Langmuir 2006, 22, 363-368
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AFM Study of Defect-Induced Depressions of the Smectic-A/Air Interface V. Designolle,† S. Herminghaus, T. Pfohl, and Ch. Bahr* Max Planck Institute for Dynamics and Self-Organization, Bunsenstr. 10, D-37073 Go¨ttingen, Germany ReceiVed September 15, 2005. In Final Form: October 20, 2005 The smectic-A/air interface of liquid-crystal droplets with antagonistic boundary conditions is studied by atomic force microscopy (AFM). The droplets are prepared on coated silicon wafers on which a planar alignment is preferred in contrast to the homeotropic alignment at the air interface. As a result, focal conic defects appear in the smectic-A phase causing a characteristic pattern of depressions in the droplet surface. The dimensions of the defect-induced depressions are measured by AFM as a function of temperature for two different compounds possessing a smecticA-isotropic and a smectic-A-nematic transition. Whereas the results are independent of temperature in the smecticA-isotropic case, reflecting the first-order nature of the transition, a pronounced temperature dependence is observed for the second compound, where the depth of the defect-induced depressions decreases continuously with increasing temperature and vanishes at the second-order transition to the nematic phase. These observations can be qualitatively explained through the behavior of the layer compressional elastic constant at the smectic-A-nematic transition.
1. Introduction Smectic-A liquid crystals are orientationally ordered fluids possessing a one-dimensional density wave (or layer structure), with the density wavelength (or layer thickness) usually corresponding to the length of the constituent molecules. The average direction of the long axis of the rodlike molecules, specified by the so-called director, is oriented parallel to the wave vector (or layer normal). With increasing temperature, the smectic-A phase transforms in many compounds into the nematic liquid-crystal phase, in which the orientational order is retained while the density modulation or layer structure has vanished. Another possible phase sequence is a direct transition from the smectic-A phase to the isotropic liquid phase. In the nematic phase, the director field is easily distorted by elastic deformations which can be described as combinations of three basic deformation types, namely splay, twist, and bend. The elastic constants associated with these deformations are of the order of 10-6 dyn. In the smectic-A phase, however, the elastic constant B describing a compression or dilation of the layers is of the order of 108 dyn/cm2; that is, the smectic layers are practically incompressible in the energy range sufficient to deform the director field. Thus, in the smectic-A phase, only those deformations of the director field can occur which leave the layer thickness unchanged: only the splay deformation is compatible with a structure of incompressible equidistant smectic layers. This results in the formation of the common focal conic defect structure in the smectic-A phase which consists of an arrangement of curved equidistant layers around two defect lines which adopt the shape of an ellipse and a hyperbola.1 In 1990, Fournier et al.2 described an experimental arrangement, in which focal conic defects appear in a thin smectic-A film floating on the isotropic liquid phase of the same material (the two phases coexisted because of a temperature gradient along the film normal). The two different interfaces (smectic-A/air and * To whom correspondence should be addressed. E-mail: christian.bahr@ ds.mpg.de. † Present address: Ecole Nationale Supe ´ rieure des Mines de Paris, 60 bd Saint-Michel, 75272 Paris Cedex 06, France. (1) Friedel, G. Ann. Phys. (Paris) 1922, 18, 273. (2) Fournier, J. B.; Dozov, I.; Durand, G. Phys. ReV. A 1990, 41, 2252.
Figure 1. Schematic representation of a single toroidal focal conic defect. At the smectic-A/air interface all molecules are aligned perpendicular to the interface whereas at the smectic-A/substrate or smectic-A/isotropic interface circular areas with diameter 2r exist, in which the molecules are oriented parallel to the interface. A defect line proceeds from the center of these areas to the air interface where a defect-induced depression of amount h exists.
