Symmetry changes and critical behavior in the smetic BE and HVI phase transitions

Solid State Communications,Vol. 23, Pp. 943—946, 1977.

Pergamon Press.

Printed in Great Britain

SYMMETRY CHANGES AND CRITICAL BEHAVIOR IN THE SMETIC B-E AND H—V1 PHASE TRANSITIONS* A. Michelson and Dario Cabib Physics Department, Technion-lsrael Institute of Technology, Haifa, Israel (Received 25 May 1977 by P.G. de Gennes) We apply for the first time the group-theoretical formalism of Landau’s theory to derive the symmetry changes that may arise in the smectic B—E and H—Vl phase transitions of second order. The critical behavior is found to be Heisenberg-like (n = 3) in the B—E transitions and Ising-like in the H—VT transitions. Criticism of Meyer’s theory of the B—E transition is given.

AS SHOWN by recent experiments [1—6], the smectic B and H phases of certain liquid crystals undergo, at some temperature Ti,, an order—disorder transition to smectic £ and VI phases, respectively. The orientations of the short molecular axes in the plane normal to the long axis (nematic director) which are random in the B phase [5] and, apparently, also in the H phase [1—4]

(b) that the freezing-of-rotation mechanism accounts correctly for the observed symmetry change. An additional purpose of this Letter is to determine the critical behavior in these transitions, as predicted by the renormalization group theory [9]. According to Landau’s theory [8], a second-order phase transition leads from a given space group G0 of

begin to order [7] below T~into a herringbone structure [Fig. 1(a)]. This structure is superposed upon the two-dimensional hexagonal (pseudohexagonal) lattice of the B(H) phase, doubling the unit cell and slightly distorting the lattice. Thus, the above transitions are characterized by a non-trivial change in symmetry, similar to those phase transitions in solid crystals in which superstructures occur. As is known, symmetry changes occurring in secondorder phase transitions in crystals are successfully treated by Landau’s theory [8]. This theory predicts the allowed symmetries below T~,given the symmetry above T~. Though the observed B—E and H—Vl transitions are of first order [3, SJ, they are nevertheless close to secondorder transitions: firstly, the latent heat of these transitions is small compared to other phase transitions in the same liquid crystals [3, 5]; secondly, there is a strong pretransitional effect [3] above T~,characteristic of second-order transitions. In view of these expenmental results, it seems reasonableto apply the grouptheoretical formalism of Landay’s theory, in order to explain the symmetry changes occurring in the above transitions. Such discussion is presented in this letter, Starting from the assumption that the B—E and H—VI transitions are driven by the freezing of molecular rotations and are of second order, we find that the herringbone symmetry is one of the allowed symmetnes of the E and VI phases. This result shows. (a) that the above transitions may in principle be of second order;

the disordered phase to a certain subgroup G1 in the ordered phase. The change G0 G1 is related to a change of some characteristic density function p(r). This change, ~p(r), must transform under the elements of G0 according to a certain irreducible representation (*k, m) of G0 (*k is the star of the representation, m is its index [10]). This representation is said to induce the phase transition in question. As distinct from a solid crystal phase, the relative positions of the smectic layers with respect to one another are random in the smectic phases. Accordingly, the symmetry breaking must be discussed in terms of a single layer and the function ~p(r) must also be restricted to a layer. Then the groups G0 and G1 in question are not the usual crystallographic space groups, but “layer space groups”: we define a “layer space group” as a set of all symmetry operations in three dimensions that carry a given smectic layer into itself (not to be confused with two-dimensional space groups, which do not contain such operations as, for example, reflection in the plane of the layer). To determine the irreducible representations of a “layer space group”, we proceed as follows: complementing this group with a translation in the direction normal to the layer, we construct a crystallographic space group, which we will term the “extended space group”. This group isD~hfor the B phase and C~h(but with a basecentered orthorombic, rather than monoclinic, lattice) for the H phase. The irreducible representations of the “layer space group” are just the irreducible representations of the “extended space group”, with stars *k parallel to the layer. According to Landau’s theory [8], the representation

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______________ *

Supported in part by the Israel Commission for Basic Research.

