Shear and flexoelectric deformation effects on the dynamic structure factor of a nematic liquid crystal

J. Non-Newtonian Fluid Mech. 165 (2010) 1033–1039

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Shear and flexoelectric deformation effects on the dynamic structure factor of a nematic liquid crystal H. Híjar a,1 , R.F. Rodríguez b,∗,1,2 a b

Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior de Ciudad Universitaria, 04510, Mexico, D.F., Mexico Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 México, D.F., Mexico

a r t i c l e

i n f o

Article history: Received 24 November 2009 Received in revised form 17 May 2010 Accepted 26 May 2010

Nematic liquid crystals Stationary states Structure factor

a b s t r a c t We use a fluctuating hydrodynamic description to calculate the orientation (director) correlation function of hydrodynamic fluctuations for a nematic liquid crystal in a non-equilibrium steady state (NESS). This state is produced by both, a planar shear flow and an external uniform electric field, which induces an elastic deformation through the so-called flexoelectric effect. We analyze the effects of these external gradients to estimate the changes on the light scattering spectrum produced by both gradients with respect to its equilibrium shape. We find that these non-equilibrium effects induce long-range effects on the orientation correlation with a k-dependence k−2 and k−3 , respectively, giving rise to an asymmetry of the spectrum. This consists of an increment of its maximum and a displacement in the frequency shift space. These changes are proportional to the magnitude of the external gradients and also depend on the relative orientation of the gradients with respect to the wave vector. We find that for MBBA, the shear flow can produce significant increases on the dynamic structure factor of 9.6%, for normalized velocity gradients of the order 0.1–0.5. The flexoelectric effect produces only small changes on the structure factor, which are about two orders of magnitude smaller with respect to those due to the presence of the shear flow. Our analysis suggests that the effects produced on dynamic correlations by the flexoelectric effect would be hard to observe experimentally, while those due to the shear flow might be detectable. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Thermal fluctuations in an equilibrium isotropic fluid always give rise to short-range equal-time correlation functions, except close to a critical point. However, when external gradients are applied, equal-time correlation functions develop long-range contributions, whose nature is very different from those in equilibrium. For a variety of systems in non-equilibrium stationary states (NESS) 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 [1]. For instance, for a simple fluid under a thermal gradient, its structure factor, which determines the intensity of the Rayleigh scattering, diverges as k−4 for small values of the wave number k. This dependence amounts to an algebraic decay of the density–density correlation function, a feature that has been verified experimentally [2–4].

∗ Corresponding author. Tel.: +52 55 56225153; fax: +52 55 56225008. E-mail addresses: [email protected] (H. Híjar), zepeda@fisica.unam.mx (R.F. Rodríguez). 1 Fellow of SNI Mexico. 2 Also at FENOMEC, UNAM, Mexico. 0377-0257/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2010.05.010

Although studies of the behavior of fluctuations in complex fluids, and in nematic liquid crystals in particular, about NESS are more scarce, some specific examples have been studied theoretically. Such is the case of the non-equilibrium situations generated by a static temperature gradient [5,6], a stationary shear flow [5,7], by an externally imposed constant pressure [8–10], or concentration gradients, [11]. For the temperature gradient and the shear flow it was found that the non-equilibrium contributions to the 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 [12]. The basic purpose of this work is to present a model which can be used to calculate the effects produced by the simultaneous presence of two external gradients, a planar shear flow and an external uniform electric field, on a measurable property of a thermotropic nematic liquid crystal, namely, its light scattering spectrum S(k, ω). To this end we use fluctuating hydrodynamic theory and study the dynamics of fluctuations of the transverse components of the velocity and director fields around a NESS produced by the simultaneous application of these two external gradients. To our knowledge, the behavior of fluctuations in this situation has not been considered

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H. Híjar, R.F. Rodríguez / J. Non-Newtonian Fluid Mech. 165 (2010) 1033–1039

by the magnetic field H is given by [14]  M ≡ a (n · H) n × H,

(3)

it produces a torque in the x-direction which tends to align the director along the z-axis. In this equation a ≡  − ⊥ is the anisotropy in the (diamagnetic) susceptibility and  and ⊥ denote the principal susceptibilities per unit volume along and perpendicular to the director axis, respectively. In usual nematics like MBBA (at 19 C) a is positive. Actually, the effect of H may also be described in terms of the associated magnetic potential energy given by 1 UM = − a H 2 cos2 , 2 Fig. 1. Schematic representation of a homeotropic nematic cell with a uniform shear flow and a constant electric field, both applied in the y direction.

