1. Introduction

For a system in thermodynamic equilibrium, there is a variational principle according to which the free energy is minimal, that is, the Helmholtz free energy when the volume, temperature, and number of particles are constant, Gibbs free energy when the pressure, temperature, and number of particles are constant, etc. On the other hand, for systems driven away from equilibrium by an external dissipative field such as a velocity gradient, temperature gradient, chemical potential gradient or an electrical potential gradient doing irreversible work that is converted to heat, there has not been any variational principle to date. However, a

theorem originally proposed by Ilya Prigogine stating that a quantity, known as the irreversible energy dissipation rate, w\_ irr, is minimal in the linear regime of a nonequilibrium steady state has recently been proven [1]. This quantity is defined as the irreversible work per unit time and unit volume that is done by a dissipative external field on the system [2]. Thus, there is a variational principle for nonequilibrium steady states.

This theorem is not only of basic scientific interest but also of technological and practical interest since shear fields, temperature gradients, concentration gradients, or chemical potential gradients and electrical potential gradients are common examples of external dissipative fields that are ubiquitous in industrial applications and in everyday life. For example, in a lubricated bearing, a planar Couette flow arises in the lubricant in the narrow space between two surfaces rotating at different angular velocities, and w\_ irr is equal to the product of the shear rate and the shear stress. Another example is the heat flow between a hot region and a cold region such as the inside and the outside of a building. Then, w\_ irr is equal to the product of the heat flow and the temperature gradient. Still another example is an electric heating element where an electric potential difference or voltage drives an electric current, and w\_ irr is equal to the product of the voltage and the current. Finally, chemical potential gradients arise when various substances are mixed and they begin to diffuse, and w\_ irr is equal to the product of the chemical potential gradients and the matter currents.

One way of testing this principle is to perform molecular dynamics simulations of microscopic model systems, but then it is hard to find a suitable model system that is easy to analyze. However, liquid crystals are particularly interesting for this purpose because the transport properties and thereby w\_ irr depend on the orientation relative to the streamlines or the temperature gradient, and at certain orientations, w\_ irr is minimal. Thus, it can be determined whether these orientations actually are attained by the liquid crystal. Moreover, it is possible to orient the liquid crystal in an experimental measurement by applying an electric or magnetic field and in molecular dynamics simulations by applying a constraint torque. This means that w\_ irr can be measured or calculated as a function of the orientation relative to the dissipative field.

liquid crystal consisting of rod-like molecules orients perpendicularly to this gradient. This means that the heat flow is minimized, since the heat conductivity is minimal in this orientation. Unfortunately, the results of these works are not wholly conclusive because the underlying experiments are very hard to carry out. On the other hand, molecular dynamics simulation of nematic phases of calamitic and discotic soft ellipsoids [15–17] clearly show that the directors orient perpendicularly and parallel, respectively, to the temperature gradient, so that the heat flow and thereby w\_ irr are minimized. However, one system, where the director definitely orients perpendicularly to the temperature gradient, is the cholesteric liquid crystal, where the cholesteric axis orients parallel to the temperature gradient, so that the director becomes perpendicular to this gradient, and the heat flow is minimized

A nematic phase of the Gay-Berne fluid undergoing planar Couette flow. The velocity gradient is directed in the vertical direction and the streamlines are directed in the horizontal direction. Note that the director forms an

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

This chapter is organized in the following way: in Section 2, the basic theory is outlined, and in Sections 3, 4, and 5, molecular dynamics simulation results and experimental measurements on the director orientation and the irreversible energy dissipation rate are presented and discussed for shear flow or planar Couette flow, planar elongational flow and heat conduction, respectively. In Section 6, the effects of the thermostat are discussed, and finally in Section 7, there is a conclusion. Some

In order to describe transport properties of a liquid crystal, we must first define the order parameter, the director, and the director angular velocity. In an axially symmetric system such as a nematic or a smectic A liquid crystal, the order param-

> <sup>u</sup>^iu^<sup>i</sup> � <sup>1</sup> 3 1

, (1)

[3, 4, 18–20], which is in agreement with the variational principle.

2.1 Order parameter, director, and director angular velocity

eter, S, is given by the largest eigenvalue of the order tensor,

<sup>Q</sup> <sup>¼</sup> <sup>3</sup> 2

1 <sup>N</sup> <sup>∑</sup> N i¼1

background theory is given in the Appendices.

2. Basic theory

125

Figure 1.

angle with streamlines of about 18°.

DOI: http://dx.doi.org/10.5772/intechopen.80977

The simplest kind of liquid crystal is the nematic liquid crystal [3, 4]. It consists of rod-like or plate-like molecules oriented in a certain direction—the director—but there is no translational order, see Figure 1. A nematic liquid crystal cannot support shear stresses, so it is by definition a liquid, but it can support torques, which is the basis for various orientation phenomena relative to external fields. A special case of a nematic liquid crystal is the cholesteric liquid crystal, where the director rotates in space around an axis perpendicular to itself—the cholesteric axis or the optical axis. The spatial rotation period or the pitch is of the order of 1 μm or about 500 molecular diameters. A cholesteric liquid crystal is different from its mirror image, and it is formed by chiral molecules.

