THESIS
2013

xiii, 120 pages : illustrations (some color) ; 30 cm

**Abstract**
The Onsager theory of liquid crystals deserves a special place among all the liquid crystal theories, not only because it was the first theory on liquid crystals, but also because its approach is based on rigorous statistical mechanics and therefore offers a relatively straightforward path for its improvement. With the assumptions of the low density and large aspect ratio, Onsager treated liquid crystal molecules as hard rods in which the free energy can be expanded as virial coefficients, in terms of the orientational distribution function. While the Onsager theory succeeded to capture the orientational transition of liquid crystals as a function of increasing density, the predictions of the transition density do not agree well with the molecular dynamics (MD) simulation results. As we...[

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The Onsager theory of liquid crystals deserves a special place among all the liquid crystal theories, not only because it was the first theory on liquid crystals, but also because its approach is based on rigorous statistical mechanics and therefore offers a relatively straightforward path for its improvement. With the assumptions of the low density and large aspect ratio, Onsager treated liquid crystal molecules as hard rods in which the free energy can be expanded as virial coefficients, in terms of the orientational distribution function. While the Onsager theory succeeded to capture the orientational transition of liquid crystals as a function of increasing density, the predictions of the transition density do not agree well with the molecular dynamics (MD) simulation results. As we show in this thesis, the higher virial coefficients in the Onsager theory framework will decrease the transition density. However, in the range of the low aspect ratio, the prediction of the transition density is smaller than that of the MD simulations. We can not improve the Onsager theory by just considering the higher virial coefficients.

The starting point of this thesis, i.e., the generalization of the Onsager theory, is to note that there can be a state in which there is very strong short-range orientational order but no long-range orientational order, obtained from hard-core molecular dynamics simulations in two-dimensional systems. Such a state obviously has a lower free energy (compared to the state of overall long-range orientational order), owing to the higher orientational entropy. In fact, in this particular case the Onsager theory’s prediction is inaccurate to say the least.

From this simple example it is clear that there is an aspect ratio regime which needs a more accurate theory. We did that by including short-range orientational order within the Onsager framework and extending the original theory along different directions. Besides the short range order, we have included the steric repulsive interaction to higher-order virial coefficients so that the behavior at high densities can be more accurate. The most important feature of our theory is that short-range orientational order introduces a mesoscale into the problem—the short-range correlation length. Within the short-range correlation length, clearly the steric repulsive interaction dominates as it forces the neighboring molecules to align so as to satisfy the density constraint. At a scale much larger than the short-range correlation length, the basic unit for long-range orientational order is now the cybotactic cluster, i.e., a group of molecules within the short-range correlation length.

We use the MD simulations as a check on the theory as well as to afford direct visualization of the short range order. The complication of MD simulations by using hard core repulsive interaction is that the collision time is essentially zero. Hence one can not use the discretized version of Newton’s law to follow the trajectory. Instead, we have to use energy and (translational and angular) momenta conservation to determine the post-collision status. In this thesis, we choose the time driven algorithm of MD simulations for the hard non-spherical model.

The result of the generalized theory is improved as compared to that of the Onsager theory. Due to the consideration of the short range orientational order, the transition density of the generalized theory is higher than that of the original theory. The results of the generalized theory agree better with the simulation results, especially in the range of the low aspect ratio.

The melting behavior in two-dimensional systems is also discussed. In the MD simulations of the hard rods-like model, the hexatic phase occurs for the low aspect ratio rods. For the small aspect ratio, such as 5.0, the melting undergoes the isotropic-hexatic-solid transition. However, for the large aspect ratio, such as 8.0, the transition becomes nematic-hexatic-solid.

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