Equilibrium and non-equilibrium dynamics of the dilute lamellar phase

Physica A 186 .992)154-159 North-Holland

Equilibrium and non-equilibrium dynamics of the dilute lamellar phase Sriram

Ramaswamy

1

Centre for Theoretical Studies and Department of Physics, Indian Institute of Science, Bangalore-560012, India

A model for the dynamics of the sterically stabilized dilute lamellar phase is constructed and studied. The model consists of a stack of flexible fluid sheets, with excluded volume, separated by macroscopic layers of solvent. The dynamics of small fluctuations of the sheets about their mean positions is found to have two distinct short-wavelength regimes in which the frequency to depends on the wavenumber q in an unusual manner. One is a singlemembrane Zimm mode, ¢ o - - i q 3, while the other is a "red-blood-cell mode", t o - - i q 6. These modes give rise to fluctuation corrections for the viscosities of the system, going as ¢o-1/3 and to 2/3, respectively. In addition, it is shown that a sufficiently rapid shear flow with velocity and gradient in the plane of the layers causes a transition into a state where regions of reduced layer spacing co-exist with regions devoid of any layer material. The critical shear-rate for this transition should go as (layer spacing) -3. Possible experimental tests of these predictions are discussed.

T h e d i l u t e l a m e l l a r p h a s e [1] in s u r f a c t a n t s o l u t i o n s consists o f a r e g u l a r l y s p a c e d s t a c k of flexible fluid m e m b r a n e s in a b a c k g r o u n d s o l v e n t (oil o r w a t e r ) w i t h viscosity r 1. T h e n o r m a l to t h e l a y e r s is t h e z axis, while d i r e c t i o n s in t h e p l a n e o f t h e l a y e r s a r e l a b e l l e d ± . E a c h m e m b r a n e is a b i l a y e r o f s u r f a c t a n t , t y p i c a l l y 20 ,~ thick, while t h e d i s t a n c e d b e t w e e n a d j a c e n t b i l a y e r s can b e as l a r g e as s e v e r a l t h o u s a n d a n g s t r o m s [2]. I n t h e limit w h e r e all o t h e r i n t e r a c t i o n s a r e e f f e c t i v e l y e l i m i n a t e d , this p h a s e o w e s its s t a b i l i t y to an i n t e r p l a y b e t w e e n t h e flexibility o f e a c h fluid b i l a y e r g o v e r n e d by a c u r v a t u r e elasticity ~¢ a n d t h e steric r e p u l s i o n b e t w e e n b i l a y e r s [3, 4]. I n this p a p e r , I p r e s e n t a s u m m a r y o f r e c e n t results o n the d y n a m i c a l p r o p e r t i e s o f t h e d i l u t e l a m e l l a r p h a s e . T h e first p a r t o f this w o r k [5a]* 1, o n t h e s h o r t - w a v e l e n g t h d y n a m i c s at e q u i l i b r i u m , was d o n e in c o l l a b o r a t i o n with J a c q u e s P r o s t ( E S P C I Paris) a n d T o m L u b e n s k y ( U n i v e r s i t y o f P e n n s y l v a n i a ) .

E-mail: [email protected] ~1 A derivation of the q3 mode but with boundary conditions appropriate to a single bilayer was given in ref. [5b]. Such a mode was also suggested by di Meglio, Dvolaitsky, Leger and Taupin in ref. [1]. 0378-4371/92/$05.00 ~ 1992- Elsevier Science Publishers B.V. All rights reserved

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We find a rich set of crossovers from single-membrane to red-blood-cell-like to bulk smectic behaviours as the wavenumber is decreased towards the hydrodynamic range. Specifically, the microscopic layer undulation mode has the dispersion relation to = --i(Kd3/16n)q~

(1)

appropriate for the fluctuations of a closed red blood cell [6], for q i d ~ 1 q~La, with qzd = -+rr (which we call regime A) and to = - - i ( K / 4 ~ ) q ~ ,

