Role of dislocation loops on the elastic constants of

Eur. Phys. J. E (2005) DOI 10.1140/epje/i2005-10042-6

THE EUROPEAN PHYSICAL JOURNAL E

Role of dislocation loops on the elastic constants of lyotropic lamellar phases E. Freyssingeas1,a , A. Martin1,b , and D. Roux2,c 1 2

Laboratoire de Physique, UMR CNRS 5672, Ecole Normale Sup´erieure de Lyon, 46 all´ee d’Italie, 69364 Lyon Cedex 7, France Centre de recherche Paul Pascal, UPR CNRS 8641, Avenue Albert Schweitzer, 33600 Pessac, France Received 17 February 2005 and Received in final form 1 August 2005 / c EDP Sciences / Societ` Published online: 24 October 2005 – ° a Italiana di Fisica / Springer-Verlag 2005 Abstract. We study the role of dislocation loops defects on the elasticity of lamellar phases by investigating the variation of the lamellar elastic constants, B and K, induced by the proliferation of these defects. We focus our interest on one particular lamellar phase made up of a mixture of C12 E5 and DMPC in water, which is already well-characterised. This lamellar phase undergoes a second-order (or weakly first-order) lamellar-to-nematic phase transition at about 19 ◦ C and dislocation loops are seen to proliferate within the lamellar structure when temperature is decreased below 30 ◦ C. The values of both elastic constants of this given lamellar phase are measured as a function of temperature, approaching the lamellar-to-nematic transition, with the help of Quasi-Elastic Light Scattering (QELS) on oriented lamellar phases. Very surprisingly we observe a strong and rapid increase in both B and K as the lamellar-to-nematic transition temperature is approached. These increases are seen to start as soon as dislocation loops can be observed in the lamellar phase. We interpret our results as being the consequence of the appearance and proliferation of dislocation loops within the lamellar structure. According to a simple model we developped we show that B and K are proportional to the density of dislocation loops in the lamellar phase. PACS. 61.30.St Lyotropic phases – 61.30.Jf Defects in liquid crystals – 82.70.Uv Surfactants, micellar solutions, vesicles, lamellae, amphiphilic systems, (hydrophilic and hydrophobic interactions) – 87.16.Dg Membranes, bilayers, and vesicles

1 Introduction Lamellar phases are lyotropic phases (i.e. mixtures) with smectic-A liquid-crystal symmetry, which are often encountered in aqueous solutions of amphiphilic molecules (also called surfactants) such as detergents (soaps) or phospholipids (constituents of cells lipidic membranes). These phases are made up of surfactants bilayers with an infinite lateral extension (also called membranes or lamella), periodically stacked along the direction perpendicular to their plane [1]. For the last thirty years a lot of work has been carried out on the study of such liquid-crystals systems and a large literature can be found on this topic [2]. Nowadays the thermodynamics of these phases is rather well understood [3]. Nevertheless there are still a few questions that have not found any answer yet, a

e-mail: [email protected] Present address: Laboratoire de M´et´eorologie Dynamique - LMD10, Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris Cedex 05, France. c Present address: Saint-Gobain - Les Miroirs, 18 avenue d’Alsace, F-92096 La D´efense Cedex, France. b

among which the consequences of topological defects on the equilibrium properties of these phases. (Topological defects are structural defects that can be either pointlike, such as pores, necks, passages, or linear, such as screw or edge dislocations.) Yet topological defects seem to play an important role in many lyotropic systems. Indeed, in several systems, one can observe a huge proliferation of these kinds of defects in lamellar phases as temperature approaches the temperature of phase transition towards a nematic, or isotropic, phase [4–34]. Besides there exists models that describe the role that these defects can play in lamellar-to-isotropic and lamellar-to-nematic phase transitions [35, 37]. On their sides, the experimental works especially concerned the appearance and proliferation of these defects in the lyotropic lamellar phases at the approach of these transitions, as well as the study of their topology. These studies do not focus much on the consequences of these defects on the equilibrium properties of lamellar phases and so these questions still remain unsolved. In the present article we investigate the influence of one particular kind of topological defects: dislocation loops (two screw dislocations; one dislocation having a positive sign, the other a negative one, connected to each other, at

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both ends, by an edge dislocation parallel to the bilayers plane) on the elasticity of lyotropic lamellar phases. Several experimental reasons can explain why the influence of topological defects on the equilibrium properties of lamellar phases has not been investigated much until now. First of all, unlike textural defects such as conic focal, topological defects are not visible in optical microscopy. Therefore the appearance and proliferation of these defects, as well as their topology, are difficult to study. These investigations are usually carried out using techniques such as: Spin Labelling [5, 22, 23], Freeze-Fracture Electron Microscopy [7–9, 21, 24, 26, 30], Birefringence Measurements [11, 27], NMR [14, 19, 20, 29, 30], Small-Angle Neutron (or X-ray) Scattering [12, 15– 17, 25, 28–30], Fluorescence Recovery After Photobleaching [33]. Furthermore in most of the lamellar systems where one can observe the appearance and proliferation of topological defects, it is often difficult to measure equilibrium properties. Equilibrium properties of lamellar phases are often obtained through the analysis of bilayers fluctuations, which are usually measured by means of scattering experiments. These experiments can be either the analysis of the structure factor measured by X-rays (or neutrons) scattering experiments [38], or the analysis of the relation of dispersion of the Baroclinic mode (a hydrodynamic mode) measured using Quasi-Elastic Light Scattering experiments [39−43]. Both techniques required very well-oriented samples of lamellar phases, which are difficult to obtain with most of the lyotropic systems and the appearance of topological defects tends to destroy the sample’s orientation. Thus to carry out our project on the influence of dislocation loops on the elasticity of lyotropic lamellar phases successfully, it is necessary to work with a lyotropic system giving lamellar phases, which can orient themselves in homeotropic anchoring, whose sample orientation is able to stand up to the defects appearance. Furthermore, to facilitate experiments it is better to use rather dilute lamellar phases, with dislocation loops over a wide range of temperatures close to room temperature (typically less than 40% of surfactant and with a temperatures range 20–30 ◦ C). The lyotropic system made up of a mixture of C12 E5 (non-ionic surfactant) and DMPC (phospholipids) in water gives lamellar phases that have all the required characteristics [32, 44]. In this system, one can observe a lamellar-to-nematic phase transition when temperature is decreased with appearance and proliferation of dislocation loops several degrees above the phase transition temperature. Using this latter system we study the influence of dislocation loops on the elasticity of lamellar phases by investigating the variation of the smectic elastic constants, B and K, induced by the proliferation of these defects. (The constant B is the smectic compression modulus at constant chemical potential and this modulus depends on the membrane-membrane interaction potential per unit area [39, 45]. The constant K is the smectic splay modulus, which accounts for the intrinsic bending elasticity properties of a single bilayer [39, 45].) The investigation is achieved with the help of Quasi-Elastic Light

Scattering (QELS) experiments performed on one particular lamellar phase of this system. With this technique we measure both elastic constants simultaneously, as a function of temperature, at the approach of the lamellarto-nematic phase transition. QELS experiments, carried out on homeotropic-oriented samples, measure the characteristic frequencies of the Baroclinic mode, which are then fitted to the theoretical relation of dispersion of this hydrodynamic mode to give B and K. Very surprisingly we observe a strong increase in the values of both elastic constants as temperature approaches that of the lamellar-to-nematic transition. This result is unexpected, the lamellar-to-nematic phase transition in this system is second order, or weakly first order, so intuitively we expect B to decrease and go toward 0 at the approach of the transition as B vanishes in the nematic phase (as it is observed in thermotropic liquid crystals [46]). We interpret this increase in B and K as being due to the proliferation of dislocation loops within the lamellar phase. In this article we present the experimental results we obtained as well as the thoughts that take us to this conclusion.

