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 approxima-

tion, into two completely independent sets of variables, Z !<sup>L</sup> <sup>X</sup> <sup>¼</sup> ð Þ <sup>z</sup>1; <sup>z</sup><sup>2</sup> <sup>t</sup> and Z !<sup>L</sup> <sup>Y</sup> <sup>¼</sup> ð Þ <sup>z</sup>3; <sup>z</sup>4; <sup>z</sup><sup>5</sup> <sup>t</sup> , as it is shown in the Subsection 3.1 of Ref. [25], or in more detail in [24]. This approximation allows us to rewrite the characteristic polynomial of longitudinal variables as pLð Þ¼ <sup>λ</sup> pL XXð Þ<sup>λ</sup> <sup>p</sup><sup>L</sup> YYð Þ<sup>λ</sup> : It should be mentioned that pL XXð Þλ and pL YYð Þλ are polynomials of second and third degree in λ, and explicitly are given by the Eqs. (44) and (45) in Ref. [26] (or Eqs. (117) and (118) in [25]).

While there is no analytical difficulty to solve the quadratic and cubic equations pL XXð Þ<sup>λ</sup> and pL YYð Þλ , the explicit form of their exact roots can be quite complicated. However, it is possible to estimate them following a procedure based partially on a method suggested in Ref. [40], which allows to identify the following quantities in the equation for pL YYð Þ<sup>λ</sup> , namely, ð Þ <sup>γ</sup> � <sup>1</sup> DTk<sup>2</sup> , σ1k<sup>2</sup> , k<sup>2</sup> c2 <sup>s</sup> and <sup>g</sup><sup>2</sup>k<sup>2</sup> <sup>∥</sup>= c<sup>2</sup> <sup>s</sup> <sup>k</sup><sup>2</sup> . They depend on the anisotropic coefficients of diffusivity DT and on the viscosity σ1: The former quantity is a function of the parallel χ<sup>∥</sup> and perpendicular χ<sup>⊥</sup> components of thermal diffusivity, while the latter depends on the nematic viscosity coefficients ν<sup>i</sup> ð Þ i ¼ 1; …; 5 (see Eqs. (23) and (24) in Ref. [26], or Eqs. (74) and (75) in [25]). In the same way, in the equation for p<sup>L</sup> YYð Þλ , the following quantities can be identified, <sup>g</sup>αβ <sup>k</sup><sup>2</sup> ⊥ <sup>k</sup><sup>2</sup> , gX<sup>β</sup> <sup>k</sup><sup>2</sup> ⊥ <sup>k</sup><sup>2</sup> , DTk<sup>2</sup> , Ωχak<sup>2</sup> , σ3k<sup>2</sup> , <sup>K</sup><sup>1</sup> γ1 k2 , <sup>Ω</sup><sup>2</sup>KI ρ0 k4 , which depend on the anisotropic coefficients of viscosity σ3, of elasticity KI, symmetry Ω (see, respectively, Eqs. (26), (28), and (30) in [26]), as well as the anisotropy χ<sup>a</sup> ¼ χ<sup>∥</sup> � χ<sup>⊥</sup> and the torsional viscous coefficient γ1. We now compare all these quantities with ω � csk, by introducing the (small) reduced dimensionless quantities:

$$a\_0 \equiv \frac{g\alpha\beta}{\overline{\alpha}^2} \frac{k\_\perp^2}{k^2}, \quad a\_0' \equiv \frac{gX\beta}{\overline{\alpha}^2} \frac{k\_\perp^2}{k^2}, \quad a\_0'' \equiv \frac{g^2 k\_\parallel^2}{\overline{\alpha}^2 c\_i^2 k^2}, \quad a\_1 \equiv \frac{D\_T k^2}{\overline{\alpha}}, \quad a\_1' \equiv \frac{\Omega \chi\_a k^2}{\overline{\alpha}}, \tag{5}$$

$$a\_2 \equiv \frac{\sigma\_1 k^2}{\overline{\alpha}}, \quad a\_3 \equiv \frac{\sigma\_3 k^2}{\overline{\alpha}}, \quad a\_5 \equiv \frac{K\_l k^2}{\chi\_1 \overline{\alpha}}, \quad a\_6 \equiv \frac{\Omega^2 K\_l k^4}{\rho\_0 \overline{\alpha}^2}.$$

