Summary
By means of a phenomenological approach, we demonstrate that the mixed splay-bend elastic constantK 13 in the free energy density of nematic liquid crystals must be considered zero, unless the bulk contributions of the squares of the distortion second-order derivatives are taken into account, together with the squares of the first-order derivatives times the second-order derivatives, and with the fourth powers of the first-order derivatives. Such contributions just reduce to one in the presence of—and close to—a threshold. Furthermore, the saddle-splayK 24-term instead is shown always to play an essential role, as the bulk first-order elasticity, in determining the distortion free energy of nematics with weak anchoring subjected to spatial deformations. Finally, the new surfacelike elastic constants are shown to have a nilpotent character: thus they behave as well asK 24 from the point of view of the variational calculus.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. W. Oseen:Ark. Mat. Astron. Fys. A,19, 1 (1925);Trans. Faraday Soc.,29, 883 (1933);F. C. Frank:Discuss. Faraday Soc.,25, 19 (1958);E. B. Priestley, P. J. Wojtowicz andPing Shen:Introduction to Liquid Crystals, (Plenum Press, New York, N.Y., 1974). p. 143, and references therein;P. G. de Gennes:The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974), p. 59, and references therein;S. Chandrasekhar:Liquid Crystals (Cambridge University Press, Cambridge, 1977), p. 98, and references therein.
J. Nehring andA. Saupe:J. Chem. Phys.,54, 337 (1971).
J. Nehring andA. Saupe:J. Chem. Phys.,56, 5527 (1972).
E. Dubois-Violette andO. Parodi:J. Phys. (Paris) Colloq.,30, C4–57 (1969).
D. W. Berreman andS. Meiboom:Phys. Rev. A.,30, 1955 (1984).
The NLC directorn (undistinguishable from −n) is the unit vector parallel to the local average orientation of the molecules. A NLC-cell has strong anchoring if the surface director is independent of the bulk distortion; in the opposite case (weak anchoring), then-orientation at the surface depends both on the bulk distortion and on the surface treatment.
C. Oldano andG. Barbero:J. Phys. (Paris) Lett.,46, L-451 (1985);Phys. Lett. A,110, 213 (1985).
G. Barbero andC. Oldano:Nuovo Cimento D,6, 479 (1985).
A. Strigazzi:Mol. Cryst. Liq. Cryst.,152, 435 (1987).
W. H. de Jue:Physical Properties of Liquid Crystalline Materials (Gordon and Breach, New York, N.Y., 1980).
H. R. Brandt andH. Pleiner:J. Phys. (Paris),45, 563 (1984);H. Pleiner:Liq. Cryst.,3, 249 (1987);H. Yokoyama;Mol. Cryst. Liq. Cryst.,165, 265 (1988).
G. Vertogen, S. D. P. Flapper andC. Dullemond:J. Chem. Phys.,76, 616 (1982).
G. Vertogen:Physica A,117, 227 (1983).
E. Govers andG. Vertogen:Phys. Rev. A,30, 1998 (1984).
For instance, the assumptionA ijkl =A klij is not restrictive, since the possible antisymmetric part of the tensorA ijkl with respect to the double change (ij, kl.) gives no contribution tof.
G. Barbero, I. Dozov, J. F. Palierne andG. Durand:Phys. Rev. Lett.,56, 2056 (1986).
G. Vertogen:Z. Naturforsch. A,38, 1273 (1983).
E. Govers andG. Vertogen:Physica A,141, 625 (1987).
A. J. M. Spencer andR. S. Rivlin:Arch. Ration. Mech. Anal.,11, 45 (1961).
S. Meiboom, J. F. Sethna, P. W. Anderson andW. F. Brinkman:Phys. Rev. Lett.,46, 1216 (1981).
J. P. Sethna, D. C. Wright andN. D. Mermin:Phys. Rev. Lett.,51, 467 (1983)
S. Meiboom, M. Sammon andW. F. Brinkman:Phys. Rev. A,27, 438 (1983).
L. Longa, D. Monselesan andH. R. Trebin:Liq. Cryst.,2, 769 (1987).
Of course, the flexoelectricity should be taken into account also in the frame of the ordinary elastic theory. But the flexoelectric contributions renormalize the elastic anisotropy (G. Barbero andM. Meuti:J. Phys. (Paris),47, 341 (1986); therefore, they vanish close to a threshold, as well as the square of the distortion angle. According to the generalized elastic theory, new flexoelectric contributions of the same order of the new elastic constants would appear, getting to further monstrous complication. Fortunately, all the more reason the new flexoelectric contributions can be shown to vanish close to a distortion threshold.
G. Barbero andA. Strigazzi:Fizika,13, 85 (1981).
A. Strigazzi:Mol. Cryst. Liq. Cryst.,179, 427 (1990).
G. Barbero andA. Strigazzi:Liq. Cryst.,5, 693 (1989).
R. S. Schechter:The Variational Methods in Engineering (McGraw-Hill, New York, N.Y., 1967).
A. Rapini andM. Papoular:J. Phys. (Paris) Colloq.,30 C4–54 (1969);A. Strigazzi:J. Phys. (Paris),46, 1507 (1985) and references therein;G. Barbero andG. Durand:J. Phys. (Paris),47, 2129 (1986) and references therein.:J. T. Gleeson andP. Palffy-Muhoray:Determination of the surface anchoring potential of a nematic in contact with a substrate; preprint.
H. P. Hinov:Mol. Cryst. Liq. Cryst.,148, 197 (1987).
N. V. Madhusudana:Experimental determination of the elastic constant K 13 of a NLC, presented at the2nd International Topical Meeting on Optics of Liquid Crystals, Torino, 1988;N. V. Madhusudana andR. Pratibha:Mol. Cryst. Liq. Cryst.,179, 207 (1990).
H. Mada:Mol. Cryst. Liq. Cryst.,51, 43 (1979);Appl. Phys. Lett.,39, 701 (1981).
J. L. Ericksen:Arch. Ration. Mech. Anal.,10, 14 (1962).
G. Barbero andC. Oldano:Mol. Cryst. Liq. Cryst.,170, 99 (1989).
Ping Sheng:Phys. Rev. A,26, 1610 (1982).
I. P. Nicholson:J. Phys. (Paris),48, 131 (1987).
H. Hsiung, Th. Rasing andY. R. Shen:Phys. Rev. Lett.,24, 3065 (1986).
M. Kléman:J. Phys. (Paris),46, 1193 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Barbero, G., Sparavigna, A. & Strigazzi, A. The structure of the distortion free-energy density in nematics: Second-order elasticity and surface terms. Il Nuovo Cimento D 12, 1259–1272 (1990). https://doi.org/10.1007/BF02450392
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02450392