We study theoretically the effect of an external field on the
nematic-smectic-A (NA) transition close to the tricritical point, where
fluctuation effects govern the qualitative behavior of the transition. An
external field suppresses nematic director fluctuations, by making them
massive. For a fluctuation-driven first-order transition, we show that an
external field can drive the transition second-order. In an appropriate liquid
crystal system, we predict the required magnetic field to be of order 10 T. The
equivalent electric field is of order $1 V/\mu m$.Comment: revtex, 4 pages, 1 figure; revised version, some equations have been
  modifie

A. Yethiraj

Anand Yethiraj

B. I. Halperin

B. R. Patton

B. S. Andereck

C. Rosenblatt

C. W. Garland

H. Hama

H. Marynissen

I. Lelidis

John Bechhoefer

M. A. Anisimov

N. Goldenfeld

N. Tamblyn

P. E. Cladis

P. G. de Gennes

Ranjan Mukhopadhyay

W. G. Bouwman

English

arXiv

Crossref

Transition

We study theoretically the effect of an external field on the nematic–smectic-A transition close to the

tricritical point, where fluctuation effects govern the qualitative behavior of the transition. An external

field suppresses nematic-director fluctuations by making them massive. For a fluctuation-driven firstorder

transition, we show that an external field can drive the transition second order. In an appropriate

liquid-crystal system, we predict the required magnetic field to be of order 10 T. The equivalent electric

