It is experimentally demonstrated that a transition from a synclinic to an anticlinic liquid crystal phase occurs via stable domain wall propagation after quenching in a uniform temperature field. Such a one-dimensional growth may be explained in terms of a nonlinear diffusion equation. The experiment provides the first example of free, one-dimensional growth in a system subjected to a pure and uniform temperature field

Fan, S.

Gurarie, Eliezer

Keast, S. S.

Kyu, Thein

Li, Jian-feng

Rosenblatt, Charles

Wang, X. Y.

English

The University of Akron

The University of AkronIdeaExchange@UAkronCollege of Polymer Science and Polymer Engineering5-1998Kinetics of Phase Transition in An Anticlinic LiquidCrystal Induced by a Uniform Temperature Field:Growth in One DimensionX. Y. WangJian-feng LiEliezer GurarieS. FanThein KyuUniversity of Akron Main Campus, tkyu@uakron.eduSee next page for additional authorsPlease take a moment to share how this work helps you through this survey. Your feedback will beimportant as we plan further development of our repository.Follow this and additional works at: http://ideaexchange.uakron.edu/polymer_ideasPart of the Polymer Science CommonsThis Article is brought to you for free and open access by IdeaExchange@UAkron, the institutional repository ofThe University of Akron in Akron, Ohio, USA. It has been accepted for inclusion in College of Polymer Science andPolymer Engineering by an authorized administrator of IdeaExchange@UAkron. For more information, pleasecontact mjon@uakron.edu, uapress@uakron.edu.Recommended CitationWang, X. Y.; Li, Jian-feng; Gurarie, Eliezer; Fan, S.; Kyu, Thein; Keast, S. S.; and Rosenblatt, Charles, "Kinetics ofPhase Transition in An Anticlinic Liquid Crystal Induced by a Uniform Temperature Field: Growth in OneDimension" (1998). College of Polymer Science and Polymer Engineering. 66.http://ideaexchange.uakron.edu/polymer_ideas/66AuthorsX. Y. Wang, Jian-feng Li, Eliezer Gurarie, S. Fan, Thein Kyu, S. S. Keast, and Charles RosenblattThis article is available at IdeaExchange@UAkron: http://ideaexchange.uakron.edu/polymer_ideas/66VOLUME 80, NUMBER 20 P HY S I CA L REV I EW LE T T ER S 18 MAY 1998Kinetics of Phase Transition in an Anticlinic Liquid Crystal Induced by a UniformTemperature Field: Growth in One DimensionX.Y. Wang,1 Jian-feng Li,2,* Eliezer Gurarie,2 S. Fan,1 T. Kyu,1 M. E. Neubert,3 S. S. Keast,3 and Charles Rosenblatt21Institute of Polymer Engineering, University of Akron, Akron, Ohio 443252Department of Physics, Case Western Reserve University, Cleveland, Ohio 441063Liquid Crystal Institute, Kent State University, Kent, Ohio 44242(Received 27 October 1997)It is experimentally demonstrated that a transition from a synclinic to an anticlinic liquid crystalphase occurs via stable domain wall propagation after quenching in a uniform temperature field. Sucha one-dimensional growth may be explained in terms of a nonlinear diffusion equation. The experimentprovides the first example of free, one-dimensional growth in a system subjected to a pure and uniformtemperature field. [S0031-9007(98)06141-9]PACS numbers: 64.70.MdLiquid crystals are among the most suitable materialsto mimic the dynamic characteristics of pattern formationand a selection mechanism. Because of their stronginteraction anisotropy, dynamic growth processes maybe limited to one or two dimensions. Many fascinatingphenomena in liquid crystals have been investigated inthe past decade [1–4], including oscillating and chaoticpatterns of nematics during directional solidification [5],spiral wave growth in quenching of cholesterics [6], anddendrite formation of discotic liquid crystals [7]. Clearly,use of these materials has fostered considerable progressin understanding many nonlinear phenomena.Compared with directional solidification [8] subjectedto a