2 “Theoretical” concepts The principle of our measurements can be summarised as follows. Quasi-Elastic Light Scattering experiments, with homodyne detection, give access to the time autocorrelation function of the scattered light intensity, namely: hI (−q, 0) I (q, t)i, where q is the scattering wave vector. This function is built, using a correlator device, from the fluctuations of the scattered light intensity measured by a detector. The scattered light intensity arises from local fluctuations of the sample’s refractive index: δn (r, t); we have [47] hI (−q, 0) I (q, t)i = N |hδˆ n (−q, 0) δˆ n (q, t)i|

2 2

+N 2 |hδˆ n (−q, 0) δˆ n (q, 0)i| ,

(1)

where δˆ n (q, t) is the spatial Fourier transform of δn (r, t) and N is the number of coherence area on the detector surface (i.e. the number of speckle spots). Assuming these fluctuations to have a Gaussian statistics, we simply have ¯2 ¯ ¯ ¯X ¯ ¯ 2 (2) Ai exp (−Ωi t)¯ . |hδˆ n (−q, 0) δˆ n (q, t)i| = ¯ ¯ ¯ i

Each exponentially decaying functions of time, Ai exp(−Ωi t), corresponds to one of the relaxation processes of the sample associated to the local fluctuations of the refractive index. The frequency characterising the time decay is the characteristic frequency of the given relaxation process. For lyotropic lamellar phases that are both binary systems and smectic (i.e. two compounds system: membranes and solvent, optically uniaxial with the optical axis perpendicular to the bilayers) the local fluctuations of the refractive index originate from two phenomena: concentration fluctuations, as in any binary system,

E. Freyssingeas et al.: Role of dislocation loops on the elastic constants of lyotropic lamellar phases

and smectic period modulations (i.e. membranes displacements) that change the direction of the optical axis. The dynamics of these relaxation processes was successfully described fifteen years ago using a linear hydrodynamic model [39]. According to this model, relaxation processes of concentration fluctuations and/or membranes displacement fluctuations are described by three hydrodynamic modes: the Second-Sound mode (2 propagative modes) and the Baroclinic mode (1 diffusive mode). Only one of these three modes, the Baroclinic mode, has a light scattering signature that is relatively easy to measure by means of QELS experiments on homeotropically oriented samples; Second-Sound modes frequencies are too high for a correlator device. The Baroclinic 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 wave vectors q are exactly perpendicular to the stacking direction (i.e. parallel to the smectic layers). In this particular limit, the membrane displacement field reduces to a collective undulation, with curvature strains only (note that in this limit the Baroclinic mode becomes the undulation mode). Quantitatively, the relaxation frequency of the Baroclinic mode is governed by the magnitude and orientation of the wave vector, by the elastic restoring forces and dissipation. Membrane-membrane interaction opposes changes in the inter-bilayer distance, membrane flexibility opposes deviations from the planar conformation and flow motions inside each solvent layer dissipate the elastic energy stored into the displacement field. In a simple model the dispersion relation of the Baroclinic mode was found to be given by the following equation [48]:

3

106

Ω [s -1 ] 105 104 103 102 10 1012

1013

[ ]

q 2⊥ m −2

1014

1015

2 Fig. 1. Frequency of the Baroclinic mode versus q⊥ for a given 2 wave vector q as described by equation (3) (q = 8.7·1014 m−2 , B = 3500 Pa, K = 8.8 · 10−13 N, ηs = 10−3 Pa s, µ = 5.4 · 10−15 m2 Pa−1 s−1 ; these values are typical values for lamellar phases).

than π/4 (i.e. q⊥ larger than qz ) the frequency is no longer 2 proportional to q⊥ and “takes off” to reach its maximum value at qz = 0 (i.e. q⊥ = q). This latter value corresponds at the end of the “candelabrum” branch, this is the undulation limit of the Baroclinic mode. Then, equation (3) amounts to Ω = Kq 2 /ηs .

3 Experimental system, procedures and techniques 3.1 Experimental system

Ω=

4 B qz2 + K q⊥

ηs q 4 +

qz2 µ

2 . q⊥

(3)

In equation (3), qz and q⊥ are the projections of the wave vector q along the stacking direction (z) and in the 2 plane of the lamellae (⊥), respectively (i.e. q 2 = qz2 + q⊥ ); ηs and µ are two dissipative parameters. For simplification, ηs is taken equal to the solvent viscosity (water in our experimental case) and µ to the surfactant mobility [39]. This latter parameter is given by the simplest available model (Poiseuille flow) and is found equal to 2 (d − δ) /12η s [45] (with d the smectic period, δ the membrane thickness and ηs the solvent viscosity). Figure 1 displays the evolution of the relaxation frequency of the Baroclinic mode for a given wave vector q as a function of its orientation with respect to the smectic layers, as described by equation (3) (i.e. for a Baro2 clinic branch Ω calculated using Eq. (3) versus q⊥ ). It can be observed that the evolution of frequency as a func2 tion of q⊥ has a characteristic “candelabrum” shape. This shape illustrates the strong anisotropy of the dynamic light scattering signal coming from an oriented lamellar phase. When the angle between q and the smectic layers is larger than π/4 (i.e. qz larger than q⊥ ) equation (3) 2 . When this angle is smaller simply restricts to Ω ≈ µBq⊥

We perform our investigation using one particular lamellar phase with a membrane volume fraction φm equal to 35% and a C12 E5 -to-DMPC molar ratio RS/L equal to 3.1. The structure of this particular lamellar phase as a function of temperature approaching the lamellar-to-nematic transition was very well characterised by Dhez et al. [32]. According to this investigation the lamellar-to-nematic phase transition takes place at about 19 ◦ C and topological defects start to nucleate and proliferate at about 10 ◦ C above the phase transition temperature; so when temperature is decreased below 30 ◦ C. The topological defects that proliferate are described as with screw dislocations with Burgers vector equal to 2. This latter observation is confirmed by recent Cryo-TEM results [49] that show that screw dislocations pairs form dislocation loops. Hence in the following we consider that topological defects proliferating in our samples are dislocation loops. In Figure 2 we show a partial phase diagram as a function of membrane volume fraction and temperature of C12 E5 , DMPC, water system at a constant C12 E5 -toDMPC molar ratio RS/L equal to 3.1. We can notice that this phase diagram is similar to that presented in reference [32]. The few little differences that can be observed between both phase diagrams are likely to be due to chemical impurities in the used products.