before for liquid crystals. Since the flexoelectric effect induces an elastic deformation of the director field, it might produce detectable effects on the light scattered by the fluid. To quantify these effects is the basic purpose of this work. To this end, we first define our model and construct the initial state in which the system is prepared. We define it so that it is physically well defined, stable and in a geometry which is used in light scattering experiments in liquid crystals. As will be argued below, for this state it is reasonable to assume statistical independence between the initial velocity and orientation fluctuations. With these features this initial state could be experimentally realizable. Then we use a perturbation scheme to solve the coupled equations for the evolution of the director and velocity fluctuations and calculate analytically the orientation correlation function of this complex fluid in both, the stationary reference state and in the NESS. Afterwards, we calculate the dynamic light structure factor in the presence of both gradients and compare their relative magnitudes. We find that the external gradients induce an asymmetry of the spectrum shifting its maximum towards positive frequency intervals by an amount proportional to the magnitude of the gradients. This effect may be of the order of ∼9.6% for shear flow, and less than 1% for the flexoelectric effect. We find that the k dependence of the normalized structure factors is S eq (k, ω)∼k0 , S flux (k, ω)∼k−2 , S flexo (k, ω)∼k−3 , in the limit of small values of the wave number (k → 0). 2. Model and basic equations Consider a quiescent, initially undistorted thermotropic, nematic liquid crystal layer of thickness d, confined between two parallel (dielectric) plates, as depicted in Fig. 1. The dimensions of the cell along the x and y directions are large compared to d. The initial orientation of the director field, no , is chosen to be homeotropic along the z-axis no = (0, 0, 1) .

(1)

If a uniform electrostatic field E is applied in the y direction, a flexoelectric effect is produced and is manifested as a distortion of the director field in the y − z plane, n = (0, sin , cos ), as shown schematically by the dashed lines in Fig. 1. For the early stages of the reorientation of no ,  << 1, the director field configuration may be approximated by [13,7] n=





0,

e3 E z, 1 , K3

(2)

where e3 is the flexoelectric coefficient and K3 is the bend elastic constant of the nematic. This initial orientation configuration may be further stabilized by applying a uniform magnetic field along the positive z-direction. Since the magnetic torque (per cm3 ) produced

(4)

which shows that H indeed lowers the total Helmholtz free energy and tends to reinforce the initial director configuration. Consider now a simple shear flow between the plates caused by moving them parallel to each other at a constant relative velocity v along the y direction with a velocity gradient along z. At low values of vy the sample retains its initial configuration, but above a certain critical velocity vC , or a critical shear rate C = vC /d, the director tilts towards the y-axis. C increases as H 2 . From the Ericksen-Leslie theory, the viscous torque exerted by the flow is [15], visc = n × g ,

(5)

where g

is the hydrodynamic (viscous) part of the intrinsic director body force. For the present geometry and taking into account that for MBBA the viscosity coefficients 2 , 3 < 0 and 4 > 0, it can be shown that the viscous torque visc,x > 0 implies  > 0. Thus, the viscous torque has a destabilizing effect on the initial orientation and above C it overcomes the elastic and magnetic torques and the system becomes unstable. Then, if  < C , the flow will not reorient no and the director configuration induced by the electric field E may be considered to be independent of the flow field. This clearly indicates that an initial state where the orientation and velocity fields are decoupled is feasible. However, when  > C and the reorientation process evolves in time this situation changes, and the director will couple with the hydrodynamic velocity field and, furthermore, the shear viscosity will generate heat dissipation. With the purpose of defining the appropriate fluctuation–dissipation theorems, we follow the strategy used in computer simulation methods, and assume that the fluid is in contact with a thermal bath which extracts energy uniformly from the fluid (thermostat), to exactly compensate for this viscous heating. This leads to a macroscopic state of isothermal uniform shear flow at constant temperature T o [16],



vo = 0, 



z+

d 2





,0 .

