Laser Physics, Vol. 14, No. 12, 2004, pp. 1516–1519. Original Text Copyright © 2004 by Astro, Ltd. Copyright © 2004 by MAIK “Nauka /Interperiodica” (Russia).
BIOPHOTONICS
Refraction Index and the Kerr Effect in Droplet Microemulsions M. Richterova, J. Tothova, and V. Lisy* Institute of Physics, P.J. Safarik University, Jesenna 5, 041 54 Kosice, Slovakia * e-mail:
[email protected] Received July 15, 2004
Abstract—In the experiments by Edwards et al. [1], the Kerr constant of droplet microemulsions was found to be proportional to the square of the volume fraction of the droplets, instead of to K ~ φ at small φ, as predicted by the theories that considered the droplet surface layer as a highly flexible membrane. In this work, we propose an alternative attempt to describe the response of a suspension of fluid (emulsion or vesicle) droplets to an electric field. We assume that, when placed in the usual electric fields, the droplets do not deform but are electrically polarized, and that the suspension behaves as a system of rigid dielectric spheres. The refraction index of such a system is given by the Lorentz–Lorenz function. For the Kerr effect, the contribution from the clusters of pairs of droplets polarized in the external field is determining at φ 0. Such an approach allowed us to calculate the refraction index with high accuracy, and it also qualitatively explains the Kerr-effect experiment. The theory is applicable in the limit when the thickness of the surfactant monolayer or lipid bilayer is negligible compared to the droplet radius.
1. INTRODUCTION The interest in vesicle and microemulsion systems has contributed a great deal to the development of soft condensed matter physics. Microemulsions, which are mixtures of two immiscible fluids and surface active substances, have numerous applications in industry. For physicists, chemists, and biologists they are attractive due to the richness of their possible structures and unusual properties. The properties of such suspensions are, to a great extent, determined by the interface between the bulk fluids. Much effort has been devoted to the study of these interfaces. It is thus surprising that for several years there has been practically no reaction to the results of the experiments by Edwards et al. [1]. At the same time, that work has far-reaching consequences for existing conceptions of the static and dynamic properties of microemulsions. The optical birefringence measurements on droplet microemulsions have shown that their Kerr constant is proportional to the square of the concentration of the droplets, including the limit c 0 when the droplets are regarded as independent. This observation disqualifies the current view of the droplet surface as a highly flexible interface with a surface tension that is 100 or more times smaller than in usual liquids [2–5]. Let us make the following estimate: the effective surface tension of the droplet is αeff = (6κ – kBT/8πξ)R–2 [6, 7], where κ is the bending rigidity of the interface, R is the equilibrium radius of the droplet, and ξ is the polydispersity in the sample. Assume that the droplet deforms. In the constant electric field E0, it takes an ellipsoidal shape with an eccentricity e, 2
9ε ε – 1 2 RE e = --------0- ----------- ---------0- , 16π ε + 2 α eff 2
ε = ε 1 /ε 0 ,
(1)
where ε1 and ε0 are the dielectric permittivities of the droplet interior and exterior, respectively. One can judge whether the droplet is deformed or not only on the basis of some observable manifestation of this deformation. In birefringence measurements, the deformation would reveal itself in different refraction indexes with respect to the directions parallel and perpendicular to the applied field E0 [6]: 2
2
n – 1 2 - φ, n || – n ⊥ = 9e n 0 ------------ n 2 + 2
(2)
where φ is the volume fraction of the droplets and n = n1/n0 is the ratio between the refraction indexes of the bulk fluids. Equations (1) and (2) follow from the Lorentz–Lorenz equation [8, 9] by minimalization of the total (surface plus electrostatic) energy of the droplet in the electric field. The estimation for the experimental fields about 105 V/m [1], when the water droplets are suspended in some of the aliphatic oils with ε0 ≈ 2, then shows that n|| – n⊥ ≈ 10–6φ/αeff (αeff is in mN/m). For φ ≈ 10–2, R ≈ 10 nm, κ on an order of kBT, and usual polydispersities [10], one gets αeff ≈ 0.2 mN/m, so that n|| – n⊥ ≈ 5 × 10–8. Such changes in the refraction index would be observable in experiments [1] that are capable of detecting ∆n ≈ 10–8 or smaller (the experiments were 10–1000 times more precise than the previous measurements of the Kerr effect in microemulsions). However, in [1] the linear dependence on φ was not observed: the only registered 2 dependence of the Kerr constant K = (n|| – n⊥)/ E 0 on φ was the quadratic one. Our proposition for the resolution of this contradiction is as follows. The deformability of the droplet interface is, in reality, much smaller than is currently
