Hydrodynamic and electrohydrodynamic properties of the smectic CM phase in liquid crystals

(1)

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Hydrodynamic and electrohydrodynamic properties of the smectic CM phase in liquid crystals

Helmut Brand, Harald Pleiner

To cite this version:

Helmut Brand, Harald Pleiner. Hydrodynamic and electrohydrodynamic properties of the smectic CM phase in liquid crystals. Journal de Physique II, EDP Sciences, 1991, 1 (12), pp.1455-1464.

�10.1051/jp2:1991162�. �jpa-00247603�

(2)

1

l9ys.

II &once 1

(1991)

1455-1464 DtCEMBRE1991, PAGE 1455

Classification

P%ysksAbsmrts

6130-0570-7700

Hydrodynandc and electrohydrodynandc properties

of the smectic CM phase in liquid crystals

Hehnut R. Brand and Harald Pleiner

FB 7, Physik, Universitat Ewen, D 43W Essen I,

liernwny

(Received3 Awe 1991, reared 22

July

1991,

accepted

16

September

1991)

Abstract. We present ^the

hydrodynamic

and

electrohydrodynamic equations

for the

liquid

_crys-

talline smectic CM phase. This phase, which is fluid in the

layers, orthogonal

and

optically

biaxial, has recently been found

experimentally

in polymeric liquid crystals. We compare the macroscopic

properties

of _the CM

phase

with those of the classical fluid smectic

phases, namely

^smectic A, which is uniaxial, and the classical smectic C phase, that is tilted and biaxial. lAh predict that the

in-plane

director in the CM phase can show flow alignment. ^We

point

out that the chiralized version of smectic

CM,

C&,

has electromechanical properties, that _are qualitatively different from those of chiral smectic

C*,

hut rather similar to those of _a classical cholesteric phase. In contrast to

C*,

_C& is not helielectric and thus does not show a linear response to an externally

applied

spatially homogeneous electric field, except for _an electroclinic effect _near

phase

transitions _to _a tilted phase. But similarly to C* and

cholesterics it is

predicted

to show

piezoelectricity.

1, Intndoction.

The field of

liquid crystalline (LC) polymers,

materials

combining

the

properties

of

liquid

_crys-

tals and of

polymers,

ha5 seen a very

rapid development

after

thermotropic liquid crystalline

side-chain

polymers

have been

synthesized

first

[I]

in 1978. In the _case of side~hain

polymers

the

mesogenic

unit is attached to the

polymeric

^backbone ^via a spacer group ^of ^variable

length (compare Refs.[~3]

for _a review of this _new class of

materials),

lb obtain _a

thermotropic

biaxial

nematic

liquid crystal,

side-on side-chain LC

polymers

^have ^been

synthesized

and characterized [4-7~. ^From ^these ^studies not

only

biaxial nematic

phases emerged,

but upon

mixing

^with a low

molecular

weight compound

^also a

layered orthogonal~

fluid and

optically

biaxial

phase

_[6,7~ was

found.

Such _a

phase

has been

predicted

to occur two decades ago

[8],

but until very

recently [6,7]

^there

has been neither _a

report

^of an observation _nor of the theoretical evaluation of the

macroscopic properties

of such a

phase.

In his

textbook~

de Gennes [8]

suggested

to call this

orthogonal,

biaxial

and fluid smectic

phase

^smectic

CM,

where M stands for Mcmillan

[8].

^This CM

phase

has _a

density

wave in the direction of the

layer normal, fi, just

like the classical fluid smectic

phases,

smectic A and smectic C. In _contrast _to smectic A,

however,

which has

complete

two-dimensional

fluidity

(3)

1456 JOURNALDEPHYSIQUEII N°12

~

CTICTZ _III / / /

CIZ CZ3 fill ^II ^II

izg czg ^/ ^/ ^II ^II

CT7 CZl ~/ _II

~

a)

^~ ^"

b)

Fig, I.

a)

