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Dynamics of the “Sponge” (L3) Phase† B. Schwarz, G. Mo¨nch, G. Ilgenfritz, and R. Strey* Institut fu¨ r physikalische Chemie, Universita¨ t zu Ko¨ ln, Luxemburger Strasse 116, 50939 Ko¨ ln, Germany Received February 15, 2000. In Final Form: June 16, 2000 The dynamics of the sponge structure of the L3-phase in the ternary H2O-C10E4-n-decanol (C10E0) system has been examined. Temperature jump relaxation employing scattered-light detection, pressure jump with conductivity detection, time-resolved electric birefringence, and dynamic light scattering experiments were performed on the same system at varying surfactant volume fraction φ at constant temperature. The observed relaxations were all found to be single exponentials. The time constants τ-1 obtained by these different methods vary over several orders of magnitude. They reveal in part rather strong dependencies on φ obeying effective scaling laws τ-1 ∼ φn, with n ) 9 for the temperature and pressure jump relaxation and n ) 1 for dynamic light scattering. The time constants from electric birefringence yield n ) 3. The origin of the various exponents can theoretically be understood in terms of passage formation, concentration fluctuations, and relaxation of shape anisotropy due to elastic deformation.
I. Introduction Surfactant molecules in water self-assemble into a variety of structures among which micelles and membranes are the most common. At comparatively high surfactant concentrations, long-range ordered mesophases of, e.g., hexagonal or lamellar structure are formed.1 For certain surfactants in the vicinity of the lamellar phases a phase of bilayers with smectic ordersan anomalous isotropic phase2 is encountered. It is nowadays usually referred to as L33 or “sponge”4 phase showing a number of remarkable phenomena. Well-known is the observation that upon dilution this phase becomes increasingly streaming birefringent and concomitantly strongly opalescent. The explanation is found in the local structure of the L3-phase consisting of fluctuating bilayerssvery similar in composition to those of the lamellar phases which are multiply connected and avoid intersections.5 The bilayers comprise two opposing surfactant monolayers, the surfactant tails touching each other at the bilayer mid-plane.4 Shear induces anisotropic order of the bilayers seen as birefringence. The strong scattering of light is attributed to fluctuations4 the nature of which needs to be discussed in detail. Less well-known is that the dynamics of the sponge structure extends over time scales of more than 10 orders of magnitude. The clarification of the dynamics is the interest of the present study. Fluctuations around the thermodynamically stable state may in the case of the “sponge” phase be related to different modes.4 In order to discuss the various modes, it is necessary to recall the structural properties of L3-phases. A typical structure as it is seen by freeze fracture electron microscopy6 is shown in Figure 1. † Part of the Special Issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millennium”.
(1) Ekwall, P. Advances in Liquid Crystals; Brown, G. H., Ed. Academic: New York, 1975; Vol. 1, p 1. (2) Lang, J. C.; Morgan, R. D. J. Chem. Phys. 1980, 73, 5849. (3) Anderson, D. M.; Wennerstrom, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243. (4) Roux, D.; Coulon, C.; Cates, M. E. J. Phys. Chem. 1992, 96, 4174. (5) Porte, G. J. Phys. Cond. Matter 1992, 4, 8649. (6) Strey, R.; Jahn, W.; Porte, G.; Bassereau, P. Langmuir 1990, 6, 1635. See also: Strey, R.; Jahn, W.; Skouri, M.; Marignan, J.; Porte, G.; Olsson, U. In Structure and Dynamics of Strongly Interacting Colloids; Chen, S.-H., et al., Eds., Kluwer: Amsterdam, 1992; p 351.
Two equivalent water subphases are separated by a bilayer, which is self-connected throughout the solution. Each of the distinct water subphases is sample-spanning continuous. For studying the dynamics of fluctuations, various relaxation methods as well as dynamic light scattering (DLS) may be employed. In relaxation experiments an intensive thermodynamic parameter is rapidly changed, and the reestablishment of the equilibrium is observed. Temperature (T) changes bring about changes in the spontaneous curvature of the individual monolayers of the bilayer and may lead to the formation of new passages. The same holds for changes in pressure (p). T-jump and p-jump relaxation experiments are therefore expected to provide information on the dynamics of the formation of passages. In time-resolved Kerr effect measurements, the applied electric field (E) may bring about a distortion of the sponge structure at unchanged topology. The relaxation of the induced anisotropy is expected to depend on the viscoelastic properties of the local L3 structure. Differently than the deliberate perturbations of the relaxation methods, the DLS method probes spontaneous fluctuations in the light scattering from the L3-phase. All processes, which are driven by changes in the external parameters, are, of course, expected to be present in the dynamic equilibrium as a main consequence of the dissipation-fluctuation theorem. However, we will see below that the dominant observable contribution seen in dynamic light scattering (in the observable q range) is connected to processes not seen by the other methods. As stated above, there is little information on the dynamics in the literature. Miller et al.7,8 applied electric birefringence measurements to the zwitterionic waterC14DMAO-n-hexanol system while Porte et al.9 studied the water-betaine-n-pentanol system. Miller et al. interpreted their data in terms of orientation of discshaped units. However, in the light of the spongelike structure (cf. Figure 1) a different interpretation of the (7) Miller, C. A.; Gradzielski, M.; Hoffmann, H.; Kra¨mer, U.; Thunig, C. Colloid Polym. Sci. 1990, 268, 1066. (8) Miller, C. A.; Gradzielski, M.; Hoffmann, H.; Kra¨mer, U.; Thunig, C. Prog. Colloid Polym. Sci. 1991, 84, 243. (9) Porte, G.; Delsanti, M.; Billard, I.; Skouri, M.; Appell, J.; Marignan, J.; Debeauvais, F. J. Phys. II Fr. 1991, 1, 1101.
10.1021/la000220q CCC: $19.00 © 2000 American Chemical Society Published on Web 09/08/2000
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Figure 1. Structural units of L3-phase: as seen by freeze fracture electron microscopy,6 they are self-connected bilayers. The magnified portion shows schematically how the surfactant molecules and the smaller alcohol molecules are thought to be arranged in the bilayer.