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assumed (i.e., αeff is larger than 1 mN/m). The droplets behave more like hard spheres than like highly flexible objects. In the electric field, however, many-particle interactions between the spheres appear as a result of their electric polarization. In the lowest approximation, this interaction can be considered as having the pair character. 2. THE MODEL OF THE HARD-SPHERE MICROEMULSION AND THE REFRACTION INDEX The formal realization of this idea involves the generalization of the Clausius–Mossotti equation, ε – ε0 ---------------- = α 1 φ [ 1 + φI ( φ, T ) ], ε + 2ε 0
(3)
where ε is now the permittivity of the suspension and α1 = (ε1 – ε0)/(ε1 + 2ε0) is the dimensionless polarizability of the droplet. The temperature-dependent quantity I(φ, T) is the statistical average of the excess (with respect to the system of two independent spheres) polarizability of two spheres. Analogously, the Lorentz–Lorenz equation for the total index of refraction has to be generalized. In the absence of the field E0, the excess polarizability is determined by an equation derived on the basis of the polarization theory for many-particle systems [11]: I ( φ, T ) ∞ 3 –1
= ( α 1opt R )
∫ (a
+ 2b 1 – 3α 1opt )g 2 ( D )D dD, 2
1
(4)
– + is the optical polarwhere α1opt = izability of one droplet, D is the distance between the droplet centers, and g2(D) is their radial distribution function. Equation (3), after the change of ε to n2 and ε0 2 ( n1
2 2 n 0 )/( n 1
Then, the Lorentz–Lorenz function obtained from Eqs. (3)–(5) has the form 2
2 2 n0 )
2
to n 0 , is also applicable for the calculation of the refraction index of microemulsion. The quantities a1 and b1 are the diagonal (zz and xx) elements of the polarizability tensor and are the solutions of the equations [11]
∑A
lj ( R/D )
l+ j+1
al ,
(5)
l=1
where Alj = (l + j)!/l!j! and Alj = –(l + j)!/(l – 1)!( j + 1)!, when aj is changed to bj . The solution of system (5) can be found numerically or in the form of a rapidly converging sum. The pair distribution function g2(D) can be determined from independent experiments or can be based on some model representation of the microemulsion. As an illustration, we give the result for the simplest case, when g2(D) is modeled by the Heaviside function, which is 0 when D < 2R and 1 when D > 2R. LASER PHYSICS
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(6)
7 3 2 = α 1opt φ 1 + ------ φα 1opt 1 + ------ α 1opt + … . 18 28 For the systems studied in [1] (the water-AOT-aliphatic oil microemulsions) the optical polarizability of the droplets is |α1opt | ~ 0.02–0.04, so that the corrections of the second and higher orders are very small. As distinct from Eq. (6), experiments show a strong dependence of the dielectric constant on the temperature [12–14]. To describe this dependence, a model was proposed in which the droplets can be bound in clusters [11, 12]. The pair correlation function then has the form g 2 ( D ) ≈ 2g 0 R exp ( – E/k B T )δ ( D – 2R ),
(7)
where E is the binding energy of the pair of droplets and g0 is a phenomenological constant. Equation (4) for the clustering spheres at a distance Ds becomes D 3 a 1 + 2b 1 I(φ, T ) ≈ g 0 exp ( – E/k B T ) ------s ------------------–1 . (8) R 3α 1opt D = Ds For the refraction index, we thus have (taking Ds = 2R) 2
n – n0 2 E -------------------2 ≈ α 1opt φ 1 + Aφα 1opt exp – --------- , 2 k BT n + 2n 0
(9)
where the constant A should be determined from experiments. The dielectric measurements on water-AOTisooctane microemulsions [13] show very good agreement with this equation in the range of temperatures from approximately 10 to 35°C, assuming that exp(−E/kBT) ≈ 1 – E/kBT. 3. THE KERR EFFECT
∞
a j = α 1opt δ ij +
2
n – n0 -------------------2 2 n + 2n 0
2
2R
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When the optical birefringence in the electric field is considered, one can act as in [11] and calculate the optical polarizabilities. Let the constant electric field E0 be parallel to the axis z. The resulting electric field of the suspension is determined by the polarization vector P that is, in its turn, given by the one-particle, two-particle, etc., optical polarizabilities. These polarizabilities are the same as found in [11], if one replaces ε and ε0 by n i and n 0 (i = ||, ⊥). The difference between the refraction indexes in the directions parallel and perpendicular to the field, n|| – n⊥, is evaluated from the 2
2
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orientations of the pairs is done using the pair distribution function (7). The result for the Kerr constant, in the lowest approximation for small φ, is
K, 10–21 m2/V2 5
3 2 3 2 K = 〈 n || – n ⊥〉 /E 0 ≈ ------------n 0 ε 0 R α 1 α 1opt φ 〈 ∆I〉 , (13) 4k B T
4
where 〈∆I〉 = 〈I|| – I⊥〉 is determined by
3
∞ 3 –1
2
〈 ∆I〉 = 3 ( α 1opt R )
∫ (a
– b 1 )g 2 ( D )D dD. 2
1
(14)
2R
1 0
Further analysis requires the functions a1 and b1. For dielectric spheres and Ds = 2R, we find from Eq. (5) that 26
28
30
32
34
The dependence of the Kerr constant on the water surfactant ratio (Eq. (17)) for the water–AOT–decane microemulsion studied in [1]. The triangles are for experimental data. The curves correspond to Eq. (16) at c = 0.01 (the lower curve) and c = 0.015.
