Chiral symmetry breaking by spatial confinementin tactoidal droplets of lyotropicchromonic liquid crystalsLuana Tortoraa and Oleg D. Lavrentovicha,b,1

aLiquid Crystal Institute and bChemical Physics Interdisciplinary Program, Kent State University, Kent, OH 44242

Edited by Randall D. Kamien, University of Pennsylvania, Philadelphia, PA, and accepted by the Editorial Board February 11, 2011 (received for reviewJanuary 3, 2011)

In many colloidal systems, an orientationally ordered nematic (N)phase emerges from the isotropic (I) melt in the form of spindle-likebirefringent tactoids. In cases studied so far, the tactoids alwaysreveal a mirror-symmetric nonchiral structure, sometimes evenwhen the building units are chiral. We report on chiral symmetrybreaking in the nematic tactoids formed in molecularly nonchiralpolymer-crowded aqueous solutions of low-molecular weightdisodium cromoglycate. The parity is broken by twisted packingof self-assembled molecular aggregates within the tactoids asmanifested by the observed optical activity. Fluorescent confocalmicroscopy reveals that the chiral N tactoids are located at theboundaries of cells. We explain the chirality induction as a replace-ment of energetically costly splay packing of the aggregates withinthe curved bipolar tactoidal shape with twisted packing. The effectrepresents a simple pathway of macroscopic chirality induction inan organic system with no molecular chirality, as the only require-ments are orientational order and curved shape of confinement.

parity breaking ∣ twisted tactoids ∣ geometrical anchoring ∣nematic–isotropic coexistence

In 1949, Onsager demonstrated theoretically that a dispersion oflong rigid rods develops an organized state, a so-called nematic

(N) with rods being parallel to each, once their concentration ex-ceeds some critical value (1). This remarkable collective behaviorwith an establishment of nonpolar axis of alignment n̂ ¼ −n̂,called the director, stems from the molecular symmetry alone,as the only requirement of the Onsager model is that the rodsdo not overlap. The theory was inspired by Bernal and Fanku-chen’s experiments (2) on aqueous suspensions of tobacco mosaicviruses (TMV) that phase separated into an isotropic (I) phasewith a relatively low concentration of TMV rods and an N phasewith a high concentration of TMV. The N domains emerged asspindle-like “tactoids” (3, 4) with the axes of rods following themeridians of the spindles. Besides TMV dispersions (2, 5), the Ntactoids have been experimentally documented in water disper-sions of many other materials, such as vanadium pentoxide(3, 4, 6, 7), bohemite rods (8), fd viruses (9, 10), f-actin (11),carbon nanotubes (12), as well as in lyotropic chromonic liquidcrystals (LCLCs) (13, 14).

Reversible chromonic assembly and ensuing LCLC mesomor-phism is displayed broadly by dyes, drugs (15–18), nucleotides(19, 20), and DNA oligomers (21, 22). Typically, the LCLCmolecule is plank-like with aromatic cores and peripheral polargroups. In water, the polyaromatic cores stack face-to-face, form-ing elongated charged aggregates. The aggregates, self-assembledthrough weak noncovalent interactions, are polydisperse, with thelength distribution that depends on concentration, temperature,ionic strength, and presence of crowding agents. At low concen-trations, the aggregates are short and orient randomly. As thevolume fraction increases, the aggregates elongate, multiply,and eventually some fraction of them separates from the parentalI phase and forms N-phase tactoids (13, 23).

The shape and internal structure of tactoids are controlled bythe balance of anisotropic surface tension at the I–N interfaceand by internal orientational elasticity of the N domain, as dis-cussed in recent publications (7, 24–28). For rod-like buildingunits, such as TMV rods, the surface orientation is typicallytangential, so that n̂ follows the meridional lines and connectstwo opposite poles of the spindle, thus establishing a bipolartactoidal structure with splay (near the poles) and bend (nearthe equatorial plane) deformations of n̂. A unifying feature ofall N tactoids reported over the 85-y history of studies is thatthe director structure is mirror-symmetric, i.e., nonchiral. Inter-estingly, when the building units are chiral and the liquid crystalis a cholesteric with a helicoidal twist, the tactoids can still appearto be mirror-symmetric, as long as the pitch of the helicoidal twistis much larger than the tactoid, which is the case of TMV (2) andfd virus (10); of course, when the tactoids are larger than thepitch, the inner structure is twisted (29–32).

