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 R k � �! =Rc , R0, where

$$R\_0 \equiv -\frac{\left[\left(\sigma\_3 - \frac{\Omega^2 K\_l}{\rho\_0 \sigma\_3}\right) - D\_T\right]^2}{4D\_T \sigma\_3},\tag{10}$$

propagation modes, this prediction suggests that it may be worth to design experi-

The real part of the nematic visco-heat modes <sup>λ</sup><sup>3</sup> and <sup>λ</sup><sup>4</sup> as a function of the Rayleigh ratio R k � �!

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

Eq. (7) and the other one remains identical to Eq. (8), but all are real and completely diffusive. In this regime, the following cases are of special interest. For instance, if

=Rc ¼ R0, then the visco-heat modes (7) reach the same value, and conse-

DTk<sup>2</sup> <sup>þ</sup> <sup>σ</sup>3k<sup>2</sup> � <sup>Ω</sup><sup>2</sup>

and λ5, that takes the same form as in Eq. (8). These visco-heat modes are

sive mode decreases, whereas the one of the shear mode increases. At the onset of

<sup>λ</sup><sup>4</sup> <sup>¼</sup> DTk<sup>2</sup> <sup>þ</sup> <sup>σ</sup>3k<sup>2</sup> � <sup>Ω</sup><sup>2</sup>

!

=Rc ≤1, the two visco-heat modes preserve the same form as in

=Rc ≤1, both are completely diffusive. For

=Rc ¼ 0, and the onset of convection occurs for

KIk<sup>4</sup> ρ0σ3k<sup>2</sup>

<sup>ρ</sup>0σ<sup>3</sup> is usually greater than DT, it can be seen from

=Rc ¼ 1, i. e., when R reaches its critical value Rc and the

KIk<sup>4</sup>

λ<sup>3</sup> ¼ 0, (12)

<sup>ρ</sup>0σ3k<sup>2</sup> , (13)

� �=Rc grows and approaches 1, the magnitude of the heat diffu-

, (11)

=Rc. When

ments to corroborate this phenomenon for nematics.

<sup>=</sup>Rc , <sup>R</sup>0, both modes are propagative; if <sup>R</sup><sup>0</sup> <sup>≤</sup> R k � �!

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

<sup>=</sup>Rc <sup>¼</sup> <sup>R</sup>0, both modes are equal. In equilibrium, R k � �!

<sup>λ</sup>3,<sup>4</sup> <sup>¼</sup> <sup>1</sup> 2

identified at the vertex of the parabola in Figure 3.

5.1.3.2 Pure diffusive modes

When <sup>R</sup><sup>0</sup> <sup>≤</sup>R k � �!

quently, the three decay rates are:

Since for nematics, <sup>σ</sup><sup>3</sup> � <sup>Ω</sup><sup>2</sup>KI

two visco-heat modes (7) are simplified to:

Eq. (7) that, as R ek

convections regime, R k � �!

R k � �!

155

Figure 3.

R k � �!

R k � �!

R k � �!

=Rc ¼ 1.

which is always negative. Thus, if we consider the orders of magnitude of the involved quantities and typical light scattering experiment values of <sup>k</sup>, DTk<sup>2</sup> � <sup>10</sup><sup>7</sup> , <sup>σ</sup>3k<sup>2</sup> � <sup>10</sup>8, and <sup>Ω</sup><sup>2</sup>KIk<sup>4</sup> ρ0 � 1014, then <sup>R</sup><sup>0</sup> ffi �10<sup>1</sup> and the visco-heat modes, Eq. (7), will be propagative when R k � �! <sup>=</sup>Rc <sup>≲</sup> � <sup>10</sup><sup>1</sup> . This situation corresponds to the propagation region indicated in Figure 3. The decay rate λ5, Eq. (8), remains to be real. It is worth emphasizing that this case corresponds to overstabilized states, where out of the three decay rates, two are propagative visco-heat modes and the other one is completely diffusive. According to Eq. (9), this occurs if the α contained in the effective temperature gradient X changes its sign and increases by several orders of magnitude, situation that may be achieved by reversing the direction in which the temperature gradient is applied, i. e., when heating from below, and by increasing its intensity. As far as we know, there are no theoretical analyses nor experimental evidence for the existence of visco-heat propagating modes in NLC under the presence of a temperature gradient and a uniform gravitational field. Given that in simple fluids, under these conditions, there are analytical [8, 37, 38] and experimental [43] studies that support the presence of visco-heat

