Generalized hydrodynamic equations for nematic liquid crystals

Transcription

1 PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalize hyroynamic equations for nematic liqui crystals Tiezheng Qian 1,2 an Ping Sheng 1 1 Department of Physics, The Hong Kong University of Science an Technology, Clear Water Bay, Kowloon, Hong Kong, China 2 Department of Physics, University of California at Berkeley, Berkeley, California Receive 9 July 1998 We present a formulation of the nematic liqui crystal hyroynamics base on the tensor orer parameter, with the spatial variations of both the irector an the scalar uniaxial an biaxial orer parameters explicitly taken into account. New length an time scales are shown to arise that can be important for the switching ynamics of thin cells subject to large external fiels. The Ericksen-Leslie theory is shown to be a special limit of our formulation. S X PACS number s : v, Jr I. INTRODUCTION Nematic liqui crystals NLC s are at the same time similar an ifferent from orinary fluis. They are similar because both have no translational long-range orer for the molecules. In aition, nematic liqui crystals possess no static shear moulus an can therefore flow like a flui. However, NLC s have long-range orientational orer, which gives rise to many of the special optical properties which are the basis of liqui crystal isplays. The existence of the orientational orer means that locally, the alignment of the NLC s is characterize by a irector n, the local axis of uniaxial orientational orer. Base on symmetry an elastic energy consierations, a continuum elastic theory of NLC s has been evelope by Oseen 1, Zocher 2, an Frank 3. The incorporation of the orientational egree of freeom in the hyroynamics of NLC s was subsequently evelope by Ericksen 4 an Leslie 5. They formulate the general conservation laws of flui mechanics for NLC s an erive the hyroynamic equations governing the time evolution of the irector fiel n(r) an the velocity fiel v(r). The Ericksen- Leslie EL theory has been wiely an successfully use in explaining various ynamic properties of thermotropic NLC s. In aition to the irector n, which gives the local axis of uniaxial symmetry, the NLC s are also characterize by the egree of local orientational orer S(r). By noticing that NLC s o not istinguish the hea an tail of the irector n, an by taking into account the egree of orientational orer S(r), e Gennes 6 has shown that the orer parameter of NLC s is a secon rank traceless symmetric tensor QJ. While the continuum elastic theory base on the irector escription of NLC is sufficient for many fiel-inuce liqui crystal behaviors, it has been shown that the consieration of substrate-nematic interaction an/or interaction with strong electric or magnetic fiel woul require the more complete theoretical framework base on the tensor orer parameter. The Lanau e Gennes LG 7 theory has been instrumental in explaining an preicting various static phenomena, such as surface wetting 8, surface-inuce bulk alignment 9,10, an efect core structure 11 13, all involving the fast spatial variations of liqui crystalline orer LCO, a term use here to enote both n an S as expresse through the tensor orer parameter QJ. Recently, it has been observe that the polyimie-coate substrates can inuce a bounary layer of strong biaxial character, escribable only through the tensor orer parameter 9. In aition, the preiction of the bulk orientational orer through the LG theory shows goo agreement with the experiment. The use of a tensor orer parameter is therefore not only necessary, but also sufficiently accurate. The purpose of this work is to explore the role of LCO, as expresse by the tensor orer parameter QJ, in hyroynamic processes. We aopt the general framework evelope by Ericksen an Leslie, meanwhile applying the LG free energy to present a hyroynamics formulation that can be use to stuy the ynamic behavior. We ientify new length an time scales associate with the spatially an temporally varying LCO in generic hyroynamic processes. Illustrative numerical results are given. It is shown that the EL theory correspons to the long-range, slow-motion limit of the general formulation. We note that the EL theory has been rewritten in the tensorial context but with S(r) treate as a constant 14,15. Our formulation shoul be istinguishe from such treatment since we take into account the variations of both S(r) an n(r), as well as the coupling between the two an/or the biaxial orer. II. STATEMENT OF RESULTS The LG free energy ensity is given by 7 F LG 1 2 Q 2 2 ij L 1 Q ij,k L 2 Q ij,j Q ik,k 4 P L 1 ijk Q il j Q kl Q ij Q jk Q ki Q 2 ij 2. Here, L 1, L 2,,, an P are phenomenological material constants, ijk is the Levi-Civita symbol, the inices i, j,k run from 1 to 3, summation over repeate inices is implie, an the comma in the subscript means erivative with respect to the spatial coorinate that follows. We consier incompressible NLC s obeying k v k 0. The central results of this work are as follows. The hyroynamic equations for the flow an LCO are X/98/58 6 / /$15.00 PRE The American Physical Society

