## Abstract

We use a fluctuating hydrodynamics (FH) approach to study the fluctuations of the hydrodynamic variables of a thermotropic nematic liquid crystal (NLC) in a nonequilibrium steady state (NESS). This NESS is produced by an externally imposed temperature gradient and a uniform gravity field. We calculate analytically the equilibrium and nonequilibrium seven modes of the NLC in this NESS. These modes consist of a pair of sound modes, one orientation mode of the director and two visco-heat modes formed by the coupling of the shear and thermal modes. We find that the nonequilibrium effects produced by the external gradients only affect the longitudinal modes. The analytic expressions for the visco-heat modes show explicitly how the heat and shear modes of the NLC are coupled. We show that they may become propagative, a feature that also occurs in the simple fluid and suggests the realization of new experiments. We show that in equilibrium and in the isotropic limit of the NLC, our modes reduce to well-known results in the literature. For the NESS considered, we point out the differences between our modes and those reported by other authors. We close the chapter by proposing the calculation of other physical quantities that lend themselves to a more direct comparison with possible experiments for this system.

### Keywords

- fluctuating hydrodynamics
- nonequilibrium fluctuations
- hydrodynamic modes
- thermotropic nematic liquid crystals

## 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 (

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*. This amounts to an algebraic decay of the density-density correlation function, a feature that has been verified experimentally [12, 13, 14]. However, there are few similar studies for

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

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 (

For simple fluids with fixed

In the case of an anisotropic system like a

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 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 so-called 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 visco-heat modes, which is quantified in them by means of the Rayleigh quotient

## 2. Liquid crystalline phases

The liquid crystal phase is a well-defined and specific phase of matter characterized by a remarkable anisotropy in many of their physical properties as solid crystals do, although they are able to flow. Liquid crystal phases that undergo a phase transition as a function of temperature (thermotropics) exist in relatively small intervals of temperature lying between solid crystals and isotropic liquids. Due to this intermediate nature, sometimes, these states are called also mesophases [32]. In general, liquid crystals are synthesized from organic molecules, some of which are elongated and uniaxial, so they can be represented as rigid rods; others are formed by disc-like molecules [35]. This molecular anisotropy is manifested macroscopically through the anisotropy of the mechanical, optical, and transport properties of these substances. The typical dimensions of the lengths of this type of structures are some tens of angstroms.

Liquid crystals are classified by symmetry. As it is well known, isotropic liquids with spherically symmetric molecules are invariant under rotational,

This preferential direction is described by a local unitary vector field,

## 3. Model

Consider a

### 3.1 Stationary state

The external gradients drive the nematic layer into a nonequilibrium steady state. We shall assume that the temperature difference

where *effective temperature gradient*

which contains explicitly the contributions of both external forces. In Eq. (2),

## 4. Nematodynamic equations

The geometry of the proposed model allows us to separate the hydrodynamic variables into transverse (

In order to take into account the effect of the intrinsic anisotropy of the fluid in the dynamics of the fluctuations, as well as to facilitate the calculation of the nematic modes and the spectrum of light scattering, it is convenient to introduce a new state variables. In the case of the present model, owing to the initial orientation of the director

where

However, in order to facilitate the calculation of the hydrodynamic modes, we define a new set of variables having the same dimensionality,

where

## 5. Hydrodynamic modes

In order to find the hydrodynamic modes, or decay rates [37], we need the Fourier transform of the linear system (4), which yields an algebraic system of equations in terms of the variables

### 5.1 Longitudinal modes

Following the method proposed by [13] for a simple fluid, it can be shown that longitudinal variables can be separated in turn and within a very good approximation, into two completely independent sets of variables,

While there is no analytical difficulty to solve the quadratic and cubic equations

The relevant point for our purpose is to realize that for most nematics at ambient temperatures,

#### 5.1.1 Sound longitudinal modes

They are the roots of the characteristic equation

where

#### 5.1.2 Visco-heat and director longitudinal modes

These modes are the roots of the characteristic equation

and

with

where

The Rayleigh-number ratio

The decay rates

#### 5.1.3 Values of R k → / R c

The three nematic modes (7) and (8) could be two propagative and one diffusive, or all of them completely diffusive; its nature depends on the values assumed by the ratio

#### 5.1.3.1 Propagative and diffusive modes

If we take into account the orders of magnitude of the small quantities (Eq. (5)), the nematic modes (7) and (8) in general are real and different. Nevertheless, it may happen that these modes may be transformed into one real and two complex conjugate roots. This occurs if

which is always negative. Thus, if we consider the orders of magnitude of the involved quantities and typical light scattering experiment values of

#### 5.1.3.2 Pure diffusive modes

When

and

Since for nematics,

while the third,

It should be noted that our expressions for these three decay rates are not in agreement with those reported for an

### 5.2 Transverse modes

As mentioned earlier,

#### 5.2.1 Shear and director transverse modes

Accordingly, the shear and director transverse modes are the roots of

It should be noted that these shear and director diffusive transverse modes also match completely with those already reported for nematics [22, 31, 32].

