Surface ordering and destruction of the first order phase transition smectic A-smectic C* in free standing smectic films

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Surface ordering and destruction of the first order phase transition smectic A-smectic C* in free standing smectic

films

E. Demikhov, U. Hoffmann, H. Stegemeyer

To cite this version:

E. Demikhov, U. Hoffmann, H. Stegemeyer. Surface ordering and destruction of the first order phase

transition smectic A-smectic C* in free standing smectic films. Journal de Physique II, EDP Sciences,

1994, 4 (10), pp.1865-1874. �10.1051/jp2:1994237�. �jpa-00248083�

J. Phv.v. II Fiance 4 (1994) 1865-1874 OCTOBER 1994, PAGE 1865

Cla,,iiication Ph _{i'ii (I} Ah-Iii _a( ii

64.711M 68.15

Surface ordering and destruction of the first order phase

transition smectic A-smectic C* in free standing smectic films

E. Demikhov. U. Hoffmann and H.

Stegemeyer

Univer,ity ol Paderborn, ln,iiiuie oi Phy,real Cheiii,try, 33115 P;tderborn, Germany

(R£>( (»led II M(fi£h 1~94. £i(££>/»e(I _m tin£d trim Iii./iii£> /994)

Abstract. The therniodynaniic behitviour ot the fir,t order phJ,e iritn,ition ,meciic A-,meciic C _in l'ree ,landing film, _i, ,tudied numeric,tlly. Profile, of the tilt angle _acro,, the film, the tree

energy ~inii ave<~tge ,iructural p;tr;ii1Jeier, ((M)) ;tad

(co,

_{(M)) tire} calculated. By dimini,hing the number of ,niectic layer, the phJ,e iritn,iiion di,Jppeitr, ii it criticitl lilm ihicLne,, ((, which _i,

~tpproxiniiitely _one order of'mitgnitude litrgcr ih~tn the peneir;ttion length of the ,urlace induced oriier.

Cunipuier,imulatitJn

re,uli, lie,cribe well _recent

e,perinientitl

ii;iiJ.

Introductit~n.

Free

,t~inding

,mectic film, _are ,tact, of,mectic

layer,

with two free boundarie;. Without _any

complication,

one can

produce

^film, of ?-I ()I)()

layer,,

^which make them _J convenient

ob_ject

for

inve,tigation,

ot'low dimen,ion eiiect,

(1-3

_(, Pha,e tran,ition, and

precritical phenomena

in the iree

,tanding

,niectic iilm, _are iniluenced

by

_a combinatiun oi the ,uriace

orderiig

and finite-,ile effect, _(,ee al,o

general

review, (4, 5ii,

In many ca,e, the

houndary layer,

oi,mectic iiieiihrane, _{are niore} ordered than the

layer,

in

the iiiidiile. Tfie,tructure of'tfie

hour<tory layer, u,uall»

reilect, order

typical

nor the lower temperature

pha,e,

c)I _a

given

material. For

e~ample

the ob,ervation oi,mectic

layer,

in a

,mectic C iiliii ha, been

reported

in (7(. A ,iiectic A-,mectic B

pha,e

tran,ition _in

boundary

layer,

ha, been iound _in

high preci,ion calorimetry

mea,urement, (8(.

In the pre,ent article _we ,hall con,ider,uriace

ordering

eiiect, in the

vicinity

^oi the iir,t

order

pha;e

trJn,ition ,mectic A-,mectic C '..

Meyer

<,t <II

(~(

have ,hown that _in the ;meciic A

pha,e

oi DOBAMBC the ,iiectic C' appear, in

boundary layer,

tar above the

pha,e

tran,ition temperature. The ieniperature

dependence

ot'the order parameter ^and the tran,ition tempera-

ture; are t'ound to be iunction, oi the iilm thickne,,. An other

intere,ting example

oi the

influence of confined diiiien,ion _on the char;tcter of the

pha,e

tran,ition, I; the

recently

ob,erved eiiect ui the

di,appearance

^oi the iir,t order

pha,e

tram,ition SmA-Sm C~

by

dimini,hing

the number oi ,iiiectic

layer,

(10,

11(.

By

iinite-,ize eiiect, the variation ot

thermodyn;tmical

parameter, on

dimini,hing

,y,tem

;ile I, under,toad (,ee ion _{a review}

II ?]).

