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The smectic A - smectic C phase transition : sense and nonsense

W.J.A. Goossens

To cite this version:

W.J.A. Goossens. The smectic A - smectic C phase transition : sense and nonsense. Journal de

Physique, 1985, 46 (8), pp.1411-1415. �10.1051/jphys:019850046080141100�. �jpa-00210085�

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The smectic A - smectic C phase transition : sense and nonsense W. J. A. Goossens

Philips Research Laboratories, P.O. Box 80000, 5600 JA Eindhoven, The Netherlands

(Reçu le 17 janvier 1985, révisé le 27 février, accepte le 28

mars

1985)

Résumé.

2014

On déduit

un

critère simple pour distinguer la transition smectique A - smectique C de la pseudo- transition, c’est-à-dire la transition qui n’existe pas, smectique A - smectique A incliné. Ce critère est utilisé pour classer et examiner les théories existantes pour la transition smectique A - smectique C.

Abstract.

2014

A simple criterion is presented to discriminate between the smectic A - smectic C phase transition

and the pseudo- (i.e.

non

existing) smectic A - tilted smectic A phase transition. This criterion has been used to

classify and examine the existing theories for the smectic A - smectic C phase transition.

Classification Physics Abstracts

64.70M - 61.30E - 61.30C

1. Introduction.

To describe the second-order smectic A - smectic C

phase transition it has been postulated in 1972 [1] that

the free energy could be written as

with a(T ) oc (T - Tc)Y and b( T)

oc

(T - Tc}y-2/J.

The complex number qf

=

wX + i(oy

= a)

e"’ is the

order parameter of the smectic C phase ; mx and OJy

are the components of the two-dimensional rotation vector w that brings the director from the layer normal

in the smectic A phase to a tilted position in the smectic C phase. The tilt angle w = I ql | is proportional to (Tc - T)O. We do not wish to participate in the

discussion on what the exponents y and ought to be ;

y

=

2 p

=

1 is satisfactory. Since then many theories, based on a molecular field approach, have been proposed to describe and explain the smectic A-smec- tic C phase transition [2-6]. These theories do not

apparently have much in common. In an attempt to

create some order and understanding,

we

will first

show in section 2 that there is

a

rather simple criterion

for the SA-SC (smectic A-smectic C) phase transition

to exist. In connection with that criterion

we

notice the possible existence of

a

tilted smectic A phase, A ;

from the thermodynamic point of view this phase is

the same as the smectic A phase.

In section 3 it will then be shown that the phase

transitions described in [2, 3 and 4] do not exist. The phases described are at best tilted A phases. The phase

transitions described in [5] and [6] do exist. In section 4

we

will show that only the phase transition described in [6] corresponds to the SA-SC phase transition albeit based upon a rather hypothetical potential.

We apologize for the fact that when quoting

formulae from the literature we had to use (in order

to avoid misunderstanding) the original notations, badly attuned to each other, which indeed may be

confusing. The tilt angle however will everywhere be

denoted by

w

as introduced in [1].

2. A criterion for the existence of the SA -Sc phase

transition.

In a molecular field approach the free energy can be written as :

with

and

the molecular field can quite generally be written

as [7],

/

where an

=

3P 2 (COS 0), cos qz and combinations thereof.

The interaction coefficients in equation (5), which

determine the strength of the various interactions,

may or may not be

a

function of the tilt angle cv,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019850046080141100

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1412

dependent on the model used. The bracketi ( )

denote

a

statistical average, that is,

with

where r denotes the configurational coordinates.

The a,, > then constitute the usual order parameters P2(COS 0), ( cos qz > and P2(COS 0).cos qz >,

which in view of the mean field approximation, ( a2 >= a )2, are independent of the tilt, i.e.,

With the above definitions the entropy can be written

as :

Consequently one finds

A potential as defined in equation (5) produces an

entropy that is independent of w, whatever the order parameters are. Consequently a mean field approach

can never produce

a

smectic A-smectic C phase

transition in terms of

w.