smectic-A/isotropic) impose antagonistic alignment conditions: at the air interface the liquid-crystal molecules are always aligned perpendicular to the interface (homeotropic alignment), whereas they try to align parallel to the second interface to the isotropic phase (planar alignment). In very thin films, the molecules show also at the smectic-A/isotropic interface a homeotropic alignment, but if the thickness of the smectic-A film exceeds a certain value, circular areas appear at the smectic-A/isotropic interface, in which the molecules are aligned planar on the interface. Each circular area corresponds to the base of a focal conic defect with curved smectic layers mediating between the two different surface orientations of the molecules. In the special film structure described above, the focal conic defects adopt a toroidal shape with the ellipse becoming a circle that borders the area of planar alignment. The second defect line, in general a hyperbola, is in this case a straight line which proceeds from the centers of the circular areas on the smectic-A/isotropic interface to the air interface on the opposite side of the smectic-A film. Each of these cylindrical focal conic defects is accompanied by a cone-like depression of the smectic-A/air surface of the film, the center of the depression coinciding with the endpoint of the straight defect line (cf. Figure 1).
10.1021/la0525224 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/19/2005
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A similar experimental arrangement can be realized with a smectic-A film prepared on a silicon wafer. If the silicon substrate is covered with a suitable alignment layer inducing a planar alignment of the liquid-crystal molecules, the same geometry with antagonistic alignment conditions results. Recently, toroidal focal conic defects of smectic-A liquid crystals on silicon substrates and the confining effects of microchannels on the defect pattern has been studied;3 in this study, it was also shown that the depth of the defect-induced surface depressions can be measured by atomic force microscopy (AFM). By means of the special sample geometry used in ref 2, Fournier et al. could vary the thickness of the smectic-A film floating on the isotropic film in the range between a few µm and ≈100 µm by adjusting the temperature within a small interval of ≈1 °C at the smectic-A-isotropic transition. This technique enabled a detailed analysis of the relation between film thickness and defect diameter at a temperature close to the transition. It is not known, however, how the focal conic defects behave when the temperature is varied over an interval of 10-20 °C within the smectic-A phase and when a transition other than smectic-A-isotropic is approached. We present here a comprehensive AFM study of the temperature dependence of the depth of the defect-induced surface depressions for two compounds possessing a smectic-A-nematic and a smectic-A-isotropic transition. Whereas in the latter case the depression depth appears to be largely temperature independent, a pronounced temperature dependence of the depression depth is observed in whole smectic-A phase range of the compound with the second-order smectic-A-nematic transition. We discuss our results in relation with continuum theory of the smectic-A phase, especially with regard to the layer compressional elastic constant B. 2. Experimental Procedures and Optical Observations The compounds under investigation are two common liquid crystals: 4-n-octyl-4′-cyanobiphenyl (8CB) and 4-n-decyl-4′-cyanobiphenyl (10CB), both obtained from Synthon Chemicals (Wolfen, Germany) and used without further purification; 8CB undergoes a second-order smectic-A-nematic transition at 33 °C, whereas 10CB shows a first-order smectic-A-isotropic transition at 50 °C. The surface of liquid materials can be imaged by tapping-mode scanning force microscopy provided that certain conditions are observed. In tapping-mode AFM, the cantilever oscillates with a frequency close to its resonance value usually in the high kilohertz range. The damping of the oscillation amplitude due to the interaction with the sample surface serves as the feedback signal. For the study of liquid surfaces, it is essential to keep the amplitude damping as low as possible, the working amplitude should not be smaller than 80% of the undamped value; details are given in refs 4 and 5. Measurements are done using a Nanoscope IIIa equipped with a Nanoscope heater and heater controller which enables us to control the temperature of the sample with a resolution of ≈0.1 K. Typical values of scan area and scan frequency were 25 × 25 µm2 and 0.1 Hz/line, respectively. Our samples are flat droplets of liquid-crystal material on coated silicon substrates. In contrast to the technique used in ref 2, where a temperature gradient divided the sample into a smectic-A film floating on an isotropic liquid film, we avoid (within our experimental possibilities) temperature gradients. Thus, in our experiment, the whole sample is in the smectic-A phase, and the thickness of the smectic-A film or droplet does not change with temperature. Whereas in ref 2 the thickness of the smectic-A film could be changed by small temperature adjustments, we can obtain results for different (3) Choi, M. C.; Pfohl, T.; Wen, Z.; Li, Y.; Kim, M. W.; Israelachvili, J. N.; Safinya, C. R. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 17340. (4) Herminghaus, S.; Fery, A.; Reim, D. Ultramicroscopy 1997, 69, 211. (5) Pompe, T.; Fery, A.; Herminghaus, S. Langmuir 1998, 14, 2585.
