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ANOMALOUS STIFFNESS AND TILT ANGLE IN NEMATICS FROM NONUNIFORM ATTACHMENT
ANGLE
D. Berreman
To cite this version:
D. Berreman. ANOMALOUS STIFFNESS AND TILT ANGLE IN NEMATICS FROM NONUNI- FORM ATTACHMENT ANGLE. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-58-C3-61.
�10.1051/jphyscol:1979313�. �jpa-00218709�
ANOMALOUS STIFFNESS AND TILT ANGLE
IN NEMATICS FROM NONUNIFORM ATTACHMENT ANGLE
D. W. BERREMAN
Bell Laboratories, Murray Hill, New Jersey 07974, U.S.A.
Abstract. — If the splay elastic constant kxi and the band elastic constant k33 in a nematic liquid crystal are unlike, and if the directors immediately adjacent to the surface are rigidly attached in nonuniform directions then the apparent bend and splay constants very near the surface will be altered by an averaging effect. If liquid crystals such as MBB A, having kxi < k33, are rigidly attached to the surface in irregular alignment that is parallel to the surface on average and coplanar they will behave under stress as if there were a thin inflexible layer above the surface. If their overall alignment is homeotropic, they will bend as if the surface attachment were somewhat weak and elastic, or as if there were a little more than the actual thickness of liquid crystal. If klt > k33, the sign of the thickness anomaly is reversed. If the average angle of surface attachment is not homeotropic or parallel, then in the absence of an applied field the tilt angle beyond the distorted surface region also depends on kl i/k33. For variations that are not coplanar results also depend on the twist constant, k22, but the trends are the same. Computer generated illustrations are presented.
1. Introduction. — We have written a computer program to find equilibrium director configurations of nematic or cholesteric liquid crystals confined within long tubes or between parallel periodic gratings with fixed boundary orientation. Solutions are res- tricted to those that do not depend on position along the tube axis or parallel to the grating lines, but the directors may have components parallel to that direction. In order to save time we have ignored flow in the liquid crystal, but have kept the rotation- resistance viscosity term yt of the Leslie-Ericksen equations [1]. The final equilibrium configuration obtained from a particular starting configuration is a correct relative or absolute minimum energy state, but the fact that flow is ignored could occasionally lead to a different final state than would result from
the full flow equations. For instance, we could not replicate the reversal effects discovered by Gerritsma, de Klerk and van Zanten with this program [2].
Since we have not assumed that the three elastic constants are alike, we obtain equilibrium solutions that do not obey Laplace's equation. The program utilizes a finite difference method with reverse-Euler time iteration to avoid instability of solutions.
There is some experimental evidence to support the idea that small-scale patches of homeotropic and parallel alignment occur when unusually thin layers of homeotropically aligning surfactant cover a glass surface that would otherwise yield parallel alignment.
It has also been observed that extremely thin layers of silicon monoxide deposited by evaporation near grazing incidence over a thicker high angle deposit
JOURNAL DE PHYSIQUE Colloque Ci, supplément au n° 4, Tome 40, Avril 1979, page C3-58
Résumé. — Si, dans un cristal liquide nématique, la constante d'élasticité de déformation en éventail ktl et la constante d'élasticité de flexion k33 sont dissemblables, et si les directeurs immé- diatement adjacents à la surface sont fixés rigidement dans des directions non uniformes, alors les constantes apparentes k33 et klx très proches de la surface seront modifiées par un effet de moyenne.
Si des molécules de cristaux liquides tels que MBBA, ayant klt < k33, sont fixées rigidement à la surface en alignement irrégulier, à savoir, parallèles à la surface en moyenne et coplanaires, elles se comporteront sous tension comme s'il p avait une couche mince rigide sur la surface. Si leur alignement macroscopique est homéotrope, elles se courberont comme si le point d'ancrage à la surface était faible et élastique, ou comme s'il y avait un peu plus que l'épaisseur réelle du cristal liquide. Si kil > k33, le signe de l'anomalie en épaisseur est inversé. Si l'angle moyen d'ancrage à la surface n'est ni homéotrope, ni parallèle, alors, en l'absence d'un champ appliqué, l'angle d'in- clinaison au-delà de la zone de surface déformée dépend aussi du rapport krljk33. Pour des variations qui ne sont pas coplanaires, les résultats dépendent aussi de la constante de torsion k22> mais les effets sont qualitativement les mêmes. Des simulations par ordinateur sont présentées en illustration.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979313
ANOMALOUS STIFFNESS AND TILT ANGLE IN NEMATICS C3-59
yield unusually low tilt angles [3]. This could also be due to incomplete coverage and patchy alignment on a very small scale.
