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Effect of elastic degrees of freedom atcolumnar-crystalline phase transition

Yu-Feng Sun, J. Swift

To cite this version:Yu-Feng Sun, J. Swift. Effect of elastic degrees of freedom at columnar-crystalline phase transition.Journal de Physique, 1984, 45 (6), pp.1039-1048. �10.1051/jphys:019840045060103900�. �jpa-00209834�

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Effect of elastic degrees of freedomat columnar-crystalline phase transition

Yu-feng Sun and J. Swift

Department of Physics, University of Texas at Austin, Austin, Texas 78712, U.S.A.

(Reçu le 25 novembre 1983, accepté le 31 janvier 1984)

Résumé. 2014 Un Hamiltonien du type Ginzburg-Landau décrivant des systèmes de molécules en forme de disquedans la phase en colonnes est utilisé pour étudier les propriétés de la transition de la phase en colonne à la phasecristalline. Près de la température de transition et dans la phase en colonnes, l’augmentation de la constanteélastique de courbure est obtenue par un calcul de perturbation. Le mécanisme de couplage avec la jauge, quimène à une transition du premier ordre dans le cas du supraconducteur ou de la transition nématique-smectique,laisse ici une transition continue. Un terme de couplage de la forme trouvée pour les aimants compressibles pourraitmener à une transition du premier ordre pour un couplage suffisamment fort Nos résultats sont brièvementcomparés avec ceux de Kats.

Abstract. 2014 A Ginzburg-Landau Hamiltonian describing systems of disc-like molecules in the columnar phaseis used to investigate the properties of the columnar-crystalline transition. Near the transition temperature in thecolumnar phase, the enhancement of the bend elastic constant is found by a perturbation calculation. The gaugecoupling mechanism which leads to a first order transition in the case of the superconductor or nematic-smectic-Atransition is found to leave the transition continuous here. A coupling of the form found in compressible magnetsmay lead to a first order transition for large coupling. Our results are briefly compared with those of Kats.

J. Physique 45 (1984) 1039-1048 JUIN 1984,

Classification

Physics Abstracts64.70M

1. Introduction.

The existence and the stability of two-dimensionalcrystals in a three-dimensional space is based on Lan-dau-Peierls theorem [1]. This phase of matter wasexperimentally found by Chandrasekhar, Sadashiva,and Suresh [2] in a thermotropic liquid crystal system.The simplest example of such a phase consists of anarray of tubes which are parallel to each other andcomposed of a stack of disc-like molecules with theirnormal directions along the tube axis. In the planeperpendicular to the tubes, there is a two-dimensionallattice structure while there is no long-range order inthe direction parallel to the tubes [3].

In the so-called columnar phase [3], the orderedtwo-dimensional structure is a triangular lattice. Theelastic properties of this phase have been discussed byProst and Clark [4]. Here, we introduce a Ginzburg-Landau free energy which includes the elastic free

energy of Prost and Clark and a coupling of theelastic strains to the order parameter in order todiscuss the transition from the columnar phase to thethree-dimensional crystalline phase [5].

In section 2 we introduce a model which includes acoupling of the gauge field type. In section 3 we usemean field theory to investigate the influence of the

coupling between the order parameter and the elasticdegrees of freedom on the order of the transition. Wealso calculate here the enhancement of the elasticconstants of the columnar phase near the transitionto the crystalline phase. In section 4 we present adiscussion of the transition based upon a renormali-zation group calculation. In section 5, we introducethe coupling mechanism present in compressiblemagnets and discuss the possibility of a first ordertransition for large coupling constant Finally, insection 6 we briefly summarize our results and com-pare them with those of Kats [5].

2. ModeL

The symmetry of columnar phase is D6h and, also, ithas translational invariance along the C6 axis whichis broken at the columnar-crystalline transition. Theorder parameter yy, for the columnar-crystallinetransition then corresponds to the appearance of adensity wave with wave vector parallel to the C6 axis.Or we can introduce a complex order parameter ql as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019840045060103900

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where qo = 2 n/d and d is the lattice constant alongC6 axis in the crystalline phase.