2 7476 TIEZHENG QIAN AND PING SHENG PRE 58 v i t j p ji ji f ji, ji JQ ij h ij h ij ij ijk k, Here is the NLC ensity, p enotes the pressure, J is the moment of inertial ensity, usually negligible, an k are the Lagrange multipliers etermine by requiring TrQJ 0 an Q ij Q ji, while /t is the total time erivative / t v. The istortion stress an elastic molecular fiel h are etermine by the LG free energy: ji F LG j Q iq, h ij F LG F LG Q k ij k Q ij. 5 In the presence of electric fiel E, there is a fiel-inuce free energy ensity F f (1/8 )D E, where D JE is the electric isplacement fiel, an J 2 3 m a QJ I is the ielectric tensor with m a being the maximum ielectric anisotropy an Tr J/3. Here the sign in (1/8 )D E means that F f takes the electric fiel or potential as thermoynamical coorinate. The fiel-inuce stress f is given by f ji 1 4 D je i 8 D 1 k E k E E ji, an h is reexpresse by aing the fiel-inuce molecular fiel (1/12 ) a m E i E j to the right-han sie RHS of Eq. 5. The viscous stress an viscous molecular fiel h are given by 1 Q Q A 4 A 5 Q A 6 Q A N 1 Q N 1 Q N, h A 1 N, where 1, 4, 5, 6, 1, an 2 are viscosity coefficients with 6 5 2, A ji 1 2 ( j v i i v j ) is the symmetric part of the velocity graient tensor, while N ji Q ji / t v k k Q ji j Q i i Q j is the time rate of change of Q ji with respect to the backgroun flui angular velocity 1 2 v. There are intrinsic length an time scales that emerge from the LG free energy. We choose the length scale LC/B 2, where B 3 /4, C 9 /4, an L 3L 1 /4 L 2 /2, as the characterisitic length unit to escribe the spatial variations of Q ji. Corresponingly, the time scale is 2 /L where is some representative viscosity coefficient. For most NLC s, 50 Å an 0.1 s. In aition to these intrinsic scales, there are also the length an time scales that arise from external impose parameters, such as those in the EL theory. Consier a liqui crystal cell of thickness with parallel, uniform substrates imposing fixe surface alignment conitions. The EL theory gives as the length scale an 0 2 /K as the time scale, where K LS 2 is a Frank elastic constant in the continuum theory. In the presence of applie voltage V, the fiel-inuce correlation length is given by E 4 K/ a E 2 ( a is the ielectric anisotropy an E V/ is the fiel strength an the associate time scale becomes E E 2 /K. In our formulation, the EL theory is recovere when E an E 0, i.e., in the limit of slow spatial an temporal variations. In fact, the EL theory can be obtaine from our formulation when Q ij is written as Q ij r S 0 3n i r n j r ij /2, i.e., the orientational orer is treate as a frozen constant. However, that means the ynamic behavior of S(r), an those involving the couple evolution of S(r) an n(r), cannot be escribe within the EL theory. We have carrie out numerical calculation for a cell of 1 m fille by a NLC of large a 15, with homogeneous substrate alignment conitions. We first minimize the LG free energy to obtain the equilibrium NLC configuration uner a large holing voltage ( 20 V). Biaxial orering in the bounary layer is note. The hyroynamic equations 2 an 3 are then solve to show the backflow effect 16,17 after the voltage is switche off. It is seen that the large elastic energy store in the biaxial bounary layers results in a very fast orientational relaxation close to the NLCsubstrate interfaces. Because of the coupling between the translational an orientational motions, a large flow fiel is inuce. The length an time scales an are clearly manifest in the resulting ynamic behavior. The paper is organize as follows. In Sec. III, the erivation of the hyroynamics equations, base on the tensor orer parameter, is presente. In Sec. IV, the new length an time scales of our formulation are iscusse, an the EL theory is obtaine as a limit of our equations. In Sec. V, the numerical results are given to illustrate the hyroynamic effects of LCO in thin cells subject to strong fiels. We conclue in Sec. VI with a iscussion of implications for future work. III. EQUATIONS OF HYDRODYNAMICS In this section we erive the hyroynamic equations for NLC s by using the LG free energy to efine the nonissipative molecular fiel an stress. The viscous molecular fiel an stress are then introuce through the consieration of entropy prouction in a issipative flowing nematic. The two inepenent fluxes for friction are ientifie by using the conservation of angular momentum, an explicit expressions for the viscous forces are obtaine, leaing irectly to Eqs. 2 an 3. Fiel-inuce effects are inclue in the erivation. A. Free energy an hyrostatics The LG free energy F LG F LG r is a functional of Q ij (r) an Q ij,k (r). The NLC can be virtually istorte by either a rotation of molecular alignment: Q ij (r) Q (r), ij or a isplacement while keeping its orientation fixe: r r r u(r), Q ij (r) Q (r ) Q ij ij (r). The elastic molecular fiel h, as efine by Eq. 5, results irectly from the virtual orientational istortion. The equilibrium state is given by the orer parameter configuration that mini- 9

3 PRE 58 GENERALIZED HYDRODYNAMIC EQUATIONS FOR mizes the free energy functional r F LG.AsQJonly has five inepenent components, the optimal configuration satisfies h ij ij ijk k, where the four an k are Lagrange multipliers ue to the constraints Tr QJ 0 an Q ij Q ji. On the other han, the istortion stress is efine by F LG ji j u i r. Here the change in free energy is cause by a change of Q ij,k : k Q ij k Q ij Q ij k u, an F LG is given by F LG r F LG Q ij,k Q ij k u, from which we obtain Eq. 4. In the presence of an electric fiel E, the total free energy ensity becomes f F LG F f, where the fiel-inuce energy ensity F f is given by (1/8 ) ij E i E j. Accoringly, the molecular fiel becomes h ji ji ji m a 12 E je i, 10 where ji F LG / Q ji an ji F LG / Q ji,. The fiel-inuce stress f, efine by F f f ji j u i, is consequently given by Eq The LG free energy is invariant uner a rigi rotation of the nematic. That is, F LG 0 if u i (r) ijk j r k, Q ij i Q j j Q i, where is a small rotation angle. F LG can be expresse as As is arbitrary, we have L 0. L r ji ji ji j k Q ki i k Q jk ji j k Q ki, i k Q