Lorentz–Lorenz equation. To second order in the density of the droplets, we have 3 3 2 3 2 2 n i ≈ n 0 1 + --- φα 1opt + --- φ α 1opt I i – --- φ α 1opt , 2 2 4
(10)
i = ||, ⊥, 3 2 n || – n ⊥ ≈ --- n 0 φ α 1opt ( I || – I ⊥ ). 2
(11)
This difference must be averaged over the spatial configuration of the droplets (in the absence of the external field) and over the Boltzmann distribution for a dielectric in the field E0 [15]. The latter average is done using the energy of the system in the electric field, –PE0 /2. The polarization vector can be expressed as a sum in which the first term corresponds to individual spheres, the second term to the pairs of spheres, etc. [11, 16]. The second term is given by the irreducible contribution to the polarizability of the pair [16], and is proportional to φ2. Thus, the corrections to the single-particle energy and to the distribution function are small. Since we are interested in the difference n|| – n⊥ that is already ~φ2, to second order in φ the two-particle contribution to the electrostatic energy plays no role in the averaging of the dielectric constants ni . The average thus results merely in the multiplying of n|| – n⊥ by the quantity R ε U el 2 – -------- = -----------0-α 1 E 0 2k B T kBT
a1 – b1 = (3/8) α 1 (1 + 37α1/162 + …). Again assuming clustering of the droplets (7), one finds the following for the Kerr constant from Eqs. (13) and (14): 2
36 ω
3
(12)
(this follows from the Boltzmann distribution, since Uel /kT is small) and in the configurational averaging. The averaging over the positions of the particles and
9 2 3 2 K ≈ ------------g 0 n 0 ε 0 α 1 α 1opt R φ 4k B T 37 × exp ( – E/k B T ) 1 + --------- α 1opt + … . 162
(15)
The correction term in the brackets is on the order of 1% [1]. The temperature dependence of this expression, K ~ (1 – E/kBT)/kBT, is expected to be essentially ~1/T, in qualitative agreement with the experiments [13, 14]. We have compared the result (15) with the experimental dependence [1] of the Kerr constant on the molar ratio ω between the water and the surfactant: ω = number of H2O molecules/number of AOT molecules; i.e., ω = M H2 O W AOT /M AOT W H2 O , where Mi and Wi are the mass and molecular weights of the i. Assuming that d is the thickness of the surface layer and that the ρi are the mass densities, ω can be expressed via the water core radius Rw: W AOT ρ H2 O Rw - ----------- ----------------------------------ω = -----------W H2 O ρ AOT ( R + d ) 3 – R 3 3
w
w
W AOT ρ H2 O R w d - ----------- ------ 1 – ------ + … . ≈ ----------- W H2 O ρ AOT 3d Rw
(16)
The measurements [1] were carried out at various mass concentrations c of the droplets: ρ H2 O M AOT + M H2 O - ≈ ----------φ, c = ------------------------------------------------M AOT + M H2 O + M OIL ρ OIL
(17)
where the approximation assumes small φ and d/Rw . As can be seen from the figure, a good agreement with the experiment occurs only for the lowest droplet concentration, when c = 0.01 (as ω changes from 26 to 36, Rw /d changes from 5 to 6.6). As was expected, for higher concentrations the theory is not able to describe the dependence of the Kerr constant on the water/surfactant ratio (or on the droplet radius) over the whole LASER PHYSICS
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REFRACTION INDEX AND THE KERR EFFECT IN DROPLET MICROEMULSIONS
range of its change. At the largest radii, the agreement is fairly good, but it becomes unsatisfactory at smaller Rw /d, when the influence of the surface layer becomes important. 4. CONCLUSION This work was inspired by the electro-optical experiments in [1] (see also the earlier work [17]), which indicate that the existing theories are unable to describe the response of droplet microemulsions to an external electric field. We propose a model that qualitatively corresponds to the experiments. When the microemulsion is placed in the electric field, the droplets do not deform (in the sense that the deformation is not detectable at the current accuracy of the experiments and the used fields). The main effects originate in their electric polarization. As a result of this polarization, many-particle long-range interactions between the droplets occur, with the pair interactions playing the dominant role. In this representation, the microemulsion in the described