In the columnar state, the mean value of ’1 is equalto zero, but near the phase transition long-wave lengthfluctuations are important, so that t/J can be consideredas a slowly varying function of spatial position.Suppose the phase transition is second order or firstorder with a small latent heat Then near the transi-tion temperature in the columnar state the free energydescribing the fluctuations of the order parameter canbe expanded as

where Cj_, CII’ and b are positive constants anda = a’(T - T,) with Tc being the mean field transi-tion temperature and a’ > 0. (See Eq. (7) in Ref [5].)

In equation (2) n is a unit vector, the director, whichdescribes the local orientation of the tubes. In thecolumnar state n has an average direction parallel tothe C6 axis and we will choose the z axis to be in thisdirection.

We now write down the elastic free energy. If u., and uy are displacements in the plane perpendicular to C6axis then the elastic energy, Fe, will be [4]

where B°, D °, and KO are bare elastic constants. The first two terms correspond to the deformations of thehexagonal lattice [4, 6] and the last term is the bend energy [4]. For our purposes it is not necessary to includethe usual bulk dilation energy and the coupling between the bulk and the tube dilations [4]. We now allow forfluctuations of the director field, denoted by bn = - (6n.,, bny, 0). Here n = no + bn with n being the local directorand no the average director. Then the total free energy can be written as [5]

where a =a+ -L q 2 C11 and b = 2 b.The relations between bn and u we take to be those given by Meyer [7] for an infinite chain polymer nematic,

namely

The relations should be compared with those given by Kats [5] in his equation (15).The coupling between the order-parameter and the fluctuations of the director-field then is included in the

term 21. I (V_L _ iqo 6n) g/ 2 in equation (4). In some other systems (e.g. the super-conductor and N-A system2

[8-10]) a similar coupling may drive the phase transition to be first order and give rise to enhancements of theelastic constants in the vicinity of the transition point These properties will be discussed for our model in thenext two sections.

3. Mean field theory.

First we calculate the enhancements of the elastic constants due to order parameter fluctuations in a meanfield theory. In the hexagonal crystal state, additional first-order elastic energy terms emerge (e.g. (ðzux)2, (O.UY)2)[6], so the second order term which indicates the bend energy in the columnar state (e. g. (ð;ux)2) is hinderedTherefore, if the transition to the crystalline state is continuous, the enhancement of K33 can be expected when Tapproaches T., but no divergence for Bo and D° are expected because (O.,,u Y)2 terms are present in both thecolumnar and crystalline state. These results can be obtained by a perturbation calculation similar to that ofChen and Lubensky [10] for the N-A-C model.

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Using a Fourier transformation, we can rewrite equation (4) as

where

and

where the terms quartic in ’" have been neglected, 1 - d3q/(2 n)3 , and u(r) = u(k) eil.r, etc.In equation (7) the subscripts 1 and 2 refer to the components of vectors in a coordinate system in which

the basis-vector el is parallel to the projection of k in x - y plane, and ê2 is in the same plane but perpendicularto e 1 as shown in figure 1.

In equation (7) a summation convention for the repeated index i from 1 to 2 is implied. Note that becauseof the assumed D6h symmetry of the columnar state the elastic free energy has a simple form in this coordinatesystem [6].

We calculate the enhancements of K3 3, B, and D by treating the terms that appear in F’ as a perturbation.The calculation follows that of Chen and Lubensky [10]. The form of the diagrams and Ward identity is thesame. The only change is that in this case we work with a complex order parameter, and

and

where the subscript naught means a statistical mechanical average with Fo alone occurring in the Boltzmannweight factor.

