REVISTA MEXICANA DE FISICA S 52 (5) 5–22 NOVIEMBRE 2006

Correlation functions and long-range order for a nematic innonequilibrium stationary states

H. Hıjar and R.F. Rodrıguez∗

Departamento de Fısica Quımica, Instituto de Fısica, Universidad Nacional Autonoma de Mexico,Apartado Postal 20-364, Mexico D.F. 01000, Mexico

Recibido el 10 de enero de 2006; aceptado el 21 de febrero de 2006

A thermotropic nematic thin film under the action of an external thermal gradient is studied theoretically. We use a fluctuating hydrodynamicapproach and a time-scale perturbation formalism to calculate the orientation, temperature and velocity autocorrelation functions as well asthe temperature-velocity fluctuations cross correlation function of the liquid crystal in the nonequilibrium state induced by the stationary heatflux. This method allows us to find, on slow time-scales, a contracted description in terms of the slow variables only, with a reduced dynamicmatrix which can be constructed by the perturbation procedure. The wave number and frequency dependence of these correlation functions isevaluated analytically and their explicit functional form in the configuration space is also calculated for both equilibrium and steady states ofthe fluid. We show that, out of equilibrium, all these correlactions are long-ranged. We calculate the effects of this long-range behavior on thelight scattering structure factor. Our results also show that the temperature and velocity autocorrelations contain two contributions, namely,a local equilibrium and a mode coupling contribution. From our quantitative estimations of these correlations, it can be established thatthe contribution due to the mode coupling mechanism is dominant over that based on spatial inhomogeneities in the fluctuation-dissipationrelation.

Keywords: Liquid crystals; fluctuations; nonequilibrium; correlation functions.

Se estudia teoricamente una pelıcula nematica termotropica sometida a la accion de un gradiente termico externo. Se usa una descripcionhidrodinamico fluctuante y un formalismo perturbativo de escalas temporales para calcular las funciones de autocorrelacion de orientacion,temperatura y velocidad, ası como la correlacion cruzada temperatura-velocidad del cristal lıquido en el estado fuera de equilibrio inducidopor el flujo de calor estacionario. Este metodo permite encontrar en la escala de tiempos lenta, una descripcion contraıda en terminos de lasvariables lentas solamente, con una matriz dinamica reducida que puede construye mediante el metodo perturbativo. La dependencia conel numero de onda y la frecuencia de estas correlaciones se calcula analıticamente y tambien su forma funcional explıcita en el espacio deconfiguracion, tanto para los estados de equilibrio como para los estados estacionarios del fluido. Se muestra que fuera de equilibrio todaslas correlaciones anteriores son de largo alcance. Tambien se calcula el efecto de largo alcance sobre el factor de estructura del sistema.Los resultados obtenidos muestran que las funciones de autocorrelacion de temperatura y velocidad contienen dos contribuciones, a saber,de equilibrio local y de acoplamiento de modos. De las estimaciones cuantitativas obtenidas de estas correlaciones, se establece que lacontribucion debida al mecanismo de acoplamiento de modos domina sobre el mecanismo basado en las inhomogeneidades espaciales en larelacion de fluctuacion-disipacion.

Descriptores: Cristales lıquidos; fluctuaciones; fuera de equilibrio; funciones de correlacion.

PACS: 24.60.Ky; 61.30-v; 78.35.+c

1. Introduction

In spite of the fact that the theory of fluctuations in nonequi-librium fluids was initiated in the late 70′s and pursued bymany authors [1–16], nowadays several questions concern-ing the nature of hydrodynamic fluctuations in stationarynonequilibrium states are still of current active interest. Oneof these issues is the long-range character of these fluctua-tions far from instability points [17]. It is well known thatthermal fluctuations in a simple fluid in equilibrium alwaysgive rise to short-range equal time correlation functions, ex-cept when close to a critical point. But when the fluid isdriven to nonequilibrium steady states (NESS) by the ac-tion of externally applied gradients, the equal-time correla-tion functions may develop long-range contributions, whosenature is very different from those in equilibrium. For simplefluids in NESS, it has been shown theoretically that the exis-tence of the so called generic scale invariance is the origin ofthe long-range observed in its correlation functions, [18–20].

Although these issues have been mostly investigated forsimple fluids, some studies for equilibrium and nonequi-librium stationary states in complex fluids also exist, in-cluding the enhancement of concentration fluctuations inpolymer solutions under external hydrodynamic and elec-tric fields [21], or polymer solutions subjected to a station-ary temperature gradient in the absence of any flow havebeen investigated [22]. Other applications have also beendeveloped [23–26]. Also, the behavior of fluctuations aboutsome stationary nonequilibrium states have been analyzedin the case of thermotropic nematic liquid crystals. Spe-cific examples are the nonequilibrium states generated by astatic temperature gradient [27], a stationary shear flow [28]generated by an externally imposed constant pressure gradi-ent [29, 30] or by means of a concentration gradient of im-purities in a dilute suspension [31]. In the first two cases, itwas found that the nonequilibrium contributions to the corre-sponding light scattering spectrum were small, but in the last

6 H. HIJAR AND R.F. RODRIGUEZ

two cases the nonequilibrium effects on the structure factormay be quite large and perhaps measurable. To our knowl-edge, however, at present there is no experimental confirma-tion of these effects, in spite of the fact that for nematics thescattered light intensity is several orders of magnitude greaterthan for ordinary simple fluids.

In this work we first present a model calculation based ona fluctuating hydrodynamic description which uses a time-scale perturbation formalism constructed upon the fact thatfor a thermotropic nematic the modes associated with thedirector relaxation are much slower than the visco-heat andsound modes [32, 33]. As a result, in the nonequilibrium sta-tionary states the hydrodynamic fluctuations evolve on threewidely separated times scales. By using this time-scale per-turbation procedure, in this work we calculate analyticallyall the equal-time correlation functions of the transverse andlongitudinal state variables for a thermotropic nematic liq-uid crystal in equilibrium and out of equilibrium states. Wederive the explicit

−→k and −→r dependence of these quantities

and show that they exhibit long-range order not only in equi-librium, as could have been expected at least for the orienta-tional correlation owing to the nematic nature of the fluid, butalso in the NESS due to the heat flux induced by a station-ary thermal gradient. We also evaluate theoretically the effectof the thermal gradient on the dynamic structure factor of thefluid. However, since to our knowledge there are no exper-imental results available in the literature for light scatteringfrom a nematic in NESS to estimate the predictions of ourmodel, we have used experimental parameter values similarto those used in light scattering experiments for an isotropicsimple fluid. In this way the effect of this long-range behav-ior on the dynamic structure factor of the fluid is examinedfor a geometry consistent with those used in light scatteringexperiments. We first calculate the director autocorrelationfunction in equilibrium and in NESS and find that the exter-nal gradient introduces an asymmetry of the spectrum shift-ing its maximum towards negative frequency intervals by anamount proportional to the magnitude of the gradient, whichmay be of the order of ∼ 7%. Also, the width at half heightmay decrease by a factor of ∼ 10%.

Then we extend our previous calculations to the trans-verse and longitudinal components of its temperature andvelocity autocorrelations and the cross temperature-velocitycorrelation function in and out of equilibrium. As before, theexistence of widely separated time-scales in the nematic isused to eliminate the fast variables from the general dynamicequations, thus obtaining a reduced description in which onlythe slow variables are involved. The method allows us to find,on the slow time-scales, a contracted description in terms ofslower variables with a reduced dynamic matrix which canbe constructed by the perturbation procedure. We derive theexplicit space dependence of the above mentioned correla-tions and show that all of them exhibit long-range order. Toestimate quantitatively the correlation functions predicted byour model calculations, we used experimental parameter val-ues similar to those used in light scattering experiments for

an isotropic simple fluid, since to our knowledge there areno reported light scattering experiments with liquid crystalsin NESS. Our results show that all these correlation func-tions exhibit long-range order. Furthermore, the temperature-temperature and velocity-velocity correlation functions con-tain two types of contributions, namely, a local equilibriumand a mode coupling contribution. From our quantitative es-timate we also verified that the contribution due to the modecoupling mechanism is dominant over that based on spatialinhomogeneities in the fluctuation-dissipation relation. Inthis way our analysis and model calculations confirm the ex-istence of the so-called generic scale invariance in a liquidcrystal [17].

2. Basic Equations

2.1. Model

Consider a thermotropic nematic liquid crystal layer of thick-ness d confined between two parallel plates in the presenceof the gravitational field and of a stationary heat flux, as de-picted in Fig.1. The transverse dimensions of the cell alongthe x and y directions are large compared to d. The long axisof the nematic molecules are oriented normal to the lowerand upper plates located at z = −d/2, z = d/2 and main-tained at the uniform temperatures T1 and T2, respectively.The hydrodynamic state of the fluid is specified by the veloc-ity field, ~v (~r, t), the unit vector, n (~r, t), defining the localsymmetry axis (director field), and the pressure p (~r, t) andthe temperature T (~r, t) fields.