In this work, we demonstrate that the bipolar N tactoids inaqueous solutions of achiral chromonic molecules and an achiraldepletion (crowding) agent such as neutral polymer poly (ethyle-neglycole) (PEG), exhibit a chiral symmetry breaking, formingoptically active left and right twisted structures. Macroscopicchirality is most often associated with the packing of chiral par-ticles (molecules) (33). Macroscopic chiral domains can occur ifthe molecules are achiral, but have a nontrivial shape that makesit easier for them to pack into the twisted structures, as demon-strated by Clark and coworkers for banana-shaped thermotropicliquid crystals (34, 35), i.e., materials that form a liquid crystal in acertain temperature range between the crystalline and isotropicphases. Twisted structures are also observed in sessile (36, 37) andspherical freely suspended droplets (38–40) of achiral thermotro-pic liquid crystals. However, for the lyotropic liquid crystals thatform in a solvent such as water, the occurrence of macroscopicchirality from achiral molecules has not been known. The findingdemonstrates that the pathway to structural chirality in watersolutions of organic molecules can be very simple, requiringessentially only the orientational order and curved geometry ofconfinement, but no molecular chirality nor macroscopic chiralforces such as mechanical stirring (41).

Experimental SystemWe study LCLC material disodium cromoglycate (DSCG) dis-solved in water at concentration cDSCG ¼ 0.34 mol∕kg (14.8wt%). DSCG molecules in water reversibly assemble formingpolydisperse elongated aggregates. Their length is influenced

Author contributions: L.T. and O.D.L. designed research, performed research, analyzeddata, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. R.D.K. is a guest editor invited by theEditorial Board.1To whom correspondence should be addressed. E-mail: olavrent@kent.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1100087108/-/DCSupplemental.

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by temperature, volume fraction, and presence of condensingagents such as PEG (23). A pure DSCG solution with cDSCG ¼0.34 mol∕kg is isotropic at elevated temperatures T > 35 °C, asthe aggregates are too short to form an orientationally orderedphase. In the range 29.3 °C < T < 35 °C the material exhibits abiphasic IþN region. At T < 29.3 °C, it is a homogeneous N.

PEG is well-known as a condensing agent for aqueous solu-tions of DNA (42) and proteins (43). In LCLCs, it serves a similarcondensing role, causing phase separation of the homogeneousI or N phase into a biphasic IþN region (23). PEG tends toremain in the I phase, exerting an osmotic pressure on the orien-tationally ordered condensed regions, promoting elongation ofaggregates and their closer packing (23). Addition of PEG atconcentration cPEG ¼ 0.005 mol∕kg to the DSCG solution withcDSCG ¼ 0.34 mol∕kg expands the temperature range of thebiphasic region to 17 °C < T < 37 °C. The condensing effect ofPEG is evidenced by an increase of the local optical birefringenceΔn ¼ ne − no measured in the condensed N regions (Fig. S1).Here, ne and no are the extraordinary and ordinary refractive in-dices of the N phase, respectively. When cPEG > 0.03 mol∕kg,PEG condensation produces inclusions of columnar hexagonalphase with morphology of bundles and toroids (23), differentfrom the N tactoids. In this work, we limit our attention to thevery low concentrations of PEG, cPEG ≤ 0.005 mol∕kg, at whichthe condensed phase is still N.

To explore the spatial location and shape of N tactoids, inaddition to the regular optical polarizing microscopy (OPM),we used the three-dimensional imaging capabilities of the confo-cal microscope Olympus Fluoview BX-50. The solutions weredoped with a tiny (<10−5 mol∕kg) amount of fluorescein isothio-cyanate PEG (FITC-PEG). The cells are filled at the elevatedtemperature with the mixture in the I phase, followed by cool-ing (5 °C∕min) toward T ¼ TI→NþI − 5 °C, allowing nuclea-tions of the tactoids; TI→NþI is the temperature at which the firstnuclei appear. The temperature is then raised and fixed atT ¼ TI→NþI − 1 °C, and the samples are left to stabilize in thebiphasic IþN state for several hours.