Equations (8a) and (8b) are to be compared with equations (3. 5a) and (3. 5b) of reference [10]. Our vertexcan be read off from equation (7) and is

where i is 1 or 2. Our computation then follows reference [10] and we find no enhancement for D and B whilethe enhancement of K33 is

where f II is the correlation length parallel to the tubes.To investigate whether the phase transition is first order or second order, in the presence of the order para-

meter-elastic interaction. We will follow the so-called generalized mean field approximation [8].If the temperature T - Tel is not extremely small, we can neglect the fluctuations of ql and suppose I ql I

to be a constant in space. Equation (4) then gives

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Now we define an effective free energy F(gr) by taking the trace over all values of u(k),

Following reference [8], we can show that

Where Q is the total volume of the system, and - - > means a statistical mechanical average with F( 1/1) enteringthe Boltzmann weight factor. From equations (5) and (11), we have

The domain of the integration is taken to be a small cylinder with 2 All as its length and A1. as its radius.We expand the integral in powers of 11/1 I. This yields

where the dots indicate higher order terms. The zeroth order term, a(O), corresponds to a shift in the transitiontemperature in which we are not interested Note that there is no linear term in the expansion of bnl > whichwould give a cubic term in the free energy.

The coefficient of I ql I’ is

where

The integral I can be performed analytically, but for the present we note that it is positive.The coefficient of the cubic term is

If we now use (15) and (13) the effective free energy which involves 4/ alone is

where the renormalized values of a and b are

and

and

where a (2) and a(3) are given in equations (16), (17), and (18).Contrary to the superconductor case [8], in which a cubic term with a negative coefficient appeared in the

effective free energy, a quintic term with a positive coefficient appears here. Thus, if the renormalized value of thequartic term is positive [11], the effect of the coupling between the elastic degrees of freedom and the orderparameter, in mean field theory is to leave the columnar-crystalline transition as second order.

We now turn to a discussion of the phase transition by means of the renormalization group.

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4. Renormafization group calculation

In this section we follow the work of Lubensky and Chen [12] as closely as possible. We employ an aniso-tropic volume-preserving rescaling procedure so that CII = C 1. = 1, and assume and u have the same cutoff

value in d-dimensional wave-vector space. Then we can define a reduced Hamiltonian H - ,2013= with

In equation (23) we have generalized 4/ to have nl2 complex components and a sum over repeated Greek

indices from I to nl2 is implied; 1 == d d q and all wave vector integrations are over a d-dimensional sphericalf, q f (2 7E)d region with radius unity; K3’ ffo, DO are redefined elastic constants after rescaling [13]. We will henceforthignore the bars over K33, B, and D for compactness. Also, u(k) is a d - 1 component-vector in the subspaceperpendicular to C6 axis, and can be resolved as u(k) = u_L(k) k, + u,(k), u _L (k) is the projection of u parallel tokl, and Dr(k) is perpendicular to k 1. and has d - 2 components. Here k.L is the part of the wave vector perpendi-cular to the C6 axis. (See Fig I where k, = ê1 and q is parallel to e2 in three dimensions.) We take the z axisparallel to the C6 and i as an index referring to d - 1 directions in the hyperplane perpendicular to the z axisand j refers to d - 2 directions perpendicular to subspace spanned by the C6 axis and the wave vector.

All the formal diagrams we are taking into account for the renormalization calculations are the same asin reference [12]. The differences are the vertex and propagators which are here

Fig. 1. - The axes of the space fixed coordinate system are labelled x, y, z with z parallel to C6 axis; k is the wave vector;el is the unit vector which coincides with the projection of k onto the x-y plane, and ’2 is in the same plane, but perpendi-cular to ê1.

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and

The scaling relations we use here are analogous to those of reference [12] and are

In (27a) ii and i2 are introduced to keep D and B unchanged through the transition, since according to ourcalculations in section 2 they receive no enhancement from order parameter fluctuation, while p jj and v 1. aredetermined by keeping C 1. = C - 1 under the R.G. transformations. The relation between flu and the restof the exponents can be given by Keeping qo unchanged under the R.G. transformations and is [12]

The dependence of K33 on ç can be obtained from (25) and (27a) and is

The subsequent calculations are analogous to those given in reference [12] with the differences noted above.The recursion relations are obtained by two steps : (1) remove the degrees of freedoms within a thin sphericalshell with its inner and outer radii being bland unity respectively. (2) Rescale all the relative quantities. If wedefine b = e’, then for b greater than but very close to unity, we have the following differential recursion rela-tions :

The exponents tj and p are determined by

where the two integrals /1 and 12 are

and

Now, we investigate the possible fixed-point values of D + B, D and K33 from (30a), (30b), and (30c).q2 q2 q2If we introduce f, = 7r 2 (D -0 +B) " f2 - n ;oD’ and f3 - n 2 qKO 33 then equation (30) yields7T(D-t-z?) 7C D 7T K