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943

944 SMECTIC B—E AND H—Vl PHASE TRANSITIONS Vol. 23, No. 12 (*k, m) inducing the phase transition must obey the state) of the short axis parallel to 7’. It is appropriate to following “selection rules”: name ê the “freezing director”. Now, ~p(r) can be (i) The symmetric cube of(*k, m) must not contain written as a tensorial “lattice function” the identity representation of G0 (ii) The antisymmetric square of(*k, m) must not contain any representation in common with the vector representation of G0. Generally speaking, (ii) is not a necessary condition for a second-order phase transition [11]. However, it becomes necessary, if we restrict ourselves (following the experimental data) to those transitions in which the periods of the superstructure arising in the £ and VT phases are commensurable with the periods of the molecular lattice [11] (as distinct, say, from helimagnetics). Rule (ii) restricts the allowed stars *k to the points, F, M, and K of the Brillouin zone for the D~hspace group, and to the points A and F for the C~hspace group with a base centered orthorombic lattice [12]. Besides rules (i) and (ii), an additional “selection rule” for (k*, m) exists [13] due to the following two reasons: (a) ~p(r) is a “lattice function”, i.e. it differs from zero only at the molecular sites; (b) Ap(r) is related to the assumed freezing-of-rotation mechanism of the B £ and H vi transitions. To clarify the second point, let us consider the molecule at the 1-th lattice site. For a unified treatment of the tilted (H’and VI) and non-tilted (B and £) phases, let us introduce fixed (Cartesian unit vectors I and ~ in the plane perpendicular to the nematic director so that I coincides with the vector ~ in the smectic layer (see Fig. 1). Assuming the molecules to be rigid, let us choose a unit vectorj’ =/~I +j~2 along one of the principal short axes of the 1-th molecule (it does not matter which one, but it has to be the same for all the molecules). The orientational ordering of this molecule can be characterized by a two-dimensional second-rank tensor QLe = x/H~cz~), (c~,j3 1,2), where ( ) denotes statistical average. —~

-*

Introducing the two parameters 11/2 1 = [~.

~

(Ql~)2

~‘

J we can write the tensor =

=

~p~(r)

=

~

Q~S(r r,)

(3)

—

where r1 = x1x + y~is the radius vector of the 1-th molecule in the layer, and the summation is over a single layer. The elements of the “layer space group” G0 act on both r and Qj,,~in (3). The set of all functions (3) transforms according to a certain reducible representation fDof this group. Thus the additional “selection rule” for (*k, m) is: (m) (*k m) must be contained in ~I)[13]. The technique of selection of the allowed (*k, m) according to rules (i)—(iii) is the same as in references [14] andB [15]; therefore the we present only the results. For the E transition, only allowed represen-+

tation of the “layer space group” coincides with the representation (*M, 4) ofD~h,and for the H VI transition, there are four allowed representations, which coincide with the representations (* F, 2), (A 1), (*A, 2) and (*F, 1) of C~h(with a base-centered orthorombic lattice) [12]. The representation (*M, 4) inducing the B £ transition is three-dimensional (n = 3), the star *M consisting of the three vectors 2~ k2 (— ~ 5’), k1 = —5, —~

-~

— —~-—

—

—

k =

—

where a is the hexagonal lattice constant. The corresponding ~p~(r) is of the form (3), with 1~~ o~+ I = Qi eik~ri~x Q e

(~~i

(

—

‘~~? I

+Q

3~

—

—

(4)

