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Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal

By Jorge Fernando Camacho and Rosalío Fernando Rodríguez

Submitted: June 26th 2018Reviewed: November 20th 2018Published: December 28th 2018

DOI: 10.5772/intechopen.82609

Downloaded: 118


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.


  • 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 (FH) of Landau and Lifshitz [1, 2]. It stems from the idea that the hydrodynamic equations are valid for any flow, including fluctuating changes in its state. Accordingly, stochastic currents are incorporated into the deterministic energy and momentum fluxes by adding fluctuating sources. This theory was put on a firm basis within the framework of the general theory of stationary Gaussian Markov processes by Fox and Uhlenbeck [3, 4, 5]. This approach has matched the theory of Onsager and Machlup with that of Landau and Lifshitz for systems where the basic state variables do not possess a definite time reversal symmetry [6, 7]. However, in spite of the fact that the theory of fluctuations for nonequilibrium fluids was initiated in the late 1970s, and was pursued by many authors [8], still nowadays several questions concerning the nature of hydrodynamic fluctuations in stationary nonequilibrium states (NESS) are of current active interest. One of these issues is the long-range character of these fluctuations, especially far away from instability points [9]. Thermal fluctuations in an equilibrium fluid always give rise to short-range equal time correlation functions, except close to a critical point. But when external gradients are applied, equal-time correlation functions can develop long-range contributions, whose nature is very different from those in equilibrium. For many models and systems in nonequilibrium states, it has been shown theoretically that the existence of the so-called generic scale invariance is the origin of the long-range nature of the correlation functions [10, 11].

In the case of a simple fluid in a thermal gradient, the structure factor, which determines the intensity of the Rayleigh scattering, diverges as k4for small values of the wave number 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 NESSof complex fluids. Among these, the enhancement of concentration fluctuations in polymer solutions under external hydrodynamic and electric fields [15], or the case of a polymer solution subjected to a stationary temperature gradient in the absence of any flow [16], has been discussed. Also, the behavior of fluctuations about some NESShas been analyzed in the case of thermotropic nematic liquid crystals. Specific examples are the nonequilibrium situations generated by a static temperature gradient [17], a stationary shear flow [18] or by an externally imposed constant pressure gradient [19, 20]. In the first two cases, it was found that the nonequilibrium contributions to the corresponding light scattering spectrum were small, but in the case of a Poiseuille flow induced by an external pressure gradient, the effect may be quite large. To our knowledge, however, at present, there is no experimental confirmation of these effects, in spite of the fact that for nematics, the scattered intensity is several orders of magnitude larger than for ordinary simple fluids.

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 τdetermined by the microscopic interactions of the system. There are, however, some other degrees of freedom of collective character, the hydrodynamic modes, which will decay much more slowly. When τ, its characteristic frequencies ω0ω1/τ, when k0.Such is the case, for example, of the propagation of sound waves and the conduction of heat in a simple fluid [21]. Hydrodynamics allows to describe these modes or degrees of freedom of greater duration, through the laws of conservation and balance of the system, and, as in the case of ordered systems, by the continuous breaking of symmetries [22, 23].

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 (NLC) in a NESS. In particular, we describe the dynamics of the fluctuations of its hydrodynamic variables induced by a stationary temperature gradient and under the influence of gravity (a Rayleigh-Bénard system) on a nematic layer confined between two parallel horizontal plates in a steady state in a nonconvective regime [24, 25, 26]. Once the dynamics of fluctuation is established, we calculate the time-dependent correlation functions in equilibrium between the fluctuating hydrodynamic variables, quantities that allow to obtain the transport properties of the system [27, 28]. One of these properties is the dynamic structure factor Skωof the system, which measures the magnitude of the changes in energy and momentum between the light beam and the fluid as functions of the wave vector kand ω.

