as N , (v,), and dhs.
Results and Discussion Variation of the Stickiness Parameter with the Micellar Volume Fraction: Depletion Model. By SAXS, the structural parameters of droplets are determined for fixed w,various &, and three solvents: isooctane, nonane, and decane. Figure 1 shows the scattering intensity versus the wave vector for AOTwater-nonane microemulsions at w = 40 and different qh. Table 1 shows that (Ip)is roughly constant in all the systems, whereas t-l increases with the length of the oil alkyl chain and decreases with the micellar volume fraction whatever the solvent is, as shown in Figure 2. Such a decrease in the stickiness parameter has been previously obtained in AOT/water/isooctane micro emulsion^'^^'^ and were not well understood. From Figure 2 it seems, as expected, that Umm(r)is not the usual potential of mean force at infinite dilution. This might suggest to treat micelles and solvent on the same footing. However in what follows we will see that a large part of the experimental results can be described by simply rescaling the solvent packing fraction according to (9). From statistical mechanics, the potential of mean force Umm( r ) between two micelles can be written as where fmm(r)is the Mayer function associated with the direct
TABLE 1: Polar Radius, (rp)(A), Stickiness Parameter, z-l, and Polydispersity, 0, Deduced from SAXS at Various Water Volume Fraction, q&, in Three Dflerent Solvents dw
(A)
t-
'
0
Isooctane
0.05
65
4.00
0.06 0.10 0.11 0.13
59
2.30 2.00 1.35 1.oo
0.21 0.20 0.21 0.24 0.25
5.90 5.00 3.35 2.30 1.10
0.22 0.20 0.21 0.22 0.22
6.25 4.15
0.20 0.20 0.22
65 65 61 Nonane
0.04 0.06
0.09 0.12 0.20
59 62 60 64
58 Decane
0.06 0.15 0.22
60 62 57
3.00
micelle-micelle interaction, while ymm(r)is a function defined in terms of diagrams and taking into account the micellemicelle interaction mediated by the solvent. The function ymm(r),which is calculated taking into account only two micelles can be splitted in two parts: (i) ymmHS(r) which represents the coupling between two micelles when the micelle-solvent and the solvent-solvent interaction are restricted to the hard-sphere interaction. (ii) A non-hard-sphere one, Ymm(r)NHS. Hence, the potential of mean forcelo is given by
In a monocomponent hard-sphere fluid, the excluded volume introduces via ymmHs(r) an effective attraction between molecules at short distances. Because of this effect the contact value of the pair distribution function is larger than 1. The effective attraction between molecules at short distance is enhanced for large hard sphere immersed in a solvent composed by small hard-sphere. That this tends to stick together the large spheres is very well-known in the case of free polymers in colloidal suspensions, where it has been called the depletion effect. We will focus now on the calculation of ymmHS(r), Le., the depletion effect. ymmHS(r)depends on (i) The ratio a = (dhdd,), where d, is obtained by assuming that the volume of one solvent molecule is equivalent to a sphere of diameter d,. (ii) The solvent packing fraction, q,, in the solution.
Cassin et al.