jk, 0, 11 which essentially expresses angular momentum conservation of the system, an shows that an h are not inepenent. Equation 11 can also take the form r ijk jk ijk Q j k Q j k r ijk Q j k k Q j k k r ijk Q j h k Q j h k m a 12 ijk Q j E k E Q j E E k, in which we replace by h m a E E /12 using Eq. 10. This leas to the term r ( m a /12 ) ijk (Q j E k E Q j E E k ) explicitly epening on E in the RHS of the above equation. Then using J 2 3 m a QJ I for the ielectric tensor, ( m a /12 ) ijk (Q j E k E Q j E E k ) is foun to be equal to (1/4 ) ijk D j E k, which is simply relate to f by f (1/4 ) ijk D j E k ijk jk. Thus, Eq. 11 can be expresse as r ijk jk f jk r ijk Q j h k Q j h k, from which we obtain ijk r j S k ijk Q j k Q j k f k Q j S k ijk r j k ijk Q j S k f k Q j h k Q j h k r, f jk 12 by using ijk ( jk ) ijk r j ( k k ) ijk r j ( f k k ). Here S is the vectorial surface element on the bounary. It is clearly seen that the boy torques are externally supplie by the surface torques. In Eq. 12, since the appearance of ( ji f ji ) is always in conjuction with ijk, which is totally antisymmetric, it follows that the result will not be altere by replacing f ji ji by f ji ji p ji, since p ji is symmetric. However, in equilibrium j ( p ji ji f ji ) 0 an h ij ij ijk k, therefore the integrant in the RHS of Eq. 12 vanishes. That is, the boy torque equal to zero, consistent with the requirement of angular momentum conservation. B. Dissipative ynamics an entropy source With, f, an h efine from the free energy expression, we are reay to write own the ynamic equations by introucing the viscous stress an the viscous molecular fiel h which are responsible for entropy prouction in flowing nematics. Knowlege of the viscous stress means the ynamical equation, expresse as conservation of linear momentum, may be written as v t i r S j ji, f 13 where the total stress ji p ji ji f ji ji. The last term, ji, represents the viscous stress tensor, which is to be erive. The time evolution of LCO is governe by the angular momentum equation J t ijk Q j Q k Q j Q k r Q j h k h k Q j h k h k r, 14 where the viscous molecular fiel h is introuce as a friction force with respect to the rotational egree of freeom, also to be erive. The ifferential equations corresponing to Eqs. 13 an 14 are Eqs. 2 an 3 state in Sec. II. Equation 3 is actually the tensorial analogue of the vectorial Oseen equation see Eq. A4 in Appenix A of the EL theory. That relation will be icusse in Sec. IV an Appenix B.

4 7478 TIEZHENG QIAN AND PING SHENG PRE 58 The entropy prouction in an isothermal flowing nematic is efine by the thermoynamic relation Ḟ Ẇ Q TṠ Ẇ T, where S enotes entropy, an T enotes the issipative part of entropy prouction. Written out explicitly, we have T t 1 2 v JQ 2 F LG F f S Q v, 15 where the first integral is the ecreasing rate of the total free energy ( Ḟ) an the secon integral is the rate of external work (Ẇ) one by the surface forces. Substituting the ynamic equations 2 an 3 into Eq. 15 yiels T r A ji ji h Q ji ji, 16 where A ji has been efine previously in relation to Eqs. 7 an 8. InEq. 16, while we have explicitly ientifie an h, yet the two quantities are correlate through angular momentum conservation. To make this clear, we note that the rate of change of angular momentum may be expresse as ijk r j S k ijk Q j S k Q j S k t ijk r j v k ijk J Q j Q k Q j Q k r. 17 Using Eqs. 2 an 3, we may express the RHS of Eq. 17 as ijk r j k Q j h k h k Q j h k h k r. We thus see that the sum of boy torques, given by i an h h, equals the sum of surface torques. However, the sum of boy torques exerte by ( f i i ) an h must equal to the sum of nonissipative surface torques, as state by Eq. 12. Subtracting Eq. 12 from 17, we obtain ijk r j k ijk r j u k ijk Q j h k Q j h k, 18 which simply means that the viscous boy torques from i an h are supplie by the surface torque from S i. Substituting Eq. 18 into 16, we obtain the entropy prouction expression T r s ji A ji h N ji ji, 19 where s ji is the symmetric part of ji. The two sets of inepenent fluxes A ij an N ij are thus ientifie as efine previously in relation to Eqs. 7 an 8, an the laws of friction can be explicitly expresse as s 1 Q Q A 4 A Q A Q A N, h A 1 N, where 1, 4, 1 2 ( 5 6 ), 2, 2, an 1 are inepenent viscosity coefficients. Accoring to the Onsager theorem 19, 2 2. Here the expressions for s an h keep the lowest-orer terms in Q ij so as to avoi the efinition of a large number of viscosity coefficients. They can be relate to the corresponing quantities in the EL formulation. This is elaborate in Sec. IV an Appenix B. Using s 1 2 ij ij an Eq. 18, we obtain as given by Eq. 7. With the explicit expressions obtaine for all the stresses an molecular fiels, the erivation of the nematoynamics equations is thus complete. IV. DISCUSSION A. Intrinsic length an time scales The short-range, fast variations of LCO can be inuce by surface interaction potentials, strong fiels, an/or tight geometric confinements. The characteristic istances associate with such variations are etermine by the NLCs elastic correlation lengths, which can be efine from F LG. In Sec. II we efine as the characteristic length unit to escribe the spatial variations of Q ji. Since the inertial effects are usually negligible, the ynamic equations 2 an 3 are essentially the balance equations between elastic forces an viscous forces, an the intrinsic time scale corresponing to the NLCs length scale is etermine by such viscoelastic effects. Since h, L/ 2 an h, /, setting L/ 2 / leas to 2 /L, where enotes some typical viscosity coefficient. Consier the NLC confine