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Figure 2. Optical micrographs showing circular focal conic defects in smectic-A droplets on PEI-coated silicon wafers; the defects are obtained by slowly cooling through the nematic to smectic-A transition of 8CB (left) or the isotropic to smectic-A transition of 10CB (right). The diameter of the largest defects is of the order of 20 µm. smectic-A thicknesses only by measurements on droplets with different thicknesses. On the other hand, we can vary the temperature over the whole smectic-A phase range, whereas in ref 2, only results for temperatures close to the transition to the isotropic phase were obtained. To prepare a substrate imposing a planar alignment on the liquid crystal, small pieces (≈4 mm2) of silicon wafers are coated with a polyethyleneimine (PEI) layer as described in ref 6. On these substrates, a tiny amount of the corresponding liquid-crystal compound is deposited. Because of the different phase sequences and temperature ranges of the two liquid-crystal compounds, different procedures for the controlled generation of focal conic defects for the AFM measurements are required. In the case of 8CB, which is at room temperature in the smectic-A phase, a tiny smectic droplet is deposited on the silicon substrate at room temperature. The substrate with the droplet (which shows after deposition an amorphous appearance) is first heated into the nematic phase, whereas it is observed in an optical microscope. In the nematic phase, the droplets equilibrate within a few minutes to a quasistatic shape (the nematic phase of n CB compounds is known to wet silicon substrates completely, but the corresponding time interval is of the order of a few days7). The sample is then cooled back to the smectic-A phase at a modest rate (≈1 °C/min). At the phase transition, an irregular pattern of circular defects with different sizes appears (Figure 2 left). On further cooling, the appearance of these samples (i.e., the location and diameter of the defects and the shape of the droplets) does not change, and the samples can be stored at room temperature until they are investigated by AFM. Contrary to 8CB, 10CB is a crystalline solid at room temperature. Focal conic defects have thus to be generated while the sample is already positioned in the AFM and the temperature must be kept above the melting point. After the deposition of a tiny 10CB crystal at room temperature, the substrate is placed into the AFM sample holder, and the temperature is raised for a few minutes above the smectic-A-isotropic transition while the solid 10CB crystal transforms into an isotropic liquid droplet. The temperature is then decreased at a slow rate (≈0.5 °C/min) until the coexistence region of the isotropic to smectic-A transition is reached and the first focal conics are observed. In contrast to the case of 8CB, the position of these first appearing focal conics is not fixed on the subrate, rather the defects are floating on a probably still isotropic liquid layer. When the temperature is slowly further decreased (by ≈0.2 °C), more focal conics appear and form a hexagonal lattice (which still is able to float as a whole) of uniformly sized defects (Figure 2 right); different-sized defects either drift to the edge of the droplet and disappear or join the lattice, with their dimensions converging in most cases toward the common size of the others (sometimes different sized focal conics are integrated causing defects in the hexagonal lattice, cf. upper left edge of Figure 2 right). These (6) Pfohl, T.; Kim, J. H.; Yasa, M.; Miller, H. P.; Wong, G. C. L.; Bringezu, F.; Wen, Z.; Wilson, L.; Kim, M. W.; Li, Y.; Safinya, C. R. Langmuir 2001, 17, 5343.
