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Langmuir 1996, 12, 2408-2411
Relation between Exchange Process and Structure of AOT Reverse Micellar System T. K. Jain,† G. Cassin,† J. P. Badiali,*,† and M. P. Pileni*,†,‡ Universite´ Paris VI, Laboratoire S.R.S.I., 4 Place Jussieu, 75252 Paris, France, and CEN Saclay, DRECAM-S.C.M, 91191 Gif-sur-Yvette, France Received September 11, 1995. In Final Form: January 10, 1996X The decreases of the exchange micellar rate constant, determined from stopped flow kinetic study, and of the sticky parameter, determined by small angle X-ray scattering, with the water volume fraction are strongly correlated. This phenomenon is explained according to the contact value of the micelle-micelle correlation function and the surfactant monolayer rigidity. It is shown that the attractive interactions govern the decrease of the exchange process with the increase in the number of droplets.
I. Introduction
nm ) 1.4 × 104 L mol-1 cm-1). Murexide solutions were used immediately after preparation since they undergo slow decomposition. II-2. Experimental Apparatus. The kinetics of the metalligand complexation reaction was followed at a fixed temperature (T ) 22 °C) by three syringes in the Biologic stopped flow instrument (SFM 3M). A Biologic modular optical system and Biokine analysis software were used for the acquisition and analysis of the kinetic data. Small angle X-ray scattering (SAXS) experiments were performed at LURE (Orsay) on the D22 diffractometer at room temperature.
The ternary phase diagram1 water/AOT/isooctane shows a large zone where the water in oil microemulsion (L2 phase) takes place. In this isotropic liquid phase the surfactant (AOT) forms a monolayer dividing water nanodroplets from oil. The ratio of water over surfactant concentrations (w ) [H2O]/[AOT]), called the water content, determines the size of the reverse micelles.2 Lower consolute points are observed in AOT-reverse micellar solutions.3 This phase separation due to a competition between internal energy favoring phase separation and entropy effects promoting miscibility appears at high temperature. This implies that the effective intermicellar potential depends on temperature.3 These droplets are submitted to the Brownian motion and to their attractive interactions. Collision between two droplets results in the formation of short-lived droplet clusters that dissociate in two isolated droplets.4-7 In the present paper, it is shown that the intermicellar exchange rate constant decreases with water volume fraction. This effect is linked to the variation of the intermicellar interaction potential. The attractive interactions govern the decrease of the exchange process with the increase in the number of droplets. The interfacial rigidity decreases with the increase in the water volume fraction.
VH2O and Voil are the water and solvent volumes, respectively. VAOT is the volume occupied by the AOT molecules in reverse micellar solutions. V0 differs from the total volume of reverse micellar solution, V; the ratio V0/V has been analyzed in ref 8. The micellar volume fraction, φm, is equal to
II. Experimental Section
φm ) φw + VAOT/V0
(2)
φm ) φw + φw (nAOTvAOT/nH2OvH2O)
(3)
II-1. Materials. Aerosol OT (AOT or sodium bis(2-ethylhexyl)sulfosuccinate) was obtained from Sigma and was used without any further purification. Isooctane was purchased from Fluka (purity 99.5%). Millipore doubly distilled water was used for all the experiments. Zinc sulfate heptahydrate (Prolabo) and murexide (Sigma) were used as recieved. The purity of murexide was checked spectrophotometrically (extinction coefficient at 522 * Authors to whom correspondence should be addressed. † Universite ´ Paris VI. ‡ CEN Saclay. X Abstract published in Advance ACS Abstracts, April 15, 1996. (1) Mitchell, D.; Ninham, B. J. Chem. Soc., Faraday Trans. 1981, 77, 2, 601. (2) Pileni, M. P. Structure and Reactivity 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) Fletcher, P. D. I.; Robinson, B. H. Ber. Bunsen-Ges. Phys. Chem. 1981, 863. (5) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1987, 83, 985. (6) Robinson, B. H.; Toprakcoglu, C.; Dore, J. C.; Chieus, P. J. Chem. Soc., Faraday Trans. 1984, 80, 413. (7) Clark, S.; Fletcher, P. D. I.; Xiling, Y. Langmuir 1990, 6, 1301.
