1448
Langmuir 1990,6, 1448-1450
Interactions between Solid Surfaces in a Presmectic Fluid P. G. de Gennes Coll2ge de France, 75231 Paris, Cedex 05, France Received February 26, 1990
Some smectic ordering often shows up, in an isotopic liquid, near a solid wall. When two such walls face each other at a distance D, and D is an exact multiple (nu)of the smectic period ( a ) ,the two walls should attract each other with an energy fo = -Q exp(-D/(), where ( is the smectic correlation length. But when D # nu,an elastic contributionf e appears: the distortion is not uniform and is located preferentially in the weaker region near the midpoint. The contribution f e is oscillatory, but the average f e over the oscillations is strongly positive. Thus the overall coarse-grainedinteraction fo + f e = +Q exp(-D/E) should be repulsive. In the present model, there are no metastable states: the transition from n - 1to n layers does not require the nucleation of a dislocation loop. These presmectic forces might be of interest for colloid stabilization or for lubrication in extreme conditions.
I. Introduction
(4'
Many liquids made with anisotropic (or amphiphilic) molecules tend to display a layered ordering of the smectic A type.' We consider here situations where the bulk phase is not yet smectic: it may be either isotropic or nematic.2 Near a wall (or a free surface), the smectic alignment is sterically favored, and a few smectic layers (each of thickness a ) extended up to a thickness [ can be observed, parallel to the wall.3 In most cases, 6 is not very large (except for the vicinity of a nearly second order smectic transition). If we disperse colloidal grains in such a liquid, do we expect stabilization or destabilization from the smectic features? As we shall see, even for the simplest case of two parallel plates separated by a distance D , the answer is not entirely trivial, because for most values of D the smectic layers do not fit exactly within the gap. Our description is based on a simple Landau theory of the smectic order,' with a free energy
(1)
+
where \k is a complex order parameter 9 = +ei@, measuring the amplitude of the smectic density modulation and r#J = ( 2 ~ / a ) au phase related to a layer displacement u. Here a is positive (no spontaneous order far from the walls), and the higher order terms (p,...) can be neglected. The correlation length is 5 = L1/2a-1/2.Finally, h, (>O) describes the coupling between the walls and the smectic order. The gradient term may be split into two parts
the second part being the elastic energy. We shall always assume that the smectic layers at both ends stick exactly to the wall. This imposes a phase difference 4 ( D / 2 )- 4 ( - 0 / 2 ) = (27r/a)(D- nu) = [41 (3) It is important to realize that the elastic deformation ~
(1) See, for instance: de Gennes, P. G. The physics of liquid crystals; Oxford, 1974. ( 2 ) If the bulk fluid is nematic, we shall assume in what follows that the preferred orientation of the molecules near the wall is normal ('homeotropic") and thus compatible with smectic A layers parallel to
____
t.he wall. ..
(3)Ocko, B.; B r a s h , A.; Pershan, P.; Ala-Nielsen, J.;Deutsch, M. Phys. Rev. Lett. 1986, 57, 94-97.
4s
--
D 2
* 0
D
"
2
t
Figure 1. (a, Top) Plot of the order parameter $ ( x ) measuring the amplitude of the density modulation in the smectic layers. Near the walls ( x = *D/2) the smectic layers are strongly developed. Near the center J, = ( p 3 1 / 2is small. (b, Bottom) Plot of the phase 4 = (2?r/a)u measuring the displacement of the smectic layers from their ideal position. Note that the strain du/dx = (a/2s)(d4/dx) is concentrated in the weak central portion. described by d ( x ) is not uniform; indeed it will turn out that the gradient dr#J/dxis nonvanishing only in a thin region (of thickness [) near the midplane. This is the weakest part of the sample. We shall see that for [$I = 1the order parameter \k is strongly reduced in this region. The whole problem is somewhat reminiscent of other inhomogeneous frustrated systems: twisted arrays of spins, or critical currents in a one-dimensional superconductor.4 But the situation discussed in ref 4 corresponds to order in the bulk, and in those cases the \k4terms in the Landau free energy play an essential role. 11. Organization of the Interplate Region
The smectic order parameter and the phase are represented in Figure 1. To obtain them, we write that the free energy f of eqs 1 and 2 is minimal with respect to variations of 4 and of 1+5and get two equations: (4) Langer, J.; Ambegaokar, V. Phys. Reu. 1967,164,498.. See also the discussion by Tinkham, M. Introduction to supraconductrurty; McGrawHill: New York, 1975; Chapter 7.
