Langmuir 2002, 18, 1511-1517
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Contact Angle between Smectic Film and Its Meniscus Andrzej Poniewierski,*,† Patrick Oswald,‡ and Robert Hołyst†,‡,§ Institute of Physical Chemistry, Polish Academy of Sciences, Department III, Kasprzaka 44/52, 01-224 Warsaw, Poland, Laboratoire de Physique de l’Ecole Normale Supe´ rieure de Lyon, 69364 Lyon cedex 07, France, and Department of Mathematics and Natural SciencessCollege of Science, Cardinal Stefan Wyszyn´ ski University, Dewajtis 5, 01-815 Warsaw, Poland Received September 25, 2001. In Final Form: December 3, 2001 We apply de Gennes model (Langmuir, 1990, 6, 1448-1450) for presmectic ordering to a freely suspended smectic film coupled to the meniscus. We show that the quartic term in the expansion of the free energy in power of the smectic order parameter cannot be neglected in the model in order to explain the full behavior of the contact angle, θm, between the film and the meniscus. As shown previously (Phys. Rev. E 2001, 63, 021705-1-021705-9), this contact angle is nonzero due to the disjoining pressure caused by the enhanced smectic ordering at the film surfaces. The temperature dependence of θm with a maximum above the bulk temperature of the nematic-smectic phase transition, TNA, is qualitatively reproduced by the model for all film thicknesses h ) Nd, where N is the number of layers and d the smectic period, although some discrepancy between theory and experiment for the temperature at the maximum still remains. The apparent change of slope of the contact angle at TNA observed experimentally in thin films cannot be reproduced by the mean-field treatment of the model and presumably critical fluctuations in the meniscus and their influence on the behavior of the surface tension close to TNA have to be taken into account.
1. Introduction Soft condensed matter systems such as, e.g., smectic liquid-crystalline phases, exhibit surface properties which are very different from ordinary liquids. For example, one can prepare smectic films of constant thickness freely suspended on a metal frame (somewhat like soap films);1,2 this is shown schematically in Figure 1. Normal liquids would flow under the influence of stress induced by the curved meniscus around the film, but smectic films are stable for months if kept free from dust particles. In ordinary liquids, the formation of meniscus can proceed via a reversible thermodynamic process, whereas in smectics the formation of the meniscus is always an irreversible process. It is due to the necessity to nucleate dislocations in order to change the thickness of the sample, and nucleation of defects is always associated with an irreversible thermodynamic process.3 Therefore, the height of the meniscus in ordinary liquids is independent of the particular process leading to its formation, whereas in smectics the height of the meniscus is set by the number of dislocations created during the process. In smectics the asymptotic shape of the meniscus (close to the film, where dislocations are elementary) is circular,3,4 whereas in ordinary liquids it is exponential. Due to the curvature of the circular meniscus, there is a pressure difference ∆P between the surrounding air and the smectic phase, which is given by the Laplace law ∆P ) γ/R, where γ is the smectic-A-air surface tension and R is the radius of †
Institute of Physical Chemistry, Polish Academy of Sciences. Laboratoire de Physique de l’Ecole Normale Supe´rieure de Lyon. § Department of Mathematics and Natural SciencessCollege of Science, Cardinal Stefan Wyszyn´ski University. ‡
(1) Friedel, G. Ann. Phys. (Paris) 1922, 18, 273. (2) Pieranski, P.; Beliard, L.; Tournellec, J.-Ph.; Leoncini, X.; Furtlehner, C.; Dumoulin, H.; Riou, E.; Jouvin, B.; Fe´nerol, J.-P.; Palaric, Ph.; Heuving, J.; Cartier, B.; Kraus, I. Physica A 1993, 194, 364-389. (3) Picano, F.; Hołyst, R.; Oswald, P. Phys. Rev. E 2000, 62, 37473757. (4) Geminard, J.-C.; Hołyst, R.; Oswald, P. Phys. Rev. Lett. 1997, 78, 1924-1927.