III. Definitions of Parameters Several parameters used to describe reverse micellar solutions are defined. The water volume fraction, φw, is defined as
φw ) VH2O/V0 with V0 ) (VH2O + Voil + VAOT) (1)
which gives
where nAOT and nH2O are the number of AOT and water molecules. By taking vAOT ) 640 Å3 and vH2O ) 30 Å3, φm is given by
φm ) φw(1 + 21/w)
(4)
The solvent packing fraction, ηS, is defined according to
ηS ) [φSFS](VO/V)vs
(5)
where FS, vs, φS ) Voil/V0 are the density of pure solvent, the geometrical volume of a solvent molecule, and the solvent volume fraction. For low micellar volume fraction, (8) Cassin, G.; Badiali, J. P.; Pileni, M. P. J. Phys. Chem., in press.
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Langmuir, Vol. 12, No. 10, 1996 2409
φS, the solvent packing fraction, ηS, is expressed as8
ηS ) φSFSVs ) FSVs(1 - φm) ) η∞(1 - φm)
(6)
where η∞ is the pure solvent’s packing fraction. The micellar hard sphere packing fraction is estimated as8
ηd ) (N/V0)(π/6)dhs3
(7)
where N and dhs are the number of micelles and the mean value of the hard sphere diameter associated to a micelle, respectively. The latter, dhs, is calculated by taking into account the size polydispersity of AOT reverse micelles. The packing fraction ηd is related to φw by the approximate relation
ηd ) φw(1 + 7/w)(1 + 12/〈rp〉)
between the radius rw and rp ) rw + 4 Å. The solvent electronic density, Fes, is estimated to 0.25 e- Å-3. P(q) can be written as
P(q) ) [((Fes - Fp) vp B(qrp)) + ((Fp - Fw) vw B(qrw))]2 (11) with
B(qri) ) 3
IV. Experimental Treatments IV-a. Stopped Flow Experiments. Previously4-7 micellar exchange rate constant has been determined by using the well-known metal-ligand complexation, described as
Pσ(q) )
b
where Zn , Mu , and [Zn-Mu] are zinc chloride, ammonium purpurate usually called murexide (Mu-) and the complex, respectively. Zinc-murexide complex, [ZnMu], is characterized by an absorption spectrum centered at 440 nm, whereas Mu- absorption spectrum is centered at 520 nm. In aqueous solution (20 °C), the forward, kf, and back, kb, kinetic rate constants are found equal to (2.8 ( 0.3) × 107 L mol-1 s-1 and (4.5 ( 0.2) × 104 s-1.9 Each reactant (Zn2+ and Mu-) is highly soluble in aqueous solution and not in isooctane. The procedure used to determine the intermicellar exchange rate constant is similar to that described by Fletcher et al.4-7 IV-b. SAXS Experiments. Analysis of scattered X-ray intensity, I(q), requires taking into account the existence of an inherent polydispersity in droplet size. According to Chen et al.10,11 an approximate relation for I(q) is
(9)
where N, q, Pσ(q), and Sm(q) are the number of micelles, the scattering wave vector, the average polydisperse form factor, and the average intermicellar structure factor, respectively. The scattering wave vector is given by
q ) (4π/λ) sin θ/2
1
x2πσ2
∫
(
P(q) exp
)
(rp - 〈rp〉)2 2σ2
Sm(q) ) 1 + β(q)[S(q) - 1]
-
I(q) ) (N/V) Pσ(q) Sm(q)
(qri)3
(10)
where θ and λ are the scattering angle and the wavelength of the X-ray beam. Form Factor. For a given micelle the form factor, P(q), depends on the water pool through its radius rw and its electronic density Fw ) 0.334 e- Å-3. The form factor also depends on the electron density of sulfonate groups of AOT molecules which is assumed uniformly distributed with the density Fp ) 0.55 e- Å-3 in a spherical shell located (9) Mass, G. Z. Phys. Chem. (Munich) 1968, 60, 138. (10) Chen, S. H.; Lin, I. L.; Huang, J. S. Physics of Complex and Supermolecular Fluids; Safran, S. A., Ed.; Wiley Interscience: New York, 1987. (11) Kotlarchyk, M.; Chen, S. H.; Huang, J. S.; Kim, M. W. Phys. Rev. A 1984, 29 (4), 2054.
