Ring Defects in a Strongly Confined Chiral Liquid Crystal

Jun-ichi Fukuda (福田順一)1,2,3,* and Slobodan Zumer2,4,3

1Nanosystem Research Institute, National Institute of Advanced Industrial Science and Technology (AIST),1-1-1 Umezono, Tsukuba 305-8568, Japan

2Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia3Center of Excellence NAMASTE, Jamova 39, 1000 Ljubljana, Slovenia

4Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia(Received 30 December 2010; published 1 March 2011)

We find numerically that a regular array of isolated ring defects can exist as a stable state in a highly

chiral liquid crystal confined in a thin cell imposing fixed planar anchoring at the parallel confining

surfaces. This peculiar defect structure can be stable when the cell thickness d is around 3=4 of the helical

pitch p. A cell of thickness 3p=4 with parallel surface anchoring is incompatible with helical alignment

that favors d ¼ mp=2 (with m being an integer). Formation of ring defects can thus be regarded as a

result of frustrations between the helical alignment with a specific pitch and the confining surfaces that

prevent it.

DOI: 10.1103/PhysRevLett.106.097801 PACS numbers: 61.30.Dk, 61.30.Jf, 61.30.Mp, 61.72.Lk

Topological defects arise as a result of broken continu-ous symmetry and are ubiquitous in nature, from micro-scopic condensed matter systems governed by quantummechanics to a universe in which gravity plays a decisiverole [1–4]. A system with topological defects is usually notin a ground state; unless kinetically arrested it relaxesthrough the annihilation of defects.

In some interesting cases, however, a regular array oftopological defects can be stable. Such examples includecholesteric blue phases in a highly chiral liquid crystal, inwhich line defects of orientational order, or disclinations,of winding number �1=2 form a regular network em-bedded in a stack of so-called double-twist cylinders[5,6]. It is now understood that cholesteric blue phasesare the consequences of frustrations between the double-twist order locally favorable over a single-twist helicalorder and the global topological constraint that double-twist order cannot fill the whole space without introducingdiscontinuities. Frustrations can thus give rise to nontrivialordered structures with topological defects.

Additional frustrations can be introduced when a systeminteracts strongly with confining surfaces. In the case of achiral liquid crystal, when the chirality is weak, a thin cellimposing normal alignment at the surfaces yields a uni-form alignment, incompatible with the preference of achiral liquid crystal for twist orientation. It is knownexperimentally that this frustrated configuration is relaxedto form various locally excited structures referred to ascholesteric fingers [7], or triple-twisted torons [8]. In ourprevious numerical study [9], we showed that a highlychiral liquid crystal can exhibit a variety of exotic defectstructures when it is confined in a thin cell whose thicknessis of the order of the helical pitch. These structures includea parallel array of double-helix line disclinations, and twosets of undulating disclinations. With high chirality, due to

strong twisting power, a uniform alignment is not allowedunless the cell thickness is extremely small. Moreover, thebulk blue-phase structure does not conform to normalalignment at the confining surfaces. Thus these exoticstructures can be regarded as being the result of frustrationsbetween the locally preferable blue-phase ordering and thesurface anchoring that does not allow it.In this study, we will focus on the structure of a highly

chiral liquid crystal confined in a thin cell whose surfacesadopt strong planar anchoring with the same preferreddirection (referred to as the x direction). In contrast tonormal anchoring cases, a helical structure with a singletwist is stable when the helical axis is perpendicular to theconfining surfaces (z direction) and the cell thickness d isan integer or half-integer multiple of the cholesteric pitch p[10]. However, if d is incommensurate with the cholestericpitch, a helical structure is frustrated with the surfacesimposing planar alignment. How this frustration can berelaxed is a nontrivial problem. Here we will show numeri-cally that when d is around 3p=4 leading to maximalfrustrations, the liquid crystal accommodates a regulararray of isolated ring disclinations, which have not beendiscussed as a stable structure in bulk liquid crystals exceptfor ones encircling a colloidal particle or found in nematicdroplets [11]. Moreover, we will demonstrate that the de-tailed structure of the ring is considerably different fromthose of ring defects in a nematic liquid crystal discussedbefore.For the calculation of the orientation profile, we adopt

the same calculation procedures as used in our previousstudy [9], and here we repeat only the essential ingredients.We employ the Landau–de Gennes theory in which theorientational order of a liquid crystal is described by asecond-rank symmetric and traceless tensor order parame-ter �ij. After an appropriate rescaling of �ij, length, and