Kerr-effect results may be needed. Porte et al. assumed the underlying structure to resemble that seen in Figure 1 and interpreted their results in terms of a distortion that relaxes. In a different set of experiments, Porte and co-workers10 used a T-jump apparatus and followed the relaxation by light scattering for the ionic system water/ NaCl-cetylpyridinium chloride-n-hexanol. A number of different time constants were observed. For the present study, we chose the nonionic surfactant system water-C10E4-n-decanol (C10E0) for the following reasons. First, it can be studied free of electric charges as necessary for (electric field) E-jumps. Second, electrolyte can be added without major changes in phase behavior to have a conducting medium for T- and p-jumps. Third, the L3-phase is already observed in the binary system waterC10E4, which may be important in cases where the structure or the relaxation processes are attributed to compositional changes. For practical purposes, it can be brought to any desired (e.g., room) temperature by the addition of an appropriate cosurfactant. For ease of interpretation we chose n-decanol the C10-hydrocarbon chain matching that of C10E4. One of the questions was whether different perturbation methods and ways of detecting the processes yield different relaxation rates. In order to answer these and other questions we use a T-jump detecting the change in intensity of scattered light and a p-jump detecting the change of electrical conductivity. In addition, we complemented the experiments by E-jump following the transient birefringence and dynamic light scattering. All experiments were performed on the same system. II. Experimental Section A. Materials. The nonionic surfactant n-decyltetraoxyethylenoxide (C10E4) was purchased from Bachem Biochemica GmbH, Heidelberg, Germany (P-1010 Lot 504238). n-Decanol (C10E0) was obtained from Merck-Schuchardt, Hohenburg, Germany (CE 203-956-9) with a purity >99%. The purity of C10E4 was judged from the critical point of the binary system H2OC10E4. We determined Tc ) 20.2 °C close to the recommended value of Tc ) 20.5 °C for purified C10E411,12 at a mass fraction γ ) 0.026. Fortunately, the purity is a less important issue for the present study because any changes in purity can be compensated by the mixing ratio of surfactant with n-decanol. Water was ultrapure Millipore water, type Milli-Q RG with a resistance of several MΩ‚cm. (10) Waton, G.; Porte, G. J. Phys. II 1993, 4, 515-530. (11) Schubert, K. V.; Strey, R.; Kahlweit, M. J. Colloid Interface Sci. 1991, 141, 21. (12) Schubert, K. V.; Strey, R.; Kahlweit, M. Prog. Colloid Polym. Sci. 1991, 84, 103.
B. Phase Diagram. The data points shown in the experimental phase diagrams were determined in a water bath (fish tank) with the samples contained in sealed tubes thermostated to (0.01 K, when necessary. The atmospheric pressure was assumed constant. Phase-transition temperatures were determined by visual inspection of the transmitted and scattered light and between crossed polarizers, if anisotropic phases like the LR phase were involved. The number and types of phases were recorded. In this paper, the mass fraction of surfactant is denoted by
γ)
mC10E4 + mC10E0 mH2O + mC10E4 + mC10E0
(1)
The composition of the bilayer may be tuned by
δ)
mC10E0 mC10E4 + mC10E0
(2)
For structural considerations, the volume fraction φ of total bilayer
φ)
VC10E4 + VC10E0 VH2O + VC10E4 + VC10E0
(3)
and the relative volume fraction of cosurfactant
δv )
VC10E0 VC10E4 + VC10E0
(4)
may be useful. When calculating these quantities we have used the following densities for H2O/0.1 N NaCl 998 kg m-3, C10E4 959 kg m-3, and C10E0 827 kg m-3 and assumed ideal mixing. The monomeric solubility of both C10E4 and C10E0 is of the order of 1 × 10-5 and hence neglected in calculating φ. For the T-jump and p-jump experiments a 0.1 N NaCl solution instead of water was used to guarantee a sufficient conductivity in the samples for fast heating and measuring the conductivity, respectively. The addition of the lyotropic salt NaCl lowers the solubility between water and surfactant. Accordingly, the phase behavior is shifted as a whole to slightly lower temperatures,13,14 in our case by 1 K, a shift that is easily compensated by adjusting δ. In the process of filling the cells for the relaxation experiments great care has been taken to avoid demixing. C. Relaxation Experiments. Relaxation techniques in principle consist of a perturbation method and a detection method. While the time constants should be independent of the perturbation method (if the same kinetic process is monitored), the (13) Kahlweit, M.; Strey, R. Angew. Chem. 1985, 24, 654. (14) Firman, P.; Haase, D.; Jen, J.; Kahlweit, M.; Strey, R. Langmuir 1985, 1, 71.
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amplitudes of the observed signal depends on the individual detection method. p-Jump. The pressure-jump apparatus used consists of two parts, an autoclave and a Wheatstone alternating current bridge.15-17 The well-thermostated autoclave filled with air-free water contains the measuring cells with about 0.4 cm3 of homogeneous L3-phase sealed with a Teflon membrane. In the measuring cells pressurizing the autoclave by a mechanical pump, a defined pressure p can be generated. At pressures p of about 130 bar, a brass membrane bursts and the pressure drops to ambient pressure p0 within τp < 50 µs. The 40 kHz Wheatstone alternating current bridge comprises the measuring cell and an adjustable decade as R1 and R2 and the potentiometers R3 and R4 of the bridge. The temperature of the autoclave is kept constant within (0.01 K. A piezoelectric crystal triggers on the down slope of the pressure the recording of the resistance difference ∆R(t) between the decade and the measurement cell using a 12-bit AD card. A hardware demodulator that detects the maximum amplitudes of the oscillator demodulates the amplified ac signal. The shortest measurable relaxation time is limited by the time the demodulator needs to register the first signal. ∆R(t) may be described by two exponential functions
∆R(t) ) ∆R∞ + A exp(-t/τc) + B exp(-t/τP)
(5)
