7. Conclusions

In this work, we have used the standard formulation of FH to describe the dynamics of the fluctuations of a NLC layer in a NESS characterized by the simultaneous action of a uniform temperature gradient α and a constant gravitational field g, which corresponds to a Rayleigh-Bénard system. The analysis carried out takes into account only the nonconvective regime. The most important results are the analytic expressions for the seven nematic hydrodynamic modes. The explicit details of several of the calculations can be found in Refs. [25, 26]. To summarize the results obtained in this work and to put them into a proper context, the following comments may be useful.

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, λ<sup>1</sup> and λ2, modes (6), as well as the triplet formed by the visco-heat pair λ<sup>3</sup> and λ4, modes (7), and the director λ5, mode (8). In addition, the transverse ones are given by the shear λ<sup>6</sup> and the director λ7, modes (14). We find that the influence of the temperature gradient α and the gravitational field g occurs only in the longitudinal modes, being greater its effect � <sup>10</sup>�<sup>9</sup> � � on the visco-heat pair <sup>λ</sup><sup>3</sup> and <sup>λ</sup>4. This effect is quantified by means of the Rayleigh ratio R k � �! =Rc, Eq. (9), where R is the Rayleigh number and Rc is its critical value above which convection sets in. The developed analysis corresponding to the nonconvective regime was carried out under the condition R k � �! =Rc ≤1:

Nevertheless, our calculated expressions for the visco-heat λ3, λ4, and director λ<sup>5</sup> modes predict both the existence of propagative modes and the form that this triplet acquires in the convection threshold, and moreover, they reduce to the corresponding modes in all the different limit cases already mentioned. In this respect, we believe that they are more general than those reported in the literature [44, 45]. As far as we know, the diffusive or propagative nature of the modes λ<sup>3</sup> and

Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

its derivation represents a relevant contribution of this work. Since in simple fluids, the existence of propagative modes has been predicted and verified experimentally, our predictions about the existence of this phenomenon in the modes of an NLC

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

, may be calculated from our FH description. In Ref. [49], an appli-

cation of this nature was developed by calculating the Rayleigh dynamic structure factor for the NLC under the NESS already mentioned, and its possible comparison with experimental studies is discussed; a preliminary analysis can be consulted in Ref. [50]. Another studies of the dynamic structure factor for an NLC in a different NESS, such as that produced by the presence of an external pressure gradient, were

We thank the UACM for the economic facilities granted to cover the total cost of

The authors declare that they have no conflict of interests.

τ relaxation time of almost all degrees of freedom

Rc Rayleigh number at the convection threshold

=Rc, was not known; therefore,

<sup>λ</sup>4, depending on the values taken by the ratio R k !

suggest the realization of new experiments.

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

factor S k!

;ω

Acknowledgements

Conflict of interest

Nomenclature

S k! ;ω

n^, n

161

k

published in the references [19, 20].

the publication of this research work.

k wave number ω angular frequency

! wave vector R Rayleigh number

R=Rc Rayleigh ratio

! or n<sup>α</sup> director field

dynamic structure factor

Oð Þ3 orientation symmetry group Tð Þ3 translation symmetry group

d thickness of the nematic layer

The analytical expressions calculated for the hydrodynamic modes of a nematic in the NESS considered exhibit behaviors that are of great interest in the following particular situations. First, if the isotropic limit of the simple fluid is taken, the NLC hydrodynamic modes reduce to those in the same state out of equilibrium, modes (17)–(19), [8, 37, 38]. If R ¼ 0, that is, in the absence of the uniform temperature gradient and the constant gravitational field, our expressions are simplified and reduce to those already reported for a nematic in the state of thermodynamic equilibrium, modes (6), (8), (14), (15), and (16), [22, 31, 46]. In this case, if we also consider the limit of the simple fluid, they agree with those of this system in equilibrium, modes (17), (19), (22), and (23), [41, 47, 48]. When R ¼ Rc, that is, at the threshold of convection, from the triplet of longitudinal λ3, λ<sup>4</sup> and λ5, the viscoheat λ<sup>3</sup> vanishes, and λ<sup>4</sup> is the sum of the thermal and shear modes, modes (12) and (13); while that of director λ<sup>5</sup> is identical to mode (8) [37, 38]. Moreover, if in this nematic threshold of convection, the limit of the simple fluid is considered, the modes of this system are recovered: one is zero, mode (24), and the other is the sum of the thermal and shear modes, mode (25), [37, 38]. Also, if R k � �! =Rc , R<sup>0</sup> ek � �,

where R<sup>0</sup> k � �! is the reference value (10), our results predict that the visco-heat pair λ<sup>3</sup> and λ4, modes (7), become propagative; in the limit of the simple fluid, under similar conditions, the corresponding modes (18) are also propagative. The latter have been predicted theoretically [8, 37, 38] and verified experimentally [43].

However, it should be mentioned that our hydrodynamic modes λ3, λ4, and λ<sup>5</sup> do not coincide with those reported in the literature for an NLC in the same NESS considered here [44, 45], which consist in one mode due to the director, another more product of the coupling of the thermal and director modes, and a shear mode. The effect of external forces α and g is only manifested in the first two modes. This triplet is reduced to the corresponding director, thermal, and shear longitudinal modes of an NLC in the state of thermodynamic equilibrium, as well as to the thermal and shear modes of a simple fluid in such state. It should be noted that from the analytical expressions of these modes, the existence of nematic propagative modes cannot be predicted; much less, in this NESS, in the simple fluid. In addition, when the threshold of convection in the nematic is considered, the director mode is canceled, another one is the sum of the thermal and director modes, and the shear mode remains unchanged; consequently, when the limit of the simple fluid is taken, they are reduced to thermal and shear modes. This last result differs completely from the already reported [37, 38] for the hydrodynamic modes of a simple fluid at the threshold of convection, where one is zero, mode (24), and the other the sum of the thermal with the shear, mode (25).

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

Nevertheless, our calculated expressions for the visco-heat λ3, λ4, and director λ<sup>5</sup> modes predict both the existence of propagative modes and the form that this triplet acquires in the convection threshold, and moreover, they reduce to the corresponding modes in all the different limit cases already mentioned. In this respect, we believe that they are more general than those reported in the literature [44, 45]. As far as we know, the diffusive or propagative nature of the modes λ<sup>3</sup> and <sup>λ</sup>4, depending on the values taken by the ratio R k ! =Rc, was not known; therefore, its derivation represents a relevant contribution of this work. Since in simple fluids, the existence of propagative modes has been predicted and verified experimentally, our predictions about the existence of this phenomenon in the modes of an NLC suggest the realization of new experiments.

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 S k! ;ω , may be calculated from our FH description. In Ref. [49], an application of this nature was developed by calculating the Rayleigh dynamic structure factor for the NLC under the NESS already mentioned, and its possible comparison with experimental studies is discussed; a preliminary analysis can be consulted in Ref. [50]. Another studies of the dynamic structure factor for an NLC in a different NESS, such as that produced by the presence of an external pressure gradient, were published in the references [19, 20].
