Theoretical

In the absence of polymer, the liquid crystal director will be a unit vector whose direction will be independent of position. The order parameter s will similarly be position-independent. The introduction of partially oriented polymer fibrils will have the effect of modifying both s and from their previously uniform values. We consider the effect of the polymer as being equivalent to that of some spatially varying applied field . Here is not a magnetic field, but is a generalized force field that changes the local energy by an amount proportional to and s, and dependent on the angle between and . The action of on the liquid crystal is then to reorient the molecules and to either reduce or to amplify the local order parameter. The immediate, or zeroth order, effect of is to tend to reorient into the volume-averaged preferred direction of the polymer fibrils. The next effect observed will be a change in the volume average of s that will be of first order in . Following this there will be some local reorientation of as molecules tend to align themselves with the local variations in orientation of this spatially varying field. The changes in energy resulting from these shifts will be at least of order , and thus of less importance than the changes in s when the field is weak. For this reason we shall adopt a model in which the elastic energy due to variation of is neglected, and in which the dominant effects are due to local variations in s. This approximation is not strictly necessary, but has the effect of reducing the number of parameters in the theory to a manageable quantity. A more thorough approach would follow the approach of Palffy-Muhoray et al.[5], who treat spatial variations in the tensor order parameter.

Our model is then one in which the director is uniform, but in which the order parameter varies under the influence of the mean field of neighboring molecules and of the applied field due to the aligning effects of the polymer fibrils. The Hamiltonian for a single molecule in these combined fields is then

where and is the angle between the molecular axis and . The mean field is taken to be

with a constant, the local average of , and a normalized function of sufficiently short range that we can rewrite this equation without significant error as

The formulation of the Helmholtz free energy of this system requires some care. We cannot simply write the partition function for a single molecule in the mean field , as this would give an incorrect expression for when is spatially varying. Instead we first evaluate the free energy for an ensemble of non-interacting molecules subject to the constraint that the order parameter be s, and then add the effects of the fields and interactions. The resulting expression can then be minimized to find .

The maximization of the entropy of an ensemble of independent liquid crystal molecules with a constrained order parameter leads to the introduction of a Lagrange multiplier in the form of a field A. The free energy in this field is given by the equation

say, where with T the temperature and Boltzmann's constant. Then

where . This relation may be inverted to give an expression of the form

Now

and so

To this expression, which is no more than the negative of the entropy of the constrained ensemble, we add the effects of the mean field h and the applied field H to find
 
Minimization with respect to s of the integral over volume of this free energy then yields the equation
 
The equilibrium state will then be the solution of this differential equation that gives the lowest value of the free energy (9).

The model that we choose for attempts to capture the essential features of the action of the network of polymer fibrils. The primary effect is to introduce a preferred direction of orientation, and this we represent by a uniform field . Because the fibrils form a cross-linked network there will also be regions in which the local fiber orientation lies at some large angle from the dominant direction, and in these regions the effect will be to diminish the order parameter. We represent this effect by adding a spatially varying component to H. For this we choose the simplest possible form, and superimpose on a term , with the wavelength characterizing the average spacing in the network.

The effect of a sinusoidal field alone acting on the liquid crystal at a temperature above the nematic-isotropic transition would be to induce a positive ordering s. The reason for this lies in the fact that the free energy is an asymmetric function of s. It increases rapidly when s becomes negative, but does so only slowly as s becomes positive. A cosinusoidal field, which consists of equally large positive and negative regions, thus has a much larger effect in perturbing s in the positive direction than it does in the negative direction. For this reason we must apply a negative value of along with the oscillatory field if we are to mimic the effect of a dispersion of polymer fibrils of random orientation. To describe a system in which the polymer fibrils have a preferred orientation, a positive component of must be added. It is not then clear whether the system studied experimentally is best described with a positive or with a negative . In our numerical solution of Eq. (10) we chose a negative value of equal to /10 simply because it appeared to give a reasonable qualitative fit to the data.

Our goal is to calculate the spatial average of the order parameter s as a function of temperature and concentration of polymer, and hence to be able to predict the birefringence of the sample. The solution of Eq. (10) will give in terms of the dimensionless quantities , , , and ql. At high and low temperatures some reasonable approximations can be made which lead to analytical solutions. At intermediate temperatures the problem is more difficult, and requires numerical solution, since there is then the possibility of the system splitting into domains of strongly and weakly ordered material. When this happens the first-order phase transition is either lost or greatly diminished. The single first-order transition disappears when ql or the ratio becomes sufficiently small.

The results of a numerical solution of Eq. (10) are shown in Fig. 2 for a range of for the case where and where ql = 0.5. From the figure we see that the discontinuity in that occurs at the transition temperature is gradually diminished as is increased, and vanishes when in units of . Because the order parameter is generally linearly related to the birefringence, we would predict that the birefringence should show a similar pattern.

These theoretical results appear to capture the qualitative features of the experimental results shown in Fig. 1, and give us some confidence that we have captured the essential physics of the processes involved. As one might expect, the effect of increasing the spatially varying field is initially to reduce the discontinuity in at the phase transition, and then to eliminate it. The theoretical results resemble the experimental ones in exhibiting a range over which varies linearly with temperature, although the theoretical curves show an increase in slope at the boundary of the linear region that is absent in the experimental curves.