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the free energy density [5,9,12,13], the total free energy ofthe system confined by parallel surfaces of distance d readsF ¼ R

dxdy½Rd0 dzð’local þ ’elÞ þ ’sjz¼0 þ ’sjz¼d�. Here

the rescaled bulk free energy density consists of a local part’local and an elastic part ’el written as

’localf���g ¼ �Tr�2 � ffiffiffi6

pTr�3 þ ðTr�2Þ2;

’elf���;�rg ¼ �2f½ð �r� �Þ�� þ ����2 þ �½ð �r � �Þ��2g;

(1)

in which the rescaled parameters �, �, � denote tempera-ture, strength of chirality, and the anisotropy of elasticity,respectively. The length is rescaled so that the cholestericpitch p equals 4� [5,9,12]. The surface anchoring energydensity is given by

’s ¼ 12wTrð�� �sÞ2; (2)

where w characterizes the anchoring strength. �sij repre-

sents the order parameter preferred by the surfaces, and wechoose �sij ¼ �s0ðexi exj � 1

3�ijÞ. Here �s0 is the degree of

orientational order and we set it so that �sij minimizes

’local. We consider the case in which the surfaces favorplanar alignment along the x direction, and ex is a unitvector along the x direction. The order parameter �ij and

the dimension of the numerical system (imposing periodicboundaries along the x, y directions) are relaxed using theprocedures employed in our previous studies [9,12] toobtain the profile of �ij minimizing the total free energy F.

In the present study, following our previous experience[13], we choose � ¼ 1 (so-called one-constant elasticity),and � ¼ 0:4 that amounts to the cholesteric pitchp ’ 280 nm, corresponding to relatively strong chirality.We set w ¼ 2:5, corresponding to the anchoring strengthW ’ 1:8� 10�3 Jm�2, which is large but experimentallyachievable. The anchoring extrapolation length is ’ 0:01p, which is small enough compared with the cellthickness d we will consider. So one can safely regardw ¼ 2:5 as strong anchoring. We have checked in a sepa-rate calculation [13] that for � ¼ 0:4, in the temperaturerange�0:61 & � & 0:23, the most stable phase in the bulkis blue phase I (BP I) with O8 (or I4132) cubic symmetry.Below and above this temperature range, the most stablephases are a helical phase with single twist of the nematic

FIG. 1 (color online). Profiles of ring defects for d=p ¼ 0:756and � ¼ �0:1 from three different viewpoints. The defectswith weaker orientational order are depicted by isosurfacesTr�2 ¼ 0:4. Thin cylinders describe the orientational orderalong the x direction at the confining surfaces (shaded planes).Here we show 2� 2 numerical cells connected by periodicboundary conditions along the x and y directions.

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(a)

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Helical alignment(rotation: π)

Helical alignment(rotation: 2π)

ring defectsdouble-helix

defects

FIG. 2 (color online). Free energy per unit areaF as a functionof d=p for � ¼ (a) �0:3, (b) �0:1, and (c) 0.1.

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director, and an isotropic phase with �ij ¼ 0, respectively.

The absolute temperature T is related to rescaled � by� ’ 1� ðT � T�Þ=K, where T� is the temperature at whichan isotropic phase becomes unstable under any small per-turbations [13]. We note here that the above correspon-dence between the rescaled (�, �, w) and the dimensional(p, T, W) parameters is derived for typical material pa-rameters given in [9,13], and may depend sensitively on thechoice of material parameters (say, coefficients in theLandau expansion and elastic constants).

A single-twist helical alignment with its pitch axis alongthe z direction adopts a 2�d=p rotation of the nematicdirector n through the cell thickness d. Therefore, whend=p ¼ m=2 with m being an integer, the m� rotation iscompatible with the imposed parallel surface alignment atthe confining surfaces. Indeed, our numerical calculationsshown below yield a stable helical alignment when d=p ’1=2 or 1. However, when d=p ’ 3=4, the ð3=2Þ� rotation isfrustrated with the surface anchoring and thus a differentprofile is possible. In Fig. 1, we show the profile of topo-logical defects for d=p ¼ 0:756 and � ¼ �0:1. The pres-ence of frustrations leads to the formation of topologicaldefects as in bulk cubic blue phases. However, the structureof topological defects in the present case, a regular array ofisolated ring defects, does not at all resemble those in bulkcubic blue phases [5], or those found in a chiral liquidcrystal confined by surfaces with normal anchoring [9].Close inspection of Fig. 1 reveals that a ring defect does notlie in a flat plane but is distorted outside it. The centers ofthe ring defects form a two-dimensional centered-rectangular lattice, with lattice constants a1 ¼ 0:875pand a2 ¼ 1:65p. In the case of � ¼ 0:1, where ring defectsare stable in a wider range of d (see Fig. 2), we observethat, with decreasing d from 0:796p to 0:637p, the shorterlattice constant a1 increases from 0:768p to 0:976p, whilethe longer one a2 shows a slight decrease from 1:42p to1:39p. Thus the variation of the cell thickness d has a largereffect on a1 than on a2.