A smectic phase, described by the order parameters ( P2 (cos 0) >1 cos qz >

and ( P2 (COS 0).cos qz >, for which

with w, determined by 0 V )lam

=

0, non zero,

must be classified as

a

tilted A phase, A. This phase

constitutes the same thermodynamic phase as does

the A phase, which it replaces; both are described by

the same order parameters. Here we encounter the

same situation as with the cholesteric or twisted nematic phase N* and the nematic phase N; at

a

given temperature the one or the other exists.

It seems reasonable to identify the smectic C1 phase, also denoted as the ordinary smectic C phase,

which has an almost temperature independent tilt angle and no corresponding A phase [8], with the

tilted A phase described above; in view of the thermo-

dynamic relationship with the A phase the latter

name is to be preferred.

Considering the above one has to conclude that the smectic A-smectic C phase transition must be due to the appearance of

a

new order parameter ( X ),

characteristic for the smectic C phase, which drives the transition; this is in agreement with the approach

of [5] and [6]. In addition the tilt must be coupled to

that new order parameter such that 0 V )law

=

0 yields a) oc X )

oc

(Tc - T)1/2 [6]. Then indeed

the free energy can afterwards be written as an expan- sion in 0)2, similar to equation (1), which eventually

may elucidate the meaning of the complex order parameter 1/1. The

«

unusual » smectic C phase or C2 phase [8], which has

a

strongly temperature dependent

tilt angle and a second order phase transition to the smectic A phase, can then be identified with the real C

phase discussed above, which indeed is

a

different

phase than the A phase.

3. Smectic C or tilted A ?

3.1.

-

In reference [2]

a

rather unusual potential

of the mean field type has been proposed, which after

some calculations, leads to the following coupled equations for the smectic order parameter p >>

and the tilt angle 0) :

Here S

=

S(p) is

a

mean field energy due to the smec-

tic order and Kp is connected with the energy due to

fluctuations of that order. The functions G and F, defined as

determine the energy due to the dipole-dipole inter-

action which depends on the tilt.

The above coupled equations, which indeed cons-

titute the base of the theory, can be solved imme-

diately to yield the exact solutions,

that is

and

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There is no phase transition. Even the existence of

a tilted A phase seems unlikely in view of equation (17) ;

the smectic order parameter p increases with increasing temperature.

The phase transition obtained in [2], however,

is the outcome of an unjustified approximation used

to solve the coupled equations (12) and (13). The spirit of the approximation can be illustrated as

follows. Multiply both sides of equation (16) with

kT

=

kTc + k(T - Tc)’ where kTc is defined by G(0) + Kpc = - gkTc/2 f

= -

KkT j2(iJS/iJp)c; this

also defines Pc and implies fK

=

g(iJS/iJp)c. Then

substitute the equations (14) and (15) and neglect

«

higher order » terms, such as g’ (J)2(T - Tc). Then

one finds with fKTc

=

g(iJS/iJp)c Tc, the following equation for (J)2,

leading to ro2(T)

=

ro2.(T - Tr

The longer course actually pursued in [2] was to

obtain from equation (12) an approximate solution for

« p - p, >> and to use that solution in the same approxi-

mate way as sketched above to recalculate

(Kp + G) (OF/Oco) in equation (13). The result is

essentially equation (18), which violates the equality

in equation (16).

3.2.

-

In reference [3] the free energy per molecule is given as :

Here E is put forward as the mean field energy v

=

vnem. + Vm., corrected for the packing entropy of rotationally symmetric, non spherical molecules.

This packing entropy for the nematic phase, as derived

in [9] is given by,

where y and a2 are numbers, dependent on the packing

fraction and the form anisotropy respectively;

P2(cos 0) > with P2 the second-order Legendre polynomial, is the order parameter for the nematic

phase. Then with y.em. > = _ Bo p2(COS 0) >2

one may write [3, 9], Fp

= -

TSP,

where Tp

=

Bo/Z kya2.

Now the averaged mean field ( V... > for the

smectic phase, derived in [3], is, written in shorthand

notation, given by :

where a = (2 A2 - B2 P2 (cos 0) ») P2(cos 0) >, b

=

2 C2 ( P2 (cos 0) ) and

c =

B4 ( P2(COS 8) ) Z are inter-

action coefficients ; ( cos qz ) is the order parameter for the smectic phase and the P.(cos ro), n

=

2, 4,

are Legendre polynomials of order n, describing the tilt-, that is the (u, dependence.