in a cell of thickness with applie voltage V. The EL theory efines two other sets of length an time scales. For a brief review of the EL theory, see Appenix A. The Frank free energy ensity F F, given by Eq. A1 in Appenix A, leas to a correlation length 20, an the cell thickness becomes the only length scale. Accoring to the EL theory, the corresponing time scale 0 is etermine by the balance between g, K/ 2 an g, / 0. That means 0 2 /K. In the presence of electric fiel applie perpenicular to the substrates, F F an the fiel-inuce free energy ensity (1/8 ) a (n E) 2 together lea to a finite correlation length E 4 K 2 / a V 2 where V is the applie voltage. The corresponing time scale E, etermine by the balance between g, K/ 2 E an g, / E, is thus 2 E /K. We emphasize that, 0 an E, E are etermine by the cell thickness an applie fiel while the new length an time scales an are intrinsic parameters that can only be obtaine by taking into account the elastic correlation of LCO. With L 10 6 yn, B 0.5 J/cm 3, C 1.0 J/cm 3, an 0.1 p for 5CB, we have 50 Å an 0.1 s. For large a 15 4-n-pentyl-4 -cyanobiphenyl 5CB has a 13, V 20 V an 1 m, the fielinuce correlation length E 100 Å, which is comparable to. Such high voltages can inuce the spatial variations as well as the time evolutions of LCO with an as the

5 PRE 58 GENERALIZED HYDRODYNAMIC EQUATIONS FOR relevant length an time scales. However, for large thickness, low voltage V, an small ielectric anisotropy a, is much smaller than an E. In that limit, the S(r) is effectively a frozen egree of freeom that cannot be activate. Consequently, the irector fiel suffices for the escription of time-evolving alignment state, an our formulation equations are reuce to the EL theory. This is shown below with more etails. B. Aiabatic S r,t an the Ericksen-Leslie theory It is straightforwar to reuce our formulation to the irector escription by taking the limit expresse by Eq. 9. Here we focus on the role of S in such limit. A etaile comparison can be foun in Appenix B. We first restrict our iscussion to the uniaxial case. From Q ij S(3n i n j ij )/2, F LG is reuce to F U LG a T T* S 2 BS 3 CS L L 2 S L 2 n S L 2 S n n S 3 4 L 2 S n n S 9 4 S 2 L L 2 n 2 L 1 n n 2 L L 2 n n 2, 22 where a(t T*) 3 /4, T* is the supercooling temperature, B 3 /4, an C 9 /4. Here the helical term is omitte for simplicity. In case S S 0 with S 0 being the uniaxial orer parameter that minimizes a(t T*)S 2 BS 3 CS 4, the free energy is effectively a functional of n r only, represente by the last three terms. The splay, twist, an ben istortions of n r are expresse as in the Frank theory. The Frank elastic constants, given by K S 2 (L L 2 ), K S 2 L 1, an K S 2 (L L 2 ), are preicte to be proportional to S 2 21, which has been experimentally verifie by measuring the temperature epenence of K i an S. The EL theory ignores both the spatial an temporal epenence of S(r). This correspons to Eq. 9 in which the uniaxial orientational orer parameter S is set equal to S 0. To justify such an approximation, it is necessary to examine the equation Q ij (h ij h ) 0, ij obtaine from Eq. 3 by neglecting the inertial term. This is actually the molecular fiel balance equation for the S egree of freeom. To make this clear, we first note from the efinition of h ij Eq. 5 that Q ij h ij S F U LG S U F LG k k S 3S 2 U F LG k Q ij k n i n j. 23 In the RHS of this equation, the secon term is negligible compare with the first term if the characteristic istance l n over which the irector variations take place is much larger than, the characteristic istance for the spatial variations of S. An estimate of orer of magnitue shows that in the RHS of Eq. 23, the first terms is on the orer of L/ 2 while the secon term is on the orer of L/ l n, with its ratio to the first one being /l n 1. Actually l n is just one of the conitions for the EL theory to be recovere as we will see below. Then, by ropping the k (n i n j ) term in Eq. 23 an using N ij Ṡ(3n i n j ij )/2 3S(n i N j n j N i )/2 an Eq. 21 for the efinition of h ij, we obtain for Q ij h ij Q ij h ij the equation F U LG S U F LG k k S 3 2 1Ṡ 3 4 2n i n j A ij. 24 From Eq. 24, it can be euce that the slow spatial an temporal variation of n effectively makes itself the only ynamic variable out of the original QJ. To see this point, we write Eq. 24 in the imensionless form F S F ns S k ks t S 3 4 2n i n j Ā ij. 25 Here F S F ns F U LG /(B 4 /C 3 ) is the imensionless free energy ensity, S S/(B/C) is the scale orientational orer, 1 1 / an 2 2 / are the imensionless viscosity coefficients, respectively, an an are chosen as the length an time units such that k k, /t (/t), an Ā ij A ij. F S an F ns are given by F S T 1 4 S 2 S 3 S L L 2 S L 2 n S 2, 26 an F ns 3 2 L 2S n n S 3 4 L 2S n n S 9 4 S 2 L L 2 n 2 L 1 n n 2 L L 2 n n 2, 27 where T ac(t T IN )/B 2 with the isotropic-nematic transition temperature T IN given by T* B 2 /4aC, L 1 L 1 /L, an L 2 L 2 /L. F S is etermine by the egree of orientational orer S an its spatial variation, while F ns comes from the coupling between k n an S. Let l n enote the characteristic istance over which the irector variations take place. It follows that n an n are on the orer of /l n. The slow spatial variation of n is efine to be l n, thus n, n 0. As a consequence, F ns is negligible as compare to F S. Now let n enote the time scale associate with the ynamics of n an the flow. It follows that Ā ij is on the orer of / n. The slowness of time evolution means n, thus Ā ij 0. By neglecting F ns an Ā ij in Eq. 25, we obtain F S S k ks t S. Thus the temporal variation of S is ecouple from the temporal variation of n. In aition, we note that the equilibrium solution of S is given by the solution to the Euler-Lagrange equation