with ∆R ) Rdecade - Rcell and ∆R∞ the long time limit (t ) ∞). A and B are the amplitudes. The limiting demodulator constant was τc ) 70.0 µs throughout the experiments. The relaxation of conducting pathways inside the L3-structure induced by the p-jump was found to be characterized by a single relaxation time τP. T-Jump. The temperature-jump apparatus consists of a highvoltage discharge unit containing the thermostated cell and an optical unit for registering the change of scattered light intensity.18 The operating principle of this setup is a heating of the sample by discharge of a high-voltage (HV) capacitor. The heart of the measuring cell is the precise thermostating of the HV electrode that on the one hand is connected via a thick copper cable to the high-voltage capacitor and on the other hand may not deviate from the correct temperature for the L3-phase to be stable. The height of a temperature jump varies with the charging voltage U of the capacitor and can be calculated from ∆T ) bU2. The constant b was determined using a critical 1,2-lutidinewater mixture, which has a similar heat capacity as the samples, to be 7.1 (( 0.5) × 10-3 K‚(kV)-2.19 Thus charging voltages between 2.5 and 11 kV result in temperature jumps between 0.04 and 0.86 K. The heating time τh ) RcellC/2, where C is the capacitance (10 nF), is effectively set by the electrolyte concentration and is kept constant. It was τh ) 2.2 µs. Because of heat exchange, the temperature inside the sample relaxes back and reaches after a few times the cooling time τc again at the initial temperature T0. The cooling time τc ) 18 (( 2) s is nearly constant and its exact value of subordinate importance for determining relaxation times less than seconds. The optical unit allowed the simultaneous monitoring of the intensity change at different angles. The relaxation of scattered light intensity was found to be single exponential. Thus, taking into account time τc characterizing the return to equilibrium, the process is quantitatively described by
Is(t) ) I0 + A{exp(-t/τc) - exp(-t/τT)}
(6)
with I0 being the scattered light at the time t ) 0. The amplitude A is determined by the intensity increase associated with the T-jump ∆T. E-Jump. The field-jump apparatus20 is designed to observe simultaneously the electric current passing through the cell and (15) Strehlow, H.; Knoche, W. Fundamentals in Chemical Relaxation; VCH: Weinheim, Germany, 1987. (16) Strehlow, H.; Becker, M. Z. Elektrochem. 1959, 63, 4. (17) Knoche, W.; Wiese, G. Chem Instrum. 1973-1974, 5, 91. (18) Mayer, W.; Woermann, D. J. Chem. Phys. 1988, 92, 2036. See also: Beckmann, V. Dissertation, Ko¨ln, 1995. (19) Uhrmeister, P. Diplomarbeit, Ko¨ln, 1998. (20) Runge, F.; Ro¨hl, W.; Ilgenfritz, G. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 1991.
the optical birefringence of the sample. Rectangular field pulses of up to 10 ms duration and up to voltages of 20 kV can be applied. While switching on the electric field E is comparatively slow, about 50 µs, switching off is done with spark gaps allowing measuring relaxation times from 1 µs on upward. The voltage pulses are monitored with an (noncompensated) Ohmic voltage divider. The Kerr effect is measured using a He-Ne laser (Uniphase, 10 mW) of λ ) 632.8 nm, two polarizers (Halle), a quarter-wave plate (Spindler & Hoyer), and an RCA IP28 photomultiplier in the standard optical arrangement: polarizer 45°, slow axis of the quarter wave plate 135° with respect to the field direction. The analyzer offset angle from the crossed position was chosen to be (5°. Measurements were carried out in a cell with cylindrical geometry with Pt electrodes 1.71 cm apart and an optical light path of 0.6 cm. Birefringence of the cell windows, glass plates of 2 mm thickness, is very small and can be neglected. All field-jump measurements were carried out at 25.1 °C. A current viewing resistor in series with the measuring cell measures the momentary electrical conductivity. In contrast to Kerr effect, no current signal can be observed after the end of the field pulse. Dynamic Light Scattering (DLS). The dynamic light scattering experiments were performed using a commercial ALV goniometer (Lange, Germany) consisting of an ALV-5000 Multiple Digital Correlator and a 50 mW He/Ne laser with a wavelength of λ ) 632.8 nm. An angular range of 30° to 150° was routinely covered to explore the q-dependence of the fluctuations where q is defined as
q)
4πn sin(θ/2) λ
(7)
Near room temperature the temperature stability is (0.02 K. The result of each dynamic light scattering experiment is an intensity-intensity autocorrelation function. The data points of this autocorrelation function are fitted with CONTIN21 and yield the q-dependent time constants Γ:
Γ)
1 ) Dq2 2τDLS
(8)
with the D being the mutual diffusion coefficient.
III. Results Since meaningful experiments require the detailed knowledge of the phase behavior we first determined the phase diagrams of the ternary system H2O-C10E4-C10E0 as function of temperature. Then various relaxation kinetic experiments detecting different relaxation processes related to structural reorganizations were performed and compared. These experiments called for dynamic light scattering experiments known to monitor thermal fluctuations. As a result, we are able to present the q, φ, and T dependencies of time constants from complementary techniques, which we will attribute to different dynamic processes in the subsequent Discussion section. As can be seen below, our measurements cover an extended range of L3 dynamics. A. Phase Diagram. As mentioned above, the exact knowledge of the phase diagram facilitates performing experiments on the L3-phase, in particular as the L3-phase only appears as a narrow band in the phase diagrams. It starts in the binary systems H2O-C10E4 as is well-known from the classic study by Lang and Morgan.2 The binary L3-phase extends over a large concentration range (γ ) 0.03 to γ ) 0.2) in a temperature range between 45 and 59 °C. At given γ, the width of the L3-phase is, however, only 1 or 2 K. The addition of the cosurfactant decanol (C10E0) results in a monotonic shift of the phase behavior to lower temperature, as we know from Jonstro¨mer and (21) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213.
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Figure 2. Three isothermal Gibbs triangles for H2O-C10E4-n-decanol (C10E0) for demonstrating the location and extent of the L3-phase. Note that a number of two- and three-phase regions have been omitted for clarity.
