The mean-field type phase transition from smectic A to smectic C

Physica 127A (1984) 634-645 North-Holland, Amsterdam

THE MEAN-FIELD TYPE PHASE TRANSITION SMECTIC A TO SMECTIC C

FROM

K. ROSCISZEWSKI Institute

of Physics, Jagellonian

University, Reymonta 4, 30-0.59 Krakow, Poland

Received

1 February

1984

The critical properties of the smectic A to smectic C phase transition are analyzed framework of the modified Ginzburg-Landau model -similar to that of Chu and McMillan. The transition shows mean-field type behaviour.

in the

1. Introduction The smectic A to smectic C (A-C) phase transition has received much theoretical and experimental attention. A decade ago de Gennes’) suggested that the phase transition should display XY critical exponents. Most of the following theoretical papers were devoted to the analysis of the Chen-Lubensky model*) for the nematic, smectic A and smectic C phase diagram. In the Chen-Lubensky model the NAC multicritical point is a Lifshitz point3). Some of the new results, however, suggest that the Lifshitz point model is not quite adequate for the satisfactory explanation of the experimental data4.5). Therefore it is reasonable to study other models which have been proposed to explain the NAC phase diagram@). Much work is still needed to examine their physical properties. Presently let us briefly review the earlier theoretical results and the experimental data on the A-C phase transition. Hornreich and Shtritkman”) working with the original de Gennes model showed that a very strong magnetic field should quench the fluctuations of the nematic director field and that the A-C phase transition under these circumstances should have mean-field exponents (with logarithmic corrections). Then, for the Chen-Lubensky model it was found that in zero magnetic field the critical behaviour of the A-C phase transition is that of the XY model in the XY critical region could, however, be d = 3 dimension”,12). Th e asymptotic very narrow. Therefore according to the Ginzburg criterion13) the A-C transition could exhibit a mean-field behaviour14). The second finding was that the nematic splay elastic constant, contrary to common beliefs, does not undergo divergent pretransitional fluctuations in the nematic phase close to the A-C transition12). 0378-4371/84/$03.00 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

MEAN-FIELD

TRANSITION

In this place let us make framework

FROM SMECTIC

a brief interruption

of the models’s*) there

A TO SMECTIC

635

C

and let us note that within

is no such a problem

concerning

the

the splay

elastic constant. Namely in the smectic A and C phases the field, which in the nematic phase is the nematic director field, becomes equivalent to the field of vectors

normal

to the smectic

layers

and

hence

does

not

show

any critical

behaviour. (The tilted position of the molecules within the layers in the smectic C phase is being taken into account by a second order parameter). Returning

to the subject,

data. The A-C exponentPs). To finish

it is necessary

phase transition

this very

short

to refer to some of the experimental

in most cases exhibits resume

let us note

that

nearly there

mean-field are some

critical vague

speculations on the possibility of a Thouless-Kosterlits type phase transition as the likely candidate for the A-C phase transition 19-*l). In the present paper using the simplest and unsophisticated arguments we will study the critical properties of the A-C phase transition in the framework of the models) - very similar to that of McMillan’). We will find mean-field type behaviour at the phase transition in agreement with the experimental data 4.5.14-18). The paper is organized as follows. In the second section we introduce the Ginzburg-Landau Hamiltonian (GLH) for the nematic, smectic A and smectic C phases. In the following part of the paper, by an elimination of certain degrees of freedom of the order parameter fields, we obtain the effective GLH describing the A-C phase transition.

2. The Ginzburg-Landau phases

Hamiltonian

The GLH- W introduced H

ws+ W=kBT=

K=‘,

for the nematic, smectic A and smectic C

in the papers’,**), reads:

w.+ w*+

w&3+

ws.+WIN,,

I dV{Kl(v.n)*+K21n.(vxn)1*+K31nx(vxn)~*}, (2.1)

636

K. ROSCISZEWSKI

(2.1) contd.