We assume that the nematic has already reached a sta-tionary state (NESS) where its state variables take on thevalues Tss, pss, ~vss and nss. We shall restrict ourselves tosteady states without convection so that ~vss = 0 and assumean initial homeotropic configuration for the director field,nss = ez . By symmetry, pss and Tss can only depend onthe z coordinate and, if the gravitational field is ~g = −gez ,the stationary state is defined by the solution of the equations

FIGURE 1. Schematic representation of a plane homeotropic ne-matic cell. A stationary thermal gradient is applied in the directionez . The light scattering geometry which will be used later is alsoshown. −→q i and−→q s denote the incident and scattered wave vectors,−→p i and −→p s the incident and scattered polarizations, θ the scatter-ing angle and −→q the scattering vector.

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 7

dpss

dz+ gρss = 0, (1)

d2Tss

dz2= 0, (2)

which are derived from the well-known general nematody-namic equations [35, 36]. In Eq. (1), ρss = ρss (z) is the sta-tionary mass density of the nematic. It should be noted that,to arrive at Eqs. (1) and (2), we have neglected the depen-dence of the thermal conductivity on Tss (z) and pss (z). (1)and (2) are subjected to the following boundary conditions:

p (d/2) = p2, T (−d/2) = T1, T (d/2) = T2, (3)

where p2 is the pressure external to the cell.The spontaneous thermal fluctuations around the station-

ary state are defined as the deviations of the actual fieldsat point ~r and time t from the average macroscopic values,that is, δp ≡ p (~r, t) − pss (z), δT ≡ T (~r, t) − Tss (z),δ~v ≡ ~v (~r, t) , δ~n ≡ n (~r, t) − nss. Note that, since thedirector is a unitary vector field, it has only two indepen-dent fluctuating components. The dynamical behavior of theabove fluctuations is governed by a complete set of linearizedLangevin equations for the time evolution of the state vector~a ≡ (δp, δT, δvx, δvy, δvz, δnx, δny)T which are derived inthe Appendix. The superscript T denotes transpose. In com-pact form, these equations may be rewritten as

∂

∂tai (~r, t) = −Mijaj (~r, t)− Fi (~r, t) , (4)

where the explicit form of the elements of the hydrodynamicmatrix Mij is read from Eqs. (A.1)-(A.7). The stochasticforce vector ~F is given by

~F =(

αc2

cp∇iQi,

1ρsscv

∇iQi,1

ρss∇iΣxi,

1ρss

∇iΣyi,

1ρss

∇iΣzi, Υx, Υy

)T

, (5)

where the quantities α, c2, cp and cv have been defined in theAppendix. Qi, Σij and Υi represent the stochastic compo-nents of the heat flux, the stress tensor and the so-called direc-tor’s quasi-current respectively, ~F has the following stochas-tic properties:

〈Fi (~r, t)〉ss = 0, (6)

〈Fi (~r1, t1) Fj (~r2, t2)〉ss = Qij (~r1, ~r2) δ (t2 − t1) , (7)

where Qij (~r1, ~r2) is the covariance matrix and its explicitform in the stationary state is obtained by replacing, in itsequilibrium expression, Qeq

ij , all the equilibrium quantitiesappearing there as parameters, by their position-dependentsteady state values. In Ref. 33 we have calculated Qeq

ij fora nematic liquid crystal for the same homeotropic geometrywe are considering in this work.

It will be convenient to write the time evolution equationsfor the fluctuations in Fourier space by defining the space-time Fourier transform of an arbitrary field f (~r, t) by

f(~k, ω

)=

∫dteiωtf

(~k, t

), (8)

where f(~k, t

)is the space Fourier transform of f defined as

f(~k, t

)=

∫d~re−i~k·~rf (~r, t) . (9)

In this way Eq. (4) becomes

∂

∂tai

(~k, t

)= −Hij

(~k)

aj

(~k, t

)− Fi

(~k, t

), (10)

where Hij(~k) is the linear hydrodynamic matrix in ~k-space.Since the matrix elements Mij may depend on z, in arriv-ing at Eq. (10) we restricted ourselves to considering onlythose fluctuations that can be described as a superpositionof plane waves within a horizontal nematic layer of thick-ness l0, so that the spatial variation of the thermodynamicquantities inside the layer can be neglected. In this way theelements Mij are considered to be constants. Furthermore,as a first approximation to a more general description of ne-matic fluctuations where boundary conditions on the nematicsample should be taken into account [16, 34], we shall cal-culate correlation functions for points −→r and −→r ′, with z andz′ such that |z − z′| < l0, which correspond also to posi-tions far away from the boundaries, that is, |z − z′| ¿ d and|z + z′| ¿ d.

2.2. Transverse and longitudinal variables

Since the cartesian components of the director and velocityfields are strongly coupled, it will be convenient to work witha different representation where this coupling is minimized.This may be accomplished by defining the transverse com-ponents of the director and velocity fluctuating fields as theirprojections along the perpendicular direction to the ~k − nss

plane, that is,

δn1 ≡ 1k⊥

nss ·(~k × δ~n

)(11)

and

δv1 ≡ 1k⊥

nss ·(~k × δ~v

), (12)

where k⊥ =(k2

x + k2y

)1/2. Similarly, the longitudinal com-

ponents of these fields are defined as

δn3 ≡ 1k~k · δ~n, (13)

δv3 ≡ 1k~k · δ~v, (14)

δv2 ≡ 1kk⊥

~k ×[~k × nss

]· δ~v, (15)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

8 H. HIJAR AND R.F. RODRIGUEZ

which correspond to independent projections of the fluc-tuating fields in the ~k − nss plane. It follows di-rectly from definitions (11)-(15) and the explicit formof Eq. (10) that the longitudinal and transverse com-ponents evolve independently. It is therefore convenientto split the state vector ~a into longitudinal and trans-verse parts, namely, ~a(l) = (δn3, δv2, δT , δv3, δp)T and~a(t) = (δn1, δv1)T , which satisfy the equations

∂

∂ta(l)i = −H

(l)ij

(~k)

a(l)j − F

(l)i , (16)

∂

∂ta(t)i = −H

(t)ij

(~k)

a(t)j − F

(t)i , (17)

where H(l)ij (~k) and H

(t)ij (~k) are the reduced longitudinal and

transverse hydrodynamic matrices. Their explicit forms maybe obtained from Mij and will be given explicitly in the nextsection for a set of variables defined below, which have thesame dimensions and are more suitable for implementing thetime scaling perturbation theory later on. In Eqs. (16), (17),F

(l)i and F

(t)i are the corresponding random forces given in

terms of the stochastic currents Σij , Q :i, Υi

−→F (l)(

−→k , t) =

1k

(kxΥx + kyΥy

)

iρss

(k

k⊥kjΣzj − kz

k⊥kkjklΣjl

)

iρsscv

kjQji

kρsskjklΣjl

i αρsscvχT

kjQj

, (18)

−→F (t)(

−→k , t) =

1k⊥

(kxΥy − kyΥx

)

ik⊥ρss

(kxkjΣyj − kykjΣxj

) . (19)

In the next section we shall solve Eqs. (16) and (17) bytaking into account that the nematodynamic variables evolveon three widely separated time-scales and using a time scaleperturbation theory which allows us to describe the dynamicsof these variables separately.

3. Time Scaling Perturbation Theory

In previous works [31, 33], we have shown that, for a typicalthermotropic nematic, the modes associated with the directorrelaxation are much slower than those of the velocity field,i.e. τorientation/τvelocity ∼ 105. Therefore, transverse (longitu-dinal) director fluctuations relax to equilibrium much moreslowly than transverse (longitudinal) velocity fluctuations.Actually, this situation is common to a variety of physicalsystems where different variables evolve on different time-scales. The existence of widely separated time-scales may be

exploited to eliminate the fast variables from the general dy-namical equations, thus obtaining a reduced description inwhich only the slow variables are involved. A time-scaleseparation procedure may be implemented for our model inorder to diminish the remaining couplings between nemato-dynamic fluctuations by using the time scaling perturbationmethod introduced by Geigenmuller et al. [37-36]. Thismethod will allow us to find, on the slow time-scales, a con-tracted description in terms of the slow variables only, witha reduced dynamical matrix which can be constructed by aperturbation procedure. Moreover, a contracted descriptionon the fast time-scales can be found in terms of the fast fluc-tuations only, and a corresponding reduced matrix given alsoby a perturbation expansion.

In order to compare the relative magnitudes of the variouselements of the hydrodynamic matrices later on, we rescalethe variables in such a way that they all have the same di-mensions. Following the same procedure as for an isotropicfluid [40], we define the rescaled pressure, temperature, ve-locity and director fluctuations as

δp(~k, t

)≡

(χT

γ

)1/2

δp, (20)

δT(~k, t

)≡

(ρsscv

Tss

)1/2

δT , (21)

δvµ

(~k, t

)≡ ρ1/2

ss δvµ, µ = 1, 2, 3, (22)

δnµ

(~k, t

)≡ γ1kz

ρ1/2ss

δnµ, µ = 1, 3, (23)

where χT denotes the isothermal compressibility coeffiecientand γ1 is the orientational viscosity coefficient of the nematic.

