Transient Electric Birefringence Study of Rod-Shaped Water-in-Oil

© Copyright 1998 American Chemical Society

JANUARY 6, 1998 VOLUME 14, NUMBER 1

Articles Transient Electric Birefringence Study of Rod-Shaped Water-in-Oil Microemulsions Francesco Mantegazza,†,‡ Vittorio Degiorgio,*,‡,§ Mario E. Giardini,‡,§ A. Louise Price,| David C. Steytler,| and Brian H. Robinson| Istituto di Scienze Farmacologiche, Universita` di Milano, 20133 Milano, Italy, INFM, Istituto Nazionale per la Fisica della Materia, Italy, Dipartimento di Elettronica, Universita` di Pavia, 27100 Pavia, Italy, and School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ, U.K. Received August 6, 1997. In Final Form: October 13, 1997X Transient electric birefringence (TEB) studies have been carried out on water-in-oil w/o microemulsions stabilized by Ni(AOT)2, the nickel salt of bis(ethylhexyl) sulfosuccinate. The system forms rod-shaped droplets at low water contents which convert to more spherical aggregates as the water content is increased. TEB data have been obtained as a function of microemulsion volume fraction, φ, water content, and temperature. Relaxation transients of the electric birefringence signal were found to be nonexponential, following asymptotically a stretched-exponential behavior. The value of the stretching exponent at low volume fraction is consistent with the assumption that the length probability distribution is exponential. A model describing the Kerr response of the microemulsion droplets is developed. By using this model we derive the specific Kerr constant as a function of the volume fraction, finding a good agreement with the experimentally observed behavior. We also use the model to derive, from the initial slope of the relaxation, the mean rod length Lm. It is found that Lm grows approximately as the square root of φ. Values for Lm obtained from TEB are in good agreement with those obtained from small-angle neutron scattering measurements.

Introduction It has recently been shown that reverse micelles and water-in-oil (w/o) microemulsions may exhibit a wide range * To whom correspondence should be addressed at Dipartimento di Elettronica, Universita` di Pavia, via Ferrata 1, 27100, Pavia, Italy. † Universita ` di Milano. ‡ Istituto Nazionale per la Fisica della Materia. § Universita ` di Pavia. | University of East Anglia. X Abstract published in Advance ACS Abstracts, December 15, 1997.

of shapes, in addition to a spherical structure. These include disks,1 rods,2-6 and reverse-vesicular assemblies.7 Quantitative structural studies of reverse-micelles and w/o microemulsions have been made using a variety of (1) Steytler, D. C.; Sargeant, D. L.; Robinson, B. H.; Eastoe, J.; Heenan, R. K. Langmuir 1994, 10, 2213. (2) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. (3) Eastoe, J.; Towey, T. F.; Robinson, B. H.; Williams, J.; Heenan, R. K. J. Phys. Chem. 1993, 97, 1459. (4) Eastoe, J. Langmuir 1992, 8, 1503. (5) Steytler, D. C.; Sargeant, D. L.; Welsh, G. E.; Robinson, B. H. Langmuir 1996, 12, 5312. (6) Feng, K.-I.; Schelly, Z. A. J. Phys. Chem. 1995, 99, 17212. (7) Nakamura, K.; Kunieda, H.; Strey, R. Langmuir 1996, 12, 3045.

S0743-7463(97)00888-3 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/06/1998

