4. Nematodynamic equations

The geometry of the proposed model allows us to separate the hydrodynamic variables into transverse (T) and longitudinal (L) variables with respect to n^<sup>0</sup> and k ! , [33]. The former set is Ψ<sup>T</sup> r !; t � f g vx; nx , while the latter reads Ψ<sup>L</sup> r !; t � <sup>p</sup>; vy; vz; <sup>s</sup>; ny . We want to describe the stochastic dynamics of the spontaneous thermal deviations (fluctuations) δΨ r !; t <sup>¼</sup> <sup>Ψ</sup> <sup>r</sup> !; t � <sup>Ψ</sup>st around the above defined stationary state. A complete set of stochastic equations for the spacetime evolution of the fluctuations is obtained by linearizing the general nematodynamic equations [20, 22, 24], and by using the FH formalism. This starting set of equations is given explicitly by Eqs. (19)–(22) in Ref. [25]. However, since for the nematic mesophase, the rotational invariance has been broken, it is convenient to rewrite these nematodynamic equations in a representation which takes into account that a symmetry breaking has occurred along the z axis.

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

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

new state variables. In the case of the present model, owing to the initial orientation of the director n^st <sup>i</sup> , the NLC exhibits several symmetries: rotational invariances around the z axis, symmetry under inversions with respect to both, the xy plane and with respect to reflections on planes containing the z axis. A proper set of variables for this purpose was proposed long ago [38, 39], in terms of the variables δp; δφ; δs; δξ; δf <sup>1</sup>; δψ; δf <sup>2</sup> � �, defined in detail in Eqs. (6)–(10) in Ref. [26] (or Eqs. (53)–(57) in [25]). In this new representation, the complete set of stochastic hydrodynamic equations for the fluctuations takes an alternative form given by Eqs. (11)–(17) in Ref. [26] (or Eqs. (58)–(64) in [25]). The matrix representation of the Fourier transformation of this set of equations is given by:

$$\frac{\partial}{\partial t}\delta \vec{X}\left(\vec{k},t\right) = -\mathcal{M}\delta \vec{X}\left(\vec{k},t\right) + \vec{\Theta}\left(\vec{k},t\right),\tag{3}$$

where δ X ! k ! ; t � � <sup>¼</sup> <sup>δ</sup><sup>X</sup> !<sup>L</sup>; δX !T � �<sup>t</sup> with δX !<sup>L</sup> k ! ; t � � <sup>¼</sup> <sup>δ</sup> <sup>e</sup>p; δφe; <sup>δ</sup>es; <sup>δ</sup>eξ; <sup>δ</sup>e<sup>f</sup> <sup>1</sup> � �<sup>t</sup> and δX !<sup>T</sup> k ! ; t � � <sup>¼</sup> δψe; <sup>δ</sup> <sup>e</sup><sup>f</sup> <sup>2</sup> � �<sup>t</sup> . The superscript t denotes the transpose, while L and T indicate, respectively, the longitudinal and transverse sets of variables. In Eq. (3), M stands for a 7 � 7 hydrodynamic matrix which is diagonal in the 5 � <sup>5</sup> <sup>N</sup><sup>L</sup> and the <sup>2</sup> � <sup>2</sup> <sup>N</sup><sup>T</sup> blocks. The explicit form of these matrices is not necessary in our discussion; however, they are given explicitly by Eqs. (21) and (22) in Ref. [26] (see also Eqs. (72) and (73) in Ref. [25]). The stochastic terms, Θ ! k ! ; t � �, in Eq. (3) are given by the column vector Θ ! k ! ; t � � <sup>¼</sup> <sup>Θ</sup> !<sup>L</sup>; Θ !T � �<sup>t</sup> which explicit form of its components can be found in Eqs. (32) and (33) in Ref. [26] (or Eqs. (84) and (85) of Ref. [25]). It is important to emphasize that as a consequence of this change of representation, in this last system, it can be clearly seen how the nematic variables are separated in two sets completely independent: the five longitudinal <sup>δ</sup> <sup>e</sup>p; δφe; <sup>δ</sup>es; <sup>δ</sup>eξ; <sup>δ</sup>e<sup>f</sup> <sup>1</sup> n o and the two transverse δψe; <sup>δ</sup> <sup>e</sup><sup>f</sup> <sup>2</sup> n o.

