Flexoelectric polarization and second order elasticity for nematic liquid crystals

Physics Letters A 180 ( 1993 ) 456-460 North-Holland

PHYSICS LETTERS A

Flexoelectric polarization and second order elasticity for nematic liquid crystals A.L. Alexe-Ionescu Department of Physics, PolytechnicalInstitute of Bucharest, Splaiul Independentei 131, R-77216 Bucharest, Romania Received 11 June 1993; accepted for publication 9 July 1993 Communicated by V.M. Agranovich

The flexoelectric properties of nematic liquid crystals are analysed. It is shown that in the frame of the usual elasticity two coefficients characterize completely the flexoelectric properties of the liquid crystal. These coefficients, in the limit of small scalar order parameter, i.e. near the clearing point, are approximately equal. More precisely they have the same linear term in the scalar order parameter, and differ for terms quadratic in this parameter. Their difference behaves, hence, as the usual nematic liquid crystal elastic constants, whereas their sum depends on the temperature, like the mixed splay-bend elastic constant. It is shown furthermore that in the frame of a second order elastic theory in the flexoelectric polarization there are no terms from the second order spatial derivatives of the nematic director, or the nematic tensor order parameter. Consequently also in the frame of the second order elasticity the flexoelectric polarization is given by the usual expression. This conclusion is important in connection with the surface polarization recently discovered in pretilted nematic liquid crystal samples.

I. Introduction The bulk elastic b e h a v i o u r o f the nematic liquid crystals ( N L C ) is well described by the c o n t i n u u m theory p r o p o s e d long ago by F r a n k [ 1,2 ]. In this theory the bulk elastic properties o f the N L C are connected with three elastic constants, Kl l,/(22 and K33 , called splay, twist a n d bend elastic constant, respectively. T h e y are the analogue o f the L a m e coefficients for an isotropic elastic b o d y [ 3 ]. According to Oseen [4] first, a n d Nehring and Saupe [5] later, two other elastic constants have to be taken into account when the anchoring energy [ 6 ] is finite. These elastic constants are connected to elastic contributions which can be integrated, and hence they give only surface contributions. They are i n d i c a t e d usually by/£=4 a n d KI3 , and are called s a d d l e - s p l a y and mixed s p l a y - b e n d elastic constants respectively. Recently a new interest in these elastic constants, not considered for a long time, has been shown by different groups [ 7 - 1 3 ]. As is easy to show [ 14 ] the K24 term does not introduce any difficulties in the analytical solution o f the elastic problem, on the contrary the i n t r o d u c t i o n o f the K~3 term in the total 456

elastic energy o f the nematic, which has to be mini m i z e d in order to find the elastic d e f o r m a t i o n o f the NLC, makes the variation problem ill posed [ 15-18 ]. In order to have a well posed variational p r o b l e m Barbero et al. [ 19-21 ] have proposed to modify the bulk elastic energy o f the N L C taking into account terms which are usually neglected in the linear elastic theory. This procedure is well known in the elastic theory o f solid materials [22]. In this paper I want to analyse the electric polarization connected to the N L C deformation, usually indicated with the term flexoelectric polarization [ 2 3 - 2 5 ] , when the elastic description o f the liquid crystal is done in terms o f the theory o f Barbero et al. [ 19-21 ]. My paper is organized as follows. First I recall, in section 2, what are the s y m m e t r y elements o f the N L C phase. After that, in section 3, the usual flexoelectricity is reconsidered and the t e m p e r a t u r e d e p e n d e n c e o f the flexoelectric coefficients deduced. Finally, in section 4, we will show that for usual N L C the new terms used in the wide elastic theory proposed in refs. [ 19-21 ] do not give any contribution to the flexoelectric polarization. The consequences o f this fact for the surface polarization in the preElsevier Science Publishers B.V.

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tilted NLC recently discovered, are discussed there. In section 5, I stress the main results of my paper.

2. Nematic liquid crystal NLC are formed by elongated molecules. In the NLC phase the major axes of the molecules a tend to be parallel. The average direction of a, in the statistical mechanical sense, is indicated by n and termed the NLC director [2]. The average quadratic fluctuation of a with respect to n, minus the value of this quantity in the isotropic phase (which is ] ), is called the scalar order parameter [26]. It is defined as S=3[ ( (n.a)2)-~]

,

(1)

and it is unity for perfect order at very low temperature, and zero in the isotropic (i.e. for a temperature larger than the nematic-isotropic phase transition) phase. Since ordinary NLC are not ferroelectric, in the bulk n and - n are equivalent. By taking into account this fact, it is possible to introduce a traceless tensor of rank two characterizing completely the symmetry of the NLC [27 ]. This tensor is called the tensor order parameter, and it is defined by Qij = 3 S ( n~nj - ~ j )

.

(2)

The tensor Q has quadrupolar symmetry. Hence, according to Barbero et al. [28], NLC may be considered as quadrupolar ferroelectric materials. When n is position independent, the NLC is in its fundamental elastic state, having zero elastic energy. If n is position dependent, according to Frank [ 1 ], the NLC elastic energy density f may be written in the form __1 ~= ]kijemni,jn . . . .