2

Langmuir, Vol. 14, No. 1, 1998

techniques,8 particularly small-angle neutron scattering (SANS) and photon correlation spectroscopy (PCS). With PCS it is possible to obtain the translational diffusion coefficient, from which dimensions and interactions of the aggregates in solution can be obtained if their shape is known. Another optical technique which is potentially very useful for the study of anisotropic aggregates is transient electric birefringence (TEB). The amplitude of the TEB signal is very sensitive to the shape anisotropy of the particles, and the temporal behavior of the signal is controlled by rotational diffusion. Since the rotational diffusion coefficient is very sensitive to the particle size. TEB is a simple and accurate technique to measure the size and polydispersity of anisotropic micellar aggregates.9 The TEB technique has previously been applied to a number of amphiphile solutions including rod-shaped ionic micelles,10-12 nonionic micelles near the cloud point,13 reverse micelles,14 droplet clustering in w/o microemulsion media,15 and deformable spherical droplets.16 It is known that the w/o microemulsions stabilized by the surfactant sodium bis(ethylhexyl) sulfosuccinate (AOT) form spherical droplets which have been extensively characterized by a wide range of experimental techniques.17 For AOT w/o microemulsions the droplet radius grows in proportion to the water-to-surfactant molar ratio. The system Ni(AOT)2 behaves in a rather different way. At very low water content, w ) [H2O]/[Ni(AOT)2], small spherical aggregates are present in the oil solution but, as the molar ratio w is increased, rod-shaped microemulsion droplets of short length, L, are first formed. The rod length slightly increases with w up to a maximum at a particular w value after which the length decreases until spherical droplets are again formed close to the solubilization boundary (wmax).3 In the region of w in which rods are formed the radius increases in proportion to w regardless of rod length. Another surfactant showing anisotropic structures is ammonium di(ethyl)hexylphosphate (NH4DEHP), which forms rod-shaped reversed micelles in cyclohexane. The response to water of this system is rather different to Ni(AOT)2 since, when very small amounts of water are added, the viscosity reduces dramatically and when w ) 5 only small spherical aggregates are present. This behavior appears to be quite general and is observed for group I metal salts of DEHP and a range of other surfactant types in hydrocarbon media, e.g., lecithin2 and DDAB.4 We have undertaken a series of TEB measurements on both NI(AOT)2 and NH4DEHP systems, over a range of volume fractions φ, water contents w, and temperatures (8) Cabane, B. In Surfactant SolutionssNew Methods of Investigation; Zana, R., Ed.; Marcel Dekker: New York, 1987. (9) Schorr, W.; Hoffmann, H. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; NorthHolland: Amsterdam, 1985; p 160. Schelly, Z. A. In Microemulsions: Fundamental and Applied Aspects; Kumar, P., Mittal, K. L., Eds.; Marcel Dekker: New York, 1997. (10) Bellini, T.; Mantegazza, F.; Piazza, R.; Degiorgio, V. Europhys. Lett. 1989, 10, 499. (11) Wu, X.-l.; Yeung, C.; Kim, M. W.; Huang, J. S.; Ou-Yang, D. Phys. Rev. Lett. 1992, 68, 1426. (12) Hoffmann, H.; Kra¨mer, U.; Thurn, H. J. Phys. Chem. 1990, 92, 2027. Oda, R.; Lequeux, F.; Mendes, E. J. Phys II 1996, 6, 1429. (13) Degiorgio, V.; Piazza, R. Phys. Rev. Lett. 1985, 55, 288. Degiorgio, V.; Piazza, R. Prog. Colloid Polym. Sci. 1987, 73, 76. (14) Koper, G. J. M.; Cavaco, C.; Schurtenberger, P. In 25 Years of Non-Equilibrium Thermodynamics; Brey, J. J., Eds.; Springer: Berlin, 1995. (15) Eicke, H.-F.; Hilfiker, R.; Thomas, H. Chem. Phys. Lett. 1985, 120, 272. Guering, P.; Cabazat, A. M.; Paillette, M. Europhys. Lett. 1986, 2, 953. Tekle, E.; Schelly, Z. A. J. Phys. Chem. 1994, 98, 7657. (16) van der Linden, E.; Geiger, S.; Bedeaux, D. Physica A 1989, 156, 130. (17) Tapas, K. D. Adv. Colloid Interface Sci. 1995, 59, 95.

Mantegazza et al.