Strey.22 The reason is that the amphiphilic character of n-decanol leads to a change of the curvature of the monolayers toward water.22-24 To compensate the curvature induced, one has to lower the temperature resulting in a stronger hydration of the surfactant headgroups. In this fashion, it is possible to stabilize the L3-phase at a preselectable constant temperature, e.g., room temperature, by using an appropriate δ. In order to bring the L3-phase down to room temperature, we started with a certain mass fraction γ in the binary system H2O-C10E4 and added n-decanol. After each addition we measured the phase transition temperatures LR + L3 f L3 and L3 f L1′ + L3. This procedure yields vertical sections through the phase prism of the ternary system from which we constructed isothermal sections. Three of these so-called Gibbs triangles for 15, 20, and 25 °C are shown in Figure 2, which illustrates the location of the one- and two-phase regions only. The other two and three-phase regions are not shown for clarity. The central miscibility gap touches the binary side system H2O - C10E4 for T > Tc ) 22 °C explaining the appearance of the two-phase region L′1 + L′′1 at low decanol concentration. The L3-phase forms a natural divider between the L1 + L2 region and the LR-phase. It shapes an opening funnel extending from the water corner toward the binary side C10E0-C10E4. The cosurfactant ratio δ needed for the L3-phase is the larger the lower the temperature. From Figure 2 one notes that at 15 °C δ ) 0.4 is needed compared to δ ) 0.3 at 25 °C. The observations parallel those of Penders and Strey25 for the system H2OC8E5-C8E0. B. Relaxation Experiments with p- and T-Jump. Time Constants τT,p. A noteworthy observation is that the relaxation curves in the L3-phase yield characteristically a single time constant both for p- and T-jump experiments. Figure 3 shows two comparable relaxation curves. Both experiments are carried out inside the temperature range of the stable L3-phase of samples with the same volume fractions. Following a pressure jump (typically from 100 to 1 bar) the difference in resistance ∆R ) Rdecade - Rcell becomes smaller until the resistance ∆R∞ is reached. Accordingly, the resistance Rcell inside the measurement cell containing the L3-phase is rising during the experiment. Qualitatively, the observation holds that the cell resistance containing the L3-phase is smaller at a high pressure than at 1 bar. So both with rising temperature and falling pressuresin both cases we are (22) Jonstro¨mer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993. (23) Kahlweit, M.; Strey, R.; Busse, G. J. Phys. Chem. 1991, 95, 3881. (24) Strey, R.; Jonstro¨mer, M. J. Phys. Chem. 1992, 96, 4537. (25) Penders, M. H. G. M.; Strey, R. A. Chem. Soc. 1995, 99, 16, 6091.
Figure 3. Experimental relaxation curves for p- (top) and T-jump (bottom) obtained on the same sample yielding the same relaxation times.
approaching the L1′ + L3-phase boundarysthe resistance is rising. The qualitative trend may be of importance when discussing the origin of the relaxation signal. The experimental data of ∆R(t) between the sample and the decade were fitted with eq 5. The fit is shown as line in the upper part of Figure 3. Following a temperature jump (typically 0.1-0.5 K) the intensity of scattered light increases. The lower part of Figure 3 shows the relaxation of the scattered light in a typical temperature-jump experiment. The intensity of the scattered light I0 before the jump and the time-dependent scattered light I are described by eq 6 with the cooling constant τc ) 18 s fixed. The fit of eq 6 to the data points is shown as full line. For both experiments we obtain identical relaxation times, that is, τT ) τP. In numbers, the relaxation time τP ) 5.75 × 10-3 s for the pressure jump at T ) 20.05 ((0.02) °C and for the temperature jump at T ) 20.04 ((0.06) °C τT ) 5.74 × 10-3 s. This observation is remarkable insofar as in the case of the pressure jump we follow the process by electrical conductivity, a function of the permeability and/or connectivity of the L3 structure, while in the case of the temperature jump we follow the scattered light thought to be scattered of the mesoscopic structural units visualized in Figure 1. The scattered light intensity depends on the q vector which for visible light wavelength
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Figure 4. q dependence of the time constants of dynamic light scattering τDLS-1 (full squares) for a sample with φ ) 0.301 and δV ) 0.397 at T ) 20.22 °C and T-jump τT-1 (open circles) for a sample with φ ) 0.372 and δV ) 0.414 at T ) 20.71 °C.
Figure 6. Relaxation time constants τP-1 (full squares) from p-jump, τT-1 from T-jump (+) cover several orders of magnitude. Note the mutual agreement and the strong φ dependence. τE-1 from electric field jump (full circles) and τDLS-1 from dynamic light scattering (full triangles) experiments. The statistical error bars correspond to 2 standard deviations.
significance of an effective activation energy Ea. For the observed process we find numerical values for Ea ) 200 ( 30 kJ/mol which is a rather strong temperature dependence. φ Dependence of τT,p. The structural relaxation times τT,p from temperature jumps carried out to 20.0 °C and pressure jumps carried out at 20.0 °C are shown in Figure 6, which reveals a strikingly strong φ-dependence of τT,p. In the log-log representation the data show a steep effective power law dependence
τT,p-1 ∝ φ9.2 Figure 5. Temperature dependence of the relaxation time constants in an Arrhenius plot. The open symbols are for T-jump (i.e., τT-1) the full symbols for p-jump (i.e., τP-1) results. The membrane volume fractions φ of the samples are indicated together with the cosurfactant in fraction δV in the membrane. Note the agreement of T-jump and p-jump for φ ) 0.301 and 0.227.
corresponds to length scales many times the diameter of the tubular elements of the bicontinuous structure (cf. Figure 1). q-Independence of τT. The observation of a single time constant describing both p- and T-jump relaxations suggests an independence of τT on the detection angle for the T-jumps. We experimentally checked this point for a sample with φ ) 0.372 and δV ) 0.414 and T ) 20.71 °C. Temperature jumps were carried out for detection angles of the scattered light θ ) 40°, 60°, 90°, 120°, and 140°. The open points in Figure 4 show the result. For a structural reorganization process, a constant, i.e., q-independent relaxation time τT, comes as no surprise. Temperature Dependence of τT,p. If both relaxation methods detect the same structural relaxation time τT,p, this should hold for the dependence on volume fraction φ and the temperature T as well. The Arrhenius plot in Figure 5, i.e., ln(τT,p-1) vs 1/T, yields a set of parallel straight lines each for a different membrane volume fraction. The straight line fits both the p-jump and the T-jump measurements. The slope of such a plot has according to
( )
τT,p-1 ) A exp -
Ea RT
(9)
(10)
If there were no jumps exactly to 20.0 °C, we extra- or interpolated the relaxation time from the data given in Figure 5. Accordingly, a small change in surfactant volume fraction by a factor of 3 from φ ) 0.13 to φ ) 0.39 causes an enormous reduction of the relaxation time τT,p from about 10 s to 1 ms, i.e., a change of several orders of magnitudes. C. Dynamic Light Scattering. φ Dependence of Fluctuations. Having detected such dramatic dependencies of the structural time constants in Figure 6, it was of interest whether dynamic light scattering experiments would yield similar spectacular variations as the T-jump experiments. Yet, the dependence of τDLS-1 on surfactant volume fraction φ was found to be rather weak
τDLS-1 ∝ φ0.8
(11)
The time constants are plotted for comparison in Figure 6. Over the whole φ range covered, the fluctuation process is faster than the T- and p-jump relaxations. The dependence of τDLS-1 on q included in Figure 4 clearly reveals a q2 dependence passing through the origin. This result holds for all φ examined. Thus, we obviously observe by DLS a hydrodynamic fluctuation process. From eq 8 a diffusion coefficient D ) 7.8 × 10-8 cm2 s-1 is obtained for this experiment. If the time constants measured by dynamic light scattering describe fluctuations inside L3phase, one might write for the mutual diffusion coefficient D, as it has successfully been done for critical fluctuations
D)
kT 6πηξ
(12)
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Figure 7. Field-jump dynamics of the L3-phase: (top) voltage pulse; (center) electric current, (bottom) birefringence signal, all as a function of time for surfactant volume fractions of 0.072 (left) and 0.219 (right).