^Side.View of smectic CM: the

board.shaped

side-chains _are stacked in layers

(with

normal fi) and orientationally ^ordered ^within ^the layers

(along di);

the backbone is _not shown, b) ^Side-view of smectic C: the rod-like molecules _are tilted away from the

layer

normal towards the I-direction.

inside the smectic

layers

and is thus

transversely isotropic

and

optically

uniaxial, smectic CM has a

preferred

direction in the

layer planes,

^which we denote

by

&. Smectic CM is thus

anisotropically

fluid in the smectic

layers

and

optically

biaxhL In

addition,

the

ground

state of smectic CM ^is invariant under the transformations

fi

_-

-fi

and di _-

-diseparate§1.

It has

locally D2h

_symmetry

[9]:

^{it is}

orthorhombic,

has three

orthogonal

twofold _axes and _a horizontal reflection

plane.

Its

macroscopic properties

can therefore be

expected

tc be

qualitatively

different bom those of the smectic A

phase

dbcussed above and from those of the classical smectic C

phase [8],

which has

locally Cn

_symmetry. In the classical smectic C

phase

the molecules are tilted _on average with

respect

^to ^the

layer

normal

fi

and this tilt b characterized

by

the director fi

(different

from

fi).

In smectic C

only

the sbnultaneous

replacements _[10,11]

d _- -d and

fi

_-

-fi (t

is the

projection

of A onto the

layers,

cf.

Fig. I)

^leave ^the

ground

state invarianL

The smectic CM

phase

is similar _to the hexatic B

phase

^with

respect

to the broken

symmetries

and the number of

hydrodynamic

variables

[12],

^{but is}

difleren~

since the latter is uniaxial due to the sixfold nature of the hexatic bond orientational order

(in

contrast to the twofold _nature of the nematic-like order in the CM

phase).

The tilted hexatic

phases (smectic

F and

I)

are biaxial like the CM

phase,

but

they

have

(in

addition _to

an1h.like)

also _a t-like

preferred

^direction in the

layers

and _are, thus, more similar _to the smectic C than to the CM

phase (compare Ref.[12]

for a

detailed discussion of the

hydrodynamic properties

^of hexatic B and smectic I and

F).

It is the purpose ^{of this} paper ^to

give

a characterization of the

hydrodynamic

and the electro-

hydrodynamic properties

^of the smectic CM

phase

and to compare ^them ^with ^those ^of ^smectic A and smectic C. In addition _we will outline the

propyrties

^of smectic

C&,

a helical

phase

that is obtained when CM ^is chiralized. It turns out that the

macroscopic properties

^of

C&

are

quite

different from those of the helielectric

[8,13]

^smectic ^C*

phase,

but that

they

are rather similar to those of cholesteric

liquid crystals

_[8]

excep~

^df course, for the

presdnce

of the smectic

layering

j~ c*

M'

The

goal

^{of thb} paper ^b to

point

out, how CM can be

distinguished

on the bash of its _macro-

scopic properties

^from ^the ^other ^two ^fluid ^smectic

^phases,

^smectic ^A ^and ^smectic ^C.

Very recently [14]

^we ^have

already given

a mean field

description

of _some

phase

transitions

involving

the _smec.

ticCM

phase.

The characterization of the defect structures that _can _occur in the CM

phase

will be

given

ⁱⁿ ^detail ^elsewhere

[15].

^{In this} paper we

dbregard

^for ^the

^dynamic aspects

^the

polymeric

nature of the CM

phase

found in reference

[7].

(4)

N°12 HYDRODWAMIC PROPERnES OF THE SMECrIC CM PHASE ¹⁴⁵⁷

2.

Hydrodynamics

of the smecdc CM

Phase.

In this section _we dbcuss the

hydrodynamic properties

^of

CM,

I.e. the response ^to

perturbations

of

sullicientfy long wavelengths

and low

frequencies [10,11,16].

^In a first

step

we have _to deter- mine the

hydrodynamic

varhbles. Aside from the conserved

quantities, namely

the

density

_{p, the}

energy

density

_e, and the

density

of linear momentum g, these _are the variables associated with the

spontaneously

broken continuous

symmetries

of the system.