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T (°C) 30

Lα Experimental line

25

L α /Ι 20

Ν 15 0.2

0.25

0.3

φ

0.35

thickness of 100 or 200 microns that we seal at both ends once filled up with the solutions. Afterwards the capillaries are set inside a water-circulated oven, which is installed under a microscope objective. The temperature in the oven is controlled within ± 0.05 ◦ C. We start our observations at 32 ◦ C and lower the temperature down to 17 ◦ C slowly. For both samples the lamellar-to-nematic phase transition is observed to take place at about 19.4 ± 0.1 ◦ C (19 ◦ C for Dhez et al.). As already pointed out by Dhez and co-workers [32], we do not observe any two-phases coexistence between the lamellar and the nematic phases in our samples. This suggests that this transition is second order, or weakly first order.

0.4

Fig. 2. Partial phase diagram of C12 E5 /DMPC/water system with a C12 E5 -to-DMPC molar ratio equal to 3.1; Lα, lamellar phase; N, nematic; I, isotropic. For our study we follow the experimental line, φm = 0.35, between T = 35 ◦ C and T = 20 ◦ C.

3.2 Samples preparation and characterisation The non-ionic surfactant C12 E5 (penta-ethyleneglycol mono n-dodecylether) is bought from Nikko Chemicals while the DMPC is purchased from Bˆale Chimie. DMPC is a chiral molecule, we used the racemic DL-β, γ-DMPC (1,2-dimyristoyl-sn-rac-glycerol3-phosphatidylcholine). Both products are used without further purification. Ultra-purified water (USF Elga) is used as the solvent. Two different mixtures are prepared the same way. First by mixing C12 E5 (a viscous liquid) and DMPC (a powder) together, then by adding ultrapurified water once the C12 E5 -DMPC mixture has become a “homogenous paste”. For each sample we make 1 g and all the compounds are weighted using a balance precise down to the microgram tenth. The mixtures compositions in weight fraction are 21% ± 0.1%, 14.1% ± 0.1% and 64.9% ± 0.1% for C12 E5 , DMPC and water, respectively (giving φm = 35% ± 0.1% and RS/L = 3.1 ± 0.05). After preparation the samples are let to equilibrate themselves for about two days at room temperature and in the dark since C12 E5 may undergo a chemical degradation with light. We finally obtain homogenous, clear solutions, which are in a lamellar state at room temperature (strongly birefringent, with oily streaks and focal conics). The temperature of the lamellar-to-nematic phase transition, which depends on DMPC hydration and so may change slightly from one sample to another, is determined for each sample individually. The transition temperature is measured using polarised-light microscopy observations. This technique is quite powerful since the kinds of textures that can be observed are very different depending on the liquid-crystal phase [46]. For non-oriented lamellar phases we observe focal conics, oily streaks and domains with different brightness while we observe disinclination lines or Schlieren patterns for non-oriented nematic phases. To achieve these observations we place the equilibrated mixtures into flat glass capillaries with a

3.3 Samples orientation To measure B and K using Dynamic Light Scattering experiments we need to work with homeotropic-oriented samples (i.e. with the smectic layers all parallel to limiting plates). Good homeotropic orientation of lamellar phases is obtained by using the directional growth technique [50]. In this technique the lyotropic lamellar phase is sandwiched between two plates and one monitors the temperature-induced growth of a lamellar phase germ from an isotropic or a nematic phase (called the A phase) by moving the sample in a temperature gradient across the A-to-lamellar phase transition temperature. This technique comes down to moving the A-lamellar interface through the sample along the direction of the temperature gradient. This technique required two temperaturecontrolled ovens with a small gap between them. Both ovens are set at different temperatures so that the temperature corresponding to the lamellar-to-A phase transition temperature (i.e. the A-lamellar front) lies in the gap between the two ovens. The sample is moved at constant velocity from the A phase oven to the “lamellar phase” oven. In this technique, the control parameters are the velocity and the temperature gradient. When these two parameters are optimised then passing the lyotropic system across the A-lamellar interface leads to an almost perfect homeotropic orientation of lamellar phases. In practice we used the following procedure. First, the lamellar solution is introduced into a thin rectangular cell with a broad side much larger than the small one. As cells we use borosilicate glass capillaries from VitroCom Inc., with cross-section 2×0.2 mm2 . The glass capillary is sealed at both ends once filled up. Then the capillary is placed in the lamellar phase-temperature oven (T ≈ 25 ◦ C) with its cross-section parallel to the lamellar-nematic front and moves at constant speed (v = 5 µm/s) to the nematic phase-temperature oven (T ≈ 16 ◦ C) across the lamellarnematic front. Once the capillary is in the nematic phasetemperature oven, it is slowly brought back to the lamellar phase-temperature oven with a speed v = 1 µm/s. Finally, the lamellar phase-temperature oven is stopped and let to cool down to room temperature (about 23–24 ◦ C) before taking out the sample. This procedure allows perfect homeotropic alignment of our lamellar samples.

E. Freyssingeas et al.: Role of dislocation loops on the elastic constants of lyotropic lamellar phases

3.4 Light scattering technique and procedures The Quasi-Elastic Light Scattering experiments are carried out using a Coherent Innova 305 ionised-argon laser light source, operating at λ = 514 nm (delivering up to 1.5 W) and linearly polarised vertically. The scattered light is collected with a photon-counting PMT set on a goniometer. The angle φ between the incident beam and the axis of the goniometer can be changed between 10 ◦ and 150 ◦ . So the modulus of the scattering wave vector q, that is given by the relation, q = (4πn/λ) sin (φ/2) with λ the wavelength and n the sample refractive index (here taken equal to that of water: n = 1.33), varies in the range 4 · 106 –3 · 107 m−1 . A polarizer is used to analyse the polarisation state of the scattered light. In the vicinity of the undulation limit (i.e. qz close to zero), the scattered signal is expected to be depolarised, thus the analyser is set perpendicular to the incident beam polarisation, in VH configuration; otherwise the analyser is maintained parallel to the incident beam polarisation, in VV configuration. The oriented lamellar sample held in its capillary is mounted on a special capillary holder and set at the centre of a temperature-regulated vat containing an index-matching liquid, the refractive index of which is nd = 1.48 (the used index-matching liquid is decahydronaphthalene). This special capillary-holder allows varying the orientation of the smectic layers with respect to the incident direction of the laser beam, i.e. changing the values of q⊥ and qz at a given q. The value of qz is given by qz = 2πn cos ψ (cos θ − cos (θ − φ))/λ [39], where ψ is the angle between the normal to the smectic layers n and the scattering plane and θ the angle between the projection of n in the scattering plane and the incident beam. With this capillary holder ψ is equal to 25 ◦ and θ can be changed between 0 and 2π. When θ is scanned within the range [φ/2, φ/2 + π/2], qz increases from zero to its maximum value that is equal to qzmax = q cos ψ ≈ 0.9q, so that q⊥ varies from q to 0.1q. The internal temperature of the vat is monitored within ± 0.05 ◦ C by means of a PC-controlled water bath. Finally, to allow the measurement of the time auto-correlation function of the scattered intensity, the measured photocurrent is processed using a PC-controlled 256-channel 4700-Malvern correlator, which is able to work either in a linear mode, or in a logarithmic mode with sample times as fast as 0.1 µs. This allows probing relaxation frequencies over six decades, typically in the range 105 –0.1 s−1 . The orientation state of the lamellar phase is very fragile, so that rapid changes in temperature (up or down) might result in the sample’s disorientation [32]. To avoid sample’s disorientation while changing temperature during our experiments we wrote a routine that permits to vary the temperature of the vat step by step with a given time delay ∆t between two steps. The temperature step ∆T can be as small as ±0.1 ◦ C. In practice we start our experiments with both the oriented sample and the vat at the same temperature (i.e. room temperature). Once the capillary has been set into the vat we rise its temperature up to 38 ◦ C with temperature steps equal to 0.1 ◦ C on every two minutes and then wait for several hours (typically