temperature gradient, free growth is a more natu-ral process of pattern formation. Free growth patternsusually show complex dendrites in three dimensions be-cause of competition between growth effects and ther-mal diffusion. Unfortunately, this makes a quantitativeanalytical description extremely difficult. A particularlyinteresting problem is whether free growth patterns ina specific liquid crystalline phase could be sufficientlysimple to facilitate an exact description. If yes, such asystem could provide an opportunity to investigate pat-tern formation and test selection mechanisms. The anti-clinic liquid crystalline (ALC) phase (the Sm-CA phase)is a prime candidate for such a study. One may imaginethat ALCs consist of smectic-C sSm-Cd layers in whichthe azimuthal orientation of the director varies by p fromlayer to layer [9]. Recently it was demonstrated experi-mentally and theoretically [10,11] that ALCs exhibit fas-cinating kinetic phase behavior in an electric field. Thepurpose of this Letter is to present the first experimen-tal evidence of one-dimensional pattern growth subjectto uniform temperature field following a rapid tempera-ture quench. The experiment reveals several novel fea-tures: (i) The growth pattern is one dimensional; (ii) thisis an example in which the kinetics of growth are de-scribed by domain wall propagation in a uniform tempera-ture field; and (iii) the kinetics represent an example of atype of nonlinear diffusion process which shows either un-stable or metastable behavior depending upon the materialparameters.Cells were constructed from two microscope slides,which were cleaned, spin coated with the polyimideCU-2012 (Merck), and baked. The slides were thenrubbed unidirectionally with a cotton cloth using adedicated rubbing machine. A pair of treated slideswas separated by Mylar spacers of nominal thicknessd › 3 mm, and cemented together with the rubbingdirections parallel to each other. Two different enan-tiomeric mixtures of the liquid crystal TFMHPOBC[4-(1-trifluoromethylhexyloxy-carbonyl) phenyl 40-octyloxybiphenyl 4-carboxylate] were used: a racemicmixture with enantiomeric excess X › 0, and a chiralmixture of R- and S-TFMHPOBC, with X › 0.3 ofS-TFMHPOBC. [We note that for X i 0.4 the synclinicSm-C phase is not present. This limits the experimentalrange of X.] Each of the two cells was filled in theisotropic phase and placed into an oven which wastemperature controlled to 25 mK.The oven was brought into equilibrium in the Sm-Cphase just above the Sm-C–Sm-CA phase transitiontemperature; for the X › 0 sample this correspondedto Tinitial › 100 ±C, and for the X › 0.3 sample thiscorresponded to Tinitial › 106 ±C. Resting nearby on apolarizing microscope stage was another oven, which wasequilibrated at some final temperature Tfinal. In order torapidly quench the sample temperature from Tinitial toTfinal, the cell was quickly transferred from the highertemperature to the lower temperature hot stage on the mi-croscope; this process generally took ,2 s. Very shortly(a few seconds) thereafter, fingers were observed to nu-cleate and travel parallel to the layers at constant velocity(Fig. 1). These fingers appear to be similar to photographsof stripes reported earlier [12]. The finger widths w weretypically 5 to 8 mm, and no correlation was observed be-tween w and enantiomeric excess X. By rotating the cellon the polarizing microscope stage, we ascertained that theoptic axis of the fingers was parallel to the rubbing direc-tion zˆ, as opposed to tilted with respect to zˆ as in the higher4478 0031-9007y98y80(20)y4478(4)$15.00 © 1998 The American Physical SocietyVOLUME 80, NUMBER 20 P HY S I CA L REV I EW LE T T ER S 18 MAY 1998FIG. 1. VCR image showing the one-dimensional nature offingers.temperature Sm-C phase. Moreover, the entire cell wastransformed to