6 7480 TIEZHENG QIAN AND PING SHENG PRE 58 F S 0. S k ks 29 Thus Eq. 28 escribes the temporal relaxation of S when it is out of equilibrium. Moreover, since the response time scale for S is much shorter than that for n, the S (r/, t ) solution may be approximately given by the instantaneous optimal configuration, which minimizes the free energy (r/ )F S n(r/, t ) (S/ 2 )U n(r/, t ), note to epen on n explicitly. Here SU is the substrate-lc interaction potential. Through time-varying n, the optimal solution S b thus obtaine epens on t explicitly. That means S b / t is on the orer of / n 0. Meanwhile, v in the bounary layer is on the orer of k v / n an S b 1, thus v S b / n also tens to vanish. With S b /t 0 justifie, the instantaneous optimal configuration S b, satisfying Eq. 29 but subject to the time-varying n configuration, is thus obtaine as the approximate solution of Eq. 28 for the bounary layer affecte by the substrate-lc intercation. Base on the above iscussion, it is reaily seen that the egree of orientational orer S, varying spatially in the bounary layer but uniform in the bulk, only aiabatically follows the slow temporal variation of n. The flow fiel thus inuce is on the orer of / n, which is negligible compare with the n-inuce flui velocity l n / n. We conclue that in the regime of l n an n, the alignment state can be escibe by n alone, an n an v can be chosen as the two ynamic variables. This constitutes the basis of the EL theory, an l n, n may be ientifie with, 0 an/or E, E. Through a more complete iscussion base on Q ij S(3n i n j ij )/2 P(l i l j m i m j )/2, with l, m, an n being the principal axes of QJ, it can be shown in a similar way that in the regime of l n an n, n is effectively the only ynamic variable from QJ. V. NUMERICAL RESULTS We have carrie out numerical calculation for a thin cell subject to a high switching voltage in orer to emonstrate the LCO hyroynamic effects. The coorinate system is efine with the z axis being the substrate normal, an all quantities are functions of only z an t. We focus on the in-plane motions with v 3 0. The bounary conition is that the velocity components v 1 an v 2 must vanish at the surfaces z 0 an z ( is the cell thickness an the surface orer parameters are fixe strong anchoring. With Q ij Q ij (z) an E is in the z irection, we have 31 f f 32 0 an Since the inertial terms are negligible, we obtain 3 3i 0, i 1,2, 30a f k f k, k 1,2,3,4,5, 30b where f k are efine by f 1 h 11 h 33, f 2 h 22 h 33, f 3 h 12 h 21, f 4 h 23 h 23, an f 5 h 31 h 13, an f k are efine similarly. Here we select five inepenent orer parameters q 1 Q 11, q 2 Q 22, q 3 Q 12, q 4 Q 23, an q 5 Q 31, while f k an f k are the molecular fiels conjugate to q k. Explicit expressions for all the quantities in Eq. 30 are given in Appenix C. The seven couple ifferential equations in Eq. 30 can be integrate to yiel the solution v i (z,t) an q k (z,t) provie the initial configuration q k (z,0) is given. Due to the neglect of inertial terms, v i (z,0) an q k(z,0) are etermine by q k (z,0). To etermine the initial configuration, which is assume to be in static equilibrium, we use the conition of f k 0, i.e., q i k q k,i F LG F f 0. Dynamics is obtaine by straightforwar time integration from the initial configuration. The cell thickness use in our computation is 100, on the orer of 0.5 microns. a is chosen to be 15, with 20 an 5, typical for the LC s of large ielectric anisotropy. The voltage unit use here is V F 4 K 1 / a,on the orer of the threshol voltage of the Fréeericksz transition. For K yn an a 15, V F 0.3 V, we calculate the temporally varying flow fiel in the cell right after a high holing voltage V H is switche off. The material constants an bounary conitions are given in Appenix D. Strong anchoring conition for the homogeneous alignment at the two substrates is assume. The pretilt angle is set to be 8 an the substrates are arrange to have parallel surface irectors. The surface value of S is fixe at S 0 B(3 33)/8C at temperature T T IN 4(T IN T*). Accoring to the EL theory, switching off V H results in a fast rotation of the irectors close to the substrates as the large elastic istortion energy in the bounary layers is suenly release 16. A large flow fiel is thus inuce by the couple evolution of n an v. This is known as the ynamic backflow effect. In particular, the EL theory gives E 2 E /K 1 here E 4 K 1 2 / a V 2 H ) as the time scale of the relaxation process right after V H is switche off. The calculation is performe for V H ranging from 10V F to 100V F for parameters suitable to 5CB, V F 0.25 V). Note that V F V H E, thus E approximately ranges from 10 (V H 10V F )to (V H 100V F ). First we want to show that our formulation oes agree with the EL theory in the regime of low voltages. This is achieve by examining the numerical results obtaine for V H 10V F. Let v Q enote the timeepenent velocity fiels obtaine from our generalize hyroynamic equations an v n enote that from the EL theory. In Fig. 1, it is seen that the initial v Q an v n possess similar spatial variation but iffer appreciably in magnitue. However, the initial ifference vanishes quickly as v Q approaches v n in a very short relaxation time scale. Subsequently, v Q merges with v n an unergoes futher relaxation with the time scale E. As the initial ifference between v Q an v n lasts only within a negligible time uration ( E ), the two solutions are thus equivalent an the EL theory is effectively applicable.