5

over 12 hours). Then we decrease the temperature to the first required temperature very slowly; using temperature steps equal to −0.1 ◦ C on every ten minutes. Once all the measurements at this temperature are done, we decrease the temperature further, at the same rate, to the next chosen temperature. We repeat this procedure to reach the lowest temperature, which we carry out experiments at. We checked this procedure on an oriented sample placed inside the oven used for the polarised-light microscopy observations, we managed to keep the homeotropic orientation of the lamellar phase all the way down to the nematic phase. Moreover, we checked that an oriented sample maintained at a constant temperature does not disorient itself with time (at least not for three weeks).

4 Quasi-Elastic Light Scattering measurements 4.1 Experimental method QELS experiments were carried out on two samples having the same composition (φm = 35% and RS/L = 3.1). Each sample was prepared from a different mixture and homeotropically oriented according to the procedure described in Section 3.3. For both samples we measured hI (−q, 0) I (q, t)i as a function of the magnitude and orientation of the scattering wave vector for seven different temperatures, all above the temperature of the lamellarto-nematic phase transition (T ∗ ≈ 19.4 ◦ C): 35 ◦ C, 32 ◦ C, 28 ◦ C, 23.4 ◦ C, 22.8 ◦ C, 21 ◦ C and 20 ◦ C, respectively. In practice, for each temperature we fully characterise several Baroclinic branches (typically 5–7). For each Baroclinic branch (i.e. for a given magnitude of q), we record hI (0) I (t)i as a function of q⊥ , for q⊥ within the range [q, 0.1q], by varying θ in the range [φ/2, φ/2 + π/2]. We analyse the behaviour of the reduced auto-correlation 2 function g2 (t) ≡ hI (0) I (t)i/hIi − 1 as the square of simple tests functions: exp (−Ωt), single exponential decay, A exp (−Ω1 t)+(1 − A) exp (−Ω2 t), double exponential decay. From perfectly oriented lamellar phases we expect the relaxation frequencies that we measure as a function of {q⊥ , q} (therefore as a function of the angles φ and θ) to be described by equation (3). For a given configuration (given by the set of q and q⊥ ), we always carry out several measurements (typically 4–5) and whatever the values of q and q⊥ , we always measured the time autocorrelation function of the scattering signal using the logarithmic mode of the correlator. Finally, for both samples, after we fully investigated the seven temperatures, we increase the temperature slowly (as slowly as we decreased it) and we check at 23.4 ◦ C, 28 ◦ C and 35 ◦ C respectively, whether we find again similar results to those we have already obtained while approaching the phase transition. 4.2 Measurements For all scattering vectors q, as long as qz is not too close to zero we measure a strong scattering signal in VV configuration that can be analysed easily. As qz draws near

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The European Physical Journal E 106

Ω

106

[s -1]

Ω

[s-1 ]

105

105

104

104

103

103

102

102

10 12 10

1013

[ ]

q⊥2 m −2

1014

1015

Fig. 3. Measured frequencies of the Baroclinic mode as a func2 tion of q⊥ for T = 35 ◦ C. The lines are the fitting curves obtained using equation (3) with B and K as adjustable parameters. (The different Baroclinic branches correspond to φ = 20 ◦ , 50 ◦ , 70 ◦ , 90 ◦ , 150 ◦ , respectively.)

zero, there is no exploitable VV signal any longer. Nevertheless, for scattering angles φ below π/2, an exploitable scattering signal may be recorded in VH configuration. This VH signal is always rather weak. For all the {q, q⊥ } configurations, the observed value of the signal-to-noise ratio, namely hI (0) I (0)i/hI (0) I (t → ∞)i, is always in the range 1.2–1.8. This indicates that the dynamic part of the scattered signal is large compared to its static part and, hence, that the signal corresponds to homodyne detection. Furthermore, it should be noticed here that the measured frequencies are very similar for both samples we investigated. For a same temperature and {q⊥ ,q} configuration, the discrepancy between the frequencies that we measure for both samples is always less than 10%. When we rise temperature back, for 23.4 ◦ C, 28 ◦ C and ◦ 35 C, we measure frequencies that are similar to those we had already obtained for these temperatures. This result suggests that we measure equilibrium properties. It also shows that we do not have samples disorientation during our experiments (at least if there is any samples disorientation this disorientation is very weak). This latter conclusion is supported by the behaviours we observe for the evolutions of the Baroclinic mode frequencies with q and q⊥ . Our experimental data are very well described by equation (3) (Figs. 3 and 4) and such behaviours would not be observed with a strong orientational disorder in our samples. a) For 35 ◦ C and 32 ◦ C, the time auto-correlation function of the scattered intensity always appears to be a single exponentially decreasing function of time. (The characteristic frequency, Ω, related to a given configuration is taken as the average of all the different frequencies that are measured for this configuration and the dispersion is always less than 10%.) For both temperatures, the characteristic frequencies we obtain from our measurements have a behaviour as a function of {q⊥ , q} that corresponds to that expected for the frequencies of the Baroclinic mode. For a given wave vector q, as long as q⊥ is smaller than

10

1012

1013

[ ]

q⊥2 m −2

1014

1015

Fig. 4. Measured frequencies of the Baroclinic mode as a func2 tion of q⊥ for T = 21 ◦ C. The lines are the fitting curves obtained using equation (3) with B and K as adjustable parameters. In this figure we can observe that Ω is larger at 21 ◦ C than at 35 ◦ C. (The different Baroclinic branches correspond to φ = 10 ◦ , 20 ◦ , 30 ◦ , 50 ◦ , 80 ◦ , 130 ◦ , respectively). 2 qz , we have Ω ≈ αq⊥ , with α independent of q. When q⊥ becomes larger than qz , Ω increases rapidly as q⊥ approaches q to reach a maximum, that depends on q, at q⊥ equal q (i.e. qz equal 0). Finally, if we plot Ω versus 2 , for every scattering wave vector, we see that the set q⊥ of measured frequencies lies on a curve that has the shape of a candelabrum branch (Fig. 3). b) For all temperatures below 28 ◦ C, surprisingly, the time auto-correlation function of the scattered intensity always appears to be bimodal. Its time decay is the square of a sum of two exponentially decreasing functions of time, which gives us two different characteristic frequencies: Ωf and Ωs . (For a given configuration, Ωf is taken as the average of all the high frequencies measured for this configuration and Ωs as the average of all the low frequencies. For the high frequencies the dispersion is always less than 10%, on the other hand, for the low frequencies the dispersion is much larger, up to 40%.) For all the measurements, the respective frequencies and magnitudes of these two modes are very different. The characteristic frequency of the fast mode is always approximately 20–30 times higher than that of the slow mode and its magnitude accounts for about 80%–90% of the overall magnitude. The characteristic frequencies of the fast mode are of the same order of magnitude of those measured for the Baroclinic mode at 32 ◦ C and 35 ◦ C. Moreover, the behaviour of these frequencies as a function of {q⊥ , q} has all the features that are expected for the Baroclinic mode (see Fig. 4). Therefore, we conclude that the fast mode corresponds to the Baroclinic mode.

