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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�

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) ^{Materials }Department, ^{UCSB, }^{Santa }^{Barbara, }^{CA }93106, ^{U.S.A.}

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

Résumé. ^{2014 } On présente ^{une }étude de la dynamique ^{du }couplage ^{entre }^{le }^{module du }paramètre

d’ordre et les degrés de liberté hydrodynamique d’un cristal liquid smectique ^{C* }^{au }voisinage ^{de}

la transition vers la phase A. On utilise une description ^{locale }^{et }^{on }suggère ^{une }expérience pour
l’observation directe de la relaxation du paramètre ^{d’ordre.}

Abstract. ^{2014 }For a smectic C* liquid crystal ^{near }^{the }transition to the A phase ^{the }dynamics, ^{which}
couples ^{the }^{order }parameter ^{modulus }^{to the }hydrodynamic degrees ^{of }freedom, is derived using ^{the}

local description ^{valid }^{on }length scales small compared ^{to }^{the }pitch. ^{An }experiment ^{is }suggested,

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 ^{for }electro-optic devices. In this
communication however, ^{we }want to concentrate on a very interesting aspect ^{of the }dynamics ^{of}

the smectic C* phase ^{near }^{the }transition e.g. ^{to }^{the }^{smectic A }phase : ^{the }coupling of the soft
mode (order parameter ^{modulus }relaxation) 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 }^{a }large pitch ^{or a }film for which the pitch ^{is }larger ^{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, ^{in }contrast to the "global" description, appropriate ^{for }length ^{scales }larger

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 }^{diûerent }from the nonchiral C phase, since it lacks

a horizontal symmetry plane, ^{i.e. in }^{terms }^{of c and }^{p, }^{the }layer normal, ^{it is }^{not }invariant under
c ^{--; }-c and p ----+ -]. This local description ^{has been }^{useful in }explaining [4, 5] ^{the }early
experiments ^{on }shear flow induced polarization [6].

In addition to the hydrodynamical ^{variable }6jJ (called -n2 ^{in Ref. }[4]), describing in-plane

rotations of c, ^{we }take also into account 6S, ^{the }deviation 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 }^{with }respect ^{to }p, while 0 ^{is the }^{azimuthal }angle ^{of }^{the }^{director.}

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

852

A similar description ^{of }^{the }smectic C* phase ^{has }^{been }given recently [7, 8] lacking, however, ^{the}
dissipative crosscoupling ^{between }^{the }^{two }variables, which will be crucial in the following.

Statics.

The statics of the systems is characterized by ^{its }(linearized) ^{free }energy 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 ^{free }energy part ^{not }^{con-}
taining ^{6S. }^{Since the }system ^{is }^{not }orthorhombic, ^{there is }^{no a }priori ^{reason, }why off-diagonal

terms in equation (2) ^{should }vanish. In the smectic C phase equation (1) is valid for _{qo }^{= }0 and

K12 ^{= }h 23 ^{= }0, where the latter two equations ^{are }required by ^{the }presence ^{of }^{a }^{horizontal}
symmetry plane ( i. e. a 0 --+ - P symmetry) .

Comparing ^{with }^{an }appropriate (nonlinear) Ginzburg-Landau expansion, ^{valid in }^{a }^{certain}
temperature ^{interval }^{near }^{the }transition (i. e. outside the critical region ^{and }outside the true hy-
drodynamic region) equation (1) follows from the Ginzburg-Landau expression ^{for }^{ôS }^{« }^{S. }^{One}

finds that Kij ^{scales }with the second power ^{of }S, ^{while the }ya are proportional ^{to }S, ^{and the }Ai are
at least 0 (,S3) ; a is related to S by a ^{= }^{bS2 }^{+ }^{d5’4 }(b, d ^{are }Ginzburg-Landau parameters), ^{i.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 - }^{,S2 }^{or }^{a - }^{84 }give ^{a }^{different }T -dependence ^{for }^{the }hy-
drodynamic coefficients. In this description ^{the }polarization ^{P }^{is }proportional ^{to }^{S }(p ^{x }c) ; ^{since}