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and

We first discuss the fixed points of equation (30) in the subspace spanned by f1 and f2. The following casesarise :

Before going further, it is useful to notice that in this model system, when the fixed point value of D or Bis equal to zero, then its physical value should be zero also if this fixed point can be approached. The reasons areexplained as follows. At T = Tr the real values of the parameters are located somewhere on the critical surfacespanned by all the irrelevant parameters, i.e., the relevant parameters should be equal to zero [14]. Accordingto (30a) and (30b), D and B are strongly relevant if their fixed point values are zero. So, their physical valuesshould be equal to zero at T = T c. Since D and B are supposed to be temperature independent constants throughthe transition, therefore, if their physical values are zero at T = Tc, they will be zero at any temperature.

Bearing this in mind, we are now in a position to discuss the above five cases.The hexagonal structure of the columnar phase corresponds to non-zero D and B. If D and B are zero,

then splay and twist are allowed and these terms should be kept in the free energy. Thus, case 2) in fact corres-ponds to the nematic-smectic A transition in a liquid crystal which was discussed by Lubensky and Chen.

Similarly, in cases 3) and 4) either one of the other of D and B is zero which does not correspond to thehexagonal order of the columnar state. In other words, cases 2) through 4) do not correspond to the columnarsystem and will not be considered further.

Case 5) gives u jj - E 2 . For small s, this is a finite number. However, the renormalization group pro-cedure [12] is to remove the degrees of freedom within a thin spherical shell, as we did above, with the assumptionof p small to first order in B. Since finite p does not correspond to the procedure used to find JlII we will discardthis case.

Now, the only case among the five which can be employed to describe the columnar system is the case 1).In the subspace spanned by fl, f2, and f3, case 1) gives following fixed points :

Case a) is so-called Heisenberg fixed point which corresponds to order parameter being completelydecoupled from the elastic field (i.e. qo = 0 or D, B, and K33 infinite).

The mean field theory as discussed in section 3 would give no enhancement of K33. This fixed point isunstable with respect to turning on the perturbation f3.

In case b), when we put the fixed point values into (32a) and (32b). we find h = 12 = 0, and from (31 a)and (31 b) we get p I I = 0, l1u = 0 (to first order in s). Linearizing (3 3) around this fixed point, we obtain

For s less than 2, this means that this fixed point is stable with respect to perturbations in fl, f2, and f3 abouttheir fixed point values. All the contributions to ro and uo from diagrams involving the u-u propagator areequal to zero. So, the correlation-length exponent v 1. is exactly the same as that in the n-vector model. Thisfixed point is only unstable with respect to a single variable (temperature). So a usual second order phase transi-

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tion is expected with exponents the same as those in the x-y model (n = 2). The physical elastic constant K33would diverge near the transition as

according to equation (29) where ç is the correlation length for the x-y model.

5. Another kind of coupling.

The above discussion is based on the coupling mechanism stemming from the fluctuations of the gauge fieldwhich exists in liquid crystals. In an elastic medium, there exists another general form for the coupling betweenthe order parameter and the elastic continuum which is not included in the above. In the columnar phase, thiscan be taken as

where g is the coupling constantWe now consider the influence of such a coupling upon the columnar-crystalline transition. For simplicity,

we ignore the coupling to the gauge field and consider H;nt alone. After a Fourier transformation, it can berewritten as

In equation (36), 0 is the volume of the system and we have already separated the homogeneous deformations,efg, from the phonon parts of the displacements [15, 16]. Suppose the system has free boundary surfaces. Thenwe can follow Sak [15] and de Moura, Lubensky, Imry and Aharony [16] by integrating out u(k) and e., toobtain an effective Hamiltonian Heff, which is

’

where vo term is the contribution from the homogeneous deformations

and

Now, if we add I v(O) tk.(ql) (qi) t/JP(Q3) t/J1(Q3) to the second term and subtract it from the third inequation (37), with v(O) == -9

2

0 [17], then2(B + D’)

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If we separate the constant v(0) from v(k), equation (40) can be furher written as

where

and we suppose F 0.