## 6. The equilibrium and simple fluid limits

From the hydrodynamic modes calculated for an

### 6.1 Nematic in equilibrium

It has been found that for an

another of shear:

and one more of the director,

### 6.2 Simple fluid in a Rayleigh-Bénard system

Given that in the isotropic limit (simple fluid limit), the degree of nematic order goes to zero,

where

In the isotropic limit of the simple fluid,

In Eq. (18), the ratio

which, in this limit case, can be derived from Eq. (9). It should be pointed out that Eq. (20) coincides with the Eq. (2.21) of reference [37]. The modes (17)–(19) are in complete concordance with those analytically calculated in [8, 37, 38].

Moreover, if in the coefficient matrix

#### 6.2.1 Values of R k → / R c

The two visco-heat mode, as in the nematic, could be propagative or diffusive. These characteristics depend on the values assumed by the ratio

#### 6.2.1.1 Propagative modes

If

#### 6.2.1.2 Pure diffusive modes

When

On the other hand, if the simple fluid is in a state of homogeneous thermodynamic equilibrium,

and the shear mode:

These decay rates are well known in the literature [8, 37, 38]. Finally, because in a simple fluid, commonly

and

These three cases are consistent with those obtained in analytical studies already reported for simple fluids in this regime [8, 37, 38]. Schematically, its behavior is very similar to that illustrated in Figure 3, and this can be seen in Figure 1 of the reference [37].

## 7. Conclusions

In this work, we have used the standard formulation of

First, in our analysis, the symmetry properties of the nematic are taken into consideration, and this allowed us to separate its hydrodynamic variables into two completely independent sets: one longitudinal, composed of five variables, and the other transverse, consisting of only two variables. From the equations that govern the dynamics of the variables in these sets, the corresponding hydrodynamic modes were calculated. The longitudinal modes are two acoustic,

The analytical expressions calculated for the hydrodynamic modes of a nematic in the

However, it should be mentioned that our hydrodynamic modes

Nevertheless, our calculated expressions for the visco-heat

Finally, it should be noted that this theory can be useful, since the description of some characteristics of our model lend themselves to establish a more direct contact with the experiment. Actually, physical quantities, such as director-director and density-density correlation functions, memory functions or the dynamic structure factor

## Acknowledgments

We thank the UACM for the economic facilities granted to cover the total cost of the publication of this research work.

## Conflict of interest

The authors declare that they have no conflict of interests.

## Nomenclature

wave number

angular frequency

relaxation time of almost all degrees of freedom

dynamic structure factor

wave vector

Rayleigh number

Rayleigh number at the convection threshold

Rayleigh ratio

orientation symmetry group

translation symmetry group

director field

thickness of the nematic layer

constant gravitational force of magnitude

Cartesian unitary vectors

Cartesian coordinates

temperature

temperature gradient of magnitude

hydrostatic pressure

pressure gradient

volumetric density of mass

flow velocity

specific density of entropy (entropy per unit mass)

position vector

temperature difference between the plates of the cell

effective temperature gradient

coefficient of thermal expansivity

specific heat at constant pressure

specific heat at constant volume

ratio of specific heats

adiabatic sound velocity

isothermic sound velocity

set of nematodynamic variables

divergence of

component

component

divergence of

component

as superscript, indicates transpose matrix

vector whose components are the spatial Fourier transform of the variables

longitudinal component of

transverse component of

coefficient matrix of the linear system for

longitudinal and transverse submatrices of

stochastic vector of the linear system for

longitudinal component of

transverse component of

variables of same dimensionality (

vector of the variables

longitudinal component of

transverse component of

coefficient matrix of the linear system for

longitudinal and transverse submatrices of

noise vector of the linear system for

longitudinal component of

transverse component of

noise components of

characteristic polynomial of the matrix

characteristic polynomial of the submatrix

characteristic polynomial of the submatrix

eigenvalues of

components of the vector

polynomials in which

anisotropic thermal coefficient

anisotropic viscous coefficients

thermal diffusivities parallel and perpendicular to

anisotropic thermal diffusivity

nematic viscosities (

anisotropic adimensional nematic coefficients

elastic coefficients of Frank

anisotropic elastic coefficients

torsion viscosity

auxiliary parameter

small dimensionless longitudinal quantities

small dimensionless longitudinal quantities

small dimensionless longitudinal quantities

acoustic propagative longitudinal modes

anisotropic sound attenuation coefficient

visco-heat longitudinal modes

director diffusive longitudinal mode

components of

unit vector of

reference value of the Rayleigh ratio below which visco-caloric modes are propagative

small dimensionless transverse quantities

shear and director diffusive transverse modes

nematic modes in the state of equilibrium (

thermal diffusivity of a simple fluid

shear and volumetric viscosities of a simple fluid

kinetic viscosity of a simple fluid

attenuation coefficient of sound in a simple fluid