PHYSICAL REVIE% A VOLUME 14, NUMBER 5 NOVEMBER 1976 Theory of piezoelectricity in nematic liquid crystals, and of the cholesteric ordering Joseph P. Straley Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506 (Received 25 June 1976) A statistical-mechanical theory of the elastic constants of a liquid crystal based on the Onsager expansion is reexamined, with emphasis on the terms linear in the gradients of the director. This gives a microscopic theory of the piezoelectric effect in nematic liquid crystals, and of the spontaneous twisting of a cholesteric. Model cases of tapered rods, bent rods, and threaded rods are considered. I. INTRODUCTION Nematic liquid crystals have uniaxial symme- try: there is locally a single preferred axis, whose orientation is conventionally represented by a vector n(the "director" ). Since these materi- als actually have quadrupole (rather than dipolar) ordering, the directions yg and — n are not physi- cally distinguishable. This implies that the prob- ability that a, given molecule has a chosen orienta- tion is a function of the direction cosines (n rn, } of the director relative to some body-fixed axes (m, . } of the molecule, but is invariant under rota- tions of the molecule about the director and is even in n. The molecules themselves are much less sym- metric. The molecules forming a cholesteric liquid crystal need have no symmetry at all, and nematogens are expected only to have chiral sym- metry (i. e. , to be nonenantiomorphic'). The rela- tively low symmetry of the molecules comprising a liquid crystal does manifest itself in certain ways. Meyer' first noted that bent or tapered molecules can exhibit a piezoelectric effect under the conditions of orientational strain. This sug- gestion has received subsequent discussion by Helfrich' and Qruler. ' A microscopic statistical- mechanical theory of this effect will be given here. This paper will also give a microscopic theory of cholesteric liquid crystals. In the usual elastic theory of cholesterics, the pitch length appears as a ratio of two elastic constants, one of which is Frank's twist constant K», and the other of which is the coefficient in the free-energy density of terms linear in the gradients of the director. ' This latter coefficient will be calculated here. The method may be regarded as an extension of Goossens's calculation' in which it is no longer necessary to assign molecules to planes. The common feature of these two problems (pie- zoelectricity and cholesteric ordering) is that both involve the linear order response to gradients in the director field. The published discussion of these aspects of liq- uid crystals have been somewhat qualitative: the relationships between molecular asymmetry and orientational order have been sought without ref- erence to the microscopic theories which have been used to discuss the liquid-crystalline order- ing. ' " This paper will extend the Onsager' the- ory to a calculation of the piezoelectric coeffi- cients of a model nematic, and of the pitch of a cholesteric liquid crystal. The extended theory is already in existence; it has been used previously"'" to calculate the Frank elastic constants for nematic liquid crys- tals; however, in these previous discussions the piezoelectric terms were ignored. A more care- ful rederivation of that theory will be given here (Sec. II). The theory will then be applied to some simple model systems, namely tapered rods (Sec. III), bent rods (Sec. 1V), and enantiomorphic par- ticles (Sec. V). II. THEORY OF THE DISTRIBUTION FUNCTION Consider a rigid molecule of arbitrary shape, and let its orientation be specified by the quantity ur, which may be represented as a triplet of Euler angles, or as the collection of direction cosines of a body-fixed axis system with respect to a labora- tory frame. Following Onsager' we shall define an orientational distribution f(ur) which gives the relative probability that a molecule has orienta- tion in the differential element d&g near &u: if dN(&u) is the number of such particles in a volume dV, then dN(e) = pf(m) da dV. Since all particles have some orientation, this distribution function is normalized f (d G(d= 1~ where the integral ranges over all orientations. In a distorted liquid crystal or in a cholesteric, f(&u) is also a function of position; for simplicity (and without great loss of generality) we will as- sume that the density remains uniform in these 14 1835