the Sm-CA phase, over several hours.To measure the velocity y of the anticlinic fingers, themicroscope image was sent to a video cassette recorderfrom a charge-coupled device camera. The recording, witha spatial resolution of ,2 mm, was played back frame byframe; the temporal resolution was 0.033 s. The positionfinger tip, which moved with constant velocity and shape,was measured on the screen, thereby yielding the velocity.Data for velocity vs quench depth dT f› Tinitial 2 Tfinalgare shown in Fig. 2. The data and error bars correspond toan average over approximately ten fingers for each quenchdepth. Note that due to the quenching technique and noise(see below) we were unable to obtain reliable data forshallow quenches, i.e., dT , 3 ±C.We also examined the latent heat of the transitionby differential scanning calorimetry. For the X › 0.3sample, the latent heat L at the isotropic–smectic-A(I–Sm-A) transition is s5.84 3 107d erg g21, and at theSm-C –Sm-CA transition it is s3.9 3 105d erg g21. Thus,FIG. 2. Finger velocity y vs quench depth dT for the X › 0sample sdd and the X › 0.3 sample smd.LSm-C Sm-CAyLI Sm-A , 0.0067. For the X › 0 sample,the ratio is even smaller. These results indicate that theSm-C –Sm-CA transition is extremely weakly first order.We now discuss a simple phenomenological modelfor the observed behavior. There are several issues toconsider. Because of the tiny latent heat and moderatefinger propagation velocity [1], thermal diffusion cansafely be neglected; the growth kinetics should thereforebe based on a free energy functional, but not on a thermalequation. Additionally, if layer-by-layer enantiomericsegregation were to occur, the time scale of mutualdiffusion between layers would definitely be much fasterthan that of domain growth, and concentration gradientswould therefore not affect the finger velocity. Finally,in our sample small chemical impurities would act asa random force, and affect only shallow quenches (seebelow), not those studied herein. Neglecting any helicalstructure, we begin with a free energy F ›P Rfi dx,where fi › felastic 1 flayer-layer 1 fanchoring:fi ›12K sin2 uµ≠wi≠x∂21U2fcosswi21 2 wid 1 cosswi 2 wi11dg 1fL 1 W sdsz 2 12 dd 1 dsz 112 dddg2sin2 wi .Here, fi is the free energy density of the ith smectic layer,K sin2 u is the effective elastic constant associated with achange in azimuthal orientation w within a smectic layer,and W is the effective anchoring strength coefficient. Thecoupling between layers is expressed in terms of thecoefficient UsTd, which has dimensions of energy pervolume, and represents both entropy and a coarseningaverage of local interactions between layers involvingmolecular dipoles and steric effects. UsTd plays the roleof the spatially uniform free energy of the system. For theanticlinic phase, T , Tc and UsTd . 0; in this case, themost stable configuration is when jwi 2 wi11j › p, i.e.,anticlinic. The term 12 L sin2 wi is a symmetry-permittedterm which involves steric and dispersive interactions andfavors both anticlinic and synclinic configurations, but notwi ›12 p.We now take the equations of motion found from fiand introduce a viscosity g associated with azimuthalrotations of the director. On integrating over d, we find,in the overdamped limit,g≠wi≠t› K sin2 u ≠2wi≠x22 Ufsinswi21 2 wid 2 sinswi 2 wi11dg 2µL21Wd∂sin 2wi .We assume the simplest model for this situation, viz., a structure in which all layers have azimuthal angle wi › 0 at4479VOLUME 80, NUMBER 20 P HY S I CA L REV I EW LE T T ER S 18 MAY 1998time t › 0; this corresponds to the Sm-C phase. As theSm-CA phase invades the Sm-C phase, alternate layershave orientation wi › 0 and 0 # wi11 # p . Long afterthe finger passes wi11 ! p , corresponding to the usualSm-CA phase. We accordingly make the approximationthat wi › 0 for all odd i, and wi › w for even i. Wethen obtain the overdamped double sine-Gordon equationg≠w≠t› K sin2 u ≠2w≠x21 2U sinw2µL21Wd∂sin 2w .We should distinquish two cases: (i) L 1 2Wyd . 