PHYSICAL REVIEW E VOLUME 53, NUMBER 3 MARCH 1996 Anchoring of nematic liquid crystals at a solid substrate A. Poniewierski and A. Samborski Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland (Received 18 July 1995} A nematic liquid crystal in contact with a solid substrate is studied by means of the Landau — de Gennes formalism. The free-energy functional is expanded around the bulk nematic order parameter up to the second-order terms. This approximation is used to obtain an explicit condition for the anchoring direction in terms of the surface and bulk coupling constants, in a serru-infinite system. Then a finite sys- tem is studied and the equilibrium free energy is found as a function of the angular deviation from the anchoring direction and the sample thickness. A geometrical measure of the anchoring strength, resem- bling the de Gennes extrapolation length, is obtained from the asymptotic behavior of the free energy for large sample thicknesses. PACS number(s): 61. 30. Gd, 68.35. Md I. INTRODUCTION Nematic liquid crystals are uniaxial media [1], and the orientation of the symmetry axis (nematic director) is ar- bitrary in the absence of external fields or limiting sur- faces. Various limiting surfaces such as: the free nematic surface, the nematic-isotropic interface, or the solid sub- strate surface, fix the orientation of the nematic director. This phenomenon is called the anchoring of the nematic liquid crystal at the interface [2 — 7]. The anchoring of liquid crystals at solid substrates attracts particularly much attention. First, it is interesting as a fundamental problem in the statistical mechanics of nonuniform, or- dered fluids. Second, it is important for the technology of liquid-crystal display devices. Three regions can be dis- tinguished in the nematic liquid crystal in contact with the substrate [8]. Close to the surface, liquid-crystal mol- ecules interact directly with the substrate, and this direct interaction determines the microscopic anchoring condi- tion. Next, there is an interfacial region in which the or- der parameters and the nematic director change. Finally, far from the substrate, there is the bulk phase with the director orientation fixed by the surface. This bulk orien- tation is called the anchoring direction or the easy axis; it corresponds to the minimum of the interfacial tension y between the nematic phase and the substrate. The struc- ture of the interfacial region determines the macroscopic anchoring condition. The energetic manifestation of the anchoring is re- ferred to as the anchoring energy or the anchoring strength. In anchoring energy measurements, the direc- tor field in the nematic phase has to be distorted. Various experimental techniques are discussed in detail by Yokoyama [3], who also gives a thermodynamic definition of y as a function of the director orientation. In order to define y, a hypothetical dividing surface is in- troduced. Above the dividing surface the nematic phase is assumed to have a bulklike behavior, and distortions of the director field are described by the Frank elastic theory [9]. Then y is defined as a function of the director orientation at the dividing surface. Yokoyama contrasts this thermodynamic definition with phenomenological formalisms, e. g. , the Rapini-Papoular formalism [10], in which one simply postulates a particular form of y. He also emphasizes a fundamental conceptual diff'erence be- tween the definitions of the anchoring direction and the anchoring energy. The anchoring direction has an unam- biguous physical meaning, independent of a model of the interface. It refers to the director in the bulk nematic phase in the absence of deformations. The anchoring en- ergy, on the other hand, is an interfacial parameter acces- sible only through an appropriate theoretical framework. It is defined as the second derivative of y with respect to the director orientation calculated at the anchoring direc- tion [2]. Therefore, it depends on the choice of the divid- ing surface. It is worth mentioning that some phenome- nological formalisms assume that y depends not only on the director orientation at the interface but also on its gradients. The surface elastic terms are usually referred to as the K24 and K, 3 terms, where K24 and K» are the surface elastic constants [11 — 13]. Consequences of the presence of the surface elastic terms in y have been stud- ied by many authors [4, 14 — 21]. As we do not discuss them in this paper, we only note that the K» term leads to some mathematical problems, the solution of which has been proposed recently by Pergamenshchik [17]. A long time ago de Gennes [1] introduced a geometri- cal measure of the anchoring strength. He considered only the simplest case of the twist deformation. In that case, the director orientation is specified by the azimuthal angle y, measured with respect to the easy axis parallel to the substrate. It results from the solution of the Euler- Lagrange equation that far from the substrate y is a linear function of the distance z, y(z) =const X(z+b), where b is called the extrapolation length. The anchoring strength can be characterized by the relation between b and the molecular dimensions. The anchoring is strong if they are comparable, and it is weak if b is Inuch larger than the molecular diInensions. In the case of short- range forces, the anchoring energy is equal to Kalb, where K2 denotes the twist elastic constant. Dubois- Violette and de Csennes [22] also studied the case of 1063-651X/96/53(3)/2436(8)/$10. 00 53 1996 The American Physical Society