with η the viscosity of the mixture and ξ is identified with the characteristic domain size. Since ξ goes as φ-1 one would expect a dependence
τDLS-1 ∝
φ η
(13)
Because η rises weakly with φ, the exponent 0.8 slightly less than 1 is reasonably explained. D. Field-Jump Experiments. Without introducing a biased opinion at this point on the structural relaxation mechanism, it can be said that the electric field acts on the dipoles of the surfactant molecules. This is the lever that induces a distortion of the L3 structure. The resulting anisotropy is observed as birefringence. Figure 7 shows the relaxation into the distorted state reaching a stationary birefringence value for a low and high membrane volume fraction φ and, after the field has been switched off, the relaxation back into equilibrium. We measure simultaneously electric current and optical birefringence. This has proven very useful for the investigation of field-induced structural changes in water-inoil microemulsions where water is the minority component.26 In the present experiments we find that Ohm’s law is fulfilled and no relaxation processes are observable in the current signal. Also, the conductivity calculated from the stationary values of voltage and current is practically independent of the field strength applied. From (26) Schlicht, L.; Spilgies, J.-H.; Runge, F.; Lipgens, S.; Boye, S.; Schu¨bel, D.; Ilgenfritz, G. Biophys. Chem. 1996, 58, 39.
the bottom curves in Figure 7, which show the timeresolved birefringence signal, it is seen that both the (fieldoff) relaxation and the amplitude depend strongly on the volume fraction of membrane. For the sample with φ ) 0.112 which is rather slow and exhibits large amplitudes the field-off relaxation has been analyzed with a stretched exponential function, i.e.
∆φ ) ∆φ0 exp(-(t/τE)β)
(14)
in order to test for a possible distribution of relaxation times. At all field strengths a consistent fit was obtained giving a stretch exponent of β ) 0.85 with time constants varying only within a few percent. Such an analysis was not feasible for the faster systems. For comparison of all curves we evaluated an average relaxation time 〈τ〉 obtained from the area under the decay curve. We also fitted a simple exponential function yielding τE. The results of both procedures yield similar results and are given in Table 1. The entries are averages from curves at different field strengths. τE is plotted for comparison in Figure 6 as a function of surfactant volume fraction φ. As is seen the field induced changes show the fastest relaxation times, the φ-dependence scaling as τE-1 ∝ φ 2.8. In Figure 8 the stationary values of the birefringence phase shift φ are plotted vs the square of the electric field strength. It can be seen that the quadratic Kerr law is fulfilled at all conditions. From the slope of the curves the Kerr constant is calculated according to B ) φ/(2πlE2), where
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Table 1. Field-off Birefringence Relaxation Times and Kerr Constants for the L3-Phase of H2O-C10E4 n-Decanol (C10E0) at 25.1 °Ca φ
δV
〈τ〉 (µs)
τE (µs)
B (m‚V-2)
0.357 0.286 0.287 0.219 0.144 0.112 0.072
0.376 0.351 0.355 0.343 0.317 0.333 0.291
6(1 8(1 8(2 11 ( 1 42 ( 8 67 ( 12 176 ( 13
2.1 ( 0.1 4.8 ( 1.2 4.2 ( 0.6 9.0 ( 0.7 40 ( 1 71 ( 3 182 ( 6
-1.09 × 10-11 -1.65 × 10-11 -1.65 × 10-11 -2.20 × 10-11 2.92 × 10-12 5.70 × 10-11 2.31 × 10-10
a
φ, membrane volume fraction; δv, cosurfactant volume ratio; 〈τ〉, mean relaxation times determined from the area under the decay curve; τ, relaxation times from the fit to a single exponential; B, Kerr constants.
of fluctuations affecting the length scale can be expected, local inside-outside (ψ) and surfactant volume fraction (φ) fluctuations.28 The energetics of the surfactant bilayer is thought to be governed by the bending energy of the surfactant films first introduced by Helfrich29
dFs ) [2κ(H - c0)2 + κjc1c2] dA
Here κ and κj are the rigidity and the saddle-splay modulus of the bilayer. For the bilayer the spontaneous curvature c0 is zero by symmetry. Both the “sponge” and the lamellar phase are characterized by an area averaged zero-mean curvature of the surfactant bilayers and similar rigidities.30-32 However, they differ in the spontaneous curvature of the monolayers forming the bilayer. The mean curvature H of an amphiphilic film is given by
1 H ) (c1 + c2) 2
Figure 8. Stationary birefringence vs square of electric field strength demonstrates the validity of the Kerr law.
