Measurement of the Membrane Flexibility in Lamellar and “Sponge

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Langmuir 1996, 12, 6028-6035

Measurement of the Membrane Flexibility in Lamellar and “Sponge” Phases of the C12E5/Hexanol/Water System E Ä . Freyssingeas,† F. Nallet,* and D. Roux Centre de recherche Paul-Pascal, CNRS, Avenue du Docteur-Schweitzer, F-33600 Pessac, France Received May 29, 1996. In Final Form: August 30, 1996X

The present work is primarily concerned with the measurement, by several different methods (using X-ray and static light scattering experiments or dynamic light scattering experiments), of the elasticity of bilayers made up of n-dodecyl pentaethylene glycol monoether (C12E5) and hexanol molecules. The membrane bending elasticity modulus, κ, has been measured for both the LR (lamellar) and the L3 (“sponge”) phase of the C12E5/hexanol/water system. For the lamellar phase, κ was measured using both the excess area and the dynamics of the fluctuations, while for the sponge phase κ was measured using only the excess area. The values of κ measured seem to be of the same order (approximately kBT) for both the lamellar phase and the sponge phase, although the molar ratio of hexanol to C12E5 is 4 times bigger in the sponge phase. The effect of temperature on κ was also studied: the slight increase in κ found for this system with temperature seems to be related to a decreasing polar head area of the C12E5 molecules. For the sponge phase, κ cannot be measured from the dynamics of the fluctuations; however, some results can be extracted from the dynamic light scattering experiments. The most interesting one was that the collective diffusion coefficient D is proportional to the membrane volume fraction, φm, and is a function of the membrane elastic constants, κ and κj, as well as of the thermal energy kBT, leading to an estimate of κj.

Introduction A wide variety of structures and behaviors are encountered in the study of the surfactant and solvent mixtures also known as lyotropic phases. In some cases the basic building unit is locally a planar bilayer of surfactant molecules that extends in two dimensions to form a membrane. Examples of such structures with different long-range or internal symmetries are, for instance, the lamellar (LR) and cubic phases, as well as the “sponge” (L3) and vesicle phases.1-5 In the lamellar phase the bilayers are stacked along an axis z with a smectic order, while in the sponge phase they form a multiply connected dividing surface. The surfactant bilayer may be described as a thin plate whose shape and behavior depend only on the competition between its elastic properties, any applied constraint, and the entropy. The stability of a phase made up of membranes is governed by, among other factors, the competition between the elastic energy of the membrane and the thermal energy, kBT. At the harmonic level, two elastic constants describe the elastic properties of the membrane: the elastic mean curvature modulus, κ, and the elastic Gaussian curvature modulus, κj.6,7 The two elastic constants, κ and κj, play very different roles.8,9 The elastic mean curvature modulus, κ, can be understood as † Present address: Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 0200, Australia. X Abstract published in Advance ACS Abstracts, November 1, 1996.

(1) Ekwall, P. Advances in Liquid Crystals; Brown, G. M., Ed.; Academic Press: New York, 1975; Vol. 1, p 1. (2) Porte, G.; Marignan, J.; Bassereau, P.; May R. J. Phys. (Paris) 1988, 49, 511. (3) Gazeau, D.; Bellocq, A.-M.; Roux, D.; Zemb Th. Europhys. Lett. 1989, 9, 447. (4) Luzzati, V. Biological Membranes; Chapman, D., Ed.; Academic Press: New York, 1968; p 71. (5) Mariani, P.; Luzzati, V.; Delacroix, H. J. Mol. Biol. 1988, 204, 165. (6) Canham, P. B. J. Theor. Biol. 1970, 26, 61. (7) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (8) David, F. Statistical Mechanics of Membranes and Surfaces; Nelson, D. R., Piran, T., Weinberg, S., Eds.; World Scientific: Singapore, 1989; p 158. (9) Porte, G. J. Phys.: Condens. Matter 1992, 4, 8649.