Experimental ResultsThe fluorescent FITC-PEG accumulates in the I phase, andthe tactoids are seen as dark voids at the fluorescent background;the horizontal xy cross-sectional confocal view of the tactoidin Fig. 1B matches the birefringent OPM view in Fig. 1A.Although some tactoids are seen as nucleating in the bulk, theyall eventually settle at the bounding top and bottom plates ofthe cell, as clearly revealed by the vertical yz confocal imageof the sample (Fig. 1C).

The surface location of LCLC tactoids at both plates suggeststhat the effect is caused by the balance of (anisotropic) surfacetension and elasticity and less so by gravity or temperature gra-dients. Confocal fluorescence microscopy allows us to determinethe shape of the sessile tactoids, which we express through thelength of major semiaxis R (parallel to the x axis) and minor semi-axis r ( parallel to the y axis), as well as the depth d of the tactoidmeasured along the z axis normal to the plates (Fig. 1D).

Addition of PEG does not modify much the ratio R∕r (Fig. 2A),but it increases the depth d of the tactoids (Fig. 2B) and thecontact angles Θxz and Θyz (Fig. 2C), measured between the sub-strates and the N–I interface in the two orthogonal planes, xz andyz, respectively (Fig. 1D). The most dramatic change caused bythe PEG addition is that the inner structure of the tactoidschanges from mirror-symmetric (Fig. 3A) to a chiral one withoptical activity (Fig. 4) once cPEG exceeds 0.0015 mol∕kg.

Nonchiral Tactoids. In all compositions with cPEG < 0.0015 mol∕kg, the central part of tactoids is extinct when observed betweenthe crossed linear polarizer P and analyzer A (Fig. 3A). Such atexture corresponds to the standard models of the axially sym-

metric bulk tactoids (26–28) in which n̂ joins the two cusps alongthe axis. Away from the axis, n̂ is curved to satisfy the tangentialboundary conditions. The interpretation of the director structurein symmetric tactoid (Fig. 3C) is confirmed by mapping the op-tical “slow axis” of the texture (Fig. 3B). Because in the N phaseof DSCG ne < no (13) (see Fig. S1), the slow axis is perpendicularto n̂ðrÞ (Fig. 3B). The light intensity transmitted through thecenter of the tactoid and plotted as a function of the angle γ be-tween the rotating analyzer and fixed (along the x axis) polarizeris symmetric with respect to the crossed arrangement γ ¼ 90°,demonstrating that the structure is not optically active (Fig. 3D).The curve TðγÞ in Fig. 3D measured for the tactoid coincideswith the similar curve measured for the I-phase background,as expected. The value of TðγÞ was normalized by intensity of lighttransmitted through the I phase for parallel polarizers.

Chiral Tactoids. When cPEG > 0.0015 mol∕kg, the tactoids showoptical activity (i.e., rotation of the plane of polarization of apropagating linearly polarized light beam). The central part ofthe tactoid is no longer extinct when viewed between the crossedpolarizers, γ ¼ 90° (Fig. 4). The most prominent evidence of op-tical activity is given by comparing the textures for two comple-mentary values of the angle γ. For example, γ ¼ 70° and γ ¼ 110°result in either increased or decreased light transmission throughthe central region of the tactoid, depending on the handednessof the director twist, as shown schematically in Fig. 4A andquantified in terms of TðγÞ in Fig. 4B. In Fig. 4 A and B, the dataare presented for the cells formed by pairs of glass plates coatedwith a rubbed aligning layer of polyimide SE-7511, to reinforcethe director orientation along the axis x of the tactoid (and therubbing direction).

The differences in TðγÞ measured in rubbed and unrubbedsamples are minor (see Fig. S2). The optical activity of thetactoids is maximum along the axis z passing through the geome-

Fig. 1. Polarizing microscopy (A) and fluorescent confocal microscopy (B andC) of sessile tactoids in the same area of the biphasic Nþ I sample formed by0.34 mol∕kg DSCG aqueous solutions doped with 0.0025 mol∕kg PEG, and≤10−5 mol∕kg of FITC-PEG. (A) In-plane OPM xy view showing tactoids atdifferent depth in the flat cell. (B) Fluorescent horizontal XY image of thesame area. (C) Vertical yz cross-section of the cell along the white line shownin parts A and B, revealing the largest tactoid at the top plate and a smallertactoid at the bottom plate. (D) Schematic view of the twisted tactoid: Therectangular Inset shows a magnified view of the chromonic aggregates nearthe N–I interface. The chemical structures of DSCG and PEG are shown at theBottom Right side.