we do not deal with an external electric field, ^{P }^{does }^{not }_{appear }explicitly ^{in }equation (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^{of }symmetrized velocity

gradients (elongational flow) ^{with }^{60 }^{and }^{6S, }respectively. ^{The },Q-tensors ^{are }generally ^{of the}

biaxial form (2), since there is no a priori ^{reason }why ^{the }principle ^{axes }^{of }3ij ^{should be }^{identical}

with p, c, ^{and }p ^{x }c. In nonchiral smectic C ,Q12 ^{= }^{0 }^{= }(323 by symmetry. ^{The }right ^{hand sides}

of (4) ^{and }(5) give ^{the }dissipative part ^{of the }dynamics ^{of }^{60 }^{and }6S, respectively, ^{which }^{are }^{most}
conveniently derived from the entropy production ^{R }^{given by}

where Rn contains all other terms not connected with hs. ^{The }thermodynamic conjugate quantities

are defined by hs = 8 f / 86S, ho = 8 f / 8"V 6/J ^{and }^{ha }^{= }8 f / 860: (here ^{a }^{stands }^{for }entropy
density, concentration (in mixtures) ^{and }layer ^{dilation }^{or }compression). For the first two of these
varibles the ei ^{are }of the form (3), while the last ^{one }only ^{contains }^{a }p -component. ^{The }^{new}
crosscoupling ^{term }proportional ^{to }(3 ^{is the }^{one }^{whose }implication ^{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 ^{on }T, which is the usual assumption,

one firids that

## 13)

^{and }

^{(3 }scale like S-1, (1

^{and }

## /?

^{are }independent

^{of }

^{S, }

^{while }

^{(2 }

^{is }proportional

to S-2, the latter relation being already ^{confirmed }experimentally [10].

It is now straightforward ^{to }derive the frequencies ^{and }eigenvectors ^{of the }coupled ^{motion }^{of}

bS and 6/J. ^{The }^{two }relaxation rates y1,2 ^{fulfil }thé 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 }^{one }relaxation mode with yi ^{= }2a(i, ^{the }soft 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 }^{the }macroscopic ^{variable }^{bS }also renders 6ljJ non-hydrodynamic ^{near }^{the}
phase transition due to dynamic dissipative cross-coupling (3. ^{The }present system ^{seems }^{to }^{be the}
first one showing ^{this }phenomenon. Of course, the limit S ^{- }0 is not really contained in the
above formulas, ^{because }of critical fluctuations very ^{close }^{to }Te and because of 6S ^{S. }^{For }k540

there are two relaxational modes, ^{one }proportional ^{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 ^{and }bolbs - S-’ ^{and }^{S, }respectively.

Experiment.

We now propose ^{an }experiment, which allows to observe the relaxation of 6S. Thke a smectic
C* film, ^{whose }thickness is smaller than its pitch. ^{If }(near ^{the }transition temperatures) ^{the }temper-

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

value S [described e.g. by ^{one }^{of the }à ’s in Eq. (1)] ^{or }^{to }phrase ^{it }differently, ^{there is }^{a new}
equilibrium ^{value }^{S }^{which the }^{old S }^{is }relaxing ^{to. }^{Viewed }^{from }^{above }^{or }^{below }(along ^{the }layer

or film normal) this would result in an achiral C phase ^{in }^{a }change ^{of the }overall light intensity
passing ^{crossed }polarizers - ^{an }^{effect }^{not }easy ^{to }detect. In a chiral smectic C* phase, ^{however, }^{this}

S relaxation is accompanied by ^{a }rotation of the c -vector due to the dissipative cross-coupling ^{(3.}

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

854

its mean value over the film thickness) ^{is }arbitrary ^{with }respect ^{to }^{a }^{fixed }laboratory ^{frame, }^{the }^{S}
relaxation ^{shows }_{up }^{also in }^{a }change ^{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 }^{smectic }C* 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 }^{the }boundaries, ^{such that }^{the }physical boundaries are in a smectic A phase allowing

for any ^{direction }^{for }c, ^{and }orient the c -vector within the film by ^{an }external 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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