curvature of the meniscus.3-5 At equilibrium, the pressure must be the same in the film and in the meniscus (the meniscus acts as a reservoir of particles and fixes the pressure); as a consequence, the film supports the pressure difference ∆P across its flat surface, which is only possible because of the elasticity of smectic layers.4 The films can be overheated a few degrees above the bulk transition temperature to the nematic phase, TNA, and still remain stable with well-defined smectic layers, thus, with a nonzero smectic order parameter.6 Recently, it has been shown experimentally that in the case of thin smectic films (typically less than 20-30 layers) the meniscus does not wet the film (at least at the micrometric scale of the microscope resolution) but makes with it a macroscopic nonzero contact angle, θm.8 Experimentally this angle is obtained by extrapolating the circular shape of the meniscus down to the thickness of the film, which is shown in Figure 2. θm is small (∼2°) for very thin (five layer thick ) 150 Å) films; nonetheless, its nonzero value calls for some theoretical explanation. The angle is independent of the radius of curvature of the meniscus, of its height, etc. It decreases with the number of layers and is practically indistinguishable from zero for films thicker than 15 layers (450 Å). It increases with temperature below TNA and reaches a maximum above TNA, showing an apparent discontinuity in its first derivative at TNA (Figure 3). The original de Gennes model7 for presmectic ordering in a nematic between parallel walls has been applied to the problem8 and solved analytically by neglecting the quartic term in the Landau free energy expansion. It was shown that it predicts the correct behavior of θm far from TNA. On the other hand, an unphysical nonmonotonic behavior of θm very close to TNA (5) Proust, J. E.; Perez, E. J. Phys. Lett. 1977, 38, L-91-94. (6) Stoebe, T.; Mach, P.; Huang, C. C. Phys. Rev. Lett. 1994, 73, 13841387. (7) de Gennes, P. G. Langmuir 1990, 6, 1448-1450. (8) Picano, F.; Oswald, P.; Kats, E. Phys. Rev. E 2001, 63, 0217051-021705-9.
10.1021/la011476e CCC: $22.00 © 2002 American Chemical Society Published on Web 02/08/2002
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Figure 1. Schematic representation of the experimental situation. h(x) defines the shape of the meniscus, which close to the film is circular (with radius R), θm and π/2 - θ0 are the contact angles at the film and at the frame, respectively, and τ is the film tension. The distribution of dislocations in the meniscus is shown in (b). “Deformation parabolas” limit the regions of strong deformation of the smectic layers associated with each dislocation.
Figure 2. Macroscopic contact angle θm obtained by extrapolating the circular shape of the meniscus. This circular profile meets the film of thickness H at xm. In the real matching region there is no angular discontinuity; however, this microscopic region is unobservable in the microscope.
was found, with a deep minimum at TNA, which is not observed experimentally. It was suggested that this failure was due to the model itself and to some 3D-2D crossover close to TNA. In this paper, we would like to clarify this point and show that the original de Gennes model,7 solved without neglecting the quartic term in the Landau expansion in power of the smectic order parameter, captures the physics of the problem and predicts qualitatively correct behavior of θm close to TNA. This is a very important conclusion since so far the de Gennes model has been applied with success to the interpretation of many different phenomena: the measurements of interactions between two
Figure 3. θm2 vs temperature obtained from the experiment8 for the following numbers of smectic layers in the film: N ) 3, 4, 5, 7, and 15.
surfaces immersed in the nematic phase,9 the thinning transition in smectic films,10-12 the behavior of dispersions of latex particles in a nematic solution,13 and the sponge(9) Richetti, P.; Moreau, L.; Barois, P.; Kekicheff, P. Phys. Rev. E 1996, 54, 1749-1762. (10) Gorodetskii, E. E.; Pikina, E. S.; Podneks, V. E. JETP 1999, 88, 35-39. (11) Pankratz, S.; Johnson, P. M.; Hołyst, R.; Huang, C. C. Phys. Rev. E 1999, 60, R2456-R2459. (12) Shalaginov, A. N.; Sullivan, D. E. Phys. Rev. E 2001, 63, 0317041-031704-9. (13) Raghunathan, V. A.; Richetti, P.; Roux, D. Langmuir 1996, 12, 3789-3792.
