A new crosscoupling in smectic C* liquid crystals near the transition to the A phase

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**A new crosscoupling in smectic C* liquid crystals near the transition to the A phase**

Harald Pleiner, Helmut R. Brand

To cite this version:

Harald Pleiner, Helmut R. Brand. A new crosscoupling in smectic C* liquid crystals near the transition to the A phase. Journal de Physique, 1989, 50 (8), pp.851-854. �10.1051/jphys:01989005008085100�.

�jpa-00210963�

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Short Communication

A new

crosscoupling

ⁱⁿ^smectic^C*

liquid crystals

^near^the^transition

to the A

phase

Harald Pleiner(1,2) ^and^{Helmut R.}Brand(1) (1) ^FB^Physik,Universität Essen, D-4300 Essen 1, ^R.F.A.

(2) ^MaterialsDepartment, ^UCSB,^Santa^Barbara,^CA93106, ^U.S.A.

(Reçu le 27 décembre 1988, accepté ^le27 février 1989)

Résumé. ²⁰¹⁴ On présente ^uneétude de la dynamique ^ducouplage ^entre^le^{module du}paramètre

d’ordre et les degrés de liberté hydrodynamique d’un cristal liquid smectique ^C*^auvoisinage ^de

la transition vers la phase A. On utilise une description ^localeêtônsuggère ûneexpérience pour l’observation directe de la relaxation du paramètre ^d’ordre.

Abstract. ²⁰¹⁴For a smectic C* liquid crystal ^near^thetransition to the A phase ^thedynamics, ^which couples ^the^orderparameter ^modulus^{to the}hydrodynamic degrees ^offreedom, is derived using ^the

local description ^validônlength scales small compared ^to^thepitch. Ânexperiment îssuggested,

which allows us to observe the order parameter relaxation directly.

Classification

Physics ^Abstracts

61.30-v - 64.70 Md

Introduction.

The chiral smectic C* liquid crystal phase [1] has become the object of intense investigations during ^the^last^decade,mainly ^because^{of its}possible applications ^forelectro-optic devices. In this communication however, ^wewant to concentrate on a very interesting aspect ^{of the}dynamics ^of

the smectic C* phase ^near^thetransition e.g. ^to^the^{smectic A}phase : ^thecoupling of the soft mode (order parameter ^modulusrelaxation) with the rotation of the c -vector [2], which allows to observe the former. We do not consider electrical effects here.

We consider a smectic C* phase ^with^alarge pitch ^{or a}film for which the pitch ^islarger ^thane

the thichness of the sample. It is then appropriate ^to^{use a}"local" description, ^i.e. choosing ^as hydrodynamic variable the rotation of the c -vector (the projection of the director onto the layers)

in the smectic layers, ⁱⁿcontrast to the "global" description, appropriate ^forlength ^scaleslarger

than the pitch, ^{where the}hydrodynamic variable is the displacement of the helix [3]. ^On^{the local}

level the smectic C* phase is biaxial [4, 5] ^but^diûerentfrom the nonchiral C phase, since it lacks

a horizontal symmetry plane, ^{i.e. in}^terms^{of c and}^p,^thelayer normal, ^{it is}^notinvariant under c ^--;-c and p ----+ -]. This local description ^{has been}^{useful in}explaining [4, 5] ^theearly experiments ^onshear flow induced polarization [6].

In addition to the hydrodynamical ^variable6jJ (called -n2 ^{in Ref.}[4]), describing in-plane

rotations of c, ^wetake also into account 6S, ^thedeviation from the equilibrium value of the order

parameter ^modulus,which becomes slow near the phase transition. Of course, S = sin 8, ^where

8 is the tilt angle ^{of the}^director^withrespect ^top, while 0 ^{is the}^azimuthalangle ^of^the^director.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005008085100

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852

A similar description ^of^thesmectic C* phase ^has^beengiven recently [7, 8] lacking, however, ^the dissipative crosscoupling ^between^the^twovariables, which will be crucial in the following.

Statics.

The statics of the systems is characterized by ^its(linearized) ^freeenergy density

where the material tensors are generally of the form

and

Here c is the position dependent preferred direction within the layers, ^êi ⁼(cos [qo z ⁺Po],

sin [qo z ⁺Po], 0) i , ^where^qo^{is the}equilibrium wave vector of the pitch (the ^{helix is}in p -direction),

and where a stands for mass density, entropy density, layer compression ^or^dilation(and ^con-

centration in the case of a mixture), respectively ; ^fc^{is the}remaining ^freeenergy part ^not^con- taining ^6S.^{Since the}system ^is^notorthorhombic, ^{there is}^{no a}priori ^reason,why off-diagonal

terms in equation (2) ^shouldvanish. In the smectic C phase equation (1) is valid for _qo⁼0 and

K12 ⁼h 23 ⁼0, where the latter two equations ârerequired by ^thepresence ôfâ^horizontal symmetry plane ( i. e. a 0 --+ - P symmetry) .

Comparing ^withânappropriate (nonlinear) Ginzburg-Landau expansion, ^{valid in}â^certain temperature înterval^near^thetransition (i. e. outside the critical region ândoutside the true hydrodynamic region) equation (1) follows from the Ginzburg-Landau expression ^for^ôS^«^S.Ône

finds that Kij ^scaleswith the second power ôfS, ^{while the}ya are proportional ^toS, ^{and the}Ai are at least 0 (,S3) ; a is related to S by a ⁼^bS2⁺^d5’4(b, d âreGinzburg-Landau parameters), î.e.

a - S2 for S’ ^---+0, ^but^{a -}S4 if b > d. The latter relation was used to describe experimen-

tal results [9]. Since a is assumed to be linear in the distance to the true transition temperature, T ⁼(Tc - T) /Te, ^the^relations^{a -}^,S2ôr^{a -}⁸⁴give â^differentT -dependence ^for^thehy- drodynamic coefficients. In this description ^thepolarization ^Pîsproportional ^to^S(p ^xc) ; ^since

we do not deal with an external electric field, ^P^does^not_appearexplicitly ⁱⁿequation (1).