The _mo,t

important

iinite-,ize eiiect, _are the

rounding

iii the

pretran,itional

characteri,tic, and

,hiiting

iii tfie tram,ition teiiiperature,.

Experimentally

it I, <iiificult _to

di,tingui,h

the finite-,ile effect, iroiii the ,unfree

ordering,

becau,e

u,ually integral propertie,

oi the iilm, _are urea,urea

II II-

In thi, wc)rL we have carried _out nuiiierical calculation, oi the ,uriace

ordering

eiiect, in iree

,tanding

it Ini, in the

vicinity

ⁱⁱⁱ the iir,t orcier

pha,e

tran,ition ,niectic A-,niectic C in the _ca;e oi _a ;trong

anchoring

iii'the tilt

angle

at the

houndary.

It _wa, iound that at ,ante critical thicLne,, _(£l~ the

pha,e

tran;ition

<it,appear,. Average

^,truciural paraiiieter, oi the ,y,tent, vary

continuously

from the smectic A state with surface ordered

boundary layers

to the smectic

C*, which correlates with recent

experiments

lo, II,

13].

The

penetration length

of the

surface order is found to be _one order of

magnitude

smaller than the critical thickness. A

surface induced

phase

transformation in _a thin

boundary region

was found close above the

phase

transition temperature ^{in thick films. In} our work _we show that the observed

experimental

results

[10,

II _can be

explained by taking

surface

ordering phenomena

into account.

Theoretical model.

In _our work _we u,eii _;i contiiuuiii _nie;in iield

lie,cription

iii'tfie t'ir,t order

ph;t,e

tr;tn,ition ,niectic A-;niectic C

.inalogou;

to tfiJt

of']13].

Tfie ,t;tte oi tfie iiIni _wa, cfi;ir~icteriled

hy

_a

<iependence

of'tfie tilt

migle

((M)) on tfie coordinate (=

perpendicular

to tfie _,iiiectic

plane, Fig,

). The

origii

oi coordinate, ha, heen taken at _one oi the houndarie,. To rind ifie

proiile,

oi the tilt

angle

we fiave

nuiiiericall»

iiiiniiiiiled (he li.ee energy oi the l'ilni taken in ;i iorni

coniiiionly

u,ed li>r tfie

de;cription

iii the

pfia,e

ir;in,ition ,iiiectic A-,iiieciic C ~. ]14]

1/

~

=

~

~

L)Z

>

where (I I, tfie thickne,, ot tfie iiIni and

F~

^{I, tfie} part ^{oi tfie li.ee} energy

den,iiy

_per unit _area iii the iiliii, wfiicfi I,

coupled

witfi the tilt oi the directt>r. The tilt

angle

_~,;i, u,eci _{a, an} artier

parJiiieter ior the

lie,cription

oi tfie

pfia,e

tram,ition. The iiee energy

lien,ity

F'~ i,

given hy

the

iollowing e~pre,,ion

>~ i

_(.~2

+ (">~ +

i

("~< +

i

j~">

~

i^~

where _<,

= t~

~ ~

<y i, ;i teiiiperature

independent

con,t;int and J'i I, the t~niperatur~ ^{of the}

T

ab,olute

in,lability

iii the,niectic A

pha,e

with re,peat to the,niectic C '

ph;i,e.

Mininii/Jtion oi the iree energy with re;peat ^tLJ (M) re,ult, _in the

iollowing

Lliiierenti;iI

e~u;itier

ion the

iunction _(M)j=

q

~

+ <, (M) + Ii ("I'+ (Ml'

=

I) 1~)

~=~

There _{are two} adLfitional

bounLf;iry

conLlition,, which _are ,utl'icient to,rive e~uation (3)

h)ji)

=

M)j~ at the ,url'ace iii the l'i[iii iii

lLl(.)

^{= I) in the niiLlLfle iii the l'i [iii j~)}

Lf= _~i1

The iir,t

bounLfary

conLlition nioLlel, the ,trong

michoring.