7 PRE 58 GENERALIZED HYDRODYNAMIC EQUATIONS FOR FIG. 1. a Spatial an temporal variations of v 1 (z,t) after the holing voltage V H 10V F is switche off at t 0. The soli lines represent the solution from the generalize formulation an the open squares represent that from the EL theory. Both solutions are obtaine for the time uration from t 0 to , with /K 1. The time interval between each two ajacent soli lines is It is reaily seen that the temporal variation epicte by the soli lines is fast an monotonic, exhibiting a ecreasing v 1 magnitue everywhere. For the open squares, they have similar spatial epenence but negligible temporal variation. b Same as a except for V H 20V F. FIG. 2. a Initial velocity fiels v 1 (z,t 0) obtaine from the EL theory for V H 50V F circles, 75V F squares, an 100V F iamons. b Initial velocity fiels v 1 (z,t 0) obtaine from the generalize formulation for V H 50V F circles, 75V F squares, an 100V F iamons. Here the spatial epenence of v 1 is only epicte in the first half-space because of the symmetry relation v 1 (z,t) v 1 ( z,t). For V H high enough so that E is comparable to, qualitative ifferences emerge between our formulation an the EL theory. First, we note that within the LG theory, the equilibrium LCO configuration maintaine by the holing voltage V H ) possesses strong biaxial character if E is on the orer of. With an initial state alreay beyon the irector escription, it is imperative to use the generalize hyroynamic equations to solve the time evolution problem. This is further justifie by consiering the ynamic time scales. Since E is actually comparable to the intrinsic time scale if E, the ynamic relaxation involves the couple evolution of QJ an v from the very beginning. This is inee confirme by our numerical results. In Fig. 2, it is seen that the initial v Q an v n still possess similar spatial variation but again are appreciably ifferent in their magnitues. But now the ifference persists as v Q an v n evolve, respectively, with the time scales an E, which are of the same orer. In aition, since z v in the central part of the cell is crucial in etermining the relaxational behavior of the alignment state 16,17, the whole ynamic process preicte from our hyroynamic equations iffers qualitatively from that obtaine within the EL theory. From Fig. 3, we see that for V H 50V F ( 13 V), the initial velocity increase with V H ) is almost flat, in sharp contrast to that preicte by the EL theory. This is ue to the fact that the elastic istortion in the bounary layers is no longer characterize by the spatially rotating irector. Instea, it exhibits preominant biaxial or- FIG. 3. Maximum initial velocity epicte as a function of the switche off V H. Soli triangles represent the solutions from the EL theory an the open triangles represent those from the generalize formulation.

8 7482 TIEZHENG QIAN AND PING SHENG PRE 58 the right-hane helix. For incompressible nematics, the basic equations of the EL theory rea i v i 0, v i t j p ji ji f ji, ji A2 A3 I 2 n i t 2 g i g i n i, A4 FIG. 4. Time variation of the central velocity graient v(z /2,t) plotte as a function of time. V H 50V F circles, 75V F squares, an 100V F iamons are switche off at t 0. The soli symbols represent the solutions from the EL theory an the open symbols represent those from the generalize formulation. It is seen that after about E , each solution from the generalize formulation merges with the corresponing EL solution for the same V H. ering with only a small rotation of the principal axes of QJ. As a result, we fin the subsequent relaxation of the velocity fiels to be much slower than that preicte by the EL theory, as epicte in Fig. 4. VI. CONCLUDING REMARKS We have obtaine the generalize hyroynamic equations base on the tensor orer parameter. Numerical results obtaine for the backflow phenomena serve to make apparent the new length an time scales inherent to our formulation an to show qualitatively ifferent ynamics in thin cells at high holing voltages. This coul be important for the simulations of twiste-nematic an super-twiste-nematic cells. Further evelopments in taking into account more realistic surface anchoring conitions are currently uner way. It is also interesting to take into account the flexoelectric effect which can be inuce by either irector istortion or orer parameter variation 23. ACKNOWLEDGMENTS Tiezheng Qian woul like to thank Professor Yuen-Ron Shen at Berkeley for helpful iscussions. This work was supporte by Hong Kong ITDC