(12)
drp
(13)
where σ is the root-mean-square deviation from the mean polar radius 〈rp〉. Structure Factor. Instead of calculating a sum of partial structure factors, the Chen et al. approximation10 introduces an effective monodisperse structure factor S(q). It is related to an average structure factor, Sm(q) through the relation
kf
Zn2+ + Mu- y\ z [Zn-Mu] k 2+
}
sin qri - qri cos qri
The polydisperse form factor, Pσ(q), is obtained from P(q) by assuming a Gaussian distribution of sizes as
(8)
The mean value of the radius of the polar volume, 〈rp〉, can be deduced from SAXS experiments.7
{
(14)
where β(q) is a corrective term taking in account the size polydispersity in the form factor. In order to calculate S(q), the McMillan-Mayer formalism12 is implicitly used. That is to say, it is assumed that the solvent structure only appears through 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 localized near the micellar radius. It is possible to mimic the physical ingredients that determine Umm(r) by using a mathematical convenient form which is the so-called sticky hard-sphere potential VAS(r) defined as13,14
VAS(r)/kBT ) limδfdhs A(r) where
A(r)
{
for r < dhs ) +∞ ) -ln[dhs/12τ(δ - dhs)] for dhs < r < δ (15) )0 for r > δ
Here dhs is the average value of the micellar hard-sphere diameter and τ-1 is the so-called stickiness parameter. The real pair potential Umm(r) and its limiting form VAS(r) are related through the second virial coefficient:
1/τ )
∫[exp(-βUmm(1,2) - 1)] dr
(16)
From VAS and the Percus-Yevick (PY) approximation, an analytical expression of S(q) can be obtained.13,14 V. Results V-a. Determination of the Micellar Exchange Rate Constant. The kinetic experiments are performed by mixing (with “stopped flow” technique) two microemulsions having the same micellar concentration, [RM], and water content, w ) 40. One contains zinc ions and the (12) McMillan, W. G.; Mayer, J. E. J. Chem. Phys. 1945, 13, 276. (13) Baxter, R. J. Aust. J. Phys. 1968, 21b, 563. (14) Baxter, R. J. J. Chem. Phys. 1968, 49 (1), 2770.
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Jain et al.
Figure 2. Variation of the stickiness parameter with the water volume fraction φw at w ) 40. Figure 1. Variation of the second-order intermicellar exchange rate constant, kex, with the water volume fraction φw at w ) 40. Table 1. Polar Radius, 〈rp〉 (Å), Stickiness Parameter, τ-1, and polydispersity, σ, Deduced from SAXS, at Various Water Volume Fractions Ow φw
rp (Å)
τ-1
σ
0.05 0.06 0.10 0.11 0.13
65 59 65 65 61
4.00 2.30 2.00 1.35 1.00
0.21 0.20 0.21 0.24 0.25
other murexide. The overall micellar concentration of murexide and zinc was kept at 50 µM and 1.5 mM, respectively. Excess zinc concentration is used to ensure pseudo-first-order conditions. Nevertheless, zinc and murexide micellar concentration were limited such that the procedure used by Fletcher et al.4-7 to determine the intermicellar exchange rate constant can be applied. The characteristic absorbance of murexide at 520 nm is very sensitive to the complex formation. A decrease in the absorbance of the microemulsion solution at 520 nm and an increase at 440 nm, corresponding to the formation of Zn-Mu complex, is observed. The ratio of absorbances at 520 and 440 nm is taken as a sensitive function corresponding to the amount of the free murexide present in the reaction mixture. This on substracting from total murexide concentration gives an estimate of the concentration of complex in the reaction mixture. In all cases the experimental transient is indistinguishable from a single exponential and the analysis of these transients gives the pseudo-first-order rate constant kobs. To determine the second-order intermicellar exchange rate constant, kex, the procedure described in refs 4 and 15 has been followed by using a Poisson distribution of species among reverse micelles. Figure 1 shows the decrease in the intermicellar exchange rate constant with the increase in the water volume fraction, at w ) 40. The intermicellar exchange rate constants determined in the present study are lower compared to those obtained by Fletcher et al.4,5,15 These differences can be attributed to the fact that for their analysis the authors worked with a lower AOT concentration range. V-b. Determination of the Reverse Micellar Size and Sticky Parameter. In absence of metal-ligand complexation, the structural parameters of droplets are determined by SAXS, at w ) 40, and various water volume fractions, φw. The mean value of the radius of the polar volume, 〈rp〉, is roughly constant (Table 1) whereas the stickiness parameter, τ-1, decreases with the micellar volume fraction (Figure 2). The SAXS experiments have been performed at room temperature and the decrease of (15) Fletcher, P. D. I. Ph.D. Thesis, 1982, University of Kent, U.K.