12 U,and (ii) L 1 2Wyd , 12 U. The first case correspondsto a stable anticlinic phase invading the metastable syn-clinic phase, for which we have a Huxley-type solution[13] with velocity y › 2UfK sin2 uyg2sL 1 2Wyddg1y2.In the second case, we have a Fisher-type solution [14],which will be considered below. For the Huxley case, weneed to examine the coefficients U and L, where U isbased on dipole-dipole interactions associated with anti-clinic ordering, a topic which has been controversial. Inthe classical picture [9,15], the polarizations P0 lie withinthe layers, perpendicular to the tilt plane, and alternatesign from layer to layer (Fig. 3). In a more recent picture[16,17], supported by infrared absorption measurements[18], the polarizations P1 lie in the boundary between lay-ers, in the tilt plane, and again alternate in sign from layeredge to layer edge. On symmetry grounds, both polariza-tions may exist concomitantly, although a nonzero P0 re-quires the system to be chiral. Thus, in a racemic mixture,the layer polarizations P0 vanish and the P1 polarizationsare largely unaffected [16].For an upper limit of U, we assume dipolar inter-actions spaced by a dielectric smectic layer. For thepure enantiomer R-TFMHPOBC, the bulk polarizationP0 › 225 esu cm22 [19]. Although this value may notbe quite the same as P1, it serves as a reasonable esti-FIG. 3. Schematic representation of the anticlinic Sm-CAphase. u and w correspond to the polar and azimuthalmolecular angles. P0 corresponds to the classical polarizationswhich are perpendicular to the tilt plane; P1 corresponds topolarizations which lie in the tilt plane.mate. Assuming a layer spacing of 25 Å and a cross-sectional area per molecule of 25 Å2, this corresponds toa dipole moment of ,1.4 3 10219 esu cm; thus, an upperlimit for U is of order 100–500 erg cm23. With an ex-perimental velocity y , 0.1 cm s21, and taking u , 20±,K , 1026 dyn, g , 0.25 poise [10], we find an upperlimit L 1 2Wyd , 5 200 erg cm23 for the Huxley so-lution. This is consistent with typical values for W [20].However, based on our estimates for the physical parame-ters above, L 1 2Wyd f~sUyyd2g is apparently less than12 U, which would seem to be inconsistent with the re-quirements of a Huxley-type solution.With L 1 2Wyd , 12 U, the Fisher case holds[10,11,14]. With small latent heats and deep quenches,the system corresponds to a stable Sm-CA phase in-vading the unstable Sm-C phase. For L 1 2Wyd › 0[21], the exact velocity is y › s2ygdp2UK sin2 u. ForL 1 2Wyd . 0, we have numerically investigatedthe behavior of n and find that it is quite insensitiveto both L and Wyd as along as L 1 2Wyd ø 12 U[22]. Although this condition is not rigorously satisfied,we may nevertheless take n ł s2ygdp2UK sin2 u as areasonable approximation for the velocity. We thereforefind y , 0.04 to 0.1 cm s21 for the estimated values ofU above. As seen in Fig. 2, this is of the same orderas observed experimentally. Additionally, we see thatthe velocity is smaller for smaller quench depths. Thisis not surprising, as UsT d would grow on decreasing thetemperature deeper into the Sm-CA phase. The apparentplateauing of n for deep quenches is due to a number offactors, including the strong temperature dependencies ofK and especially g. Finally, for Fisher-type behavior, theselection mechanism plays an important role. Accordingto results based on exact solutions of a class of nonlin-ear diffusion equations [23], the selection mechanisminvolves the competition between nonlinear eigenmodes.When the system is driven to an unstable state, an infinitenumber of nonlinear modes, each with a different decayrate, is excited. They interact with each