For simple fluids with fixed k, Skωconsists of three well-separated Lorentzian features: a line or central peak (Rayleigh peak) located at ω=0and two Brillouin peaks symmetrically located with respect to the central one [29, 30]. These three lines are directly related to the hydrodynamic modes of the simple fluid, and from them, it is possible to obtain relevant information about transport properties. For instance, the Rayleigh line, associated with a thermal diffusive mode, is due to the fluctuations of the entropy (or temperature) that diffuse in the fluid and its width is proportional to the thermal diffusivity. On the other hand, the Brillouin lines are related to two acoustic propagative modes and are the result of the coupled dynamics of the pressure fluctuations and a component of the flow velocity that are transmitted with the speed of sound in the medium. Their widths are proportional to the absorption of sound.

In the case of an anisotropic system like a NLC, fluctuating hydrodynamic theories have recently been proposed [22, 31] based on the methodology proposed by Landau and Lifshitz [1]. However, this analysis of the fluctuations of the nematic hydrodynamic variables is not precise, since it does not take into account the parity with respect to time reversal, so their description using the Onsager-Machlup formalism would be strictly inadequate. The correct theoretical framework should be the more general theory of Fox and Uhlenbeck [3, 4, 5, 20, 24]. However, although a NLCdisperses light by several orders of magnitude more than an ordinary fluid [32], from both a theoretical and experimental point of view, the studies corresponding to the behavior of the fluctuations in these media around stationary states out of equilibrium are rather scarce. From the theoretical point of view, and only for the case of the transverse hydrodynamic variables [33], some studies of the behavior of orientational fluctuations have been carried out when analyzing the effect produced in the light scattering spectrum of a NLCin NESSinduced by the presence of uniform temperature gradients [17] and by the action of a shear flow [18]. In both cases, it has been found that the effect of fluctuations in the light scattering spectrum is small, being difficult to detect experimentally. On the other hand, as far as we know, no theoretical study has been carried out on the behavior of the longitudinal variables of a nematic and much less on its spectrum of light scattering, both in states of thermodynamic equilibrium and outside of it. This is an open research topic. Nor have been performed analyzes of stationary states generated by other types of external gradients in these systems, with which could be obtained qualitatively and quantitatively much greater effects than those reported so far in the literature for simple fluids. It should be mentioned that although preliminary attempts have been made to calculate the transverse hydrodynamic modes of a nematic [34, 35], there are few studies that also involve the corresponding longitudinal modes [31]. Unfortunately, a clear and systematic method to derive the set of complete, transverse, and longitudinal hydrodynamic modes of a NLCis still lacking.

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 R/Rc, where Ris the number of Rayleigh and Rcthe value that it reaches when in the nematic initiates the convection.

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, O3, and translational, T3, transformations. Thus, the group of symmetries of an isotropic liquid is O3×T3.However, by decreasing the temperature of these liquids, the translational symmetry T3is usually broken corresponding to the isotropic liquid-solid transition. In contrast, for a liquid formed by anisotropic molecules, by diminishing the temperature, the rotational symmetry O3is broken, which leads to the appearance of a liquid crystal. The mesophases for which only the rotational invariance has been broken are called nematics. As shown, the centers of mass of the molecules of a nematic have arbitrary positions, whereas the principal axes of their molecules are spontaneously oriented along a preferred direction. If the temperature decreases even more, the symmetry T3is also partially broken. The mesophases exhibiting the translational symmetry T2are called smectics [36].

This preferential direction is described by a local unitary vector field, n̂, called the director. This vector is easily distorted by the presence of electric and magnetic fields, as well as by the surfaces of the containers of the liquid crystals if they have been prepared properly [32]. With respect to NLC, it is important to point out that the director’s orientation does not distinguish between the n̂and n̂directions (nematic symmetry). A schematic representation of the order presented by the molecules in a nematic is shown in Figure 1.

Figure 1.

Representation of the average orientation of the molecules of a thermotropic NLC by means of the director vector n̂.

3. Model

Consider a NLCthin layer of thickness dunder the presence of a constant gravitational field g=gẑ, where gdenotes its magnitude and ẑthe unit vector along the zaxis. The initial configuration of the layer is homeotropic with a preferential orientation n̂0along the zaxis, as depicted in Figure 2. The nematic is confined between two parallel flat plates kept at fixed temperatures T1and T2(T1<T2), so that a uniform temperature gradient zTαẑis established downward in the layer. The situation where the temperature gradient goes from bottom to top can also be considered, and in this case, zTαẑ. The gravitational force induces a pressure gradient, zp=ρgẑ, where ρis the mass density.