The relevant point for our purpose is to realize that for most nematics at ambient temperatures, <sup>ρ</sup><sup>0</sup> and <sup>Ω</sup> are of order of magnitude 1, <sup>γ</sup><sup>1</sup> � <sup>10</sup>�<sup>1</sup> , χ<sup>i</sup> and ν<sup>i</sup> are of order 10�<sup>2</sup> –10�<sup>3</sup> , Ki � <sup>10</sup>�<sup>6</sup>–10�<sup>7</sup> , while <sup>β</sup> � <sup>10</sup>�<sup>4</sup> [32]. If we consider that <sup>α</sup> <sup>≲</sup>1 and

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

<sup>g</sup> � <sup>10</sup><sup>3</sup> , and knowing that in a typical light scattering experiments <sup>k</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup> cm�<sup>1</sup> and cs <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup><sup>5</sup> cms�<sup>1</sup> [41], the quantities given in Eq. (5) have the following orders of magnitude: <sup>a</sup><sup>0</sup> � <sup>10</sup>�21, <sup>a</sup><sup>0</sup> <sup>0</sup> � <sup>10</sup>�21, <sup>a</sup><sup>00</sup> <sup>0</sup> � <sup>10</sup>�24, <sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>3</sup> , a<sup>0</sup> <sup>1</sup> � <sup>10</sup>�<sup>3</sup> , <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>2</sup> , <sup>a</sup><sup>3</sup> � <sup>10</sup>�<sup>2</sup> , <sup>a</sup><sup>5</sup> � <sup>10</sup>�<sup>5</sup> and <sup>a</sup><sup>6</sup> � <sup>10</sup>�6. We now follow the method of Ref. [40] and the solutions of the polynomial pL YYð Þλ may be obtained by a perturbation approximation in terms of these small quantities. However, in what follows, we improve this approximation by using its exact roots and by expressing them in terms of the reduced quantities (Eq. (5)) of order k<sup>2</sup> [24].

## 5.1.1 Sound longitudinal modes

explicit form of the components ζm, m ¼ 1…7, as well as their fluctuation-

respectively, in Appendix A of [25].

Non-Equilibrium Particle Dynamics

equations in terms of the variables k

longitudinal variables as pLð Þ¼ <sup>λ</sup> pL

the same way, in the equation for p<sup>L</sup>

<sup>a</sup><sup>2</sup> � <sup>σ</sup>1k<sup>2</sup>

, Ki � <sup>10</sup>�<sup>6</sup>–10�<sup>7</sup>

, Ωχak<sup>2</sup>

<sup>0</sup> � gX<sup>β</sup> ω2

5. Hydrodynamic modes

roots are calculated below.

5.1 Longitudinal modes

<sup>Y</sup> <sup>¼</sup> ð Þ <sup>z</sup>3; <sup>z</sup>4; <sup>z</sup><sup>5</sup> <sup>t</sup>

XXð Þ<sup>λ</sup> and pL

the equation for pL

<sup>a</sup><sup>0</sup> � <sup>g</sup>αβ ω2 k2 ⊥ <sup>k</sup><sup>2</sup> , a<sup>0</sup>

Z !<sup>L</sup>

pL

<sup>g</sup>αβ <sup>k</sup><sup>2</sup> ⊥ <sup>k</sup><sup>2</sup> , gX<sup>β</sup> <sup>k</sup><sup>2</sup> ⊥ <sup>k</sup><sup>2</sup> , DTk<sup>2</sup>

10�<sup>2</sup>

152

–10�<sup>3</sup>

and pL

dissipation relations (FDR), can be found in Eqs. (169)–(175) and Eqs. (176)–(186),

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

by calculating its eigenvalues λ ¼ iω, given by the roots of the characteristic equation <sup>p</sup>ð Þ¼ <sup>λ</sup> pLð Þ<sup>λ</sup> <sup>p</sup>Tð Þ¼ <sup>λ</sup> <sup>0</sup>, where pLð Þ<sup>λ</sup> and <sup>p</sup>Tð Þ<sup>λ</sup> are the characteristic polynomials of fifth and second order in λ of the matrices N<sup>L</sup> and NT, respectively. These

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 approxima-

YYð Þλ are polynomials of second and third degree in λ, and explicitly are given

YYð Þλ , the explicit form of their exact roots can be quite complicated.