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

### Figure 3.

The Rayleigh-number ratio R k

Non-Equilibrium Particle Dynamics

thermal DTk<sup>2</sup> and shear <sup>σ</sup>3k<sup>2</sup> � <sup>Ω</sup><sup>2</sup>KIk<sup>4</sup>

� �! =Rc

� �!

experimentally for nematics.

5.1.3.1 Propagative and diffusive modes

conjugate roots. This occurs if R k

ρ0

will be propagative when R k

<sup>σ</sup>3k<sup>2</sup> � <sup>10</sup>8, and <sup>Ω</sup><sup>2</sup>KIk<sup>4</sup>

154

order 10�16.

5.1.3 Values of R k

by the ratio R k

� �!

due to the presence of the effective temperature gradient X, which depends on both, the temperature gradient α and the gravity field g. The second term is entirely a contribution due to α and the nematic anisotropy χa. For typical nematics and conventional light scattering experiments, both contributions are approximately of

The decay rates λ<sup>3</sup> and λ<sup>4</sup> for an inhomogeneous nematic given by Eq. (7) are called visco-heat modes, because they are composed of the coupling between the

The nature of these modes may be propagative or diffuse, as will be shown below.

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

retically and corroborated experimentally, but to our knowledge, not for an NLC. In this sense, the following results suggest that it might be feasible to be also verified

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

=Rc , R0, where

h i<sup>2</sup>

4DTσ<sup>3</sup>

� DT

� 1014, then <sup>R</sup><sup>0</sup> ffi �10<sup>1</sup> and the visco-heat modes, Eq. (7),

. This situation corresponds to the

, (10)

,

<sup>σ</sup><sup>3</sup> � <sup>Ω</sup><sup>2</sup>KI ρ0σ<sup>3</sup> � �

which is always negative. Thus, if we consider the orders of magnitude of the involved quantities and typical light scattering experiment values of <sup>k</sup>, DTk<sup>2</sup> � <sup>10</sup><sup>7</sup>

propagation region indicated in Figure 3. The decay rate λ5, Eq. (8), remains to be real. It is worth emphasizing that this case corresponds to overstabilized states, where out of the three decay rates, two are propagative visco-heat modes and the other one is completely diffusive. According to Eq. (9), this occurs if the α

contained in the effective temperature gradient X changes its sign and increases by several orders of magnitude, situation that may be achieved by reversing the direction in which the temperature gradient is applied, i. e., when heating from below, and by increasing its intensity. As far as we know, there are no theoretical analyses nor experimental evidence for the existence of visco-heat propagating modes in NLC under the presence of a temperature gradient and a uniform gravitational field. Given that in simple fluids, under these conditions, there are analytical [8, 37, 38] and experimental [43] studies that support the presence of visco-heat

<sup>=</sup>Rc <sup>≲</sup> � <sup>10</sup><sup>1</sup>

� �!

R<sup>0</sup> � �

� �!

=Rc. For simple fluids, these features have been predicted theo-

=Rc contains two contributions: the first term is

� �! =Rc.

<sup>ρ</sup>0σ3k<sup>2</sup> diffusive modes through the ratio R k

The real part of the nematic visco-heat modes <sup>λ</sup><sup>3</sup> and <sup>λ</sup><sup>4</sup> as a function of the Rayleigh ratio R k � �! =Rc. When R k � �! <sup>=</sup>Rc , <sup>R</sup>0, both modes are propagative; if <sup>R</sup><sup>0</sup> <sup>≤</sup>R k � �! =Rc ≤1, both are completely diffusive. For R k � �! <sup>=</sup>Rc <sup>¼</sup> <sup>R</sup>0, both modes are equal. In equilibrium, R k � �! =Rc ¼ 0, and the onset of convection occurs for R k � �! =Rc ¼ 1.

propagation modes, this prediction suggests that it may be worth to design experiments to corroborate this phenomenon for nematics.