(6)

It has been shown that these non-equilibrium computer simulation methods can be put into close correspondence with microscopic descriptions where non-conservative forces are used to maintain constant temperature states [17]. Since to our knowledge the full non-linear equations for this model have not been derived in the literature in the presence of both, shear and an external electric field, in the Appendix we sketch their derivation and show that the chosen non-equilibrium state, Eqs. (2) and (6), do satisfy (approximately) these non-linear equations. It is essential to emphasize that in the present model we are not interested in the time evolution of the hydrodynamic fields themselves, but in the time development of the orientation and velocity fluctuations of the transverse modes around them. For this reason we do not specify the boundary conditions on these fields, but instead give the initial values of the fluctuations. As a consequence, for the above defined initial state, it is plausible to assume that the thermal fluctuations in the director field may be considered to be

H. Híjar, R.F. Rodríguez / J. Non-Newtonian Fluid Mech. 165 (2010) 1033–1039

statistically independent of the fluctuations in the velocity field. As we shall see later on, the reorientation process will generate coupling of modes between these fluctuations. In Fig. 1 we have also defined the light scattering geometry to be used below for the calculation of the dynamic light structure factor S(k, ω). The intensity of the scattered radiation depends on the geometry of the experiment including the orientation of the director in the sample, and the angle of detection. The scattering plane y − z contains the wave vectors ki and kf of the incident and scattered light, the scattering vector k ≡ ki − kf , and the scattering angle . The unit vector i and f specify the polarization of the incident and scattered beams, respectively. This geometry corresponds to one of the common geometries used in light scattering experiments in liquid crystals, namely, the Mode 1 Bend geometry [12,18]. It should be pointed out that for the early stages of reorientation, in our model k ⊥ no , i ⊥ no , which corresponds to the same geometry used to measure the absorption coefficients which yield the Frank elastic constants of MBBA, the same thermotropic nematic that we shall use below to estimate the light structure factor, [19–21]. Also, intensity contours of Rayleigh scattering from MBBA provided the bias angles in nematic cells [22]. We believe that all the above described features of the model render it consistent, feasible and amenable to be treated by Rayleigh scattering theory. The spontaneous thermal (hydrodynamic) fluctuations of the director and velocity fields around the stationary state defined by Eqs. (2) and (6), are defined as ın (r, t) = n (r, t) − no , ıv (r, t) = v (r, t) − vo . A complete set of stochastic equations for the spacetime evolution of these fluctuations is obtained by linearizing the general hydrodynamic equations of a nematic [23,10] and by using the well established fluctuating hydrodynamics formalism [24,25,4]. It is well known that the set of hydrodynamic variables for a nematic can be split into two orthogonal  groups  [26], which for the present model are the transverse ıvx , ınx and the lon-





gitudinal fluctuations ı , ıvy , ıvz , ıny , ınz , ı , with respect to the plane defined by no and k. ı and ı are the fluctuations in the mass density and the entropy density, respectively. It should be stressed that in this work we will only consider the dynamics of the transverse fluctuations. The inclusion of the longitudinal fluctuations and their contribution to the light scattering spectrum can also be considered, but it substantially complicates the description of the dynamics, and they will not be dealt with here. We use the general form of the fluctuating, linearized, hydrodynamic equations as discussed in Refs. [27,5,10,28], which for the geometry under consideration read

∂ 1 1  K2 ∇ 2y + K3 ∇ 2z − a H 2 ınx ınx = ( + 1) ∇ z ıvx + 2 1 ∂t − −

o



d z+ 2



2 e3 E K3

z

1

1 e3 E ∇ y ınx − (K2 − K3 ) ∇ y ınx 1 K3

(K2 − K3 ) ∇ z ∇ y ınx −

 ∂ ıvx = 2 ∇ 2y + 3 ∇ 2z ıvx ∂t −

 1 ( + 1) K2 ∇ 2y + K3 ∇ 2z − a H 2 ∇ z ınx 2 

+  (3 − 2 ) ∇ y ınx − o



d z+ 2



∇ y ıvx

 e3 E + (3 − 2 ) ∇ y ıvx + 2z ∇ y ∇ z ıvx K3 −

1



2

e3 E 3 ( + 1) (K2 − K3 ) ∇ y ∇ z ınx − ∇ j xj . K3 2

Here, 1 , 2 , 2 , 3 with = 1 /2 , are viscosity coefficients of the nematic; K2 is the twist elastic constant; o is the mass density in the stationary state defined by (2) and (6), and ϒx and xj are zero average Gaussian, stochastic processes defined through their fluctuation–dissipation relations