S0743-7463(96)00524-0 CCC: $12.00

the energy that has to be provided to bend the membrane around its equilibrium position. When κ is comparable to the thermal energy, kBT, thermal fluctuations give rise to significant displacement fluctuations of the membranes around their equilibrium position. These fluctuations have important consequences on the existence of these phases and on their static and dynamic properties.10-20 On the other hand, the elastic Gaussian curvature modulus, κj, is the driving parameter for the membrane topology. Theoretically,21 if -2κ < κj < 0 (assuming κ positive), the membrane prefers to be a plane and this has a tendency to favor the lamellar phase. If κj < -2κ, the membrane prefers to curve into a spherical shape and the vesicle phase is then favored. If κj > 0, the membrane prefers to have a saddle form tending to favor the cubic or the sponge phases. It results that κj plays a role each time a structural transformation involves a topological change for the membrane22 (like the transition between lamellar and sponge phases, for instance), but it has no effect as long as the curvature fluctuations take place at constant topology or degree of connectivity. Therefore, a knowledge of the elastic constants is important for the understanding of the physical behavior of membrane phases. Experimentally, the bending modulus κ can be extracted with scattering experiments,23-28 while it is more difficult to measure κj.29 (10) Helfrich, W. Z. Naturforsch. 1978, 33a, 305. (11) Larche´, F. C.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, J. Phys. Rev. Lett. 1986, 56, 1700. (12) Safinya, C. R.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A.-M. Phys. Rev. Lett. 1986, 57, 2718. (13) Bassereau, P.; Marignan, J.; Porte, G. J. Phys. (Paris) 1987, 48, 673. (14) Roux, D.; Safinya, C. R. J. Phys. (Paris) 1988, 49, 307. (15) Bassereau, P.; Marignan, J.; Porte, G.; May, R. Europhys. Lett. 1991, 15, 753. (16) Nallet, F.; Roux, D.; Milner, S. T. J. Phys. (Paris) 1990, 51, 2333. (17) Roux, D.; Cates, M. E.; Olsson, U.; Ball, R. C.; Nallet, F.; Bellocq, A.-M. Europhys. Lett. 1990, 11, 753. (18) Di Meglio, J.-M.; Bassereau, P. J. Phys. II 1991, 1, 247. (19) Halle, B.; Quist, P.-O. J. Phys. II 1994, 4, 1823. (20) Auguste, F.; Barois, P.; Fredon, L.; Clin, B.; Dufourcq, E. J.; Bellocq, A.-M. J. Phys. II 1994, 4, 2197. (21) Helfrich, W. Physics of Defects (les Houches XXXV); NorthHolland Publishing Company: Amsterdam, 1981; p 716. (22) Huse, D.; Leibler, S. J. Phys. (Paris) 1988, 49, 605.

© 1996 American Chemical Society

Membrane Flexibility

The present work is primarily concerned with the study, by several different scattering methods (X-ray and static light scattering experiments, or quasi-elastic light scattering experiments), of the membrane elasticity for both the LR phase and the L3 phase of the n-dodecyl pentaethylene glycol monoether (C12E5)/hexanol/water system.30 For this system, not only the temperature but also the membrane composition (i.e., the molar ratio hexanol/C12E5) control the lamellar-to-sponge transition. The system is therefore ideally suited to observe the influence of the membrane composition on the membrane elasticity and, indirectly, the influence of the membrane elasticity on the stability of sponge and lamellar phases. The first section of this article describes the phase diagram in the dilute region and the experimental setup. The second section describes our study of the membrane elasticity in the lamellar phase at fixed or variable temperature, while the third section shows a similar study in the sponge phase at 24 °C. The effects of the membrane fluctuations on the sample structures were studied using X-ray and static light scattering experiments, leading to a measurement of the membrane elastic modulus κ for both the lamellar and the sponge phases. The dynamics of the lamellar and sponge phases was studied by quasielastic light scattering experiments. Because the spatial and time ranges probed with this technique are, respectively, 0.3-3 µm and 10-5-10-1 s, up to moderately dilute lamellar or sponge phases (with structural sizes typically less than 0.1 µm), the dynamical signal originates in the coupled motions of a large number of membranes. At larger dilution, the dynamics probed corresponds to single membrane motions. For both the lamellar and the sponge phase, these experiments give information more or less directly related to the membrane elasticity.

Langmuir, Vol. 12, No. 25, 1996 6029

Figure 1. Phase diagram of the C12E5/hexanol/water system at 24 °C: abscissa, C12E5 mass fraction; ordinate, hexanol mass fraction. I is an isotropic phase, LR is the lamellar phase, and L3 is the “sponge” phase.

I. Phase Diagram and General Information Both the dilute lamellar phase and the “sponge” phase may be simultaneously found in the C12E5/water system, which has the advantage of being truly binary.23 The phase diagram exhibits these interesting phases only above 50 °C, however. Recently, a closely-related systems with hexanol added as a cosurfactantshas been shown to exhibit the same kind of behavior but at room temperature.30 Figures 1 and 2 show the locations of the lamellar, sponge and micellar phases at 24 °C. A lamellar structure can be obtained with as little as 0.15 wt % of C12E5 (i.e., a membrane volume fraction φm ) 0.2%), while the sponge phase can be diluted to 0.2 wt % of C12E5 (i.e., a membrane volume fraction φm ) 0.25%). The membrane is presumably made up of a bilayer of C12E5 and hexanol molecules while the solvent is a mixture of water with a small amount of hexanol (approximately 99.7% water and 0.3% hexanol). Note that if the amount of C12E5 remains constant, increasing the quantity of hexanol favors the sponge phase against the lamellar phase. The stability of these phases with respect to temperature was investigated along two dilution lines differing by the (23) Strey, R.; Schoma¨cker, R.; Roux, D.; Nallet, F.; Olsson, U. J. Chem. Soc., Faraday Trans. 1990, 86, 2253. Roux, D.; Nallet, F.; Freyssingeas, E Ä .; Porte, G.; Bassereau, P.; Skouri, M.; Marignan, J. Europhys. Lett. 1992, 17, 575. (24) Freyssingeas, E Ä .; Roux, D.; Nallet, F. J. Phys. Condens. Matter 1996, 8, 2801. (25) Skouri, M.; Marignan, J.; Appell, J.; Porte, G. J. Phys. II 1991, 1, 1121. (26) Bassereau, P.; Appell, J.; Marignan J. J. Phys. II 1992, 2, 1257. (27) Brochard, F.; de Gennes, P.-G. Pramana Suppl. 1975, 1, 1. (28) Nallet, F.; Roux, D.; Prost J. Phys. Rev. Lett. 1989, 62, 276. Nallet, F.; Roux, D.; Prost J. J. Phys. (Paris) 1989, 50, 3147. (29) Bolthenhagen, P.; Lavrentovich, O.; Kle´man, M. J. Phys. II 1991, 1, 1233. (30) Jonstro¨mer, M.; Strey, R. J. Phys. Chem. 1992, 96, 5993.

