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44178-AC7
Theory of Long-Range Interactions in Smectic Liquid Crystals
Philip L. Taylor, Case Western Reserve University
Some smectic
liquid crystals exhibit a series of phases, including ferroelectric, antiferroelectric, and ferrielectric
commensurate structures as well as one or more incommensurate
phases. A long-standing
problem in the physics of liquid crystals has been to understand the origin of
the long-range interaction that must be present between distant layers in order
for this rich variety of phases to exist.
We have investigated in depth a model that incorporates thermal
fluctuations in the flexing of layers, as a preliminary study has shown that it
leads to a long-range interlayer interaction sufficiently strong to support a
variety of commensurate and incommensurate structures. The vibrational
entropy of the sequence of layers is maximized when the c-directors of all
layers are parallel or antiparallel. This tendency to alignment competes
with an assumed interaction between nearest-neighbor layers that favors a
helical arrangement of c-directors.
As a consequence, an increase in temperature leads to an unwinding of
the helix that proceeds at first through commensurate phases and then into an
incommensurate phase. This result
is consistent with the experimentally observed "distorted clock model". It has been our goal to explore this
new phenomenon in all its aspects by performing calculations on a wide variety
of systems and over a wide range of material parameters. Our results should be of value both for
direct comparison with experiment and as input to some existing
phenomenological models.
The
origin of the effect that we have studied lies in the fact that the elastic
constant for bending a layer of smectic-C liquid
crystal along its c-director differs from the value for bending in the
perpendicular direction. This
gives rise to interactions between distant layers. The effect of this
entropy-induced interaction is to favor a parallel or antiparallel
alignment of the c-directors in these non-adjacent layers. We have calculated in detail the range
and strength of this interaction in both infinite and finite samples, and find
the results to depend mainly on the ratio of the average layer bending elastic
constant to the layer compression modulus. At low values of this ratio, the
interlayer interaction is of long range in a bulk sample, while at high values
of the ratio it decays as the inverse cube of the interlayer distance. For a
sample confined between rigid substrates parallel to the layers, the
interaction is greatly reduced.
For a free-standing film the interaction may be enhanced if the surface
tension is weak, but may be diminished if the surface tension is strong.
The
absolute strength of the interaction that we have calculated can be expressed
as a free energy per molecule that is equal to the product of the thermal
energy kBT with three other factors, all
of which are less than unity. Of these, the most important is an anisotropy
parameter that describes the difference in elastic constants for bending a
layer around an axis parallel or perpendicular to the c-director. A second
factor is approximately the square of the ratio of molecular diameter to its
length. The third factor is a function of the layer thickness and the ratio of
the average bending elastic constant to the layer compression modulus, and has
its maximum at intermediate values of this ratio.
It
might at first be thought that such a perturbation would be too small to modify
the phase diagram. However, it
turns out that the differences in free energy between the various phases of antiferroelectric liquid crystals are very small indeed. One can appreciate just how small they
are by calculating the difference between the free energies of the
ferroelectric and antiferroelectric phases in a
typical material far from its transition temperature. We obtained this number by considering the electric field
strength necessary to switch a material from its antiferroelectric
phase, which has no net dipole moment per unit volume, to a ferroelectric phase
having a significant dipole moment per unit volume. The free energy difference between the helical phases can be
shown to be much smaller than this, and hence comparable to our calculated
amount.
The
magnitude of the effect that we have studied is thus in the right range to have
a significant effect.
Nevertheless, a definitive identification of the effects of correlations
in layer fluctuations may have to await realistic calculations of the
interactions between adjacent layers, and this is a much more difficult task.
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