Contact Angle between Smectic Film and Its Meniscus
lamellar transition in confined systems.14 We note that in a recent study on smectic films, Shalaginov and Sullivan12 explicitly included the quartic term in the free energy and also showed that singularities would otherwise occur when T f TNA. The paper is organized as follows. In the next section, we recall the origin of a nonzero contact angle, which is related to the thickness-dependent free energy and the disjoining pressure. In section 3, we apply to the problem the de Gennes model for presmectic ordering with the quartic term included and solve it numerically. In section 4, we present the results for the contact angle obtained from the numerical solution of the model and compare them with the experiment and also discuss the limitations of the mean field approximation. 2. The Contact Angle In ordinary liquids, the meniscus approaches the liquid surface exponentially with a decay length equal to the gravitational capillary length.15 The line of contact between the meniscus and the film is at infinity, and the contact angle θm ) 0. In the smectic case, the circular profile of the meniscus (at both free surfaces of the system) approaches and crosses the film surface at a finite distance and with a macroscopic finite contact angle θm * 0. To find the equation for the contact angle, we consider the mechanical stability of the contact line. In equilibrium, the free energy must be stationary with respect to any translation dx of the line position. To be precise, we should assume that dx is sufficiently small that it does not affect the shape of the meniscus, which means that it must be small compared to the radius of curvature. Now, we follow de Gennes16 by assuming that the change of area of each interface is observed in the far field, i.e., far from the contact line, where the in-plane gradients of the smectic order parameter can be neglected. This assumption is necessary since in the vicinity of the contact line (core region) the fluid is strongly inhomogeneous in all directions perpendicular to it. Thus, the distance from the contact line, l, where the changes of the interfacial areas are measured, must be large compared to the size of the core, rc. At the same time this distance must be small compared to the radius of curvature, to measure a well-defined contact angle; hence, rc , l , R must hold. It is important to note that the argumentation presented above can be formulated in a more formal mathematical language, which in the context of simple fluids has been done by Kerins and Boiteux,17 and a generalization to nematic liquid crystals can be found in ref 18. Let us forget for a moment about the layer structure of smectics and proceed in a similar way as in the case of simple fluids; possible complications will be discussed later. Then it is argued16 that a shift of the contact line by dx affects neither the energy of the core, since the core is simply translated, nor the bulk energy of the fluid. This means that the changes of the free energy (per unit area) of the meniscus and of the film are only due to the changes of area of the far-field interfaces, and they are given by dFmeniscus ) 2γ cos θm dx and dFfilm ) -[2γ + f(h)] dx, respectively. In the film case, the additional term f(h) is the excess free energy due to a finite thickness of the film, hence, f(h) f 0 when h f ∞. The origin of f(h) is the (14) Antelmi, D. A.; Kekicheff, P.; Richetti, P. Langmuir 1999, 15, 7774-7788. (15) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon: Oxford, 1982. (16) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827-862. (17) Kerins, J.; Boiteux, M. Physica A 1983, 117, 575-592. (18) Poniewierski, A. Liq. Cryst. 2000, 27, 1369-1380.
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modification of the fluid structure near one surface by the presence of the other surface. Since in equilibrium the total free energy must be stationary, we have dFfilm + dFmeniscus ) 0. This condition expresses the balance of forces acting on the contact line in the x direction, analogous to the Young equation, i.e.
2γ cos θm ) 2γ + f(h)
(1)
It is clear from the presented derivation that a nonzero contact angle is not specific to liquid crystals but is a property of any confined system (in particular, the notion of contact angle is well-known in the physics of soap films19). So far we have treated the smectic meniscus as if it was a homogoneous medium sufficiently far from the interfacial regions and from the contact line. In other words, we have not taken into account inhomogeneities produced by the complex defect structure of the meniscus (see Figure 1b), described and discussed in detail in ref 3. For our present purpose, this seems justified since the experimentally measured contact angle is very small. Therefore, we expect that in the asymptotic region relevant for the measurement of the contact angle, i.e., rc , l , R, the defects are relatively far away from each other. We also note that it was shown in the previous studies3,8 that the inclusion of the energy of defects in the free energy functional, with the density of defects proportional to the slope of the meniscus dh/dx, does not modify the equation for the meniscus shape, although another functional dependence of the free energy on dh/dx could change this conclusion. Since θm is small, we believe that possible corrections to eq 1 due to the presence of defects would probably be of higher order than θm2, where the latter results from the approximation 2 cos θm ≈ 2 - θm2. However, to verify this, more information about the distribution of defects in the interesting region is necessary. Therefore, with all these provisos, we assume in what follows that θm can be determined from eq 1. We note that in ref 8, eq 1 was derived in a different way from a more general condition of mechanical equilibrium, which applies both to the meniscus and the film
∆P - Πd - γ/R ) 0
(2)
where Πd ) -df/dh is the disjoining pressure. In the film region (R ) ∞), ∆P ) Πd, far in the meniscus (h f ∞) ∆P ) γ/R, and the matching condition between these two regions leads to eq 1. 3. de Gennes Model for Smectic Ordering in Films There are two contribtuions to f(h) which have to be taken into account. The first one is due to the van der Waals attraction between surfaces,20 and the second one is due to the structure of smectics near surfaces. The first contribution is given by
fvdw ) -A/12πh2
(3)
where A is the Hamaker constant. Here we assume that A ) 10kBTNA, where TNA is the temperature of the bulk continuous nematic-smectic-A transition. However, as pointed out by Picano et al.,8 the van der Waals interaction alone cannot explain either the magnitude of θm or its (19) Toshev, B. V.; Platikanov, D. Adv. Colloid Interface Sci. 1992, 40, 157-189. (20) Israelashvili, J. Intermolecular and Surface Forces; Academic Press: London, 1991.