(3)

where k is the elastic tensor characterizing the NLC phase, and n~.j=Oni/Oxj [29]. In (3) the Einstein convention has been used. The generalization of (3) proposed in refs. [ 1921 ] is to consider also terms quadratic in the second order derivatives of n, i.e. f =_ _ 1~koem ni,j n ~,,, + ~1 mijkemp nijk n ~.mp ,

(4 )

where m is the new elastic tensor connected with the second order derivatives. In this manner the elastic

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problem connected to the Kl3 elastic term is well posed, as shown in refs. [ 19-21,30]. In (3) or (4) k and m have to be decomposed in terms of the unity tensor of components 3u and of n, following the usual procedure [ 31-33 ]. In this decomposition it is necessary to take into account that f ( n ) = f ( - n) o r f ( n ) = f ( - n) and hence the tensors k and m have to be even in n. It is possible to rewrite (2) and (4) by using the tensor Q defined in (2). In this manner the symmetry of the NLC phase is automatically taken into account [ 34]. In this frame (4) is rewritten as __1 f = ~gijkemp Qij,k aern,p + I Mijkempqr aij, ke amp,qr .

(5)

In (5) the new elastic tensors K and M have to be decomposed in terms of the unity tensor and of the tensor order parameter Q. As shown in refs. [26,29 ] in this manner it is possible to derive the temperature dependence of the usual elastic constants of NLC.

3. Usual flexoelectric polarization of NLC The flexoelectric polarization in NLC is the analogue of the piezo-electric polarization in solid crystals [35 ]. It has been discovered long ago by Meyer [23]. He has shown, with a simple model, that to a NLC deformation characterized by n~j, corresponds an electric polarization given by Pi =eia#rla,# •

(6)

The tensor e is called the flexoelectric tensor, and it is temperature dependent. As shown by Helfrich [ 36 ], the tensor e depends on the molecular shape. More recent investigations performed by Prost and Marceron [ 37 ] allow one to connect a also with the quadrupolar properties of the NLC phase. The tensor e has to be odd in n, for the abovementioned symmetry of the nematic phase. Simple calculations, performed according to Vertogen's rules, give e~c~=elnit~c~p+e2nc~i~+e3na~i~+e4ninc~np,

(7)

where e, (i = 1, ..., 4) are temperature dependent. By taking into account that n is a vector of modulus one, and hence n~n~= 1, it follows that n~ni.,~= O. By substituting (7) into (6), and by observing that 457

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e~ ni~,~a =e~ nino~,o~ =el ni d i v n ,

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E3 ~ir Q,~aQ,~a,r = E3 Q,~aQ,~p,, ,

e2 na ~i# na,o = ez n,~ n,~,i = 0 , e3 naO,~ n,~,# = e3 npn,,p = - e3 (n × curl n ) ~,

½E5 ( ,~,. ()/j~, + ~ia Q.p ) Q.a,': = E5 Qa~ Qip,~,,

e4ninan#na.#=O ,

½E6 ( Q,. tS#r q- Q i a ~ e ) Q.#.~, = E6 Qic~Q,~y,y,

one obtains [23] P = e l n d i v n - e 3 n X c u r l n . The phenomenological coefficients e~ and e3 are usually indicated with ell and e33 respectively. Hence the usual flexoelectric polarization is given by

since (2 is traceless, one obtains for the flexoelectric polarization the expression

P = e l l n div n - e 3 3 n × curl n .

Pi = El Qo,,./ + E3 Q-a Q.a,, + E5 Qm Qip, r

+ E6Q,.Q,w,/+o( 3 ) .

(12)

(8) Expression (12) in the first order in S reduces to

The influence o f the flexoelectric polarization on the deformation induced by an external electric field on a N L C has been analysed by different authors [ 3 8 41 ]. Barbero and Durand have also shown that the back electric field connected to the flexoelectric polarization may change the effective surface energy [42]. Now I want to show that in the limit of small order parameter the two flexoelectric coefficients are equal. To do this let us rewrite (6) in terms of the tensor order parameter O, instead o f n (see eqs. (4) and (5) ). In this manner the flexoelectric polarization is P, = E~,~v Q,~p,y,

(9 )

where the tensor E has now to be decomposed by means o f the unity tensor and of the tensor order parameter. As follows from (2), (1 is a symmetric tensor. Consequently the tensor E is symmetric with respect to the exchange of a and fl, i.e.

E , ~ , = E,~,~ .

(10)

By taking into account (10) simple calculations give, at the first order in S,

p}l) = E , Q,~,~,+O(2) .

(13)

As it follows from (2), in the hypothesis of scalar order parameter position independent, one obtains 3 Qi~,r = ~S( ninr,e + n~,ni:,)

=3S[ni divn. (n×curlnL] . Consequently the flexoelectric polarization in first order in S is given by

P~l)=3SEl[ndivn-n×curln]

.