4.3 Results First of all, our results show that the appearance and proliferation of dislocation loops within the lamellar phase do not change the structure of the Baroclinic mode that still

E. Freyssingeas et al.: Role of dislocation loops on the elastic constants of lyotropic lamellar phases

7

2 10-12

4 104 3.5 104

K [N]

B [Pa ] 3 104 1.5 10-12

2.5 104 2 104 1.5 104

1 10-12

1 104 5 103 0

0

2

4

6

8 10 ∆T [°C]

12

14

16

Fig. 5. Values of B obtained from the fit of experimental data to equation (3) (dark circles) and estimated values of B calculated using equation (7b) (line), both as a function of ∆T .

follows the same relation of dispersion. However, the scattering signal is strongly modified. As soon as dislocation loops appear in the sample we observe a second relaxation process that adds to the Baroclinic mode and that is much slower. This second mode cannot be due to samples’ disorientation during the experiments and therefore we believe that this mode originates from the presence of dislocation loops in the lamellar phase. We observe that the characteristic frequencies of the Baroclinic mode increase as temperature decreases below 30 ◦ C and approaches that of the lamellar-to-nematic transition (see Figs. 3 and 4). This suggests that µB and K/ηs increase at the approach of the lamellar-to-nematic transition. For each temperature, we fit our experimental data to equation (3), with B and K as adjustable parameters, which returns the values of both elastic constants. (The values of the surfactant mobility µ we use to fit our data are calculated using the bilayers thickness δ and smectic periods d that were measured by Dhez et al. using X-ray experiments [32].) All the values of B and K that are extracted from our data are displayed as a function of ∆T (∆T = TM easurement − T ∗ ) in Figures 5 (B) and 6 (K), respectively. These values are of the order of magnitude of what we expect for the elastic constants of this lamellar phase (assuming κ, the bilayer bending elasticity modulus, of the order of a few kB T ). Each value of B and K that is reported in Figures 5, or 6, is the average of the values, which have been obtained for the two different samples (in any cases these two values are always close; discrepancy within 10%). (Here, it should be noticed that a weak orientational disorder always remains in the oriented lamellar samples despite all the care taken for their orientation. Such a disorder can cause a significant broadening of the signal spectral-width in the vicinity of qz equal zero [43]. As a consequence, it is likely that our measurements underestimate the “true” characteristic relaxation frequency of the Baroclinic mode in the undulation limit and therefore it is likely that we underestimate the values of K we measure.)

5 10-13

0

2

4

6

8 10 ∆T [°C]

12

14

16

Fig. 6. Values of K obtained from the fit of experimental data to equation (3) (dark circles) and estimated values of K calculated using equation (8) (line), both as a function of ∆T .

5 Discussion In Figures 5 and 6 we see that B and K have the same kind of evolution with temperature. For ∆T larger than 10 ◦ C (so for T above 30 ◦ C) both elastic constants seem to be constant, they start to increase as ∆T goes below 10 ◦ C approximately (thus as soon as dislocation loops nucleate and proliferate within the lamellar phase) to increase strongly close to the lamellar-to-nematic phase transition. These results are very interesting, but nevertheless very surprising. Intuitively, we expected both elastic constants to decrease at the approach of the nematic phase. Especially, we expected B to decrease and go to 0 since B vanishes in the nematic phase [46] and the transition is second order, or weakly first order [32]. 5.1 Comparison of our results with the observations of Dhez and co-workers First of all, before further discussing our experimental results, it is important to make sure that the evolutions of B and K we observe as a function of temperature are in agreement with the results that Dhez et al. obtained on the same lamellar phase [32]. For that purpose, we compare the values of the Caill´e exponent, η, that were measured by Dhez et al. on the same lamellar system [32], with those we can calculate using the values of B and K we obtained from our experimental data. The Caill´e exponent η quantifies the amount of diffuse scattering around the Bragg peaks of the lamellar phase and it is related to both B and K as follows [51]: η=

πkB T √ . 2d2 BK

(4)

In Figure 7 we show, as function of ∆T , the different values of η that we calculate with the help of equation (4), from our experimental values of B and K and using the values of d given in reference [32]. We display these values

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The European Physical Journal E

where f is the free energy density of the lamellar phase. Here, we are not in the case of a very dilute lamellar phase (the smectic period d 6 9 nm, the membrane thickness δ ≈ 3 nm), hence we cannot neglect the van der Waals attractive interactions between membranes and therefore we must write f as follows [52]: µ ¶ δ2 3π 2 kB T kB T f = − kB T χ 2 + (6) 2 . d 128 κ d (d − δ)

η

1 0.8 0.6 0.4 0.1

1

10 ∆T [°C]

Fig. 7. Calculated (dark circles) and measured (open circles) values of η as a function of ∆T . The lines are guide for eyes.

together with the experimental values of η that were directly obtained by Dhez and co-workers using the analysis of X-ray spectra [32]. There is a factor of about 1.5–1.6 between both set of values, however we see that both behaviours as a function of ∆T are quantitatively exactly the same. Both seem to be constant for ∆T larger than 10 ◦ C and decrease in the same way as soon as ∆T goes below 10 ◦ C. Therefore we conclude that the respective evolutions of B and K that we measure at the approach of the lamellar-to-nematic transition are in agreement with the results that were obtained by Dhez et al. [32] on the same lamellar phase, which validates our results. (The numerical coefficients of Eq. (4), as well as those used to extract B and K from QELS data, or η from X-ray data, are approximate and their roughness is likely to be the explanation for the observed discrepancy between both sets of values.)