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temperature dependence close to TNA. The second contribution to f(h) is due to the variations of the smectic order parameter Ψ ) ψ exp(iφ), with ψ and φ denoting its amplitude and phase, respectively. According to the Landau-de Gennes theory,7 the free energy functional has the following generic form
˜fLG[Ψ] ) h/2 1 dψ 2 1 dφ 2 + Lψ2 + fL(ψ) - fL(ψb) dz + L -h/2 2 dz 2 dz fs(ψ(-h/2)) + fs(ψ(h/2)) (4)
∫
[ ( )
]
( )
where N is the number of smectic layers and d is the layer thickness in an unstressed smectic. Equations 7 and 8 have to be solved with the four boundary conditions
dψ (0) ) 0 dz and
φ(0) ) 0
1 1 aψ2 + bψ4 2 4
(5)
where a ) a0(T - TNA) and a0, b, and L are positive phenomenological constants. In general, the equilibrium value of ψ in the middle of the film differs from its bulk value ψb. fs(ψs) denotes the contribution to the free energy due to the presence of a free surface, and we assume that
fs(ψs) ) -hsψs +
1 gψ2 2 s s
(6)
where hs and gs are the surface coupling constants, and ψs ) ψ(z ) (h/2). In ref 12, it is argued that the condition hs ) 0 should apply to free-standing films. Since, in our opinion, there are no fundamental reasons to neglect the linear term in fs, we use the general form given by eq 6. In the bulk system, fL describes a second-order nematicsmectic-A transition, and TNA is the mean-field transition temperature. The equilibrium Landau-de Gennes free energy, corresponding to the minimum of ˜fLG with respect to ψ and φ at fixed h, is denoted by fLG(h). Thus, ˜fLG(∞)/2 is the contribution to the surface tension due to the appearance of the smectic order. We assume that this is always small compared to the contribution due to the density difference between a liquid and the air. In ref 8, the quadratic approximation for ˜fLG was used, which led to easily solvable linear Euler-Lagrange equations. However, this approximation was responsible for an unphysical temperature dependence of θm very close to TNA. Moreover, fs was not included in the free energy functional, but instead a fixed surface value of the smectic order parameter, ψs, was assumed. Here, we apply to the problem ˜fLG given by eq 4 without any additional approximations, which means that we have to solve the nonlinear Euler-Lagrange equations:
L
dφ 2 ∂fL d2ψ )0 - Lψ 2 dz ∂ψ dz
( )
d dφ Lψ2 )0 dz dz
(
)
(7) (8)
It follows from eq 8 that
ψ2
dφ )Γ dz
(9)
where Γ is an integration constant. The integration of dφ/dz over the whole film gives the phase difference
∆φ ) φ(h/2) - φ(-h/2) )
2π (h - Nd) d
∂fs dψ (h/2) ) ) hs - gsψs dz ∂ψs
L
with z axis normal to the film and with
fL(ψ) )
(10)
(11)
(12)
and
π φ(h/2) ) (h - Nd) d where we have used the symmetry z f -z, which requires that ψ(z) ) ψ(-z) and φ(z) ) -φ(z). In principle, one can use the following procedure. First, dφ/dz is expressed in terms of ψ from eq 9. Due to the energy conservation principle, this leads to a first-order differential equation
1 1 dψ 2 L ) fL(ψ) - fL(ψm) - LΓ2(ψ-2 - ψm-2) (13) 2 dz 2
( )
where ψm ) ψ(0), and we have used eq 11. Assuming that ψs > ψm, we find the free energy of the smectic film of thickness h and a fixed number of smectic layers N, for ψ(z) satisfying eq 13:
∫ψψ
[f (ψ) - f (ψ ) - 21 LΓ (ψ )] dψ + LΓ∆φ + h[f (ψ ) - f (ψ ) 1 LΓ ψ ] + 2f (ψ ) (14) 2
˜fLG ) 2(2L)1/2 ψm-2
-2
2
s
m
L
L
m
1/2
L
2
m
-2 m
L
b
s
s
Then, to obtain fLG(h), ˜fLG(ψm,ψs,Γ;h) is minimized at constant h, which leads to a set of three nonlinear equations for ψm, ψs and Γ. Note also that
dfLG ∂f˜LG 2π 1 ) ) LΓ + fL(ψm) - fL(ψb) - LΓ2ψm-2 dh ∂h d 2 (15) In ref 12, to find the equilibrium free energy the set of nonlinear equations for ψm, ψs, and Γ is solved at a fixed value of the film thickness. To avoid problems with possible multiple roots, we do not follow this route, however. Instead, we integrate numerically the set of four firstorder differential equations for ψ(z), φ(z), and their derivatives, resulting from the second-order EulerLagrange equations (eqs 7 and 8), with the boundary conditions specified in eqs 11 and 12. The solution is obtained by means of a standard relaxation method21 for two boundary value problems. Compared to ref 12, this approach seems to us more straightforward as it gives explicit forms of the profiles ψ(z) and φ(z); hence ψm, ψs, and Γ follow. Then, to compute dfLG/dh, we use identity 15 with ψm and Γ obtained directly from the numerical solution. The total excess free energy of the film as a function of h is defined as follows (21) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Veterling, W. T. Numerical Recipes; University Press: Cambridge, 1986.