xj (r, t) xl r , t 



ϒx (r, t) ϒx r , t 











o = 2kB T o xjxl ı r − r ı t − t  ,

= 2kB T o

(9)

 1  ı r − r ı t − t  , 1

(10)

where kB is Boltzmann’s constant and T o is the temperature in the reference stationary state. Assuming that the nematic is incomo , is the pressible, i.e., 4 = 2 and 5 = 0 [14], the tensor xjxl linearized part of the stress tensor of the nematic,





o xjxl = 2 ıjl + ıxj ıxl + (3 − 2 ) noj nol ,

(11)

where ıij is the usual Kronecker delta and the angular brackets denote an equilibrium average. 3. Light scattering spectrum For an electromagnetic plane wave incident on a non-magnetic, non-conducting and non-absorbent nematic, with an average electric permittivity 0 , the dynamic structure factor at a large distance, R, is completely determined by the correlation function of the fluctuations of the dielectric tensor, ıij , of the fluid, S(k, ω) =

k4f |E0 |2 323 20 R2



∞





e−iωt ı(i,f ) (k, t)ı(i,f )∗ (k, 0) dt,

(12)

−∞

where * denotes complex conjugate [12,29]. E0 = |E0 | is the magnitude of the incident field and ı(i,f ) (k, t) ≡ i · ı(k, t) · f. For this model the incident beam is polarized in the x direction, while the dispersed beam has a polarization in the yz plane and perpendicularly to kf . Thus, the unit vector for the initial polarization is i =





(1, 0, 0), while that for the final polarizations is f = 0, sin , cos  . Furthermore, since the nematic is uniaxial ıij = a (noi ınj + noj ıni ), where a is the dielectric anisotropy and the dielectric tensor fluctuations become





ıif = a  sin  + cos  ınx ,



 (k, ω) = 2a  sin  + cos 

(13)

(7)

2



ınx (k, ω) ın∗x k , ω



.

(14)

To calculate this correlation function we have to solve first the coupled system of Eqs. (7) and (8) for the fluctuating variables ıvx and ınx . To this end we use a perturbation approach and assume that the external gradients are small enough so that they may be considered as perturbations. More precisely, we shall only consider small values of the dimensionless gradients  F ≡ (/C )  1 and  E ≡ (E/EC )  1, where EC and C , denote, respectively, the critical values of the electric field and of the shear rate [5]. Then, up to first order in the gradients (0)

F(1)

ıvx (r, t) = ıvx (r, t) +  F ıvx



(8)

and yield the orientation correlation function



+1 ∇ y ıvx − ϒx , 2

+

1035

(0)

F(1)

ınx (r, t) = ınx (r, t) +  F ınx

E(1)

(r, t) +  E ıvx

(r, t) ,

E(1)

(r, t) +  E ınx

(r, t) .

(15) (16)

The superscript 0 denotes zeroth order in the gradients and iden-



e3 E z ( + 1) K2 2∇ 2z − ∇ y + 2K3 ∇ 2z ∇ y ınx K3 2

tifies the stationary contribution, whereas the superscripts F(1) and indicate first order terms in the presence of flow and electric field, respectively.

E(1)

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Substitution of Eq. (16) into (14) gives the following orientation correlation function,





 (k, ω) = ınx (k, ω) ın∗x k , ω flexo

+



= eq (k, ω) + flux (k, ω)

(k, ω) .