5.2 First attempt to explain the observed evolutions Both elastic moduli are functions of the inverse of the lamellar period: B ∝ 1/d3 , K ∝ 1/d, and it was observed [32] that the smectic period of this particular lamellar phase decreases as temperature approaches that of the lamellar-to-nematic transition (d is constant for temperature above 30 ◦ C and varies from 9.2 nm down to 7.2 nm between 30 ◦ C and 19.5 ◦ C). Therefore, at that point, the first idea that comes up to our mind for explaining the respective evolution of B and K, is that these evolution are due to the decrease of the lamellar period at the approach of the lamellar-to-nematic transition. a) Effect of the decrease in the lamellar period on B. The smectic compression modulus at constant chemical potential B is related to the membrane-membrane interaction [39, 45] and according to Nallet et al. B is defined as follows [39]: B=

µ · 2 ¸¶ ∂ f , d2 ∂d2 eq

(5)

The first term describes the contribution of the van der Waals attractive interactions between membranes; the second represents the Helfrich contribution [54]. κ is the intrinsic bending elasticity modulus of a bilayer and χ is a parameter, which is the correction to the hard-wall result for the virial coefficient (so we have χ > 0). This latter coefficient can be estimated from the inflection point of f (φ) since for this particular value of the membrane volume fraction, φc , we have B (φc ) = 0 (spinodal line). This concentration φc is equal to 2φ∗ /3, where φ∗ is the concentration at which a first-order phase separation between a lamellar phase of concentration φ∗ and pure water occurs (i.e. binodal line: the dilution limit of the lamellar phase). Hence, B can be written as a function of the parameters κ, d, δ and φc : " # µ ¶ 4 φc 9π 2 kB T kB T (d − δ) B= . 4d 1 − 64 κ δd3 (1 − φc )4 (d − δ) (7a) According to the phase diagram [32, 44], the dilution limit of the lamellar phase in the system we use is found for a membrane volume fraction in the range 15–20%; so we take φ∗ = 0.18. We know [32] that between 20 ◦ C and 40 ◦ C the membrane thickness remains almost constant. Moreover, measurements performed on different kinds of bilayers: C12 E5 /hexanol [42], or pure DMPC [53], exhibit very weak evolutions of κ with temperature; approximately 10−2 –10−3 kB T /K. Therefore, we assume that κ can be taken as constant. So we can use equation (7a) to calculate the evolution that B should have due to the decay of the lamellar period between 35 ◦ C and the lamellarto-nematic transition temperature: ¶2 µ ¶4 µ d35 ◦ C − δ d35 ◦ C B (T ) ≈ B 35 ◦ C dT dT − δ ! Ã 4 3 δdT − 0.28 (dT − δ) . (7b) × 4 δd335 ◦ C − 0.28 (d35 ◦ C − δ) B 35 ◦ C is the measured value of B at 35 ◦ C, d35 ◦ C and dT are the lamellar periods at 35 ◦ C and T , respectively. This “theoretical” evolution is displayed in Figure 5, as expected we see that B increases as ∆T decreases. However this “theoretical” increase is much weaker than that we observe experimentally. Hence the decay of the lamellar period does not allow accounting for the experimental evolution of B. b) Effect of the decrease in the lamellar period on K. K the smectic splay modulus originates simply in the bending elasticity properties of free membranes [39, 45] and is

E. Freyssingeas et al.: Role of dislocation loops on the elastic constants of lyotropic lamellar phases

simply given by the following relation: K=

κ . d

(8)

We still assume that κ is constant with temperature and we calculate the evolution of K that is due to the decay of the lamellar period. This “theoretical” evolution is displayed in Figure 6 and we see that the expected increase in K is much weaker than that we measure. Again, we conclude that the decrease in the lamellar period cannot explain our experimental observations. c) Effect of a variation of κ on the elastic constants. One can imagine an important variation of κ as temperature decreases (such a variation could be due to surfactants segregation with temperature; C12 E5 molecules going preferentially into the defects, i.e. regions of high curvature, DMPC molecules remaining in the flat regions). However an important variation of κ cannot be the explanation for the evolutions we observe, since then B and K would have variations in opposite directions. Indeed we have B ∝ 1/κ, while K ∝ κ. 5.3 Effect of dislocation loops Here we assume that the unexpected increase in the elastic constants is due to the proliferation of dislocation loops in the lamellar phase. We imagine that these dislocations couple membranes through the sample over a scale much larger than the smectic period making so the lamellar phase much stiffer. With this picture, which is likely to be “naive”, we can make a simple microscopic model that links the dislocation loops proliferation and the increase in B and K. We assume that dislocation loops are at thermal equilibrium (this seems to be reasonable since the evolutions of B and K with ∆T are reversible), thus the lengths of screw and edge dislocations, lS and lE respectively, correspond to equilibrium values. a) Effect of dislocation loops on B. Let us consider a volume of lamellar phase V , V = S0 N d, containing one screw dislocation of length lS = N d and radius r (for simplification, we assume that S0 À πr2 ). We assume that screw dislocations are fixed to the bilayers and cannot slide up, or down, along the perpendicular axis to the smectic layers. Thus compressing, or dilating, the lamellar phase results in a change of lS , the energy cost of which superimposes to that required for overcoming the membranemembrane interactions. Hence the free energy density can be written as follows: µ ¶2 d − deq ε , (9) f = flam + 2N dS0 deq where flam is the usual free energy density of the lamellar phase that is given by equation (6) and ε the energy required to change the screw dislocation length upon compression or dilatation. Using equation (5), we obtain that B is the sum of two terms. One that corresponds to the compression modulus of the lamellar phase without screw

9

dislocations, B lam (so given by Eq. (7b)) and, another one, B d , that accounts for the dislocation loops contribution. Therefore, in this framework, we expect the contribution of dislocation loops to the compression modulus of the lamellar phase to be proportional to their density ρ: B d ∝ ρε. b) Effect of dislocation loops on K. Let us consider a volume V of lamellar phase containing one dislocation loops (i.e. two screw dislocations of opposite sign connected to each other by two edge dislocations) and let us imagine a deformation that bends this lamellar structure without compression. As surfactants bilayers are in a liquid state, we assume that upon this deformation screw dislocations do not bend and remain locally perpendicular to the smectic layers. Therefore, to be able to bend the lamellar structure with the dislocation loop, it is necessary to change the length lE of both edge dislocations. That costs energy, which superimposes to that already required for bending the lamellar phase. The free energy density can be written as follows: Ã µ ! 2 eq ¶2 elSeq lE − l E Klam + , 2 = eq lE R2 4V R2 (10) where e is the energy required for changing the edge dislocation length, R the radius of curvature of the bending deformation that is applied to this lamellar structure. Hence the effective bending modulus of the lamellar phase K appears to be the sum of two different terms; one that corresponds to the bending modulus of the lamellar phase without dislocations loops, Klam (so given by Eq. (8)), the other one accounting for the dislocation loops contribution, Kd . Again, in this framework, we expect the contribution of dislocation loops to the bending modulus 2 to be proportional to their density: Kd ∝ elSeq ρ. e Klam + f= R2 2V

To summarise, in this simple framework, the contribution of dislocation loops to both elastic constants is expected to be proportional to their density in the lamellar phase. Since we assume that dislocation loops are at thermal equilibrium their density is given by the Boltzmann law. Therefore, we have µ ¶ Ed ρ (T ) = ρ (0) exp − , kB T

(11)

where Ed is the energy of the dislocation loops. As the number of defects increases upon approaching the lamellar-to-nematic transition [32, 49], this suggests that Ed decreases. The simplest assumption we can make for Ed is that of linear behaviour with ∆T over the investigated range: Ed = α (T − T ∗ ) + E0 .

