8822
J. Phys. Chem. B 2000, 104, 8822-8829
Thin Films of Liquid Crystals Confined between Crystalline Surfaces Vladimir Kitaev and Eugenia Kumacheva* Department of Chemistry, UniVersity of Toronto, 80 St. George Street, Toronto, Ontario, Canada, M5S 3H6 ReceiVed: August 24, 1999; In Final Form: June 23, 2000
This paper reports on a study of thermotropic liquid crystals (LC’s) confined between two crystalline surfaces studied using the surface force balance technique. A quantitative approach based on measurements of the wavelength splitting (WS) of fringes of equal chromatic order was proposed, which allows the study of orientational transitions in LC films. Reasonable agreement was found between predicted and experimentally measured WS in moderately thick LC films. Optical properties of the LC’s were studied experimentally for different orientations of the confining surfaces as a function of the film thickness, humidity, and duration of the experiment. It was shown that upon progressive confinement the medium-range forces in the layer noticeably increase when the twisting angle between confining surfaces increases. It was demonstrated that increase in humidity diminishes anchoring of the LC’s to the surface.
Introduction Many important applications of liquid crystals (LC’s) are governed by their properties in thin films.1 The thickness of such films may exceed hundreds of nanometers or even microns, since the orientational and positional order induced in a LC by a substrate can propagate through many molecular layers. The insertion of LC’s into narrow capillaries or the progressive confinement of LC layers between two flat solid surfaces further increases the range of film thicknesses in which surface effects should be appreciated. Extensive studies of anchoring and alignment of LC’s by various substrates have been carried out, leading to the acquisition of a comprehensive body of empirical knowledge to date.1-9 Still, no clear predictions can be made on a solid theoretical background for orientations of molecules of a particular LC at the interface with a given solid surface. To characterize the alignment of the LC on a particular substrate, its surface has to be clean, smooth, and well characterized. When the LC are confined between two crystalline surfaces, their alignment should be controlled, since the orientation of LC’s on crystalline surfaces is determined by the surface symmetry of a crystal cleavage plane. The surface force balance (SFB) technique10 represents a very beneficial method for studies of thin layers of LC’s. In this method, the sample under study is placed between the two atomically smooth mica surfaces of the same thickness, and normal and lateral forces in the layer are measured as a function of various system parameters, e.g., the distance between the surfaces, the composition of the sample, the temperature, or the humidity. In addition, the SFB method allows one to measure the refractive index of the medium confined between the surfaces. The possibility of following the change in the birefringence of the LC represents a very useful way to study orientational transitions in the film. The first experiments with thermotropic LC’s using the SFB method were carried out by Horn et al.11 Recently, several other groups have reported on experimental12-15,16 and theoretical17,18 studies of the confined nematic and smectic thermotropic LC’s in the configuration of the SFB. Most of experiments involved measurements of normal and shear forces in 4-cyano-4′alkylbiphenyls in which the alkyl group contained five, six, or eight carbon atoms in it, i.e., 5CB, 6CB, or 8CB, respectively.
In several studies,11,16 ordinary and extraordinary refractive indexes of LC’s were examined as a function of the thickness of the film. It has been confirmed11,12,14-16 that cyanobiphenyls confined between bare mica surfaces adopt a planar orientation for which birefringence of LC’s is observed. By contrast, when the mica surface is modified with surfactants, LC’s adopt a homeotropic orientation.11,15 In this configuration, a long axis of the LC molecules is aligned parallel to the direction of light propagation and the sample performs as an isotropic nonbirefringent material. The properties of 6CB confined between the mica plates which were either aligned, or twisted by 90° were studied by Janik et al.16 For parallel orientation of the mica sheets, the authors observed a planar birefringent orientation of the LC, whereas when the crystallographic axes of the mica plates formed a right angle, a planar-twisted nonbirefringent molecular orientation was found. The interpretation of experimental results is complicated by the lack of a theoretical basis for the analysis of optical properties of LC’s in the geometry of a SFB. The only attempt to date has been made by Rabinovitz,18 who obtained numerical solutions for the resonance and polarization states for an anisotropic medium placed in the Fabry-Perot interferometer. However, that study made no comparison between the predicted and measured values of birefringence as a function of the thickness of the LC layer or of the mutual orientation of the LC and mica. In this work, we analyze the optical properties of moderately thick layers of LC’s confined between two symmetric crystalline mica surfaces. We propose a quantitative approach to study orientational transitions of LC’s in the configuration of the SFB. We examine the variation in the wavelength splitting in fringes of equal chromatic order as a function of the thickness of the LC layer for different mutual orientations of the mica surfaces, the presence of water vapors, and the duration of the experiment. The experimental and predicted results are in reasonable agreement. Along with giving a better understanding of the properties of low-molecular LC’s, our approach provides a methodology for the study of more complicated LC structures produced by high molecular weight molecules in constrained geometries.