experiments behaves as a system of hard dielectric spheres. The refraction index of such a system is given by the Lorentz–Lorenz function (n2 – 2 2 n 0 )(n2 + 2 n 0 )–1, which depends on the optical polarizability of the droplets, α1opt, and on their volume fraction φ. In the first approximation, this function equals φα1opt. This corresponds to the known results for the dielectric properties of suspensions [15]. The next corrections contain higher powers of φ. Calculating the Kerr constant, the linear dependence on φ disappears, so that essentially K ~ φ2. This is in qualitative agreement with experiments [1, 17]. For the Kerr effect, the contributions from clusters of droplets bound in pairs polarized by the external field are determining. Good agreement with experiment is also found for the dependence of the Kerr constant on the droplet radius, in the range of low concentrations of the droplets and small thicknesses of their surfaces as compared to the droplet radius. In spite of the fact that the theory can predict with high accuracy the refraction index of the droplet microemulsions and that it gives a phenomenological description of the Kerr electro-optical effect, in its current form it has significant shortcomings. First of all, the influence of the surfactant monolayer on the optical properties of the droplets is not taken into account. For small droplets, the different optical properties of the interfacial layer and the droplet core should be revealed both in the refraction index and in the Kerr constant of the whole suspension. In the case of the refraction index when the main contribution comes from individual droplets, the necessary generalization is straightforward. In the Lorentz–Lorenz equation, only the dominant term ~φ will be changed by the substitution of α1opt with the known polarizability of a sphere covered by a membrane [9]. The next contributions are relatively LASER PHYSICS
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small at low concentrations of the droplets. For the Kerr effect, however, this approach is inapplicable. In this case, we need to know the optical polarizabilities of the pairs of interacting droplets covered by the surface layer. The solution of this problem is not known. Our model is thus applicable only for systems of droplets with relatively large radii when the influence of the surface layer, which is about 1 nm in thickness, is small. This excludes from consideration such an interesting phenomenon as the negative Kerr effect, which is observed in systems of dispersed droplets with radii of a few nanometers. ACKNOWLEDGMENTS V. Lisy thanks the NWO (Dutch Research Council) for the grant that enabled him to stay at the Leiden Institute of Chemistry, where a part of this work was done. He is grateful to Prof. D. Bedeaux and Dr. A.V. Zvelindovsky for their kind hospitality and for fruitful discussions. This work was supported by grant no. VEGA 1/0429/03, Slovak Republic. REFERENCES 1. M. E. Edwards, X. L. Wu, J. S. Huang, and H. Kellay, Phys. Rev. E 57, 797 (1998). 2. M. Borkovec, Adv. Colloid Interface Sci. 37, 195 (1992). 3. U. Seifert, Adv. Phys. 46, 13 (1997). 4. Micelles, Microemulsions, and Monolyars, Ed. by D. O. Shah (Marcel Dekker, New York, 1998). 5. Modern Characterization Methods of Surfactant Systems, Ed. by B. P. Binks (Marcel Dekker, New York, 1999). 6. M. Richterova and V. Lisy, J. Biol. Phys. 29, 55 (2003). 7. M. Richterova and V. Lisy, Laser Phys. 13, 1301 (2003). 8. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975; Inostrannaya Literatura, Moscow, 1965). 9. E. Van der Linden, S. Geiger, and D. Bedeaux, Physica A (Amsterdam) 156, 130 (1989). 10. T. Hellweg and D. Langevin, Phys. Rev. E 57, 6825 (1998). 11. D. Bedeaux, Z. Phys. B 68, 343 (1987). 12. M. A. van Dijk, J. G. H. Joosten, Y. K. Levine, and D. Bedeaux, J. Phys. Chem. 93, 2506 (1989). 13. G. J. M. Koper, W. F. C. Sager, J. Smeets, and D. Bedeaux, J. Phys. Chem. 99, 13291 (1995). 14. D. Bedeaux, G. J. M. Koper, and J. Smeets, Physica A (Amsterdam) 194, 105 (1993). 15. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (GIFML, Moscow, 1959; Pergamon, Oxford, 1960). 16. Yu. A. Arkatov and I. Z. Fisher, Fiz. Zhidk. Sostoyaniya, No. 7, 71 (1979). 17. M. E. Edwards, Y. H. Hwang, and X.-L. Wu, Phys. Rev. E 49, 4263 (1994).
