Introduction

 

In nearly all applications of liquid crystals, the interaction of the material with the walls of the container plays an important role. In display devices, the transparent surface of the cell holding the liquid crystal is generally treated in some manner in order to encourage a particular orientation of the layer of molecules in contact with it.

There is now a very large body of empirical knowledge concerning this interaction between liquid crystal and substrate. Surfaces may be reproducibly and reliably produced on which the liquid crystal director will assume a predetermined orientation. The methods used for inducing a given orientation include a wide variety of chemical and mechanical treatments.[1] Our purpose in this paper is to study by simulations the microscopic mechanisms by which this ordering is induced in the simplest experimental configurations.

The fundamental effects of the introduction of a surface into a nematic liquid crystal system are to break the translational symmetry and to reduce the rotational symmetry of the liquid crystal.[2, 3] The positions of liquid crystal molecules near a surface are restricted by the presence of the surface, and this causes the molecules near the surface to form layers. These layers are similar to those found in a smectic structure if the molecules do not lie in a plane parallel to the surface. The layered structure persists for several molecular lengths into the bulk nematic phase.[2, 4]

In addition to breaking the translational symmetry, the surface selects a subset of the degenerate orientational states. In the absence of competing forces such as those that might be generated by an electric field, the orientation selected by the interaction with the surface propagates into the bulk liquid crystal as a result of intermolecular interactions. This adoption of a particular orientation by the liquid crystal in contact with a surface is known as anchoring. In general, the anchoring direction is specified by the angle between the director and the surface normal and by the azimuthal angle . In some cases, such as that of a structureless surface, the anchoring is degenerate with respect to the angle . The anchoring is called homeotropic if the preferred orientation is for , homogeneous for and a preferred , planar if the preferred orientation is for for any , and tilted for .

A number of physical mechanisms can contribute to anchoring phenomena (see, for example, references 2, 5, and 6). Entropy is one factor: structureless walls tend to induce planar anchoring simply as a result of entropic considerations. However, liquid crystal molecules with intrinsic dipoles tend to align with their dipoles perpendicular to the surface even if the surface is structureless and free of charge.[7] The presence of charged impurities at the surface further complicates the matter.[6, 7] Often, mixtures of liquid crystal molecules are used,[8, 9] and the anchoring properties of these mixtures are not necessarily just an interpolation of the properties of the components. Surfaces used to contain liquid crystal systems are frequently rubbed, thus generating grooves that tend to cause the molecules to be aligned with the grooves.[10, 11] Finally, if a liquid crystal is in contact with a rough surface, the anchoring behavior can be frustrated. By this we mean that close to the nematic-isotropic transition, the isotropic liquid crystal can wet the surface;[12] anchoring then takes place between the nematic and isotropic phases, not between the nematic phase and the surface.

There are several approaches that have been taken to studying the anchoring problem from a theoretical perspective: lattice models,[3, 13][15] Landau-de Gennes theories,[7, 16][19] mean-field density-functional theories,[7, 19][26] and computer simulations.[4, 13][15, 27][31] Somewhat surprisingly, lattice models have not received much attention from an analytical point of view.[3] There have been a few Monte Carlo simulations of the Lebwohl-Lasher lattice model.[13][15] In this model, the nematic liquid crystal-forming molecules are taken to be located on a simple cubic lattice. The interaction energy between nearest neighbor molecules is

where , is the angle between the symmetry axes of the molecules at sites i and j, and is the second Legendre polynomial. Lattice models of surface effects such as anchoring can yield information such as shifts in transition temperatures due to surface effects and changes in the degree of order as a function of distance from the surface. However, the anchoring direction itself must be supplied quite directly in the specification of the interaction at the boundary. Even in the case of free boundary conditions used to describe the nematic-vapor interfaces, the simple fact that a lattice model is being used can introduce artifacts into the theory. Specifically, the lattice is a crystalline solid whereas liquid crystals are fluids; consequently, the direction of the spins at the surface of a Lebwohl-Lasher model may depend on which plane of the crystal is chosen as the surface.[13]