Contact Angle between Smectic Film and Its Meniscus
f(h) ) fvdw(h) + fLG(h) - fLG(∞)
Langmuir, Vol. 18, No. 5, 2002 1515
(16)
and the equilibrium value of h is determined from the condition of mechanical equilibrium
df ) -Πd ) -∆P dh
(17)
Since it is found experimentally that θm is very small, we use for the contact angle the formula (see eq 1)
θm2 ≈ -
f(h) γ
(18)
In eq 18, we also neglect the temperature dependence of γ in a small temperature range around TNA, as it comes mainly from the temperature dependence of the smectic order parameter. This is justified because the contribution to the surface tension due to the appearance of the smectic order is expected to be always small compared to the contribution due to the density difference between the liquid crystal and the air. In our model, we can verify this assumption a posteriori by comparing the value of fLG(h) with γ; the ratio of these quantities was usually around 10%. In our calculations, we have applied in eq 4 the following scaling: z f z/d, ψ f (L/bd2)1/2ψ, h f h/d, and hsf (d-2(L3/b)1/2)hs. Now, the dimensionless free energy density is obtained by putting in eq 4 and eq 5: L ) 1, b ) 1, and a ) (d/ξ0)2(T - TNA)/TNA, where ξ0 has the sense of the amplitude of the correlation length, which in the smectic phase is given by ξ ) ξ0[(TNA - T)/TNA]-1/2. The Landaude Gennes free energy scales as fLG(h) f (L2/bd3)fLG(h/d), where L2/bd3 has the dimension of energy per unit area. After the scaling transformation eqs 17 and 18 assume the following forms, respectively
A h d d ) -rfLG′(h/d) R 6π h
3
( )
(19)
and
θm2 ) -r[fLG(h/d) - fLG(∞)] +
A h d 12π h
2
()
(20)
h ) A/γd2 are dimensionless where r ) L2/bd3γ and A parameters. Assuming γ ) 26.5 erg/cm2 and d ) 30 Å, we obtain for the dimensionless Hamaker constant A h ) 10kBTNA/γd2 ≈ 0.18. The radius of curvature of the meniscus in the experiment is of order R ) 0.3 cm. Finally, ξ0, r, hs, and gs are the fitting parameters. The best fit to the experimental dependence of θm2 on the number of layers is obtained for ξ0 ≈ 8 Å, in accord with an earlier estimate by Picano et al.8 4. Results and Discussion First of all it is very important to realize where the results come from. Normally one calculates the full free energy of the system and next subtracts its bulk and surface parts in order to get the excess free energy due to a finite thickness (see eq 16). In Figure 4, we present θm2 vs temperature as a difference of two terms (see eq 20): θm2 ) Fh - F∞. The first term (explicitly: Fh ) -rfLG(h/d) + (A h /12π)(d/h)2) is due to the free energy of a finite film (with the bulk part subtracted), and the second term (F∞ ) -rfLG(∞)) corresponds to the part of the surface tension 2γ (with minus sign) coming from the enhancement of the smectic order at the free surfaces in an infinitely thick film. For thick films (h g 15d), both terms exhibit a
Figure 4. θm2 vs temperature (points) presented as a difference of two terms: Fh (solid line) and F∞ (dashed line), corresponding to the films of finite and infinite thickness, respectively (see text), for (a) N ) 3, (b) N ) 5, and (c) N ) 15. Here ξ0 ) 8 Å, R ) 3 mm, hs ) 0.4, and r ) 0.215. The vertical dashed line marks the bulk nematic-smectic transition temperature, TNA.