W,,,, = i

I

dV{clll(n

*

V - iq&12+ c&tl-

+ c12lO2 * V - iq2)412 + d,l(n

V - iq&12

x V)>‘$l’} ,

where ke is the Boltzmann constant, V is the volume, T is the temperature, II is the nematic director (or the vector normal to the layers in the smectic A and C phases) and ti, r2 are orthogonal unit vectors perpendicular to II. They point along local crystal axes, according to the hypothesis that there is some amount of a local crystal-like short range order present in liquid crystalsaS”); C, coincides with the direction of the tilt in the smectic C phase. The McMillan order parameter p is equal to Poti where in mean-field approximation PO = /tan(w)/ and where w is the tilt angle’). Finally $ is the de Gennes complex order parameter which describes density modulations in smectics’). Within the smectic A phase ti and t2 are fields strongly varying in space but in the smectic C phase their direction should be more or less stabilized. In the following we will study the smectic A to smectic C phase transition. All the calculations will be performed at the smectic C side of the phase transitions. (The nematic to smectic A phase transition was studied in paper 22.) The GLH (2.1) presents many difficult problems. Therefore it is necessary to make numerous assumptions and simplifications. First of all we restrict ourselves to the region on the NAC phase diagram corresponding to the smectic C phase, far away from the NAC multicritical point. Thus we assume that (-a)% 0. The second reasonable assumption is that the field of vectors n (normal to the layers) does not play a significant role in the A-C phase transition (compare ref. 22). We remind that in the smectic C phase II does not coincide with the average direction of the long axes of the molecules. Thus we neglect W. and take A to be parallel to the z-axis. Then following de Gennes’) we neglect fluctuations in the amplitude cc/,,of density

modulations

+=+0e

i0

.

Only fluctuations of @ will be considered. Looking at the form of W, we expect that in the smectic

(2.2)

C phase where the

MEAN-FIELD TRANSITION FROM SMECTIC A TO SMECMC C

63-l

direction of ti is stabilized

@W= qoz+ 91x

47.y +

+

v(r)

(2.3)

9

provided cL is large and d, is very small”). In the mean-field approximation cp = 0 and pi does not fluctuate at all. It is necessary to stress that the form (2.3) does not imply the truly long range crystal-like order in the x, y directions. In reality the fluctuations of cp smear out such an order on a global scale. On the other hand, for small c, and very large d, we expect that even for t, well stabilized the crystal lattice in the x, y directions is virtually non-existent:

W-1 = q0z + v(r) .

(2.4)

Again in the mean-field approximation cp = 0. The possibility (2.4) will be discussed later. Now we will consider the former case. Thus we assume that (2.3) is valid. The phase factor cp takes account of all the density fluctuations. For the model in which c,i = cl2 and qi = q2 we obtain (we recall that c,(q,)* % d,(q,r):

W*-;

I

d V{c,,&VZ’p)* + cJ(V,cp )* + (v,cp )* + 2%(v,cp)(I - o - 7~)

+ 2q*(V,40)(1- (+- r) + 4q?(l-

@)I

+ dl$$(Vz,cp )’ + (Q y + 2(% )@‘y(P) + (VXCP r + (VYP>” + 4q:(V,rp ) + 4q:(V,cp ) + W’,cp )*q: + W,q )*q: + 4(vM)Ql+

(2.5)

4@,(P Y41+ 2q41) 7

t, = (0, TTT, 0) 3 t*=(-qTT,,O).

The model with cl1 f cL2 and q1 f q2 cannot be presented in a transparent way. Nevertheless, it can be shown that small deviations from the limit cl1 = cl2 and q1 = q2 do not change the overall picture presented below. At the proper place we will comment on this problem once more. Now the last term from (2.9, i.e.

can be incorporated

into

W,

and the resulting

mean-field

value of I,/J~is

638

K. ROSCISZEWSKI

(compare

ref. 1,2,7):

t,b;= -6tilb,

h = a + 2d,qf.

As the consequence

(2.6)

of all the above

assumptions

we obtain

(2.7)

%l

12g

---

ha'

41=

4!

46’

Then for Wpm in the case when the direction down the following form:

of t1 is well stabilized

I

W,, = 4 d WLL(V,PCXV,PO) + WzPo)*+ P&(bWpa)

we can write

+ L,,(V,a)*]l , (2.8)

where tl = (u, 7r, 0) = (cos (Y,sin (Y,0) = (1, (Y,0)) t2=(-7r,~,0)=(-sin~,~0~~,0)=(-~,1,0),

where we keep the Einstein summation convention b = x, y, without z), and where to simplify further calculations we assumed L, = L, = L, and L, = L,,=) and where we assumed that v = l(~ = 1 - i7r2 - ir” - . . . ). (For details see the appendix.) The GLH for the smectic