× (17)

Here eq (k, ω) is an equilibrium contribution, while flux (k, ω) and flexo (k, ω), represent the non-equilibrium contributions due to the shear flow and the flexoelectric deformation, respectively. These contributions are defined by eq (k, ω) ≡

 (0)∗

(0)

ınx (k, ω) ınx

flux (k, ω) ≡  F +



k , ω

  

F(1)∗

(0)

ınx (k, ω) ınx

flexo (k, ω) ≡  E

F(1)

ınx



(0)∗

(k, ω) ınx



E(1)∗

(0)

E(1)

(0)∗

(k, ω) ınx

,



(18)

k , ω

k , ω

ınx (k, ω) ınx

+ ınx







k , ω

k , ω

,

(19)





.

(20)



iω − ˆ 2 ınx + = i F

(21)

we

1 o

+ 3 kz2



2 ky2

eq (k, ω) =

2kB T o ω ˜ , ˜ 2 + ω2 Kˆ 2 ω



ωω ˜ ω ˜ 2 + ω2

 3

ˆ4 − ˆ 2  ˆ2



2 ,

(25)

where we have introduced the characteristic frequency ω ˜ ≡ ˆ 2 + ˆ 4 / ( ˆ 2 ). Note that these quantities contain the magnetic field H through Kˆ 2 . Moreover, since the dispersion relation implies that ω∼k, it follows that in the k → 0 limit, eq ∼k−4 , flux ∼k−6 , flexo ∼k−7 . In order to simplify the subsequent analysis, it will be convenient to introduce normalized correlation functions eq (k, ω) , eq (k, 0)

S flexo (k, ω) ≡

S flux (k, ω) ≡

flux (k, ω) , eq (k, 0)

flexo (k, ω) , eq (k, 0)

(26)

which are defined with respect to the maximum of the equilibrium contribution eq (k, 0) = (2kB T o /Kˆ 2 )(1/ω) ˜ obtained from (23). The total normalized spectrum, S (k, ω), is then of the form S (k, ω) = S eq (k, ω) + S flux (k, ω) + S flexo (k, ω) ,

(27)

where the equilibrium normalized spectrum reads S eq (k, ω) =

S

flux

1 1 + ωo2

(k, ω) = o

o ≡ − ≡ (22)

have defined Kˆ 2 ≡ K2 ky2 + K3 kz2 − a H 2 ,  ˆ2 ≡ 2 2 ˆ ˆ ,  ≡ (1/1 )K . These equations may be significantly simplified by noting that for a typical thermotropic nematic like MBBA, which is nematic in the temperature range 20◦ C-47◦ C, its material parameters are such ˆ 4 / ˆ 4 ≡ (( + 1)2 /4 o )Kˆ 2 k2 . that (ˆ 2 / ˆ 2 )  1, ( ˆ 2 ˆ 2 )  1,with z Therefore  ˆ 2 ∼105 s−1 and (ω/ ˆ 2 ) << 1. By using these relations, Eqs. (21) and (22) can be solved up to first order in external gradients, and by using the fluctuation–dissipation relations (9) and (10), the equilibrium and non-equilibrium contributions to the eq (k, ω) are explicitly obtained, namely, where

(24)

,

(28)





ωo 1 + ωo2 ωo

1 + ωo2

2 + 

a 1 + ωo2

2 ,

(29)

2 ,

(30)

˜ Here we have used the abbreviations with ωo ≡ ω/ω.

3 e3 EC +  E ( + 1) o (K2 − K3 ) ky kz ınx 2 K3 ikj e3 EC − i o (3 − 2 ) ky ıvx + o xj , K3

2 ,

4kB T o e3 EC K2 − K3 ky K3 Kˆ 2 Kˆ 2

S flexo (k, ω) =  E 

dC (3 − 2 ) C ky ıvx ky ınx + i F o 2

E

+ ω2



 ( + 1) ˆ 2 K kz ınx + iω −  ˆ 2 ıvx i o 2 = −i F

×

ω ˜ ω ˜2



and the non-equilibrium normalized contributions have the forms

i ( + 1) kz ıvx 2

dC e3 EC ky ınx + ϒx + i E (K2 − K3 ) ky ınx , 2 1 K3



flexo (k, ω) =  E

S eq (k, ω) ≡

Note that the Fourier transforms of the terms in Eqs. (7) and (8) which contain z-dependent coefficients lead to a shift (k ± q) in the k-dependence of the transform. However, these terms do not appear in the following Eqs. (21) and (22) because it can also be proven (see Ref. [27]) that they do not contribute (to linear order) to the correlation functions, which is the main concern in this work, and may be neglected for this purpose. Thus, the relevant of part of the Fourier’s transform of Eqs. (7) and (8) is





+ 1 (3 − 2 ) 2kB T o C ky ωd − ω ˜ kz 2 o  ˆ2 Kˆ 2

flux (k, ω) =  F

(23)

+ 1 (3 − 2 ) kz , 2 o d ˆ2

e3 EC ky K2 − K3 K3 ω ˜ Kˆ 2



6

(31)

ˆ4 − 2ˆ 2  ˆ2

 .