Figure 2.

The NLC cell subject to a constant gravitational field g→ and an external uniform temperature gradient ∇T. k→ is the scattering vector.

3.1 Stationary state

The external gradients drive the nematic layer into a nonequilibrium steady state. We shall assume that the temperature difference T1T2amounts only to a few degrees, so that there are no nematic layer flows vist=0, nor instabilities of the Rayleigh-Bénard type. In this NESS,we choose as the nematodynamic variables the set Ψ=ρsvini, where srtis the specific entropy density (entropy per unit mass), the hydrodynamic velocity is virtand nirtis the director field. It is to be expected that in this steady state, the changes in Ψstwill only occur in the zdirection, so that Ψst=ΨpzTz, where pis the local pressure. We assume that Ψstadmits an expansion of the Taylor series around an equilibrium state T0p0at z0=0, and we consider only first-order terms in the gradients. Thus, by setting the values of the temperature at the plates, T1=Tz=d/2and T2=Tz=d/2,the steady temperature profile is completely determined by:


where T0Tstz=0=T1+T2/dand αΔT/d, with ΔTT1T2. In what follows, we shall only consider T03×102K, and it will be convenient to introduce the effective temperature gradient zTstXẑas [37],


which contains explicitly the contributions of both external forces. In Eq. (2), cpis the specific heat at constant pressure, βis the thermal expansion coefficient, which satisfies the relationship β2γ1cp/T0cs2,where csis the adiabatic sound velocity in the nematic, γcp/cv=cs2/cT2, being cvthe specific heat at constant volume and cTthe isothermic sound velocity in the nematic.

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 n̂0and k, [33]. The former set is ΨTrtvxnx, while the latter reads ΨLrtpvyvzsny. We want to describe the stochastic dynamics of the spontaneous thermal deviations (fluctuations) δΨrt=ΨrtΨstaround the above defined stationary state. A complete set of stochastic equations for the space-time evolution of the fluctuations is obtained by linearizing the general nematodynamic equations [20, 22, 24], and by using the FHformalism. 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 zaxis.

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 n̂ist, the NLCexhibits several symmetries: rotational invariances around the zaxis, symmetry under inversions with respect to both, the xyplane and with respect to reflections on planes containing the zaxis. A proper set of variables for this purpose was proposed long ago [38, 39], in terms of the variables δpδφδsδξδf1δψδf2, 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:


where δXkt=δXLδXTtwith δXLkt=δpδφδsδξδf1tand δXTkt=δψδf2t. The superscript tdenotes the transpose, while Land Tindicate, respectively, the longitudinal and transverse sets of variables. In Eq. (3), Mstands for a 7×7hydrodynamic matrix which is diagonal in the 5×5NLand the 2×2NTblocks. 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, Θkt,in Eq. (3) are given by the column vector Θkt=ΘLΘTtwhich 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 δpδφδsδξδf1and the two transverse δψδf2.

However, in order to facilitate the calculation of the hydrodynamic modes, we define a new set of variables having the same dimensionality, δzjkt=M1/2L1/2t(j=1,, 7): z1ρ0cs21/2δp, z2ρ0k21/2δφ, z3ρ0T0cp11/2δs, z4=ρ0k41/2δξ, z5ρ0cs2k21/2δf1, z6ρ0k21/2δψ, z7ρ0cs2k21/2δf2. In terms of these new variables, the system of equations (3) is rewritten in the more compact form as:


where Zkt=ZLZTtwith ZLkt=z1z2z3z4z5tand ZTkt=z6z7t. In Eq. (4), Nstands for a 7×7hydrodynamic matrix which is diagonal in the 5×5NLand the 2×2NTblocks. 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), Ξkt=ΞLΞTtis the stochastic term, composed by the longitudinal ΞLkt=ζ1ζ2ζ3ζ4ζ5tand transverse ΞTkt=ζ6ζ7tnoise vectors. The explicit form of the components ζm, m=17, as well as their fluctuation-dissipation relations (FDR), can be found in Eqs. (169)–(175) and Eqs. (176)–(186), respectively, in Appendix A of [25].