T. Our primary concern is the determination of the length of the rod-shaped aggregates formed. In this paper we concentrate on the description of TEB measurements obtained with Ni(AOT)2 solutions, and we discuss both the steady-state birefringence ∆ns and the birefringence decay after the applied electric field is switched off. The rod lengths obtained from TEB are compared with those derived from SANS measurements made on the same system.18 Experimental Section (a) Materials. The Ni(AOT)2 surfactant was prepared from AOT (Sigma) using a liquid-liquid ion exchange process, as reported previously.19 A saturated solution of nickel nitrate was shaken with AOT in absolute ethanol (1 mol dm-3) and extracted with diethyl ether. The organic phase was washed with water at least eight times, evaporated, and dried for 48 h in a vacuum oven at 40 °C. The surfactants were then stored in a desiccator at room temperature. Total nickel substitution of sodium was confirmed by UV-VIS spectroscopy. In all the measurements we have used analytical grade cyclohexane (BDH, AnalaR) as solvent (oil) and analytical grade water (BDH, AnalaR). Analysis of the surfactant showed the presence of trace quantities of Ni(NO3)2, which is difficult to remove in the washing stage. This has the effect of reducing the rod length of w/o microemulsions formed. Since the main aim of this work was to test TEB as a technique for determining microemulsion droplet dimensions, care was taken to use the same batch of surfactant for all TEB (and SANS) measurements. (b) TEB Measurements. The TEB experiments were performed by applying a rectangular voltage pulse to the electrodes of a Kerr cell containing the microemulsion and observing the time dependence and the steady-state value of the birefringence signal ∆n(t) arising from induced anisotropy of the sample. Details of the apparatus employed have been presented previously.20 The rate of rise and decay of this signal is controlled by the rotational-diffusion time τr of the droplets. The pulse duration was chosen to be longer than τr, so that the induced birefringence can reach the stationary value ∆ns. Our investigation was limited to the Kerr regime where the stationaryinduced birefringence ∆ns is proportional to the square of the applied electric field E. The Kerr constant is defined as: B ) ∆ns/(λE2), where λ is the wavelength of the laser beam used as birefringence probe. The results described in this work were obtained by applying pulses with amplitude in the range 10-100 V and duration 10100 ms to a cell presenting a gap between the electrodes of 1 mm and a geometrical path length of 6 cm. The optical source was a He-Ne laser emitting at λ ) 0.633 µm. Since the sample is slightly absorbing at that wavelength, the power of the laser beam was kept at a very low level (about 0.2 mW) to prevent heating of the sample. Heating effects give rise to refractive index gradients inside the cell which may distort the profile of the laser beam and may induce convective motion. It should be noted that, unlike TEB measurements on aqueous solutions, Joule heating is negligible for w/o microemulsions since they present a very low electrical conductivity. Signal averaging over many pulses was performed by a Data Precision DATA 6100 transient digitizer. The measurements on the Ni(AOT)2 w/o microemulsions were made at 25 °C over a range of droplet volume fractions, φ, from 0.005 to 0.07 and w values ranging from 10 to 40. To estimate the effect of temperature, measurements were also made at 15 °C for a w value of 18.

Theoretical In principle, it is easily possible to obtain the length L, of monodisperse, rigid rod particles (or aggregates) using (18) Price, A. L. Ph.D. Thesis, University of East Anglia, U.K., 1997. Price, A. L.; Robinson, B. H.; Steytler, D.; MacDonald, I. Unpublished. (19) Partridge, J. A.; Jensen, R. C. J. Inorg. Nucl. Chem. 1969, 31, 2587. (20) Piazza, R.; Degiorgio, V.; Bellini, T. Opt. Commun. 1986, 58, 400. Piazza, R.; Degiorgio, V.; Bellini, T. J. Opt. Soc. Am. B 1986, 3, 1642.

Transient Electric Birefringence Study of Microemulsions

TEB in the limit of extreme dilution, where there are no inter particle interactions. In such an idealized case the normalized time-dependent birefringence decay is an exponential function exp(-t/τr) with a time constant τr ) (6Dr)-1, Dr being the rotational diffusion coefficient of the individual rod. Dr is related to the rod length L, by the expression:21

Dr )

(

[

3kBT 1 L ) log - 1.45 + 7.5 6τr πηL3 r

1

L log r

- 0.27

)] 2

(1) where r is the rod radius, kB is the Boltzmann constant, T is the temperature, and η is the solvent viscosity. We see that, apart from logarithmic corrections, Dr ∝ L-3. Once Dr is measured, L can be obtained by numeric solution of eq 1 if the rod radius is known. In the case of the rodlike droplets studied in this work, r is a quantity which depends on the length of the surfactant molecule and is linearly dependent on the water content but does not depend on the volume fraction. For the interpretation of our data, we have used the values of r derived from SANS measurements.18 In particular, ref 18 gives r ) 1.6 nm at w ) 18 and r ) 1.8 nm at w ) 22. Theories of self-aggregation of surfactants22 predict that rod-shaped aggregates should exhibit a polydispersity in length described by an exponential distribution P(L)