0743-7463/90/2406-1448~02.50/0 0 1990 American Chemical Society
Solid Surfaces in a Presmectic Fluid
Langmuir, Vol. 6, No. 9, 1990 1449 (4)
ft
where p = $%), and \
\ \
(5)
\ \
Equation 4 can be integrated in the form
where a is the elastic component of the stress between the plates (a > 0 corresponding to an attractive force). Equation 5 may then be rewritten as
and has the first integral
where C is an integration constant. Multiplying both sides by qZ,we obtain
where pm is the density at the midpoint ( x = 0), where dp/ dx = 0. The parameters pm and b are related to r and C via
c = 2(b - p,) (r5)2= P"
- P,)
(loa) (lob)
Equation 9 has a simple solution p
= pm
+ b[cos h ( 2 ~ / 5 -) 11
(11)
We may obtain b from the boundary condition, derived from the optimization of the free energy (eq 1)with respect ) I to the order parameter at the surface $(D/2) = ,
Figure 2. Free energy versus distance (continuous line) and free energy averaged over the oscillations (dotted fine) for $, = 1.5a. P, = b ( l + cos[$])
(16) Thus when [$] increases from 0 to A,the smectic "density" a t the middle of the plane, Pm, drops from 2b t o 0. Equation 10 fixes the reduced stress
r = t-'b sin $ (17) where b is given explicitly by eq 14 (for D >> t). Let us now turn to a calculation of the free energy (per unit area) f . For [$I = 0 we get f
+
fo = -2Sgh,k dh, = -h;[a[
tan h(D/2[)]-'
= -Q[1 +
(D >> 5 ) with Q = hs2 (at).Thus fo is attractiue. For [$] # 0 we can write
f, f - fo = at2L[% d[$]
giving
(18)
= aSb(1- cos [$I)
= 2Qe-D/E(1- cos [$I)
(19)
111. Discussion From now on we shall restrict our attention to the limit D >> 5. Then from eq 11 we see that P,
-
b[cos W / 5 ) - 11
and eq 13 leads to b
= 2(a)zexp(-D/t)
In this limit, b and ps are thus independent of [$I. We now need a relation between the reduced stress r and the phase [$I: this is obtained from eqs 6 and 11, giving
or equivalently
1. Three Main Features. A qualitative plot of the free energy f as a function of the interplate distance D is shown in Figure 2. This figure shows the following: (a) The interaction energy oscillates. (b) The average of the energy over one oscillation (superposing (18) and (19)) is f = -Q
+ Qe-D/t
and is repulsive; the frustration features dominate. ( c ) The energy plot is continuous when [$] = A (maximal frustration). At this point pm = 0: the two plates become uncoupled. This has an important physical consequence: there are no metastable states corresponding to [$] > ?r (Le., to a wrong number of layers between the plates). This means that when we decrease the separation between the plates, we do not need to nucleate a dislocation loop to adjust the number of layers from n + 1 to n. 2. Comparison with a Smectic Matrix. Our problem is very different from the situation obtained when two plates (or two solid particles) are separated by a bulk smec-
de Gennes
1450 Langmuir, Vol. 6, No. 9, 1990 tic phase as discussed long ago5 in a thermotropic smectic. The nucleation of dislocations is thermally activated and should result in strong forces opposing compression. Lyotropic smectics-in particular highly swollen lamellar phases of lipid + water-have been studied more Here again, we are dealing with bulk phases. The most interesting features of the forces are related to bilayer undulations. Presmectics-as opposed to bulk phases-may be of some practical interest for the following reasons: (a) They do give repulsive forces, and they can provide stabilization for colloids (or foams). (b) Far from the grains, the liquid is not smectic and thus not too viscous. Similar arguments suggest that presmectics may be of interest for lubrication in extreme conditions. 3. Limitations of the Model. On the theoretical side, our discussion based on the Landau free energy1 suffers from (at least) three weaknesses. (a) Fluctuations in the layer position are omitted. This is probably correct for thermotropic smectics end also for compact lipid systems. (b) The continuous model with gradient terms (d\k/ dx)2does not give a full representation of the layers if 5 is comparable to a . Indeed the X-ray reflectance data of ref 3 on a single wall can be interpreted in terms of a constant smectic amplitude ($ = constant) in a certain ( 5 ) Orsay group on liquid crystals J. Phys., Colloq. 1976, C1, 305313. (6) Kekicheff, P.; Christenson, H. K. Phys. Rev. Lett. 1989, 63,2823. (7) Abillon, 0.;Perez, E. To be published.
region (0 < x < L ) plus a sharp end at x = L = nu. This differs from the exponential decay J, = $, exp(-x/[) which is expected from eq 1. Thus sharp boundaries may possibly occur because 5 < a (or for other reasons discussed under c below). But there are cases with E >> a where our model clearly holds; for instance, with the smectogens 80 CB and 9 CB, the reflectance data obtained by Pershan and co-workers8do show “critical adsorption”, Le., a slow, exponential decay of the order. (Note that our .$ = 511 in the notation of ref 8.)
(c) If the bulk phase is an isotropic liquid, we may have an induced nematic order parameter S(x) # 0 near the wall. If the nematic director tends to lie normal to the wall, nematic order and direct wall effects conspire to promote smectic layers. If S(x) is larger than a certain threshold S,, smectic ordering may be locally favored even if the direct influence of the wall (described by h, in eq 1)is negligible. The possible importance of this S effect is pointed out in ref 3. It may lead (for small values) to a sharp smectic interface, occurring at a distance z from the wall such that S ( t ) = S,. If we now have two plates separated by a distance D < 22, we would then return to the problem of colloid stabilization with a bulk smectic described in ref 5. (8)Pershan, P. Liquid crystal surfaces. Proceedingsof the intamtianal conference on Surface and Thin Film studies, using glancing incidence X-ray and neutron scattering (to be published).