(14)

By comparing (14) and (8) I can derive that in the first order in S

el{)=e~=3E,

S,

(15)

i.e. the flexoelectric coefficients in this limit are equal and linear in the scalar order parameter. In the second order in S the last three terms in (12) have to be considered. By observing that

E 3 Q,~pQ.p,, = o , EsQa~Q,a,y = - 3S212(n ×curl n) + n div n ] i , E6Qic~Qc~r,y = ]$212n div n + n ×curl n ] , , in the second order in S the flexoelectric polarization is

+ Ea t~a# Qie -t- ½E5 ( t~,r Q#r + 6,a Q,~r ) + ½E6(Qi,~Ot~ +QiaO,~r) ,

(11)

p~2)= [ 3 E i S + 3 ( 2 E 6 _ E s ) S 2 ] n d i v n _ [ 3 E ~ S + 3 (2E5 - E 6 ) ]n × c u r l n .

( 16 )

where now E, ( i = 1..... 6) have to be considered temperature independent. By substituting ( 11 ) into (9) and by observing that

By comparing (16) with (8) I derive that in the second order in S

½e, (0,,~c~#r+ cT#~6,~r)Q,,a,r = E, Q,r,r,

e~2) II = 3 E I S + ~ ( 2 E 6 - E s ) S 2

E~ 6~Q~#.~ = E~ Q,~,,~ = 0 ,

e~Z3)= 3 El S + ] (2E5 - E6)S 2 ,

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i.e. the flexoelectric coefficient differs for terms of second order in S. This implies that e(2)_e~3 2 ) ~ S , ~11 .(2)±~(2) T~33 ~ S , I1

(18)

near the clearing point To. From (18) it is possible to deduce that near T¢, ~,1"~2)_e~) behaves like the usual elastic constants [26], whereas el 2) +e~] ) behaves like the mixed splay-bend elastic constant [29].

4. Flexoelectric coefficients and second order elasticity In the previous section I have considered the usual flexoelectric polarization, in the sense that P~ was connected with n~j. In the case that the elastic deformations characterizing the deformed state are the first and second order spatial derivatives of the NLC director, or of the tensor order parameter, a question arises: what is the contribution to P, coming from n,,.a~. Instead of (6), I have to write Pi = e~,~pn,,.~ + d~,~n~,,~, ,

( 19 )

where the tensor e is again given by (7), whereas the tensor d has to be decomposed in the usual manner. Since n~,t~= n,,,rp, it follows that d,,,~ = d,,~ra.

(20)

Furthermore, for the symmetry of the NLC phase, the tensor d has to be odd in n. But by means of the unity tensor of components 8 o and of the components of the NLC director ni, it is impossible to build a tensor odd in n, having the symmetry property (20). Consequently d is identically zero for the NLC. The conclusion is that there is no flexoelectric polarization coming from second order spatial derivatives of n. It is possible to reach the same conclusion by writing P~ in terms of the tensor order parameter. In this case, instead of (19) I write Pi = Ea~, Q,~#a + Di,,~,u Q,,#.ru -

(21 )

From the symmetry of the tensor order parameter and for the theorem relevant to the inversion of the order of the spatial derivatives, the tensor D has the following symmetry properties, Di,~u

=

D i ~ r u = Di,~aur =

Di#aur

•

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The tensor D has to be decomposed in terms of the unity tensor and of the tensor order parameter. But since 13 is of fifth order, whereas the unit tensor and O are of second order, it follows that it is impossible to decompose D in the usual manes. Hence it has to be identically zero as deduced before. This fact may have relatively important consequences on the surface polarization in pretilted NLC samples recently discussed by Barbero and Kosevich [43]. These authors show that in pretilted NLC samples a sharp variation of the tilt angle over a semimacroscopic length is expected near the bounding wall. The source of this "surface discontinuity" is the mixed splaybend elastic constant gl3. In the analysis reported in ref. [43] the surface polarization is obtained by evaluating the flexoelectric polarization, in the usual manner, and integrating it over the surface layer, in which the tilt angle changes abruptly. As discussed in ref. [ 20 ] the surface variation of the tilt angle can be connected with the second order elasticity. It follows that, in principle, the surface polarization should be evaluated by using eq. (21 ), instead of eq. (6). But as we have shown before the tensors d or 13 are identically zero. Consequently the analysis reported in ref. [43] remains valid even in the frame of the second order elasticity.

5. Conclusion I have shown that the usual flexoelectric coefficients in the limit of small S are equal and they depend linearly on the scalar order parameter. They differ for terms of the second order in S. In particular e ~ - e 3 3 tends to zero, near the clearing point, as S 2, i.e. like the usual elastic constant. On the contrary el i + e33 tends to zero, near the clearing point, as S, i.e. like the K,3 elastic constant [29]. Furthermore I have shown that there are no contributions to the flexoelectric polarization from the second order spatial derivatives of the nematic director. This implies that the flexoelectric polarization is given by the usual expression even if the second order elasticity is used to describe the elastic behaviour of the NLC. This is true also for surface polarization expected in pretilted NLC samples recently discovered.

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References

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