10.1021/jp992996+ CCC: $19.00 © 2000 American Chemical Society Published on Web 08/25/2000
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J. Phys. Chem. B, Vol. 104, No. 37, 2000 8823
Figure 2. Schematics of fringes of equal chromatic order (FECO) observed for (a) parallel and (b) perpendicular orientation of the crystallographic axes of the mica sheets.
Figure 1. The scheme of the surface force balance setup.
Background The SFB technique is described in detail elsewhere.10 Here we outline the essentials of the method only. In the SFB method, two mica sheets of equal thickness of ca. 2-3 µm are silvered on their backsides and glued onto cylindrical lenses mounted in a cross-cylinder configuration (Figure 1). The silvered mica sheets and the intervening medium form a Fabry-Perot interferometer.19 When white light passes through the interferometer, only constructively interfering wavelengths are transmitted. As emerging light is passed through a monochromator, these wavelengths split up and appear as an array of fringes of equal chromatic order (FECO). The distance D between the plates, i.e., the thickness of the intervening layer, is measured using an interferometric technique from the relative position of the FECO using an equation of a three-layer interferometer10
tan(2πnD/λDk ) ) 2µsin (1 + µ2)
[
[
1 - λ0k /λDk
0 1 - λ0k /λk-1
1 - λ0k /λDk π 0 1 - λ0k /λk-1
]
π
]
( (µ2 - 1)
µ)
nm (1) n
0 where k is the order of the interference fringe; λ0k and λk-1 are the positions of the kth and (k - 1)th fringe measured when the surfaces are in contact, λDk is the position of the kth fringe when the surfaces are separated by a distance D, nm is the refractive index of mica at λDk , and n is the refractive index of the medium between the mica plates at λDk ; “+” or “-” refers to odd and even order fringes, respectively. Mica is an anisotropic material with the properties of a negative biaxial crystal.20 In the SFB, the direction of light propagation almost coincides with the axis R of the smallest refractive index of mica (Figure 2); therefore, mica exhibits properties of a uniaxial crystal with no,m ) nγ and ne,m ) nβ, where no,m and ne,m are the refractive indexes for ordinary and extraordinary rays, respectively. When white light passes through the aligned mica plates, an array of closely spaced sets of orthogonally plane-polarized FECO is observed, as shown in Figure 2a. The fast axis in mica coincides with the extraordinary ray; therefore no,m > ne,m, and λγ ) λo,m > λe,m ) λβ, where λo,m and λe,m are the positions of the resonance wavelengths corresponding to ordinary and extraordinary refractive indexes, respectively. When the crystallographic axes of the two mica sheets are perpendicular to each other, the ordinary
and extraordinary rays in each sheet interchange, and the positions of FECO are then determined by the average positions of the β and γ wavelengths obtained at parallel mica alignment. The resulting FECO appear as singlets, as shown in Figure 2b. When the twisting angle, θ, between the crystallographic axes of mica plates varies from 0 to 90°, the wavelength splitting (WS) of the FECO defined as ∆λ ) λo,m - λe,m gradually changes from the maximum value to zero, respectively; thus the value of θ can be found by measuring the WS. The equation for the WS for mica sheets brought into contact is based on the equation of a single-layer interferometer10,19 as
4dm[no,m(λ) - ne,m(λ)] ) k(λo,m - λe,m)
(2)
where dm is the thickness of a single mica sheet. The wavelength dependence of the refractive indexes of mica can be found from a Sellmeier equation20
n2(λ) ) 1 +
Aλ2 (λ - B) 2
(3)