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trical center of the tactoid. Along this axis, n̂ is expected to beeverywhere perpendicular to the direction of light propagation[because n̂ is aligned tangentially at both the bounding plate(13) and at the N–I interface (23)], and the measured light trans-mission can be compared to the analytical expression derived fora twisted director (44):

T ¼ cos2 β −τmax

2δsin 2δ cos 2β½τmax

δtan δþ tan 2β�; [1]

where δ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ2max þ

�πΔndλ

�2

r, β ¼ γ − τmax, λ is the wavelength of

light (550 nm in all OPM experiments), and τmax is the twist anglebetween the director at the two bounding surfaces measured atthe center of tactoid. The experimental data are well fitted byEq. 1 (Fig. 4B), which yields the sign and the absolute valueof τmax as a function of PEG concentration (Fig. 4C). Note thatbecause the material is weakly twisted, propagation of polarizedlight corresponds to the so-called “Mauguin regime”; i.e., thepolarization of light follows the twisted director structure.

The chiral structure of tactoids is caused by the specific curvedgeometry of the N–I interface (the footprint of the tactoids is

Fig. 2. Tactoid shape as a function of cPEG. (A) Major semiaxis R vs. minorsemiaxis r. (B) Depth d vs r. (C) Contact angle Θxz near the cusp and Θyz nearthe equator.

Fig. 3. Mirror-symmetric tactoids in the Iþ N biphasic region with cDSCG ¼0.34 mol∕kg and cPEG ¼ 0.0015 mol∕kg. (A) As seen under an OPM withcrossed polarizer P and analyzer A (shown by arrows). (B) As seen underthe LC-Polscope, which maps the direction of “slow axis” that is perpendicu-lar to n̂. (C) Schematic view of the n̂ field. (D) Light intensity transmittedthrough the central part of the tactoid as the function of angle γ that theanalyzer makes with the fixed polarizer of OPM.

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essentially flat as it rests on the glass substrate). If the tactoids areallowed to grow (say, by decreasing the temperature) so that theycontact both bounding plates and adopt a shape of a flat pancake,the optical activity disappears. The chirality of the tactoids is notrelated to any mechanical disturbances; two closely located tac-toids often have an opposite handedness, and the overall numberof left- and right-handed tactoids in the sample is roughlythe same.

DiscussionThe twisted director structure has been previously observed in thedroplets of thermotropic N liquid crystals; namely, in circular ses-sile droplets (36, 37) and in spherical droplets with tangentialboundary conditions (38, 39). In both cases, the twist occurs asan energetically less-costly replacement of confinement-inducedsplay and bend deformations (45, 46). In the case of sphericaldroplets, a naive twistless structure would be the one with thedirector lines joining the two poles of the droplet along the mer-idional lines, with splay deformations near the two point defects—boojums at the poles and bend deformations in the equatorialregion. Experimentally, however, the director lines at the surfaceare closer to the loxodromes (lines that intersects the meridians ata constant angle), forming a twisted structure (38, 39). As shownby Williams (45) and experimentally verified in ref. 39, such astructure is energetically preferable if the elastic constant K22

of twist is sufficiently small as compared to the splay K11 andbend K33 constants, K22∕K11 ≤ 2.32ð1 − K33∕K11Þ. As noted byPrinsen and van der Schoot (28), the Williams condition is muchharder to fulfil for slender axially symmetric tactoids, because thesolid angle in which the director is splayed near the pole is small.The slender shape results from a relatively small interfacial ten-sion in lyotropic systems (as compared to thermotropic liquidcrystals that usually form spherical inclusions when placed in iso-tropic fluids) and might explain why the twisted structures havenot been observed so far in N tactoids of lyotropic liquid crystals.

The analytical description of bipolar tactoids in our case isimpossible because the surface placement eliminates the axialsymmetry of the problem, and the director field depends on allthree spatial coordinates. However, one can still point to a simplequalitative mechanism as the source of the twisted structure, theso-called geometrical anchoring and the relative smallness of thetwist constant in the condensed regions (46). The sessile tactoid isshaped by a curvilinear N–I interface and the glass substrate.According to Fig. 2C, the angles Θxz and Θyz that the N–I inter-face makes with the glass substrate are large, approximately50–70°. The director experiences a strong splay near the cuspsof the tactoid. This splay is caused by two factors: (i) the tangen-tial alignment of n̂ at the interfaces and (ii) the tilt of the N–Iinterface. The energy cost of splay deformations in a tiltedgeometry of confinement can be partially relieved by twist, as illu-strated below (see Fig. 5).