Minimizing V,.. > yields ro2

=

(4/7) (1 + (3(b- a)/

10 c)), which for ro2 > 0 may give rise to a tilted A phase but not to a phase transition.

The appearance of a phase transition in [3] however

is the result of the daring assumption, prompted by

the form of equation (21), that the mean energy

density E,.. > defined by E,.. > = ( V... > +

2 Fp,,..IN, can be obtained from the averaged mean

field ( V... > by multiplying a part of it by (1 + T/Tp),

that is ( E... > is assumed to be :

Minimizing this ( E... > leads to (02(T) = co2.((Te - 1)IT c).(T p/(T p + T)), where T,

oc

Tp oc

Bo, the strength of the interaction giving rise to nema-

tic order.

Equation (23) however implies the introduction of a packing entropy for the smectic phase of the very

special form, Sp

= -

Fp/T

which indeed is the essential part of ( E... > for producing a phase transition. Such packing entropy, however, if it exists, ought to be derived in a proper way, with due regard to the tilt dependence, and not

to be invented on an unjustified analogy of equa- tion (21).

3. 3.

-

In reference [4] the free energy is written as

where F. is in fact the mean potential energy density

E = V(co) >. Based upon assumptions and trun-

cated calculations, the biaxiality y in fact being put

equal to zero, F. is presented as,

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1414

leading to

The arguments concerning the different dependences

of D’ and C’ on qro and therefore on T, q

=

q(T) =

2 n/d can only imply that it is possible to have

a

non vanishing Q)2, i.e. a tilted A phase. The statement that

the above leads to a

Q)

oc (Tc - 7)1/2 is without any foundation as is the statement that it is possible for

a

second order SA -Sc transition to occur if the ratio of the interaction coefficients 14/,2 is large enough.

4. The smectic A - smectic C phase transition.

4. l.

-

The phase transition described in [5] is based

upon a two particle potential of the form,

where

n =

1, 2, 2-bis denotes the three perpendicular

molecular dipoles, Jln

=

(Pn,x, 0, 0), i.e. Jln,x =’ Jln.

Here the X-axis is a molecular axis perpendicular to

the long molecular 3-axis, which is taken along the layer normal, i.e. the Z-axis. The X-axis is then in the

smectic plane. Then pn,x

=

Jln,X.XX

=

It.. cos cp, where Xx

=

X. is the direction cosine of the X-axis with respect to the X-axis. In equation (28) only the inter-

action between dipoles on the same level, defined with respect to the Z-axis, is taken into account. The

neglect of the interaction between dipoles on different

levels seems rather unrealistic and hardly to justify.

However, according to the model, the averaged mean

field can be written as

where vo

=

(nn)312/g . and

The distribution functions fl(2) are defined by

where

The order parameters

a

and P are defined selfconsis-

tently by above equations. There are three ordered

phases, i.e., C with

a =

0, P -# 0, C, with ot :0 0, fl

=

0 and C2 with

a

0 0, p :0 0. We do not wish to

discuss the physical relevance of these dipole-ordered

smectic phases, but only to comment on what happens

when the outboard dipoles P2 have a component Jl2,3

=

6-92 along the molecular 3-axis. In [5] it is

stated that with these dipoles ordered antiparallel in

the smectic plane there

are

two competing torques,

namely 3 fl 6kT,,, and KúJ, where

m

is the tilt angle,

cos m = 3z and K an elastic constant. Equating torques yields

m =

3 fl 6kTr IK oc (T,, - T) . 112 The

existence and special form of these torques have not been argued nor have they been derived; we are afraid

it can not be done. Doing things properly, i.e. repeating

the above derivation with IZ2,X

=

112X’XX + Jl2,3.3x,

there are two possible solutions. If one considers cp

as the azimuthal angle with respect to the layer normal,

i.e. the Z-axis one has xx

=

cos cp cos

m

and 3x

=

cos 9 sin

co.

The relevant part of the averaged mole-

cular field is then given by - vo lt2 #2(COS

ro

+

6 sin ro)2 leading to

(o =

6

=

Jl2;J Jl2.