other in a dissi-pative process. Finally, only the mode with the smallestdecay rate survives, and thus determines the selectedvelocity. Our computer simulations indicate that thermalnoise also may significantly affect the selected velocity.This was apparent experimentally, as the finger velocityassociated with shallow quenches sdT , 3 ±Cd wassusceptible to noise and varied with time. The physics ofshallow quenching may also involve metastability of thehigher temperature phase, and is qualitatively differentfrom the nearly noise-free deep quench results reportedherein. Our shallow quench results will be reportedelsewhere in the context of nucleation [22].The seeming similarity between the X › 0 and theX › 0.3 data requires comment. If only the classicalpolarization P0 mechanism were operative, P0 wouldbe zero for the X › 0 sample and no solitary wavebehavior would occur. On the other hand, the P14480VOLUME 80, NUMBER 20 P HY S I CA L REV I EW LE T T ER S 18 MAY 1998polarizations which lie in the anticlinic tilt plane arelargely unaffected by chirality [16]. Therefore, if theparameter U were indeed dominated by dipole-dipoleinteractions, the chirality-independent P1 polarizationswould drive the transition and dominate U in the anti-clinic phase. This would apply to both the racemate andthe X › 0.3 samples. The small differences betweenthe two velocity profiles (Fig. 2) may be due to severalfactors, including a nonzero P0 polarization componentin the anticlinic and synclinic phases of the chiralX › 0.3 sample, and chirality-related differences in stericinteractions.In summary, we have observed and studied a Sm-C toSm-CA phase transition by growth of a one-dimensionalsolitary wave. Although the data indicate that the systemis in the Fisher regime of our model equation, we cannotcompletely rule out Huxley-type behavior. Finally, theinteraction seems to be dominated by polarizations whichlie in the tilt plane of the anticlinic phase.We thank Dr. Helen Gleeson for critically reading themanuscript. This work was supported by NSF’s ALCOMcenter under Grant No. DMR-8920147 and by NSF’sSolid State Chemistry program under Grant No. DMR-9505473.*Present address: Reveo Corp., Hawthorne, N.Y.[1] See the recent review, K. Kassner, in Pattern Formationand Diffusion-Limited Crystal Growth (World Scientific,Singapore, 1996).[2] A. Buka and N. Eber, Europhys. Lett. 21, 477 (1993).[3] J. Bechhoefer, P. Oswald, A. Libchaber, and C. Germain,Phys. Rev. A 37, 1691 (1988).[4] F. Melo and P. Oswald, Phys. Rev. Lett. 64, 1381 (1990).[5] K. Kassner, C. Misbah, and H. Mullerkrumbhaar, Phys.Rev. E 49, 5477 (1994).[6] R. Meister, H. Dumoulin, M. Halle, and P. Pieranski,J. Phys. II (France) 6, 827 (1996).[7] Z.M. Sun and X.Y. Wang, Acta Sci. Sin. 3, 263 (1990).[8] P. Oswald, J. Bechhoefer, and A. Libchaber, Phys. 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Fukuda, M. Kakimoto, and T. Kitazume, Phys. Rev. E53, R4295 (1996).[19] Y. Suzuki, H. Hagiwara, I. Kawamura, N. Okamura, T.Kitazume, M. Kakimoto, Y. Imai, Y. Ouchi, H. Takezoe,and A. Fukuda, Liq. Cryst. 6, 167 (1989).[20] L.M. Blinov, A. Yu. Kabayenkov, and A.A. Sonin, Liq.Cryst. 5, 645 (1989).[21] J. E. Maclennan, N. A. Clark, and M.A. Handschy, inSolitons in Liquid Crystals, edited by L. Lam and J. Prost(Springer-Verlag, New York, 1992).[22] X. Y. Wang, J.-F. Li, E. Gurarie, S. Fan, T. Kyu, andC. Rosenblatt (to be published).[23] X. Y. Wang, S. Fan, and T. Kyu, Phys. Rev. E 56, R4931(1997).4481

Kinetics of Phase Transition in An Anticlinic Liquid Crystal Induced by a Uniform Temperature Field: Growth in One Dimension

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Kinetics of Phase Transition in An Anticlinic Liquid Crystal Induced by a Uniform Temperature Field: Growth in One Dimension

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