l is the optical path length. The most pronounced feature of the Kerr amplitudes is the change in sign with increasing membrane volume fraction. From Figure 8 one finds that B starts at low volume fraction with large positive values, decays to zero at φ ) 0.155 and then passes through a shallow minimum. The results of the analysis are included in Table 1. A detailed discussion follows below. IV. Discussion A. Some Theoretical Considerations. Our experiments show that various relaxation methods applied probe different fluctuation modes of the underlying structure of the L3-phase. In discussing the various processes, we envisage the L3 structure as composed of randomly connected units shown in Figure 1. The L3-phase is characterized by a length scale set by the total area per unit volume of the bilayer mid-plane A/V
ψ(1 - ψ) A/V
ξ)a
(15)
where a is a dimensionless geometric factor and
A/V )
1 φ a 2 vs s
(16)
is given by the volume fraction of surfactant φ incorporated in the membrane and the mean area as and volume vs of the surfactant molecules, which may be combined into an effective surfactant length ls ) vs/as. For symmetric sponge phases considered here, the equilibrium volume fractions of water of the inside ψ and outside (1 - ψ) water subphases are equal, i.e., ψ ) 0.5. Numerical values for the effective coordination number a are typically close to 6 for simple models.27 From eq 15 immediately two types
(17)
(18)
with c1 and c2 being the two principal curvatures. For micelles or microemulsion droplets both principal curvatures are approximately equal (c1 ≈ c2) and of the same sign. For cylindrical structures one of the principal curvatures is approximately zero (c1 ≈ 0, c2 * 0). For the bilayers in a lamellar structure, both (c1 ≈ c2 ≈ 0) and H ) 0. For the bilayers of the dilute L3-phase the principal curvatures are small but nonzero and, importantly, have opposite sign (c1 ≈ -c2). In this fashion they still add up to H ) 0. The individual monolayers of the bilayer have a curvature toward water, that is Hmono< 0. This fact is a consequence of the smaller area of the surface at the surfactant headgroup position than at the bilayer midplane. The smaller area at the location of the surfactant headgroups is consistent with the differential geometry of surfaces parallel to a saddle-shaped base surface.3 This is a characteristic feature of the L3-phases manifesting the frustration5,33 of the monolayers that is partially relieved by forming the distorted 3D structure seen in Figure 1. On a local scale, fluctuations of the surfactant film around the mean curvature and peristaltic and baroclinic modes34 of the film may occur. By changing the spontaneous curvature of the monolayers or by dilution, the transition from the LR-phase to the L3-phase may be induced. In the latter case the stability and order of the LR-phase, which is controlled by the steric repulsion between the lamellae due to undulations,35 is overwhelmed by entropic orientation fluctuations.4 For the nonionic system we are concerned within this study, the spontaneous mean curvature of the monolayers can be changed by varying the temperature or pressure. The spontaneous curvature of the monolayers inside the L3-phase is directed toward water and becomes more strongly curved, i.e., more negative, when the pressure drops or temperature increases until a water-rich phase separates out. Porte5 derived the bending elastic properties (27) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (28) Le, T. D.; Olsson, U.; Wennerstrom, H.; Schurtenberger, P. Phys. Rev. E 1999, 60, 4300. (29) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (30) Sottmann, T. Dissertation, Go¨ttingen, 1997. (31) Tlusty, T.; Safran, S. A.; Menes, R.; Strey, R. Phys. Rev. Lett. 1997, 78, 2616. (32) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (33) Sadoc, J. F.; Charvolin, J. J. Phys. 1986, 47, 683. (34) Nallet, F.; Roux, D.; Prost, J. J. Phys. (Fr.) 1989, 50, 3147. (35) Helfrich, W. Z. Naturforsch. 1978, 33, 305.
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of the bilayer from those of the monolayers. Specifically he obtained for the saddle-splay modulus of the bilayer
κjbil ) 2κjmono - 4c0,monoκmono
(19)
while c0,bil ) 0 and κbil ) 2κmono. Here 2 is the thickness of the hydrophobic part of the bilayer. This result is remarkable insofar as it states that for decreasing c0,mono eventually κjbil turns positive and the Gaussian term in eq 17 drives the transition from LR to L3. The stability of the L3-phase against shrinking toward extremely small length scales is either thought to be provided by the entropy of fluctuations5 or by higher order, anharmonic terms.28 Morse36 argued that both views might deserve consideration, the entropy being important for high dilutions while the anharmonic terms are deemed relevant for highly curved structures. In another paper, Morse37 predicts from the statistical mechanics of fluctuating membranes the L3-LR transition to occur on a line in the phase diagram described by
ln
()
6πκjbil φ )φ0 5kT
(20)
where φ0 is a constant of order unity. This result has been independently confirmed by Gompper and Kroll38 using MC simulation of triangulated membranes. Independent experimental results on nonionic bilayer systems lead to L3-LR transition lines in binary phase diagrams consistent with the functional form of eqs 19 and 20. If the typical near-linear temperature dependence of c032,39 of these systems is taken into account in eq 19 the phase transition temperatures are expected to be nearly linear in ln(φ) which is indeed observed.19,40,43 The equilibrium structure of the L3-phase may be understood in terms of a competition between elastic energy of the surfactant film and entropy. The transition from LR to L3 is seen as a consequence of passage formation between bilayers affecting both energy and entropy. Consequently, there must be an equilibrium distribution of passages in the L3-phase around which fluctuations may occur or which relaxes back into equilibrium after a perturbation. B. p- and T-Jump. φ Dependence. Figure 6 compares on double logarithmic scale the φ dependence of the time constants τDLS with the relaxation time τT,p at T ) 20 °C. The φ dependence of τT,p is comparatively strong and seems to behave as
τT,p-1 ∝ φ9.2
(21)
In contrast, the φ dependence for τDLS-1 shows a weak φ 0.8 dependence. It is obvious that we are monitoring two different processes. This statement is further corroborated by the q independence of τT,p and the q2 dependence of τDLS. The q2 dependence is typical for fluctuation processes. On the other hand, the q independence of τT,p is an indication for the detection of structural changes inside (36) Morse, D. C. Curr. Opin. Colloid Interface Sci. 1997, 2, 365372. (37) Morse, D. C. Phys. Rev. E 1994, 50, R2423. (38) Gompper, G.; Kroll, D. M. Phys. Rev. Lett. 1998, 81, 2284. (39) Sottmann, T.; Strey; R. J. Chem. Phys. 1997, 106, 8606. (40) Strey, R.; Schoma¨cker, R.; Nallet, F.; Roux, D.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86 (12), 2253. (41) Strey, R.; Pakusch, A. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Publishing Corp.: New York, 1987; Vol. 4, p 465. (42) Le et al. Manuscript in preparation. (43) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182.