In the last section _we have

already

clarified the invariance

properties

of the

ground

state in smectic

CM. SitnilarJy

as in smectic A and in smectic

C,

_one has _as _an additional

hydrodynamic

variable

[17,10,tl~ll]

the

layer displacement

in the direction of the

layer

normal _u ₌ _u

fi. Inspect- ing

further the

ground

state of smectic CMit is also clear that _one has _one _more

degree

of freedom

as a

hydrodynamic variable, namely

the deviation bm ₌

(fi

_x

di)

.bm of the

in-plane

^director from its

preferred

direction in the

plane.

This varhble is the

in-plane

^rotation

angle characterizing

the

broken rotational synlmetry ⁱⁿ ^the

layer planes.

After

having

^clarified the

question

of the

hydrodynamic

variables

applicable

in the case of smec-

tic CM we make _use of

general

synlmetry arguments

~behaviour

under

parity

and time reversal, Galilean invariance

etc.)

and linear irreversible

thermodynamics [18]

to derive the

hydrodynamic equations

for smectic CM. Close to local

thermodynamic equilwrium

we have the Gibbs relation

Tda ₌ de

pdp v;dg;

4ldu hdm

(2.1)

relating

the

entropy density

_a to the other

hydrodynamic

varhbles. This

equation

defines the ther-

modynamic conjugate quantities temperature T,

^chemical

potential

_p,

velocity

_v;, and the thermo-

dynamic

forces related to the varhbles

characterizing

broken

symmetries, namely

the molecular fields 4l and h.

Using

du and dm instead of the true

hydrodynamic

variables dV;u and

dV;m (via partial integration) simplifies

the

presentation

and b sufficient for the

present

purposes.

The

hydrodynamic

variables

satisfy

the balance

equations

.

b + V;

if

=

R/T (2.2)

#

+

V;g;

₌ 0

(2.3)

#;

+

V;a;;

₌ 0

(2.4)

& ₊ Y'~ ₌ 0

(2.5)

fi + X" ₌ 0

(2.6)

thus

defining

the _currents

if,

_g;, and a;; and the

quasi-currents

Y'~ and X" for the conserved

quantities

and the variables associated with broken

synlmetries,

lb close the system ^of

hydrody-

namic

equations

^b ^then a two

step procedure [10].

^The ^statics

provides

the connection between the

thermodynamic

forces defined via

equation (2.I)

and the

hydrodynamic

varhbles; and in the

dynamics

the currents and

quasi~urrents

^defined ^via

equations (12) (2.6)

are linked ~vith the

thermodynamic

forces.

lb derive the statics it b convenient to start from the

generalized

_energy F

F =

Fo

+

/dr( if ^{((fi V)u)~}

+

(71bp

+

72ba)(fi V)u

+

I' (Vi'l')(NJ'l')

+

DiJk (Vivi ")(Vk'l') (~'~)

+

~(~~ (T7;T7;u)(T7k

Vi

u))

(5)

1458 JOURNAL DE PHYSIQUE II N°12

where Fo ^contains ^all the contributions

already present

ⁱⁿ a

simple

fluid. The

property

tensors

A;;ki, B;;,

and

D;;

in

equation (2.7)

take the form

A;;ki

₌

Aii;I;ikii

+

A2&;&;&k&1+ A3(I;I;&k&1

+

(2.8) B;;

₌

Bi#;#;

+

82&;&;

+

B3i;I; (2.9) D;;k

₌

D(I;&;#k

+

I;&k#;

+

I;&;#k

+

I;&k#;

+

ik&;#;

₊

ik&;#;) (2.10)

where I; ₌

(fi

_x

&);

and where stands for all the other combinations

containing

two l's and two m's. In

writing

down

equations (17) (2,10)

_we have confined ourselves to terms

quadratic

in the variables

(and

thus to linearized

hydrodynamics)

and to lowest

non-vanishing

order in the

gradients.