Figure 2. Dilute part of the phase diagram of the C12E5/ hexanol/water system at 24 °C: abscissa, C12E5 mass fraction; ordinate, hexanol mass fraction. I is an isotropic phase, LR is the lamellar phase, and L3 is the “sponge” phase.

membrane composition (hexanol/C12E5 ) 0.27 or 0.41 by weight, corresponding to a 1/1 C12E5 to hexanol mole fraction, or to a 1/1.6 mole fraction, respectively) (Figures 3 and 4). The ordered phase (LR) is stable at low temperatures, which is the commonly observed behavior: increasing the amount of hexanol in the membrane or increasing the temperature give the same effect; this favors the sponge phase against the lamellar phase. Note that the return to thermodynamic equilibrium after such perturbations as gentle shearing or small changes in temperature is sometimes very slow. This is especially true for mixtures with a low surfactant content (typically, less than 4 wt % of C12E5, corresponding to a membrane volume fraction φm less than 5%). So, care was taken to allow enough time (usually a few hours) for

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Freyssingeas et al.

the incident beam and the axis of the goniometer can be changed between 20° and 160°. A polarizer is used to analyze the polarization state of the scattered light. For the dynamic light scattering experiments, the photocurrent is processed with a 72-channel Brookhaven Instruments digital correlator that may be operated in multiplesample-time mode. II. Measurement of the Bending Modulus in the Lamellar Phase

Figure 3. Evolution as a function of temperature of the dilution line with hexanol to C12E5 mass ratio equal to 0.27. Abscissa is C12E5 mass fraction. I is an isotropic phase, LR is the lamellar phase, and L3 is the “sponge” phase.

II.1. Excess Area Method. Upon addition of solvent to the lamellar phase, the periodicity d of the stack of membranes increases. On geometrical grounds (when the membranes can be viewed as rigid objects) a simple law d ∝ 1/φm is expected upon dilution. However, it has been predicted that, in the absence of direct, long-range interactions between membranes, a logarithmic correction to this simple dilution law is required because of the short wavelength fluctuations of the membrane.31 This correction has been experimentally observed for systems made of flexible enough membranes (i.e., κ in order of kBT).23-26 According to the theoretical argument, and in agreement with experimental observations, the logarithmic correction is writtensfor dilute enough lamellar phasessas follows:

dφm ≈ A - B ln φm

(1)

The best theoretical estimates for the parameters A and B appearing in eq 1 are, respectively32

[

A)δ 1+

B)δ

Figure 4. Evolution as a function of temperature of the dilution line with hexanol to C12E5 mass ratio equal to 0.41. Abscissa is C12E5 mass fraction. I is an isotropic phase, LR is the lamellar phase, and L3 is the “sponge” phase.

the sample to relax prior to any experiments. Reproducibility of the static light scattering spectra was used as a criterium for thermodynamic equilibrium. The X-ray experiments were performed using an inhouse (copper rotating anode) high-resolution X-ray spectrometer. The high resolution is obtained by using both a monochromator and an analyzer, each made up of two face-to-face monocrystals of germanium with three Bragg reflections for the Cu KR1 line. This setup leads to a very good in-plane resolution function, both narrow and well localized (half width at half maximum of the order of 10-3 Å-1). The light scattering experiments were performed using a Coherent IK-90 ionized krypton laser light source, operated at wavelength λ ) 647.1 nm and polarized vertically. The samples held in glass tubes (1 mm diameter) or flat capillaries (0.2 × 2 mm2) are put at the center of a temperature-regulated tank containing an index-matching liquid (carbon tetrachloride). The scattered light is collected with a photon-counting PMT (Hamamastu) set on a goniometer. The angle φ between

(

)]

kBT 16κδ3 ln 8πκ 3πkBTvs kBT 4πκ

(2)

