1. Introduction

When a fluid is in thermodynamic equilibrium, its state variables always present spontaneous microscopic fluctuations due to the thermal excitations of its molecules, producing deviations around the state of equilibrium. The theory of fluctuations for fluids in states close to equilibrium was initiated long ago by Einstein and Onsager, and it has been reformulated in several but equivalent ways. The first more systematic approach to introduce thermal fluctuations into the hydrodynamic equations was the fluctuating hydrodynamics (FH) of Landau and Lifshitz [1, 2]. It stems from the idea that the hydrodynamic equations are valid for any flow, including fluctuating changes in its state. Accordingly, stochastic currents are incorporated into the deterministic energy and momentum fluxes by adding fluctuating sources. This theory was put on a firm basis within the framework of the general theory of stationary Gaussian Markov processes by Fox and Uhlenbeck [3–5]. This approach has matched the theory of Onsager and Machlup with that of Landau and Lifshitz for systems where the basic state variables do not possess a definite time reversal symmetry [6, 7]. However, in spite of the fact that the theory of fluctuations for nonequilibrium fluids was initiated in the late 1970s, and was pursued by many authors [8], still nowadays several questions concerning the nature of hydrodynamic fluctuations in stationary nonequilibrium states (NESS) are of current active interest. One of these issues is the long-range character of these fluctuations, especially far away from instability points [9]. Thermal fluctuations in an equilibrium fluid always give rise to short-range equal time correlation functions, except close to a critical point. But when external gradients are applied, equal-time correlation functions can develop long-range contributions, whose nature is very different from those in equilibrium. For many models and systems in nonequilibrium states, it has been shown theoretically that the existence of the so-called generic scale invariance is the origin of the long-range nature of the correlation functions [10, 11].

time-dependent correlation functions in equilibrium between the fluctuating hydrodynamic variables, quantities that allow to obtain the transport properties of the system [27, 28]. One of these properties is the dynamic structure factor S k!

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

of the system, which measures the magnitude of the changes in energy and

! , S k! ;ω

and ω.

For simple fluids with fixed k

DOI: http://dx.doi.org/10.5772/intechopen.82609

to the absorption of sound.

modes of a NLC is still lacking.

147

momentum between the light beam and the fluid as functions of the wave vector <sup>k</sup>

Lorentzian features: a line or central peak (Rayleigh peak) located at ω ¼ 0 and two Brillouin peaks symmetrically located with respect to the central one [29, 30]. These three lines are directly related to the hydrodynamic modes of the simple fluid, and from them, it is possible to obtain relevant information about transport properties. For instance, the Rayleigh line, associated with a thermal diffusive mode, is due to the fluctuations of the entropy (or temperature) that diffuse in the fluid and its width is proportional to the thermal diffusivity. On the other hand, the Brillouin lines are related to two acoustic propagative modes and are the result of the coupled dynamics of the pressure fluctuations and a component of the flow velocity that are transmitted with the speed of sound in the medium. Their widths are proportional

In the case of an anisotropic system like a NLC, fluctuating hydrodynamic theories have recently been proposed [22, 31] based on the methodology proposed by Landau and Lifshitz [1]. However, this analysis of the fluctuations of the nematic hydrodynamic variables is not precise, since it does not take into account the parity with respect to time reversal, so their description using the Onsager-Machlup formalism would be strictly inadequate. The correct theoretical framework should be the more general theory of Fox and Uhlenbeck [3–5, 20, 24]. However, although a NLC disperses light by several orders of magnitude more than an ordinary fluid [32], from both a theoretical and experimental point of view, the studies

corresponding to the behavior of the fluctuations in these media around stationary states out of equilibrium are rather scarce. From the theoretical point of view, and only for the case of the transverse hydrodynamic variables [33], some studies of the behavior of orientational fluctuations have been carried out when analyzing the effect produced in the light scattering spectrum of a NLC in NESS induced by the presence of uniform temperature gradients [17] and by the action of a shear flow [18]. In both cases, it has been found that the effect of fluctuations in the light scattering spectrum is small, being difficult to detect experimentally. On the other hand, as far as we know, no theoretical study has been carried out on the behavior of the longitudinal variables of a nematic and much less on its spectrum of light scattering, both in states of thermodynamic equilibrium and outside of it. This is an open research topic. Nor have been performed analyzes of stationary states generated by other types of external gradients in these systems, with which could be obtained qualitatively and quantitatively much greater effects than those reported so far in the literature for simple fluids. It should be mentioned that although preliminary attempts have been made to calculate the transverse hydrodynamic

modes of a nematic [34, 35], there are few studies that also involve the corresponding longitudinal modes [31]. Unfortunately, a clear and systematic method to derive the set of complete, transverse, and longitudinal hydrodynamic