N° II) SURFACE ORDERING IN FREE STANDING FILMS 1867

x

y

,

8

, ~i

, ZmL

i

to

t~

z.o

Fig, I. Schematic

dr~iwiig

iii the,iruciuie _{i~ ~i} lice,mn<fi~g,meciic A Ii Iii _m ~icciird~mce with re,uli, iii IL)(

To finLf red,enable nuniericJl ,olution,

by

a,iiiiul;iurn

proceLfure

we u,eLf the coel'iicieni,

<y. />, iinLf _g which _were calculated from the

typical

value, of'the

following

parameter,

. jT~ ^T ), where _T~ I, the l'ir,t orLfer

pha,e

tram,itiiin teiiiperature

. the tilt

angle

_F)~ ,ii the iron,turn teiiiperature ^,niectic A-,iiiectic C

. the pen~tratiiin

length

iii ^of the ,unlace inLfuceLl orLfer ;inLf the tran,ition

enthalpy

~l-I.

The ,hilt of the fir,i orLler

pha,e

^tram,ition tempermur~

(l~

^with re,peat to the ab,olute

in,tability

teiiipermure ^al'the ,niectic A

pha,e

II ^' cmi be

e~pre,,eLf by

3]/>]~(

~T T~ ^T 16)

the tilt

angle

at the tran;ition temperature _(M), can be written _;i,

<m)2

)

j7)

~ 4 _<.

In _a iir;t

approximation

the variation oi the tiff

angle

in the

boundary region

can be de,cribed

by

an

exponential

iunctiLJn oi ₌

(M)j= ₌ ^M)j~ + _{(M)jj (M)~} exp(

=lij~

). (8)

where

(M)~ i, the value of the tilt iar away irom the

bounLiary.

ij~ ^I,

given by

ii)

=

~)(

^(~)

A

typical

^{value iii the}

penetration length

I~J, been e,timated in

(13(,

Value, iii the tr;in,ition

enthalpy

have been taken irLJm

(15,

l~(. Nuiiierical par;iureter, ^u,eLf in the ,iniulation

procedure

_are ;ummJrized a, fLJllow,

t~ ₌ 6, _x 11)~ Jni ^~ />

=

(),75 _x

lo"

_{Jiii ~,}

<. = 4,4 _x lll'~ _{Jiii ~, g}

=

6 _x lo ^'~ Jm

(M)jj ⁼ 41i°, T1

=

300 K,

T,

=

301.17 K.

(M), ⁼ ?0°. (ii ₌ 8 _nni, ~Hl bulk1= 4 _x 11)~ Jm _~.

The iiith parameter ^{iii the numerical}

procedure

_wa, the

boundary

contrition ior

(M), In [hi, paper we report J

,impler

ca,e oi the temperature

inLfepenLient boundary

cLJnLfition,.

thereiore _we ha~e taken the tilt

angle

LJn the

boundary

to be _a con,t;int (")j, m (M)~, Numerical

procedure

with teiiiper;ilure

dependent boundary

I; _{now in} prLJgre,,. Differential

e~u;ition

3

was ,olved

according

to _J nunierical

procedure

de,cribeLl in

ii 7(.

Results and discussion.

PROFILES OF THE TILT ANGLE. Prairie, oi the tilt

angle

calculated iur _une half of the

,ample

for three film thicknesses

(400nm~

160nm~ 70nm) _are shown in

figures

2-4. All

plots

correspond

to the minimum of the free energy

(E~,

)) (the ,table ;olution;). In the ,imulation

procedure

the

boundary

condition, for the tilt

angle

were

kept

con,t;int for all temperature, ^and thicLne,,e,,

m.67

3ot.28

o.4

z

/ 100

_nm

Fig. ~. Till migle prattle, _{in ~i} 4()() _nn> t'ilni ~ldili'erenlleniper~ilure,. The iii,i order ph~,e ^irmi,ition Sni A/Siii ^C' i>ccLir, al T~ = I()I ? K.

N° II) SURFACE ORDERING IN FREE STANDING FILMS 1869

Although

we

applied

_a continuou, model ;ometime; we ,hall _u;e

«

layer;

_» 3,5 _nm in

thickne;, I,ee () to illu;trate the relevance of the computer ^{;imulation result,} to known

experiments.