We _now

briefly

_compare the

generalized

_energy of CM with that of smectic A and C.

The _terms in the first line of

equation (2.7)

_are identical _to those [8]

present

ⁱⁿ ^smectic ^A and

C. Due _to the nonexistence of _an orientational

degree

^of freedom the terms in the second line of

equation (17~

do not exist in smectic A

~ltef.[8]);

in smectic C the tensors B and D have four and _two coefficients

[19,10,8], respectively, reflecting

the fact that the

synlmetry

^{of the} ^classical ^C

phase

^b ^lower

(monoclinic),

^when

compared

to that of the CM

phase (orthorhombic).

The last term of

equation (2.7)

contains in smectic A _one _term

only,

since the

system

^b

uniaxial,

whereas in smectic C there b _an

equal

number of _terms

(3) [20]

^due ^to ^the restriction to

in-plane gradients (gradients along

the

layer

normal

already

_occur to lower

order,

cf, first line of

Eq.(2.7)).

The

thermodynamic conjugates

_can be obtained from the

generalized

_energy

by taking

variational

derivatives,

for

example,

for the molecular fields 4l and h

bfj ₍₂₁₁₎

° j~

and *

bF

(2.12)

~ bm

where the

ellipses

^indicate that all other

hydrodynamic

varhbles _are

kept

^fixed ^while

taking

the varhtional derivative with

respect

to one varaible.

lb close the

system

^of

hydrodynamic equations

we have to link the currents and

quasi~urrents

to the

thermodynamic

forces. We do thb for the reversible

parts (superscript R)

^of the currents

(giving

rise to

vanbhing

_entropy

production,

R

=

0)

and for the irreversible parts

(superscript D)

of the _currents

~yielding positive entropy production,

R >

0) separately.

We obtain

Y'~~

=

A;;V;v; (2.13)

and

a(

=

A;;h (2.14)

where

iii

₌

-)e;>k#k (>(iii>

⁺

iii;) _(2.15)

We thus see that _we have one flow

alignment _parameter

in smectic

CM,

whereas there are two in classical smectic C

~ltefs.[10,21]). Analyzing equations (2.14)

and

(2.15)

_we find that flow

align-

ment _as a consequence ^of

balancing _torques

_on the director field is

posslle

ⁱⁿ ^smectic CM and _we find for the flow

alignment angle

CDS(2e)

₌

(2.16)

(6)

N°12 HYDRODWAMIC PROPERnES OF niE SMECrIC CM PHASE 1459

for _a shear flow inside the

layer planes (I;&;V;v;

₌

const) aligning

the

in-plane

director &.

Equation (l16)

looks

structurally

_[8] identical _to the

expression

for flow

alignment

familiar from uniaxial nematic

liquid c1ystaIs. Intuitively

one can view the situation encountered to be that of

a two-dimensional nematic _as

long

_as the

layers

are

kept

fixed. Thb b the first time that the

posslility

of flow

alignment

_emerges for _a fluid

orthogonal

smectic

phase,

lb demonstrate flow

alignment

in smectic CM

experimentally,

it will be

probably

most convenient to use the geometry ^of

a

freely suspended film,

_a

geometry

^that ^has

already

been found to be useful for the demonstration of flow

alignment

in the classical tilted smectic C

phase [22].

lb derive the

dbsipative

contrlutions _to the currents, we start from the

entropy production

R.

We have for R in smectic CM

2R ₌

/ dr(q;;kiA;;Aki

+ «;;

(V;T)(V;T)

+

7/~h~

₊

(lb~

+

2f4l(fi V)T) (2.17)

where

A;;

₌

)(V;v;

₊

_V; v;).

In

writing

down

equation (2,17)

we have focussed _on the terms to lowest order in the

gradients.