Fig. 4. (A) Tactoids with opposite twist deformation: OPM images at threedifferent values of the analyzer rotation angle γ. The polarizer is parallel tothe tactoid major axis x. The schematic drawing on the right-hand side showsthe top view of the clockwise (top raw) and counterclockwise (second raw)twisted sessile N tactoids. (B) Light transmission TðγÞ for sessile N tactoidstwisted clockwise (closed symbols) and anticlockwise (open symbols); thelines represent Eq. 1 with a single fitting parameter τmax. (C) The twist angleτmax as a function of PEG concentration for the family of tactoids with asimilar aspect ratio r∕R.

Fig. 5. Schematic view of the N structure along the z axis, confined betweentwo surfaces imposing tangential orientation on n̂. (A) Two surfaces parallelto each other result in a degenerate in-plane orientation of n̂. (B) Tiltingof the top plate by an angle Θ around the y axis in absence of in-plane phy-sical anchoring produces uniform orientation of n̂ along the y axis. (C) Thebalance between the geometrical anchoring imposed by the surface tilt andthe physical anchoring along the x axis at the bottom substrate results in atwisted structure.

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Consider a one-dimensional N structure along the z axis, lim-ited by two surfaces at z ¼ 0 and z ¼ d (Fig. 5A). Both surfacesimpose a tangential orientation of n̂. If the surfaces are parallel toeach other, n̂ can adopt any in-plane orientation, n̂ ¼ ðnx;ny;0Þ ¼constant. If one plate is tilted, say by a rotation around the y axisby an angle Θ, then there is only one orientation, along the axis y,in which the director is uniform and the elastic energy is zero(Fig. 5B). This alignment along the y axis results exclusively fromthe tilted confinement, because there is no “physical” axis such asa rubbing direction at either of the two plates. Suppose now thatone of the two surfaces (the bottom one in Fig. 5C) favors such a“physical” axis of alignment along the axis x, either because ofthe rubbing or because of the orientation of the tactoid’s majoraxis. This physical anchoring along the axis x competes with the“geometrical” anchoring along the axis y, establishing a structurewith both splay and twist by some angle τ. As shown in SI Text, theelastic energy per unit area of such a structure, for a small tilt Θ,can be written

F ¼ K11

2dΘ2ð1 − τ2Þ þ K22

2dτ2: [2]

If the structure is not twisted, τ ¼ 0, the elastic energy is that ofsplay, F ¼ K11

2d Θ2. When the twist is introduced, the total energydecreases, provided that the following condition is satisfied:

K22∕K11 < Θ2: [3]

Because our consideration neglects the in-plane deformationsand the fact that Θ varies in a broad range from 0 at the centerto approximately 50–70° near the cusps of the tactoid, the lastcriterium is only a qualitative indicator of the possibility of twist,as compared to the Williams criterium (45), but both considera-tions lead to the requirement of a small K22∕K11.

For an LCLC of a given composition, the condition K22∕K11 <Θ2 is expected to be satisfied more easily when the concentrationof PEG increases. In the crowded solutions of DSCG, PEGfacilitates phase separation into I and N phases, accumulatingpredominantly in the I phase and inserting an osmotic pressureon the N domains (23). As the concentration of PEG increases,the aggregates in the condensed N phase are packed more closelyand their length increases; the extreme outcome is the PEG-induced formation of the columnar hexagonal phase (23). Thecloser packing and elongation of aggregates lead to (i) the in-crease of Θ and (ii) decrease of K22∕K11, as discussed below.

i. As seen in Fig. 2C, the angles Θxz and Θyz that the N–I inter-face makes with the solid substrate both increase upon the ad-dition of PEG. Such a behavior is consistent with the expectedincrease in the N–I surface tension σNI (which at the triple lineis balanced by I-substrate and N-substrate surface tensions aswell as by N elasticity) in the presence of PEG, because (47)σNI ∝ p, where p is the osmotic pressure of PEG.

ii. The elongation of aggregates facilitated in the PEG-condensed N regions should affect K11 but not so much K22

(48, 49). As predicted by Meyer (49), in a well-ordered LCLC,K11 ¼ kBT

4aLa, where L ¼ L0 exp E

2kBTis the average aggregate

length (50), L0 is the intermolecular distance in the aggregate,E is the scission energy for cutting an aggregate into two, kBis the Boltzmann constant, and a is the diameter of the aggre-gate. Therefore, one expects that the ratio K22

K11∝ a

L in the PEG-condensed N domains should decrease upon the addition ofPEG, thus facilitating the twisted structures. Note that inthe columnar phase that is formed in place of the N phaseat high PEG concentrations, the splay constant is expectedto diverge, K11 → ∞, which is another argument supportingthe idea of a PEG-induced decrease of K22∕K11.

