Langmuir 1991, 7, 1861-1863
1861
Membrane Fluctuations in Dilute Lamellar Phases F. Nallet CNRS, Centre de recherche Paul-Pascal, Avenue du Docteur-Schweitzer, F-33600 Pessac, France Submitted to Symposium Chairman April 2,1990. Received November 15, 1990 The impact of membrane fluctuations on elastic and hydrodynamic properties of dilute lamellar phase is investigated. Special attention is devoted to the us_e of dynamic light scattering in measuring two smectic elastic constants, the compressibility modulus, B , and the splay modulus, K. These constants are extracted from the anisotropic dispersion relation to the undulationlbaroclinic mode, which couples concentration and layer displacement fluctuations at constant chemical potential. Surfactant molecules in solution may often associate into two-dimensional sheets, or membranes, periodically stacked in space, and build lamellar phases. When there is no long-range ordering of the molecules inside the membranes, the resulting structure is that of a smectic A liquid crystal.' As compared to thermotropic smectics, these lyotropic smectics are intrinsically multicomponent systems; there are at least the surfactant and the solvent species. The amount of surfactant in the system can thus be continuously changed and, in some cases, the lamellar phase exists all along the dilution line, until very low surfactant contents are r e a ~ h e d . ~In- ~the dilute limit, the membrane thickness 6 is much smaller than the smectic repeat distance d and it becomes reasonable to describe macroscopic properties of the smectic phase in terms of local properties of the surfactant membrane. According to this scheme, it is important to consider intermembrane interactions and membrane flexibility. Interaction? fix the magnitude of the smectic compressibility modulus B,s-8 and flexibility, characterized by the Helfrich bending ~ the smectic splay modulusK, both constant K , determines control membrane fluctuations. Because the membrane fluctuations are restricted in the stacked structure of a smectic, owing to short-range steric repulsions, there is an entropic, universal part to the intermembrane interactions, the Helfrich's repulsion.'O This repulsion is long-ranged; it is strong enough to overcome van der Waals attractions when the membrane bending constant is of the order of IteT;'JOit is presumably the only intermembrane interaction relevant, in the dilute limit, in the absence of electrostatics (screened out or uncharged msmbranes). In this limit, both smectic elastic moduli, B and K , are reducible to the membrane flexibility K . In the following, we recall briefly the main theoretical background required to describe membrane fluctuations in dilute lamellar phases, i.e. the elasticity and hydrodynamic theories of the two-component smectic A. We then show how scattering experiments can be interpreted with (1) Ekwall, P. In Adoances in Liquid Crystals; Brown, G . M., Ed.; Academic: New York, 1975. (2) Dvolaitzky, M.; Ober, R.; Billard, J.;Taupin, C. Charvolin, J.; Hendrix, Y. C. R. Hebd. Seances Acad. Sci. 1981,-1145, 295. (3) Benton, W. J.; Miller, C. A. J. Phya. Chem. 1983,87, 4981, (4) Bellocq, A,-M.; Roux, D. In Microemulsions; Frieberg; Bothorel, Ede.; CRC Press: Boca Ratan, FL, 1986. ( 5 ) Larch& F.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, L. Phys. Reu. Lett. 1986,56,1700. (6) Safinya, C. R.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A.-M. Phys. Reu. Lett. 1986,57, 2718. (7) Leibler, S.; Lipowsky, R. Phys. Reo. B Condens. Matter 1987,35,
such theories and emphasize the case of dynamic light scattering. Experimental results obtained with this technique are presented. Elasticity of the Two-Component Smectic A The elastic theory starts from an expansion, up to second order in the relevant fluctuations, of the free energy density. In the case of an isothermal, incompressible system, the variables we have to consider are a composition variable (for instance, the surfactant mass fraction c) and a strain variable, the layer displacement field u. The elastic energy density is given by'' 1 1 1 f = -B(azu)2 -K(v,%)~ -6c2 cazusc (1) 2 2 2X In eq 1, the z axis is oriented along the normal to the
+
+
+
smectic layers (i.e. along the smectic director) and the I plane is perpendicular to the director, B is the smectic compressibility modulus (at constant concentration), K is the splay modulus, x is the osmotic compressibility (at zero elastic strain), and C is a coupling between layer displacement and concentration fluctuations. Of particular significance is the combination B = B - C2x,which is the compressibility modulus at constant chemical potential. It corresponds to a strain where the internal state of the surfactant membranes is left unaffected, whereas strains at constant surfactant concentration (modulus B ) have to stretch or crumple the ~ h e e t s . ' ~ItJ ~ is also directly related to the intermembrane interactions by the relation B = d -a2v ad2 where Vis the interaction potential energy per unit area.M Besides, since the splay modulus K is simply related to the membrane constant K byK = K/d, membrane flexibility and intermembrane interactions can entirely be traced from measurements of the smectic moduli B and K. In the particular case of nonelectrostatic dilute lamellar phases of flexible membranes, where Helfrich's interactions play the crucial role, all the elastic Constants are related to the bending modulus K . ' ~ From eq 1,the correlation functions that are appropriate to interpret static scattering experiments can be derived: the layer displacement correlation function S ,,
7004. . ....