There is some theoretical and experimental evidence indicating that the director comes to rest in an orientation where the irreversible energy dissipation rate is minimal in accordance with the variational principle. More specifically, such orientation phenomena have been observed in simulations of shear flow or planar Couette flow [5, 6], in experimental measurements of the viscosity [7] in this flow geometry, and in simulations of planar elongational flow [8]. In the latter case, it is actually possible to prove that the energy dissipation rate must be either minimal or maximal in a steady state in the linear or Newtonian regime by using the linear phenomenological relations between the velocity gradient and the shear stress.

In the case of a nematic liquid crystal subject to a temperature gradient, there are quite a few early experimental works [9–14] that might imply that the director of a

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

### Figure 1.

theorem originally proposed by Ilya Prigogine stating that a quantity, known as the irreversible energy dissipation rate, w\_ irr, is minimal in the linear regime of a nonequilibrium steady state has recently been proven [1]. This quantity is defined as the irreversible work per unit time and unit volume that is done by a dissipative

This theorem is not only of basic scientific interest but also of technological and practical interest since shear fields, temperature gradients, concentration gradients, or chemical potential gradients and electrical potential gradients are common examples of external dissipative fields that are ubiquitous in industrial applications and in everyday life. For example, in a lubricated bearing, a planar Couette flow arises in the lubricant in the narrow space between two surfaces rotating at different angular velocities, and w\_ irr is equal to the product of the shear rate and the shear stress. Another example is the heat flow between a hot region and a cold region such as the inside and the outside of a building. Then, w\_ irr is equal to the product of the heat flow and the temperature gradient. Still another example is an electric heating element where an electric potential difference or voltage drives an electric current, and w\_ irr is equal to the product of the voltage and the current. Finally, chemical potential gradients arise when various substances are mixed and they begin to diffuse, and w\_ irr is equal to the product of the chemical potential gradients and the

One way of testing this principle is to perform molecular dynamics simulations of microscopic model systems, but then it is hard to find a suitable model system that is easy to analyze. However, liquid crystals are particularly interesting for this purpose because the transport properties and thereby w\_ irr depend on the orientation relative to the streamlines or the temperature gradient, and at certain orientations, w\_ irr is minimal. Thus, it can be determined whether these orientations actually are attained by the liquid crystal. Moreover, it is possible to orient the liquid crystal in an experimental measurement by applying an electric or magnetic field and in molecular dynamics simulations by applying a constraint torque. This means that w\_ irr can be measured or calculated as a function of the orientation relative to the

The simplest kind of liquid crystal is the nematic liquid crystal [3, 4]. It consists of rod-like or plate-like molecules oriented in a certain direction—the director—but there is no translational order, see Figure 1. A nematic liquid crystal cannot support shear stresses, so it is by definition a liquid, but it can support torques, which is the basis for various orientation phenomena relative to external fields. A special case of a nematic liquid crystal is the cholesteric liquid crystal, where the director rotates in space around an axis perpendicular to itself—the cholesteric axis or the optical axis.

The spatial rotation period or the pitch is of the order of 1 μm or about 500

molecular diameters. A cholesteric liquid crystal is different from its mirror image,

comes to rest in an orientation where the irreversible energy dissipation rate is minimal in accordance with the variational principle. More specifically, such orientation phenomena have been observed in simulations of shear flow or planar Couette flow [5, 6], in experimental measurements of the viscosity [7] in this flow geometry, and in simulations of planar elongational flow [8]. In the latter case, it is actually possible to prove that the energy dissipation rate must be either minimal or maximal in a steady state in the linear or Newtonian regime by using the linear phenomenological relations between the velocity gradient and the shear stress.

There is some theoretical and experimental evidence indicating that the director

In the case of a nematic liquid crystal subject to a temperature gradient, there are quite a few early experimental works [9–14] that might imply that the director of a

external field on the system [2]. Thus, there is a variational principle for

nonequilibrium steady states.

Non-Equilibrium Particle Dynamics

matter currents.

dissipative field.

124

and it is formed by chiral molecules.

A nematic phase of the Gay-Berne fluid undergoing planar Couette flow. The velocity gradient is directed in the vertical direction and the streamlines are directed in the horizontal direction. Note that the director forms an angle with streamlines of about 18°.

liquid crystal consisting of rod-like molecules orients perpendicularly to this gradient. This means that the heat flow is minimized, since the heat conductivity is minimal in this orientation. Unfortunately, the results of these works are not wholly conclusive because the underlying experiments are very hard to carry out. On the other hand, molecular dynamics simulation of nematic phases of calamitic and discotic soft ellipsoids [15–17] clearly show that the directors orient perpendicularly and parallel, respectively, to the temperature gradient, so that the heat flow and thereby w\_ irr are minimized. However, one system, where the director definitely orients perpendicularly to the temperature gradient, is the cholesteric liquid crystal, where the cholesteric axis orients parallel to the temperature gradient, so that the director becomes perpendicular to this gradient, and the heat flow is minimized [3, 4, 18–20], which is in agreement with the variational principle.

This chapter is organized in the following way: in Section 2, the basic theory is outlined, and in Sections 3, 4, and 5, molecular dynamics simulation results and experimental measurements on the director orientation and the irreversible energy dissipation rate are presented and discussed for shear flow or planar Couette flow, planar elongational flow and heat conduction, respectively. In Section 6, the effects of the thermostat are discussed, and finally in Section 7, there is a conclusion. Some background theory is given in the Appendices.