If the three elastic constants were alike the surface irregularities would average out over a short decay length equal to 1/2w times the wavelength of each Fourier component of the surface irregularity, in accordance with solutions to Laplace's equation.
The final orientation could be obtained by simple superposition of decaying sinusoidal orientation modes. If the liquid crystal above the surface were subject to reorientation by an electric or magnetic field, as in the Freedericks transition [4], the orienta- tional irregularity at the surface would not affect the average orientation beyond a few decay lengths because the bend and splay produced by the field would be superposed upon the surface strains.
If the three elastic constants are not alike, however, superposition of strains is no longer correct. In that more usual case, if the average tilt is not zero or 90 degrees the tilt beyond a few decay lengths from the patchy aligning surface will depend on the relative values of the elastic constants. In addition, the change in tilt angle at a given distance from the surface pro- duced by an applied field will depend on the size of the surface patches (or the decay lengths) as well as the relative values of elastic constants. These are the anomalous tilt and stiffness effects caused by inequa- lity of elastic constants that are to be described in more detail.
It was recently observed that director orientation adjacent to obliquely evaporated silicon monoxide layers depends in a regular way on the ratio of compo- nents in liquid crystal mixtures
[S].
We hypothesize that this effect may be due to the fact that the ratio of elastic constants is changing regularly, thus changing the distant orientation, cp,, resulting from equivalent irregular orientation immediately adjacent to the surface. In particular, if the splay constant exceeds the bend constant the average alignment should be more nearly parallel to the surface than if the ratio is revers- ed. Elastic constant ratios were not given for the liquid crystal mixtures in which regular variation of tilt with component ratio were reported. Measurement of those elastic constant ratios would test our hypo- thesis.2. Numerical examples, stiffness anomaly. - If the orientation at the surface is homeotropic on average but is locally distorted as shown in the center of figure 1, there will be a strain energy associated with the deformation that decays approximately expo- nentially with distance from the surface. The integral of this strain energy, per unit surface area, is designated U,. If the surface alignment is uniform and homeotro- pic but if the directors have a finite tilt at some distance h above the surface, as shown on the left in figure 1, then there will be a strain energy per unit area U, between the surface and the plane at h that is pro-
FIG. 1. - Illustration of coplanar strains : (left) pure bend at surface, storing energy U,; (center) pure surface distortion, storing
energy U, ; (right) both, storing energy U,.
portional to the square of the tilt angle at that plane.
If both distortions are imposed as shown at the right, the energy per unit area is designated UB. If all elastic constants were alike, we would be able to add U, and U, to get UB. When the constants are unlike, the region near the surface where the distortions are large may require either more or less energy to superimpose a given tilt at h than if it were not distorted. Hence the liquid crystal layer behaves, in an applied field, as if the surface orientation were undistorted but as if the surface were slightly above or below the actual surface level. The apparent displacement, b, of the surface for Freedericks transition experiments is given by the following formula when the surface orientation varies periodically over a distance I :
Values of b/l for sinusoidal variations of amplitude A in orientation in both the homeotropic-average case and the parallel-average case are shown in table I.
The numbers in the top half are for strains that are all coplanar with the direction of variation in tilt as shown in figure 2. Those on the bottom half are for twist strains at the surface as shown in figure 3, rather than bend and splay, and solutions are not exactly coplanar.
Numerical examples of stzfness anomaly
0 = 0 : co-planar distortions; kll/kZ2 (*)/k33 kll/kZ2 (*)/k33 0, = 0, = 900 : out-of-plane = 1/0.3/0.5 = 0.5/0.3/1
- - -
e 'PO A bll 611
0 0 450 0.027 - 0.017
0 900 450 - 0.017 0.027
0 0 900 0.131 - 0.044
0 90° 900 - 0.044 0.131
(*) k,, not involved in co-planar distortions.
C3-60 D. W. BERREMAN
FIG. 2. - Increase of tilt cp from 0 to 40° with distance from sur- face, combined with sinusoidal surface distortion of amplitude 450 and mean tilt zero ; coplanar in 6 = 0 plane. Only bend and splay
strains occur.
FIG. 3. - Non-coplanar distortion and tilt variation with 6 = 900.
Strains include twist, bend and splay.
If the distortions are all coplanar with the direction of tilt variation then only the splay constant, k , , ,
and the bend constant, k,,, are involved. In that case, if average surface-orientation is changed from homeo- tropic to parallel and if, k , , / k , , is inverted, the result- ing values of b/l are alike, as illustrated on the upper half of table I and in figure 4. This symmetry is evident from the planar expansion of the Oseen-Frank equations [6]. It is lost when twist is introduced into the problem, as in the bottom of table I1 and figure 3, for example.
SOFT STIFF
HOMEOTROPIC
STIFF SOFT
PARALLEL
FIG. 4. -Stiffness anomaly, represented by thickness b, for various mean initial orientations and elastic constant ratios.