The ;imulation re;ult; for a 400 _nm film _are shown in

figure

~. In the ,mectic A

pha;e

the tilt

angle

decrea;e,

abruptly

in 3-4

layer;

,tarting ^from a value of 6)j, ^to (M) = 0 in the middle of the

film, The

penetration length fjj

^is

independent

of the film thicLne,, and temperature far from

the tran,ition. The

=-dependence

of the tilt

angle

ha, _no

peculiar point, (Fig,

?, T

=

31~.34 and

~0?,84 K).

At temperature; ^{where the ,mectic C~}

pha,e

I, ,table the tilt

angle

^{varie, from}

6)j, to _{a nonzero} temperature

dependent a;ymptotic

bulk value

(Fig.

~, T

=

301? K and

?tl6.67 K). In thi, _ca,e the

pha,e

tran,ition I,

abrupt

and the

profile change;

^from the ;mectic

A _to the ,mectic C" within several tenth, of

degree.

The

pha;e

tran;ition temperature ^I, a

function of the number of

layers

and I; ,hifted

by approximately

1.6 K to the

higher

temperature, in the 70

layer

^{film with} re;pect to the bulk.

An

intere;ting

feature of the 40(J _nm film in the

vicinity

of the transition temperature ^I; a

«,urface

pha,e

turn,ition

» in the ,mectic A

boundary

_region

of'approximately

8

layer,

in

thicLne,, IT

=

3~Jl ?8 K _curve in

Fig.

?). The

dependence

oi the iilt

angle

_on =-coordinate in

thi; _ca,e ha, two inflexion

point,,

which I,

Ljualitatively

diiierent irom the re,ult; at all other temperature;. The

change

^of the 6)

profile,

with temperature ^{indicate; that the}

pha;e

tram,ition

;mectic A-,mectic C~ I, _J di;continuou, _one.

By

definition the fir,t order

phase

tran;ition

take,

place

at the temperature ^where the tilt

angle prof'ile

with two inilection points ^tran;iorm;

into _a monotonou, ;mectic C~

prairie.

The ,imulation re;ult, for _a 160 nm iilm and a 70 nm iilm _in

figure,

3 and 4 ;how that the

tran,ition temperature ^increa,e; and the temperature interval where the

profile

varies from the

,mectic A to the ;mectic C." become, broader

by decrea;ing

the number of

layer,.

In the

vicinity

of'the critical thicLne,, the tilt

angle profile e,,entially depend,

on the iilm thicLne;;.

The

« boundJrie,

» oi the,mectic C ",urface

region

at T

=

31)1.46 K in

iigure

3 _are

practically

not ob,ervable, but the _main ieature of'thi, ,late (two inflexion

point,)

remain,

unchanged.

In

a 7() _nm f'ilm

(Fig.

4) the tilt

angle prairie,

with _two inilection

point,

were not iound and the

pha,e

tran,ition

Lii,appear,.

The ,late of'[hi, f'ilm I,

continuou,ly changed

irom ;mectic A-like

296.67

xi-w

0.2 0.6 0.8

z

/

nm

Fig. 3, Tilt angle prol'ile, _i~,1 161J _nm film I'

= 31)1.4 K,

296.67

3oi.92

31~34 302.84 302.09

0.21 0.28 0.35

z

/ 100

_nm

Fig. 4. Tilt angle profile, in _a 71J _nm film. The ph~i,e iron,ition ,iiectic A-,mectic C di,~ippear, _in ihi, film.

tLJ ,iiiectic C' -liLe

by Lfecre;i,ing

the teiiiperature. ^{For iiliii, thinner than 71)} nm no

Lju;ilitative change,

of the teniper;ilure

depenLfenc~

iii the tilt

angle profile,

were <>b,erved. It I, iiiiportant

to

enipha,ile

that the critical I'[[iii thicLne,, where (hi, behaviour _occur, l'ir,t I, about _one artier tit

iiizgni(uLle larger.

^thmi the

pene1rMiiin length

ij~. ^ThJ, i, a nontrivial re,ult, hecau,e triiiii a

naive

point

of view _one coulLl expect that the thicLne,, where the fir,t order

pha,e

tran,iiion ,mectic A-,iiiectic Cl

Lfi,appear,

,houlLf be

appriixiiiiJtely

eLju;iI to ^~

iii.