The tilt angle is a constant and the corresponding phase is a tilted, ordered-dipole phase but not the

smectic C phase. One also may consider 9 as the azi- muthal angle with respect to the long molecular axis,

i.e. the 3-axis ; then Xx

=

cos 9 cos

m

and 3jf

=

sin

m.

For the averaged molecular field

we

now find

-

vo 2(p cos

Co

+ 5 sin ro)2, leading to tg 2

cv =

2 61(p’ _ 6 2) with fl

oc

((T,, - T)/Tc)1/2.

Whatever the meaning of this exotic temperature dependence may be, it seems rather unphysical and is

not in agreement with the law predicted in [1]. There-

fore we conclude that these dipole-ordered smectic phases cannot be classified as the smectic C phase.

4.2.

-

The phase transition described in reference [6]

is based on a steric model giving rise to a two-particle

«

zig-zag interaction » which is assumed to be of the form

Here R., = 31. 1:J1, where 3 and 1:J refer to a molecular fixed coordinate system and a, fl

=

X’, Y’, Z’ refer

to a coordinate system fixed to the smectic layer; the

Z’-axis coincides with the layer normal; 5z,, etc., is

the direction cosine of the 3-axis with respect to the Z’-axis;

r

is the absolute value of the intermolecular vector

r =

(rx,, ry,, rz,). Taking a statistical average with a properly defined one-particle distribution func- tion, the averaged molecular field is of the form

With the transformation Z’ - Z cos

OJ -

Y sin

co

where the Z-axis defines the tilted director one finds

( v(O)(OJ) > = V(O) > + Cl.(o’ and Rz,

z, =

Rzz -

(Rzy + Ryz).(o. Minimizing V yields

(6)

The tilt angle

m

is proportional to the new order parameter ( RzY.cos qZ’ ), which drives the smectic A-smectic C phase transition; then

m oc

(T c - 7-)1/2.

From the thermodynamic point of view the description

and derivation of the phase transition and the tempe-

rature dependence of a) are satisfactory and up to now the only consistent derivation.

Nevertheless there is a serious objection to the physical picture. There is no connection between the

physical model proposed and the potential used.

According to the model the tilting term is due to the zig-zag interaction between zig-zag molecules in the smectic plane, favouring optimum packing in that plane. This interaction, however, is assumed to be

-

A2 r- 2 ra Rp rp, leading to - V2 OC2 cos qZ’ >

x

( Rz, z, cos qZ’) in the smectic phase; it describes the interaction of one molecule with

a

rather symmetric field, A2 oc exp( - (r/rm)2) which becomes propor- tional to the smectic density wave. In spite of the story,

this interaction is not of the zig-zag type between two zig-zag molecules in search of optimum packing. It is

the proper symmetry put in that leads to an appro-

priate solution.

Indeed it can be shown that the order parameter for the smectic C phase that comes to the fore quite naturally is the averaged second-order tensor element

(YZ) >, giving in leading order the invariant 6 1Jz 3z ) = ð sin qJ sin () cos () with 6 - a II

-

al, the molecular anisotropy [9].

5. Conclusion.

Existing molecular theories except for Wulf’s fail to describe the smectic A-smectic C phase transition.

The zig-zag tilting interaction underlying the latter, however, has no conceivable connection with

a

real steric molecular model.

References

[1] DE GENNES, P. G., C. R. Hebd. Séan. Acad. Sci. B 274

(1972) 758.

[2] CABIB, D. and BENGUIGUI, L., J. Physique 38 (1977)

419.

[3] MEER, B. W.

v.

d. and VERTOGEN, G., J. Physique Colloq. 40 (1979) C3-222.

[4] PRIEST, R. G., J. Physique 36 (1975) 437.

[5] MCMILLAN, W. L., Phys. Rev. A 8 (1973) 1921.

[6] WULF, A., Phys. Rev. A 11 (1975) 365.

[7] PRIESTLY, E. B., WOTJOWICZ, P. J. and PING SHENG,

Introduction to Liquid Crystals (Plenum Press, New York, London) 1975.

[8] DE VRIES, A., J. Physique Colloq. 36 (1975) C1-1.

[9] YPMA, J. G. J. and VERTOGEN, G., Phys. Lett. 61A (1977) 45.

[10] GOOSSENS, W. J. A., to be published.

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