Depressions of the Smectic-A/Air Interface
Figure 3. AFM topographical image of defect-induced surface depressions of a 10CB droplet on a PEI-coated silicon wafer at 35 °C. The lateral dimension of the shown area is 25 µm. The depth of the depressions is of the order of 250 nm. Note that the surface between the depressions is not flat but curved as can be seen at the front edge of the image.
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Figure 5. Temperature dependence of the depth h of the defectinduced surface depression for two different defects in droplets of 10CB on PEI-coated silicon substrates; TAI is the smectic-A-isotropic transition temperature. The radii of the two defects are r ) 8.8 µm (for the deeper depression) and r ) 6.1 µm; as h, r is essentially independent of T.
Figure 4. Cross section taken from the AFM image shown in Figure 3. The symmetry of the depressions confirms the toroidal shape of the focal conic defects. observations are very similar to those described by Fournier et al.,2 who also studied 10CB. The spatially separated coexistence of the smectic-A phase and the isotropic liquid phase might be due to a temperature gradient or promoted by the free surface which induces smectic layering already above the isotropic-smectic-A transition.8 When the temperature is further decreased by 0.2 °C, the position of the focal conics becomes fixed on the substrate. When the isotropic-smectic-A transition of 10CB is crossed with a fast cooling rate, the same irregular pattern of different sized defects as in the case of 8CB is obtained. The rate, with which the transition to the smectic-A phase is crossed, influences the homogeneity of the defect size distribution in 10CB (in 8CB, there is only little influence of the cooling rate), but once the defects are formed, their behavior is independent of the rate used for their production. We have studied for each compound about 20 individual defects (partly in different droplets, partly in different regions of a given droplet) and observed in all cases the same behavior (for a given compound) which is described in the following section. The equilibration of the droplets in the isotropic or nematic phase leads to a smooth droplet surface which is in the droplet center nearly parallel to the substrate surface. Thus, on the lateral scale of the defect diameters ( 3/2r0. Applying eq 3 to eq 5 gives
h2 )
3∆γ r(r - r0) 2γ
Figure 7. AFM topographical image of defect-induced surface depressions of an 8CB droplet on a PEI-coated silicon wafer at 23 °C. The lateral dimension of the shown area is 25 µm. The depth of the central depression amounts to 350 nm, and that of the depression at the left front edge 300 nm; the depths of the two smaller depressions in the rear are of the order of 100 nm. This surface image was obtained for a sample showing in an optical microscope an appearance as in Figure 2 (left); that is, there is a coexistence of defects with large and small diameters.
(6)
Except for the vicinity of r0, eq 6 decribes a nearly linear relation between h and r and we can easily fit our experimental data using eq 6. The solid line in Figure 6 is obtained with r0 ) 4.1 µm and ∆γ/γ ) 3.3 × 10-3, these values are similar to those already obtained in ref 2. 3.2. 8CB and the Smectic-A-Nematic Transition. Figure 7 shows an AFM image of the surface of an 8CB droplet at room temperature. As is already observed in the optical microscope (cf. Figure 2), there is no lattice of uniform sized defects but rather an irregular pattern of different sized focal conics. In contrast to the case of 10CB, there is no floating state, the defects are fixed on the substrate already when they appear at the nematic-smectic-A transition. The broad size distribution at given sample thickness H might be due to inhomogeneities in the coated
Figure 8. Temperature dependence of the depth h of the defectinduced surface depression for two different defects in droplets of 8CB on PEI-coated silicon substrates; TAN is the smectic-A-nematic transition temperature. The solid lines correspond to power law fits with an exponent of 0.4, only the data close to the transition (T TAN > - 1.5 °C) were taken into account for the fits; it is obvious that the data far from TAN are not adequately described by these fits.
silicon substrate: the alignment properties, and thus the value of ∆γ (cf. eq 4), may vary on the substrate, and pinning effects may prevent the formation of equilibrium sized defects with a lateral dimension according to eq 5. The second prominent difference to 10CB is a pronounced temperature dependence of the depth h of the defect-induced surface depressions. Figure 8 shows the temperature dependence of h for two different focal conics of 8CB droplets. With increasing temperature, the value of h decreases continuously to zero at the transition to the nematic phase. Close to the transition, the data may be described by a power law (cf. Figure 8).