(8)Roux, D.; Safinya, C. R. J. Phys. (Paris) 1988,49,307. (9) Helfrich, W. Phys. Lett. A 1973, A43, 409. (10)Helfrich, W. 2.Naturforsch., A: Phys., Phys. Chem., Kosmophys. 1978,33A, 305.
(11) Brochard, F.; de Gennes, P.-G. Pramana, Suppl. 1976,1,1. (12) Nallet, F.; Rous, D.; Prost, J. J. Phys. (Pans) 1989, 50, 3147. (13) Lubensky, T. C.; Proat, J.; Ramaswamy, S . J. Phys. (Paris) 1990, 51, 933.
0743-7463/91/2407-1861$02.50/0 0 1991 American Chemical Society
Nallet
1862 Langmuir, Vol. 7,No. 9, 1991
the concentration correlation function Scc,
and the crossed correlation function S,,
#'
I ~
'
"
20
0
Hydrodynamics of the Two-Component Smectic A The hydrodynamic theory describes the relaxation toward equilibrium of long wavelength, externally applied (small) fluctuations in the hydrodynamic variables. Its general foundations14and its specific applications to oneand two-component s m e c t i c ~ ~ ~ can J ~ Jbe~ found J ~ elsewhere. When we restrict ourselves to the case of an incompressible, athermal two-component smectic A, then the following four hydrodynamic modes are present:12 There is one uncoupled diffusive mode, associated with the relaxation of the component gy of the momentum densityg which is transverse with respect to both the wave vector and the smectic director; it has a weakly anisotropic dispersion relation, wy = -iqqz/p (7, shear viscosity; p, total mass density). There are two propagative second sound waves, which mostly couple layer displacement and transverse component gt of the momentum density, in the wave vector/ director plane, with a strongly anisotropic dispersion relation; the real parts of the second sound frequencies are finite for oblique wave vectors q, wz = ~ ( B / P ) ' / ~ * (Iqzqll/q), but zero (the waves are overdamped) when q is close to the directions parallel or perpendicular to the smectic layers (see Figure 1,where the imaginary parts of the frequencies are also displayed). There is one diffusive mode, the undulation/ baroclinic mode,l1JZwhich couples chiefly layer displacement and concentration, with again a strongly anisotropic dispersion relation
"
"
"
"
0
40
00
60
Figure 1. Real and imaginary parts of the second sound frequencies, w2, at a fixed modulus q of the wave-vector q (q 5 X 106 m-l),when the angle 8 between q and the smectic director varies from Oto 90°: dashed line, absolutevalue of the real parts; heavy lines, imaginary parts. For 8 below about 15' or above about 7 5 O , the second sound is overdamped; for middle values of 8, the two propagative second sound waves have the same damping. Elastic and hydrodynamic parameters (B = 1 X 106 Pa, g = 1 x Pes, p = 1 X 109 kg.m-9) are typical of dilute, Helfrich-stabilized,lamellar phases. I
'
'
'
'
I
'
'
'
'
1
'I '
'
'
'
d
1. l o 4 Ob
L5-7
5 . lo3
t 10'~
0
Q~'
2 loL' [m-'~
3 10"
Figure 2. Relaxation frequency wb of the undulation/baroclinic mode as a function of qL2: symbols, dynamic light scattering data from an oriented sample, at various qz and q I values; lines, fit to the theoretical dispersion relation. The insert (Wb scale unchanged; qL2scale blown up by a factor 10) shows the sharp baroclinic to undulation crossover.
XZ
OU =
-&Yq
when qz = 0, and
when q L is close to zero. The parameter p in eq 4b is a dissipative parameter related to the shear viscosity q by the relation p = d2/127." The complete dispersion relation is shown in Figure 2, together with dynamic light scattering data (see below). The time-dependent correlation functions can be extracted from the elastic and hydrodynamic theories by the general methods of the linear response theory." For simplicity, and because of its experimental relevance in dynamic light scattering (vide infra), we restrict ourselves to the undulation/baroclinic contribution of the layer displacement S,,(q, t), concentration S,(q, t ) ,and crossed Suc(q,t), correlation functions, which are displayed, at a fixed wave-vector modulus, as a function of the wave(14) Martin, P. C.; Parodi, 0.; Pershan, P. s. Phys. Rev.A: Gen. Phys. 1972, 6, 2401. (15) de Gennes, P.-G. J. Phys. (Paris) 1969,30, C-65. (16) Xin Wen; Meyer, R. B. J. Phys. (Paris) 1989,50,3043. (17) F6mter, D. In Hydrodynamics Fluctuations, Broken Symmetries and Correlation Functions; W. A. Benjamin; Reading, MA, 1975.