P(L) )

( )

1 L exp Lm Lm

(2)

with an average rod length Lm proportional to the square root of the volume fraction of rods:

( )

Lm ∝ φ1/2 exp

W 2kBT

(3)

where W is the energy required to create a pair of end caps in the middle of a rod of infinite length.23 As a consequence of polydispersity, the specific Kerr constant of the system can be written as

B ) φ

∫2r∞K(L) P(L) dL

(4)

where K(L) is the specific Kerr constant of a system of monodisperse rods having length L. Note that the lower limit of integration is the particle diameter 2r and not 0. The explicit expression of K(L) will be given in connection with the discussion of the experimental data. The normalized relaxation function after the electric field is switched off is given by

∆n(t) ) ∆ns

[

]

t ∫2r∞K(L) P(L) exp - τ (L) r

∫2r∞K(L) P(L) dL

dL (5)

We call τ1 the time constant derived from the initial slope of the logarithm of the relaxation function (21) Broersma, S. J. Chem. Phys. 1960, 32, 1626 and 1632. (22) Israelachvili, J. N. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; p 24. (23) Cates, M. E.; Marques, C. M.; Bouchaud, J.-P. J. Chem. Phys. 1991, 94, 8529.

Langmuir, Vol. 14, No. 1, 1998 3

[

τ1-1 ) -

]

1 dL ∫2r∞K(L) P(L)τ (L)

d(ln ∆n(t)) dt

r

)

t)0

∫2r K(L) P(L) dL ∞

(6)

The time constant τ1 can be easily derived from the TEB experiment and can be related to τrm, the rotational time of the rod having length Lm, once K(L) is known. Bellini et al.10 have examined the effect of polydispersity on the TEB relaxation. By assuming stiff rods with an exponential probability distribution for the rod length as given by eq 2, ∆n(t) is found to behave asymptotically for long times as a stretched exponential

[ ( )]

∆n(t) ∝ ∆ns exp -

t τSE

R

(7)

where the stretching exponent takes the value R ) 1/4. The time constant τSE appearing in the stretched exponential is proportional to τrm. Therefore, in principle, the average length Lm can be derived from either the initial slope or the long-time tail of the relaxation function. The interpretation of the experimental data may require consideration of scission effects and of interparticle interactions. In fact, a self-assembled rod may undergo a reversible breaking anywhere along its length. Cates et al.23 assume a random scission scheme which assigns a uniform probability for scission at any point along the length of the rod. The effect of scission is that of shortening the rotational relaxation times. Interactions give rise to hindered rotation: as the concentration of rods is increased above a specific “overlap” concentration (entanglement), φ* ∝ 6r2/L2, the excluded volume interaction between rods has the effect of slowing down the rotational Brownian motion so that the effective rotational diffusion coefficient D′r becomes smaller than Dr.23 By using the Doi-Edwards tube model and noting that the volume of a rod is proportional to L, it can be shown that, for a concentrated (φ . φ*) system of monodisperse rods, D′r ∝ Drφ-2 L-4. In the case of a polydisperse system with an average rod length given by eq 3, it can be predicted that the rotational diffusion coefficient of the average entangled rod, D′rm, scales as Lm-11.23 Cates et al.23 proposed a very interesting extension of the treatment of ref 10 by including both entanglement and reversible scission. Scission is characterized by a breaking time τb which is taken as inversely proportional to the average rod length. The treatment by Cates et al. gives explicit solutions only in the limiting cases of very short or very long breaking times relative to the rotational time. Calling τbm the breaking time of the rod of length Lm, the relaxation function of the dilute polydisperse solution is given by eq 7 only in the limit τbm . τrm. In the opposite limit, that is, in the situation in which the rods are disrupted during the time scale of the rotational relaxation, an exponential decay is predicted in the dilute regime with τbm and τrm contributing to a different extent to the observed relaxation time23

[(

∆n(t) ∝ exp -

t τrm2/3

τbm

1/3

)]

for τbm , τrm

(8)

A qualitatively similar behavior is found for the entangled regime. For long breaking times the decay is predicted to follow asymptotically a stretched exponential behavior with an exponent of 1/8

4


[( ) ]

∆n(t) ∝ exp -

t 1 / τ′SE 8

for τbm . τ′rm

Mantegazza et al.