characteristic change of curvature at the bulk transition temperature TNA (Figure 4c) and they are practically indistinguishable from each other. For thin films, however, their difference is much larger, and the change of curvature of Fh at TNA is much smaller than in the case of F∞ (Figure 4a,b). We will come back to this point later. θm2 is usually very small and practically vanishes for film thickness larger than 15d. In Figure 5, the experimental θ2m as a function of the number of smectic layers in the film is shown together
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Figure 5. θm2 as a function of N at T ) 28.7 °C, for ξ0 ) 8 Å and R ) 3 mm, and for (a) hs ) 0.3, r ) 0.305 and (b) hs ) 0.4, r ) 0.215. The experimental points are represented by pluses.
with the fit from the de Gennes model, for two sets of the fitting parameters. Since both fits are very good, we conclude that in the framework of the present model we cannot determine all the fitting parameters in a direct and an unambiguous way. However, this does not concern the bulk correlation length parallel to the film, ξ0, as its change from the value of 8 Å does worsen the fitting. Finally, in Figure 6 we present the theoretical predictions of variation of the contact angle with temperature for different film thicknesses. The theoretical curves in parts a and b of Figure 6 have been obtained for hs * 0 and gs ) 0. For comparison, the case of hs ≈ 0 and gs * 0 is shown in Figure 6c. We note that qualitatively the theory agrees with the experimental results presented in Figure 3, which means that the dependence of θm on temperature does not exhibit the unphysical behavior close to TNA predicted in ref 8. Thus, the inclusion of the quartic term in the free energy removes the main defect of the model. However, there is still a significant discrepancy between theory and experiment for the temperature at the maximum of the contact angle. The theoretical maxima shown in Figure 6 are much less sensitive to the number of layers than those found experimentally. Moreover, our results suggest that this disagreement cannot be removed simply by manipulation of the parameters of the model, in particular, by changing the surface coupling constants. The present theory seems also unable to explain the apparent discontinuity in the slope of θm2 at TNA, which is presumably because of its mean-field character. This behavior cannot also be explained by the fluctuation-induced interactions
Figure 6. θm2 vs temperature obtained from the Landau-de Gennes model, for N ) 3, 4, 5, 7, and 15. The values of the parameters in (a) and (b) are the same as those in parts a and b of Figure 5, respectively. In (c), ξ0 ) 8 Å, R ) 3 mm, hs ) 0.001, gs ) -0.6, and r ) 0.2. The vertical dashed line marks the bulk nematic-smectic transition temperature, TNA.
(Casimir effect), which for this model have been studied recently,22 because they are too small (smaller than the van der Waals attraction). We think that the full free energy of the film is only weakly modified by fluctuations, which are strongly suppressed in thin smectic films even at TNA, but these fluctuations should be large in the meniscus. Indeed, the maximum of θm2 above TNA (see Figure 6) is due to the characteristic behavior of the surface (22) Ziherl, P. Phys. Rev. E 2000, 61, 4636-4639.
Contact Angle between Smectic Film and Its Meniscus
tension, and we think that the change of slope of θm2(T) at TNA could be explained as the effect of fluctuations of the smectic order parameter in an infinitely thick film. This would mean that the smectic-air surface tension has a discontinuous temperature derivative at TNA. Finally, we note that the model also predicts that θm2 does not depend on the radius of curvature of the meniscus, in agreement with experimental observations.8 Indeed, we have varied R between 0.3 and 0.6 cm without apparent changes of θm2. Conclusions We have shown that the unphysical behavior of the contact angle close to the nematic-smectic bulk transition temperature obtained in ref 8 does not follow from the
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deficiency of the de Gennes model but from the linearization of the equations for the equilibrium smectic order parameter. The de Gennes model explains qualitatively, and to some degree quantitatively, the variation of the contact angle with temperature and with the number of smectic layers in the film. Further studies of the behavior of the smectic-air surface tension close to TNA, both theoretical and experimental, are required. Acknowledgment. This work has been supported by the KBN Grant 5P03B01121. R.H. acknowledges with appreciation the stipendship of the French Ministry of Education and hospitality of Ecole Normale Superieure in Lyon. LA011476E