field is of order 1 V�mm

Mukhopadhyay, Ranjan

Yethiraj, Anand

Bechhoefer, John

Memorial University Research Repository

VOLUME 83, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 6 DECEMBER 19994796External-Field-Induced Tricritical Point in a Fluctuation-DrivenNematic–Smectic-ATransitionRanjan Mukhopadhyay, Anand Yethiraj, and John BechhoeferDepartment of Physics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada(Received 24 March 1999)We study theoretically the effect of an external field on the nematic–smectic-A transition close to thetricritical point, where fluctuation effects govern the qualitative behavior of the transition. An externalfield suppresses nematic-director fluctuations by making them massive. For a fluctuation-driven first-order transition, we show that an external field can drive the transition second order. In an appropriateliquid-crystal system, we predict the required magnetic field to be of order 10 T. The equivalent electricfield is of order 1 Vmm.PACS numbers: 64.70.Md, 61.30.Gd, 64.60.KwOne of the seminal developments in the theory ofphase transitions was the understanding of how thermalfluctuations can change the apparent analytic propertiesof the free energy and thereby render the predictions ofmean-field theories invalid. The most widely appreciatedconsequence of thermal fluctuations is the shift in criti-cal exponents from their mean-field values [1]. Anotherkind of consequence, less widely known, was predictedover two decades ago by Halperin, Lubensky, and Ma(HLM) [2]. They argued that the coupling between thefluctuations of a gauge field with the order parameter canconvert a second-order transition to a first-order one (the“HLM effect”). Such fluctuation-induced first-order tran-sitions are expected in two systems: the BCS transition intype-1 superconductors and the nematic–smectic-A (NA)transition of liquid crystals. In superconductors, the HLMeffect is immeasurably weak; however, in liquid crystals,Anisimov, Cladis, and co-workers [3,4] have found ex-perimental evidence that is consistent with it. But, aswe discuss below, that evidence is indirect. In this pa-per, we show that applying a modest external field alongthe preferred orientation of the nematic leads to effectsthat may be unambiguously attributed to the couplingproposed by HLM, providing a direct test of the HLMscenario.In liquid crystals, the HLM effect is sensitive tothe ratio TNATNI [TNA and TNI being the nematic–smectic-A (NA) and nematic-isotropic (NI) transitiontemperatures, respectively]. When the nematic rangeis large, i.e., when the NI transition is sufficiently farfrom the NA transition (TNATNI ø 1), the transition isexpected to be second order. Indeed, experiments on suchsystems yield critical exponents consistent with the 3DXY model [5,6] and show no detectable discontinuities.For small nematic range (TNATNI ! 1), the nematicorder parameter, which increases sharply on coolingbelow TNI, has not yet saturated when the NA transition isreached. The nematic phase is thus only partially orderedat the NA transition, and the emerging smectic orderparameter c is intrinsically coupled to both the nematic0031-90079983(23)4796(4)$15.00order parameter magnitude S (dS-c coupling) and thefluctuations in the director nˆ (dn-c coupling).There is some evidence for a crossover from a first-order transition driven by dS-c coupling to one driven bydn-c coupling, as predicted by the HLM theory. Experi-mentally, TNATNI can be tuned by mixing liquid crystalswith slightly different aliphatic chain lengths. Based onthe dS-c coupling, one expects that, in a mixture withmole-fraction x, the latent heat L at the NA transitiongoes as L ~ x 2 x, where x is the mole fraction fora mixture at the Landau tricritical point (LTP). In onesuch mixture (of two cyanobiphenyls, 8CB and 10CB),Marynissen and co-workers [7] have found such a lineardependence. The extrapolation of this linear behaviorgives the position of the LTP to be at 40% mole fractionof 10CB in 8CB. However, the latent heat does not goto 0 at the LTP. Instead, there is a quadratic crossoverto a smaller value for x , x. Anisimov and co-workersinterpreted this result [3] as evidence supporting the HLMtheory [2]. Independent work measuring the capillarylength of 8CB-10CB mixtures by Tamblyn et al. [8]shows a similar crossover. Moreover, even for pure8CB (x  0), Cladis et al. [4] have deduced from frontpropagation experiments that the transition is first order.This has been confirmed by Yethiraj and Bechhoefer [9],who measured nematic fluctuations directly in real space.Although these experiments suggest the existence of thenonanalytic cubic term in the smectic free energy, theydo not show unambiguously that this effect arises fromthe HLM mechanism. One can directly probe the effectof director