By introducing an alternative set of state variables that takes into account the asymmetry presented by both, the velocity and the director fields due by their mutual coupling, two groups of fluctuating variables, namely, longitudinal and

consists of three well-separated

;ω 

←

In the case of a simple fluid in a thermal gradient, the structure factor, which determines the intensity of the Rayleigh scattering, diverges as k�<sup>4</sup> for small values of the wave number k. This amounts to an algebraic decay of the density-density correlation function, a feature that has been verified experimentally [12–14]. However, there are few similar studies for NESS of complex fluids. Among these, the enhancement of concentration fluctuations in polymer solutions under external hydrodynamic and electric fields [15], or the case of a polymer solution subjected to a stationary temperature gradient in the absence of any flow [16], has been discussed. Also, the behavior of fluctuations about some NESS has been analyzed in the case of thermotropic nematic liquid crystals. Specific examples are the nonequilibrium situations generated by a static temperature gradient [17], a stationary shear flow [18] or by an externally imposed constant pressure gradient [19, 20]. In the first two cases, it was found that the nonequilibrium contributions to the corresponding light scattering spectrum were small, but in the case of a Poiseuille flow induced by an external pressure gradient, the effect may be quite large. To our knowledge, however, at present, there is no experimental confirmation of these effects, in spite of the fact that for nematics, the scattered intensity is several orders of magnitude larger than for ordinary simple fluids.

When a hydrodynamic system relaxes from a state of thermodynamic equilibrium to another, almost all its degrees of freedom will return to that equilibrium value in a short, finite time τ determined by the microscopic interactions of the system. There are, however, some other degrees of freedom of collective character, the hydrodynamic modes, which will decay much more slowly. When τ ! ∞, its characteristic frequencies ω ! 0 ð Þ ω � 1=τ , when k ! 0: Such is the case, for example, of the propagation of sound waves and the conduction of heat in a simple fluid [21]. Hydrodynamics allows to describe these modes or degrees of freedom of greater duration, through the laws of conservation and balance of the system, and, as in the case of ordered systems, by the continuous breaking of symmetries [22, 23].

The central purpose of this work is to briefly review the general procedure developed by Fox and Uhlenbeck and show that it may be employed to treat fluctuating complex fluid systems like a thermotropic nematic liquid crystal (NLC) in a NESS. In particular, we describe the dynamics of the fluctuations of its hydrodynamic variables induced by a stationary temperature gradient and under the influence of gravity (a Rayleigh-Bénard system) on a nematic layer confined between two parallel horizontal plates in a steady state in a nonconvective regime [24–26]. Once the dynamics of fluctuation is established, we calculate the

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal DOI: http://dx.doi.org/10.5772/intechopen.82609

time-dependent correlation functions in equilibrium between the fluctuating hydrodynamic variables, quantities that allow to obtain the transport properties of the system [27, 28]. One of these properties is the dynamic structure factor S k! ;ω of the system, which measures the magnitude of the changes in energy and momentum between the light beam and the fluid as functions of the wave vector <sup>k</sup> ← and ω.

For simple fluids with fixed k ! , S k! ;ω consists of three well-separated Lorentzian features: a line or central peak (Rayleigh peak) located at ω ¼ 0 and two Brillouin peaks symmetrically located with respect to the central one [29, 30]. These three lines are directly related to the hydrodynamic modes of the simple fluid, and from them, it is possible to obtain relevant information about transport properties. For instance, the Rayleigh line, associated with a thermal diffusive mode, is due to the fluctuations of the entropy (or temperature) that diffuse in the fluid and its width is proportional to the thermal diffusivity. On the other hand, the Brillouin lines are related to two acoustic propagative modes and are the result of the coupled dynamics of the pressure fluctuations and a component of the flow velocity that are transmitted with the speed of sound in the medium. Their widths are proportional to the absorption of sound.