a phase as Pakusch41 has shown for the related subject of micelle kinetics. From other experiments19,42 performed earlier than the present study we expect for φ > 0.45 to see only fluctuations, because the expected structural relaxation time is faster than τDLS and the local structure adjusts instantaneously to respective fluctuation state. At this point it should be mentioned that these earlier experiments were performed as part of an international collaboration between the Institute for Physical Chemistry of the University of Cologne and the Center for Chemistry and Chemical Engineering of the University of Lund. We tried to confirm and extend the T-jump results by Porte and co-workers10 for the ionic system cetylpyridinium chloride/ hexanol/brine. However, the dynamics of the L3-phase of the ternary system H2O-n-decane-C12E5 chosen revealed a completely different behavior than the ionic system.10 Therefore, the present H2O-C10E4-C10E0 system was chosen to check whether we are dealing with a generally different behavior. A paper describing the previous experiments is in preparation.42 In that work, a single relaxation mode was observed, but it displayed an even stronger dependence of the relaxation time on membrane volume fraction than in the present system. Since the nonionic L3-phase required a change of the experimental temperature, we looked for a system that permits isothermal measurements and preferably consisted only of surfactant bilayers. Our result agrees with the preliminary evaluation Uhrmeister19 performed on H2O-n-decaneC12E5 system. For that system the φ dependence of the structural relaxation time τT,p is
τT,p-1 ∝ φ8.6
(22)
if one extrapolates the time constants to a constant temperature of T ) 40 °C. This agreement with our present findings (cf. Figure 6) allows the conclusion that the kinetic phenomena described are characteristic for L3-phases of nonionic surfactant systems and is not a special property of a single system. Moreover, effective exponents have been established much larger than the exponent 3. This result is at variance with the general opinion of Porte et al.10 From the bending and scaling properties of the L3phase they argued for a theoretical φ 3 or φ 4 dependence of the structural relaxation time, which they indeed found for ionic surfactants. T Dependence. A similar strong temperature dependence as observed in Figure 5 is known for the characteristic length scale ξ and the interfacial tension σab in microemulsions.30,39,43 The activation energy Ea is depending neither on the volume fraction φ nor on the length scale ξ which is the size of the passages inside the sponge-like structure of the L3-phase. This is interesting insofar as the scale invariance of the bending energy of vesicles or L3 structures would predict such behavior. The activation energy Ea might accordingly be associated with the formation of new passages between adjacent bilayers inside the structure of the L3-phase. Recently, Sens and Safran44 derived a theoretical expression for the energy barrier of the formation of a single passage between two membranes. They calculated a value of 1.2 × 10-18 J per passage which is an energy of 700 kJ/mol. These authors thus arrive at an order for Ea, which, as the rigidity of the lipid membranes they considered was 10 times higher than the rigidity of membranes in our system, supports the presumption of passage formation.39 (44) Sens, P.; Safran, S. A. Europhys. Lett. 1998, 43, 95.
Dynamics of the “Sponge” (L3) Phase
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C. Field Jump. Time Constants. Comparison of the time constants from field jump with those obtained from T- and p-jump relaxation shows that the electric field initiates structural changes with much higher relaxation rates. The electrical field is thought to act on the dipoles of the surfactant molecules and to bring about a distortion of the elementary units. We presume that the time scale is too short for topological changes. We interpret the observed relaxation dynamics as the viscoelastic response of the L3 structural units. Viscoelastic theory gives the relaxation time τ as a ratio of viscosity η and elastic modulus G of the viscoelastic fluid (Maxwell element).45 Then
τE-1 ) G/ηs
(23)
where ηs is defined through Newton’s law as the ratio of stress and rate of strain, the shear elastic modulus G is according to Hooke’s law given as the ratio of stress and relative distortion. For fluid droplets the restoring force is determined by the interfacial tension σ and G has to be replaced by σ/R, where R denotes the droplet radius.46,47 Thus, the elastic modulus for a dispersion of droplets is given by the surface energy density
G)
1 Fs 3 V
(24)
with FS ) ∫dS σ ) 4π R2σ and V ) (4π/3)R3. The shear modulus, with a factor 4/15 instead of 1/3 in eq 24, has directly been derived by Derjaguin and by Stamenovic for foams stabilized by surfactant films in the presence of surface tension.48,49 If the droplets are governed by bending elasticity of the surfactant film, the surface tension has to be replaced by σ ) 2κ/R2,27 where κ denotes the bending rigidity constant and the relaxation time constant is given by τ-1 ) 2κ/ηsR3. A more detailed investigation based on the fluctuation analysis of Milner et al.50 has been given by van der Linden et al.51 for water-in-oil microemulsion droplets coated with an AOT-surfactant monolayer. For the random structure of L3-phases no rigorous theory for an elastic distortion is known. For obtaining estimates of the viscoelastic relaxation time let us assume that to first order the complex L3-phase may be conceived as a dispersion of vesicles with a radius equal to the characteristic correlation length ξ. We then have for the relaxation time
τE-1 =
2κ ηsξ3
(25)
We note that the saddle-splay modulus κj does not appear because the E-jump is expected to induce only distortions, i.e., changes at constant topology, for which terms proportional to κj cancel. In numbers (45) Macosco, C. Rheology; VCH: Weinheim, Germany, 1994; Vol. 8, p 117. (46) Moriya, S.; Adachi, K.; Kotaka, T. Langmuir 1986, 2, 155. (47) Adachi, K.; Tanaka, M.; Shikata, T.; Kotaka, T. Langmuir 1991, 7, 1281. (48) Derjaguin, B. Kolloid Z. 1933, 64, 1. (49) Stamenovic, D.; Wilson, T. A. J. Appl. Mech. 1984, 51, 229. (50) Milner, S.; Safran, S. A. Phys. Rev. A 1987, 36, 4371. (51) van der Linden, E.; Bedeaux, D.; Hilfiker, R.; Eicke, H. F. Ber. Bunsen-Ges. Phys. Chem. 1991, 95, 876.