, σ1k<sup>2</sup> , k<sup>2</sup> c2

<sup>s</sup> <sup>k</sup><sup>2</sup> , a<sup>1</sup> � DTk<sup>2</sup>

<sup>γ</sup>1<sup>ω</sup> , a<sup>6</sup> � <sup>Ω</sup><sup>2</sup>

, while <sup>β</sup> � <sup>10</sup>�<sup>4</sup> [32]. If we consider that <sup>α</sup> <sup>≲</sup>1 and

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

However, it is possible to estimate them following a procedure based partially on a method suggested in Ref. [40], which allows to identify the following quantities in

depend on the anisotropic coefficients of diffusivity DT and on the viscosity σ1: The former quantity is a function of the parallel χ<sup>∥</sup> and perpendicular χ<sup>⊥</sup> components of thermal diffusivity, while the latter depends on the nematic viscosity coefficients ν<sup>i</sup> ð Þ i ¼ 1; …; 5 (see Eqs. (23) and (24) in Ref. [26], or Eqs. (74) and (75) in [25]). In

coefficients of viscosity σ3, of elasticity KI, symmetry Ω (see, respectively, Eqs. (26), (28), and (30) in [26]), as well as the anisotropy χ<sup>a</sup> ¼ χ<sup>∥</sup> � χ<sup>⊥</sup> and the torsional viscous coefficient γ1. We now compare all these quantities with ω � csk,

> <sup>0</sup> � <sup>g</sup><sup>2</sup>k<sup>2</sup> ∥ ω<sup>2</sup>c<sup>2</sup>

<sup>ω</sup> , a<sup>5</sup> � KIk<sup>2</sup>

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

[24]. This approximation allows us to rewrite the characteristic polynomial of

XXð Þ<sup>λ</sup> <sup>p</sup><sup>L</sup>

by the Eqs. (44) and (45) in Ref. [26] (or Eqs. (117) and (118) in [25]).

YYð Þ<sup>λ</sup> , namely, ð Þ <sup>γ</sup> � <sup>1</sup> DTk<sup>2</sup>

, σ3k<sup>2</sup> , <sup>K</sup><sup>1</sup> γ1 k2 , <sup>Ω</sup><sup>2</sup>KI ρ0 k4

by introducing the (small) reduced dimensionless quantities:

k2 ⊥ <sup>k</sup><sup>2</sup> , a<sup>00</sup>

<sup>ω</sup> , a<sup>3</sup> � <sup>σ</sup>3k<sup>2</sup>

temperatures, <sup>ρ</sup><sup>0</sup> and <sup>Ω</sup> are of order of magnitude 1, <sup>γ</sup><sup>1</sup> � <sup>10</sup>�<sup>1</sup>

and ω: The hydrodynamic modes are obtained

!<sup>L</sup>

YYð Þ<sup>λ</sup> : It should be mentioned that pL

<sup>s</sup> and <sup>g</sup><sup>2</sup>k<sup>2</sup>

YYð Þλ , the following quantities can be identified,

<sup>ω</sup> , a<sup>0</sup>

KIk<sup>4</sup> <sup>ρ</sup>0ω<sup>2</sup> :

<sup>1</sup> � <sup>Ω</sup>χak<sup>2</sup> <sup>ω</sup> ,

, χ<sup>i</sup> and ν<sup>i</sup> are of order

, which depend on the anisotropic

, as it is shown in the Subsection 3.1 of Ref. [25], or in more detail in

<sup>X</sup> <sup>¼</sup> ð Þ <sup>z</sup>1; <sup>z</sup><sup>2</sup> <sup>t</sup> and

<sup>∥</sup>= c<sup>2</sup>

<sup>s</sup> <sup>k</sup><sup>2</sup> . They

XXð Þλ

(5)

!

tion, into two completely independent sets of variables, Z

They are the roots of the characteristic equation pL XXð Þ¼ λ 0. Its roots are complex conjugate and are given by (see Eqs. (47) and (48) in [26], or Eqs. (128) and (129) in [25]):

$$
\lambda\_1 \simeq \Gamma k^2 + i\varepsilon\_\nu k, \quad \lambda\_2 \simeq \Gamma k^2 - i\varepsilon\_\nu k,\tag{6}
$$

where <sup>Γ</sup> � <sup>1</sup> <sup>2</sup> ½ � ð Þ γ � 1 DT þ σ<sup>1</sup> is the anisotropic sound attenuation coefficient of the NLC. This result shows that the sound propagation modes, λ<sup>1</sup> and λ2, are in complete agreement with those already reported in the literature for NLC [31, 34].