(9)

where τ′rm is the rotational time of the average entangled rod. For short breaking times, an exponential relaxation is predicted

[(

∆n(t) ∝ exp -

t τ′rm2/7

τbm

5/7

)]

for τbm , τ′rm

(10)

Results and Discussion When the electric field is switched on, the measured birefringence grows monotonically and finally attains the steady-state value ∆ns. After the electric field is switched off, the induced birefringence decays with a fast initial relaxation followed by a slower nonexponential tail. A typical experimental trace is shown in Figure 1. A qualitatively similar behavior was previously observed for rod-shaped ionic micelles in water by Bellini et al.10 and by Hoffmann et al.9 Bellini et al. have focused their attention on the slow tail which was found to be well described by a stretched-exponential function. In our case, we also find that the slower part of the relaxation can be fitted accurately by a stretched-exponential function. In the work by Hoffmann et al. the transients were fitted by the superposition of two exponentials with the fast relaxation ascribed to an initial free rotation of the micelles before collisions ensue and the slow relaxation attributed to entanglement effects. However, both the data of ref 10 and the present data show that the nonexponential tail appears also at volume fractions which are certainly below φ*. This confirms the validity of the interpretation of ref 10 which attributes the nonexponential behavior to polydispersity and not to interactions. We present in the Figures 2-5 the quantities which can be derived from the measured TEB transients, namely, the specific Kerr constant B/φ obtained from the steadystate value ∆ns, the decay time τ1 of the initial part of the relaxation, the time constant τSE, and the exponent R derived from the stretched-exponential fit of the slow relaxation. From Figure 2 it can be seen that B/φ is always negative and takes rather large values. Its absolute value is an increasing function of w and a decreasing function of T. The data at fixed w and T can be approximately described by a power law relation: -B/φ ) φγ with γ ) 1.2. We recall that for a system of noninteracting monodisperse particles having fixed size and shape, the specific Kerr constant represents a single particle property and should be independent of φ. As we show below, the increase of the quantity |B|/φ with φ can be attributed to the increase in length of the microemulsion droplets and not to a collective effect caused by interparticle interactions (entanglement). It should be noted that a rod made of isotropic material always presents a positive Kerr constant. The negative value found for B can be explained only by assuming that the droplet has not only a shape anisotropy but also an intrinsic anisotropy due to the surfactant molecules. As shown in Figures 3 and 4, both τ1 and τSE are rapidly increasing functions of the volume fraction φ. Qualitatively, this represents a clear indication of the growth of the average rod length with φ. However, in order to derive the average rod length from the relaxation times, it is necessary to take into account the size polydispersity and establish whether the scission dynamics and entanglement effects also contribute to the observed TEB relaxation.

Figure 1. Normalized birefringence decay ∆n(t), for a w ) 18, φ ) 0.02, w/o Ni(AOT)2 microemulsion in cyclohexane (T ) 25 °C). Inset: Semilogarithmic plot of the normalized decay: the continuous line represents a linear fit on the first part of the decay.

Figure 2. log-log plot of the measured specific Kerr constant (B/φ) as a function of φ. The lines correspond to a fit with a power law B/φ ∝ φγ. The dashed, full, and dotted-dashed line are the results of a least-squares linear fit of the data at T ) 15 °C, w ) 18 (diamonds), T ) 25 °C, w ) 22 (squares), and T ) 25 °C, w ) 18 (circles). The best fit values of γ obtained are 1.3, 1.2, and 1.1, respectively.

Figure 3. log-log plot of the time constant τ1 measured as a function of φ for the data shown in Figure 2.