τE-1 =
2κ φ3 ) 3 ηs(2aψ(1 - ψ)ls) 2 × 4 × 10-21 0.13 s-1 ) 3 × 104 s-1 (26) 0.01 × 33 × 10-27
for φ ) 0.1, where we used for ξ in eq 14 ψ ) 0.5, a ) 6,27 κ ) 1 kT,28 and ls ) 1 nm and η ) 10 mPas. We therefore reproduce the experimental result τE-1 ) 1 × 104 s-1 (cf. Figure 6) to better than the order of magnitude. More importantly, the analysis proposes a φ 3 dependence of the relaxation rate as experimentally observed (cf. Figure 6). The slightly lower experimental exponent 2.8 may be attributed to the viscosity η expected to increase with φ. Thus, this result is in accordance with the scaling law proposed.4 It is also in support of experimental results by refs 7 and 8. If the analysis can be refined to give more rigorously the exact numerical prefactor, time-resolved birefringence measurements may turn into a convenient method for determining the bending elasticities of surfactant and lipid bilayer membranes. Regarding a theory for the elastic modulus of the L3phase, one is attempted to apply directly eq 24. An estimate of FS can be made by a straightforward integration of the Helfrich expression with c1 ) -c2 ) ξ/2. A different expression has been used by Wennerstro¨m and coworkers.28 However, it turns out that the dominant term in the various expressions contains the saddle-splay modulus. While this may explain the stability of the L3phase, it appears to be physically inconsistent to use it for explaining field-jump results because we assume that the topology does not change. In that case the Gauss-Bonnet theorem predicts the Gaussian term in the free energy to remain constant and the saddle-splay modulus should not appear for pure distortion. It seems that the Derjaguin relation cannot directly be applied to the bicontinuous L3-phase. Previous workers had concluded,7,8 due to the similarity of the birefringence time constant for rotational diffusion of a sphere with a diameter of ξ
τrot-1 )
6 kT π ηξ3
(27)
that the L3 structure be one of oblate ellipsoids, which have effectively the same ξ3 scaling. The applicability of eq 27 is, however, not proof of a particular mechanism or structure. Kerr Constants. As noted before in connection with Figure 8, the Kerr law is fulfilled in the field strength range applied with no indication of saturation effects. Most pronounced is the dependence of the Kerr constants on the membrane volume fraction: Starting from large positive values at low volume fraction the birefringence decreases with increasing membrane fraction, changes sign, and then passes through a shallow minimum (see Figure 9). This behavior seems to be characteristic of compartmentalized surfactant systems. Water-in-oil microemulsions in the droplet regime show a similar behavior as a function of the droplet radius which can be understood as contributions of “intrisic birefringence” of the membrane due to the optical anisotropy of the surfactant molecules, and “form birefringence” of the structural units. The general existence of these two contributions to the total birefringence becomes most clearly apparent in the Peterlin-Stuart theory for suspensions of rigid particles.52,53 An outline of this theory is given by Schorr and Hoffmann.53 Van der Linden et al.54 have developed a
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Figure 9. Specific Kerr constants as a function of membrane volume fraction change sign. Solid line: fit to eq 28.
theory for deformable drops, showing that the dominant term of the positive form contribution scales with the third power of droplet radius while the negative intrinsic contribution depends quadratically on droplet radius. We can describe our field jump data for the random structure of the L3-phase very well with the same functional dependence for the specific Kerr constant, whereby the scaling ξ ∼ 1/φ is incorporated
B/φ ) R(φ-3 - φ0-1φ-2)
(28)
with R ) 2.2 × 10-12 m/V2, φo ) 0.155 being the volume fraction where B changes sign. Equation 28 reproduces very well the minimum in Figure 9. Although the observed dependence is well described by eq 28, it remains unclear how a distorted unit of the L3-phase, which contains water both inside and outside can give rise to such an extremely high form birefringence. Equation 28, however, is at variance with the scaling discussed by Porte et al.9 These authors find a very similar curve for the L3-phase of the C12-betain system, where, however, the existence of a minimum is not obvious. Further investigations on L3phases are needed, possibly in ternary systems where the membrane can deliberately be swollen with oil which allows gradual changes in thickness on which the rigidity and the form birefringence sensitively depends. V. Conclusions We examined the relaxational behavior of bicontinuous sponge-like structures of the L3-phase in the pseudobinary system H2O-C10E4-n-decanol (C10E0). Inside the stability range of the isotropic L3-phase, four different methods yield relaxation curves, which can be ascribed to different relaxation processes. T- and p-jump experiments yield single-exponential relaxation curves. The time constants τT-1 from T-jump (52) Peterlin, A.; Stuart, H. A. Z. Phys. 1939, 112, 129. (53) Schorr, W.; Hoffmann, H. Physics of amphiphiles; NorthHolland: New York, 1985; p 160. (54) van der Linden, E.; Geiger, S.; Bedeaux, D. Physica A 1989, 156, 130.
experiments are independent of the q vector of scattered light. The time constants from p-jump τP-1 are equal to those from T-jump. They cover a range of several orders of magnitude when the volume fraction of membrane φ is only changed by a factor of 3. Here obviously a reorganization of the overall structure takes place. This interpretation is further supported by the fact that we were able to detect the same process by electrical conductivity of the whole sample in the p-jump while only collecting scattered light from a small volume in the T-jump. The strong temperature dependence yields independently of φ an effective activation energy of about 200 ( 30 kJ/mol. This indicates that the same rate-determining step is involved at different compositions. For the description of the T- and φ-dependence of τT,p-1, the mean curvature H of the monolayers forming the bilayers has to be considered. Associated with a change either in temperature or pressure is a change in spontaneous curvature c0 of each monolayer differing from the mean curvature H in equilibrium. This difference after each perturbation results in a driving free energy difference ∆Fs that the system tries to minimize by a stronger curvature. It is probable that this process leads to the formation of new passages inside the bicontinuous structure of the L3-phase. This in turn affects both the scattering of light and the electrical conductivity. Much less dramatic is the variation of the time constants of the thermal fluctuations inside the L3-phase τDLS-1 from dynamic light scattering. Only a weak φ-dependence is found which may be related to the mutual diffusion coefficient D of objects of length scale ξ in the hydrodynamic regime. This explanation is supported by the q2dependence of the time constants. The fastest process is seen in field-jump experiments. In this case the L3 structure is deformed under the action of an external electric field. The numerical value as well the φ3 dependence of relaxation time constants τE-1 can be explained as an elastic deformation of the L3 structure if one considers the bending energy of the bilayers. In addition, a sign change of the Kerr constants obtained from the steady state amplitudes is observed upon increasing φ. Comparison with literature results on related systems shows that this seems to be a general property for compartmentalized surfactant systems. When a rigorous treatment of the topological deformations becomes available, birefringence relaxation might turn into a convenient tool for determining the bending rigidities of surfactant and lipid bilayers. Acknowledgment. In addition to devoting the present article to the four gentlemen honored by this special issue, we would like to express our gratitude to Prof. D. Woermann. Without his careful development and assembling of the various techniques employed, this work would have been more difficult to accomplish. We thank O. Lade, H. Leitao, B. Rathke, T. Sottmann, C. Stubenrauch, P. Uhrmeister, and A. Wehling for helpful discussions. LA000220Q