Grant No. AF/155/94 an NSF Grant No. DMR APPENDIX A: THE ERICKSEN-LESLIE THEORY Base on the irector escription of alignment state, the Frank free energy ensity is given by F F 1 2 K 1 n K 2 n n 2 2 P 1 2 K 3 n n 2, A1 where K 1, K 2, an K 3 are the splay, twist, an ben elastic constants, respectively, an P is the helical pitch, positive for where is the ensity constant, p is the pressure,, f, an are the stress tensors inuce by elastic istortions, external electric an magnetic fiel, an viscosity effects, respectively, I is the inertial momentum ensity that is usually negligible, g an g are the elastic an viscous molecular fiels, an is the Lagrange multiplier impose by n 2 1. an g are etermine by the Frank free energy through ji F F j n k in k, g i F F F F n k i k n i. A5 A6 The entropy source of the isothermal, issipative processes is given by T r ji j v i g ṅ. A7 The conservation of angular momentum relates g to the antisymmetric part of, an this fact leas to the expression T r s ji A ji g N, A8 where s ji is the symmetric part of ji, A ji 1 2 ( j v i i v j ) is the symmetric part of the velocity graient tensor, an N ṅ n is time-changing rate of the irector with respect to the backgroun flui with angular velocity 1 2 v. The inepenent fluxes contributing to the entropy prouction are thus ientifie. For A ji an N i, which are weak on the molecular scale, the viscous stress an molecular fiel are linear functions of the fluxes: ji 1 n j n i n n A 4 A ji 5 n j n A i 6 n i n A j 2 n j N i 3 n i N j, g i 2 n j A ji 1 N i, A9 A10 where the six i are the Leslie coefficients, with an APPENDIX B: COMPARISON OF EL THEORY WITH THE GENERAL FORMULATION We equate the LG free energy with the Frank free energy in the limit of Eq. 9. From the efinitions 5 an

9 PRE 58 GENERALIZED HYDRODYNAMIC EQUATIONS FOR A6, the vector molecular fiel g is relate to the tensor fiel h by g j 3S 0 2 n ih ij n i h ji. B1 Here the fiel-inuce terms in g an h are inclue because (1/12 ) m a Q ij Q i E j (1/8 ) a (n E) 2 constant, with a S 0 m a. The equilibrium conition h ij ij ijk k is thus equivalent to g i 3S 0 n i as obtaine by minimizing the Frank free energy with 3S 0 being the Lagrange multiplier ue to n 2 1). The istortion stress ji j Q,i is equal to ji with 3 2 S 0 (n j kij n j kji ) F F / n i,k. The fiel-inuce stresses f an f are the same in Eq. 6 with J expresse in terms of QJ an n, respectively. As for the viscous forces, g is relate to h by g 3S 0 j 2 n ih n ij i h, ji B2 which is of the same form as Eq. B1. Equation B2 an N ij 3S 0 (n i N j N i n j )/2 yiels the same orientational entropy prouction equation h N ij ij g k N k. From Eqs. B1, B2 an 2n j Q ij 3S 0 n i 3S 0 n j n jn i, we see the vectorial Oseen equation A4 can be constructe from the tensorial Eq. 3 with I 9 2 JS 2 0 an 3S JS 2 0 n j n j. I 9 2 JS 2 0 is also obtaine from the energy relation 1 2 In JQ 2 ij or the angular momentum relation I(n n ) i ijk J(Q j Q k Q j Q k ). an are require to be ientical in the limit of Eq. 9 since they both serve in the same ynamic equation v i j ji. This fact, combine with Eq. B2, can be use to establish the relation between the Leslie coefficients an the viscosity coefficients efine in an h. From Eqs. 8 an A10, we get 1 2S an S 0 2. Equating ji with ji yiels 1 9S /4, 2 3S 0 2 /4 9S /4, 3 3S 0 2 /4 9S /4, 4 4 S /2, 5 3S 0 5 /2 3S /4, 6 3S 0 6 /2 3S /4, B3 which give an with the Paroi relation The six Leslie coefficients linke by the Paroi relation can be use to completely etermine the value of six viscosity coefficients 1, 4, 5, 6, 1, an 2 linke by 6 5 2, provie S 0 is known. Note that other terms in higher orers of Q ij can be ae to the expansions of s ij an h ij in Eqs. 20 an 21. However, aitional viscosity coefficients woul have to be introuce 14,15, an they can not be etermine by the Leslie coefficients. In summary, our formulation can be reuce to the EL theory with the relations B1, B2,, f f, an establishe in the limit of homogeneous, uniaxial orering with uniform S S 0. APPENDIX C: EXPLICIT EXPRESSIONS For planar flui motion (v 3 0), we only nee to use 31 an 32. From Eq. 7, we have 31 1 Q 31 Q A 4 A 31 5 Q 3 A 1 6 Q 1 A N 31 1 Q 3 N 1 1 Q 1 N 3, 32 1 Q 32 Q A 4 A 32 5 Q 3 A 2 6 Q 2 A N 32 1 Q 3 N 2 1 Q 2 N 3. C1 Introucing the coefficients U ki an V ki with 3k U ki q i V kj v j (k 1,2, i 1,2,3,4,5, an j 1,2), we have U q 5, U 12 1 q 5, U 13 1 q 4, U 14 1 q 3, U q 1 q , V 11 1 q q 1 q 2 6 q q 1 q q 3 2 q 4 2 4q 5 2 2q 1 q 2 2, V 12 1 q 4 q q q q 4 q 5 q 1 q 2 q 3, U 21 1 q 4, U q 4, U 23 1 q 5, U 24 1 q 1 2q , U 25 1 q 3, V 21 1 q 4 q q q q 4 q 5 q 1 q 2 q 3, V 22 1 q q 1 q 2 6 q q 1 2q q 3 2 4q 4 2 q 5 2 q 1 2q 2 2. C2 Here q i t q i an v j z v j. The expressions for f k are relatively simple. From Eq. 8, we have