(12)

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The European Physical Journal E 3 104

1 10-12

2.5 104 8 10-13

Kd [ N ]

B d [ Pa ] 2 104

6 10-13

1.5 104 4 10-13

1

104 2 10-13

5000 0

0 0

4

8 ∆T [°C]

12

16

Fig. 8. Estimated values of B d as a function of ∆T (B d = B M easured − B lam ). The line is the fitting curve obtained using 0 equation (13) with α and B d as free parameters.

As a consequence B d and Kd should have the following evolutions as a function of ∆T : µ ¶ µ ¶ E0 α∆T B d (∆T ) = ερ (0) exp − exp − = kB T kB T µ ¶ α∆T 0 B d exp − , (13) kB T µ ¶ µ ¶ 2 E0 α∆T exp − = Kd (∆T ) = elSeq ρ (0) exp − kB T kB T µ ¶ α∆T Kd0 exp − . (14) kB T We can estimate the values of B d (∆T ) and Kd (∆T ) by subtracting the values of B lam (∆T ) and Klam (∆T ) that we have already calculated, to the experimental values we obtained from the fits of the candelabrum curves. In Figures 8 and 9 we plot the obtained values of B d and Kd as a function of ∆T and we fit these data to equations (13) and (14), respectively. We see that the respective evolutions of B d and Kd as a function of ∆T are very well described by exponentially decreasing functions of ∆T . From the fit of B d 0 to equation (13) we obtain: B d ≈ 2.65 · 104 Pa and α ≈ (111±15)kB , whereas the interpolation of Kd to equation (14) gives: Kd0 ≈ 9.7 · 10−13 N and α ≈ (114 ± 15)kB . Therefore, as expected from our model, the fits of B d and Kd yield the same value of α: α ≈ (113 ± 15)kB , for both elastic constants. Our crude model, which is likely to be the simplest that we can make, accounts very well for the evolutions of B d and Kd as a function of ∆T . Hence we believe that the increase of the elastic constants of the lamellar phase at the approach of the temperature of the lamellar-to-nematic phase transition is simply the consequence of the proliferation of dislocations loops within the lamellar phase. Finally, It should be noticed that the evolution of the dislocation loops density with temperature that we measure at the approach of the lamellar-to-nematic phase

0

4

8 ∆T [°C]

12

16

Fig. 9. Estimated values of Kd as a function of ∆T (Kd = KM easured −Klam ). The line is the fitting curve obtained using equation (14) with α and Kd0 as free parameters.

transition: ρ (T ) ∝ exp (−[113 ± 15]kB ∆T /kB T ), is very similar to the evolution of the density of topological defects with temperature at the approach of the lamellar-toisotropic phase transition that has been obtained by Constantin et al. [33] in lamellar phases made up of C12 E6 and water. So with a non-ionic surfactant that belongs to the same “surfactants family” as C12 E5 they measured ρneck (T ) ∝ exp (−100kB ∆T /kB T ). In their case, however, it seems that the topological defects that proliferate in the lamellar phase when temperature is decreased and approaches that of the lamellar-to-isotropic phase transition are necks rather than dislocations loops.

6 Conclusion We studied the role of dislocation loops on the elasticity of lamellar phases by investigating the variation of the lamellar elastic constants, B and K, induced by the proliferation of these defects. We carried out this project working with one particular lamellar phase made of C12 E5 /DMPC/water, which was already well characterised [32]. (This lamellar phase undergoes a lamellar-tonematic phase transition at about 19 ◦ C and dislocation loops made up with screw dislocations of Burgers vectors 2 are seen to proliferate within the lamellar structure when temperature is decreased below 30 ◦ C.) Both elastic constants were measured as a function of temperature approaching the lamellar-to-nematic transition by means of Quasi-Elastic Light Scattering (QELS) experiments on oriented samples. This technique made possible measurement of B and K simultaneously through the analysis of the Baroclinic mode. The appearance and proliferation of dislocation loops in the lamellar phase do not change the structure of the Baroclinic mode that still follows the same relation of dispersion. However, the scattering signal is strongly modified. When dislocation loops are present in the lamellar structure we observe two relaxation modes, the second relaxation process, which is much slower than the Baroclinic

E. Freyssingeas et al.: Role of dislocation loops on the elastic constants of lyotropic lamellar phases

mode is unknown. We believe that this second mode is not due to samples’ disorientation during the experiments. We have not identified this second mode yet, nevertheless we presume that this second mode corresponds to relaxation process that is related to the presence of dislocation loops within the lamellar structure. We observe a strong and rapid increase in B and K as temperature gets closer to that of the lamellar-to-nematic transition. These increases are seen to start as soon as dislocation loops can be observed in the lamellar phase. The enhancements of both elastic constants that we measure approaching the transition temperature are in agreement with the results that were already obtained on the same lamellar phase by another group [32]. The decay of the lamellar period that was observed while decreasing temperature [32], likewise a hypothetical variation of κ with ∆T , can be ruled out as explanations for the evolutions of B and K we observe. Therefore we interpret our results as being the consequence of the appearance and proliferation of dislocation loops within the lamellar structure. A simple model we developped, whose main ingredient is the proliferation of dislocation loops at the approach of the transition, accounts for the measured increases in the elastic constants. Accordingly, B and K are proven to be proportional to the density of dislocation loops in the lamellar phase. The authors are very grateful to Doru Constantin, J´erˆ ome Crassous, Olivier Diat, Jean-Christophe G´eminard, Fr´ed´eric Nallet for fruitful discussions (with a special thanks to Olivier Diat for his advices and Jean-Christophe G´eminard for his help with the directional growth technique) and to Patrick Moreau and Laurence Navailles for sharing their Cryo-TEM results with us before publication.