fluctuations on the nature of the transitionby expanding the parameter space of the free energy toinclude an external magnetic (or electric) field. As weshall see below, the HLM theory, thus modified, givesrise to a peculiar form for the external-field dependenceof measured quantities. An experimental observation ofthis specific form would be hard to attribute to any othermechanism.In addition, applying an external field affords an experi-mentalist two other opportunities: first, direct suppression© 1999 The American Physical SocietyVOLUME 83, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 6 DECEMBER 1999of fluctuation effects provides a continuously variableparameter with which to study the approach to thetricritical point in a single material. In contrast, each datapoint in mixtures corresponds to a different concentrationand is therefore a different experiment. More important,mixtures may differ in properties other than simply theratio of TNATNI, which complicates the comparison ofdifferent experiments. Second, the external field providesa way of suppressing the anisotropic coupling that givesthe correlation-length exponents at the NA transition theirweak anisotropy. In what follows, we will show that thesubtle fluctuation effects at play at the NA transition canbe tuned by modest magnetic (or electric) fields, makingconcrete predictions that can be checked experimentally.We start with the free energy proposed by de Gennes.Because the nematic phase from which the smectic con-denses is only partially ordered, the free-energy ex-pansion must consider the effects of both nematic andsmectic ordering. As the density modulation in smec-tic liquid crystals is nearly harmonic, one may writerr  r0r 1 1 r1r sinq0 ? r 2 f and define thesmectic order parameter by Cr  r1reifr. Assum-ing that nˆ fluctuates about the z axis, one can write thefree energy as [10]FNA Zd3x fNAC,dn 12Zd3xΩr¯jCj2 1 u2jCj4 1 CkÇ≠C≠zÇ21Cj= 2 iq0dnCj21 K1= ? dn2 1 K2zˆ ? = 3 dn2 1 K3µ≠≠zdn∂2æ, (1)where dn  dnx,dny, 0. We rescale lengths in the zˆdirection relative to other directions so that C  Ck C and thus the three Frank constants K1,2,3 are alsorescaled. Close to the NA transition, r¯ has the usual formaT 2 T0T0. The last three terms in the free energycorrespond to the splay, twist, and bend contributionsto the Frank elastic energy for nematics. Note that wehave not explicitly included the coupling between thesmectic order parameter C and nematic order parameterS, as its main effect is to shift r¯ and u. We thususe effective values of r¯ and u. In the absence of dnfluctuations, u  0 corresponds to the tricritical point, andu . 0 implies a second-order transition. However, whendn fluctuations are taken into account, nothing specialhappens at u  0: we merely cross over from a mean-field first-order transition to a fluctuation-driven first-ordertransition. Here, we assume u $ 0.Next, we consider the effect of an external field alongthe director nˆ (assumed to lie along the z axis). Weassume that the field reinforces the nematic ordering andneglect its much smaller effects on smectic ordering. Forconcreteness, we consider a magnetic field H. Then theLandau free energy becomesFHNA  FNA 2Zd3x12xaH ? nˆ2 FNA 212VxaH2 1Zd3x12xaH2dn2, (2)using n2z  1 2 dn2, and expanding in dn2. Here, Vis the sample volume. Thus, the magnetic field makesthe nematic director fluctuations “massive.” Because thefield also couples to the nematic order parameter S, thefree-energy-expansion coefficients also have a magnetic-field dependence [11,12]; however, one would need afield of several hundred tesla to change u appreciably,which is a much weaker effect than the one we shall beconsidering here.Recall that we are working in the regime wheremassless nematic director fluctuations, by coupling to thesmectic order parameter, induce a first-order transition.Adding an external field adds mass to these directorfluctuations, thus suppressing their effect. When themagnetic field is sufficiently strong, director fluctuationscan be ignored, resulting in a 3D XY , second-order tran-sition. To estimate the required critical field, we recallthat nematic twist and bend distortions are expelled bythe smectic phase over a length scale l (defined to bethe penetration depth). At a mean-field level, and in theone-constant approximation K1  K2  K3  K , thepenetration length is given byl µKC∂12 1q0jC0j . (3)When a field is added, we introduce a new length, themagnetic coherence length jH, which measures thedistance over which elastic deformations decay inthe nematic phase. One finds [10]jH µKxa∂12 1H. (4)At zero field, if the transition is first order, we can imaginesmectic droplets in the nematic phase at the coexistencetemperature. Bulk twist and bend excitations penetrate adistance l into the smectic droplets. When H