(3)

where δ is the geometric membrane thickness, vs the surfactant molecular volume, and κ the membrane bending elasticity modulus. The lamellar structure was monitored upon dilution (with a water/hexanol mixture in respective amounts 99.7 and 0.3 wt %), keeping constant the membrane hexanol/ C12E5 mole ratio (equal to 1) and the temperature (24 °C). For these conditions the volume fraction can be varied continuously by a factor of about 200, namely from φm ) 0.72 down to φm ) 0.004 (Figure 3). The periodicity d of the lamellar structure was extracted from the position of the first-order Bragg diffraction peak, q0, using d ) 2π/q0. From φm ) 0.72 to φm ) 0.38, q0 is accurately measured by high-resolution X-ray scattering. As the volume fraction is decreased, the peak intensity decreases12-14 and becomes commesurate with background intensity for φm less than 0.38; see Figure 5. Below φm ) 0.02 the periodicity of the lamellar structure, of the order of several thousand angstroms, was measured by static light scattering (Figure 6). In Figure 7, the measured values dφm over the whole dilution range (0.72 g φm g 0.004) are shown as function of the membrane volume fraction φm in a semilogarithmic plot. These data combine both X-ray and light scattering results. From a fit to eq 1 we get the two parameters A and B and we can extract a value for κ (in this case κ ) 0.8kBT) and a value for the membrane thickness δ (δ ) 24.5 Å). Note that taking only the very concentrated samples, for which the logarithmic correction is presumably negligible (experimentally, this occurs for φm g 0.5), (31) Helfrich, W.; Servuss, R.-M. Nuovo Cimento D 1984, 3, 137. (32) Golubovic, L.; Lubensky, T. C. Phys. Rev. B 1989, 39, 1211.

Membrane Flexibility

Figure 5. High-resolution X-ray scattering patterns for all the samples with a membrane volume fraction φm g 38%. For more clarity the intensities at the Bragg diffraction peaks are normalized to 1. The increase in the scattering around the Bragg diffraction position q0 can be clearly seen.

Langmuir, Vol. 12, No. 25, 1996 6031

this has little effect on the value for κ: taking δ ) 28.6 Å, we get (using eq 3 alone) κ ) 0.95kBT. II.2. Quasi-Elastic Light Scattering Measurement. Quasi-elastic light scattering experiments on oriented lamellar phases allow, in principle, simultaneous information about membrane-membrane interactions and membrane flexibility in measuring the relaxation frequency of the baroclinic/undulation mode.27,28 This mode corresponds to a membrane displacement fluctuation at constant membrane thickness, i.e. (in general), to a smectic period modulation associated with concentration fluctuations, except in the special case where the mode wave vector q is exactly perpendicular to the stacking direction. In this particular limit, the membrane displacement field reduces to a collective undulation, with curvature strains only. Membrane-membrane interactions oppose changes in the intermembrane distance, and membrane flexibility opposes deviations from the planar conformation; flow motions inside each solvent layer dissipate the elastic energy stored into the displacement field. Quantitatively, the relaxation frequency ω of the baroclinic/undulation mode is found to be related to the magnitude and orientation of the wave vector, to elastic restoring forces, and to the dissipation as follows33

ω)

Figure 6. Static light scattering pattern for a lamellar phase with a membrane volume fraction equal to 1.13%.

Figure 7. Experimental values of dφm (measured in scattering experiments: X-ray and light) shown as a function of the membrane volume fraction φm in a semilogarithmic plot. The solid line is a fit of the experimental data which gives A and B, eq 1.

another membrane thickness, namely δ ) 28.6 ( 0.5 Å, is obtained from the simple dilution law: d ) δ/φm. This second value of δ is closer to the thickness of a C12E5 bilayer obtained by Strey et al.,23 δ ) 30 Å, and may be a better estimate. In our opinion this difference between the two values for δ illustrates the lack of a proper theoretical description of the dilution law in the intermediate regime where neither φm , 1 nor φm ≈ 1 holds. Nevertheless,

B h qz2 + Kq⊥4 qz2 ηq4 + µ

q⊥2

(4)

In eq 4, qz and q⊥ are the projections of wave vector q along the stacking direction axis (z) and in the plane (⊥) of the lamellae; B h and K are smectic elastic constants: B h, the smectic compressibility modulus (at constant chemical potential), is related to membrane-membrane interactions and K, the smectic bending modulus, to membrane flexibility; dissipation is described by two dissipative parameters: η, the solvent viscosity, and µ, a mobility found equal to d2/12η in a simple model.27 The dynamic light scattering experiments were carried out on three different samples in the lamellar phase, with volume fractions φm ) 0.38, φm ) 0.2, and φm ) 0.075, located along the dilution line corresponding to an hexanol/ C12E5 weight ratio equal to 0.27. The samples are aligned in 200 µm thick planar capillaries by means of a thermal treatment. The smectic periodicities, measured in X-ray scattering for the most concentrate sample and estimated from the dilution law for the other two are respectively d ) 74 Å, d ≈ 155 Å, and d ≈ 440 Å. For all three samples, the scattered intensity time autocorrelation function has been measured as a function of the magnitude and orientation of the scattering wave vector, according to the procedure described in ref 28. In most cases, the light scattering signal I(t) is an approximately monoexponentially decreasing function of time. The observed values for the signal-to-noise ratio, namely, I(t)0)/I(tf∞) in the range1.2-1.6 indicate that the dynamic part of the scattered light is large compared to its static part and, hence, that the experiment corresponds to homodyne detection. All the measured relaxation frequencies are compiled, as a function of q⊥2, in parts a to c in Figure 8 for the lamellar phases at φm ) 0.38, φm ) 0.2, and φm ) 0.075, respectively. For the sample with φm ) 0.38 (Figure 8a), there is no data at any wave vector modulus near qz ) 0 because the light scattering signal was too weak. At φm ) 0.2 and φm ) 0.075, the relaxation frequency of the (33) Sigaud, G.; Garland, C. W.; Nguyen, H. T.; Roux, D.; Milner, S. T. J. Phys. II 1993, 3, 1343.