Numerical examples, tilt anomaly (coplanar configurations)
Surface k11 k33 qi (~f(Umin) distortion
- - - -
0.5 1.0 450
l o
1
450 amplitude1.0 0.5 450 49.90 sin wave
0.5 1.0 750 1 /6 parallel
1.0 0.5 750 780 690
1
516 homeotropic Although the values of b/l are quite small even for large surface distortion amplitudes, they might lead to some difficulty in making accurate measurements of inherent elasticity of surface anchoring using the Freedericks transition, as was proposed by Guyon and Urbach [7].ANOMALOUS STIFFNESS AND TILT ANGLE IN NEMATICS C3-61
3. Numerical examples, tilt anomaly. - One might achieve an average tilt of about 450 by making alternate stripes of surface material, one giving homeotropic and the other parallel alignment. If the splay and bend elastic constants are unlike, however, equal stripe- widths would not give exactly 450 tilt. Approximating this configuration with a sinusoidal surface tilt of amplitude 450 and average tilt of 450, we find that energy is lowest when the tilt angle A beyond 3 or 4 decay lengths is less than 450 (nearer to homeotropic) when the bend constant, k,,, is larger than the splay constant, k,
,,
as it is in MBBA [8]. It is greater than 450 (nearer parallel) when k,, is thank,, a s it is in hexylazoybenzene, for instance [9]. The results are illustrated in figure 5 and numerical values are given in table 11.To illustrate a more typical tilt angle for twist cells [lo, 111 we have also computed the lowest energy angle when the homeotropic alignment stripes are only 116 and the parallel alignment stripes are 516 of the period length, 1. The interval was broken into 24 segments with parallel alignment on 19, 450 tilted alignment on one, homeotropic on three and 450 again on the last of the series. Again, as shown in table 11, if the bend constant is larger, the final angle is more nearly vertical than if it is smaller than the splay constant.
Although the surface attachment on obliquely
/ / / / / / / / '///////
/ / / I / / / / /'//////
/ / / / / / / / ////////
I / / ' / / / / / ' 0 0 / / / /
/ / / / / / I / /I,--////
I / / - / / I
/ 1 / / - / /I /
-
k -K 3 3
>
K11 K 3 3 < K I IFIG. 5. - Tilt anomaly for coplanar distortions for contrasting ratios of k , ,/k,,.
evaporated silicon monoxide is not all in one plane but is probably some complicated arrangement with disclinations around small tilted needle-like crystals [12, 131, there still exists a flat plane above which bulk crystal elastic effects describe the configuration. Again we may expect that the tilt well above this surface will depend on the ratio of elastic constants in the same direction as with the simplified models described in this paper. In general, if alignment is tilted and if surface orientation is irregular, a relatively large splay constants favors more nearly parallel alignment, and a relatively large bend constant favors more nearly homeotropic alignment.
References
[I] LESLIE, F. M., Arch. Ration. Mech. Anal. 28 (1968) 265.
[2] GERRITSMA, C. J., DE KLERK, J. J. M. J. and VAN ZANTEN, P., Solid State Commun. 17 (1975) 1077.
[3] RAYNES, E. P., ROWELL, D. K. and SHANKS, I. A,, Mol. Cryst.
Liq. Cryst. 34 (1976) 105.
[4] FREEDERICKS, V., ZOLINA, V., Trans. Am. Electrochem. Soc.
55 (1929) 85.
[5] MORRISSY, J. H., CROSSLAND, W. A. and NEEDHAM. B.. J.
Phys. D. Appl. Phys. 10 (1977) L-175.
[6] OSEEN, C. W., Ark. Mat. Astron. Fys. 19 (1925) 1.
[7] GUYON, E. and URBACH, W., Nonemissive Electrooptic Dis- plays, A. R. Kmetz and F. K. von Willisen, editors (Plenum Press, New York) 1976, 121.
[8] HALLER, I., J. Chem. Phys. 57 (1972) 1400.
[9] DE JEU, W. H. and CLAASSEN, W. A. P., J. Chem. Phys. 67 (1977) 3705.
[lo] SCHADT, M. and HELFRICH, W., Appl. Phys. Lett. 18 (1971) 127.
[I 11 See, for example, TORIYAMA, K. and ISHIBASHI, T., Non emis- sive Electrooptic Displays (ibid), p. 145ff.
[12] GUYON, E., PIERANSKI, P. and BOIX, M., Lett. Appl. Eng. Sci.
1 (1973) 19.
[13] GOODMAN, L. A,, MCGINN, J. T., ANDERSON, C. H. and DIGERONIMO, F., ZEEE Trans. E.D. 24 (1977) 795.