In the case of an anisotropic system like a NLC, fluctuating hydrodynamic theories have recently been proposed [22, 31] based on the methodology proposed by Landau and Lifshitz [1]. However, this analysis of the fluctuations of the nematic hydrodynamic variables is not precise, since it does not take into account the parity with respect to time reversal, so their description using the Onsager-Machlup formalism would be strictly inadequate. The correct theoretical framework should be the more general theory of Fox and Uhlenbeck [3–5, 20, 24]. However, although a NLC disperses light by several orders of magnitude more than an ordinary fluid [32], from both a theoretical and experimental point of view, the studies corresponding to the behavior of the fluctuations in these media around stationary states out of equilibrium are rather scarce. From the theoretical point of view, and only for the case of the transverse hydrodynamic variables [33], some studies of the behavior of orientational fluctuations have been carried out when analyzing the effect produced in the light scattering spectrum of a NLC in NESS induced by the presence of uniform temperature gradients [17] and by the action of a shear flow [18]. In both cases, it has been found that the effect of fluctuations in the light scattering spectrum is small, being difficult to detect experimentally. On the other hand, as far as we know, no theoretical study has been carried out on the behavior of the longitudinal variables of a nematic and much less on its spectrum of light scattering, both in states of thermodynamic equilibrium and outside of it. This is an open research topic. Nor have been performed analyzes of stationary states generated by other types of external gradients in these systems, with which could be obtained qualitatively and quantitatively much greater effects than those reported so far in the literature for simple fluids. It should be mentioned that although preliminary attempts have been made to calculate the transverse hydrodynamic modes of a nematic [34, 35], there are few studies that also involve the corresponding longitudinal modes [31]. Unfortunately, a clear and systematic method to derive the set of complete, transverse, and longitudinal hydrodynamic modes of a NLC is still lacking.

By introducing an alternative set of state variables that takes into account the asymmetry presented by both, the velocity and the director fields due by their mutual coupling, two groups of fluctuating variables, namely, longitudinal and

This approach has matched the theory of Onsager and Machlup with that of Landau and Lifshitz for systems where the basic state variables do not possess a definite time reversal symmetry [6, 7]. However, in spite of the fact that the theory of fluctuations for nonequilibrium fluids was initiated in the late 1970s, and was pursued by many authors [8], still nowadays several questions concerning the nature of hydrodynamic fluctuations in stationary nonequilibrium states (NESS) are of current active interest. One of these issues is the long-range character of these fluctuations, especially far away from instability points [9]. Thermal fluctuations in an equilibrium fluid always give rise to short-range equal time correlation functions, except close to a critical point. But when external gradients are applied, equal-time correlation functions can develop long-range contributions, whose nature is very different from those in equilibrium. For many models and systems in nonequilibrium states, it has been shown theoretically that the existence of the so-called generic scale invariance is the origin of the long-range nature of the

In the case of a simple fluid in a thermal gradient, the structure factor, which determines the intensity of the Rayleigh scattering, diverges as k�<sup>4</sup> for small values of the wave number k. This amounts to an algebraic decay of the density-density correlation function, a feature that has been verified experimentally [12–14]. However, there are few similar studies for NESS of complex fluids. Among these, the enhancement of concentration fluctuations in polymer solutions under external hydrodynamic and electric fields [15], or the case of a polymer solution subjected to

a stationary temperature gradient in the absence of any flow [16], has been

the case of thermotropic nematic liquid crystals. Specific examples are the nonequilibrium situations generated by a static temperature gradient [17], a stationary shear flow [18] or by an externally imposed constant pressure gradient [19, 20]. In the first two cases, it was found that the nonequilibrium contributions to the corresponding light scattering spectrum were small, but in the case of a Poiseuille flow induced by an external pressure gradient, the effect may be quite large. To our knowledge, however, at present, there is no experimental confirmation of these effects, in spite of the fact that for nematics, the scattered intensity is

several orders of magnitude larger than for ordinary simple fluids.

discussed. Also, the behavior of fluctuations about some NESS has been analyzed in