3.1. Transverse variables

The dynamics of the transverse variables (δn1, δv1) is givenby the linear stochastic equation

∂

∂t

(δn1

δv1

)= −

(H

(t)11 H

(t)12

H(t)21 H

(t)22

) (δn1

δv1

)

−(

F(t)1

F(t)2

), (24)

where H(t)ij and F

(t)i represent the normalized transverse hy-

drodynamic matrix and stochastic force elements. It followsfrom Eqs. (17), (22), (23), that

H(t)

=

(1γ1

(K2k

2⊥ + K3k

2z

) −i (1+λ)γ12ρss

k2z

−i 1+λ2γ1

(K2k

2⊥ + K3k

2z

)1

ρss

(ν2k

2⊥ + ν3k

2z

) ,

)(25)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 9

with F(t)1 = γ1kzρ

−1/2ss F

(t)1 , F

(t)2 = ρ

1/2ss F

(t)2 . Here K2

and K3 represent the twist and bend elastic constants, respec-tively, while ν2 and ν3 denote two shear viscosity coefficientsof the nematic and λ is a non-dissipative coefficient associ-ated with director relaxation.

We now classify these transverse variables as slowor fast by estimating the magnitude of the matrix ele-ments H

(t)ij according to the method introduced in Refs. 37

to 39. To this end, note that H(t)11 ∼H

(t)21 ∼Kk2/ν and

H(t)12 ∼H

(t)22 ∼νk2/ρss, where ν and K denote any one of the

viscosity coefficients and elastic constants of the nematic, re-spectively. For a typical thermotropic nematic K ∼ 10−7

dyn, ν ∼ 10−1 poise, ρss ∼ 1 g cm−3, the inequalityK/ν ¿ ν/ρss holds, [33, 41], and H

(t)11 and H

(t)21 are small

compared with H(t)12 , H

(t)22 . If we follow the dynamics of δn1

and δv1 for a time of order τ1f = 1/H(t)22 , the contribution

of H(t)22 is order unity, while that of H

(t)11 is smaller, say of

order ε1 ¿ 1. More precisely, ε1 is the largest of the ratiosof H

(t)11 ∼ H

(t)21 with respect to H

(t)22 , that is, ε1 ∼ Kρss/ν2.

This observation allows us to identify δn1 as a slow variableand δv1 as a fast variable. There remains, however, a strongdynamical coupling between these variables because H

(t)12 is

also of order unity. Physically this means that the slow vari-able is forced to participate in the fast motion, although ithardly influences the motion of the fast one. It is possible todecrease this coupling by using a transformation, T(t), whichtransforms the state vector (δn1, δv1) and random force vec-tor (F (t)

1 , F(t)2 ) according to [37]- [39]

(δn′1δv′1

)= T(t)

(δn1

δv1

),

(F

(t)′1

F(t)′2

)= T(t)

(F

(t)1

F(t)2

), (26)

so that the new dynamical matrix[H

(t)]′

= T(t)H(t)

[T(t)

]−1

has the form[H

(t)]′

=(

0 00 F (t)

)+ ε1

(A(t) B(t)

C(t) D(t)

), (27)

where the quantities F (t), A(t), B(t), C(t) and D(t) are all oforder νk2/ρss. Following Geigenmuller et al., T(t) is chosenas

T(t) =

(1 − H

(t)12

H(t)22

0 1

)

=

(1 i1+λ

2γ1k2

z

ν2k2⊥+ν3k2

z

0 1

). (28)

As a consequence, δv′1 = δv1, F(t)′2 = F

(t)2 ,

δn′1 = δn1 + i1 + λ

2γ1k

2z

ν2k2⊥ + ν3k2

z

δv1 (29)

and

F(t)′1 = F

(t)1 + i

1 + λ

2γ1k

2z

ν2k2⊥ + ν3k2

z

F(t)2 . (30)

From (28) it can be shown that [H(t)

]′ has the desiredform (27). Thus after the transformation T(t), Eq. (26) takeson the form

∂

∂t

(δn′1δv1

)=

−{(

0 00 F (t)

)+ ε1

(A(t) B(t)

C(t) D(t)

)}(δn′1δv1

)

−(

F(t)′1

F(t)2

). (31)

This equation may be rewritten as a set of independent equa-tions for δn′1 and δv1 by applying the time scaling perturba-tion theory presented in Sec. 5 of Ref. 39. The dynamics ofδn′1and δv1 is described by two reduced hydrodynamic coef-ficients ωred

n1 (~k) and ωredv1 (~k), respectively, such that

∂

∂tδn′1 = −ωred

n1

(~k)

δn′1 − F(t)′1 (32)

and

∂

∂tδv1 = −ωred

v1

(~k)

δv1 − F(t)2 . (33)

It is important to emphasize that these equations are validfor different time scales. Eq. (32) describes the dynam-ics of δn′1 in the slow-time scale at times much larger thanτ1f , while (33) describes the evolution of δv1 in the fast timescale, that is, for times of order τ1f . Up to the first order in ε1,we have ωred

n1

(~k)

= ε1A(t) and ωred

v1

(~k)

= F (t) + ε1D(t).

The explicit expressions for these first order corrections arethen

ωredn1

(~k)

=1γ1

(K2k

2⊥ + K3k

2z

)

×[1 +

14

γ1 (1 + λ)2 k2z

ν2k2⊥ + ν3k2

z

], (34)

ωredv1

(~k)

=1

ρss

(ν2k

2⊥ + ν3k

2z

)

×[1−

(1 + λ

2

)2 ρss

(K2k

2⊥ + K3k

2z

)k2

z

(ν2k2⊥ + ν3k2

z)2

]. (35)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

10 H. HIJAR AND R.F. RODRIGUEZ

3.2. Longitudinal variables

Let us now consider the longitudinal variables (δn3, δv2, δT , δv3, δp) and use the same procedure as that followed for thetransverse variables in the previous subsection. However, we notice that in this case the time scaling perturbation methodmay be implemented twice, because Eq. (16) describes director relaxation as well as visco-heat and propagating modes whichevolve on three widely different time scales [33]. Thus, we shall apply the time scaling procedure in two steps in orderto systematically decouple the fast variables. For this purpose it will be convenient to start by grouping the longitudinalnormalized variables in the following vectors:

~w(~k, t

)≡

δn3

δv2

δT

(36)

and

~z(~k, t

)≡

(δv3

δp

). (37)

From Eqs. (16) and (20) - (23), it follows that they satisfy the stochastic equation

∂

∂t

(~w~z

)= −

(Hww Hwz

Hzw Hzz

)(~w~z

)−

(~Fw

~Fz

)(38)

with

Hww =

ωd3

(~k)

iγ1kzk⊥ρss

λ(~k)

0

i k2

kzk⊥λ

(~k)

ωd3

(~k)

ωv2

(~k)

−(

γ−1γ

)1/2γgc

k⊥k

−iρssDT

a

γ1

kkz

ω∇Tk⊥k ω∇T ωT

(~k)

, (39)

Hwz =

−iλγ1ρss

k2⊥k2

z

k2 0

ωv

(~k)

−(

γ−1γ

)1/2γgc

k⊥k ,

i(

γ−1γ

)1/2

ck + kz

k ω∇T 0

, (40)

Hzw =

−iλωd3

(~k)

ωv

(~k)

−(

γ−1γ

)1/2γgc

kz

k

−i(

γ−1γ

)1/2ρssDT

a

γ1

kkz

ω∇T −k⊥k

gc

(γ−1

γ

)1/2

ωT

(~k)

, (41)

Hzz =

(ωv3

(~k)

ick + γgc

kz

k

ick − gc

kz

k 0

), (42)

where c denotes the isentropic sound speed of the nematic as presented in the Appendix, and we have introduced the following

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 11

abbreviations

ωd3

(~k)

=K1k

2⊥ + K3k

2z

γ1, (43)

λ(~k)

=

[(1 + λ) k2

z + (1− λ) k2⊥

]

2k2, (44)

ω∇T =(cv

T

)1/2(

dTss

dz

), (45)

ωv2

(~k)

=ν3k

4⊥ + 2 (ν1 + ν2 − ν3) k2

⊥k2z + ν3k

4z

ρssk2, (46)

ωT

(~k)

= DT‖ k2

z + DT⊥k2

⊥, (47)

ωv

(~k)

=kzk⊥

[(2ν3 + ν5 − ν4 − ν2) k2

⊥ + (2ν1 + ν2 + ν5 − 2ν3 − ν4) k2z

]

ρssk2, (48)

ωv3

(~k)

=(ν2 + ν4) k4

⊥ + 2 (2ν + ν5) k2⊥k2

z + (2ν1 + ν2 + 2ν5 − ν4) k4z

ρssk2. (49)

In Eqs. (43)-(49), K1 represents the splay elastic con-stant, ν1, ν2, ... , ν5 are viscosity coefficients, and DT

‖ andDT⊥ denote the thermal diffusivities of the nematic along nss

and along the direction perpendicular to it, respectively; theyare related to the thermal diffusion coefficients by the usualexpression

DT‖(⊥) =

κ‖(⊥)

ρsscp. (50)

The stochastic force vectors ~Fw and ~Fz are given by

~Fw =

F(l)1

F(l)2

F(l)3

=

γ1kz

ρ1/2ss

F(l)1

ρ1/2ss F

(l)2(

ρsscv

Tss

)1/2

F(l)3

, (51)

~Fz =

(F

(l)4

F(l)5

)=

ρ

1/2ss F

(l)4(

χT

γ

)1/2

F(l)5

. (52)

As for the transverse variables, ~w and ~z may be classi-fied as slow and fast variables by estimating the magnitudeof the elements Hww, Hwz , Hzw and Hzz . However, be-fore carrying out this separation, we impose the condition ofconsidering only wave vectors ~k with magnitudes such that

k1 ¿ k ¿ k2, (53)

where k1 = c−1 (cv/Tss)1/2 (dTss/dz) and k2 = c/DT .

