In order to insure that the simulations were performed in a temperature range in which nematic order is to be expected in the model system, initial simulations of 32 5CB molecules were performed without the polyethylene surface. The order parameter as a function of temperature for these initial simulations is shown in Figure 3. The nematic-isotropic transition is not sharp in such a small system, but T=300 K is seen to be within in the nematic range.
The study of anchoring began with the system in the planar alignment
shown in Figure 1, for which the order parameter S is unity and
for which 
lies along the x-axis.
In the absence of any influence from the polymer surface,
we should expect S to decrease to the bulk equilibrium value
shown in Figure 3 for 300 K.
The director should then remain approximately constant
in the absence of any externally applied torque.
In contrast to this picture, the effect of the substrate on the director
orientation was clear in all the simulations.
An example of the endpoint of a 10 ps run for 16 5CB molecules on
amorphous polyethylene is shown in Figure 4.
A reorientation of the director into a more nearly homeotropic state in
which is nearly normal to the polymer surface appears to have occured.
This is confirmed by an examination of the eigenvector corresponding
to the largest eigenvalue of the order-parameter tensor.
In most of the 14 simulations performed on
different ensembles of 16 or 32 5CB molecules, the
final orientation of was found to be lying within
thirty degrees of the y-axis.
This appears to demonstrate that the anchoring in this system is
nearly homeotropic.
While this tendency to homeotropic anchoring emerges clearly from the
simulations, the route taken in transforming from planar to homeotropic
orientation differed appreciably.
Two distinct mechanisms were observed.
In some cases the director remained in the plane of the substrate
while the ordering decreased to the point where the liquid crystal
was essentially in the isotropic phase.
The ordering then gradually increased with the new director along
the y-axis.
In other cases the ordering remained strongly nematic while the
director rotated from the x-z plane into the y direction.
The second mechanism was thus a collective rotation of the
molecules in which varied while S remained roughly
constant, in contrast to the first process, in which S varied
at constant .
This becomes apparent in the eigenvalues and eigenvectors of the order
parameter matrix 
.
Figure 5 shows the three eigenvalues of the order parameter
matrix as a function of time during one simulation.
The most obvious feature of this graph is that a pair of eigenvalues
crosses during the course of the simulation.
This means that the order within a plane has momentarily vanished before
increasing in a perpendicular direction.
In order to determine to which direction the orientation has changed,
it is necessary to associate eigenvectors with the eigenvalues,
the eigenvector of the largest eigenvalue being the nematic director.
The eigenvalue that is largest at the beginning of the simulation
corresponds to an eigenvector that lies primarily along the x-axis.
The other eigenvalue that is involved in the crossing that occurs 7 ps into the
simulation corresponds to an eigenvector that lies primarily along
the y-axis,
and is perpendicular to the surface.
The identification of eigenvalues with
eigenvectors along certain axes thus remains constant in this case.
A visualization of this is shown in
Figure 6,which indicates the normalized eigenvectors every 0.1 ps as points
on the surface of a unit sphere.
Because the normalized eigenvectors are undetermined to a factor of 
,
each eigenvector appears as a pair of clusters of points at opposite
sides of the unit sphere.
Because the eigenvectors appear as stationary clusters instead of sweeping
out paths across the surface of the unit sphere,
Figure 6 shows that the eigenvectors remain essentially
fixed during this simulation.
The alternative path from planar to homeotropic orientation was taken
in other simulations that started from the same initial conditions
as before.
In these cases there was no crossing of eigenvalues.
Instead a cooperative rotation of the molecules led to a rotation of
the principal axes of .
An example is given in
Figure 7, which shows the eigenvalues of the order parameter
matrix as a function of time during one simulation.
Although two of the eigenvalues approach each other at times, an
examination of the associated eigenvectors indicates that the
largest eigenvalue remains assiciated with a particular direction
in space which varies continuously.
The normalized eigenvectors for this simulation
are again shown as points on a unit sphere in Figure 8.
We now see features that are absent from the corresponding figure
for the simulation described earlier in Figure 6.
The arcs along the surface of the unit sphere indicate motion
of the eigenvectors as the simulation progresses.
This means that the 5CB molecules rotate more or less as a single unit
to the homeotropic orientation.
Our conclusions from this work are that at an amorphous polyethylene surface the anchoring of 5CB molecules at 300 K is predicted to be nearly homeotropic. However, the path by which an initially planar orientation changes to a homeotropic orientation may follow either one of two different routes. In one of these the surface layer initially becomes isotropic before developing homeotropic order; in the other, there is a cooperative rotation of the nematic director. These results were based on simulations on rather small samples. They also were performed at fixed volume, which resulted in a liquid crystal-vacuum interface area which may have had some influence on the director orientation. For these reasons, it would be desirable to extend the computations to larger systems and to constant-pressure conditions at some later date.
Acknowledgement: This work was supported by the NSF ALCOM Science and Technology Center under Grant DMR89-20147.