As in smectic A there b _one

cross~oupling

term bebveen

temper-

ature

gradients

and the force associated with the

layer displacemenL

In smectic C there _are _two

such _terms. The contribution of heat diffusion

(«;;

to R has two coelli dents in _a smectic A

phase

[8,10] (uniaxial symmetry),

tlwee in _a smectic CM

(orthorhombic synlmetry)

and four in smectic C

(monoclinic synlmetly) _[10]. Similarly

the vbcous _tensor q~;ki has five coefficients in smectic A

~ltef§.[8,10]),

^nine in CM and thirteen in smectic C

(Ref.[10]).

Director relaxation

brings along only

_one

dissipative

coefficient in C _as well as in CM. Viscous

cross~oupling

terms of the molecu-

lar field assochted with the

in-plane

director _to

temperature gradients

and to the molecular field of the

layer dbplacement

_occur

only

as

higher

order

gradient

terms of the type ^VhVT. ^Other

higher

order

gradient

terms can be constructed

easily along

the lines of reference

[23].

lb

get

the

dbsipative parts

^of ^the currents one takes _a variational derivative of the

entropy production,

for

example

for

Y'~°

_and

X"°

Y'~°

=

()(... ^(2.18)

and

~~ JR

(2.19)

~

b4l~"'

where the

ellipses

indicate that all other

thermodynamic

forces _are

kept

constant,

except

^the one

~vith

respect

to which the variational derivative b taken.

3.

Electrohydrodynamlcs

of the CM

Pham

Until _now all electrical effects have been discarded.

However,

_to describe

electromechanical

_ef-

fects and also

electrohydrodynamical insta§ilities

a

coupling

to the

equations

of

electromagnetism

b

important.

We concentrate here on the effect of electrical fields in the

macroscopic domain,

that is

long wivelengths

and low

frequencies.

But in contrast to pure

hydrodynamics

_we also

keep

the electrical variables

relaxing

in a

large,

but finite time _rc, which _we _assume to be

long compared

to all

microscopic

^collision ^times. ^We ^follow the

procedure

^outlined in references

[8, 24, 25].

First _we discuss the influence of static electric fields in the

generalized

_energy

given

in

equation (2.7).

^{We find} ^the

following

additional contributions

FE

to the

generalized

_energy

FE _"

/dr((;;kE;V;VkU+($E;V;m+e;;E;E;) ^(3.1)

(7)

where the flexoelectric contrwutions assodated with the

layering ((;;k)

and with the

in-plane

director

((Q ⁾

^take ^the ^form

(ijk

_"

(I#I#j #k

+

(2#I fi;

fik +

(3#it;

ik +

(4(#j I;ik

+

#ki;I;

+

(5(#j

_fi;fik ₊

#kfi; fi; (3.2)

and

~ ~ ~

(ij (I 'l'i~j

~

(2 'l'j~i (~.~)

In the electrostatic limit, where curl E ₌ 0,

only

the

symmetrical

forms of

(;;k

and

(Q

enter

(3,I)

and _one _can

put (2

₌

(5, (3

₌

(4, (?

₌

(i'

without loss of

generality. Thus, only

three combinations of the

l's (and

one of the

l'~'s

are

truly

flexoelectric coefficients, while the others are related to

tempor%I changes

^of

magnetic

fields. The dielectric tensor e;; ⁱⁿ

equation (3,I)

h of

orthorhombic form and has thus three different

eigenvalues.

The discussion of the consequences for the orientation in _an external static electric field is thus identical to that

given

for

rectangular

discotics and orthorhombic biaxial nematics

[26]. Comparing

these results with those for smectic A and C we note that there are three

(two)

flexoelectric coefficients associated with the

layering

in

smectic A

~Ref.[17~)

^and ten

(six)

ⁱⁿ ^smectic ^C

(Ref. [27]),

whereas the CM

phase

has five

(three),

where here and in the

following

sentence the numbers in brackets refer to the number of the

truly

flexoelectric

coefficients,

which _are the

only

ones in the electrostatic

approximation,

curl E ₌ 0.