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 kand ω.The hydrodynamic modes are obtained by calculating its eigenvalues λ=, given by the roots of the characteristic equation pλ=pLλpTλ=0,where pLλand pTλare the characteristic polynomials of fifth and second order in λof the matrices NLand NT, respectively. These roots are calculated below.

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, ZXL=z1z2tand ZYL=z3z4z5t, 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 pLλ=pXXLλpYYLλ.It should be mentioned that pXXLλand pYYLλare 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 pXXLλand pYYLλ, 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 pYYLλ, namely, γ1DTk2, σ1k2, k2cs2and g2k2/cs2k2. They depend on the anisotropic coefficients of diffusivity DTand on the viscosity σ1.The former quantity is a function of the parallel χand perpendicular χcomponents of thermal diffusivity, while the latter depends on the nematic viscosity coefficients νii=15(see Eqs. (23) and (24) in Ref. [26], or Eqs. (74) and (75) in [25]). In the same way, in the equation for pYYLλ, the following quantities can be identified, gαβk2k2, gXβk2k2, DTk2, Ωχak2, σ3k2, K1γ1k2, Ω2KIρ0k4, 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 χa=χχand the torsional viscous coefficient γ1. We now compare all these quantities with ω¯csk, by introducing the (small) reduced dimensionless quantities:


The relevant point for our purpose is to realize that for most nematics at ambient temperatures, ρ0and Ωare of order of magnitude 1, γ1101, χiand νiare of order 102103, Ki106107, while β104[32]. If we consider that α1and g103, and knowing that in a typical light scattering experiments k=105cm1and cs=1.5×105cms1[41], the quantities given in Eq. (5) have the following orders of magnitude: a01021, a01021, a01024, a1103, a1103, a2102, a3102, a5105and a6106. We now follow the method of Ref. [40] and the solutions of the polynomial pYYLλmay 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 k2[24].

5.1.1 Sound longitudinal modes

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


where Γ12γ1DT+σ1is the anisotropic sound attenuation coefficient of the NLC. This result shows that the sound propagation modes, λ1and λ2,are in complete agreement with those already reported in the literature for NLC[31, 34].

5.1.2 Visco-heat and director longitudinal modes

These modes are the roots of the characteristic equation pYYLλ=0. In Ref. [26] (or in [25]), it is shown that, up to first order in the small quantities (Eq. (5)), these roots can be written approximately as:






where k̂2k2/k2. In Eq. (7), RβgΔTd3σ3χis the Rayleigh number and Rcdenotes its critical value above which convection sets in. It should be emphasized that our results are expressed in terms of the ratio Rk/Rcand are, therefore, independent of the value of the separation dbetween the plates. However, the appropriate value of din an experiment should be chosen with an experimental criterion [42].

The Rayleigh-number ratio Rk/Rccontains two contributions: the first term is due to the presence of the effective temperature gradient X, which depends on both, the temperature gradient αand the gravity field g. The second term is entirely a contribution due to αand the nematic anisotropy χa. For typical nematics and conventional light scattering experiments, both contributions are approximately of order 1016.

The decay rates λ3and λ4for an inhomogeneous nematic given by Eq. (7) are called visco-heat modes, because they are composed of the coupling between the thermal DTk2and shear σ3k2Ω2KIk4ρ0σ3k2diffusive modes through the ratio Rk/Rc. The nature of these modes may be propagative or diffuse, as will be shown below.

5.1.3 Values of Rk/Rc

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 Rk/Rc. For simple fluids, these features have been predicted theoretically 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 experimentally for nematics. 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 Rk/Rc<R0, where


which is always negative. Thus, if we consider the orders of magnitude of the involved quantities and typical light scattering experiment values of k, DTk2107, σ3k2108, and Ω2KIk4ρ01014, then R0101and the visco-heat modes, Eq. (7), will be propagative when Rk/Rc101. 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 αcontained in the effective temperature gradient Xchanges 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 NLCunder 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 propagation modes, this prediction suggests that it may be worth to design experiments to corroborate this phenomenon for nematics.