To check whether ring defects are really stable, wecalculate the free energy of the system per unit area;F ¼ ðF� f0VÞ=A, where A is the area of one confiningsurface, f0 is the free energy density of bulk BP I as areference, and V ¼ Ad is the volume of the system. InFig. 2, we plot F as a function of d=p for several tem-peratures �. As mentioned above, when the cell thickness dis around p=2 or p, a helical structure is the most stable.The rotation of n is � and 2�, respectively. However, thecell thickness d around 3p=4 yields relatively high F forthe helical structures due to the frustrations mentionedabove, and ring defects indeed give lower F . We alsofind that the range of d in which ring defects is the moststable depends on the temperature �; higher � yields largerstability region of ring defects; for � & �0:3, ring defectscan be no longer stable. We note that for higher tempera-tures close to the bulk isotropic–BP I transition point(� ’ 0:23), a different structure with a parallel array ofdisclinations with double-helix structure can be stable forlarger d, as seen in Fig. 2(c). In the present Letter we willfocus on the ring structure and the double-helix disclina-tions will be discussed elsewhere.In additional calculations not shown here with the varia-

tion of chirality, we observe that stronger chirality (larger �or smaller p) is more favorable for ring defects. In particu-lar, for � ¼ 0:2 or p ’ 560 nm, the temperature range ofits stability is quite limited to the vicinity of the bulkisotropic–BP I transition temperature (� ’ 0:2); ring de-fects are no longer stable for � � 0:1.For a more profound understanding of how the liquid

crystals are ordered, we plot in Fig. 3 the profile of thedirector n around a ring defect. Figure 3(a) indicates thatthe director outside the ring defect exhibits a (frustrated) �rotation along the z direction. On the contrary, inside thering the director is almost uniform pointing along the zdirection, though a slight twist along the y direction isobservable. In Figs. 3(b) or 3(c), we also see twist defor-mation along the x direction in the vicinity of the center of

FIG. 3 (color online). Profile of the director n in the vicinity of a ring defect (shown in translucence) at (a) a yz plane (x ¼ const),(b) an xz plane, and (c) an xy plane for d=p ¼ 0:756 and � ¼ �0:1. (d) and (e) give magnified views in the vicinity of defects in (b)and (c), respectively. In (c), the x direction is along the vertical direction. In (a), (b), and (d), parts of confining surfaces are shown asshaded planes. Director profile n is determined as the direction of the eigenvector of �ij with the largest eigenvalue.

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the ring. From these observation, we conclude that aroundthe center of the ring defect the liquid crystal adopts adouble-twist configuration, thus relaxing the frustrationdue to the incommensurability between the intrinsic helicalpitch and the cell thickness.

It is interesting to see the ‘‘winding number’’ of a dis-clination line. From Figs. 3(b) and 3(d), we find that in thevicinity of confining surfaces, the defect can be clearlyidentified locally as a wedge disclination [14] with windingnumber �1=2, in which the director stays on one planeperpendicular to the defect line. On the other hand, the crosssection along the xy plane shown in Figs. 3(c) and 3(e)demonstrates that the director around the defect line doesnot lie in that plane, contrary to the profile in Fig. 3(d).Moreover, while in Fig. 3(d) the radial orientation from thedefect core is observed towards the ring center, in Fig. 3(e) itgoes away from the ring. This nonuniform profile makes asharp contrast with ring disclinations in a nematic liquidcrystal discussed previously [11] whose structure is essen-tially uniform along themselves, or the line disclinations inbulk blue phases that can be regarded as a �1=2 wedgedisclination throughout [5]. It has been stressed [15–18] thatin three-dimensional liquid crystals with nematic order, awinding number characterizing a line disclination is not atopologically conserved quantity, and thus can be variedspatially and temporally.