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Freyssingeas et al. Table 1 φm

d (Å)

B h (Pa)

κ(B h )/kBT

K (N)

κ(K)/kBT

0.38 0.20 0.075

74 155 440

4625 502 24

3 3 2.8

1.5 × 10-13 7 × 10-14

0.60 0.73

A fit of eq 4 to the data is possible, with the elastic constants B h and K as adjustable parameters. For the φm ) 0.38 sample (Figure 8a), only B h is really contained in the data. From the φm ) 0.2 and φm ) 0.075 data (Figure 8b,c), it appears that eq 4 adequately describes the strongly anisotropic dispersion relation of the baroclinic/undulation mode in our system. Both the elastic constants B h and K can then be extracted (Table 1). These results may be turned into a value for the membrane bending elasticity modulus κ with the help of simple models. In the absence of intra- or intermembrane long range interactions, it is reasonable to assume that the smectic bending elasticity originates simply in the (free) membrane elastic properties. This argument leads to the relation K ) κ/d. Similarly, the Helfrich model for the steric repulsion between thermally excited flexible membranes10 results in the relation12,14,34

B h )

Figure 8. (a) Relaxation frequencies, ω, as a function of q⊥2 for an aligned sample with membrane volume fraction equal to 38%. The solid line is a fit of the experimental data (taken at qz much greater than q⊥) to the asymptotic form of the dispersion relation: ω ) µB h q⊥2. (b) Relaxation frequencies, ω, as a function of q⊥2 for an aligned sample with membrane volume fraction equal to 20%. “Candelabrum” branches correspond to q⊥ scans at different fixed moduli q. The solid line is a fit of the experimental data to the general form of the dispersion relation of the baroclinic/undulation mode (eq 4). (c) Relaxation frequencies, ω, as a function of q⊥2 for an aligned sample with membrane volume fraction equal to 7.5%. “Candelabrum” branches and solid lines as in (b).

baroclinic mode was measured varying qz for different fixed values of the wave vector modulus q, all along the range from q⊥ , qz ≈ q down to qz ) 0. For a fixed wave vector modulus it is observed, in agreement with the theoretical expression, eq 4, that the relaxation frequency ω is a linear function of q⊥2 for q⊥ small compared to qz and that it increases rapidly when qz approaches 0. This gives to the curves the characteristic shape of a “candelabrum” as shown in Figure 8b,c.

2 9π2 (kBT) 64 κd3

(5)

Therefore, estimates for κ may be extracted from both elastic constants B h and K. From our three values for B h we get κ(B h ) ≈ 3kBT; on the other hand the two K values lead to κ(K) ≈ 0.7kBT. As far as orders of magnitude are concerned, the light scattering measurements of κ are quite comparable to the excess area ones. Moreover, the h and K appear to obtain, hinting 1/d3 or 1/d scalings for B at the relevance of the Helfrich mechanism for steric interactions between membranes in our system. Nevertheless, it remains that there is a quantitative difference between κ(B h ) and κ(K). This discrepancy is so far not understood but has frequently been observed in other dilute lamellar systems.26,28 The precise value of the poorly known and controversial numerical coefficient in eq 5 (here 9π2/64) may be questionedsfor instance in the presence of attractions between membranes35sthough it was found compatible with the universal value for the KB h product measured in other experiments.12-14 The discrepancy could also arise from inadequacies in the model for the mobility parameter µ,27 from the roughness of the approximations made to calculate the baroclinic/undulation mode dispersion relation28 or from the use of solvent viscosity instead of one of the smectic viscosities in eq 4.33 On the experimental side, one should also consider that measurements of the relaxation frequency “at qz ) 0” are quite sensitive to sample misalignment. II.3. Temperature Effects on the Structure and Dynamics of the Lamellar Phase. In the lamellar phase region of the phase diagram X-ray scattering experiments as well as dynamic light scattering experiments were performed as a function of temperature, for the sample at φm ) 38% with hexanol/C12E5 weight ratio equal to 0.27. The SAXS experiment is used to record the Bragg peak position, while relaxation frequenciess ultimately, elastic constantssare obtained with dynamic light scattering. The position of the Bragg peak was measured as a function of temperature in the range 20-41 °C (Figure 9), the latter temperature corresponding to the lamellar-to(34) Lubensky, T. C.; Prost, J.; Ramaswamy, S. J. Phys. (Paris) 1990, 51, 933. (35) Milner, S. T.; Roux, D. J. Phys. I 1992, 2, 1741.