(2)

which corresponds to the dynamics of a single membrane with hydrodynamic self-interaction, for q i d >>1, and for all qz (which will be termed regime B). In the above, L d - - d ( K / T ) 1/2, as discussed in detail below, is the scale below which the in-plane curvature elasticity dominates over the compressional elasticity. Moreover, the contribution of these novel modes to the spontaneous stress fluctuations at thermal equilibrium is predicted to lead to a strong frequency dependence in the real and imaginary parts of measured viscosities of the phase. This takes the form to-2/3 for frequencies to corresponding to regime A, and to-~/3 for regime B. In the second part [7], I study the effect of a shear flow with velocity and gradient in the layers (to be contrasted with that studied by Bruinsma and Rabin [8], who look at the geometry where layers slide over one another). I show that for shear rates O larger than a critical value 0 c ( ~ d -3) the layer spacing should decrease as d ( J 2 ) - J2-1/3. In a constant-concentration picture, this means that the system will phase-separate into regions with smectic layer spacing d(O), coexisting with regions devoid of membrane (d = ~). The onset of this transition should be marked [9], in scattering experiments, by a reduced in-plane (the _1_ direction) and enhanced out-of-plane (the z direction) smallangle signal. Strictly speaking, on the basis of the present calculation, I cannot tell whether 12 --- g2c(d) marks a real transition or a crossover. Nonetheless I conjecture that a realistic calculation including attractive forces between the membranes will produce a sharp transition. Let us begin by recalling Helfrich's theory [3] of the compressional elasticity of these phases. Consider one of the surfactant bilayers (fluid membranes) in a lamellar phase with layer spacing d. Its fluctuations can be described by a height function h(s) (where s is a 2D coordinate on the membrane) and are governed to harmonic order by the curvature energy

/4 = ½ f(V h) d s.

(3)

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In addition, it is constrained not to pass through its neighbouring layers. It is easy to see that this steric repulsion is effective only on large enough length scales in the plane of the membrane. Begin by noting that, in the absence of steric constraints, the Hamiltonian (3) leads by equipartition, at a temperature T, to the result that the variance in the fourier component of h with w a v e n u m b e r q is (Ihq [2) = T/Kq4. Thus the typical fluctuations in h on a length scale L have variance (h2)L = ( T / K ) f d2q/q 4 - ( T / K ) L 2. When the steric constraint is included, it is natural to define a length scale L a ~ ( K / T ) 1/2d such that 1 are, at least as far as static properties are concerned, essentially those of an isolated fluid membrane. The scale L a can be thought of as the mean in-plane distance between successive encounters of a membrane with its neighbours. There is thus one encounter in each elementary volume L a × L a × d. The loss of configurational entropy associated with these encounters contributes a free energy density T / L 2 d ~ T2/Kd 3, which is Helfrich's estimate [3] of the compressional elastic constant of the lamellar phase. The purpose of that brief review was to emphasize the significance of the scale L a, which marks the crossover from isolated membrane to bulk smectic behaviour, and the essential role of fluctuations. Let us now examine the dynamics of the lamellar phase [5, 10, 11], and derive qualitatively the mode structure of eqs. (1) and (2). We shall first discover the red-blood-cell ( R B C ) mode of regime A. It should be clear from the discussion above that it is enough to focus on the motion of one membrane relative to its neighbours, for q±La > 1, where steric hindrance can be ignored. This relative displacement mode is simply the limit qzd = +rr of a macroscopic layer displacement field. The only energy cost then is bending energy, governed by the effective free energy (3). The dynamical properties are nonetheless not those of a single, isolated membrane, because the region between the m e m b r a n e is filled with a viscous, incompressible solvent. The relative height fluctuations of adjacent layers must therefore conserve the quantity of solvent in the region they bound. Thus, they must behave like a conserved mode [12]. Now on general grounds, we except the linearised equation for h to be of the form Ohq/O t = - F ( q ) ~H/Ohq + fq where F ( q ) is a kinetic coefficient and fq is a thermal noise with variance ~ F ( q ) . The fact that h behaves like a conserved m o d e over the range of q we are interested in means that F ( q ) = 3"q2 so that a m o d e of wavenumber q has a relaxation rate i o J - q 6 [13, 5a]. Next, let us determine 3' in terms of d and the viscosity "Oof the surrounding solvent. Purely on dimensional grounds 3" must be of order d3/*h SO that ito ct(kd3/T1)q 6, as claimed in (1). What about the single-membrane mode of regime B? We have already noted