When a hydrodynamic system relaxes from a state of thermodynamic equilibrium to another, almost all its degrees of freedom will return to that equilibrium value in a short, finite time τ determined by the microscopic interactions of the system. There are, however, some other degrees of freedom of collective character, the hydrodynamic modes, which will decay much more slowly. When τ ! ∞, its characteristic frequencies ω ! 0 ð Þ ω � 1=τ , when k ! 0: Such is the case, for example, of the propagation of sound waves and the conduction of heat in a simple fluid [21]. Hydrodynamics allows to describe these modes or degrees of freedom of greater duration, through the laws of conservation and balance of the system, and, as in the case of ordered systems, by the continuous breaking of symmetries

The central purpose of this work is to briefly review the general procedure developed by Fox and Uhlenbeck and show that it may be employed to treat fluctuating complex fluid systems like a thermotropic nematic liquid crystal (NLC) in a NESS. In particular, we describe the dynamics of the fluctuations of its hydrodynamic variables induced by a stationary temperature gradient and under the influence of gravity (a Rayleigh-Bénard system) on a nematic layer confined between two parallel horizontal plates in a steady state in a nonconvective regime [24–26]. Once the dynamics of fluctuation is established, we calculate the

correlation functions [10, 11].

Non-Equilibrium Particle Dynamics

[22, 23].

146

transverse, can be clearly identified. Both set of variables are completely decoupled: there are five in the first and two in the second group. The longitudinal variables in turn can be separated into two mutually independent sets. The first is composed of two variables whose dynamics determine the existence of acoustic propagation modes; while the second, formed by three variables, giving rise to three hydrodynamic modes: one related to the orientation of the director and two more, the socalled visco-heat modes, that result from the coupling of the thermal diffusive and shear modes due by the presence of the gradient thermal and the gravitational field. As will be discussed later on, from the set of transverse variables, two hydrodynamic modes emerge: one due to the orientation of the director and another one due to shearing. Altogether, there are seven nematic hydrodynamic modes: five longitudinal and two transversal. As will be shown below, the applied gradient of temperature and gravitational field produce their greatest effect in the pair of viscoheat modes, which is quantified in them by means of the Rayleigh quotient R=Rc, where R is the number of Rayleigh and Rc the value that it reaches when in the nematic initiates the convection.

translational, Tð Þ3 , transformations. Thus, the group of symmetries of an isotropic liquid is Oð Þ� 3 Tð Þ3 : However, by decreasing the temperature of these liquids, the translational symmetry Tð Þ3 is usually broken corresponding to the isotropic liquid-solid transition. In contrast, for a liquid formed by anisotropic molecules, by diminishing the temperature, the rotational symmetry Oð Þ3 is broken, which leads to the appearance of a liquid crystal. The mesophases for which only the rotational invariance has been broken are called nematics. As shown, the centers of mass of the molecules of a nematic have arbitrary positions, whereas the principal axes of their molecules are spontaneously oriented along a preferred direction. If the temperature decreases even more, the symmetry Tð Þ3 is also partially broken. The mesophases exhibiting the translational symmetry Tð Þ2 are

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

This preferential direction is described by a local unitary vector field, n^, called the director. This vector is easily distorted by the presence of electric and magnetic fields, as well as by the surfaces of the containers of the liquid crystals if they have been prepared properly [32]. With respect to NLC, it is important to point out that the director's orientation does not distinguish between the n^ and �n^ directions (nematic symmetry). A schematic representation of the order presented by the

Consider a NLC thin layer of thickness d under the presence of a constant

induces a pressure gradient, ∇zp ¼ �ρg^z, where ρ is the mass density.

along the z axis. The initial configuration of the layer is homeotropic with a preferential orientation n^<sup>0</sup> along the z axis, as depicted in Figure 2. The nematic is confined between two parallel flat plates kept at fixed temperatures T<sup>1</sup> and T<sup>2</sup> (T<sup>1</sup> , T2), so that a uniform temperature gradient ∇zT � �α^z is established downward in the layer. The situation where the temperature gradient goes from bottom to top can also be considered, and in this case, ∇zT � α^z. The gravitational force

!¼ �g^z, where g denotes its magnitude and ^z the unit vector

! and an external uniform temperature gradient ∇T. k

! is

called smectics [36].

3. Model

Figure 2.

149

the scattering vector.

The NLC cell subject to a constant gravitational field g

gravitational field g

molecules in a nematic is shown in Figure 1 .

DOI: http://dx.doi.org/10.5772/intechopen.82609