Membrane Flexibility

Figure 9. X-ray scattering patterns at T ) 20 °C (b), T ) 30 °C (4), and T ) 40 °C (1) for a lamellar phase with φm ) 38% oriented in a cylindrical capillary.There is a noticeable displacement of the first- and second-order Bragg peaks toward smaller scattering wave vectors as the temperature is increased.

Langmuir, Vol. 12, No. 25, 1996 6033

Figure 11. Relaxation frequencies, ω, as a function of q⊥2, for different temperatures (lamellar phase with φm ) 38%). Scattering conditions such that qz . q⊥. The solid line is a fit of the experimental data using the asymptotic form of the dispersion relation: ω ) µB h q⊥2. Table 2

Figure 10. Evolution of the lamellar period, d (lamellar phase with φm ) 38%), as a function of temperature T, far from the lamellar-to-“sponge” transition.

“sponge” phase transition, T*. The sample is held oriented inside a cylindrical capillary, with a “leek-like” (i.e., membranes parallel to the capillary surface) orientation. The periodicity of the lamellar phase, d, is found to increase with temperature over the whole range studied (Figure 10). Far from the transition temperature (T* - T > 2 °C), the periodicity is found to vary linearly with temperature. Note that this increase of d is quite uncommon; the period is usually a decreasing function of temperature.36,37 The dynamics of the membrane fluctuations was investigated on the same sample, at four different temperatures (24, 30, 35, and 39 °C) using quasi-elastic light scattering. The relaxation frequencies were measured as described in section II.2 as a function of the modulus and orientation of the scattering wave vector, in the limit where qz is much greater than q⊥. The data are displayed in Figure 11 as a plot of the relaxation frequency of the baroclinic mode vs q⊥2. The slope of the straight line observed at each temperature gives the experimental determination of the product µB h and allows calculation of a value for κ, using eq 5 and the appropriate, temperature dependent bulk water shear viscosity η. The results in Table 2 suggest that κ increases slightly with temperature. In order to, tentatively, explain the increase with temperature in both periodicity and membrane bending (36) Luzzati, V.; Mustacchi, H.; Skoulios, A.; Husson, F. Acta Crystallogr. 1960, 13, 660. Luzzati, V.; Mustacchi, H.; Husson, F. Acta Crystallogr. 1960, 13, 668. (37) Ke´kicheff, P. The`se d’E Ä tat; universite´ Paris-Sud, 1987.

T (°C)

d (Å)

24 30 35 39

74 75 76 77

µB h

(m2

s-1)

2.319 × 10-11 2.630 × 10-11 2.947 × 10-11 3.014 × 10-11

B h (Pa)

κ (J)

κ/kBT

4625 4475 4300 4100

1.23 × 10-20 1.28 × 10-20 1.34 × 10-20 1.37 × 10-20

3 3.06 3.12 3.16

rigidity, we propose to associate these phenomena to a decrease of the polar head area Σ with temperature. With the assumption that the membrane volume fraction does not depend on temperature (in other words vs, the specific volume of the surfactant molecule, and vw, the specific volume of the solvent molecule, are considered as constants over the temperature range36-37), a decrease in Σ obviously leads to an increase in smectic period d, because of the increase in membrane thickness δ ≡ 2vs/Σ. The increase in κ is then easily understood, as one theoretically expects a scaling relation κ ∝ 1/Σn between κ and Σ, with n an exponent in the range 2-3.38 Combining the values for κ extracted from light scattering with the X-ray data, reexpressed in terms of polar head areas, Σ leads to a relation compatible with a scaling form κ ∝ 1/Σ3.8, not too far from the theoretical expectations considering the boldness of our assumptions. It would be useful to confirm the validity of our explanation by other experiments, for example, looking directly at the membrane thickness behavior as a function of temperature through membrane form factor studies with neutron or X-ray scattering. III. Some Properties of the “Sponge” Phase III.1. Measurement of κ: the Excess Area Method. Similar to the lamellar phase, the bilayer-based “sponge” phase structure is associated to a characteristic length, ξ, that increases upon adding solvent to the system.2,3 Experimentally, ξ may be determined as a function of the membrane volume fraction φm from the location q* of the correlation hump that appears in X-ray, neutron, or light scattering spectra, with the definition ξ ≡ 2π/q*.2,3,23,25,39 Models for the sponge phase, for instance, the Cates et al. model40 (CRAMS) also predict the ξ(φm) functional dependence. For moderately dilute sponge phases, experiments and theory agree on the form of the dilution law, i.e., ξ ) Rδ/φm, where δ is as previously the membrane (38) Slzeifer, I.; Kramer, D.; Ben-Shaul, A.; Roux, D; Gelbart, W. M. Phys. Rev. Lett. 1988, 60, 1966 and references therein. (39) Vinches, C.; Coulon, C.; Roux, D. J. Phys. II 1994, 4, 1165. (40) Cates, M. E.; Roux, D.; Andelman, D.; Milner, S. T.; Safran S. Europhys. Lett. 1988, 5, 733.