However, in order to facilitate the calculation of the hydrodynamic modes, we define a new set of variables having the same dimensionality, δzj k ! ; t h i � � <sup>¼</sup> M<sup>1</sup>=<sup>2</sup> L�1=<sup>2</sup> <sup>t</sup> (<sup>j</sup> <sup>¼</sup> 1,…, 7): <sup>z</sup><sup>1</sup> � <sup>ρ</sup>0c<sup>2</sup> s � ��1=<sup>2</sup> <sup>δ</sup>ep, <sup>z</sup><sup>2</sup> � <sup>ρ</sup>0k�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> δφe, <sup>z</sup><sup>3</sup> � <sup>ρ</sup>0T0c�<sup>1</sup> p � �<sup>1</sup>=<sup>2</sup> δes, <sup>z</sup><sup>4</sup> <sup>¼</sup> <sup>ρ</sup>0k�<sup>4</sup> � �<sup>1</sup>=<sup>2</sup> <sup>δ</sup>eξ, <sup>z</sup><sup>5</sup> � <sup>ρ</sup>0c<sup>2</sup> <sup>s</sup> <sup>k</sup>�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> <sup>δ</sup>e<sup>f</sup> <sup>1</sup> , <sup>z</sup><sup>6</sup> � <sup>ρ</sup>0k�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> δψe, <sup>z</sup><sup>7</sup> � <sup>ρ</sup>0c<sup>2</sup> <sup>s</sup> <sup>k</sup>�<sup>2</sup> � �<sup>1</sup>=<sup>2</sup> δ ef <sup>2</sup> . In terms of these new variables, the system of equations (3) is rewritten in the more compact form as:

$$\frac{\partial}{\partial t}\vec{Z}\left(\vec{k},t\right) = -N\vec{Z}\left(\vec{k},t\right) + \vec{\Xi}\left(\vec{k},t\right),\tag{4}$$

where Z ! k ! ; t � � <sup>¼</sup> <sup>Z</sup> !<sup>L</sup>; Z !T � �<sup>t</sup> with Z<sup>L</sup> ! k ! ; t � � <sup>¼</sup> ð Þ <sup>z</sup>1; <sup>z</sup>2; <sup>z</sup>3; <sup>z</sup>4; <sup>z</sup><sup>5</sup> <sup>t</sup> and Z !<sup>T</sup> k ! ; t � � <sup>¼</sup> ð Þ <sup>z</sup>6; <sup>z</sup><sup>7</sup> <sup>t</sup> . In Eq. (4), N stands for a 7 � 7 hydrodynamic matrix which is diagonal in the 5 � <sup>5</sup> <sup>N</sup><sup>L</sup> and the 2 � <sup>2</sup> <sup>N</sup><sup>T</sup> blocks. Again, the explicit form of these matrices is not necessary in our discussion, but they are given explicitly by Eqs. (39)–(41) in Ref. [26] (see also Eqs. (94)–(96) in [25]). In Eq. (4), Ξ ! k ! ; t � � <sup>¼</sup> <sup>Ξ</sup> !<sup>L</sup>; Ξ !T � �<sup>t</sup> is the stochastic term, composed by the longitudinal Ξ !<sup>L</sup> k ! ; t � � <sup>¼</sup> <sup>ζ</sup>1; <sup>ζ</sup>2; <sup>ζ</sup>3; <sup>ζ</sup>4; <sup>ζ</sup><sup>5</sup> ð Þ<sup>t</sup> and transverse <sup>Ξ</sup> !<sup>T</sup> k ! ; t � � <sup>¼</sup> <sup>ζ</sup>6; <sup>ζ</sup><sup>7</sup> ð Þ<sup>t</sup> noise vectors. The