We find two

(one)

flexoelectric coefficients associated with the

in-plane

director in CM, which is also the _case for smectic C

~ltef.[24])

and for unhxial nematics

[8],

while for bhxial nematics

(where

three varhbles of the _m

type exist),

one has six

(three)

[27~.

lb

incorporate

^electric effects into the

dynamics,

the GWbS relation

(2,I)

has to be

supple-

mented

by

the _term

) D;dE;

_on the

right

hand side. The conservation law for the

charge density

reads

#ei

+17;j;

_# 0

(3.4)

and; discarding

the effects of

magnetic

fields in the

following,

we have from Maxwell's

equations

V;D;

₌

4«p~i (3.5)

V _x E ₌ 0

(3.6)

16 lowest order in the

gradients

_we find _no

a§ditional

reversible currents, whereas for the

entropy production

R _we have for the terms assochted with electric effects the contribution

RE

2RE

₌

/ _dr(a(E;E;

+

xi; E;V;T

+

~#;E;4l

+

~fi~(&;I;

+

&;I;)hV;I; _(3.7)

The second rank tensors for the electric

conductivity'(a()

and for

Peltier-type

effects.

(«);)

are

isostructural tc that of thermal

conductivity («;;).

We find that an electric field

couples

dbsi-

patively

in the _same way ^to the molecular field of the

layering

as in smectic A

(Ref. [10]).

As in smectic C

~ltef. [24] ),

we find _a

dissipative cross-coupling

between

spatially inhomogenous

electric fields and the molecular field of the

in-plane

director characterized

by

one coefficient

(~f)

within the

approdmation

curl E ₌ 0.

(8)

N°12 HYDRODWAMIC PROPERnES OF THE SMECrIC CM PHASE 1461

4.

Macroscopic Prope«lesofthec~

Phase.

In _a cholesteric

liquid cqstauine phase

_[8] chiral molecules

spiral

ⁱⁿ a helical fashion about _a

preferred

axis, the Mb of the helix. The director i1 and the Mb of the helix

fi

_are

orthogonal

and

general

_symmetry considerations do not allow for the _occurence of _a

macroscopic polarization

in any

plane perpendicular

to the helical axis.

The situation

changes

when _a smectic C

phase,

in which the molecules _are tilted _on average with

respect

to the

layer

normal, are

chiralize@.

In this _case _a local synlmetry argument

[28,8]

shows that within each smectic

layer

_a

macroscopic polarization

arises in smectic C°. In C° both, the tilted director fi and the

macroscopic polarization

P

spiral

in _a helical fashion around the helical

axis,

which is

parallel

to the

layer

normal

fi.

As a consequence, ^smectic ^{C° is} ^uniaxhl on

length

scales

large compared

to the helical

pitch

and has _no

spontaneous polarization.

After these

introductory

remarks about the cholesteric and chiral smectic C°

phases,

we will now characterize the

macroscopic properties

of the chiral smectic

C~ phase.

As discussed above, CM has

locally D2h

_synlmetry and

represents

a

fluid, orthogonal

and

optically

^biaxial smectic

phase. Upon

chiralization _we _assume the

in-plane

director

fir,

which is

perpendicular

to the

layer normal,

to rotate in _a

spiraling

fashion around the helical

axis,

which is

parallel

to the

layer

_nor-

mal

[29].

Local

symmetry arguments

do not allow for _a

macroscopic polarization

P -in the

layer planes

in this _case. Therefore smectic

C~j

^does ^not ^show an effect that is linear in _an

externally applied spatially homogeneous

electric field in _contrast _to smectic C°.

Correspondingly

the fast

switching

_process between _two _states of different

optical

contrast observed for chiral smectic C*

in thick

samples [30]

^does not exist in smectic C~j.

We _note that _near the

phase

^transition to a tilted smectic

phase

an external static electric field

can

produce

an electroclinic

effect,

I,e, it _can induce _a transition _to _a tilted

phase,

in materials

composed

of chiral molecules. Since _we _are

dealing throughout

the

present

paper ^with ^bulk

properties

of the

phase

_away from any

phase transitions,

_we do _not consider electroclinic effects any ^further.