^Our ,iniulation de,cribe, _J

Ljualit;itively

diiferent behaviour anLf deiiion,tr;ite, the

iniportance

^{iii the tail, oi the} (M)j=)

prairie,

I;ir iroiii the

boundary.

Let _u, compare our,iiiiulation re,ult, i&.ith the

experinient, reported

_in II), (. The author, iii lo have urea,ureLl the average tilt

angle

in iree

,tanding

iiliii, _{;i, ;i} iunction iii teiiiperature and nuniber LJi

layer,.

^{In the}

optical

thicLne,, oi iiliii, ha, been LfeterniineLf which give, an

average value of _co, (")i=), ii _one

neglect,

^the teiiiper;itune

depenLlence

oi the reiractive inLfex.

llji _~°°o~

~~

~j

+ o~

iA++

o~

§~j ~

^{+ o}

_°o

+ °o

o_~

°o~

+~ °o

~ +

x~

++~

~~

~

+++~

A

o~x~~ ₊₊₊₊

~

_6~ °°o(([[jjxx~

~6666666666666

°°

~DaDDaDaDaDDDDDa°

305 T

/

°C

Fig. 5, Temperature dependence tit the JveiJge tilt angle

((9)

[or film thicLne,,e, 41)li _nni (U), fill _nm (Zii, ~0 _nm lo). 711 nm (x). 5(1nm (+) an~l 30 _nni lo).