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11. M. Allain, P. Oswald, J.-M. di Meglio, Mol. Cryst. Liq. Cryst. B 162, 161 (1988); P. Oswald, M. Allain, J. Colloid Interface Sci. 126, 45 (1988). 12. Y. Ran¸con, J. Charvollin, J. Phys. Chem. 92, 6339 (1988). 13. G. Porte, J. Appel, P. Bassereau, J. Marignan, J. Phys. (Paris) 50, 1335 (1989). 14. N. Boden, J. Clements, K.W. Jolley et al., J. Chem. Phys. 93, 9096 (1990); N. Boden, G.R. Hedwig, M.C. Holmes et al., Liq. Cryst. 11, 311 (1992). 15. M. Clerc, A.-M. Levelut, J.-F. Sadoc, J. Phys. II 1, 1263 (1991). 16. S.S. Furani, M.C. Holmes, G.J.T. Gordon, J. Phys. Chem. 96, 11029 (1992). 17. M.S. Leaver, M.C. Holmes, J. Phys. II 3, 105 (1993); M.C. Holmes, A.M. Smith, M.S. Leaver, J. Phys. II 3, 1357 (1993). 18. H. Morgans, G. Williams, G.J.T. Tiddy et al., Liq. Cryst. 15, 899 (1993). 19. M.C. Holmes, P. Sotta, Y. Hendrikx, B. Deloche, J. Phys. II 3, 1735 (1993). 20. W. Schnepp, S. Disch, C. Schmidt, Liq. Cryst. 14, 843 (1993). 21. P. Boltenhagen, M. Kleman, O.D. Laventrovich, J. Phys. II 4,1439 (1994). 22. P.O. Quist, B. Halle, Phys. Rev. E 47, 3374 (1993); Phys. Rev. Lett. 78, 3689 (1997). 23. P.O. Quist, K. Fontell, B. Halle, Liq. Cryst. 16, 235 (1994). 24. R Strey et al., in Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution, edited by S.-H. Chen, J.S. Huang, P. Tartaglia (Kluwer, Dordrecht, 1994). 25. M.C. Holmes, M.S. Leaver, A.M. Smith, Langmuir 11, 356 (1995). 26. R. Strey, Ber. Bunsenges. Phys. Chem. 100, 182 (1996). 27. L. Sallen, P. Sotta, P. Oswald, J. Phys. Chem. B 101, 4875 (1997). 28. E. Buhler, E. Mendes, P. Boltenhagen et al., Langmuir 13, 3096 (1997). 29. C.E. Fairhurst, M.C. Holmes, M.S. Leaver, Langmuir 13, 4964 (1997). 30. J. Gustafsson, G. Oradd, M. Almgren, Langmuir 13, 6956 (1997); J. Gustafsson, G. Oradd, M. Nyden et al., Langmuir 14, 4987 (1998). 31. M.C. Holmes, Curr. Opin. Colloids Interface Sci. 3, 458 (1998). 32. O. Dhez, S. K¨ onig, D. Roux, F. Nallet, O. Diat, Eur. Phys. J. E 3, 377 (2000) 33. D. Constantin, P. Oswald, Phys. Rev. Lett. 85, 4297 (2000). 34. D. Constantin, P. Oswald, M. Clerc, P. Davidson, P. Sotta, J. Phys. Chem. B 105, 668 (2001). 35. W. Helfrich, J. Phys. (Paris) 39, 1199 (1978). 36. J. Toner, Phys. Rev. B 26, 462 (1982); D.C. Morse, T.C. Lubensky, J. Phys. II 3, 531 (1993). 37. R. Holyst, Phys. Rev. Lett. 72, 4097 (1994). 38. F. Nallet, D. Roux, S.T. Milner, J. Phys. (Paris) 51, 2333 (1990); F. Nallet, R. Laversanne, D. Roux, J. Phys. II 3, 487 (1993). 39. F. Nallet, D. Roux, J. Prost, J. Phys. (Paris) 50, 3147 (1989). 40. P. Bassereau, J. Appel, J. Marignan, J. Phys. II 2, 1257 (1992).

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41. F. Nallet, D. Roux, C. Quillet, P. Fabre, S.T. Milner, J. Phys. II 4, 1477 (1994). 42. E. Freyssingeas, F. Nallet, D. Roux, Langmuir 12, 6028 (1996). 43. E. Freyssingeas, D. Roux, F. Nallet, J. Phys. II 7, 913 (1997). 44. S. K¨ onig, P. M´el´eard, D. Roux, Nuovo Cimento D 16, 1585 (1994). 45. S.F. Brochard, P.G. de Gennes, Pramana Suppl. 1, 1 (1975). 46. a) P.-G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edition (Oxford University Press, Oxford, 1993). b) P. Oswald, P. Pieransky, Nematic and Cholesteric Liquid Crystals, Concepts and Physical Properties Illustrated by Experiments (Taylor & Francis, CRC Press, London, 2005). 47. B.J. Berne, R. Pecora, Dynamic Light Scattering: with Applications to Chemistry, Biology and Physics (John Wileys and Sons, New York, 1976). 48. G. Sigaud, C.W. Garland, H.T. Nguyen, D. Roux, S.T. Milner, J. Phys. II 3, 1343 (1993). 49. P. Moreau, L. Navailles, J. Giermanska-Kahn, O. Mondain-Monval, F. Nallet, D. Roux, submitted to Europhys. Lett. (2005). 50. P. Oswald, M. Moulin, P. Metz, J.-C. G´eminard, P. Sotta, L. Sallen, J. Phys. III 3, 1891 (1993); L. Sallen, P. Oswald, J.-C. G´eminard, J. Malthˆete, J. Phys. II 5, 937 (1995). 51. A. Caill´e, C. R. Hebd. Acad. Sci. (Paris) Ser. B 274, 891 (1972). 52. S.T. Milner, D. Roux, J. Phys. II 2, 1741 (1992). Because of the undulation fluctuations of membranes the total bilayer-bilayer interaction potential in lamellar phases cannot be written as simply as the sum of the different interaction potentials as explained by Lipowski and Leibler [53]. The idea developed by Milner and Roux to describe the competition between the various interactions between bilayers is, first, to consider the lamellar phase as a perfect gas of bilayers of thickness δ not being able to interpenetrate each other (this is the two-dimensional analogue of a perfect

gas of hard spheres). The entropy of mixing of such a system gives a term proportional to φ3 , which is nothing but the “Helfrich contribution” [54]. This is the result of the hard-wall virial expansion. Then, they add short-range interactions between the membranes, which they take into account by adding a correction to the Helfrich part by making a perturbative expansion in power of φ (as it is done for hard spheres). The first term of this expansion, which is equivalent to the virial expansion for a perfect gas of hard objects, is proportional to φ2 . (This is Eq. (6).) In this case, the parameter χ is given as the function of the interactions by (this is Eq. (5) in Ref. [52]): Z 1 χ=− 2 d3 r (1 − exp (−βUν (r))) , 2ν where Uν (r) is the potential of interaction between bits of surface of volume ν; ν = a2 δ, where a2 is the in-plane cutoff. The integral is limited to positions such that the bits of surfaces do not overlap. (I.e., if Uν (r) is the hard-wall potential, then χ is equal to 0.) Thus χ is the correction to the hard-wall result for the virial coefficient. As in our case the interactions are attractive the parameter χ is positive. For membranes concentrations smaller than a concentration φ∗ , this model predicts a first-order phase separation between a lamellar phase of concentration φ∗ and pure water. This concentration φ∗ , which is the limit ¡of dilution ¢ of the lamellar phase, is equal to φ∗ = 64κχδ 3 / 3π 2 kB T . The compression modulus of the lamellar phase B does not vanish in this point, but at the inflection point of f (φ) (spinodal line); the concentration φc for which B vanishes ¡ ¢ is equal to φc = 128κχδ 3 / 9π 2 kB T . The ratio φ∗ /φc is exactly equal to 3/2. As it should be, the limit of the diagram of phase, the binodal extremity occurs before the spinodal, where we have the divergence of the compressibility, i.e. the cancellation of B. 53. R. Lipowsky, S. Leibler, Phys. Rev. Lett. 56, 2541 (1986); erratum of Phys. Rev. Lett. 56, 2541 (1986). 54. W. Helfrich, Z. Naturforsch. 33a, 305 (1978).