0
02
04
06
08
1
QdQ Figure 3. Normalized dynamic light scattering cross sections from the undulation baroclinic mode, at constant wave-vector modulus (q = 2 X lO/m-l), varying the wave-vector orientation. The heavy line corresponds to the concentration correlation function S,, the dotted line to the layer displacement correlation
function S,,, and the dashed line to the crossed correlation , Adielectric anisotropy cz - cI 10-2 times the dielectric function.S increment de/& has been assumed. vector orientation in Figure 3 (the exponential relaxation in time is factorized out).
Scattering Experiments Scattering experiments are a convenient way of measuring various equal-time or time-dependent correlation functions. The electron (or nucleon) density correlation function is the accessible quantity in an X-ray (or neutron)
Membrane Fluctuations in Dilute Lamellar Phases
Langmuir, Vol. 7, No. 9, 1991 1863
scattering experiment. In the case of a two-component smectic A, the experiment yields both Bragg peaks and small-angle ~ c a t t e r i n g . ~ J ’ JThe ~ J ~measured intensity can be readily related to the layer displacement correlation function, S,,(q), in the vicinity of the Bragg peaks,20and to the concentration correlation function, S,,(q), in the small-angle range.21 Such experiments can in principle x , and B. be used to measure B and K,6J3122 Light scattering experiments reveal, with a proper choice of the polarizations, the correlation functions of various components of the dielectric tensor and are thus ultimately sensitive to all the three correlation functions (equal time, in static experiments; time dependent, for quasi-elastic light scattering) S,,, Scc,and S,, of two-component smectics A. In practice, one has to work with oriented samples and to align properly the smectic director with respect to the scattering plane. For instance, owing to optical selection rules, layer displacement fluctuations do not contribute to the scattered intensity, whatever the polarizations, when the director lies in the scattering plane; such a configuration renders, e.g., the undulation mode unobservable.12 On the other hand, and provided that an out-of-plane alignment of the director is ensured, inspection of Figure 3 shows that the undulation/baroclinic mode scatters light for all orientations of the wave-vector q. The elastic constants B and K can therefore be measured through a dynamic light scattering study of the relaxation frequency of this mode.l2 Figure 2 shows typical data obtained on an aligned
sample of a quasi-binary (membrane, sodium dodecyl sulfate and l-pentanol; solvent, water with 20 g/L NaCl; membrane thickness, 6 = 2.3 nm; repeating distance, d = 25 nm) dilute lamellar phase. The undulation/baroclinic mode relaxation frequency is measured varying both qz and q I . All data points, plotted as function of q12, fall on the same straight line, eq 4b, when q I is much smaller than qz. The enlargement in Figure 2 illustrates the very sharp crossover from baroclinic to undulation behavior in the vicinity of qz = 0. The continuous lines come from a single fit of all the data points to the complete dispersion relation in the qz-ql plane.12 A complete study along a dilution line of this system shows that the lamellar phase is Helfrich-stabilized, with a membrane flexibility K about 2.5k~T.
(18) Porte,G.; Marignan, J. Bassereau, P.; May, R. In Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; SpringerVerlag: Berlin, 1987. (19) Porte, G.; Marignan, J.; Baseereau, P.; May, R. Europhys. Lett.
Acknowledgment. It is a real pleasure to thank here the organizers, Dr. W. J. Benton and Dr. L. A. Turkevich, for allowingme to participate to the delightful symposium Fluctuations in Lamellae and Membranes, ACS Fall Meeting 1989. Much of the work presented here has been made in collaboration with D. ROUX,J. Prost, and S. T. Milner; I am deeply indebted to them.
1988, 7, 713. (20) CaillB, A. C.R. Hebd. Seances Acad. Sci., Ser. B 1972,274,891. (21) Nallet, F.; Roux, D.; Milner, S.T. J. Phys. (Paris) 1990,51,2333. (22) Als-Nielsen,J.;Litater, J. D.; Birgenau, R. J.;Kaplan, M.; Safinya,
C. R.; Lindegaard-Andersen, A,; Mathiesen, S.Phys. Reu. B: Condens. Matter 1980,22, 312.
Conclusion Membrane fluctuations manifest themselves, in particular, through the elastic and hydrodynamic properties of dilute lamellar phases. As an intrinsic membrane property, they are quantified by a single elastic parameter, the membrane bending modulus K. When the membranes are stacked in a lamellar phase, they condition the entropic, Helfrich part of the intermembrane repulsion; they control the line shapes and bandwidths of the experimentally relevant intensities in static or dynamic scattering measurements.