C phase,

incorporating

all the previously

discussed

transformations and approximations, in the momentum space is pretty complicated. Luckily it will turn out that the most complicated terms can be neglected. To anticipate this situation we underline such terms. The GLH form in the momentum space is the following: W=

w,,+ u’@m+ W,+AW,+

w+,= ;

W,,

c (C(K)+fC){&K:+ R,,K;+ h(K: + 64)) > K

MEAN-FIELD

TRANSITION

w, = #,V

c {274474-K,)

Wpn

{(Sd

=

;

2

(S LK I&?+

-

h W,

=

+

iR,vi 2

+

A TO SMECI-IC

+ t~(K1)~(K*)~(K3~(-K1-

K2-

639

C

K3)

+

. ’

*} ,

S,,‘$h(K)~o(-K)

$K

tzK2z)a

&(K)(K,

+ h:~

FROM SMECTIC

-

(KI)~

(K~&I(K~)~I(-

KI

+ $~T(K~)~T(K~)~T(K&T(-K~

(K3)(P

+

4i[KWx(K~x

+

K2x)

K3))

,

(2.9)

-

~2 -

- Hz)

K~ -

+ ku c {- [KIx’W&IX+ Kzx+ K3x) ‘P(K&‘(K2)(P

K2 -

K,,)?T(-K))

b(h)(K~x + K&(Kz)+-KI

x

-

(-

KI +

-

K2 -

K3)

KI~Kz~(KI~

+

K~) +

+

. .

*II

KlyK2yK3y(Kly

K2&P(K$P(K;)(p

+

(-K,

K2y +

-

K3y)]

K2)),

where i2=

-1

0 =

vqoq: >

,

KII =

RL=$

kljqo>

R,,

_

K, =

k&l

(CL=

X,

y) , lK,lls f

9

\‘$I s i >

cl& 0 2 4. (ql) ,

p _ dLd&: 40 ’ I

and where k = (k,, k,, kJ is the normal wave-vector. We assume that the Brillouin zone (BZ) in the dimensionless variables K is the unit cube (compare ref. 22). The sum over a single variable K can be replaced as follows

F

...=

&

d3K.

. . .

I BZ

The Fourier transformed

fields in the

K

space do not carry any extra subscripts.

640

K. ROSCISZEWSKI

The explicitly written real or in K space.

arguments

will distinguish

all gaussian

contributions

to (1 - a) (compare

proportional

A W,, all terms

(in PO), the second,

V,cp, V,cp and also all non-gaussian

we are working

in

about W. (see eq. (2.7)) and W,,. The correspond to Wnrl. The first one, W,,

There are no comments necessary terms W,, A W, and W, altogether collects

whether

terms

linear

in

W,, is the term

in cp. The last one,

the last term in the first square

bracket

in eq.

(2.5)).

3. The elimination

of the 9 variable

We recall that we are studying the area of the phase diagram which corresponds to the smectic C phase. To study the A-C phase transition we eliminate the variable cp and obtain Wefil-the first effective GLH - according to the formula

J 97{p}9{cr}

emw =

where 9{cp} and integrals. Using the obvious

9{a}

J 9{a} denote

emwee

the

(3.1)

normal

Gaussian

measures

in the

field

formula:

(~P(KI)(P(-Kz))~ = &q.q~-~G&d, (3.2) GJKJ’

= (R,,K~ + R&

for calculating

averages

+ P&

+ 64)))

over cp; keeping

only the lowest order

terms is rr (and

in (.y) we obtain:

W ~ITI ^I WI&)+ Gil(~)

I% =

=

R,{2-

&

Wfin(po,a)+; RlG4(~)(~,

c

c

G,l(K)+)a(-K)+

- K~)~+ 3P,R,

Int,

K:(K,

. * .> -

K,)*G~(K)} , (3.3)

d3K$,(K), BZ

Thus we omitted the higher ones.

the K-dependent term of the order i~rrr (fig. lc) and also all In this place it is worth to make an interruption and notice

MEAN-FIELD

TRANSITION

FROM SMECTIC

t

Fig. 1. The Feynman

diagrams contributing

A TO SMECTIC

641

C

-rl

to the effective

Ginzburg-Landau

Hamiltonian.