Dynamics.

The dynamical equations ^{for the}^two^variables^6jJ^and^6S^{have the}(linearized) ^form

and

Here _w=is the vorticity, ^while

/?

^and

"

characterize the coupling ^ofsymmetrized velocity

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gradients (elongational flow) ^with⁶⁰^and^6S,respectively. ^The,Q-tensors ^aregenerally ^{of the}

biaxial form (2), since there is no a priori ^reasonwhy ^theprinciple âxesôf3ij ^{should be}îdentical

with p, c, ^andp ^xc. In nonchiral smectic C ,Q12 ⁼⁰⁼(323 by symmetry. ^Theright ^{hand sides}

of (4) ând(5) give ^thedissipative part ^{of the}dynamics ôf⁶⁰ând6S, respectively, ^whichâre^most conveniently derived from the entropy production ^R^{given by}

where Rn contains all other terms not connected with hs. ^Thethermodynamic conjugate quantities

are defined by hs = 8 f / 86S, ho = 8 f / 8"V 6/J ând^ha⁼8 f / 860: (here â^stands^forentropy density, concentration (in mixtures) ândlayer ^dilationôrcompression). For the first two of these varibles the ei âreof the form (3), while the last ôneonly ^containsâp -component. ^The^new crosscoupling ^termproportional ^to(3 ^{is the}ône^whoseimplication ^we^{want to}investigate ^{here. It}

vanishes in the smectic C phase, ^because^{of the}existence of a horizontal symmetry plane.

Comparing equations (3) - (5) with the usual nematodynamic equations for the director and

assuming that the kinetic coefficients there do not depend ^onT, which is the usual assumption,

one firids that

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^and⁽³scale like S-1, (1 ^and

/?

âreindependent ôf^S,^while⁽²îsproportional

to S-2, the latter relation being already ^confirmedexperimentally [10].

It is now straightforward ^toderive the frequencies ^andeigenvectors ^{of the}coupled ^motion^of

bS and 6/J. ^The^tworelaxation rates y1,2 ^fulfilthé relation

where ( == (1(2 - Ç), ^{I{2 ==}Iijkikj,.À ⁼Àiki, and k is the wave vector of the mode. ^For homogeneous excitation (k=0) ^there^is^onerelaxation mode with yi ⁼2a(i, ^thesoft mode with 1’1 l"tJ bs2 + dS4. Due to the dissipative crosscoupling, ^however,this mode contains in smectic C* liquid crystals also rotations of the c -vector with

Thus taking ^into^account^themacroscopic ^variable^bSalso renders 6ljJ non-hydrodynamic ^near^the phase transition due to dynamic dissipative cross-coupling (3. ^Thepresent system ^seems^to^{be the} first one showing ^thisphenomenon. Of course, the limit S ^-0 is not really contained in the above formulas, ^becauseof critical fluctuations very ^close^toTe and because of 6S ^S.^Fork540

there are two relaxational modes, ^oneproportional ^to^{k 2and}^one& independent. (Note, ^however,

that the limites k ^--;0 and S ^---+0 are not interchangeable.) ^{In these}^two^modes^6ljJ^and^S,S

are coupled ^andbolbs - S-’ ^and^S,respectively.

Experiment.

We now propose ^anexperiment, which allows to observe the relaxation of 6S. Thke a smectic C* film, ^whosethickness is smaller than its pitch. ^If(near ^thetransition temperatures) ^thetemper-

ature is changed homogeneously, this results in a homogeneous ^deviation^6S^{from the}equilibrium

value S [described e.g. by ône^{of the}à ’s in Eq. (1)] ôr^tophrase îtdifferently, ^{there is}^{a new} equilibrium ^value^S^{which the}^{old S}îsrelaxing ^to.^Viewed^fromâboveôr^below(along ^thelayer

or film normal) this would result in an achiral C phase ⁱⁿâchange ^{of the}overall light intensity passing ^crossedpolarizers - ânêffect^noteasy ^todetect. In a chiral smectic C* phase, ^however,^this

S relaxation is accompanied by ^arotation of the c -vector due to the dissipative cross-coupling ^(3.

Thus, in the absence of any boundary ^condition^on^thedirector, Le. if the direction of c (or ^rather

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854

its mean value over the film thickness) îsarbitrary ^withrespect ^toâ^fixedlaboratory ^frame,^the^S relaxation ^shows_up^{also in}âchange ^{of the}direction for the maximal intensity. Near the transition

temperature this should be a large ^effect(cf. Eq. (8)). ^{The main}experimental difficulty ^seems

to be to prepare ^a^smecticC* film with no boundary conditions on the director in a state with a

definite (but arbitrary) averaged direction for c. Probably one can use a film with a temperature gradient ^at^theboundaries, ^{such that}^thephysical boundaries are in a smectic A phase allowing

for any ^direction^forc, ^andorient the c -vector within the film by ^anexternal field. After removal of the field the film should be in the desired state and a subsequent homogeneous temperature change should result in a rotation of c .

Acknowledgements.

We thank the Deutsche Forschungsgemeinschaft for financial support.

References

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