The introduction of the helix in

Cj~

^is ^thus ^rather ^similar ^to ^the ^helix ⁱⁿ cholesterics. Corre-

spondingly

we find that the

macroscopic properties

of

C~j

are

essentially

_a

supejposition

^of those of cholesterics

(due

to the

helix)

and of those of smectics A

(due

to the

layering).

We present ^here a brief summary ^{of the}

hydrodynamic equations

for

smectiq

_Cj~ valid _on

length

scales

large compared

to the

pitch. Macroscopic equations applicable

on

length

scales small _com-

pared

to the

pitch,

on which

Cj~

^is

biaxia(

_can also be written down

using

the local

approach

as it has been

applied

before to cholesteric

[31-33]

^and ^to ^smectic ^C°

(Ref§. 34, 33] liquid c1ystaIs.

We discuss first the

purely hydrodynhmic equations

without the

coupling

to electric fields. In this case we have, in addition to the conserved

quantities density

_p,

density

of linear momentum g;, and energy

density

_e, two variables

characterizing spontaneously

broken continuous

synlmetries, namely

the combined

displacemint

of the smectic

layers

and the

helix, uA

₌

#; vt,

_and the

displacement

^of ^the ^helix

only,

UC ₊

#; uf.

We note that both varhbles _are identical to the

corresponding

variables in smectic t°

(reference [24]).

^As in cholesterics and in smectic C° a

rotation of the helix about its axis is

equivalent

to a translation

along

this axis.

From this

analysis

it emerges, ^that ^the

hydrodynamic

variables without the

coupling

to electric fields _are identical in smectic C° and in smectic

Cj~.

^We ^thus ^conclude ^that ^section ^{2 of} ^reference

[24], giving

^the

hydrodynamic equations

^for smectic

C°, applies equally

well to smectic

Cj~

^and

can thus be taken _over

unchanged.

This situation

changes,

when _we take into account the effect of electric fields. Then the discussion

given

for smectic C° in section 3 of reference

[Ml

must be modified. First _we discuss static electromechanical effects, I,e,

changes

of the

generalized

_energy F, FE> due to electric fields. We

(9)

find for FE

FE

₌

/ _dr(((~E;V;

_Vk

_uA

+

((~E;V; Vk

UC +

(;;E;V;

_{UC +} _e;;

_E;E; (4.1)

where the flexoelectric tensors

((~

and

((~

^assochted with combined

displacement

and the helix

dbplacement, respectively,

^take the form

(~i

"

(l~~#I#j#k

+

(#~~#ibj~

~

(~'~(#j~(I

+

#k~~) (4.2)

with the transverse Kronecker

symbol

_b(j ₌

_b;; #;#;.

^In the limit curl E

= 0,

only

the combinations

()>~

+

(#'~

_enter

_equation _(4.2).

The

piezoelectric

tensor

£;;

^contains

only

one coefficient

(;;

=

#o(#;#; (4.3)

with qo =

2«/po

where po is the

pitch

of the helix in smectic

Cj~.

^The

piezoelectric

term is thus

isomorphic

to that of cholesteric

[Ml

and smectic C°

(Ref. [24]) liquid crystals.

The dielectric tensor e;; is of uniaxhl

form,

since

Cj~

^is ^uniaxhl on

length

scales

large compared

to the helical

pitch.

We note the absence of the term P E in smectic

Cj~,

since there is, in contrast to smectic

C°,

_no

in-plane equilibrium polarization.

lb

incorporate dynamic

effects due to electric fields _we

proceed

in

analogy

to section 3

far

nonchiral smectic CM.

Equations (3.4) (3.6)

and their treatment can be taken _over

unchanged.

As in sections 2 and 3 we deal With the reversible and the irreversible

parts

^{of the}

dynamics

_sepa-

rately.

For the reversible contributions _to the currents we find

(in

the

approximation

curl E

=

0)

it

=

(;>kqoi7kv; (4.4)

and

aij "

(kij~0Ek (4.5)

with

(;;k

₌

(i#;#;#k

+

+(2(#;b][

+

#kb[J

+

(3#;bS (4.6J Thus,

_as in the _case of smectic C°

~Ref.[24]),

there _are reversible

coupling

terms between the electrical current and

velocity gradients

and vice _versa between electric fields and the stress tensor.