3.1 Stationary state

Non-Equilibrium Particle Dynamics

The external gradients drive the nematic layer into a nonequilibrium steady state. We shall assume that the temperature difference T<sup>1</sup> � T<sup>2</sup> amounts only to a

the Rayleigh-Bénard type. In this NESS, we choose as the nematodynamic variables

It is to be expected that in this steady state, the changes in Ψst will only occur in the <sup>z</sup> direction, so that <sup>Ψ</sup>st <sup>¼</sup> <sup>Ψ</sup>½ � p zð Þ; T zð Þ , where <sup>p</sup> is the local pressure. We assume that Ψst admits an expansion of the Taylor series around an equilibrium state T0; p<sup>0</sup>

z<sup>0</sup> ¼ 0, and we consider only first-order terms in the gradients. Thus, by setting the values of the temperature at the plates, T<sup>1</sup> ¼ T zð Þ ¼ �d=2 and T<sup>2</sup> ¼ T zð Þ ¼ d=2 , the

dT

where <sup>T</sup><sup>0</sup> � <sup>T</sup>stð Þ¼ <sup>z</sup> <sup>¼</sup> <sup>0</sup> ð Þ <sup>T</sup><sup>1</sup> <sup>þ</sup> <sup>T</sup><sup>2</sup> <sup>=</sup><sup>d</sup> and <sup>α</sup> � <sup>Δ</sup>T=d, with <sup>Δ</sup><sup>T</sup> � <sup>T</sup><sup>1</sup> � <sup>T</sup>2. In

<sup>X</sup> � �<sup>α</sup> <sup>þ</sup> <sup>g</sup>βT<sup>0</sup>

is the specific heat at constant pressure, β is the thermal expansion coefficient,

<sup>s</sup> =c<sup>2</sup>

The geometry of the proposed model allows us to separate the hydrodynamic variables into transverse (T) and longitudinal (L) variables with respect to n^<sup>0</sup>

above defined stationary state. A complete set of stochastic equations for the space-

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

. We want to describe the stochastic dynamics of the

!; t 

time evolution of the fluctuations is obtained by linearizing the general nematodynamic equations [20, 22, 24], and by using the FH formalism. This starting set of equations is given explicitly by Eqs. (19)–(22) in Ref. [25]. However, since for the nematic mesophase, the rotational invariance has been broken, it is convenient to rewrite these nematodynamic equations in a representation which takes into account that a symmetry breaking has occurred along the z axis.

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

cp

!; t 

dz <sup>z</sup> <sup>¼</sup> <sup>T</sup><sup>0</sup> <sup>1</sup> � <sup>α</sup>

<sup>i</sup> <sup>¼</sup> <sup>0</sup> , nor instabilities of

is the director field.

, (1)

at

is the specific entropy density (entropy per

!; t 

and ni r

T<sup>0</sup> z 

K, and it will be convenient to

, (2)

<sup>s</sup> , where cs is the adiabatic sound

<sup>T</sup>, being cv the specific heat at constant

� f g vx; nx , while the latter reads

¼ Ψ r !; t 

� <sup>Ψ</sup>st around the

!; t 

few degrees, so that there are no nematic layer flows vst

steady temperature profile is completely determined by:

what follows, we shall only consider <sup>T</sup>0≈<sup>3</sup> � <sup>10</sup><sup>2</sup>

which satisfies the relationship <sup>β</sup><sup>2</sup> � ð Þ <sup>γ</sup> � <sup>1</sup> cp=T0c<sup>2</sup>

velocity in the nematic, <sup>γ</sup> � cp=cv <sup>¼</sup> <sup>c</sup><sup>2</sup>

4. Nematodynamic equations

� p; vy; vz; s; ny

, [33]. The former set is Ψ<sup>T</sup> r

spontaneous thermal deviations (fluctuations) δΨ r

and k !

Ψ<sup>L</sup> r !; t 

150

<sup>T</sup>st <sup>¼</sup> T zð Þ¼ <sup>T</sup><sup>0</sup> <sup>þ</sup>

introduce the effective temperature gradient <sup>∇</sup>zTst � <sup>X</sup>^<sup>z</sup> as [37],

volume and cT the isothermic sound velocity in the nematic.

the set <sup>Ψ</sup> <sup>¼</sup> <sup>ρ</sup>; <sup>s</sup>; vi f g ; ni , where s r!; <sup>t</sup>

unit mass), the hydrodynamic velocity is vi r

explicit form of the components ζm, m ¼ 1…7, as well as their fluctuationdissipation relations (FDR), can be found in Eqs. (169)–(175) and Eqs. (176)–(186), respectively, in Appendix A of [25].