12944 J. Phys. Chem., Vol. 99, No. 34, 1995
TABLE 2: Properties of the Pure Solvents: Packing Fraction r,~, Ratio a = (d$dhs)and the Stickiness Parameter at Infinite Dilution rm-lin the Three Different Solvents ra L-' isooctane nonane decane
0.41 0.42 0.44
21.3 20.6 19.8
As it is shown in (9), q, = q,(l - &). ymmHS(r) depends on qh. According to (17), we can write
7.1 8.3 11
d log(l/z)/d&, = -(lAog e)(3/,aqw)
(HNCP) (22b)
This indicates that
where the index 00 refers to quantities calculated with the pure solvent number density Bs. Here we do not discuss the physical origin of the adhesion effect which is related to the first bracket in the relation (18). It could results from an entropy contribution's or the effect predicted in ref 16; in any case the relation (18) holds. Because the introduction of $+, does not produce a qualitative change in the solution, it can be assumed that [exp(-PVmm(r))], behaves like a sticky potential of stickiness parameter zm-' and the ratio bmmHS(r)]/[ymmHS(r) 1, depends on +m. This sticky potential associated with z,-' selects the distance r = dhs and therefore the stickiness parameter, t-l , corresponding to exp(-PV,,,,(r)) can be written as1'
where gmmHS(dhs)is now the contact value of the pair correlation function for a system in which there are only the two micelles under consideration. The change of l/z with &,, depends on the approximations used in the calculation of gmmHS(dt,,).Two approximations have been chosen: (i) gmm(PY)results from the PY closure.'* (ii) gmm(HNCP)is based on an improvement of the HNC closure. l 9 These two approximations are expressed as gmm(py>= [1 + r,/2
Hence the dependence of #, with l/z is considered from the derivatives
+ 3/2rs(a- 1>1/[1- qS12
In Figure 3, we have reported the experimental values of l/z and theoretical predictions deduced from eqs 21 versus & for the three solvents. As it is shown in Figure 3, experimental data are systematically framed by theoretical curves. For isooctane, nonane and decane the experimental slope correspond to -4.0, -3.0, and -2.3, respectively. The theoretical values are roughly solvent independent, they correspond to - 1 for PY and -5.5 for HNCP. Of course the question concerning the sensitivity of our results to the approximate eq 10 arises. Figure 4 shows a good agreement, at low q value, between the simulated structure factor, Sm(q), calculated with the sticky hard sphere model according to (14) and the experimental structure factor deduced from Figure 1. Whereas at high q value a phase shift is observed. Such a disagreement is not easily explained. It could be due to the fact that the calculation has been made for an average droplet's size and not for various size due to polydispersity. Robertus et al. have performed a sophisticated analysis of their S A X S experiment^.'^.^^ The polydispersity is taken into account by considering the micellar solution as a real mixture. The partial structure factors have been calculated in the PY approximation assuming that z is size independent. The same kind of results is obtained, the decreasing of l/z with &, is roughly independent of the water content w and the slope d log(l/z)/d& has been estimated to be -2.4. No significantly differencies from our own results are observed.
(20a)
The result (20b), derived by adding some bridge diagrams in the standard HNC, has a simple physical meaning. It can be obtained by integrating the depletion force calculated in ref 20 over distances located between dhs and dhs f d,, assuming that the solvent is uniformly distributed in this region with a number density es. The validity of (20b) is determined only by this last approximation. Table 2 gives the values obtained for q, obtained from the values of d, and ps,the ratio a obtained by assuming dhs = 128 A, and the stickiness parameter l/t, at infinite dilution deduced from Figure 2. Since a value is larger than unity, from (19b) and (20) we get the PY and HNCP approximations:
I
0.10
0.20
0.30
I
Figure 3. Stickiness parameter evolution's in function of the reverse micellar volume fraction. (A) Isooctane; (B)nonane; (C) decane. (-1 PY, (- - -) HNCP; (0)experimental data.
AOT Reverse Micelles: Depletion Model
J. Phys. Chem., Vol. 99, No. 34, 1995 12945 TABLE 3: Values of the Micellar Volume Fraction Corresponding to the Percolation Threshold Obtained from Eqs 21a and 24 for Water/AOT/Alcane Reverse Micellar Systems: Water Content, w, Kept Constant and Equal to 40 (Experimental Values from Ionic Conductivity Measurements) isooctane nonane decane
0.37 0.33 0.28
0.49
0.35 0.22
3. Similar results have been previously obtained from permitivity measurements by Robertus et aLZ3 From a theoretical point of view, the percolation process is related to the formation of an infinite cluster in the system. Chiew et al. have defined the percolation condition for a sticky hard sphere ~ y s t e m : ~ ~ . ~ ~
0
0.04
0.08
where v d p is the packing fraction at the percolation threshold and 1,a dimensionless parameter. From its expression we get
0.12
Figure 4. Structure factor obtain from the scattered intensity, as it is explain in text, versus wave vector for AOT-water-nonane microemulsions at w = 40 varying water volume fraction, &. (A) & = 0.04; (B)qL = 0.06; (C) I& = 0.12. (- - -) Experimental structure factor; (-) polydisperse sticky hard-sphere structure factor.
By a combination of (21) and (24), an equation for vdp(PY) and vd,,(HNCP) have been deduced. vdp(PY) has been determined numerically while no solution exists with HNCP approximation. From (7), the micellar volume fraction at the percolation threshold q$,,p(PY) can be obtained. The comparison between the experimental and calculated results is given in Table 3. The values of q$,,p(PY) are in a rather good agreement with the experimental results. However, to accept a spherical model for reverse micelles at the percolation threshold is an approximation that can lead to some discussions.