(32)

Thus, the total normalized spectrum can be expressed in the more compact form S (ωo ) =

1 1 + ωo2



1+

1 1 + ωo2



o (ωo + a) +  E ωo

 

.

(33)

4. Results and discussion In order to plot the light scattering spectrum we consider the thermotropic MBBA, which is nematic in the temperature range 20-47 C. The required parameters are the following [14]: = 1.03, o = 1.029 g cm−3 , 2 = 41.6 × 10−2 g cm−1 s−1 , 3 = 23.8 × 10−2 g cm−1 s−1 , 1 = 76.3 × 10−2 g cm−1 s−1 , K2 = 2.2 × 10−7 g cm s−2 , K3 = 7.45 × 10−7 g cm s−2 , H = 50 kG. We consider two degrees of departure from equilibrium through the following values of the dimensionless gradient o = 0.1 and o = 0.5. For

H. Híjar, R.F. Rodríguez / J. Non-Newtonian Fluid Mech. 165 (2010) 1033–1039

1037

one finds the k−3 behavior. The flexoelectric correlation function is also an odd function of the frequency, reflecting the broken timereversal symmetry due to the external force. This non-equilibrium effect contributes to the asymmetry in the shape of the structure factor by shifting the maximum towards the region of positive values of ω; the size of the shift increases with the magnitude of the gradient. However, the flexoelectric effect produces only small changes on the structure factor, which are about two orders of magnitude smaller with respect to those due to the presence of the shear flow.

5. Conclusions

Fig. 2. The equilibrium and non-equilibrium structure factors S eq (ωo ) and S flux (ωo ) plotted as a function of ωo (arbitrary units) for MBBA. Here the dimensionless shear rate is o ≡ 0.1.

o = 0.1 we get the spectrum shown in Fig. 2, where the equilibrium structure factor is also shown. It is apparent from this figure that for this value of  the difference between both curves practically vanishes. For o = 0.5 we get the spectrum in Fig. 3 which shows that for the present model, the external gradient tends to increase its height and to shift the spectrum towards positive values of the frequency shift ω. To quantify the non-equilibrium effects it is convenient to define S (ωo ) =

S flux (ωo ) + S flexo (ωo ) . S eq (ωo )

(34)

For a fixed value of k, the maximum of this quantity occurs for ωo,max 0.208383 if o = 0.5, which corresponds to S (ωo,max ) 9.6%. As shown by Eq. (30), the flexoelectric effect changes sign upon change of the wave vector. A similar situation arises for a thermotropic nematic when the non-equilibrium stationary state is due to an external temperature gradient (see Ref. [5]), in which case the normalized (with respect to its stationary value) transverse orientation correlation function varies as k−1 , and it also changes sign upon change of sign of the wave vector. Taking this into account it is not too surprising that with the presence of the flexoelectric effect

Summarizing, we have investigated theoretically the influence of the effects produced by the simultaneous application of a shear rate and a uniform electric field, on a measurable property of a thermotropic nematic, namely, its light dynamic structure factor. The model proposed was constructed so that it corresponds to a homodyne experimental geometry [12]. By using a perturbation fluctuating hydrodynamic approach we calculated the orientation correlation function both, in equilibrium and in the non-equilibrium steady state induced by the gradients. The k-dependence of the normalized non-equilibrium orientation correlations Eqs. (29)–(30) shows long-range effects, S flux (k, ω)∼k−2 and S flexo (k, ω)∼k−3 . As shown in Eq. (33), these non-equilibrium corrections are odd function of ω and introduce an asymmetry in the shape of the structure factor, shifting the maximum towards the region of positive values of ω. The size of the shift increases with the magnitude of the gradients. We found that for MBBA, the shear flow can produce significant changes on the dynamic structure factor of about 9.6%. In contrast, the flexoelectric effect produces only very small changes on the structure factor, which are about two orders of magnitude smaller with respect to those due to the presence of the shear flow. Our analysis suggests that the effects produced on dynamic correlations by the flexoelectric effect would hardly be observed in experiments, while those due to the shear flow could be detected; however, this remains to be assessed.