Figure 5 shows that the stretching exponent R takes the value 0.25 at low φ. With increasing φ the value of the exponent R grows in a systematic fashion from 0.25 to a value around 0.8, indicating an approach to a simple exponential behavior. According to the data presented in Figure 5, the increase of R seems to be independent of w or T. The presence of the stretched-exponential asymptotic time-decay reflects the size polydispersity of the



become significant only when φ/φ* is larger than 10. Since the largest investigated volume fraction is 0.07, it is reasonable to expect that entanglement is not influencing the rotational motion in the case of our experiments. In order to derive Lm from the measured τ1 we use eq 6. The specific Kerr constant K(L) is given by the following expression25

∆Ro∆Re 30λvpns0kBT

K(L) )

Figure 4. log-log plot of the time constant τSE measured as a function of φ for the data shown in Figure 2. The lines correspond to a fit with a power law τSE ∝ φβ. The best fit values of β are 2.10, 1.96, and 2.06, respectively.

where ∆Ro ) R1o - R2o and ∆Re ) R1e - R2e are the anisotropies in the optical and electrical polarizabilities of the particle, vp is the particle volume, ns is the refractive index of the solvent, and 0 is the vacuum permittivity. Rio and Rie represent the optical and electrical polarizabilities along the symmetry axis of the particle (i ) 1) or perpendicular to it (i ) 2). The calculation of Rio and Rie for the rod-shaped microemulsion droplets is not trivial because the droplets have a core/shell structure with a water core surrounded by a surfactant shell of thickness D. Since the AOT molecules are aligned perpendicularly to the surface, the surfactant layer is anisotropic, that is, the dielectric constant and the refractive indices perpendicular and parallel to the surface of the layer are different.16 We have performed an approximate calculation of K(L) in the following way. We note first that an isotropic dielectric sphere of dielectric constant 1 and radius R, coated with a layer of dielectric constant 2 and thickness D, immersed in a medium of dielectric constant 3, is equivalent, from the electric point of view, to homogeneous dielectric sphere of radius R + D and effective dielectric constant ′2 given by the following expression26

Figure 5. Stretching exponent R measured as a function of the volume fraction φ for the data shown in Figure 2.

droplets. In particular, the observation that at low volume fractions the exponent R takes the value 0.25 represents a confirmation of the assumption that the probability distribution of rod lengths is an exponential, as discussed in ref 10. Within the formalism of Cates et al. described in the theoretical section, the observed behavior of R vs φ could suggest a crossover from the regime (described asymptotically by eq 7) in which the breaking time of the droplets is longer than the rotational time to the fast breaking exponential relaxation (described by eq 8). Such a behavior could be plausible on the grounds that τbm scales as the inverse of Lm and τrm (approximately) as Lm3 (eq 1). The ratio τbm/τrm is therefore expected to scale as φ-2, suggesting that it is indeed possible to approach the fast breaking regime on increasing φ. However, as will be shown at the end of this section, it is unlikely that this interpretation applies for our case, because the measured length of the rodlike aggregates is rather small in the investigated range of volume fractions. The shape of the observed relaxation does not contain evidence of entanglement effects: entanglement would give a stretched-exponential behavior with values of R decreasing as φ increases (see eq 9), whereas we find that R grows with φ. We have performed an approximate calculation of the entanglement volume fraction for the case w ) 18, T ) 25 °C, finding φ* ) 0.03. It is known from studies of rotational relaxation of rigid polymer chains24 that entanglement effects on rotational times (24) Teraoka, I.; Ookubo, N.; Hayakawa, R. Phys. Rev. Lett. 1985, 55, 2712.

(11)

′2 ) 2

{

( ) ( ( ) (

) )

3

1 - 2 1 + 22 1 - 2 R 3 R+D 1 + 22

R R+D

+2

}

(12)

In order to discuss data referring to spherical microemulsion droplets, the calculation of the polarizabilities was extended by van der Linden et al.16 to the case in which an isotropic core is surrounded by an anisotropic layer, provided that the core radius is much larger than the thickness of the layer. In the case of a spheroidal prolate dielectric particle of anisotropic refractive index npi (i ) 1, 2) and anisotropic dielectric constant pi (i ) 1, 2), immersed in a dielectric medium of refractive index ns and dielectric constant s, ∆Ro and ∆Re can be derived by using the following expressions for the polarizabilities25

Ri0 ) vp0

(

npi2 - ns2

1 + Li

Ri0 ) vp0

)

npi2 - ns2 ns2

i - s i - s 1 + Li s

(13)

(14)

(25) O’Konsky, C. T., Ed. Molecular Electro-optics; Marcel Dekker: New York, 1976. (26) Jones, T. B. Electromechanics of Particles; Cambridge University Press: Cambridge, 1995.