### 5.1.3.2 Pure diffusive modes

When <sup>R</sup><sup>0</sup> <sup>≤</sup>R k � �! =Rc ≤1, the two visco-heat modes preserve the same form as in Eq. (7) and the other one remains identical to Eq. (8), but all are real and completely diffusive. In this regime, the following cases are of special interest. For instance, if R k � �! =Rc ¼ R0, then the visco-heat modes (7) reach the same value, and consequently, the three decay rates are:

$$
\lambda\_{3,4} = \frac{1}{2} \left( D\_T k^2 + \sigma\_3 k^2 - \frac{\Omega^2 K\_I k^4}{\rho\_0 \sigma\_3 k^2} \right),
\tag{11}
$$

and λ5, that takes the same form as in Eq. (8). These visco-heat modes are identified at the vertex of the parabola in Figure 3.

Since for nematics, <sup>σ</sup><sup>3</sup> � <sup>Ω</sup><sup>2</sup>KI <sup>ρ</sup>0σ<sup>3</sup> is usually greater than DT, it can be seen from Eq. (7) that, as R ek � �=Rc grows and approaches 1, the magnitude of the heat diffusive mode decreases, whereas the one of the shear mode increases. At the onset of convections regime, R k � �! =Rc ¼ 1, i. e., when R reaches its critical value Rc and the two visco-heat modes (7) are simplified to:

$$
\lambda\_3 = \mathbf{0},
\tag{12}
$$

$$
\lambda\_4 = D\_T k^2 + \sigma\_3 k^2 - \frac{\Omega^2 K\_I k^4}{\rho\_0 \sigma\_3 k^2},
\tag{13}
$$

while the third, λ5, is identical to Eq. (8). This behavior for the decay rates λ<sup>3</sup> and λ<sup>4</sup> is also shown in Figure 3.

modes of a nematic in the state of equilibrium and those of a simple fluid under the same nonequilibrium regime. Both situations are of physical interest and are

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

It has been found that for an NLC in a NESS, the effects of the external gradients

=Rc ¼ 0: Thus, the hydrodynamic modes of a nematic, in the state of equilib-

<sup>2</sup> given by Eq. (6); as well as by the three diffusives, which consist

KIk<sup>2</sup> ρ0σ<sup>3</sup>

<sup>3</sup> and λ<sup>e</sup>

<sup>3</sup> η þ ζ, where η and ζ denote, respectively, the shear and

<sup>3</sup> <sup>η</sup> <sup>þ</sup> <sup>ζ</sup> , <sup>σ</sup><sup>2</sup> ! 0, <sup>σ</sup><sup>3</sup> ! <sup>ν</sup>, <sup>σ</sup><sup>4</sup> ! <sup>ν</sup>, where <sup>ν</sup> � <sup>η</sup>=ρ<sup>0</sup> is the kinematic viscosity,

, (15)

<sup>5</sup>, which is the same as Eq. (8). The longitudinal

<sup>6</sup> and λ<sup>e</sup>

(16)

<sup>7</sup> which are equal

<sup>i</sup> ð Þ i ¼ 1…7 are well

<sup>4</sup> are shown in the middle part

<sup>k</sup><sup>2</sup> � icsk, (17)

α and g are only manifested in the coupling of the thermal diffusive and shear longitudinal modes, which gives rise to the visco-heat modes λ3,<sup>4</sup> indicated, respectively, by means of Eq. (7). If the nematic layer is in a state of homogeneous thermodynamic equilibrium, g ¼ 0 and α ¼ 0, and therefore X ¼ 0 and

rium (denoted by the superscript e), are composed of five longitudinal and two transverse modes. The longitudinal modes are integrated by the two acoustic prop-