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the subtle aspects of the dynamical properties of the sponge phase arising from the so-called “in/out” symmetry,41,42 we aim at describing in simple terms the relaxation of concentration fluctuations in this system. We simply treat the sponge phase as an incompressible, athermal binary fluid. Following, for instance, ref 43, the evolution equation for small concentration fluctuations δc(x,t) reduces to the standard diffusion equation

∂tδc(x,t) )

Figure 12. Experimental values of ξφm (measured in light scattering experiments) shown as a function of the membrane volume fraction φm in a semilogaritmic plot. The solid line is a fit according to eq 6.

thickness and R a numerical coefficient. For the spatial arrangement of the membranes proposed in CRAMS, R ) 1.5,40 whereas the experimental value is found in the range 1.2-1.6.2,3,23 In the case of the C12E5/hexanol/water system we get R ) 1.6 from the ratio q*/q0 measured in sponge and lamellar phases at the same temperature and membrane volume fraction, assuming the invariance of the membrane thickness in the two phases. In analogy with lamellar systems, it is to be expected that short wavelength membrane fluctuations in the sponge phase are apt to store excess area and should therefore induce corrections to the simple dilution law ξ ) Rδ/φm. For dilute enough sponge phases, i.e., ξ . δ, the correction is logarithmic and the dilution law is written as25

ξ)

(

[ ])

kBT Rδ Rδ 1+ ln φm 4πκ xΣ φm

(6)

with Σ, playing the role of a cut-off parameter, the polar head area. In this work, the sponge phase was monitored by static light scattering along the dilution line with hexanol/C12E5 weight ratio equal to 0.41 (hexanol/C12E5 mole ratio 1.6) and constant temperature (24 °C). The membrane volume fraction φm varies between 0.02 and 0.004. The location q* of the correlation hump is obtained with light scattering since the samples are highly dilute. In Figure 12 we display our results as ξφm vs φm in a semilogarithmic plot, illustrating the relevance of the logarithmic correction to the simple dilution law in sponge phases. Assuming δ ) 28.6 Å as in the lamellar phase and R ) 1.6, we get from the slope of the fitted straight line in Figure 12 the following estimate for the membrane bending elastic modulus in the sponge phase: κ ) 1.17kBT. Here again, the result is affected by the somewhat arbitrary choice for the membrane thickness. However, κ remains in the range 0.8kBT-1.2kBT while δ varies between 22 and 29 Å. The sponge phase value of κ measured by this method looks to be of the same order as the value of κ extracted for the lamellar phase. So the amount of hexanol in the membrane (mole ratio 1 for the LR phase; mole ratio 1.6 for L3 phase) does not seem to have a great effect on κ. This may be considered as an indirect indication of the role of κj in the lamellar-to-sponge transition, if it were to be driven by the elastic properties of the two phases. III.2. Time-Dependent Concentration Fluctuations in the “Sponge” Phase. Ignoring here on purpose

[ ]

Rm ∂2f F2 ∂c2

∇2δc(x,t)

(7)

eq

where Rm is the dissipative coefficient for mass diffusion in the sponge phase, f the sponge free energy density, F the total mass density, and c the surfactant mass fraction. Considering the similarities in local structures for both the sponge and lamellar phases, we propose that Rm is of the same order as the corresponding coefficient R⊥ (for mass diffusion along the smectic layers) in the lamellar phase.28 We then have Rm ≈ δ2Fs2/12η, with δ the membrane thickness, η the solvent viscosity, and Fs the surfactant mass density. Our simple schemesquite similar to the one described in ref 44simplies an exponentially-decaying dynamic light scattering signal, with relaxation frequencies proportional to the square of the scattering wave vector. From eq 7, and using the relation between mass and volume fractions, the collective diffusion coefficient D may be explicitely expressed as

D)

[ ]

δ2 ∂2f 12η ∂φ 2 m

(8)

eq

Equation 8 is compatible with the scaling form D ∝ φm proposed and observed experimentally by Porte and coworkers,44 since on general grounds the free energy density scales as f ∝ φm3,45 up to logarithmic corrections here neglected. The CRAMS model for the free energy density f of the sponge phase40 leads to

f)

2φm3 (8πκ - 4πκj - 4kBT ln 2) 27δ3

(9)

which allows calculation of an expression for D as a function of the membrane elastic constants and volume fraction

D)

φm (8πκ - 4πκj - 4kBT ln 2) 27ηδ

(10)

As expected, the collective diffusion coefficient D is proportional to the membrane volume fraction. Furthermore, eq 10 gives an explicit dependence on membrane elastic moduli κ and κj. Our dynamic light scattering experiments were carried out on three sponge samples, at a constant temperature of 24 °C and hexanol/C12E5 weight ratio 0.41, with membrane volume fractions φm ) 5.3%, 4.3%, and 3.3% (light scattering to characteristic sponge wave vector ratio q/q* in the range 0.1-0.6). For the three samples, and whatever the scattering wave vector, the decay of the (41) Granek, R.; Cates, M. E. Phys. Rev. A 1992, 46, 3319. (42) Milner, S. T.; Cates, M. E.; Roux, D. J. Phys. (Paris) 1990, 51, 2629. (43) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Oxford, 1980. (44) Porte, G.; Delsanti, M.; Billard, I.; Skouri, M.; Appell, J.; Marignan, J.; Debauvais, F. J. Phys. II 1991, 1, 1101. (45) Porte, G.; Appell, J.; Bassereau, P.; Marignan, J. J. Phys. (Paris) 1990, 50, 1335.