by measuring no,m and ne,m at selected wavelengths and fitting experimental data to eq 3 to find parameters A and B. In this work, we used A ) 1.5154, B ) 9255 for no,m; and A ) 1.5044, B ) 8478 for ne,m.21,22 For a particular experiment, dm ) const and k ) const. The WS for mica is determined by the difference in the dispersion of no,m and ne,m. For wavelengths varying from 4500 to 5500 Å, the difference no,m - ne,m calculated from eq 3 changes from 0.0055 to 0.00484; i.e., it undergoes a decrease of ca. 14%. Liquid crystals such as alkylcyanobiphenyls have properties of positive crystals with ne,LC > no,LC. An equation similar to eq 2 can be written for a hypothetical single-layer interferometer formed by a LC layer only:
2dLC [ne,LC(λ) - no,LC(λ)] ) k(λe,LC - λo,LC)
(4)
In the configuration of the SFB, a layer of the LC and crystalline mica plates form a complex anisotropic medium. A general approach to calculation of birefringence in a complex medium consisting of two layers of anisotropic uniaxial materials is developed in Appendix A. This approach can be utilized in studies of LC’s in the configuration of a Fabry-Perot interferometer. In the SFB experiments, after introducing a LC between the mica surfaces the optical axes of the system change, one has to account for the change in effective refractive indices of the mica and the LC, as well as for the change in λ0k . The algorithm for finding D from eq 1 is given in Appendix B. In this work, we simplified eq 1 for moderately thick LC films (D > λ/2 ≈ 2000 Å) assuming that (a) the effect of a phase change on reflection of light at the silver surfaces is small
8824 J. Phys. Chem. B, Vol. 104, No. 37, 2000
Figure 3. Variation of (λγ - λβ)/λβ vs the distance between the surfaces D calculated for 5CB using eq 5. dm ) 3.00 µm, k ) 47, θ ) 0. The orientation of the LC with respect to the mica: (-O-) 0°, (-0-) 30°, (-4-) 45°, (-3-) 60°, (-]-) 90°.
and (b) the intervening medium is uniform. In this case, the wavelength shift due to multiple reflection can be neglected and the total WS can be calculated using eq A-9 derived for the path difference between ordinary and extraordinary rays in a complex anisotropic media. Several possible arrangements of the mica and LC in the SFB can be considered. (i) Parallel Alignment of the Mica Sheets; the Fast Axis of the LC Forms Angle φ with the Fast Axis of Mica, i.e., θ ) 0°, 0° e φ e 90°. In this configuration, two mica plates are treated as a single anisotropic phase with thickness 2dm and R ) φ,. Since the mica and the LC are positive and negative crystals, respectively, eq A-9 is valid when φ is the angle between the fast axes of the materials. The WS can be found as
Kitaev and Kumacheva
Figure 4. Variation of (λγ - λβ)/λβ vs D calculated for 5CB using eqs 5 and 11 for various twisting angles θ between the mica plates. dm ) 3.00 µm, k ) 47, φ ) 30°. The twisting angle: (-O-) 0°, (-0-) 30°, (-4-) 45°, (-3-) 60°, (-]-) 90°. The value of φ ) 30° is assigned in retrospect, based on the results of polarization microscopy experiments and ref 8.
2Rmcos(2φ - arctan{sin(2φ)[cos(2φ) + RLC/2Rm]-1}) + RLCcos(arctan{sin(2φ)[cos(2φ) + RLC/2Rm]-1}) ) k(λγ - λβ)θ)0,0eφe90/2 (5) where Rm is dm(no,m - ne, m) and RLC is dLC(ne,LC - no,LC). The variation of (λγ - λβ)/λβ vs D ()dLC) calculated from eq 5 for different values of φ is given in Figure 3. When φ ) 0°, i.e., when the fast axis of the LC is parallel to the fast axis of mica, eq 5 simplifies to
2dm[no,m(λ) - ne,m(λ)] + dLC[ne.LC(λ) - no,LC(λ)] ) [k(λγ - λβ)θ)0,φ)0]/2 (6) and the maximum WS is achieved, as shown by the top curve in Figure 3. When φ ) 90°, i.e., when the fast axis of the LC is parallel to the slow axis of mica, eq 5 becomes
Figure 5. Variation of (λγ - λβ) vs θ for the mica surfaces brought into contact measured experimentally (empty symbols) and calculated from eq 8 (filled symbols). (0,9), (O,b), and (4,3,2) are obtained for three different series of experiments in which mica sheets with different dm were used. Experimental points obtained using a micrometerpositioned grid (0,O,3); using manually operated spectrometer (4). The spectrometer grating was 600 grooves/mm.