ConclusionThere is an active current search for routes to create chiralstructures from achiral objects and for mechanisms that separateracemic mixtures with an equal number of “left” and “right” en-antiomers, into homochiral domains (26, 34, 35, 41, 51–60). Thescale of interest is nanometers and micrometers, which revealsthe most interesting features of chirality in biology, chemistry,physics, and materials sciences (61, 62). Chirality has been in-duced through packing frustrations in clusters of magnetic colloi-dal particles of dumbbell shape in presence of the magnetic field(56) and in systems of cylindrically confined spheres (58). Numer-ous examples of chirality inductions are known for orientationallyordered materials formed by anisometric (but nonchiral) build-ing units, such as twisted spherolytes of lamellar polymers (53),twisted droplets of thermotropic nematics (36, 38), twisted rib-bons and filaments of bent core mesogenic molecules (35),J-aggregates of water-soluble molecules (41), Langmuir filmsof surfactants (57), etc. The mechanisms of chirality inductionin these systems range from the local frustrations in packing(35), to the imprinting of a macroscopic chiral force (such asstirring) onto the molecular scale assembly (41, 57).

The simplicity of our system, an aqueous solution of low-molecular weight nonchiral organic molecules, in which the col-lective chiral symmetry breaking is observed, makes the findingrelevant not only to liquid crystals, but also within a broader con-text of homochirality in biology and materials sciences (61–64).The mechanism is also particularly simple. The observed chiralitybreaking requires only a nonflat confinement of an orientation-ally ordered system of molecules with a certain anisotropy of theelastic properties.

Further research of interest is whether similar conditions forthe twist can be met in axisymmetric (bulk) tactoids as opposedto the described tactoids located at the boundary of the systemand whether the tactoids can be made homochiral through chiraladditives or external forces [such as an electric or magnetic fieldtilted with respect to the rubbing direction, or an asymmetricsubstrate such as quartz (64)]. It is also of interest to use themacroscopically twisted tactoids in separation of chiral enantio-mers of organic molecules in aqueous solutions.

The tactoids studied in this work are close to the classic spindleN tactoids observed in the pioneering works of Zocher (3) andothers, with the director tangential to the N–I interface. Inlyotropic systems, one can also observe nuclei with perpendicularsurface anchoring of the director (see, for example, refs. 65 and66), especially when the building units are disc-like (as opposed torod-like). We expect that the mirror symmetry breaking can beobserved also in this perpendicular geometry, as the main typeof director distortions, splay, can be replaced with a twist; similareffects have been already observed in thermotropic liquid crystaldroplets (67).

Materials and MethodsDSCG with purity 98% was purchased from Spectrum and used withoutfurther purification. PEG of molecular weight 3,350 was purchased fromSigma-Aldrich. FITC-PEG of molecular weight 3,400 was purchased fromNanocs. For OPM, we used a Nikon Polarized microscope, equipped with a550-nmwavelength filter. The samples were prepared by filling either rectan-gular capillary 200-μm thick with no alignment layer or surface-treated glasscell with a fixed gap hmeasured by interference technique. The typical thick-ness of cells used for OPM experiments was between 50–100 μm. To promotetactoid alignment, the cell glass substrate was coated with polyimide SE-7511(Nissan, Inc.) and rubbed. The temperature was controlled by using a hotstage TMS94 (Linkam). Abrio LC-PolScope (68) was used to measure opticalretardance R ¼ hjΔnj at wavelength 546 nm and to map orientation of theslow axis, which was normal to n̂ for DSCG solutions.

ACKNOWLEDGMENTS.We thank Yu. Nastishin for useful discussions. The workwas supported by National Science Foundation Grants MWN DMR 076290and ARRA DMR 0906751.

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