~

The coefficients Q (i = 1, 2, 3) form the th.ree-component order parameter of the B E transition. The transformations of ~.p~(r) under the elements of G0 reduce to the transformations of Q~according to the represen-

arctan

-~

Q~in the form

Q’(cos 2Ø’o~+ sin 2Ø’a~p)= Q’(e’~e~ —

(2) where o~and o~are the z- and x-matrices of Pauli, and = e’1 I + e~2is a unit vector lying in the plane of rotation and making an angle 0’ with 1. Obviously, Q’ reflects the degree of orientational ordering of the l-th molecule (for free rotation, i.e. at T> Ti,, Q’ = 0 for all 1), and ê’ defines the average orientation (in the ordered

tation (*M, 4). We can now write down the Landau expansion of the free energy including all the fourthorder invariants of the Q’s: F

= ~(7’)

~

Q7 + ~ (u + v~51~)Q?Q?,

(5)

with r(T) = c(T T0), where T0 is the mean-field critical temperature. Mimization ofF for T< 7’0 yields two different structures, depending on the sign of v. If v < 0, —

Vol. 23, No. 12

2

(‘~

SMECTIC B—E AND H—V1 PHASE TRANSITIONS

/ /~~&~:f

.

~

(b)

any 4)

-~

311 (d)

-~

/ / / ~-~4

/---~‘#~

(a)

X1

-

(c)

/

-/ ~

945

/

3

4)

(e)

1

~-

$

(f)

Fig. 1. Orientational order in the smectic E and VI phases. The lines represent the “freezing directors” of the molecules (see the text). For the VI phase,the value of 0 = ir/4 corresponds to the case of complete absence of rotational freezing in the H phase and refers to the plane normal to the nematic director. The structures are as follows: (a) Herringbone structure in the E and VI phases. (b) Another possible structure in the £ phase. The molecules denoted by solid points do not undergo freezing. (c)—(f) Other possible structures in the VI phase. one of the Q1’s differs from zero, and the rest are zero. Taking for defmiteness Q, * 0, Q~= Q~= 0, we obtain for ~ in equation (3): k1. r1..x 2lriy,/a.13_x 6~ Q~ Qi e Q1 e u~. If the 1-th molecule lies in the m-th molecular row tmQparallel to the x-axis, equation (6) yields Q~= (— l) 1 a~. In view of equations (1) and (2), this means that 0’ = 0 + (— l)~ir/2, with 0 = ir/4. Thus, the density function (3) with Q~ given by (6) describes the experimentally observed herringbone structure [Fig. l(a)1. The “extended space group” becomes D~h.The new symmetry admits expansion (or contraction) of the lattice along the x andy-axes. Taking into account the coupling between the order parameter and the lattice strain in the expansion ofF, one can show that (in the mean field approximation) the distortions are proportional to a T. If v > 0, the minimization ofF yields Q~ = = Q~. The corresponding structure is shown in Fig. 1(b): it has not been observed experimentally. In a similar fashion one can show that the one-dimensional (n = I) representation (*A, 2) of C~,,leads to a herringbone structure in the VI phase. Experimentally [4, 51 one does not observe freezing of rotation in the H phase down to the VI phase. However, the symmetry of this phase admits, in general, orientational ordering rn which the vectors ê’ defined above are all parallel to I (or 2). In this case the H VI transition leads to an additional herringbone ordering which is superposed upon the ordering of the H phase, giving rise to 0 ~ ir/4 in Fig. 1(a). The other three allowed representations of C~hlead to orderings shown in Fig. l(c—f); these have not been observed. —