Figure 3.

The real part of the nematic visco-heat modes λ3 and λ4 as a function of the Rayleigh ratio Rk→/Rc. When Rk→/Rc<R0, both modes are propagative; if R0≤Rk→/Rc≤1, both are completely diffusive. For Rk→/Rc=R0, both modes are equal. In equilibrium, Rk→/Rc=0, and the onset of convection occurs for Rk→/Rc=1. Pure diffusive modes

When R0Rk/Rc1, 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 Rk/Rc=R0, then the visco-heat modes (7) reach the same value, and consequently, the three decay rates are:


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, σ3Ω2KIρ0σ3is usually greater than DT, it can be seen from Eq. (7) that, as Rk/Rcgrows 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, Rk/Rc=1, i. e., when Rreaches its critical value Rcand the two visco-heat modes (7) are simplified to:


while the third, λ5,is identical to Eq. (8). This behavior for the decay rates λ3and λ4is also shown in Figure 3.

It should be noted that our expressions for these three decay rates are not in agreement with those reported for an NLCin 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 λ3vanishes, the director mode λ5is virtually unchanged, while λ4has 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 Rreaches 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, pTλis the characteristic polynomial of second order in λof the matrix NT. The corresponding transverse hydrodynamic modes are the roots of this equation pTλ=0.

5.2.1 Shear and director transverse modes

Accordingly, the shear and director transverse modes are the roots of pTλ=0, and are given by Eq. (63) in Ref. [26] (or by Eq. (157) in [25]). Following again the approximate method of small quantities used previously, the quantities σ4, KIIk2/γ1and λ+KIIk2k2/ρ0, may be identified in this equation. In terms of them, we have another set of anisotropic coefficients given by the viscosity σ4, the elasticity KII, and symmetry λ+(see, respectively, Eqs. (27), (29), and (31) in [26]). We also define the small or reduced dimensionless quantities, analogous to those defined in Eq. (5), namely, a4σ4k2ω¯, a5KIIk2γ1ω¯, a6λ+2KIIρ0ω¯2k2k2, where again ω¯csk. It should be noted that the viscous coefficient σ4only depends on the viscous coefficients ν2,ν3,while the elastic coefficient KIIdepends on the two Frank elastic constants K2and K3.Since for typical nematics λ+1, γ1101, σ4102, KII106[32], and also by taking into account that cs105,k105, g103, the quantities a4, a5and a6have the orders of magnitude a4102, a5105, and a6106. According to Eqs. (64) and (65) in Ref. [26] (or Eqs. (167) and (168) in [25]), up to first order in such small amounts, these two roots can be written as:


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 NLCin a NESSdetermined by a Rayleigh-Bénard system, it is possible to obtain, as limit cases, the corresponding 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 discussed below.

6.1 Nematic in equilibrium

It has been found that for an NLCin a NESS, the effects of the external gradients αand gare only manifested in the coupling of the thermal diffusive and shear longitudinal modes, which gives rise to the visco-heat modes λ3,4indicated, respectively, by means of Eq. (7). If the nematic layer is in a state of homogeneous thermodynamic equilibrium, g=0and α=0, and therefore X=0and Rk/Rc=0.Thus, the hydrodynamic modes of a nematic, in the state of equilibrium (denoted by the superscript e), are composed of five longitudinal and two transverse modes. The longitudinal modes are integrated by the two acoustic propagatives λ1eand λ2egiven by Eq. (6); as well as by the three diffusives, which consist of one thermal:


another of shear:


and one more of the director, λ5e,which is the same as Eq. (8). The longitudinal diffusive modes (15) and (16) are obtained precisely from Eq. (7), since in this, the Rayleigh ratio, given by Eq. (9), is zero if αand gvanish. Moreover, the pair of transverse modes consist of the shear and director modes λ6eand λ7ewhich are equal to the Eq. (14). It is necessary to mention that the decay rates λiei=17are well known in the literature [22, 31, 46]. Note that λ3eand λ4eare shown in the middle part of Figure 3.