λe

λe

transverse modes consist of the shear and director modes λ<sup>e</sup>

known in the literature [22, 31, 46]. Note that λ<sup>e</sup>

6.2 Simple fluid in a Rayleigh-Bénard system

simple fluid has the two acoustic propagative modes:

λ<sup>1</sup> ≃Γ<sup>0</sup>

to the Eq. (14). It is necessary to mention that the decay rates λ<sup>e</sup>

<sup>3</sup> <sup>¼</sup> DTk<sup>2</sup>

<sup>4</sup> <sup>¼</sup> <sup>σ</sup>3k<sup>2</sup> � <sup>Ω</sup><sup>2</sup>

diffusive modes (15) and (16) are obtained precisely from Eq. (7), since in this, the Rayleigh ratio, given by Eq. (9), is zero if α and g vanish. Moreover, the pair of

Given that in the isotropic limit (simple fluid limit), the degree of nematic order goes to zero, ni is no longer a hydrodynamic variable, and the elastic constants Ki (for i ¼ 1, 2, 3) and the kinetic parameters γ1, λ vanish. Also, χ<sup>⊥</sup> and χ<sup>∥</sup> are reduced to the coefficient of thermal diffusivity χ and χ<sup>a</sup> ¼ 0. On the other hand, the nematic viscosities are reduced in the following way: ν<sup>1</sup> ! η, ν<sup>2</sup> ! η, ν<sup>3</sup> ! η,

volumetric viscosities of the simple fluid. As a result, from Eqs. (23)–(31) in Ref. [26] (or Eqs. (74)–(82) in [25]), it follows that in the isotropic limit DT ! χ,

whereas KI ! 0, KII ! 0, and Ω ! 0. Consequently, by making the identifications indicated above, the corresponding hydrodynamic modes of a simple fluid can be obtained when it is in a Rayleigh-Bénard system. Thus, according to Eq. (6), a

<sup>k</sup><sup>2</sup> <sup>þ</sup> icsk, <sup>λ</sup><sup>2</sup> <sup>≃</sup> <sup>Γ</sup><sup>0</sup>

discussed below.

R k !

agatives λ<sup>e</sup>

of Figure 3.

<sup>ν</sup><sup>4</sup> ! <sup>ζ</sup> <sup>þ</sup> <sup>1</sup>

<sup>σ</sup><sup>1</sup> ! <sup>1</sup> ρ0 4

157

<sup>3</sup> <sup>η</sup>, <sup>ν</sup><sup>5</sup> ! � <sup>2</sup>

of one thermal:

6.1 Nematic in equilibrium

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

<sup>1</sup> and λ<sup>e</sup>

another of shear:

and one more of the director, λ<sup>e</sup>

It should be noted that our expressions for these three decay rates are not in agreement with those reported for an NLC in the literature [44, 45]. In these works, the director mode tends to zero, the shear mode does not change and there is an additional mode which is the sum of the thermal and director modes. In contrast, we have found that the thermal mode λ<sup>3</sup> vanishes, the director mode λ<sup>5</sup> is virtually unchanged, while λ<sup>4</sup> has contributions from the thermal and shear diffusive modes. We know that this phenomenon also occurs in the simple fluid, where there are two diffusive modes, the thermal mode also vanishes and the other one has contributions from the shear and thermal modes. In other words, our results reduce to the corresponding one for a simple fluid as R reaches its critical value Rc. Because for a simple fluid, these features have been predicted theoretically, our results suggest that it might be feasible to verify them experimentally also for nematics [8, 37, 38].

### 5.2 Transverse modes

As mentioned earlier, pTð Þ<sup>λ</sup> is the characteristic polynomial of second order in <sup>λ</sup> of the matrix N<sup>T</sup>. The corresponding transverse hydrodynamic modes are the roots of this equation pTð Þ¼ <sup>λ</sup> 0.