N° I() SURFACE ORDERING IN FREE STANDING FILMS 1871

f~~iiiiiiiiiiiiii

A

~jf~~~~~~~~~~~~~

/

^# ~+

~~~ooo

/f

⁺

^~o°

+ ~o

V

_j ~i(R++ _j++

_~o

^~o°

))'++(o°°°

~

~o°

oo

300 305 310

T

/

°C

Fig, 6. Teniperiiture dependence al (he average

(co,

_(")) for l'ilii 1hicLne,,e, 41)() _wt, 161) _nni, X() _nni. 7() _nni, 5() _nni >id 3() _nni. The _,~inie nol~ilion, ;~, in I'[pure 5.

The critical thicLne,, in I()( _i&a,

approxiiiiately

5~ nm and in Ii 315 _nni.

Figure;

5 and 6

,how the calculateLf teiiiperature

LfepenLience,

of

(("))

anLf

(co,

(M)) l'or Lfiiierent iilm

thicLne,,e,. In the thick iiIni, both value, reveal Lfi,continuitie, _at the

pha,e

^trJn,ition

teniperature,. ^In ;i 7() _nni iiliii the

<fi,continuity, Lfi,appear,.

^Our calculation, _{give a} critical thicLne,,, which

L~ualitatively carte,ponLf,

to the ie,ult, of II)(. An morel,e of'the

penetration length

in the nunierical

proceLfure

leLf iii an i~cieJ,e al'the critical thicLne,,, Thu,, the different

critical thicLne,,e, f'ounLl in I()( anti _can be

explmneLl by

Llif'f'erent

penetration length,

oi the ,urface orLfer _at the

bounLiary

_liLjuiLf

cry<,tat-;iir,

Ii I,

inteie,ting

to point out that the critical

thicLne,, liiund in the calculation,

of'([pure,

5 anLf 6 coincide, with that oi

figure,

~-4,

although

the <fef'inition LJi the tram,ition teiiiperature, i, diif'erent, To obtain _a

Ljuantitative de,cription

iii the experiiiient,

of'(

II), the value, of the LanLf;iu

exp;in,ion

coeiiicient, iiiu,t

be known,

FREE _{E~ERI;' DENSITY.} A

,tuLfy

of'tlie temperature

depenLfence

of'tlie free energy

den,ity

give,

an

iiiiport;int

i~f'ormJtion about the

change

of'the chin.deter oi the

ph;i,e

tr;in,ition cau,eLf

by

_~i Lfecrea,e of the l'ilm thicLne,, Fruit the

general theory

^of'the iir,t orLfer tran,ition IX it I,

well-known that the iree _{energy can} be Lle,cribe<f

by

a double well f'unction oi the order

pm.Jiiieter ^{in the}

viciiity

iii the

pha,e

tr;in,ition temperature. ^{The local miiimJ}

corre,pond

to the

ph;i,e, coexi,ting

in the _cour,e al the

pha,e

trill,ition (ii our c;i,e the ,niectic A and C.' ).

At certmn teiiiperature, ^{T ' above the}

pha,e

^tran,ition the Sin C '

mini~ium

di,;ippear,,

The

lower

bounLlm.y

iii the teiiiperature interval where the free energy exhibit; two local minima I, noteLf

by

7 ~. At [hi, teniperature ^the SW A ~ii~iniuni

Lfi,appear,.

The iree energy oi the ,y,tern

varie,

continuou,ly

at the tran,ition poi~t ^IX(, thereiore the condition ior the

pha,e

tr;Ln,itit~n

teiiiperature i, given

by

F, =Fj

II())

Figure,

7a-d

Lfi,plJy

the teiiiperature

LfepenLfence

of'the ~ib,olute iree energy minimum

(F ior iour f~Ini thicLne,,e, 4()(), 161). 7() anLf 50 _nni. In the ca,e iii the fir,t order

pha;e

tran,ition _,mectic A-,iiiectic C' _m the thick iiIni, (4()0 anLf 16(1nit) the fir,t derivative

~z

i, di,continuou,, iihich _i, ,hewn in

figure,

7;i anLf 7b. The location iii the Link

point

d

« «

b b

~

~

hi

~

4

, ,

v- v-

301.25 100

a) b)

« «

b b

I I

4 4

,

~

,

v- v-

305 T/

ci d)

Fig 7, Temperature dependence ^{ot'the free} energy niinmiuni for the following ^l'ilm ihicLne,,e, 4()() _nni (al. 160 _nm (b). 70 _nm (c) and 5() _am (di,

F/(orb. units)

40

_j ^.' ",, 1' ._ 30

1 ~ ~

~ ~~~

_~~

~ 5 6 ~

~

_~

~~l~~

/

₇

Fig. X, Free energy J, a junction oi t'ilm thicLne,, and the average tilt

angle ((")).

N° lo SURFACE ORDERING IN FREE STANDING FILMS 1873

corre,pond;

to the turn,ition temperature. ^The iir,t and the ;econd derivative, oi the iree energy become continuou, at thicLne,;e, lower than </, = 70 _nm

(Fig,,

7c and 7d),

indicating

that the

phase

tran;ition

di,appear,

at thi; thicLne,;. Thi, _{i; an}

expected

re,ult becau,e the ,ymmetry ^{of the whole film} i,

already

broken in the ;mectic A

pha;e

due to the tilted

boundary layer,

and the thicLne,, of the tilted

boundary region

become,

comparable

^with the film

thickne,;

(Fig~.

?-4).

The

di,appeamnce