(ii) The Crystallographic Axes of the Mica Plates Are Twisted with Respect to Each Other, the Fast Axis of the LC Forms Angle φ with the Fast Axis of Mica, i.e., θ° > 0, 0° e φ e 90°. First, consider two symmetric mica plates brought into contact and twisted by θ () R), i.e., R1/R2 ) 1 (see eq A-9). Using trigonometric relations sin 2R ) 2 sin R cos R and cos 2R ) cos2 R - sin2 R, we get tan 2γ ) tan R and 2γ ) R. Then from eqs A-9 and 5 we calculate
-2dm[no,m(λ) - ne,m(λ)] + dLC[ne,LC (λ) - no,LC(λ)] ) [k(λγ - λβ)θ)0,φ)90]/2 (7)
k∆λ(θ) ) 4dm(no,m - ne,m)cos θ or ∆λ(θ) ) ∆λmaxcos θ (8)
In this situation, the value of WS will depend on the thickness of the mica and the LC layer. In thin layers, the WS will be dominated by the birefringence of the mica, while in thick films the birefringence of the LC will determine the wavelength splitting.23 When the mica plates are separated from contact and the LC flows into the gap between the surfaces, the value of WS will diminish until it reaches zero and then it will start to increase again. For dLC > 2000 Å the change in (λγ - λβ)/λβ for φ ) 90° is shown in Figure 3, bottom curve.
where ∆λmax is the value of WS at θ ) 0° in eq 2. The WS calculated from eq 8 is in agreement with experimental results obtained by McGuiggan et al.24 and those in the current work (see Figure 5).25 When the LC is introduced into the gap between the mica surfaces, the situation is quite different from that considered in (i). Following twist of the mica sheets the molecules of the LC will undergo gradual rotation in the lateral plane from the bottom mica surface (θi ) 0°) to the top surface (θi ) θ) where θi is
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J. Phys. Chem. B, Vol. 104, No. 37, 2000 8825
the varying angle of the director of the LC. The structure of the LC in this configuration is similar to that of a twisted nematic phase1,3 and can be considered as optically equivalent to a large number of slices with the director constant in each slice but undergoing slight rotation from a slice to a slice. The wavelength splitting ∆λLC(θ) in the LC layer can then be obtained by summing up the wavelength separations, ∆λi, in individual slices; i.e.,
∆λLC(θ) ) Σ∆λi ) Σdi(ne,LC - no,LC)
(9)
The value of di is given by dLC/m, where m is the number of slices of the LC layer. The angle of rotation of the director between two adjacent slices is δθi ) θ/m. Then, di ) (dLC/θ) δθi. In the limit of infinitely thin slices, retardation in the twisted LC layer is obtained using eq 8 by integrating the wavelength splitting ∆λi in each slice from θi ) 0 to θi ) θ as
∫θ (dLC/θ)(ne,LC - no,LC)cos θi dθi )
0
dLC(ne,LC - no,LC)sin θ/θ (10)
The total wavelength splitting for the LC placed in a FabryPerot interferometer for D > 2000 Å is then determined as
2Rmcos θcos(2φ - arctan{sin(2φ)[cos(2φ) + (RLCsin θ/θ)/ (2Rmcos θ)]-1}) + RLC[sin(θ)/θ]cos(arctan{sin(2φ) × [cos(2φ) + (RLCsin θ/θ)/(2Rmcos θ)]-1}) ) [k(λγ - λβ)0