10 7484 TIEZHENG QIAN AND PING SHENG PRE 58 f 1 h 11 h q 1 v 1 q 5 1 q 2 v 2 q 4, f 2 h 22 h q 2 v 2 q 4 1 q 1 v 1 q 5, f 3 h 12 h q 3 v 1 q 4 v 2 q 5, C3 f 4 h 23 h v 2 1 2q 4 v 1 q 3 v 2 q 1 2q 2, f 5 h 31 h v 1 1 2q 5 v 2 q 3 v 1 2q 1 q 2. Introucing another set of coefficients u ki an v kj with f k u ki q i v kj v j (k,i 1,2,3,4,5 an j 1,2), we have u , u 12 1, u 13 0, u 14 0, u 15 0, v q 5, v 12 1 q 4, u 21 1, u , u 23 0, u 24 0, u 25 0, v 21 1 q 5, v q 4, u 31 0, u 32 0, u , u 34 0, u 35 0, v 31 1 q 4, v 32 1 q 5, C4 u 41 0, u 42 0, u 43 0, u , u 45 0, v 41 1 q 3, v q 1 2q 2, u 51 0, u 52 0, u 53 0, u 54 0, u , v q 1 q 2, v 52 1 q 3. Using the free energy expression an the efinitions of h ij an f k, we obtain f 1 2q 1 q 2 3 q 1 2 q 3 2 q 5 2 q 1 q q 1 q 2 R L 1 2q 1 q 2 L 2 q 1 q 2 2 L 1 4q 3 /P m a 12 V z/ 33 2, f 2 q 1 2q 2 3 q 2 2 q 3 2 q 5 2 q 1 q q 1 2q 2 R L 1 q 1 2q 2 L 2 q 1 q 2 2 L 1 4q 3 /P m a 12 V z/ 33 2, f 3 2 q 3 6 q 1 q 3 q 2 q 3 q 4 q 5 8 q 3 R 2L 1 q 3 2 L 1 4q 1 4q 2 /P, C5 f 4 2 q 4 6 q 2 q 4 q 3 q 5 q 4 q 1 q 2 8 q 4 R 2L 1 q 4 L 2 q 4 2 L 1 4q 5 /P, f 5 2 q 5 6 q 1 q 5 q 3 q 4 q 5 q 1 q 2 8 q 5 R 2L 1 q 5 L 2 q 5 2 L 1 4q 4 /P, with q k z 2 q k an R Q ij Q ij q 1 2 q 2 2 (q 1 q 2 ) 2 2q 3 2 2q 4 2 2q 5 2. APPENDIX D: MATERIAL PARAMETERS Here we give the numerical values of the various parameters use in our calculation. The material constants in the free energy expansion are a J/cm 3 K, B 0.53 J/cm 3, C 0.98 J/cm 3, L J/cm, an L 2 /L1 1, suitable for 5CB 24. The temperature is T T IN 4(T IN T*). The Frank elastic constants are given by K S 2 0 (L L 2 ), K S 2 0 L 1, an K S 2 0 (L L 2 ), with S 0 B(3 33)/8C at temperature T. Since very few measure viscosity coefficients are available, here the Leslie viscosity coefficients are taken from Ref. 16, suitable for p-methoxybenzyliene-p- n-butyl aniline MBBA. They are liste in Table I. Using Eq. B3, the viscosity coefficients in Eq. 29 can be obtaine. Other parameters are 20, 5, 100, an P 2. The voltage unit V F 4 K 1 / a is V. The pretilt angle of substrate alignment is s 8 an the bounary conition for QJ is QJ (0) QJ() S 0 (3n s n s I)/2 with n s (cos s,0,sin s ). TABLE I. The viscosity coefficients. 1 / / / cgs 5 / / 4 ( )/ 4

11 PRE 58 GENERALIZED HYDRODYNAMIC EQUATIONS FOR C. W. Oseen, Trans. Faraay Soc. 29, H. Zocher, Trans. Faraay Soc. 29, F. C. Frank, Discuss. Faraay Soc. 25, J. L. Ericksen, Arch. Ration. Mech. Anal. 4, ; J.L. Ericksen, Phys. Fluis 9, F. M. Leslie, Q. J. Mech. Appl. Math. 19, ; F.M. Leslie, Arch. Ration. Mech. Anal. 28, P. G. e Gennes, Phys. Lett. A 30, ; P. G. e Gennes, Mol. Cryst. Liq. Cryst. 12, P. Sheng an E. B. Priestley, in The Lanau e Gennes Theory of Liqui Crystal Phase Transitions, Introuction to Liqui Crystals, eite by E. B. Priestley, P. J. Wojtowicz, an P. Sheng Plenum Press, New York, 1975, p P. Sheng, Phys. Rev. A 26, ; T. J. Sluckin an A. Poniewierski, Phys. Rev. Lett. 55, ; W. Chen, L. J. Martinez-Mirana, H. Hsiung, an Y. R. Shen, ibi. 62, X. Zhuang, L. Marrucci, an Y. R. Shen, Phys. Rev. Lett. 73, T. Z. Qian an P. Sheng, Phys. Rev. Lett. 77, ; T. Z. Qian an P. Sheng, Phys. Rev. E 55, N. Schopohl an T. J. Sluckin, Phys. Rev. Lett. 59, P. Palffy-Muhoray, E. C. Gartlan, an J. R. Kelly, Liq. Cryst. 16, A. Kilian an S. Hess, Z. Naturforsch., A: Phys. Sci. 44, ; A. Sonnet, A. Kilian, an S. Hess, Phys. Rev. E 52, H. Imura an K. Okano, Jpn. J. Appl. Phys. 11, A. C. Diogo an A. F. Martins, J. Phys. Paris 43, C. Z. van Doorn, J. Appl. Phys. 46, ; D. W. Berreman, ibi. 46, I. Dozov, M. Nobili, an G. Duran, Appl. Phys. Lett. 70, ; T. Z. Qian, Z. L. Xie, H. S. Kwok, an P. Sheng, ibi. 71, L. D. Lanau an E. M. Lifshitz, Electroynamics of Continuous Meia, 2n e. Pergamon Press, New York, S. e Groot an P. Mazur, Non-equilibrium Thermoynamics North-Hollan, Amsteram, In a fiel theory with the free energy ensity of the form k( ) 2 m 2, the elastic correlation length is k/m. The Frank energy only consiers the graient terms an the elastic correlation length is thus infinite. 21 T. C. Lubensky, Phys. Rev. A 2, O. Paroi, J. Phys. Paris 31, R. B. Meyer, Phys. Rev. Lett. 22, ; G. Barbero, I. Dozov, J. F. Palierne, an G. Duran, ibi. 56, P. Sheng, B. Z. Li, M. Y. Zhou, T. Moses, an Y. R. Shen, Phys. Rev. A 46,