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that the RBC mode makes sense only for qzd = -+~r and q±L d ~ 1. In addition I now show that it is restricted to qxd ~ 1, and that for q±d >>1, the q3 mode of eq. (2) takes over. To see this, note that we have ignored the long ranged hydrodynamic interaction of each membrane with itself through the solvent. For an isolated membrane, this is never justified. A fluctuation with wavenumber q± relaxes nonlocally as a result of this interaction, with an equation of motion [~q ~ _q-1 ~H/~hq, so that ito -- q3 [5a, 13]. Now consider the present case. An undulation with wavenumber q± penetrates (via Navier-Stokes) a distance q~l into the solvent. If q ~ 1 > d, the effect is thus blocked by the neighbouring membranes and our earlier argument is justified. If q~ld, however, the hydrodynamic self-interaction of the membrane is unimpeded, and the dynamics is therefore ito ~ q3 as described above The entire argument above is justified by a detailed calculation [5a]. The frequency-dependent viscosities can be obtained by autocorrelating the contribution of the above modes to the thermally excited stress [5a]. Let us turn next to the shear flow problem [7]. A velocity gradient O will affect the Helfrich mechanism if O is larger than the relaxation rate of those crucial modes with q±L d --1, because L d is the scale at which the membranes collide. For wavenumbers q± somewhat larger than L d 1, the effective equation of motion for the layer fluctuation is that for the RBC mode: Oh/Ot = yKV6h + f ,

(4)

where f is a conserving noise, with variance 2TTq 2 at wavenumber q. It is straightforward to see that this gives (Ihql 2) = T/Kq 4 for the equal time correlation. We must see how shear affects these correlations. Let us introduce a plane shear flow v = g2yl in the layers. This will advect the layer fluctuations, giving a modified equation of motion Oh/Ot = - g 2 y Oh/Ox + ~/KV6h + f .

(5)

The equal time correlations of h are now not given by equipartition on (3) but must be extracted from eq. (5). Assuming a stationary state, and defining C(q) by the equation ( h q ( t ) h q , ( t ) ) = C ( q ) 6 ( q + q'), which is justified because equal time correlations are translational invariant in plane Couette flow [14, 15], we find from (5) that

OC/Oqy + (2yKq6/g2qx)C( q) = 2TyqZ/g2qx .

(6)

The solution to this linear first order O D E in qy is straightforward, and I shall not describe it here. What emerges is that the q-4 singularity which leads to the

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divergent variance (h e ) - - L 2 is replaced by an anisotropic and, in general, less singular behaviour (e.g. C ( q ) ~ q x 4/7 for qy = 0 ) . This is because a new length enters the problem, namely = ( 2 y K / g 2 ) '/6 .

(7)

Modes with wavelength shorter than ~ decay too quickly to be affected by the shear flow, while those with wavelength longer than ~ are sheared before they decay. Now we find [7] that (h 2) crosses over from L 2 for L ~ ~: to ~2 for L ~> ~. Shear renders the "crumpling" fluctuations of the layers finite, even in the absence of steric repulsion by neighbouring layers. What does this imply? Nothing at all, if ~ > La, because then (h 2) > d 2 so that steric repulsion, far from being negligible, will enter to cut off the fluctuations and stabilize the lamellar phase well before shear can play any role. All that shear will be able to do is to limit [15, 6] the long wavelength undulation mode by cutting off the Caill6-Landau-Peierls [16] divergence of the Debye-Waller factor. An experiment at fine enough resolution will see the algebraic singularities of the smectic cross over to genuine delta-function peaks [17]. But as 12 increases, ~ will eventually decrease below L a. At that point, typical layer fluctuations will be s m a l l e r than d, steric repulsion will be ineffective, and the layers will move closer together until they reach a new layer spacing of o r d e r (T/K)1/2~. NOW steric repulsion will prevent further collapse. Of course, my simple treatment cannot prove that there is a real nonequilibrium analogue of a phase transition at O c rather than a gradual crossover in that general vicinity, but the effect is present and interesting in either case. The predictions of this part can be summarized as follows. Start with a lamellar phase with layer spacing d 0 at O = 0. If O is increased abruptly beyond a value De(d0) - do 3, there is a tendency to collapse to a smaller layer spacing. This will be signalled by two effects [9]; (a) a sudden enhancement of small-angle scattering for qz ~ 0 ( q . = 0), because the effective compressional elastic constant is momentarily zero; (b) reduced in-plane (q±--~0, qz = 0) small angle scattering, because the correlation length for in-plane fluctuations is now ~ which is less than L a. It should be possible to test the present predictions, for both the mode structure and the shear-induced collapse, by neutron, light and x-ray scattering experiments in a shear flow. I should emphasize that these must be performed with velocity and gradient mutually perpendicular and in the plane of the smectic layers. The necessary rates of shear can be made as low as a few kHz by working with a highly swollen (d - 1000 A) phase with K of the order of the room temperature.