Following the similar arguments as in the appendixin reference [16], we can show that

where

Since the Co term is non-analytic it rescales under therenormalization group as [16].

So far our discussions are very similar to that in thesmectic A-smectic C system (see section 5 in Ref [16])but in the A-C system v = 0.For d > 3 (i.e. E 1), Co is irrelevant. Thus, if the

heat capacity exponent a is negative (for n = 2, thisis the case), v turns out to be irrelevant, and thereexists a stable fixed point, with the fixed point valueu* > 0, F* = 0 [15]. For ü > 0, this fixed point canbe approached, so it corresponds to the usual secondorder phase transition. On the contrary, for u = uo -

2(BO 9 + DO) - 0 (if g large enough), this fixed point2( + D )can never be approached In other words, the renorma-lization group transformations run away out of theattractive regions of the fixed point. These conclusionscan be generalized to our case (d = 3, n = 2) if the

influence from Co, which is marginally relevant ford = 3 [16], can be neglected. With this assumption,we get a possible mechanism which gives rise to thefirst order phase transition.

6. Discussion.

The early work on the columnar-crystalline phasetransition was by Kats [5]. His model is different fromours in the consideration of elastic free energy (seeEq. (8) in Ref. [11]). In Kats’ model Kil is infinite sothat the splay deformation is inhibited; however,the twist and the bend deformations still exist Weuse Prost and Clark’s elastic free energy which indi-cates that both the splay and the twist deformationsare hindered by the first order terms. This increasein the cost in energy of elastic distortion when compar-ed to Kats’ consideration leads to a decrease in theeffect of the elastic degrees of freedom on the phasetransition. In particular we find that the transition inour model, which includes a coupling of only thegauge field type, is second order whereas Kats founda first order transition. Moreover, we find that thecritical exponents are unaffected by the coupling ofthe order parameter to the gauge fieldWhen we consider the kind of coupling discussed

in section 5, for small coupling constant (u > 0), thephase transition is second order; whereas for largecoupling constant (u 0) the phase transition isfirst order. We note that the transition was found

experimentally to be first order [2] for the compoundsin which the symmetry of the columnar phase wasapparently hexagonal.

In our calculations we have assumed the symmetryof the columnar phase to be hexagonal. If the symme-try is instead rectangular, then the form of the couplingin equation (35) will be unchanged. However, theform of the elastic free energy will lead, as in refe-rence [16], to anisotropy (in the plane perpendicularto the columns) in the potential OJ introduced in

equations (41) and (42). The effect of such anisotropyis presently under investigation.

Acknowledgments.

This research was supported in part by the RobertA. Welch Foundation under Grant No. F-767.

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References

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[3] PROST, J., Proceedings of the Colloque Pierre Curie(Paris, France IDSET 1982) p. 159.

[4] PROST, J. and CLARK, N., Hydrodynamic Properties ofTwo Dimensionally Ordered Liquid Crystals, pre-print.

[5] KATS, E. I., Sov. Phys. JETP 48 (1978) 916.[6] LANDAU, L. D. and LIFSHITZ, E. M., Theory of Elas-

ticity (Pergamon Press, Oxford) 1970, p. 40.[7] MEYER, R. B., in Polymer Liquid Crystals, Ciferri, A.,

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[12] LUBENSKY, T. C. and CHEN, J. H., Phys. Rev. B 17(1978) 366.

[13] Explicitly,

r0 = (C0)-2 kB T 039Bd-4 bu0 = (C0)-2 kB T 039Bd-4 b

$$

$$$$

$$[14] MA, SHANG-KENG, Modern Theory of Critical Pheno-

mena (W. A. Benjamin, Reading Mass.) 1976,pp. 136 through 143.

[15] SAK, J., Phys. Rev. B 10 (1974) 3957.[16] DE MOURA, MARCO, A., LUBENSKY, T. C., IMRY,

YOSEPH, AHARONY, AMNON, Phys. Rev. B 13

(1976) 2716.[17] IMRY, Y., Phys. Rev. Lett. 33 (1974) 1304.