6


Mantegazza et al.

Figure 6. Plot of K(L) as function of L calculated with the model described in the text. Inset: log-log representation of the same data: the dashed line represents a fit with a power law: K(L) ∝ L2.2.

Figure 7. Plot of the calculated values of f(Lm) ) τ1/τrm as a function of Lm. The overlap concentration corresponds to L ) 30 nm.

where Li (i ) 1, 2) is the depolarization factor of the ellipsoidal particle. In order to perform an explicit calculation, we need the index of refraction, npi (i ) 1, 2) and the dielectric constant pi (i ) 1, 2) of the anisotropic particle. Since in our case the particle is not homogeneous because it consists of an aqueous core surrounded by an amphiphile layer, we treat the problem in an approximated way by defining an equivalent homogeneous anisotropic ellipsoid having an effective value of npi and pi (i ) 1, 2), which take intermediate values between those of AOT and water. By using eq 12 we have calculated p1 (p2) as the equivalent dielectric constant of a sphere of radius L/2 (r) and of dielectric constant H2O, with a layer of thickness D and dielectric constant β (or γ), immersed in a solvent of dielectric constant s. Here we call β and γ, respectively, the dielectric constants of the AOT layer perpendicular and parallel to the surface, D is the thickness of the AOT layer, and H2O and s are the dielectric constant of water and cyclohexane. The numerical values of β and γ are taken from ref 16. Finally, we calculate the refractive indices as the squares of the dielectric constants. The results of the calculation are presented in Figure 6 where K(L) is plotted as a function of L. We see that the calculation reproduces correctly the sign and the order of magnitude of the Kerr constant measured in our microemulsions. It should be noted that the magnitude of the Kerr constant is very sensitive to the detailed structure of the droplet: a droplet model as an isotropic water particle or as an anisotropic particle of oriented AOT molecules would present values for K(L) which are several orders of magnitude lower than the experimental ones. The inset of Figure 6 presents a log-log plot of K(L). In the range of L explored, the slope of the plot is 2.2, that is, K(L) ∝ L2.2. Therefore, by substituting the obtained approximate power law into eq 4, we find

Figure 8. log-log plot of the values of Lm versus φ. Lm values are obtained from the experimental values of τ1 presented in Figure 3. The dashed, full, and dotted-dashed lines correspond to a fit with a power law Lm ∝ φδ of the data at T ) 15 °C, w ) 18 (diamonds), T ) 25 °C, w ) 22 (squares), and T ) 25 °C, w ) 18 (circles). The best fit values of δ are 0.48, 0.45, and 0.51, respectively. In this plot we present some data obtained with SANS measurements at different situations: T ) 25 °C, w ) 18 (filled circles); T ) 25 °C, w ) 22 (filled squares).

B ∝ φ

∫0∞L2.2 L1m exp(LLm)dL ∝ Lm2.2

(15)

Considering that Lm is proportional to φ1/2, eq 15 predicts a power-law dependence of B/φ on φ with an exponent γ ) 1.1. This is in good agreement with the value γ ) 1.2, which we derive from the experimental data of Figure 2. It is very interesting to note that Koper et al.14 derive, by using a scaling approach, γ ) 1.3, and measure for elongated lecithin inverted micelles values of γ in the range 1.1-1.3.

By using the expressions for K(L) and τr(L) given, respectively, in eqs 11 and 1, we have calculated numerically the integrals appearing in eq 6. We show in Figure 7 the ratio f(Lm) ) τ1/τrm plotted as a function of Lm. The fact that the ratio is larger than 1 for a polydisperse system is reasonable because TEB gives more weight to the longer rods. By using the experimental values of τ1, the values of the rod radius given above, and the plot in Figure 7, we have finally derived the average rod length Lm as a function of φ. More precisely the values of Lm have been extracted from TEB data by solving numerically the equation

τrm )

τ1 f(Lm)

(16)