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I t h a n k J a c q u e s P r o s t , D i d i e r R o u x , C y r u s Safinya, T o m L u b e n s k y a n d R a h u l P a n d i t for useful discussions. I also t h a n k C R P P ( P e s s a c ) , E S P C I (Paris) a n d t h e C N R S for t h e i r s u p p o r t w h i l e p a r t o f this w o r k was d o n e , a n d A b h i j i t M o o k e r j e e a n d t h e o r g a n i s e r s o f t h e W o r k s h o p on D i s o r d e r e d Solids, P o l y m e r s a n d G l a s s e s , C a l c u t t a ( 1 9 9 1 - 1 9 9 2 ) for t h e o p p o r t u n i t y to p r e s e n t it.

References [1] P. Ekwall, in: Advances in Liquid Crystal Physics, G.M. Brown, ed. (Academic Press, New York, 1975); M. Dvolaitzky, R. Ober, J. Billard, C. Taupin, J. Charvolin and Y. Hendrix, C. R. Acad. Sci. Paris 1145 (1981) 295; W.J. Benton and C.A. Miller, J. Phys. Chem. 87 (1983) 4981; J.M. DiMeglio, M. Dvolaitzky and C. Taupin, J. Phys. Chem. 89 (1985) 871; J.M. DiMeglio, M. Dvolaitzky, L. Leger and C. Taupin, Phys. Rev. Lett. 54 (1985) 1686; A.M. Bellocq and D. Roux, Microemulsions, S. Friberg and P. Bothorel, eds. (CRC Press, Boca Raton~ 1986). [2] F. Larch6, J. Appetl, G. Porte, E Bassereau and J. Marignan, Phys. Rev. Lett. 56 (1986) 1700; F. Lichterfeld, T. Schmeling and R. Strey, J. Phys. Chem. 90 (1986) 5762; N. Satoh and K. Tsujii, J. Phys. Chem. 89 (1987) 6629. [3] W. Helfrich, Z. Naturforsch. 30c (1975) 841; 33a (1978) 305. [4] T.C. Lubensky, J. Prost and S. Ramaswamy, J. Phys. (Paris) 51 (1990) 933. [5] (a) S. Ramaswamy, J. Prost and T.C. Lubensky, J. Phys. (Paris) (1992), in preparation. (b) R. Messager, E Bassereau and G. Porte, J. Phys. (Paris) 51 (1990) 1329. [6] F. Brochard and J.F. Lennon, J. Phys. (Paris) 11 (1975) 1035. [7] S. Ramaswamy, Phys. Rev. Lett. (1991), submitted. [8] R. Bruinsma and Y. Rabin, UCLA preprint (1991). [9] C. Safinya, private communication. [10] F. Brochard and EG. DeGennes, Pram~na Suppl. 1 (1975) 1. [11] F. Nallet, D. Roux and J. Prost, J. Phys. (Paris) 50 (1989) 3147. [12] EC. Martin, O. Parodi and P.S. Pershan, Phys. Rev. A 6 (1972) 2401. [13] Y. Marathe and S. Ramaswamy, Europhys. Lett. 8 (1989) 581. [14] A. Onuki, Phys. Lett. A 70 (1979) 31. [15] S. Ramaswamy, Phys. Rev. A 29 (1984) 1506. [16] L.D. Landau, in: Collected Papers of L.D. Landau, D. ter Haar, ed. (Gordon and Breach, New York, 1965) p. 209; R.E. Peierls, Helv. Phys. Acta Suppl. 7 (1934) 81; A. Caill6, C. R. Acad. Sci. Ser. B 274 (1972) 891. [17] D. Roux and C. Safinya, personal communications.