h ∝ (TAN - T)φ
(7)
For the exponent φ, we obtain ≈0.4, but the limited resolution of our heater controller (0.1 K) prevents us from a more accurate measurement. In contrast to the depth h, the radius r of a given defect is essentially independent of temperature. The secondorder nature of the smectic-A-nematic transition of 8CB suggests that the observed behavior of h is due to some material parameter decreasing continuously to zero at the transition. An appropriate candidate is the layer compressional elastic constant B, which has been measured by light scattering9,10 and second sound
Depressions of the Smectic-A/Air Interface
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∆x )
∫3λH-3λ(x) dx )
(
20H -
)
2λ2 H - 3λ λ λ2 log - + (10) H 3λ 3 H - 3λ
The value of must remain smaller than 1, including the vicinity of the defect cores: this imposes the condition 0 < λ/H.15 Assuming 0 ≈ λ/H and keeping only the dominant termes in eq 10 we get the approximation
∆x ) λ log Figure 9. Small dilation in a focal conic. As represented for the upper layers, the structure is slightly distorted when compared to a defect with completely equidistant layers which would cause a depression depth hgeom: the dashed lines show the dilated layers. ∆x is the variation of the global thickness along the line OM. This will change the observed depth of the defects from hgeom to the actually measured value h.
techniques.11-13 Most experiments yield at the transition to the nematic phase a decrease of B according to a power law with a critical exponent in the range of 0.3-0.5. Some second sound measurements indicate very close to the transition a deviation from the power law behavior and a finite value of B at the transition but this might be a dynamical effect.14 Far from the smectic-A-nematic transition, the value of B is of the order of 108 dyn/cm2, and in the description of macroscopic smectic defects, it is assumed that the smectic layer thickness is constant. If B decreases continuously to zero when the transition is approached, we can expect that along the central defect line a dilation of the smectic layers, and thus a decrease of the depth of the surface depression, occurs in order to diminish the smecticA/air interface area. Fournier15 has considered the dilation of the smectic layers in the vicinity of the singular lines (ellipse and hyperbola) of a bulk focal conic defect theoretically. Defining the local dilation as
a - a0 ),1 a0
(8)
with a0 designating the equilibrium layer thickness and a the real thickness, a general expression for the value of as a function of the distance to the defect lines is derived;15 in our case, where ellipse and hyperbola degenerate to a circle and a straight line, one gets
x ln H2 H 2 H-x -λ - λ2 (x) ) 0 x(H - x) x(H - x) x(H - x)2
(9)
where λ ) [K/B]1/2 with K being the splay elastic constant and B the layer compressional elastic constant; x is the distance from the circle along the line OM in Figure 9, and 0 is a constant. Equation 9 holds in a range outside of the immediate vicinity of the defect lines, i.e., 3λ < x < H - 3λ.15 To obtain the total change ∆x of the sample thickness, we integrate (x) (9) Davidov, D.; Safinya, C. R.; Kaplan, M.; Dana, S. S.; Schaetzing, R.; Birgeneau, R. J.; Litster, J. D. Phys. ReV. B 1979, 19, 1657. (10) Lewis, M. E.; Khan, I.; Vithana, H.; Baldwin, A.; Johnson, D. L.; Neubert, M. E. Phys. ReV. A 1988, 38, 3702. (11) Fisch, M. R.; Pershan, P. S.; Sorensen, L. B. Phys. ReV. A 1984, 29, 2741. (12) Benzekri, M.; Marcerou, J. P.; Nguyen, H. T.; Rouillon, J. C. Phys. ReV. B 1990, 41, 9032. (13) Beaubois, F.; Claverie, T.; Marcerou, J. P.; Rouillon, J. C.; Nguyen, H. T. Phys. ReV. E 1997, 56, 5566. (14) Rogez, D.; Collin, D.; Martinoty, P. Eur. Phys. J. E 2004, 14, 43. (15) Fournier, J. B. Phys. ReV. E 1994, 50, 2868.