Acknowledgments We acknowledge partial financial support from grant DGAPAUNAM IN102609, Mexico. H.H. also acknowledges financial support from DGAPA-UNAM.

Appendix A. In this appendix we use the formulation of the nematodynamic equations given in Ref. [23] to sketch the derivation of the nonlinear equations for the velocity, vi (r, t), and director, ni (r, t), fields for the present model. We argue that for the early stages of the reorientation of no ( << 1) they reduce (approximately) to the assumed non-equilibrium reference steady state defined by Eqs. (2) and (6). For an incompressible thermotropic nematic the equation of motion for vi (r, t) is ∂t vi + vj ∇ j vi + ∇ j ij = 0,

(A.1)

and the relaxation equation for ni (r, t) reads [23], ∂t ni + vj ∇ j ni + Yi = 0, Fig. 3. The same as in Fig. 2 for o ≡ 0.5.

where (r, t) is the local mass density.

(A.2)

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H. Híjar, R.F. Rodríguez / J. Non-Newtonian Fluid Mech. 165 (2010) 1033–1039

The momentum current ij and the quasi-current Yi are defined, respectively, as

Using Eqs. (A.4)–(A.7) its explicit form turns out to be C()

1

ij = pıij + lj ∇ i nl − kji hk − ijkl ∇ l vk , 2

(A.3)

1 1 ⊥ E Yi = − kji ∇ j vk + ı h − ijk ∇ j Ek , 2 1 ik k

(A.4)



dvy 1 + 1 dz

K1 D() − K3 F() + a EG() + a I()H 2

+ E [e1 M() + e3 N()]

where p is the pressure. The viscous tensor ijkl is

+ (1 − 2 )ıij ıkl + (2 − 1 + 2 )(ıij nk nl + ıkl ni nj ),

(A.5)

where 1 , 2 , 3 , 1 , 2 are the five nematic viscosity coefficients in the notation of Harvard. Here the incompressibility condition (div v = 0) is implemented by assuming 2 = 4 and 5 = 0, [14]. The tensor ij is defined by ij = Kijkl ∇ l nk + eijk Ek ,

(A.6)

(A.17)

+1 cos  − sin2  cos , 2

D() ≡ sin2  cos2 

+ (3 − 2 )(nj nl ıik + nj nk ıil + ni nk ıjl + ni nl ıjk )

= 0,

where C() ≡

ijkl = 2 (ıjl ıik+ ıil ıjk ) + 2(1 + 2 − 23 )ni nj nk nl





d2 d2 (sin ) + sin  cos  2 (cos ) , dz 2 dz

d2

F() ≡ cos2  sin2  − 1





dz 2

(sin ) + cos2 

(A.7)

≡ ıik − ni nk is a projection operator and ıij is the usual where ı⊥ ik Kronecker’s delta. The third order tensor eijk contains the flexoelectric coefficients e1 and e3 , eijk ≡ e1 ı⊥ n + e3 ı⊥ n, ij k jk i

(A.21)

I() ≡ sin  cos ,

(A.22)







J() ≡ 2 sin2  − 1 cos  sin 

(A.9)

depends on the orientational viscosities through ≡ −1 /2 . hk denotes the molecular field,



∂ ∂ Kpjkl − Kqjkl ∂nq ∂nq



∇ l nk ∇ j np ,

∇ j yj = 0,

(A.10)

(A.11)

which is explicitly given by dz 2

+ B()

dvy d = 0, dz dz

(A.12)

with A() ≡ 3 +

1 (1 + 2 − 3 ) cos2 2, 2

B() ≡ 2(1 + 2 − 3 ) sin  cos .

