Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics… DOI: http://dx.doi.org/10.5772/intechopen.80977

linear regime of a nonequilibrium steady state. Therefore, we have reviewed molecular dynamics simulations and experimental work on director orientation phenomena in nematic liquid crystals and in cholesteric liquid crystals under external dissipative fields such as velocity gradients and temperature gradients. A general observation that we have made is that in all the examples studied, the director of the liquid crystals seems to attain precisely that alignment angle relative to the external dissipative field that minimizes the irreversible energy dissipation rate.

In a nematic liquid crystal, the director orientation is in the first place determined by a mechanical stability criterion, namely, that the external torques acting on the system must be zero at mechanical equilibrium. This makes it possible to derive an exact relation between the alignment angle relative to the streamlines and the viscosity coefficients in the linear or Newtonian regime of planar elongational flow and of planar Couette flow. Both simulations and experimental measurements imply that the irreversible energy dissipation rate is minimal at this mechanically stable orientation.

It can be shown that the elongation direction is the stable orientation of flow stable calamitic nematic liquid crystals undergoing elongational flow in the linear regime. It can also be shown that the value of the energy dissipation rate is the same in the contraction direction and in the elongation direction, and that this value is either the maximal or the minimal value by using the linear phenomenological relations between the strain rate and the pressure. Simulations of the calamitic soft ellipsoid fluid have shown that the irreversible energy dissipation rate is minimal in the elongation direction.

In calamitic nematic liquid crystals, the heat conductivity is larger in the direction parallel to the director than in the perpendicular direction, and the reverse is true for discotic nematic liquid crystals. Thus, the irreversible energy dissipation rate due to the heat flow depends on the angle between the director and the temperature gradient. When a nematic liquid crystal is subjected to a temperature gradient, a torque is exerted on the molecules. Due to symmetry, this torque must be proportional to the square of the temperature gradient and it must be zero when the director is parallel or perpendicular to this gradient.

In simulations of nematic phases of soft ellipsoids under a temperature gradient, it turns out that the director of a calamitic nematic liquid crystal aligns perpendicularly to the temperature gradient, whereas the director of a discotic nematic liquid crystal attains the parallel orientation. In both cases, the irreversible energy dissipation rate is minimal. These simulation results are probably supported by some experimental measurements, but they are difficult to carry out in practice so they are not fully conclusive.

Finally, one system where there is definitely a conclusive experimental evidence for the fact that the director attains the orientation that minimizes the energy dissipation rate due to a temperature gradient is the cholesteric liquid crystal. The cholesteric axis of droplets of cholesteric liquid crystals orient parallel to a temperature gradient and the director rotates. This is a well-established phenomenon observed in studies of thermomechanical coupling, and since the heat conductivity is lower in the direction of the cholesteric axis than in the perpendicular direction, the energy dissipation rate is minimal in this case.

Thus, the director orientation relative to a temperature gradient also follows the variational principle even though there is a quadratic coupling between the torque and the temperature gradient. However, the temperature gradients are rather low so we are still in the linear regime.

There are quite a few experimental studies available on this phenomenon, where it has been found in a conclusive way that the cholesteric axis remains parallel to the temperature gradient, so this orientation seems to be stable, and thus the irrevers-

We can consequently conclude that the orientation of the director relative to the temperature gradient is consistent with the variational principle [1] even though the coupling between the torque and the temperature gradient is quadratic rather than linear and the system is inhomogeneous. However, the temperature gradient is

In the above simulations of shear flow and elongational flow, the velocity gradient does work on the system that is converted to heat, which must be removed in order to keep the temperature constant and to maintain a steady state. In a real macroscopic system, this takes place by heat conduction to the system boundaries and this could in principle be arranged in a microscopic simulation cell as well. Unfortunately, this is inconvenient because a temperature gradient of molecular dimensions would make the system inhomogeneous, and thus make it difficult to define the thermodynamic state. Therefore, the temperature is kept constant by forcing the kinetic energy to be a constant of motion by applying a Gaussian thermostat, see Eq. (5). This thermostat was originally devised independently by Hoover et al. [32–34] and by Evans [22]. The equilibrium ensemble averages of the phase functions and time correlation functions generated when this thermostat is applied are essentially canonical [35]. Away from equilibrium, it can be shown that the effect of the Gaussian thermostat on the ensemble averages is proportional to the square of the external field, whereas the thermodynamic fluxes driven by the field are directly proportional to the field in a linear transport process. Thus, the corresponding linear transport coefficients that are equal to the ratio of the flux and the field in the limit of zero field are independent of the thermostat. Therefore, transport coefficients obtained from the simulations of shear flow and elongational flow are independent of the thermostat since there is a linear relation between the velocity gradient and the shear stress in the Newtonian regime and since we are interested in the limit of zero velocity gradient. Neither is the correctness of the variational principle affected by the thermostat since it is valid in the linear regime. The situation is different in the heat flow simulations because here we actually want a temperature gradient. This gradient is obtained by applying two bar thermostats at different temperatures acting over a limited range and separated by a distance that is long compared to this range, see Figure 7 and Eq. (14). Therefore, the movements of only a small fraction of the molecules are affected by the thermostats, whereas the movements of the majority of the molecules away from the bar thermostats are governed by the ordinary Newtonian equations of motion. Thus, it is reasonable to assume that the influence of the details of the thermostat on

the ensemble averages of the phase functions is limited in this case too.

The purpose of this work has been to test a recently proven variational principle according to which the irreversible energy dissipation rate is minimal in the

ible energy dissipation rate is minimal.

Non-Equilibrium Particle Dynamics

6. Effects of the thermostat

7. Conclusion

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rather weak, so we still remain in the linear regime.