Conclusion The SAXS experiments show that the stickiness parameter is a decreasing function of the micellar volume fraction &. From an expression of the potential of mean force, we have shown that the sticky potential depends on the solvent packing fraction in the solution, due to this fact the depletion force between two micelles depends on &,, and finally we predict that t-' decreases with &,. The predictions of the model depend on the contact value for the correlation function of two large hard spheres in a solvent of small hard-spheres. The approximations used suggest that the effect considered could contribute for a large part to the decrease of t-l with &. The same model has been used for predicting the position of the percolation threshold observed by conductivity measurements. The PY approximation gives rather satisfying results. t-l
10
30
50
Figure 5. Variation of conductivity in AOT-water-oil microemulsions w = 40, with water volume fraction, w = 40. (0)Isooctane, (0) nonane; (0)decane.
Due to the crudeness of the model, we may consider these results as rather satisfying. We may conclude that the decrease of the stickiness parameter with @,, can be attributed for a large part to the modification of the depletion effect with the micellar volume fraction. Determination of the Percolation Thresholds and Comparison with the Predictions of the Depletion Model. Conductivity measurements have been performed with AOT reverse micelles in various bulk solvents. The water content is kept constant and equal to 40, the results are reported in Figure 5. The conductivity strongly depends on the water volume fraction, &. At low &, the conductivity is very low and remains in the nanosiemen range. With an increase of the water volume fraction suddenly, a drastic increase in the conductivity, about 3 orders of magnitude, is observed. This is due to percolation The percolation onset corresponds to the value of & for which the conductivity suddenly increases. By decreasing the length of the oil alkyl chains the percolation onset takes place at higher & as shown in Figure 5 and Table
References and Notes ( 1 ) Mitchell, D.; Ninham, B. J. Chem. Soc., Faraday Trans. 1981, 77, 601. (2) Pileni, M. P. Srrucrure and reactivify in Reverse Micelles; Elsevier: Amsterdam, 1989; Chapter 1. ( 3 ) Degiorgio, V. Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Proceedings of the International School of Physics, North Holland, 1985. (4) Wilcoxon, J. P. Phys. Rev. A 1991, 43, 1857. (5) Huruguen, J. P.; Authier M.; Greffe J. L.; Pileni, M. P. J. Phys. Condens. Matter 1991, 2, 865. (6) Buff, F. P.; Brout, R. J. Chem. Phys. 1955, 23, 3. (7) Huang, J. S. J. Chem. Phys. 1985, 82, 1, 480. (8) Chen, S. H.; Lin, I. L.; Huang, J. S. Physics of Complex and Supermolecular Fluids; Safran, S. A,, Ed.; Wiley-Interscience: New York, 1987.
12946 J. Phys. Chem., Vol. 99, No. 34, 1995 (9) Kotlarchyk, M.; Chen, S.H.;. Huang J. S.; Kim, M. W. Phys. Rev. A 1984,29, 2054. (10) McMillan, W. G.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276. (11) Baxter, R. J. Aust. J. Phys. 1968, 21, b, 563. (12) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. (13) P i t 6 F.; Regnault C.; Pileni M. P. Lungmuir 1993, 9, 2855. (14) Robertus, C.; Philipse, W. H.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1989, 90, 8, 4482. (15) Lemaire, B.; Bothorel, P.; Roux, D. J. Phys. Chem. 1983,87, 1023. (16) Biben, T.; Hansen, J. P. Europhys. Left. 1990, 12, 347. (17) Hansen, J. P.; Mc Donald, I. R. Theory of simple liquid: Academic Press: New York, 1986.
Cassin et al. (18) Leibowitz, J. L. Phys. Rev. 1964, 133, 895. (19) Attard, P.; Patey, G. N. J. Chem. Phys. 1990, 92, 4970. (20) Kekicheff,'P.; Richetti, P. frog. Colloid Polym. Sci. 1992, 88, 8. (21) Lagues, M. J. Phys. Lett. 1979, 40, 331. (22) Grest, G . ;Webman, I.; Safran, S. A,; Bug, L. Phys. Rev. A 1986, 33, 2842. (23) Robertus, C.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1990, 93, 1293. (24) Chiew, Y. C.; Glandt, E. D. J. Phys. A: Math. Gen. 1983, 16, 2599. (25) Seaton, N. A.; Glandt, E. D. J . Chem. Phys. 1985, 87, 1785.
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