<sup>g</sup> � <sup>10</sup><sup>3</sup>

<sup>a</sup><sup>3</sup> � <sup>10</sup>�<sup>2</sup>

cs <sup>¼</sup> <sup>1</sup>:<sup>5</sup> � <sup>10</sup><sup>5</sup>

(129) in [25]):

[31, 34].

and

with

where ^ k2 <sup>⊥</sup> � <sup>k</sup><sup>2</sup> ⊥=k<sup>2</sup>

153

where <sup>Γ</sup> � <sup>1</sup>

magnitude: <sup>a</sup><sup>0</sup> � <sup>10</sup>�21, <sup>a</sup><sup>0</sup>

the solutions of the polynomial pL

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

5.1.1 Sound longitudinal modes

reduced quantities (Eq. (5)) of order k<sup>2</sup> [24].

5.1.2 Visco-heat and director longitudinal modes

R k � �!

Rc

results are expressed in terms of the ratio R k � �!

roots can be written approximately as:

∓ 1 2

, and knowing that in a typical light scattering experiments <sup>k</sup> <sup>¼</sup> <sup>10</sup><sup>5</sup>

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

mation 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

plex conjugate and are given by (see Eqs. (47) and (48) in [26], or Eqs. (128) and

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

(or in [25]), it is shown that, up to first order in the small quantities (Eq. (5)), these

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

vuut ,

KIk<sup>4</sup> ρ0σ3k<sup>2</sup>

its critical value above which convection sets in. It should be emphasized that our

of the value of the separation d between the plates. However, the appropriate value of d in an experiment should be chosen with an experimental criterion [42].

!

These modes are the roots of the characteristic equation pL

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

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

!<sup>2</sup>

� � <sup>g</sup>β^ k 2 ⊥ DTσ3k<sup>4</sup> <sup>X</sup> <sup>þ</sup> <sup>α</sup>Ωχ<sup>a</sup>

. In Eq. (7), <sup>R</sup> � <sup>β</sup>gΔTd<sup>3</sup>

<sup>λ</sup><sup>5</sup> <sup>≃</sup> KIk<sup>2</sup> γ1 þ Ω2 KIk<sup>4</sup>

<sup>0</sup> � <sup>10</sup>�21, <sup>a</sup><sup>00</sup>

They are the roots of the characteristic equation pL

cms�<sup>1</sup> [41], the quantities given in Eq. (5) have the following orders of

, a<sup>0</sup>

YYð Þλ may be obtained by a perturbation approxi-

<sup>λ</sup><sup>1</sup> <sup>≃</sup>Γk<sup>2</sup> <sup>þ</sup> icsk, <sup>λ</sup><sup>2</sup> <sup>≃</sup> <sup>Γ</sup>k<sup>2</sup> � icsk, (6)

KIk<sup>4</sup> ρ0σ3k<sup>2</sup>

<sup>σ</sup>3k<sup>2</sup> <sup>1</sup> � <sup>R</sup>

<sup>ρ</sup>0σ3k<sup>2</sup> , (8)

� �, (9)

<sup>σ</sup>3<sup>χ</sup> is the Rayleigh number and Rc denotes

=Rc and are, therefore, independent

Rc � �

� <sup>4</sup>DTk<sup>2</sup>

DTσ<sup>3</sup>

ð Þ σ<sup>3</sup> þ DT

<sup>2</sup> ½ � ð Þ γ � 1 DT þ σ<sup>1</sup> is the anisotropic sound attenuation coefficient

<sup>1</sup> � <sup>10</sup>�<sup>3</sup>

XXð Þ¼ λ 0. Its roots are com-

YYð Þ¼ λ 0. In Ref. [26]

(7)

<sup>0</sup> � <sup>10</sup>�24, <sup>a</sup><sup>1</sup> � <sup>10</sup>�<sup>3</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

cm�<sup>1</sup> and

,

, <sup>a</sup><sup>2</sup> � <sup>10</sup>�<sup>2</sup>