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, niis no longer a hydrodynamic variable, and the elastic constants Ki(for i=1,2,3) and the kinetic parameters γ1, λvanish. Also, χand χare reduced to the coefficient of thermal diffusivity χand χa=0. On the other hand, the nematic viscosities are reduced in the following way: ν1η, ν2η, ν3η, ν4ζ+13η, ν523η+ζ, where ηand ζdenote, respectively, the shear and 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 DTχ, σ11ρ043η+ζ, σ20, σ3ν, σ4ν, where νη/ρ0is the kinematic viscosity, whereas KI0, KII0, and Ω0. 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 simple fluid has the two acoustic propagative modes:


where cscorresponds to the adiabatic velocity of the sound in this medium and Γ12γ1χ+1ρ043η+ζis the corresponding coefficient of sound attenuation. On the other hand, according to the Eq. (7), the longitudinal visco-heat modes are:


In the isotropic limit of the simple fluid, λ5=λ7=0,so that, according to the Eq. (14), the only transverse mode of this substance in a Rayleigh-Bénard system is:


In Eq. (18), the ratio Rk/Rcis defined as:


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 Mof the stochastic system given by Eq. (20) in Ref. [26], the simple fluid limit is taken, it reduces to a matrix that is a generalization of the one given by the Eq. (6) in [38]. Additionally, if in the corresponding matrix Mfound for the simple fluid, the equilibrium limit is now considered, i. e., when αand gvanish, the resulting matrix is also reduced to that given by Eq. (4) of [38].

6.2.1 Values of Rk/Rc

The two visco-heat mode, as in the nematic, could be propagative or diffusive. These characteristics depend on the values assumed by the ratio Rk/Rc. For simple fluids, these have been predicted theoretically and corroborated experimentally. Propagative modes

If Rk/Rc<R0, where R0νχ2/4χν<0,the visco-heat modes (18) will be propagative. According to Eq. (20), this occurs again if the αcontained in Xchanges its sign and increases by several orders of magnitude, a situation that is achieved by inverting the temperature gradient (when heated from below and its intensity is increased). There are analytical [8, 37, 38] and experimental [43] studies that report, for simple fluids in these conditions, the presence of visco-heat propagative modes. Pure diffusive modes

When R0Rk/Rc1, the visco-heat modes preserve the form (Eq. (18)), they are real and completely diffusive. In this regime, there are again three cases of special interest. If Rk/Rc=R0, then both visco-heat modes (18) are identical and equal to:


On the other hand, if the simple fluid is in a state of homogeneous thermodynamic equilibrium, g=0and α=0, so that X=0and Rk/Rc=0;consequently, in this equilibrium state (identified by the superscript e), there is a thermal diffusive mode:


and the shear mode:


These decay rates are well known in the literature [8, 37, 38]. Finally, because in a simple fluid, commonly νis greater than χ, according to Eq. (18), and as Rk/Rcgrows and approaches to 1, the magnitude of the thermal diffusive mode decreases, while the shear mode grows. At the threshold of the convective regime (when Rkreaches its critical value Rc), Rk/Rc=1, and the two visco-heat modes (18) acquire the values:




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 FHto describe the dynamics of the fluctuations of a NLClayer in a NESScharacterized by the simultaneous action of a uniform temperature gradient αand 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, λ1and λ2, modes (6), as well as the triplet formed by the visco-heat pair λ3and λ4, modes (7), and the director λ5, mode (8). In addition, the transverse ones are given by the shear λ6and the director λ7, modes (14). We find that the influence of the temperature gradient αand the gravitational field goccurs only in the longitudinal modes, being greater its effect 109on the visco-heat pair λ3and λ4. This effect is quantified by means of the Rayleigh ratio Rk/Rc, Eq. (9), where Ris the Rayleigh number and Rcis its critical value above which convection sets in. The developed analysis corresponding to the nonconvective regime was carried out under the condition Rk/Rc1.