Landau-de Gennes models of anchoring have also been investigated.[7, 16][19] In these models, the interaction with the surface is usually assumed to be independent of the azimuthal angle , and so the portion of the free energy that represents the interaction with the surface will contain terms of the type , where Q(z) is the tensor order parameter at a point with z-coordinate z, and is a unit vector normal to the surface. These models have the strengths and weaknesses common to all models of the Landau-de Gennes type. While these theories allow investigation of questions such as what kinds of terms are required in the free energy to describe a phenomenon or a transition, the interpretation of the coefficients of the terms in the free energy is problematic. Attempts have been made to interpret the coefficients of Landau-de Gennes models in terms of microscopic theories such as mean-field density-functional theories.[7, 19] In the mean field density functional theories,[7, 19][26] the thermodynamic potential is written as a functional of the single-particle distribution function , which is taken to be a function of position and orientation . Sometimes,[24] the surface is introduced (for example, in the case of the nematic-isotropic boundary) by assuming with for z<0 and for z>0. This assumption puts a sharp interface between the isotropic phase in the z<0 half-space and the nematic phase in the z>0 half-space. Other times,[23] a term is included in the expression for that represents an interaction with an impenetrable wall. There is also variation in the potential chosen for the interaction between molecules. In some cases, the molecules have been chosen to be hard spherocylinders;[23, 24] in other cases,[22] the interaction has both a spherical hard core and an anisotropic attractive part.

The simulation studies of liquid crystal anchoring have mainly been Monte Carlo simulations of lattice models,[13][15] or molecular dynamics simulations using hard spherocylinders or the Gay-Berne potential to represent the interaction between molecules.[4, 27][31] The Gay-Berne potential is essentially an anisotropic version of the Lennard-Jones potential:

The anisotropy of this potential is in the functions and . The function controls the depth of the potential. The function is the distance at which the potential is zero. Both of these function depend on the relative orientations of the molecules. In the work of Steltzer et al.,[4] the focus is on investigating the formation of smectic-like layers near the solid surface, the interaction with which was represented by a simplified Gay-Berne potential in which the only spatial dependence was in the direction normal to the surface, and the homeotropic anchoring direction was favored. E. Martín del Río et al.[27, 28] performed simulations of Gay-Berne molecules and attempted to obtain the favored anchoring directions for the case of the interface of the nematic phase and the vapor phase. They began by forming a nematic phase of Gay-Berne molecules subjected to periodic boundary conditions. The periodic boundary conditions were then lifted in one direction, creating a slab of Gay-Berne molecules in the nematic phase with empty space on either side. The order parameter was calculated as a function of distance from the surface and it was found that the anchoring direction depends on the degree of anisotropy in the depth of the potential, . Bates and Zannoni[31] examined the nematic-isotropic interface in a Gay-Berne system in a molecular dynamics study, and found a tendency to planar orientation at this interface.

In an atomistic simulation of liquid crystal properties, the detailed structure of the molecules is taken into account. This makes the demands on computer resources much greater than is the case for Gay-Berne or Lebwohl-Lasher models. A simple molecule of 5CB (4-n-pentyl-4'-cianobiphenyl), for example, consists of thirty-eight atoms, and so the number of intramolecular interactions that must be taken into account is large, even before intermolecular interactions are considered. It is perhaps for this reason that there seems to have been very little work done on atomistic simulations of anchoring and other surface behavior at nematic interfaces, although a greater number of atomistic simulations of bulk liquid crystals have been performed.[32][36]

The present work uses the methods of atomic level molecular modeling to study the anchoring behavior of nematic liquid crystals. In particular, we study the anchoring of 5CB at an amorphous polyethylene surface. The molecule 5CB is a commonly used liquid crystal, and atomistic molecular modeling simulations on 5CB in the bulk have been performed previously.[32][36] Amorphous polyethylene is the simplest of polymers, and has, on average, no favored directions in the surface plane. This will simplify our task, which then reduces to an examination of the tilt angle favored by this system.