that if we are working with the model cI f cl2 but still cl1 = cl2 then the result (3.3) is virtually unchanged. (There is a big difference in the K-dependent terms of the order Returning

errs.) to the subject-the

Feynman

diagrams

used

to obtain

(3.3) are

shown at figs. la, b. The neglect of higher order terms in T (U = 1) is justified because the function Gil(~) is positive. Let us note that if d, were equal to zero then G,‘(K) would be zero for ~~~= 0 and K, = -K~. If such were the case indeed (d, = 0) our approximation (3.3) would have been erroneous. In the present case however G,’ > obtain a very small dangerous region in principle destroy the

0. Only if we take the limit lim_KX_ry+OlimK,_, G,‘(K) we value (still positive) of the order PL (or d,). This is a which an other term (see fig. lc) of the order P1 can in validity of our approximation (3.3) and as a consequence

the K-dependent terms of the order ~7~rr and higher ones could not be neglected. Fortunately in this region the graph from fig. lc can be neglected (for K, = -K~ the K-dependent term in front of c~(K&(K&Y(-K~fez) in the formula (2.9) is zero). The other K-dependent terms are not dangerous thanks to the presence of the positive definite terms TSTTT, . . . , from W, (compare (2.9)).

642

K. ROSCISZEWSKI

To finish these considerations

we notice

mass to the field 7~ (or a). This mass is example G;’ is not defined for K = 0). The physical

interpretation

K

that G,’ plays the role of a positive dependent

of the G;’ > 0 condition

and nonanalytic

in

K

(for

is that the crystal-like

lattice in the smectic C phase stabilizes the direction of the tilt (the tilt is a byproduct of the lattice -compare ref. 22). Any variation of t, distorts the lattice and this costs much energy. Once more we stress that G;’ > 0 does not imply truly long range order in the field n. (In certain regions of the K space G;’ is pretty small -of the order d,). If we stick up to the Gaussian approximation for the description of fluctuations of ST (or a) then this is nothing else but the well known spin-wave approximationz4). Now there is a serious question. Is (2.9) good enough for the temperatures very close to the A-C phase transition point? The answer seems to be yes. The reason is that the mass-like term is relatively independent on temperature as long as the direction of t, is at least approximately stabilized (and (-a)% 0). Probably even in the smectic A phase this term does not change very much in comparison to the smectic C phase. If so, it is thus possible to ignore higher order terms in n and in a, to integrate over (Y and to obtain the final form of the effective GLH valid for the entire region of the smectic C phase. The final effective GLH is:

(3.4)

2gefi_ 2g h& 4! -

4,

.

-

4b

-

cud

Int3,

d3/cG,(K)(SIK:

Int3

=

&

J

d3K{G,(fc)(SIrc:

+

s,,K;)

+

s,,K$}’

>

0,

>

0

.

BZ

The Feynman

graphs

used to obtain

(3.4) from (3.3) are shown

at figs. Id, e.

MEAN-FIELD

TRANSITION FROM SMECTIC A TO SMECI-IC C

643

The effective GLH-Wer seems to look like a ordinary Ising-like GLH. One, however, must not make too hasty conclusions. It is necessary to remember that PO in (3.4) is by definition (PO= d/P:+ p’,) positive. This makes a big difference. For PO without any additional condition the GLH (3.4) would indeed have belonged to the Ising universality class. For &SO, however, it is mean-field behaviour which is to be expected (no possibility of two different kinds of domains with opposite (up and down) polarizations; symmetry is broken from the very beginning). Thus we believe that all the experiments should show classical critical exponents at the smectic C side of the transition and that the temperature of the phase transition TAc can be obtained from the equation (3.5) provided the explicit form of fJT) as the function of T is known and provided g,lr>O. For geff= feR= 0 we obtain tricritical behaviour and for geff< 0 we expect a first order phase transition. All this is a simple consequence of a synergetic interplay between the direction of the tilt and the crystal lattice stabilizing crystal in the smectic C phase.

4. The effective GLH for d, large and small c, We recall that for d,(qrr S Gus

W) = 402

+

the formula (2.4) reads:

q(r) .