Membrane Flexibility

Langmuir, Vol. 12, No. 25, 1996 6035

Figure 13. Relaxation frequencies, ω, as a function of q2 for a “sponge” phase with a membrane volume fraction equal to 5.3%. The slope of the straight line drawn gives the collective diffusion coefficient of the “sponge” phase.

with φm is observed for both systems but with different slopes, showing that D* still depends on the membrane elastic constants. The ratio between the slopes for the C12E5/hexanol/water system and the AOT/brine system is equal to 2.3, whereas the value of κ, as derived from the excess area method in both cases, is of the order of 3kBT for the AOT/brine system44 and of the order of kBT for the C12E5/hexanol/water system (section III.1). Therefore, as suggested by eq 10, D* should also depend explicitly on κj and not only on κ. This result allows us to see the importance of the competition between the elastic free energy and the entropy and, indirectly, the role of κ and κj (as also shown in section III.1) on the sponge phase stability. Using the slopes of D* as a function of φm and eq 10, it is possible to estimate the value of κj for both systems. For the C12E5/hexanol/water system κj ≈ 1.5kBT, and for the AOT/brine system κj ≈ 5.7kBT. However, it should be reminded that these numerical values are strongly linked, in particular, to the expression for the free energy density we have used in deriving D: the κj values have to be taken with reasonable care. Conclusion

Figure 14. Reduced diffusion coefficient D* (eq 11) as a function of the membrane volume fraction for the C12E5/hexanol/water (O) and AOT/brine (2) systems. For the AOT/brine system D* was calculated using the data from ref 44 with authors’ permission.

signal with time is not exactly a single exponential. A better fit is obtained using a stretched exponential (i.e., using the functional form exp[-(ωt)R]). However, the exponent R extracted from the fit always remains close to 1 (between 0.92 and 0.82). This small deviation from the single exponential behavior expected from eq 7 may come from the effects of the “in/out” fluctuations on the concentration fluctuation dynamics.41 The relaxation frequencies of the light scattering signal seem to be proportional to the square of the scattering wave vector, as illustrated in Figure 13. This scaling, observed here with the C12E5/hexanol/water system, was also observed in the AOT/brine system44 and allows the definition of the collective diffusion coefficient D from the slope of the ω(q2) dispersion relation. According to our prediction, eq 10, it is convenient in order to compare different systems to define a reduced diffusion coefficient D* as follows

D* ) ηδD/kBT

(11)

So, D* is a function of the membrane volume fraction φm and elastic constants κ and κj only. In Figure 14, D* is plotted as a function of the membrane volume fraction for both the C12E5/hexanol/water system and the AOT/brine system (results of Porte et al.44). A linear variation of D*

This work is primarily a study about the elasticity of membranes made up with C12E5 and hexanol molecules. The effect of temperature was studied in the LR phase. An increase in temperature leads to a slight increase in the membrane bending elasticity modulus κ, coming from an increase in the membrane thickness due to a lengthening of the polar head of the C12E5 molecules with temperature. The effect of hexanol was also studied by measuring the membrane bending elasticity modulus κ, for both the LR and the L3 phase using the nonideal swelling behavior associated with membrane crumpling. This method, though not very precise gives an estimate of κ. The values of κ are very nearly the same for both the lamellar and the sponge phase (respectively 0.9kBT ( 0.1kBT and 1.1kBT ( 0.1kBT). Therefore, the amount of hexanol in the membrane does not significantly affect κ. Since there is nevertheless a phase transition at constant temperature from lamellar to sponge structures induced by hexanol, one may indirectly infer that the Gaussian curvature modulus κj plays a leading role in this transition. Furthermore, for the lamellar phase, κ was also measured by quasi-elastic light scattering experiments. The values for κ obtained by light scattering are of the same order of magnitude as the value obtained by the excess area method, along the same dilution line. For the sponge phase, and as long as the “in/out” fluctuations do not affect the concentration fluctuations (i.e., the shape of the correlation function remains Lorentzian), the results of dynamic light scattering experiments show the following: the time decay of the signal is a slightly stretched exponential with an exponent close to 1; the relaxation frequencies of the concentration fluctuations scale as the square of the scattering vector; finally, the collective diffusion coefficient of the sponge phase, Dsproportional to the membrane volume fractionsis also a function of the elastic constants of the membrane, κ and κj. On comparison of the experimental value of D and its expression obtained using the CRAMS free energy, an estimate of κj is given. Acknowledgment. The authors are very grateful to R. Strey, P. Ke´kicheff, D. Antelmi, and A. Sood for their helpful discussions. We acknowledge the financial help of the IFCPAR through Grant 607-1. LA9605246