is turnedon, as long as jH is much larger than l, the nematic-smectic interface is not much affected. But when jH ismuch smaller than l, nematic fluctuations are suppressedin both the nematic and smectic phases. They thenplay no role at the transition, which becomes second-order XY . Thus, a rough estimate of magnetic field Hcneeded to reach the tricritical point can be obtained bysetting jHc  l. In reality, the different values ofK1, K2, and K3 lead to different penetration depths andmagnetic coherence lengths for the twist and bend modes,somewhat complicating the above arguments.4797VOLUME 83, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 6 DECEMBER 1999We can study the effect of a magnetic field withinthe Halperin, Lubensky, and Ma formalism, where fluc-tuations in jCj are ignored (the strongly “type-1” limit).In order to decouple fluctuations in the phase of the or-der parameter, C, from director fluctuations, we carryout the gauge transformation [13], dn  A 1 =L, andc  Ce2iq0L, where = ? A  0. Under this transforma-tion, j= 2 iq0dnCj2 goes to = 2 iq0Acj2. Detailswill be given in a longer paper [14]. Following HLM, wewritee2FckBT ZD A	e2FNAc ,AkBT . (5)Differentiating with respect to jcj givesdfdjcj  r¯jcj 1 ujcj3 1 Cq20jcj A2 . (6)For simplicity, we will assume that we are in the limitK1 ø K2,K3, and, hence, set K1 to zero; however, the4798results would not change much for finite K1. Treating jcjas a constant, we obtaindfdjcj  rjcj 1 ujcj3 2 w1jcjqjcj2 1 a2HH21 w2aH jcjjHj ln"qjcj2 1 a2HH2 1 aH jHj2aH jHj#,(7)where w1  kBTp C32q302K123  1K3 1 1K2,w2  kBTp C32q302K323 , aH pxaCq20,and r corresponds to a shift of r¯ . At this point, itis convenient to introduce the scaled (dimensionless)variables jc 0j  uw jcj, r 0  uw2r , H 0  uaHwH,f 0NA  u3w4fNA. In terms of these variables, thescaled effective free-energy density takes the formf 0 µr 022b4jH 0j∂jc 0j2 1 14jc 0j4 2 13qjc 0j2 1 H 023 1 b2H 02qjc 0j2 1 H 021b2jH 0j jc 0j2 ln"pjc 0j2 1 H 02 1 jH 0j2jH 0j#, (8)where b  K2K2 1 K3, and 0 , b , 1. Qualita-tively, one sees that as H ! 0 there is a negative jcj3term, indicating a first-order transition; for large jHj thelast three terms in Eq. (8) give only corrections to jcj2and jcj4, implying a second-order transition. Thus, weexpect a tricritical point at Hc  c0aH  1aHwu .At the coexistence point, using fjc0j  f0 and dfdjcj cc0  0, we obtainjc 00j2 8<:229bjH 0j1pD9 for 0 , H 0 ,13229bjH 0j2pD9 for13 , H0 , H 0c ,(9)whereD  9bjH 0j 2 22 2 1081 2 3b23 H 021 2 b2 2 2jH 0j . (10)Here H 0c 12 1 2 b2 is the critical field; at H  Hcwe have c0  0 at the coexistence point. Thus, thetricritical point corresponds to H  Hc; for larger mag-netic fields the NA transition is second-order. Our ear-lier informal argument giving lH0  jHc correspondsto H 0c 23 . Note that, despite appearances, c0H is ana-lytic at H 0  13. (See Fig. 1 inset.) The coexistencetemperature rNA satisfiesr 0NA  2jc 00j2 1qjc 00j2 1 H 022 bjH 0j ln"qjc 00j2 1 H 02 1 jH 0j2jH 0j#. (11)Because the spinodal temperature T changes linearlyin jHj, there is a cusp at H  0 in the function t0H.(See Fig. 1.) The behavior of t0H near zero field is thenonanalytic “signature” of the field-driven HLM effectin the same way that a jcj3 term is the signature of thezero-field HLM effect. Recall that only H2 figures in theoriginal free energy. It should also be noted that t0 as afunction of HHc does not depend significantly on b.To estimate the magnitude of magnetic field requiredto drive the transition second order, we consider thematerial 8CB, where the NA transition appears to be inthe HLM fluctuation-driven first-order regime. It is usefulto express Hc in terms of the measured value of t0 at zerofield. In the HLM formalism, t0  2w29au, and wehaveHc ∑98auCq20xa∏121 2 b2pt0  H0pt0 . (12)Using C  2 3 1027 dyn, q0  2 3 107 cm21,xa  1027 cgs, au  1, we estimate H0  35001 2b2 T, which is the field required to quench fluctuationsat molecular scales. Using t0  6 3 1026 [9], we obtainthe critical field Hc  5 10 T. For an electric field, thecritical electric field is roughly 0.5 1 Vmm.These figures are encouragingly low, but one shouldbe cautious since smectic fluctuations, which we haveignored, are important for such weak first-order transi-tions. The calculation of the critical magnetic field is onfirmer ground in the vicinity of the Landau tricritical point(u  0), where the neglect of c fluctuations is more valid.Close to the tricritical point, we retain a yjcj66 term inthe Hamiltonian. Then the critical field for a second-ordertransition is 1aH wy 13. The expression for Hc in thisVOLUME 83, NUMBER 23 P H Y S I C A L R E V I E W L E T T E R S 6 DECEMBER 1999FIG. 1. Plot of the reduced temperature t0H  TNA 2T TNA as a function of the scaled magnetic field H, at b  0.Note the cusp at H  0. In the inset, we