Figure 3. Schematic representation of the intermicellar exchange process: (step 1) collision between reverse micelles governed by stickiness parameter τ-1; (step 2) opening of the interfacial layer governed by surfactant layer rigidity.
the stickiness parameter, given Table 1, is too wide to be attributed to small temperature variations. Such a decrease previously observed16,17 is explained in taking into account the micelle-micelle interactions mediated by the solvent (depletion model8). When large hard spheres are immersed in a solvent composed by small hard spheres, the excluded volume between large spheres introduces an effective attraction that depends on the solvent concentration. It has been shown that an increase of the micellar volume fraction φm leads to a decrease of solvent packing fraction, ηS, according to (6) and then to a decrease of 1/τ with φm or φw. The theoretical estimation of this effect is in a rather good agreement with the experiments.8 VI. Discussion The intermicellar exchange process can be expressed as the sum of two equilibriums (Figure 3): (i) the formation of a dimer made of two micelles in contact; (ii) the water pool micellar exchange. The first equilibrium is related to the attractive interactions between droplets whereas the second one is associated to the interface rigidity. The intermicellar exchange rate constant, kex, can be expressed as
kex ) C0k1k2 where k1 and k2 are the probabilities of collision of two micelles and of the interface opening when two micelles are in contact, respectively. The constant, C0, includes all the other processes (in particular, the details of the kinetic processes). The rate constant k1 is related to the intermicellar structure. The probability to have two micelles separated by a distance between r and r + dr is given by 4π g(r) r2 dr where g(r) is the radial distribution function for two micelles. Depending on the nature of the micellar solution, the exchange takes place when dmin < r < dmax and k1 can be expressed as (16) Pitre´, F.; Regnault, C.; Pileni, M. P. Langmuir 1993, 9, 2855. (17) Robertus, C.; Joosten, J. G. H.; Levine, Y. K. J. Chem. Phys. 1990, 93 (10), 7293.
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k1 ) 4π g(r) r2 dr
Langmuir, Vol. 12, No. 10, 1996 2411
(17)
Since the micellar solutions are described by so-called sticky hard-sphere potential VAS(r) defined in eq 15, it is tempting to assume that dmin ) dhs and dmax f dhs. Then g(r) is restricted to its singular part gsing (r) which can be calculated analytically in the (PY) approximation13,14 as the following:
gsing(r) ) λ(dhs/12)δ(r - dhs)
(18)
with
λ ) (1/τ)[(1 + ηd/2)/(1 - ηd)2 - ηdλ/(1 - ηd) + ηdλ2/12] (19a) λ ) (1/τ) F(ηd)
(19b)
When ηd goes to zero F(ηd) tends to unity and λ tends to (1/τ). If λ is small compared to unity, F(ηd) is roughly equal to (1 + ηd/2)/(1 - ηd)2. This is the value of g(r ) σ) for a pure hard sphere fluid, in the PY approximation. Hence F(ηd) increases with increasing ηd. The rate constant, k2 is related to the dynamical properties of the water-surfactant-oil interface that are related18,19 to the bending elastic modulus of the interface, κ. Since the bending energy is proportional to κ, it is clear that as κ is larger, the fluctuations in the shape of the interface are smaller and then the exchanges of the water pool content are smaller. Thus, we may assume that k2 is proportional to 1/κ. The rate constants k1 [from (18 and 19)] and k2 are proportional to λ and to 1/κ, respectively. Hence, the intermicellar exchange rate, kex, can be expressed as
kex ) C(1/τ)(1/κ) F(ηd)
(20)