Direct observation of this array of ring defects will be anexperimental challenge because one has to deal with a thincell (3p=4 ’ 420 nm and 210 nm in the cases of � ¼ 0:2and 0.4, respectively, though in the former, more accuratecontrol of temperature is required as mentioned above).The centered-rectangular nature of the lattice of ring de-fects could be observed by a careful scattering experiment.Recent progress in the experimental techniques for real-space structural observation such as confocal microscopyhas made possible the clarification of three-dimensionalcomplex structures of liquid crystals [8,19] and thus mightfacilitate direct observation of defect structures in thinliquid crystalline systems.

We have investigated numerically the configuration of ahighly chiral liquid crystal strongly confined in a cell withparallel surfaces imposing planar anchoring with a fixedpreferred direction. We have found that, as well as a helicalorientation profile with its pitch axis perpendicular to theconfining surfaces, a configuration with a regular array ofisolated ring defects can be stable. The stability of ringdefects depends on the cell thickness, and the emergence ofring defects can be understood as the result of the relaxationof frustration arising from the mismatch between the helicalpitch and the cell thickness. Inside the ring defect, a double-twist configuration locally favorable energetically is real-ized, which compensates for the free energy loss due to thecreation of defects. We have also shown a nonuniform struc-tural nature of the ring defect along itself, in contrast touniform ring defects in nematic liquid crystals discussed

before. Though in the present study we have focused on thecase in which the confining surfaces impose parallel anchor-ing of the same direction, a twist cell could introduce frus-trations to the bulk helical alignment in a more controlledmanner, thus facilitating the formation of ring defects. Webelieve thatwehaveprovidedan intriguing example of exoticdefect structures arising from frustrations.Most of this work was carried out while J. F. was staying

at the Department of Physics, University of Ljubljana. Hethanks the Slovenian Research Agency (ARRS researchprogram No. P1-0099 and project No. J1-2335) for gen-erous financial support for his stay. J. F. is in part supportedalso by KAKENHI (Grant-in-Aid for Scientific Research)on Priority Area ‘‘Soft Matter Physics’’ from the Ministryof Education, Culture, Sports, Science and Technology ofJapan.

*fukuda.jun-ichi@aist.go.jp[1] N. D. Mermin, Rev. Mod. Phys. 51, 591 (1979).[2] H.-R. Trebin, Adv. Phys. 31, 195 (1982).[3] P. Chaikin and T. Lubensky, Principles of Condensed

Matter Physics (Cambridge University Press,Cambridge, 1995).

[4] T.W.B. Kibble, J. Phys. A 9, 1387 (1976).[5] D. C. Wright and N.D. Mermin, Rev. Mod. Phys. 61, 385

(1989).[6] P. P. Crooker, in Chirality in Liquid Crystals, edited by

H.-S. Kitzerow and C. Bahr (Springer-Verlag, Berlin,2001).

[7] P. Oswald, J. Baudry, and S. Pirkl, Phys. Rep. 337, 67(2000).

[8] I. I. Smalyukh, Y. Lansac, N. A. Clark, and R. P. Trivedi,Nature Mater. 9, 139 (2010).

[9] J. Fukuda and S. Zumer, Phys. Rev. Lett. 104, 017801(2010); Liq. Cryst. 37, 875 (2010).

[10] A single-twist alignment gives way to blue-phase struc-tures with double twists if d is much larger than p and thesurface anchoring is ineffective in the bulk. But this case isbeyond the scope of the present study.

[11] H. Stark, Phys. Rep. 351, 387 (2001).[12] J. Fukuda, M. Yoneya, and H. Yokoyama, Phys. Rev. E 80,

031706 (2009).[13] J. Fukuda, Phys. Rev. E 82, 061702 (2010).[14] P. G. de Gennes and J. Prost, The Physics of Liquid

Crystals (Oxford University Press, Oxford, 1993), 2nd ed.[15] Y. Bouligand, in Physique des Defauts/Physics of Defects,

Proceedings of the Les Houches Summer School ofTheoretical Physics, 1980, Session XXXV, edited by R.Balian, M. Kleman, and J.-P. Poirier (North-Holland,Amsterdam, 1981).

[16] M. Kleman and O.D. Lavrentovich, Soft Matter Physics:An Introduction (Springer-Verlag, New York, 2003).

[17] J. Fukuda, Phys. Rev. E 81, 040701(R) (2010).[18] H. Yokoyama, in Proceedings of the 23rd International

Liquid Crystal Conference, Krakow, 2010 (unpublished).[19] K. Higashitani, K. Yasui, and H. Kikuchi, J. Am. Chem.

Soc. 130, 6326 (2008).

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