Here DT = κ/ρsscp and κ ∼ κ‖ ∼ κ⊥. The upper boundk2 guarantees that the terms of order DT k2 are small com-pared to ck. Similarly, the lower bound k1 ensures that allthe gradient terms in Eqs. (38) are also much smaller thanck. However, notice that for typical parameter values of athermotropic nematic, DT ∼ 10−3 cm2 s−1, and k2 probesspatial distances of the order 10 A, which are too small to bedescribed hydrodynamically. On the other hand, typically

k1 ∼ l−1, where l À l0 is the distance over which the vari-ations of the gradient are significant [40]. Since we are re-stricted to considering distances |−→r ′ −−→r | ¿ l0 ¿ l, whichimplies k À l−1, we notice that (53) is not a new restrictionin our analysis, but is consistent with the hydrodynamic treat-ment and our previous assumptions. Thus, we note that theelements of Hαβ , α, β = w, z, proportional to ck and whichare contained only in Hzw and Hzz , are much larger than allthe other elements.

If we follow the dynamics of ~w and ~z for a time of orderτ2f = 1/ck, the contribution of Hzz is order unity, while thatof Hww is small, say of order ε2 ¿ 1. More precisely, ε2 isthe largest of the ratios of the elements in Hww and Hzw withrespect to ck. In our case, ε2 ∼ νck/ρss. Furthermore, Hzw

is of order ε2. This observation allows us to identify the twogroups of variables, ~w and ~z, as slow and fast, respectively.There remains, however, a strong dynamic coupling between~w and ~z because Hzw is of order unity. As before, to decreasethis coupling we introduce a transformation T, such that

( −→w ′−→z ′

)= T

(~w~z

),

(~F ′w~F ′z

)= T

(~Fw

~Fz

), (54)

which, again, can be chosen so that the new dynamic matrixhas the form proposed by Geigenmuller et al.,

(H′ww H

′wz

H′zw H

′zz

)=

(0 00 F

)+ε2

(A BC D

), (55)

where F, A, B, C and D are all of order ck. For the presentmodel, the explicit form of T is

T =(

1 c0 1

), (56)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

12 H. HIJAR AND R.F. RODRIGUEZ

with

c = −

0 00 0

0(

γ−1γ

)1/2

. (57)

From (56) and (57) we obtain −→z ′ = ~z, ~F ′z = ~Fz ,

−→w ′ =

δn3

δv2

δT ′

, (58)

with δT ′ = δT − [(γ − 1)/γ]1/2δp, and

~F ′w =

F(l)1

F(l)2

F(l)′3

, (59)

with F(l)′3 = F

(l)3 − [(γ− 1)/γ]1/2F

(l)5 . Therefore, T

transforms Eq. (38) into

∂

∂t

( −→w ′

~z

)=

−{(

0 00 F

)+ ε2

(A BC D

)}( −→w ′

~z

)

−(

~F ′w~Fz

). (60)

If we now apply the time-scaling perturbation theory ofGeigenmuller et al., we may rewrite Eq. (60) as a set of in-dependent equations for −→w ′ and ~z, in terms of the reducedmatrices H

red

ww and Hred

zz , namely,

∂

∂t−→w ′ = −H

red

ww−→w ′ − ~F ′w (61)

and

∂

∂t~z = −H

red

zz ~z − ~Fz. (62)

Up to the first order in ε2 we have Hred

ww = ε2A andH

red

zz = F + ε2D, whose explicit expressions read

Hred

ww =

ωd3

(~k)

iγ1kzk⊥ρss

λ(~k)

0

i k2

kzk⊥λ

(~k)

ωd3

(~k)

ωv2

(~k)

−(

γ−1γ

)1/2γgc

k⊥k

−iρssDT

a

γ1

kkz

ω∇Tk⊥k

[ω∇T +

(γ−1

γ

)1/2gc

]1γ ωT

(~k)

, (63)

Hred

zz =

ωv3

(~k)

ick + gkz

ck

ick − gkz

ckγ−1

γ ωT

(~k)

. (64)

Thus we have eliminated from the description of the slowvariables −→w ′, the coupling with the fast ones. At this point itis essential to point out that, if the elements of H

red

ww are eval-uated for typical parameter values of a thermotropic, we findthat some of them are much larger than others in the samesense as in the case of the transverse variables. The domi-nant elements are of order νk2/ρss and are contained in thematrix elements which describe normalized velocity fluctua-tions only. This result implies that some of the slow variables−→w ′evolve in time faster than others and suggests that, to de-scribe the dynamics of the slower variables, it is convenientto apply the time-scaling perturbation theory again. Thus,following the same line of reasoning implemented in the pre-vious cases, we group the components of −→w ′ as

−→w ′ =(

δn3

~y

), (65)

with ~y =(

δv2

δT ′,

)and introduce the transformation

T(l) =

1 i2

γ1kzk⊥[(1+λ)k2z+(1−λ)k2

⊥]ν3k4

⊥+2(ν1+ν2−ν3)k2⊥k2

z+ν3k4z

00 1 00 0 1

, (66)

which converts Eq. (61) to the form in which time scalingperturbation can be applied for the slow longitudinal variable

δn′3 = δn3 +i

2γ1kzk⊥

[(1 + λ) k2

z + (1− λ) k2⊥

]

ν3k4⊥ + 2 (ν1 + ν2 − ν3) k2

⊥k2z + ν3k4

z

δv2

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 13

and the semi-slow variable ~y, with the resulting equations:

∂

∂tδn′3 = −ωred

n3

(~k)

δn′3 − F(l)′1 , (67)

with

ωredn3

(~k)

=1γ1

(K1k

2⊥ + K3k

2z

){

1 +14

γ1

[(1 + λ) k2

z + (1− λ) k2⊥

]2ν3k4

⊥ + 2 (ν1 + ν2 − ν3) k2⊥k2

z + ν3k4z

}, (68)

F(l)′1 = F

(l)1 +

i

2γ1kzk⊥

[(1 + λ) k2

z + (1− λ) k2⊥

]

ν3k4⊥ + 2 (ν1 + ν2 − ν3) k2

⊥k2z + ν3k4

z

F(l)2 ; (69)

∂

∂t~y = −H

red

yy ~y −−→F y, (70)

where

Hred

yy =

ωv2

(~k)− λ2(~k)γ1k2

ρssωv2(~k) ωd3

(~k)

−(

γ−1γ

)1/2γgc

k⊥k

k⊥k

{[1 +

λ(~k)DTa k2

ωv2(~k)

]ω∇T +

(γ−1

γ

)1/2gc

}1γ ωT

(~k)

, (71)

−→F y =

(F

(l)2

F(l)′3

). (72)

Equations (67) and (70) are correct in the slow and semi-slowtime-scales up to the first order in ε1. Therefore, we may thensay that the time evolution of the nematic’s longitudinal vari-ables at the hydrodynamic level is described by Eqs. (62),(67) and (70) for the slow variable δn′3, the semi-slow vari-able −→y , and the fast variable −→z , respectively.

4. Orientational Correlation Functions

4.1. Long-Range Order

In previous work we have calculated the correlation functionsof the director fluctuations δn1 and δn3 in both the equilib-

rium state and the stationary state induced by the thermal gra-dient [32]. Since those calculations were performed by usingthe approach discussed in the previous section, for the sake ofcompleteness in this section we shall briefly review those re-sults. The space-time Fourier transform of Eqs. (32) and (67)in the slow time-scale, may be written in the form

∂

∂tδnµ = −ωred

nµ

(−→k

)δnµ − σnµ,

µ = {1 transverse, 3 longitudinal (73)

where ωredn1 and ωred

n3 are given by Eqs. (34) and (68). Thefluctuating sources σnµ are obtained from Eqs. (30), (69) andare defined by

σn1 =1

k⊥

[kxΥy − kyΥx +

12

(1 + λ) kz

ν2k2⊥ + ν3k2

z

(kxkjΣyj − kykjΣyj

)], (74)

σn3 =1k

[kxΥx + kyΥy +

12

(1 + λ) k2z + (1− λ) k2

⊥ν3k4

⊥ + 2 (ν1 + ν2 − ν3) k2⊥k2

z + ν3k4z

(k2kjΣzj − kzkikjΣij

)]. (75)

As previously discussed, the stochastic components of the quasi-current of the director field and the stress tensor, Υi andΣij , respectively, are zero averaged 〈Υi〉 = 〈Σij〉 = 0 and in equilibrium they satisfy the fluctuation dissipation relations [33]

〈Υi (−→r , t)Υj (−→r ′, t′)〉 = 2kBT1γ1

δ⊥ijδ (−→r −−→r ′) δ (t− t′) , (76)

〈Σij (−→r , t)Σlm (−→r ′, t′)〉 = 2kBTνijlmδ (−→r −−→r ′) δ (t− t′) , (77)

where kB is Boltzmann’s constant, T is the equilibrium temperature of the nematic, δ⊥ij = δij − ninj is a projection operatorand νijlm is the viscous tensor as given in Ref. 33. Relations (76) and (77) describe stationary Gaussian Markov processes.