where C is a constant which depends both on C0 and on dhs but it is assumed to be independent of ηd. For a pure sticky hard sphere fluid, the parameter 1/τ that determines the strength of the micelle-micelle interaction at infinite dilution is independent of ηd. In the domain of water volume fraction considered here, F(ηd) is an increasing function of ηd and then, from eq 8, with φw (Figure 4A). In addition, there are some experimental facts20 suggesting that (1/κ) should be also an increasing function of ηd. Thus, for a pure sticky hard-sphere fluid, kex as given by (20) should be an increasing function of the polar volume fraction, opposite to the experimental results (Figure 1). A micellar solution is not a pure sticky hard sphere fluid. Analysis of the SAXS experiments shows that the stickiness parameter 1/τ depends on ηd or φw (Figure 2). If φw goes to zero, kex and τ tend to their limit kex0 and τ0 that are determined from Figures 1 and 2. If (1/κ) has a limit (1/κ0), from (20), the micellar exchange rate constant at infinite dilution, φw f 0, is
kex0 ) C(1/τ0) (1/κ0)
(21)
From (20) and (21) it is deduced:
ln[kex/kex0] ) ln[τ0/τ] + ln F(ηd) + ln[κ0/κ]
(22a)
or (18) Helfrich, W. Z. Z. Naturforsch. 1973, C28, 693. (19) Milner, S. T.; Safran, S. A. Phys. Rev. A 1987, 36, 4371. (20) Petit, C.; Holwarth, J.; Pileni, M. P. Langmuir 1995, 11, 2405.
Figure 4. Variation of ln[kex/kex0] determined from Figure 1 and ln[K(ηd)] defined in (22) with the water volume fraction φw at w ) 40. The variations of ln[τ0/τ] and ln[F(ηd)] are also represented: b, X ) F(ηd); O, X ) kex/kex0; +, X ) K(ηd); 0, X ) τ0/τ.
ln[kex/kex0] ) ln K(ηd) + ln[κ0/κ]
(22b)
The values of ln[kex/kex0] and ln[τ0/τ] are determined from experimental data (Figure 4B,C) while those of ln F(ηd) are calculated (Figure 4A) from the analytical expression of F(ηd) (see eq 19). The decreasing of ln[τ0/τ] with φw is partially compensated by the increasing of ln F(ηd). However, the net result is that ln K(ηd) decreases with φw as shown on Figure 4D. Moreover, ln K(ηd) is not too far from the experimental determination of ln[kex/kex0]. If we want to reproduce the experimental results by using (20), it is possible to assume that ln[κ0/κ] is an increasing function of φw. In other words, the bending elastic modulus κ should decrease when φw increases, i.e., when the number of droplets increases. From T-jump experiments20 the decrease in the bending elastic modulus with the water volume fraction has been observed. However the observed variation in these experiments is smaller than that obtained from relation 20. However, the estimate of κ from T-jump experiments depends on the model used. VII. Conclusion In this paper it is shown that the decrease with the water volume fraction of the exchange micellar rate constant, determined from stopped flow kinetic study, and of the sticky parameter, determined by small angle X-ray scattering, are two strongly correlated phenomena. Both of them are explained by taking into account the contact value of the micelle-micelle correlation function. The comparison between these two techniques suggests that the exchange process is governed by the intermicellar attractive interactions. Furthermore, surfactant monolayer rigidity decreases with increasing the number of droplets. Acknowledgment. The authors thankfully acknowledge Dr. Paul Fletcher of University of Hull, U.K., for providing the various mechanistic steps of the reaction mechanism that was used for computer simulation to calculate the value of kex. Also T. K. Jain acknowledges the financial assistance from CNRS, France, in the form of a postdoctoral fellowship. Thanks are due to Dr. Viossat for allowing us use of a computer program similar to that used by Fletcher et al. LA950756P