of the

pha,e

tran;ition I,

qualitatively

illu,trated in

figure

8 where the free energy

den;ity

I; ,hown _{J, a} function of the average tilt

angle

and the film thickne;s at the

tran,ition temperature ^{determined for each thickne;s in} an

analogy

to

figure,

7J and 7b. The

,mectic A and ,mectic C'. _{,tates are}

,eparated by

_{an energy} barrier. The energy barrier

decrea,es and both minima

approach

each other

by

a decrea;e of the number of

layer;.

At the

critical thicLne;s any difference between above mentioned ,tates

di;appear;,

Conclusions.

Surface

ordering

effect; in free

;tanding

film; _are de,cribed

u;ing

a computer ;imulatiLJn ba;ed

on the Landau

theory

with ,trong

boundary

condition;.

By

^the decrea,e of the nuniber oi

layer;

the

pha,e

tran,ition _,mear; out, the tran;ition temperature I; ,hifted to

higher

temperature, ^and the di,continuitie, of the

integral

,tructural parameters

( 6))

and

(co, 6))

_a, well J; the energy

barrier between ;mectic A and ,mectic C '.

pha,e;

become ,mailer.

The fir,t order

pha,e

tran;ition ,mectic A-,mectic C ^'

di,appear;

at ,ome critical thickne;,

which

depend,

on the coeiiicient; of the Landau

expan,ion, e;pecially

_on the value oi the

paranieter g, A nontrivial re,ult oi [hi, paper I, that the critical thickne;, I,

approximately

one

order oi

magnitude larger

than the

penetration length iii

^{oi the ,uriace order}

characterizing

the variation oi the tilt

angle

^{irom it, value} at the

boundary

_6)jj to the bulk

(M)~). In iilm, thinner than the critical thicLne,, all a~er;ige ^,tructural parameter, ^{and the iir,t derivative oi the iree}

energy become continuou, iunctiLJn, LJi temperature, ^the energy barrier between the ,mectic A

anti ,mectic C

pha,e,

and the ,uri;ice

ph;i,e

tran,ition

di,appear,

l'he

computer,imul;ition

re,ult, de,cribe

~ualitatively

cLJrrect recent

experiniental

re,ult,

ill),

II (.

Acknow~ledgement~.

The aufhor, _are

grateiul

to Prof. P. Pieran,Li

(Or,ay)

and Dr. M.

O,ipov (Southampton)

fur

many

,tiniulating

di,cu,,ion,. Thi, work ha, been

,upported by

the Deut;che

For;chung,geiiiein,chait

and the Fond; der Chenii,chen Indu,trie.

Reierences

Pier;~n,Li P.. PI <,/ P/,i _{S,< <,} A 194 119~3 364.

]~] Huang C. C, and Siiebe T. A<fi. Pfivs 4211993j 343

(3 Gee' R.. Sioebe T. ~m(I Huaiig C C., PfiSS Rei E 48 l~~3j 40X.

(4( Bmder K.. Pha,e Tr~in,ition, wit) Critical Phenomena. C. Dr~mb and J. L. Lebr~wiiz ELI,.. Vol.

(Academic Pre,,. London, l~X~) p. 2.

(5] Forgac, G.. Lipow,Ly R anti Nieuwenhuizen Th. M.. Pha,e Tran,ition, mid Critical Pheiiomeiia.

C. Donib ~nd J. L. LebtJwitz Ed,., Vol, 14 (Academic Pre,,. LonLlon, l~~ Ii p. 136.

(6( O,LtJ B. M.. BrJ,l~iu A.. Per,han P. S., Al,-Niel,en J. anti Deui,ch M.. P/JVS Ret L<,11 57 lL)X61

~4.

(7( Siroia E. B.. Per,han P. S., AniadtJr S and Soren,en L. B., Plivs Rei A 35 l~X7j 22X3,

(~( Stoebe T.. Geer R.. Huang C. CT,ind Goodby J. W.. P/Jv.S. R<>I L<>11 69 l~~2j 2(N0.

(~( HeineLamp S., PelLovit, R.. Fonte, E.. Yi Chen E.. PindaL R, and Meyer R., Fiji Rci Leii 52 (t9X4j 1017.

(lo( Bahr Ch. ~md Fliegner D.. P/JVS R<.i A 46 (l~~~) 7657.

II Kr;iu, I., Pier;in,Li P.. DeniiLhov E.. Stegemeyer H.. P/JVS Rci E 48 (lL)~3) l~16.

? Barher M. N.. PhJ,e Tran,turn, and Critical Phenomena, C. Domb and J. L. Lebowitz Ed,.. Vol.

(Academic Pre,,. London, t~X9) p. ^146.

ii3( DemiLhov E.. Krau, I.. Piemn,Li P..

Stegemeyer

H..

Goodby

J.. Sl;iney A. ,ind O,ipov M.. B<>J.

BJIJJ,ciJqe, Fiji., C/J<>fir 97(19L)3)1376.

(14( PILI~ S. A. ind

O,ipov

M. A.. Ferroelecinc Lit~uid

Cry,tal,

lGorLion and Breach.

Philadelphia.

lL)L)I p. ~49,

(15( Bahr Ch.. HeppLe G. and SabJ,chu, B.. LJ<j CJ.i,J<,/ _II 1199~) 41.

[16] Demikhov E,. Kraus I, and Pieran~ki P., ,I. Phi-I- (Condensed Matter) 6 (1994) A415,

(17( Pre,, W. H.. Fl~mnery B, P., TeuLol,Ly ^S. A. ~ind Vetterli~g W. T., Numerical Recipe,

lc~imhridge Univer,iiy Pre,,. C~imhridge, 19L)~) _p. 645.

Xl Binder K., R<>p. ^Pi _<>qi Pfiv, 51) II 9X71 7X3.