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

14 H. HIJAR AND R.F. RODRIGUEZ

Now, we shall assume that in the non-equilibrium steady state the fluctuation dissipation relations for Υi and Σij canbe obtained from (76) and (77) by replacing the equilibrium temperature T by the local z-dependent stationary temperatureTss (z) (see Eqs. (A.8) - (A.10)). This assumption leads to

⟨Υi

(−→k , ω

)Υj

(−→k ′, ω′

)⟩ss

= 2 (2π)4 kBT01γ1

δ⊥ij

(1 + iB

∂

∂kz

)δ(−→

k +−→k ′

)δ (ω + ω′) , (78)

⟨Σij

(−→k , ω

)Σlm

(−→k ′, ω′

)⟩ss

= 2 (2π)4 kBT0νijlm

(1 + iB

∂

∂kz

)δ(−→

k +−→k ′

)δ (ω + ω′) . (79)

Here T0 = (T1 + T2) /2 is the average temperature in the celland B ≡ (dTss/dz) /T0. The space-time Fourier transformof Eqs. (32) and (67) may be written in the form

δnµ

(~k, ω

)=

σnµ

(~k, ω

)

−iω + ωnµ

(~k) , (80)

where µ = 1, 3, and for simplicity we have omitted the su-perscript red in the reduced frequencies ωnµ. The directorauto-correlation functions

Xµµ

(~k,~k′;ω, ω′

)=

⟨δnµ

(~k, ω

)δnµ

(~k′, ω′

)⟩ss

(81)

can be constructed in terms of the corresponding fluctuation-dissipation relations for σn1 and σn3,

Xµµ =

⟨σnµ

(−→k , ω

)σnµ

(−→k ′, ω′

)⟩ss[

−iω + ωnµ

(−→k

)] [−iω′ + ωnµ

(−→k ′

)] , (82)

where no summation over the repeated index µ is implied.The fluctuation-dissipation relations satisfied by σn1 and σn3

may be found from Eqs. (74), (75), (78) and (79). By insert-ing the resulting expressions into Eq. (82), we obtain explicitforms for Xµµ. First we shall examine the spatial limitingbehavior of these correlations, which is given by

Xµµ (−→r ,−→r ′; t, t′)

=1

(2π)8

∫d−→k d−→k ′dωdω′ei

(−→k ·~r+−→k ′·−→r ′−ωt−ω′t′

)

× Xµµ

(−→k ,−→k ′;ω, ω′

), (83)

when (t′− t) −→ 0. Since the calculation of X11 and X33 isformally the same, we shall only describe the procedure forX11. Calculating 〈σn1(

−→k , ω)σn1(

−→k ′, ω′)〉ss with the help

of Eqs. (74), (78) and (79), replacing the result into (82), andintegrating over ω′,

−→k ′ and ω, we find that

X11 (−→r ,−→r ′; t, t′) = X(1)11 (−→r ,−→r ′; t, t′)

+X(2)11 (−→r ,−→r ′; t, t′) , (84)

where

X(1)11 = −kBTss (z)

(2π)3

∫d~k

ei~k·(−→r −−→r ′)−|t−t′|ωn1(~k)

K2k2⊥ + K3k2

z

(85)

and

X(2)11 = − ikBT0B

(2π)3 γ1

×∫

d~kkz

[− γ1K3

(K1k2⊥ + K3k2

z)2− 2 |t− t′| b1

(~k)]

× ei~k·(−→r −−→r ′)−|t−t′|ωn1(~k). (86)

Here we have considered t′ > t and defined

b1

(~k)

=(

1 + λ

2

)2γ1ν2k

2⊥

(ν2k2⊥ + ν3k2

z)2

+K3

K2k2⊥ + K3k2

z

[1 +

14

γ1 (1 + λ)2 k2z

ν2k2⊥ + ν3k2

z

]. (87)

From Eqs. (85) and (86), we can find the spatial rangeorder of transverse director fluctuations by considering thelimiting behavior at small times |t− t′| or large distances|−→r −−→r ′|, that is, when ξ = K |t− t′| /ν |−→r −−→r ′|2 ¿ 1,which corresponds to the limit of static correlation functions.In this limit we find

limξ→0

|X11| = kBTss (z)

4π (K2K3)1/2

× 1[|−→r ⊥ −−→r ′⊥|

2 + K2K3

(z − z′)2]1/2

×[1−

(dTss

dz

)z − z′

2Tss (z)

], (88)

where |−→r ⊥ −−→r ′⊥|2 = (x− x′)2 + (y − y′)2.Similarly, for the longitudinal director fluctuations we ob-

tain

limξ→0

|X33| = kBTss (z)4π (K1 −K3)

×

1|−→r −−→r ′| −

(K3/K1)1/2

[|−→r ⊥ −−→r ′⊥|

2 + K1K3

(z − z′)2]1/2

[1−

(dTss

dz

)z − z′

2Tss (z)

]. (89)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 15

Note that, in equilibrium, B = 0, Tss (z) = T , a constant, and the transverse orientational correlation function reduces to

limξ→0

|Xeq11 | =

kBT

4π (K2K3)1/2

1[|−→r ⊥ −−→r ′⊥|

2 + K2K3

(z − z′)2]1/2

, (90)

while the corresponding correlation function for the longitudinal director components is

limξ→0

|Xeq33 | =

kBT

4π (K1 −K3)

1|−→r −−→r ′| −

(K3/K1)1/2

[|−→r ⊥ −−→r ′⊥|

2 + K1K3

(z − z′)2]1/2

. (91)

Note that both Xeq11 and Xeq

33 are long range as could havebeen anticipated due to the well-known property of a nematicwhich spontaneously exhibits a macroscopic orientational or-der. They decay algebraically as |−→r −−→r ′|−1. In Fig. 2,we plot the normalized static correlation in equilibriumX

eq

11 ≡ 4πd (K2K3)1/2

Xeq11/kBT0 for x−x′=y−y′=z = 0,

T = T0 and as a function of normalized distance z′/d. Inthe stationary state, B 6= 0, the behavior of both correla-tions is modified by the presence of a term proportional tothe temperature gradient which does not decay, in the di-rection of the temperature gradient. This is also depictedin Fig. 2, where we plot the normalized static correlationX11≡4πd(K2K3)1/2X11/kBT0 for the same conditions asin the equilibrium case and a value of the normalized thermalgradient β′ = dB = 0.5 > 0. Note that, because of the pres-ence of dTss/dz, X11 becomes asymmetric. When comparedto its equilibrium value, correlations increase in the directionof lower temperatures and decrease in the opposite direction.Indeed, for a plate separation d = 10−2 cm, the differencebetween both curves becomes significant for z′/d ∼ 10−1.This means that the wave vectors sensitive to this differenceare of the order of q ∼ 103 cm−1. Since for light scatteringthe wave vector k and the scattering angle θ are related byq = 2qi sin θ/2, where qi ∼ 105m−1 is the incident wavenumber, this implies very low scattering angles θ ∼ 0.1◦.A quantitative evaluation of this nonequilibrium effect on ameasurable property will be discussed in the next section forthe structure factor of the fluid.

4.2. Light Scattering Spectrum

As an application of the above theory, we now calculate thelight scattering spectrum of the nematic in the stationary stateand for the scattering geometry defined in Fig. 1. For a ne-matic dielectric tensor, fluctuations come mainly from direc-tor fluctuations, and for the present model the spectral in-tensity of the scattered light is proportional to the dynamicstructure factor

S (−→q , ω) = −ε2a cos2 θ

Vsts

×Re {〈δn1 (−→q , ω) δn1 (−−→q ,−ω)〉} , (92)

where Vs and ts are the scattering volume and scattering time,respectively,−→q = −→q i−−→q s is the scattering vector (qy = 0),

and ω = ωs−ωi is the frequency shift. εa = ε‖−ε⊥ denotesthe dielectric constant anisotropy. Evaluating this expressionwith the help of Eqs. (78), (79), we obtain

S (−→q , ω) =2ε2

akBT0 cos2 θ

γ1

α (−→q )ω2 + ω2

n1 (−→q )

×{

1− 1T0

(dTss

dz

)2ωqzβ (−→q )

ω2 + ω2n1 (−→q )

}. (93)

In this result we have considered the spectrum produced bya scattering volume located at the center of the cell. Thenonequilibrium contribution has been written up to the small-est power of the wave number q, that is, up to the leadingterm in the hydrodynamic limit q → 0; ωn1 (−→q ) is given byEq. (34), and the functions α (−→q ) and β (−→q ) contain angularinformation through

α (−→q ) = 1 +(

1 + λ

2

)2γ1q

2z

ν2q2x + ν3q2

z

(94)

and

β (−→q ) =1γ1

[K3α (−→q )

+(K2q

2x + K3q

2z

) γ1ν2q2x

(ν2q2x + ν3q2

z)2

], (95)

FIGURE 2. (- - -) Decay of Xeq11 as a function of z′/d. (—–) Decay

of X11 as a function of z′/d for β′ = 0.5. The elastic constantvalues are K2 = 4.4× 10−7 dyn , K3 = 8.9× 10−7 dyn.