H λ
(11)
To obtain finally the change of the depth of the surface depression, we should project ∆x onto the substrate normal. However, considering the previous approximations and since r , H, we can neglect this correction here. Thus, we estimate the depth h of the defect-induced surface depressions as
h ) hgeom - λ log
H λ
(12)
where hgeom being the value which corresponds to the case of completely equidistant layers. With typical K and B values far from the transition (K ) 10-6 dyn, B ) 108 dyn/cm2) and with H ) 100 µm, we obtain ∆h ) hgeom - h ≈ 10 nm. In first-order smectic-A-isotropic compounds such as 10CB, λ does not depend on temperature and the above correction of h corresponds essentially to a small constant (∆h depends only weakly on H) offset which does not affect our analysis for 10CB. Near a second-order smectic-A-nematic transition, however, λ becomes large, since B vanishes at the transition (whereas K does not show an anomaly). In the following, we will explore to what extent our results for 8CB can be described by eq 12. It is obvious that eq 12 cannot be correct at the temperature TAN of the second-order transition to the nematic phase since the divergence of λ ) [K/B]1/2 would lead to an arbitrarily large rise of the surface instead of a depression. However, according to eq 12 there is a temperature T0 < TAN where h becomes zero and the defect-induced surface depressions vanish. Using K ) 10-6 dyn and the temperature dependence of B measured for 8CB in12
B ) B1(TAN - T)φ
(13)
with φ ) 0.4 and B1 ) 7 × 107 dyn/cm2, we obtain TAN - T0 ≈ 10-5 K; that is, eq 12 predicts the surface depressions to vanish at a temperature which is in our experiments not distinguishable from the real transition temperature. We consider now the temperature dependence of h close to T0. Designating the values of λ and B at T0 with λ0 and B0, we can write first-order approximations λ ≈ λ0(1 - u) and B ≈ B0(1 + 2u) where u , 1. If we approximate eq 13 as B ) B0 + B1(T0 - T)φ, we can obtain from eq 12 the following approximation for h near T0
h≈
λ0 H log B1(T0 - T)φ 2B0 λ0
(14)
i.e., close to T0 the depth h of the defect-induced depressions should show the same power law behavior as B, which is indeed observed in our experiments (cf. Figure 8). However, far from the transition, eq 12 predicts an almost constant value of h, whereas we observe a clear, almost linear increase of h with decreasing temperature even 10 K below the transition. Obviously, eq 12 provides only a partial description of our experimental observations.
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4. Conclusion We have presented an AFM study of smectic-A droplets on coated silicon substrates. Antagonistic alignment conditions at the smectic-A/substrate and the smectic-A/air interface cause the generation of focal conic defects which lead to characteristic depressions of the smectic-A/air interface. The dimensions of the defect-induced depressions are determined by AFM for the two compounds 10CB and 8CB, which possess a first-order smectic-A-isotropic or a second-order smectic-A-nematic transition, respectively. The results obtained for the first-order compound (10CB) are independent of temperature in the whole smectic-A range, and the depth and the lateral extension of the surface depressions are in agreement with a model defect structure based on curved equidistant smectic layers. For the second-order compound (8CB), a pronounced temperature dependence of the
Designolle et al.
depression depth h is observed; h decreases continuously with increasing temperature and vanishes at the transition to the nematic phase. This observation can be qualitatively explained by taking the behavior of the layer compressional elastic constant B into account, but some aspects, like the temperature dependence of h far from the transition, are not in agreement with the simple model presented here. The differences which occur during the formation of the defects while cooling through the isotropicsmectic-A or nematic-smectic-A transition, namely the floating of the defects and formation of a regular lattice in 10CB and the formation of an irregular pattern of different-sized defects in 8CB, are also subjects of future research. LA0525224