(A.13) (A.14)

It then follows that in the limit  << 1 (A.12) reduces to d2 vy dz 2

= 0,

(A.15)

and it is satisfied by (6). On the other hand, for the geometry under consideration and in the stationary state (∂t ni = 0), the relaxation equation for the director is Yy = 0.

d d (cos ) + sin  cos  (sin ) , dz dz



N() ≡ J() − 2 1 + sin2 

to 0=

dvy 1 +1 + cos  2 1 dz

d dz

(cos ) − sin  cos 

(A.16)



 −K3 cos 

(A.24)

d (sin ) . dz (A.25)

2



d d2  − sin  dz dz 2

d + a E sin  − a sin H − e3 E sin  dz 2

which obeys the relation n · h = 0.   Recalling that v = [0, v(z), 0] and n = 0, cos (z), sin (z) = [0, sin (z), cos (z)], E = Ey, H = Hz, where y and z denote unit vectors in the y and z directions, and by setting ∂t vi = 0, from Eqs. (A.2)- (A.10) it can be shown that the equation of motion (A.1) is of the form

d2 vy

M() ≡ J() + cos2 



d d (sin ) + cos  (cos ) , dz dz (A.23)

For the early stages of reorientation ( << 1) Eq. (A.17) reduces

kji ≡ ( − 1)ı⊥ n + ( + 1)ı⊥ n, kj i ki j

A()

(A.20)

(A.8)

and the third order tensor kji ,

hk = −Kkjnl ∇ j ∇ l nn + ı⊥ kq

d2 (cos ) , dz 2

(A.19)

G() ≡ sin  sin2  − 1 ,

where the fourth order tensor Kljmn depends on the elastic constants K1 (splay), K2 (twist) and K3 (bend) and it is defined in terms of the Levi-Civitta tensor, ijk , by Kljmn ≡ K1 ı⊥ ı⊥ + K2 np plj nq qmn + K3 nj nn ı⊥ , lj mn lm

(A.18)

2 



.

(A.26)

As discussed in Section 2, in the stationary state the torques due to the shear flow and the bend elastic deformation should be balanced, respectively, by the magnetic and flexoelectric torques. This implies that the first term cancels the third term within the curly brackets if H 2 is chosen to be ∼( + 1)/(2a ), which is verified by direct substitution of Eqs. (2) and (6). This substitution also verifies that the first term within the curly brackets cancels the last one. This then shows that our NESS defined by Eqs. (2) and (6) satisfies (approximately) the non-linear Eqs. (A.12) and (A.17). References [1] J.R. Dorfman, T.R. Kirkpatrick, J.V. Sengers, Generic long-range correlations in molecular fluids, Annu. Rev. Phys. Chem. 45 (1994) 213–239. [2] D. Beysens, Y. Garrabos, G. Zalczer, Experimental evidence for Brillouin asymmetry induced by a temperature gradient, Phys. Rev. Lett. 45 (1980) 403. [3] B.M. Law, P.N. Segrè, R.W. Gammon, J.V. Sengers, Light-scattering measurements of entropy and viscous fluctuations in a liquid far from thermal equilibrium, Phys. Rev. A 41 (1990) 816–824. [4] J. Ortíz de Zarate, J.V. Sengers, Hydrodynamic Fluctuations, Elsevier, Amsterdam, 2006. [5] H. Pleiner, H.R. Brand, Light scattering a nematic liquid crystal in a nonequilibrium state, Phys. Rev. A 44 (1983) 1177–1183. [6] H. Híjar, Ph.D. Dissertation, Universidad Nacional Autónoma de México, 2004 (in Spanish). [7] H. Híjar, B.Sc. Dissertation, Universidad Nacional Autónoma de México, 2002 (in Spanish). [8] R.F. Rodríguez, J.F. Camacho, Nonequilibrium effects on the light scattering spectrum of a nematic under an external pressure gradient, Rev. Mex. Fis. 48 (S1) (2002) 144. [9] R.F. Rodríguez, J.F. Camacho, Nonequilibrium light scattering from nematic liquid crystals, in: A. Macias, E. Díaz, F. Uribe (Eds.), Recent Developments in

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[10] [11]

[12]

[13] [14] [15] [16] [17] [18] [19]

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