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

16 H. HIJAR AND R.F. RODRIGUEZ

since they do not depend on the magnitude of −→q but only onits orientation. In equilibrium, (93) reduces to

Seq (−→q , ω) =2ε2

akBT cos2 θ

γ1

α (−→q )ω2 + ω2

n1 (−→q ), (96)

and exhibits a q−4 dependence which is responsible for thewell-known long-range order spatial behavior of the orienta-tional correlations exhibited spontaneously by a nematic inequilibrium. On the other hand, the nonequilibrium contribu-tion is

Sneq (−→q , ω) = −2ε2akB sin2 θ

γ1

(dTss

dz

)

×2ωqzα (−→q )β (−→q )

[ω2 + ω2n1 (−→q )]2

, (97)

which also shows a long-range order decaying as q−5. Its odddependence on ω introduces an asymmetry in the shape of thestructure factor, shifting the maximum towards the region ofnegative values of ω, as shown in Fig. 3. Note that closeto equilibrium the size of the shift is indeed proportional todTss/dz.

For a fixed scattering vector−→q , we define the dimension-less dynamic structure factor, S0 (ω0), in terms of the normal-ized thermal gradient β′ = dB and the normalized frequencyω0 = ω/ωn1 (−→q ), by

S0 (ω0) =S (−→q , ω)Seq (−→q , 0)

=1

1 + ω20

{1− β′

2Aω0

1 + ω20

}, (98)

where A = qzβ (−→q ) /ωn1 (−→q ) d. In Fig. 3, we compareS0 (ω0) and Seq

0 (ω0) for low scattering angles, θ ∼ 0.1◦,qi ∼ 105 cm−1, β′ = 0.5 and typical values of the materialparameters of a thermotropic nematic [41]. From this resultit follows that the increase in the maximum is about 7% andthat the decrease in the half width at half height is 10%. Thisshows that the nonequilibrium state may induce changes inthe dynamic structure factor which could be detected experi-mentally.

FIGURE 3. (- - -) Normalized structure factor at equilibriumSeq

0 (ω0). (—–) Nonequilibrium structure factor S0 (ω0). We useβ′ = 0.5, d = 10−1 cm, θ ∼ 0.1◦, qi ∼ 105 cm−1. The valuesof the elastic constants are the same as in Fig. 2, ν2 = 0.41 poise,ν3 = 0.24 poise, γ1 = 0.76 poise, λ = 1.03.

5. Semi-slow variables correlation functions

In this section, we use the same method developed in theprevious section to calculate the correlation functions of thesemi-slow variables δv1, δv2 and δT . In the time-scale whereEqs. (33) and (70) are valid, the fast variables δp and δv3

have already decayed and average zero. Therefore, we canmake the approximations

y1 = δv2 = k−1⊥ (kδvz − kzδv3) ' k−1

⊥ kδvz (99)

and

y2 = δT − γ−1/2 (γ − 1)1/2δp ' δT , (100)

where δvz = ρ1/2δvz is the normalized z component of thevelocity field. On the other hand, in the same time-scale, theslow variables only slightly perturb the motion of the semi-slow ones through the different powers of the small quantityε1 ∼ Kρss/ν2. In Eqs. (33) and (70), we have explicitlywritten out the first of such perturbation corrections. How-ever, for a typical thermotropic ε1 ∼ 10−5 and in a first anal-ysis, these perturbation terms can be neglected.

Also notice that we can approximate (Hred

y )21'k⊥ω∇T /kin Eq. (71), since for a typical thermotropic nematic and typi-cal experimental values of the thermal gradient the followingrelations hold:

λ(~k)

DTa k2

ωv1

(~k) ∼ ρssD

Ta

ν∼ 10−3 ¿ 1 (101)

andg

c¿ ω∇T . (102)

Under these conditions, the stochastic equations (33)and (70 ) in ~k − ω space in terms of the non-scaled variablesδv1, δT and δvz , respectively, read

δv1 =1

−iω + ωv1

(~k) σv1, (103)

−iω + ωv2

(~k)

−k2⊥

k2 gα(

dTss

dz

) −iω + ωT

(~k)

×(

δvz

δT

)= −

(σv2

σT

), (104)

where we have introduced the following abbreviations for thestochastic force terms:

σv1 =i

ρss

(kjΣxj − kjΣyj

), (105)

σv2 =i

ρss

(kjΣzj − kz

k2kikjΣij

), (106)

σT =i

ρsscpkjQj . (107)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 17

The correlation functions of these stochastic terms are ob-tained from Eqs. (105)-(107) and the Fourier transform ofthe fluctuation dissipation theorems of the original noisesΣij and Qi in the stationary state, which are given byEqs. (A.8)-(A.10). In abbreviated form they read

⟨σv1

(−→k , ω

)σv1

(−→k ′, ω′

)⟩ss

= 2 (2π)4 kBT0fv1

(−→k ,−→k ′

) (1 + iBez · ∇−→k

)

× δ(−→

k +−→k ′

)δ (ω + ω′) , (108)

⟨σv2

(−→k , ω

)σv2

(−→k ′, ω′

)⟩ss

= 2 (2π)4 kBT0fv2

(−→k ,−→k ′

) (1 + iBez · ∇−→k

)

× δ(−→

k +−→k ′

)δ (ω + ω′) , (109)

⟨σT

(−→k , ω

)σT

(−→k ′, ω′

)⟩ss

= 2 (2π)4 kBT0fT

(−→k ,−→k ′

)

×[1 + iBez · ∇−→k − B2

(ez · ∇−→k

)2]

× δ(−→

k +−→k ′

)δ (ω + ω′) , (110)

where the functions fv1(−→k ,−→k ′), fv2(

−→k ,−→k ′) and

fT (−→k ,−→k ′) are given by

fv1

(−→k ,−→k ′

)= − 1

ρ2ss

kik′j

× (νxixj − νxiyj − νyixj − νyiyj) , (111)

fv2

(−→k ,−→k ′

)= − 1

ρ2ss

(kik

′jνzizj − k′z

k′2kik

′jk′lνzijl

+kz

k2kikjk

′lνijzl +

kzk′z

k2k′2kikjk

′lk′mνijlm

), (112)

fT

(−→k ,−→k ′

)= − 1

ρ2ssc

2p

kik′jκij . (113)

From Eqs. (103) and (108) we get an expres-sion for the transverse velocity correlation function〈δv1(

−→k , ω)δv1(

−→k ′, ω′)〉ss. Taking the inverse Fourier trans-

form of this expression and following a similar procedure tothe one described in Sec. 3 of Ref. 32, we arrive at

limζ¿1

〈δv1 (−→r , t) δv1 (−→r ′, t′)〉ss

=kBTss(z)

ρssδ (−→r −−→r ′) , (114)

where we have taken the limit of large distances or smalltimes defined by ζ¿1 with ζ=ν|t−t′|/ρss|−→r −−→r ′|2, since

in this work we are only interested in the spatial behaviorof the correlations. Notice that the non-equilibrium correla-tion function of transverse velocity fluctuations correspondsto a local version of its short-ranged equilibrium counterpart.This local equilibrium behavior is obtained from the assump-tion of the validity of the local version of the fluctuation dis-sipation relations.

We shall now construct the correlations of semi-slow lon-gitudinal variables by solving Eq. (104) for δT , δvz . Fur-thermore, these correlations will be expressed as the sum oftwo parts, namely, one arising from the dynamics of the vari-ables of interest and the other coming from the coupling withthe other variables. The first contribution will be identifiedby the superscript LE (local equilibrium) and is obtained bysolving Eq. (104) when the off-diagonal elements of the re-duced hydrodynamic matrix are neglected. The second con-tribution will be denoted by the superscript MC (mode cou-pling), because it contains the effects of the coupling termsof the reduced hydrodynamic matrix. The MC contributionwill be calculated up to the smallest power in the temperaturegradient. This procedure is similar to that used in Ref. [34]to compare the non-equilibrium long-range order effects pro-duced by the local equilibrium version of the fluctuation dis-sipation theorems and the mode coupling mechanism.

For the temperature autocorrelation, this procedure leadsto

⟨δT

(~k, ω

)δT

(~k′, ω′

)⟩ss

=⟨δT

(~k, ω

)δT

(~k, ω′

)⟩LE

ss

+⟨δT

(~k, ω

)δT

(~k′, ω′

)⟩MC

ss, (115)

with⟨δT

(~k, ω

)δT

(~k′, ω′

)⟩LE

ss

=

⟨σT

(~k, ω

)σT

(~k′, ω′

)⟩ss[

−iω + ωT

(~k)] [

−iω′ + ωT

(~k′

)] (116)

and⟨δT

(~k, ω

)δT

(~k′, ω′

)⟩MC

ss

= −k⊥k′⊥kk′

(dTss

dz

)2

×

⟨σv2

(~k, ω

)σv2

(~k′, ω′

)⟩0[

−iω + ωv2

(~k)] [

−iω + ωT

(~k)]

× 1[−iω′ + ωv2

(~k′

)] [−iω′ + ωT

(~k′

)] . (117)

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

18 H. HIJAR AND R.F. RODRIGUEZ

where the subscript 0 indicates evaluation at (dTss/dz) = 0, that is,⟨σv2

(~k, ω

)σv2

(~k′, ω′

)⟩0

=⟨σv2

(~k, ω

)σv2

(~k′, ω′

)⟩ ∣∣(dTss/dz)=0 .