Thus we should not expect any transverse density modulations (neither globally nor locally). The procedure of obtaining the effective GLH in such a case should proceed via a different way. From the technical point of view it is better to assume that II, = (4j~~ + &) eiqoz

and that & is the fluctuating part of I/J.This assumption allows one to simplify the gradient terms in the GLH (2.1). If cI = 0 we obtain the effective GLH in the universality class of the two component Heisenberg model (just like in refs. 11 and 12). If one, however, includes small non-zero cI then the situation changes. The precise analysis is difficult. The partial calculations indicate that

K. ROSCISZEWSKI

644

the

direction

Heisenberg

of the

tilt is strongly

type form of effective

In conclusion

we see that,

destabilized

and

as a consequence

the

GLH can be lost.

for smectic

substances

which

in the smectic

C

phase do possess distinct transverse density modulations, the phase transition from smectic A to smectic C can be described at the smectic C side of the transition by mean-field theory. For the substances above mentioned class the situation is unclear.

which do not belong

to the

Acknowledgements The author would helpful remarks.

like to thank

Dr. L. Longa

for numerous

discussions

and

Appendix If the fluctuations therP):

WS” -+

where notice

of at are are ignored

the operators that

div and rot are defined

p, Y = x, y (without

only on the x, y-plane.

expressions

(A.11 It is easy to

(A.2)

z) and where

out from the volume

(VXP,)’+ ” = (V&J’ + PWX~)’ + (VP)*1

(Vlp)*+

the z-axis

+ 2 Di ,

is the full divergence which can be dropped Then we notice that

(and similar

II along

I dVW,Wv PI’ + L,(rot PI’+ ~,,(~,p)‘j,

(div p)’ + (rot j3)’ = (V,&)(VJ$) where

and if we take

for y-and

(VX7r)’ = s==

z-derivatives)

(V,(Y)* for ff = 1.

integral

(A.l).

(A.4)

and

(A.5)

To discard in (AS) checked

MEAN-FIELD

TRANSITION

higher

terms

order

is a very crude a posteriori.

FROM

SMECTIC

A TO SMECTIC

and keep only the simplest

approximation.

Whether

it really

gradient works

C

terms

645

of (Y

can only be

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)

P.G. de Gennes, Mol. Cryst. Liq. Cryst. 21 (1973) 49. J. Chen and T.C. Lubensky, Phys. Rev. A 14 (1976) 1202. R. Hornreich, M. Luban and S. Shrikman, Phys. Rev. Lett. 35 (1975) 1678. R. de Hoff, R. Biggers, D. Brisbin, R. Mahmood, C. Gooden and D. Johnson, Phys. Rev. Lett. 47 (1981) 664. C.R. Safinya, R.J. Birgeneau, J.D. Litster and M.E. Neubert, Phys. Rev. Lett. 47 (1981) 668. L. Bengui, J. Phys. (Paris) 40 (1979) C3-419. K.C. Chu and W.L. McMillan, Phys. Rev. A 15 (1977) 1181. K. Rosciszewski, Acta Phys. Pol. A62 (1982) 379. B.W. Van der Meer and G. Vertogen, J. Phys (Paris) 40 (1979) C3-222. R. Hornreich and S. Shritkman, Phys. Rev. Lett. 63A (1977) 39. M.A. de Moura, T.C. Lubensky, Y. Imry and A. Aharony, Phys. Rev. B 13 (1976) 2176. G. Grinstein and R. Pelcovitz, Phys. Rev. A 26 (1982) 2196. V.L. Ginzburg, Sov. Phys. Solid State 2 (l%O) 1824. J.D. Litster, R.J. Birgeneau, M. Kaplan and CR. Safinya, in: Ordering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum, New York 1980) p. 357. J.D. Litster, C.W. Garland, K.J. Lushington and R. Schaetzing, Mol. Cryst. Liq. Cryst. 63 (1981) 145. CR. Safinya, M. Kaplan, J. Ah-Nielsen, R.J. Birgeneau, D. Davidov, J.D. Litster and D.J. Johnson, Phys. Rev. B 21 (1980) 4149. Y. Galerne, Phys. Rev. A 24 (1981) 2284. S. Kumar, Phys. Rev. A 23 (1981) 3207. M. Matushita, J. Phys. Sot. Jap. Lett. 47 (1979) 331. S.T. Lagerwall, B. Stebler, in: Ordering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum, New York 1980) p. 383. D.R. Nelson, in: Fundamental Problems in Statistical Mechanics V, E.G. Cohen, ed. (NorthHolland, Amsterdam, 1980). K. Rosciszewski, Physica 125A (1984) 412. There are reasons to expect that the difference Lt - L2 is an irrelevant variable. Compare a similar problem studied by D. Nelson, R. Pelcovitz, Phys. Rev. B 16 (1977) 2191. V.L. Brezinsky, Sov. Phys. JETP 32 (1970) 493.