plot the (scaled)smectic order parameter at the transition as a function of H.case isHc ∑µay∂12Cq20xa∏12t140 . (13)In the 8CB-10CB system studied by several groups [7,8],the LTP occurs at a mole fraction of roughly 40% 10CBin 8CB. In this system, one of us has measured t0 to beroughly 1024 [15]. Unfortunately, the t140 dependencethen results in a much higher critical field, on the order of300 T (or roughly 30 Vmm).Lelidis and Durand have extensively studied the effectsof large electric fields on the NA transition [16,17].In his Ph.D. thesis, Lelidis looked for evidence of anelectric field-induced tricritical point at the NA transitionof 8CB. The experiments, which measure S, give someevidence for a tricritical point at an external electric fieldsomewhere between 5 and 20 Vmm. Unfortunately, thetemperature resolution was 25 mK. Since the zero-fielddiscontinuity is only 2 mK, this does not rule out amuch smaller critical field for 8CB, and better temperatureresolution will be needed to confirm these results.External fields may have other interesting, observableeffects. In the type-2 limit, where the NA transitionshould be second order, applying a field should be a rele-vant perturbation that changes the universality class ofthe transition to second-order XY . In the superconduc-tor analogy, adding a field in the liquid-crystal systemcorresponds to adding mass to the gauge fluctuations insuperconductors. For massless fluctuations, in type-2 su-perconductors, magnetic vortices are screened by currentloops. The interaction between the current loops (andnot the vortices) is long range, giving rise to the invertedXY transition. However, when the gauge fluctuations be-come massive, the interaction between current loops de-cays exponentially while that between vortices becomeslong range, leading to the usual XY transition. It would bevery interesting to probe the experimental consequencesof this crossover to the XY fixed point.One such consequence would be the suppression ofspatial anisotropy in the critical region. It has beenproposed that experiments probe the crossover regionbetween an isotropic, high-temperature region and thetrue critical region governed by a renormalization groupfixed point [18]. The nature of the fixed point is stillunder debate. Since with a magnetic field we could tunethe strength of nematic fluctuations, it would give us abetter understanding of the role of these fluctuations inthe crossover region, and the observed weak anisotropy.This is currently under investigation.In conclusion, we have shown that the HLM effect forthe NA transition leads to an unusual, nonanalytic formfor the effective smectic free energy in the presence of anexternal field. Measurements by Lelidis and Durand seemconsistent with the predicted effects. More precise experi-ments on the field dependence would be an extremelypromising way to probe these unusual effects of thermalfluctuations.This work was supported by NSERC (Canada). Weacknowledge useful discussions with Ian Affleck andJacques Prost.[1] N. Goldenfeld, Lectures on Phase Transitions and theRenormalization Group (Addison-Wesley, Reading, MA,1992), 1st ed.[2] B. I. Halperin, T. C. Lubensky, and S. K. Ma, Phys. Rev.Lett. 32, 292 (1974).[3] M. A. Anisimov et al., Phys. Rev. A 41, 6749 (1990).[4] P. E. Cladis et al., Phys. Rev. Lett. 62, 1764 (1989).[5] W. G. Bouwman and W. H. de Jeu, Phys. Rev. Lett. 68,800 (1992).[6] C. W. Garland, G. Nounesis, and K. J. Stine, Phys. Rev. A39, 4919 (1989).[7] H. Marynissen, J. Thoen, and W. van Dael, Mol. Cryst.Liq. Cryst. 124, 195 (1985).[8] N. Tamblyn, P. Oswald, A. Miele, and J. Bechhoefer,Phys. Rev. E 51, 2223 (1995).[9] A. Yethiraj and J. Bechhoefer, Mol. Cryst. Liq. Cryst. 304,301 (1997).[10] P. G. de Gennes and J. Prost, The Physics of LiquidCrystals (Clarendon Press, Oxford, 1993), 2nd ed.[11] C. Rosenblatt, J. Phys. (Paris), Lett. 42, L9 (1981).[12] H. Hama, J. Phys. Soc. Jpn. 54, 2204 (1985).[13] B. I. Halperin and T. C. Lubensky, Solid State Commun.14, 997 (1974).[14] R. Mukhopadhyay (unpublished).[15] A. Yethiraj, Ph.D. thesis, Simon Fraser University, 1999.[16] I. Lelidis, Ph.D. thesis, Université de Paris-Sud U.F.R.Scientifique D’Orsay, 1994.[17] I. Lelidis and G. Durand, Phys. Rev. Lett. 73, 672 (1994).[18] See, for example, B. R. Patton and B. S. Andereck, Phys.Rev. Lett. 69, 1556 (1992); B. S. Andereck and B. R.Patton, Phys. Rev. E 49, 1393 (1994).4799

External-Field-Induced Tricritical Point in a Fluctuation-Driven Nematic–Smectic-ATransition

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External-field-induced tricritical point in a fluctuation-driven
  nematic-smectic-A transition

External-field-induced tricritical point in a fluctuation-driven nematic-smectic-A transition

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