Similarly, for the temperature-velocity cross correlation we get⟨δvz

(~k, ω

)δT

(~k′, ω′

)⟩LE

ss= 0, (118)

⟨δvz

(~k, ω

)δT

(~k′, ω′

)⟩MC

ss= −k⊥

k

(dTss

dz

) ⟨σv2

(~k, ω

)σv2

(~k′, ω′

)⟩0[

−iω + ωv2

(~k)] [

−iω′ + ωv2

(~k)] [

−iω′ + ωT

(~k′

)] , (119)

and for the velocity autocorrelation we obtain

⟨δvz

(~k, ω

)δvz

(~k′, ω′

)⟩LE

ss=

k⊥k′⊥kk′

⟨σv2

(~k′, ω

)σv2

(~k′, ω′

)⟩ss[

−iω + ωv2

(~k)] [

−iω′ + ωv2

(~k′

)] (120)

⟨δvz

(~k, ω

)δvz

(~k′, ω′

)⟩MC

ss= −k⊥k′⊥

kk′gα

(dTss

dz

) ⟨σv2

(~k, ω

)σv2

(~k′, ω′

)⟩0

× k2

⊥k2

[−iω + ωv2

(~k)] [

−iω + ωT

(~k)] +

k′2⊥k′2

[−iω′ + ωv2

(~k′

)] [−iω′ + ωT

(~k′

)] . (121)

It should be pointed out that in order to arrive at expres-sions (116)-(121) we have used the same typical parametervalues for a thermotropic nematic and taken into account re-lations (101) and (102).

We now write the above correlation functions in ~r − tspace. For this purpose we first substitute the fluctuation-dissipation relations (108)-(110) in Eqs. (116)-(121) and takethe inverse Fourier’s transform according to (9). Moreover, ifwe also take the limit ζ → 0, the LE part Eq. (116) reducesto

limζ¿1

⟨δT (−→r , t) δT (−→r ′, t)

⟩LE

ss=

kBTss (z)ρsscp

δ (−→r ′ −−→r )

+(

dTss

dz

)2kB

4πρsscp

×

(DT‖ /DT

⊥)1/2

[|−→r ⊥ −−→r ′⊥|

2 +DT‖

DT⊥

(z − z′)2]1/2

, (122)

which shows a short-range contribution proportional toδ (−→r ′ −−→r ) and an algebraically decaying anisotropic long-range part arising from the thermal gradient through the as-sumption of the validity of a local version of the fluctuationdissipation relations.

It can also be shown that, in the same limit, the modecoupling contribution Eq. (117) reads

limζ¿1

⟨δT (−→r , t) δT (−→r ′, t)

⟩MC

ss

=(

dTss

dz

)2kBT0

2πρssDT‖

(ν3 + DT

‖)

×[

Ia1 (−→r ,−→r ′)(a1 − a2) (a1 − a3)

+Ia2 (−→r ,−→r ′)

(a2 − a1) (a2 − a3)

+Ia3 (−→r ,−→r ′)

(a3 − a1) (a3 − a2)

], (123)

where να = να/ρss, α = 1, 2, 3, are kinematic viscositycoefficients, a3 = DT

⊥/DT‖ , and a1 and a2 depend only on

material parameters through

a1

a2

}=

1

2(ν3 + DT

‖)

{2 (ν1 + ν2 − ν3)

+ DT⊥ + DT

‖ ±[(2(ν1 + ν2 − ν3) + DT

⊥ + DT‖ )2

− 4(DT‖ + ν3

) (DT⊥ + ν3

) ]1/2}. (124)

In (123) the functions Ia (−→r ,−→r ′), where a is an arbitrarypositive constant, are

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 19

Ia (−→r ,−→r ′) ≡ 1√a×

[|−→r ⊥ −−→r ′⊥|2 + a (z − z′)2

]1/2

. (125)

Eqs. (123) and (125) show that the mode coupling contribution to the temperature autocorrelation increases anisotropicallywith the distance |−→r −−→r ′| with a simple power law. This result is similar to the one obtained for an isotropic fluid [43].

Similarly, in ~r − t space Eqs. (120) and (121) are, respectively,

limζ¿1

〈δvz (−→r , t) δvz (−→r ′, t)〉LEss =

kBTss (z)4πρss

(3 (z − z′)2

|−→r −−→r ′|5− 1

|−→r −−→r ′|3)

− kBT0B

4πρss

z′ − z

2

(3 (z − z′)2

|−→r −−→r ′|5− 1

|−→r −−→r ′|3)

, (126)

limζ¿1

〈δvz (−→r , t) δvz (−→r ′, t)〉MCss = − gαkBT0

4πρssν3

(ν3 + DT

‖)

(dTss

dz

)1

(b2 − b1) (a1 − a2)

×{[

Ib2 (−→r ,−→r ′)a1 − b2

− Ib2 (−→r ,−→r ′)a2 − b2

]+

[Ib1 (−→r ,−→r ′)

a2 − b1− Ib1 (−→r ,−→r ′)

a1 − b1

]

+[Ia2 (−→r ,−→r ′)

a2 − b2− Ia2 (−→r ,−→r ′)

a2 − b1

]+

[Ia1 (−→r ,−→r ′)

a1 − b1− Ia1 (−→r ,−→r ′)

a1 − b2

]}. (127)

Here

b1

b2

}=

ν1+ν2−ν3

ν3±

[(ν1+ν2−ν3

ν3

)2

− 1

]1/2

(128)

and Ib1 , Ib2 are given by (125). Note that in contrast toEq. (122), (126) apparently does not show a short-range be-havior proportional to δ (−→r −−→r ′), but decays as |−→r −−→r ′|−3.

These terms arise because we have extrapolated the correla-tion (120) for t < t′, to t = t′. But in doing this we haveneglected the effects due to the fast variables; this is actuallythe physical significance of approximation (99). However, itcan be shown that, if the fast variable δv3 is also taken intoaccount in the construction of the correlation (120), the firstterm in (126) reduces to δ (−→r −−→r ′) [42].

Finally, in configuration space the cross-correlationEq. (119) turns out to be

limζ¿1

〈δvz (−→r , t) δT (−→r ′, t′)〉MCss =

(dTss

dz

)kBT0

4πρss

(ν3 + DT

‖)

(a2 − a1)

×

1√

a1

[|−→r ⊥ −−→r ′⊥|

2 + a1 (z − z′)2]1/2

− 1√

a2

[|−→r ⊥ −−→r ′⊥|

2 + a2 (z − z′)2]1/2

, (129)

which shows an anisotropic long-range decay as1/ |−→r −−→r ′|.

Eqs. (123) and (127) are valid in the range of distances|−→r −−→r ′| such that

|−→r −−→r ′| ¿ Λ, (130)

where Λ is a characteristic length defined by

Λ =

∣∣∣∣∣αg

ν3DT‖

(dTss

dz

)∣∣∣∣∣

−1/4

. (131)

The limit (130) guaranties that the calculated correlationsare produced in the bulk of the nematic and in a region wherethe changes to the material parameters induced by the thermalgradient are not significant.

In order to illustrate the non-equilibrium behavior ofthese correlations, we consider the long-range order contri-butions to the temperature auto-correlation function as givenby the second term on the right hand side of Eq. (122) andby Eq. (123). We define the normalized long-range order

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

20 H. HIJAR AND R.F. RODRIGUEZ

contributions by

FLE (−→r ,−→r ′)

= β′2d

[|−→r ⊥ −−→r ′⊥|

2 +DT‖

DT⊥

(z − z′)2]1/2

, (132)

where β′ = dB is the normalized temperature gradient and,accordingly,

FMC (−→r ,−→r ′) = β′22dT0cp

DT‖

(ν3 + DT

‖)

[Ia (−→r ,−→r ′)

(a− b) (a− c)

+Ib (−→r ,−→r ′)

(b− a) (b− c)+

Ic (−→r ,−→r ′)(c− a) (c− b)

]. (133)

It should be emphasized that the former arises from the localversion of the fluctuation dissipation relations while the latteroccurs due to the coupling between velocity and temperaturefluctuations in the presence of the thermal gradient. In Fig. 4we plot these contributions for −→r = 0, −→r ′ = z′ez , values ofthe material parameters involved for a typical thermotropicnematic, γT ∼ 10−4 and d ∼ 1 cm, as a function of the nor-malized distance z′ = z′/d. Fig. 4 shows that, for values ofz′ such that z′ ∼ 10−6 ¿ Λ/d, we have FMC À FLE , thatis, in the region where the calculated correlations are valid,the mechanism of mode coupling is much stronger than thelong-range order induced by the local equilibrium version ofthe fluctuation-dissipation relations.