We note that the

coupling

terms are of the same structure as for C°.

For the part ^of ^the

entropy production RE

associated with electric field effects _we have

2RE ₌

/ _dr(a(E;E;

+

«(; E;V;T

+ ~fiA#;E;WA + ~fiCfi;E;_WC)

(4.7)

where the electric

conductivity

tensor

(a(

_and _the _tensor

_containing Peltier-type

effects

(«(;

are

of uniaxial

form,

and where WA" and WC _are the

thermodynamic conjugate

^forces to

uA

_and _UC,

respectively.

As in sections 2 and

3,

the

dissipative

currents can be obtained

by taking

the proper

varhtional derivative of the

entropy production.

From _the

description oftheelectrohydrodynamic properties

of smectic

C~j

we conclude that these _are shnilar to those of

cholesterics, except

^for the

additional

degree

of freedom due to the smectic

layering.

Since in smectic

C&

^the

displacement

of the helix UC is not

coupled rigidly

to rotations of the

polarization

P

(as

for smectic C*

),

both the helix

displacement

and the

polarization

are

independent macroscopic

varhbles.

(10)

N°12 HYDRODWAMIC PROPERnES OF THE SMECrIC CM PHASE 1463

5. Discussion and conclusions.

We have discussed the

hydrodynamics

and

electrohydrodynamics

of the smectic CM

phase

and

we have found that it is in between that for the smectic A and that for the smectic C

phase.

As

a rather

interesting

result it _turns _out that smectic CM _can show flow

alignment

and that it is the first

orthogonal

and fluid smectic

phase

to offer such _a

posslility.

lb

distinguish experimentally

on a

macroscopic

scale, whether _one has a smectic CM

phase

or a

classical tilted smectic C

phase,

it is convenient _to chira&e the material either

by adding

_a chiral

agent

or

by attaching

_an

asymmetric

carbon to the molecule in

question.

In contrast to smectic

C*,

smectic

C&

^does not show _a response ^that ^b ^linear ⁱⁿ an

applied spatially homogeneous

elec-

tric

field,

since its

synlmetry

^does not allow for _an

in-plane polarization

that could be oriented.

Correspondingly

we

predict

that smectic

C~

does _not s~vitch like smectic C° in thick

samples [30].

We have also

pointed

ou~ that smectic

C~j

^is

piezoelectric,

_a

property

^that ^it shares with both, cholesteric

[24]

and smectic C°. Since

piezoelectricity

has been demonstrated to be

easily

ex-

perimentafly

detectable in cholesteric

[35]

^and chiral smectic

liquid crystalline

elastomers

[Ml,

^it

seems most

interesting

to crosslink _a

po1ynler showing

the smectic

C~j phase

and to

investigate

its electromechanical

properties.

In this connection we note, that the extension of

hydrodynam-

ics and

electrohydrodynamics

to

polymeric [37,38]

^and elastomeric

[39,40]

systems ^has ^started

only recently.

We will _use thb

approach

^for ^the

description

of the

dynamic properties

of smectic

po1ynleric

and elastomeric

liquid crystals

^such as

polymeric

smectic CM

~Refs.[6,7])

in the _near future.

Another

interesting possibfiity

to

distinguish experimentally

a smectic CM

phase by polarizing microscopy

from smectic A and smectic C _are the defect structures, which _turn _out to be

qualita- tiveJy

difierenL A detailed

exposition

on this

topic

will be

given

elsewhere

[15].

Acknowledgements.

It is _a

pleasure

for H.R.B, _tc thank Hartmann Leube and Heino Finkelmann for

stimulating

dis- cussions.

Support

of this work

by

the Deutsche

Forschungsgemeinschaft

is

gratefully

acknowl-

edged.

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(A25)

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+

(2fh;

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((3i;fh;

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((si;&;

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