The analytical expressions calculated for the hydrodynamic modes of a nematic in the NESSconsidered exhibit behaviors that are of great interest in the following particular situations. First, if the isotropic limit of the simple fluid is taken, the NLChydrodynamic modes reduce to those in the same state out of equilibrium, modes (17)(19), [8, 37, 38]. If R=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=Rc, that is, at the threshold of convection, from the triplet of longitudinal λ3,λ4and λ5, the visco-heat λ3vanishes, and λ4is the sum of the thermal and shear modes, modes (12) and (13); while that of director λ5is 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 Rk/Rc<R0k, where R0kis the reference value (10), our results predict that the visco-heat pair λ3and 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 λ5do not coincide with those reported in the literature for an NLCin the same NESSconsidered 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 αand gis only manifested in the first two modes. This triplet is reduced to the corresponding director, thermal, and shear longitudinal modes of an NLCin 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).

Nevertheless, our calculated expressions for the visco-heat λ3,λ4,and director λ5modes 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 λ3and λ4, depending on the values taken by the ratio Rk/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 NLCsuggest 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 Skω, may be calculated from our FHdescription. In Ref. [49], an application of this nature was developed by calculating the Rayleigh dynamic structure factor for the NLCunder the NESSalready 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 NLCin a different NESS, such as that produced by the presence of an external pressure gradient, were published in the references [19, 20].


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.



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

n̂, nor nα

director field


thickness of the nematic layer


constant gravitational force of magnitude g


Cartesian unitary vectors


Cartesian coordinates




temperature gradient of magnitude zT


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 δv


component zof the rotational of δv


component zof the double rotational of δv


divergence of δn


component zof the rotational of δn


as superscript, indicates transpose matrix


vector whose components are the spatial Fourier transform of the variables δp,δφ,δs,δψ,δξ,δf1and δf2


longitudinal component of δXkt


transverse component of δXkt


coefficient matrix of the linear system for δXkt

MLand MT

longitudinal and transverse submatrices of M


stochastic vector of the linear system for δXkt


longitudinal component of Θkt


transverse component of Θkt


variables of same dimensionality (i=1,, 7)


vector of the variables zikt


longitudinal component of Zkt


transverse component of Zkt


coefficient matrix of the linear system for δZkt

NLand NT

longitudinal and transverse submatrices of N


noise vector of the linear system for Zkt


longitudinal component of Ξkt


transverse component of Ξkt


noise components of ΞL(i=1,, 5) and ΞT(i=6,7)


characteristic polynomial of the matrix N


characteristic polynomial of the submatrix NL


characteristic polynomial of the submatrix NT


eigenvalues of pλ

ZXLktand ZYLkt

components of the vector ZLkt

pXXLλand pYYLλ

polynomials in which pLλis broken down


anisotropic thermal coefficient


anisotropic viscous coefficients


thermal diffusivities parallel and perpendicular to n


anisotropic thermal diffusivity


nematic viscosities (i=1,,5)

Ωand λ+

anisotropic adimensional nematic coefficients

K1,K2and K3

elastic coefficients of Frank


anisotropic elastic coefficients


torsion viscosity


auxiliary parameter

a0,a0, and a0

small dimensionless longitudinal quantities

a1,a1, and a2

small dimensionless longitudinal quantities

a3,a5, and a6

small dimensionless longitudinal quantities

λ1and λ2

acoustic propagative longitudinal modes


anisotropic sound attenuation coefficient

λ3and λ4

visco-heat longitudinal modes


director diffusive longitudinal mode

kand k

components of kperpendicular and parallel to n̂0


unit vector of k̂


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

a4,a5, and a6

small dimensionless transverse quantities

λ6and λ7

shear and director diffusive transverse modes


nematic modes in the state of equilibrium (i=1,,7)


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


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Jorge Fernando Camacho and Rosalío Fernando Rodríguez (December 28th 2018). Equilibrium and Nonequilibrium Hydrodynamic Modes of a Nematic Liquid Crystal [Online First], IntechOpen, DOI: 10.5772/intechopen.82609. Available from:

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