6. Discussion

In summary, by using a fluctuating hydrodynamic descrip-tion, in this work we have investigated theoretically the spa-tial behavior of all the fluctuation correlation functions for athermotropic nematic liquid crystal. To this end we have

FIGURE 4. Normalized long-range order contributions to thetemperature-temperature autocorrelation function for −→r = 0,−→r ′ = z′ez , as given by Eqs. (132) and (133), vs. the normalizeddistance z′ = z′/d. The dashed line (- - -) denotes the contributionarising from the mechanism of inhomogeneities in the fluctuation-dissipation relation, Eq.(132). The continuous line (—) representsthe mode coupling contribution, Eq. (133).

explicitly used the fact that for a thermotropic nematicthe modes associated with the director relaxation are muchslower than the visco-heat and sound modes. The method de-scribed in Refs. 37 to 39 allowed us to find, on the slow time-scales, a contracted description in terms of the slow vari-ables only, with a reduced dynamic matrix which can be con-structed by the perturbation procedure. To clarify and elabo-rate on some of our results, the following comments may beuseful.

First, we showed that the director, temperature and veloc-ity autocorrelations, as well as the temperature-velocity crosscorrelation functions exhibit long-range order when the sys-tem is in a nonequilibrium steady state (NESS) induced bythe action of an external temperature gradient and the gravityfield. This is shown in Eqs. (88), (91), (122), (123), (126),(127) and (129), respectively. These results also show explic-itly that these correlation functions contain two contributions,namely, a local equilibrium and a mode coupling contribu-tion. For the temperature-temperature correlation, these con-tributions are plotted in Fig. 4 for typical material parametervalues and for a value of the gradient which is used exper-imentally for simple fluids, (dTss/dz) = 50 Kcm−1 [43].From these curves one can clearly see that the contributiondue to the mode coupling mechanism is dominant over thespatial inhomogeneities in the fluctuation-dissipation rela-tion. We also estimated the influence of the stationary heatflux on the light scattering spectrum of a thermotropic ne-matic. The analysis carried out in this work included onlythe orientation correlation functions and the model’s geome-try has been constructed so that it corresponds to those usedin an experimental arrangement appropriate for detecting theso-called mode 2 of the spectrum [44]. However, it shouldbe emphasized that this is a model calculation and since toour knowledge there are no experimental results to comparewith, the correctness of our choice of experimental parame-ters such as d, θ or β′, remains to be assessed. However, asfound in other nonequilibrium states for liquid crystals [33],their magnitude suggests that they could be experimentallydetected.

Secondly, this behavior of the correlation functionsis in agreement with the one obtained for the director-director fluctuation correlation function obtained in previouswork [32]. Therefore, all functions for a liquid crystal inNESS exhibit long-range behavior. Thus these results con-firm the existence of the so-called generic scale invariance ina liquid crystal, a property shared by systems in nonequilib-rium steady states, which has been proposed as the origin ofthe long-range nature of the correlation functions [18–20].

Acknowledgments

We acknowledge partial financial support from DGAPA-UNAM IN112503 and from FENOMEC through grantCONACYT 400316-5-G25427E, Mexico. One of us (H. H.)acknowledges a scholarship from CONACYT, Mexico.

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

CORRELATION FUNCTIONS AND LONG-RANGE ORDER FOR A NEMATIC IN NONEQUILIBRIUM STATIONARY STATES 21

A Appendix: Fluctuating Nematodynamics

The hydrodynamic equations for the small deviations of the state vector ~a≡ (δp, δT, δvx, δvy, δvz, δnx, δny)T in the geometryunder consideration shown in Fig. 1, are derived by linearizing the general conservation equations of mass, momentum, energyand the relaxation equation for the director field of a thermotropic nematic liquid crystal given by Eqs. (A.1)-(A.9) in Ref. 33.The result is

∂

∂tδp =

αγ

χT

(DT‖∇2

z + DT⊥∇2

⊥)

δT − γ

χT∇iδvi + gρss(z)δvz + DT

a

α

χT

dTss

dz∇iδni − α

χT cvρss(z)∇iQi, (A.1)

∂

∂tδT = γ

(DT‖∇2

z + DT⊥∇2

⊥)

δT − γ − 1α

∇iδvi − dTss

dzδvz +

dTss

dzDT

a∇iδni − 1cvρss(z)

∇iQi, (A.2)

ρss(z)∂

∂tδvx =

[(ν2 + ν4)∇2

x + ν2∇2y + ν3∇2

z

]δvx + (ν3 + ν5)∇x∇zδvz + ν4∇x∇yδvy

−∇xδp− 1 + λ

2(K1 −K2)∇x∇y∇zδny − 1 + λ

2(K1∇2

x + K2∇2y + K3∇2

z

)∇zδnx −∇jΣxj , (A.3)

ρss(z)∂

∂tδvy =

[ν2∇2

x + (ν2 + ν4)∇2y + ν3∇2

z

]δvy + (ν3 + ν5)∇y∇zδvz + ν4∇x∇yδvx −∇yδp

− 1 + λ

2(K1 −K2)∇x∇y∇zδnx − 1 + λ

2(K2∇2

x + K1∇2y + K3∇2

z

)∇zδny −∇jΣyj , (A4)

ρss(z)∂

∂tδvz =

[(2ν1 + ν2 + 2ν5 − ν4)∇2

z + ν3∇2⊥

]δvz + (ν3 + ν5) (∇x∇zδvx +∇y∇zδvy)−∇zδp

− gρss(z)χT δp− λ− 12

(K1∇2

⊥ + K3∇2z

)∇iδni + gρss(z)αδT −∇jΣzj , (A.5)

∂

∂tδnx = − 1

γ1

(K1∇2

x + K2∇2y + K3∇2

z

)δnx +

1 + λ

2∇zδvx − 1− λ

2∇xδvz

+1γ1

(K1 −K2)∇x∇yδny −Υx, (A.6)

∂

∂tδny =

1γ1

(K2∇2

x + K1∇2y + K3∇2

z

)δny +

1 + λ

2∇zδvy − 1− λ

2∇yδvz

+1γ1

(K1 −K2)∇x∇yδnx −Υy. (A.7)

In these equations, α = − (1/ρ) (∂ρ/∂T )p is the thermal expansion coefficient, χT is the isothermal compressibility, DT‖

and DT⊥ are thermal diffusivities along the parallel and perpendicular directions with respect to nss and DT

a = DT‖ − DT

⊥ isthe corresponding anisotropy, and c, cp are the specific heat at constant volume and pressure with γ = cp/cv . Eqs. (A.1)-(A.7)can be written in terms of the isentropic sound speed, c, by using the thermodynamic relation c2 = γ/ρχT . ν1, ν2, ν3, ν4, ν5

denotes the five nematic viscosity coefficients of a nematic, γ1 is the orientational viscosity and K1, K2 and K3 are the splay,twist and bend elastic constants [35].

∇jΣij (~r, t), ∇iQi (~r, t), Υi (~r, t) are the fluctuating components of the momentum current, the heat current and therelaxation quasi-current of the orientation of the nematic, respectively. These stochastic currents are chosen so that they arezero averaged stochastic processes 〈Σij(~r, t)〉 = 〈Qi(~r, t)〉 = 〈Υi(~r, t)〉 = 0 satisfying fluctuation-dissipation relations of theform [31]

〈Σij (−→r , t)Σkl (−→r ′, t′)〉 = 2kBTss(z)νssijklδ (−→r −−→r ′) δ (t− t′) , (A.8)

〈Υi (−→r , t)Υj (−→r ′, t′)〉 = 2kBTss(z)

γ1δss,⊥ij δ (−→r −−→r ′) δ (t− t′) , (A.9)

〈Qi (−→r , t)Qj (−→r ′, t′)〉 = 2kBT 2ss(z)κss

ij δ (−→r −−→r ′) δ (t− t′) . (A.10)

Here kB is Boltzmann’s constant, δ (t− t′) denotes Dirac’s delta function, δss,⊥ij = δij − nss,inss,j is a linearized projection

operator and δij is the usual Kronecker’s delta. κssij is the linearized thermal conductivity tensor, κss

ij = κ⊥δij + κanss,inss,j ,

Rev. Mex. Fıs. S 52 (5) (2006) 5–22

22 H. HIJAR AND R.F. RODRIGUEZ

where κa = κ‖ − κ⊥ is the anisotropy in the thermal conductivity of the nematic where κ⊥, and κ‖ denote its perpendicularand parallel components with respect to the director field respectively. The linearized stress tensor νss

ijkl is given by

νssijkl = ν2(δjlδik + δilδjk) + 2(ν1 + ν2 − 2ν3)nss,inss,jnss,knss,l

+ (ν3 − ν2)(nss,jnss,lδik + nss,jnss,kδil + nss,inss,kδjl + nss,inss,lδjk)

+ (ν4 − ν2)δijδkl + (ν5 − ν4 + ν2)(δijnss,knss,l + δklnss,inss,j). (A.11)

∗. Fellow of SNI, Mexico. Also at FENOMEC. Correspondenceauthor. E-mail: zepeda@fisica.unam.mx.

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