The results are reported in Figure 8. It is clear that the data follow the law Lm ∝ φ0.5, consistently with the theoretical model and also with the experimentally observed stretched-exponential behavior. Some SANS results are also plotted in Figure 8: the SANS and TEB lengths are in very good agreement. From the data we can derive the scission energy W which appears in eq 3. We find W ) 12kBT, a value which is comparable to that (W ) 20kBT) found for sodium dodecyl sulfate micelles. The fact that the rod length is in the range of few nanometers casts some doubt on the interpretation of the



increasing function of w. By characterizing the relation between φ and τSE with a power-law, τSE ∝ φβ, we find β ≈ 2. Such a value is not unreasonable considering that, at low volume fraction, the system consists of rigid nonbreakable noninteracting rods and, as expressed by eq 8, the time constant τSE is proportional to the cube of Lm (i.e., β ) 3/2). As a final point, it is interesting to discuss whether our experiment gives clear evidence for the formation of discrete rods rather than for a clustering phenomenon of the type observed for spherical droplets close to a phase transition boundary. The first evidence is represented by the absolute value of the Kerr constant which in our system is larger by 2 orders of magnitude with respect to values found for NaAOT droplets at similar volume fractions. A second fact which favors the interpretation in terms of rods is that the nonspherical structures are observed over a wide range of temperature and concentration conditions, whereas droplet clustering occurs only approaching well-defined transition lines. Figure 9. (a) Time constant τ1 derived from the initial slope measured as a function of w at φ ) 0.03 and T ) 25 °C. (b) Specific Kerr constant B/φ measured as a function of w at φ ) 0.03 and T ) 25 °C. (c) Values of Lm as a function of w at φ ) 0.03 and T ) 25 °C, as obtained from TEB data (open squares) and SANS data (filled squares).

R vs φ behavior discussed at the beginning of this Section. If we consider the case of elongated cetyltrimethylammonium bromide (CTAB) micelles in aqueous solutions, where measurements of the breaking time have been performed,27 the rod length at which the breaking time becomes equal to the rotational time is above 0.5 µm. Although the breaking time of our inverted micelles is not known, it seems unlikely that the situation may be so different from that of CTAB micelles. We have also investigated the dependence of the average rod length on the water-to-surfactant ratio w. We present in Figure 9 the behavior of τ1, -B/φ, and Lm as a function of w for Ni(AOT)2 at T ) 25 °C and φ ) 0.03. The three plots are qualitatively very similar. We find that Lm presents a maximum for a value of w between 30 and 35, in agreement with viscosity measurements.18 In the case of Lm we present in Figure 9 also the SANS data which appear to be fully consistent with the TEB data. We shortly comment also on the data of Figure 4. We see that τSE increases with increasing φ, decreases with T, and increases with w at fixed volume fraction. This can be understood if we consider eq 2 and the assumption that, in this experimental range, the size of the rods is an (27) Candau, S. J.; Merikhi, F.; Waton, G.; Lemare´chal, P. J. Phys. (Paris) 1990, 51, 977. Faetibold, E.; Waton, G. Langmuir 1995, 11, 1972.

Conclusions We have applied TEB for the measurement of the average length of rod-shaped w/o microemulsion droplets of relatively short length. In the volume fraction range φ ) 0.005-0.07 the TEB relaxation shows a nonexponential decay which follows asymptotically a stretchedexponential behavior. The value of the stretching exponent R at low volume fraction is consistent with the assumption that the length probability distribution is exponential. A model describing the Kerr response of the microemulsion droplets is developed. By using this model we derive the specific Kerr constant as a function of the volume fraction, finding a good agreement with the experimentally observed behavior and also with literature results referring to a different system. We apply the model to the interpretation of the dynamic data, deriving, from the initial slope of the relaxation, the mean rod length Lm. It is found that Lm grows approximately as the square root of φ. Values for Lm obtained from TEB exhibit essentially the same dependence on φ, w, and temperature as those obtained from SANS measurements. Acknowledgment. We thank G. Fragneto, T. Jenta, I. MacDonald, and T. Towey for the help in measurements and sample preparations and T. Bellini for useful discussions. B.H.R., D.C.S., and A.L.P. acknowledge financial support for travel and consumables from BBSRC to undertake TEB experiments at Pavia (Italy) and EPSRC for SANS measurements at ISIS (U.K.). A postdoctoral fellowship from BBSRC is gratefully acknowledged by A.L.P. LA970888D

Transient Electric Birefringence Study of Rod-Shaped Water-in-Oil

Recommend Documents