—

—

—

—

To determine the critical behavior in the B—E transition, we shall use the result of Brézin eta!. [16]. According to this work, the isotropic fixed point of the renormalization group equations for an n-component vector model in d = dimensions is stable for n ~ 3. Since in our case n = 3, we infer that the B—E transition should exhibit Heisenberg-like critical exponents. The H—Vl transition (with herringbone ordering) involves n = I ; it should therefore exhibit Ising-like exponents. It should be of great interest to check these predictions experimentally. In conclusion, we have derived the symmetry of the £ and VI phases from the symmetry ofB and H phases. Such derivation is a novel contribution not treated in recent theoretical works [17, 181 discussing these phase transitions. In the papers by Meyer [17, 18], the symmetry of E and VI phases (mainly their two-sublattice structure) was an “input”, rather than “output”, of the theory. Furthermore, because of the inability of the theory [19] of reference [18] to relate the B E transition to a certain irreducible representation of the space group G0 of the B phase, the following incorrect results were obtained there: (1) the number of components of the order parameter was found to be n = 2 instead ofn = 3. On going beyond the mean-field approximation (e.g. on applying the renormalization group theory) this leads to incorrect critical exponents (XY-like instead of Heisenberg-like); —

-+

-+

(2) in the absence of coupling to the lattice, the B E transition was found to be completely isotropic. This leads to an arbitrary angle 0 instead of 0 = ir/4, in Fig. 1; -,

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SMECTICB—EANDH—Vl PHASE TRANSITIONS

(3) a coupling term linear in both the order parameter and the strain components was assumed; this leads to the incorrect conclusion that B—E transitions are necessarily first order. However, such coupling is forbidden by symmetry, because the strain components transform under G0 according to representations with k

=

Vol. 23,No. 12

(4) another conclusion due to the above assumption of linear coupling is that monoclinic distortions of the lattice are possible in the E phase. This conclusion is also wrong, in the light of our results.

0, contrary to the order parameter; REFERENCES

1. 2. 3.

DOUCET J., LEVELUT A.M. & LAMBERT M.,Phys. Rev. Len. 32, 301 (1974). HERVET H., VOUNO F., DIANOUX AJ. & LECHNER R.E.,Phys. Rev. Lert. 34,451(1975). VOLINO F., DIANOUX A.J. & HERVET H., Solid State Commun. 18, 453 (1976).

4.

DVORJETSKI D., VOLTERRA V.& WIENER-AVNEAR E.,Phys. Rev. A12, 681 (1975).

5. 6.

DOUCET J., LEVELUT A.M., LAMBERT M., LIEBERT L. & STRZELECKI L., J. de Phys. (Paris) 36, Cl-13 (1975). DE VRJES A. & FISHEL D.,Mol. Cryst. Liq. Cryst. 16, 311(1972).

7 8.

This type of ordering is commonly referred to as freezing of molecular rotation. LANDAU L.D. & LIFSCHITZ E.M., Statistical Physics ch. XIV. Pergamon, New York (1968).

9.

WILSON K.G. & KOGUT J., Phys. Rept. 12C, 75 (1974).

10

LYUBARSKII G.Ya., The Application ofGroup Theory in Physics. Pergamon, London (1960).

II 12

14

DZYALOSHJNSKII I.E., Soy. Phys.—JETP 19, 960 (1964). HAAS C.,Phys. Rev. l4OA, 863 (1965). The notation of the points in the Brillouin Zone and the enumeration of the irreducible representations is according to ZAK J., The Irreducible Representations ofSpace Groups. Benjamin, New York, (1969). This “selection rule” is a particular case of rule (E) in BIRMAN J.L., Phys. Rev. Lett. 17, 1216 (1966) and similar to the “selection rule” in reference [14]. MICHELSON A.,Phys. Rev. B12, 1071 (1975).

15

MICHELSON A., Phys. Rev. B12, 3964 (1975).

16.

BREZIN E., LE GILLOU F.C. & ZINN-JUSTIN J., Phys. Rev. BlO, 892 (1974).

17. 18.

MEYER R.J.,Phys. Re~.A12, 1066 (1975). MEYER R.J.,Phys. Rei’. A13, 1613 (1976).

19

The Landau theory used in reference [18] is not the one used in this letter, but an earlier theory of antiferromagnetism [CollectedPapers bj’ L.D. Landau, (Edited by TER HAAR D.). pp. 73—76. Gordon and Breach, New York (1967)].

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