Exploiting imperfections: Directed assembly of surface colloids viabulk topological defects

Marcello Cavallaro Jr.,∗ Mohamed A. Gharbi,∗ Daniel A. Beller,∗ Simon Copar,

Zheng Shi, Tobias Baumgart, Shu Yang, Randall D. Kamien, and Kathleen J. Stebe†

The Laboratory for Research on the Structure of Matter, University of Pennsylvania,3231 Walnut Street, Philadelphia, Pennsylvania 19104, USA

We exploit the long-ranged elastic fields inherent to confined nematic liquid crystals toassemble colloidal particles trapped at the liquid crystal interface into reconfigurable struc-tures with complex symmetries and packings. Spherical colloids with homeotropic anchoringtrapped at the interface between air and the nematic liquid crystal 5CB create quadrupolardistortions in the director field causing particles to repel and consequently form close-packedassemblies with a triangular habit. Here we report on complex, open structures organized viainteractions with defects in the bulk. Specifically, by confining the nematic liquid crystal inan array of microposts with homeotropic anchoring conditions, we cause defect rings to format well-defined locations in the bulk of the sample. These defects source elastic deformationsthat direct the assembly of the interfacially-trapped colloids into ring-like assemblies, whichrecapitulate the defect geometry even when the microposts are completely immersed in thenematic. When the surface density of the colloids is high, they form a ring near the defectand a hexagonal lattice far from it. Since topographically complex substrates are easilyfabricated and liquid crystal defects are readily reconfigured, this work lays the foundationfor a new, robust mechanism to dynamically direct assembly over large areas by controllingsurface anchoring and associated bulk defect structure.

topology | 5CB | elastic interaction | nematic | micropost

Abbreviations: LC, liquid crystal; 5CB, 4-cyano-4’-

pentylbiphenyl; DMOAP, 3-(Trimethoxysilyl)propyl-

octadecyldimethylammonium chloride; POM, polarized

optical microscopy; FCPM, fluorescence confocal polariza-

tion microscopy; LdG, Landau-de Gennes

Significance Statement

In this research, we develop new means of direct-ing colloids at an interface to assemble into com-plex configurations by exploiting defects in a liquidcrystal (LC). Through confinement of a nematicLC over a topographically patterned surface, wedemonstrate the formation of defects at precise lo-cations in the LC bulk. These defects source elasticdistortion fields that guide the assembly of colloidsconstrained to the LC-air interface. This work sig-nificantly extends prior work in which LCs confinedin film or droplet geometries guide colloidal assem-bly beyond simple triangular lattices and chains.

∗These authors contributed equally to this work.†To whom correspondence should be addressed. E-mail:kstebe@seas.upenn.edu

Here, we demonstrate colloidal assembly at preciselocations, with particle rich and poor regions, de-termined remotely by defects deliberately seeded inthe LC bulk. Experimental results are supportedby numerical and analytic investigation.

Introduction

Classically, the bulk of a material system is wherethe action is: the interface is oft relegated to a set of“boundary conditions.” However, crystal faceting[1], the quantum hall effect [2], and even the AdS-CFT correspondence [3] fundamentally reverse thisrelationship – the bulk properties can be read offfrom their effects on the boundaries. In this con-tribution, we demonstrate migration and organiza-tion of colloids constrained to a liquid crystal-airinterface, driven remotely by the elastic distortioncreated by the presence of topological defects inthe liquid crystalline bulk. Just as phantoms areemployed in magnetic resonance imaging [4], it isnecessary for us to prepare bulk defects in knownconfigurations in order to verify our bulk/boundaryconnection. To do this, we prepare a substrate pat-terned with microposts that, with appropriate sur-face treatment, seed a reproducible defect complex-ion. Colloidal spheres on the interface experience

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an attraction to the regions above the submergeddefects, as well as an elastic repulsion from eachother, leading to complex new assemblies. The longrange of these elastic interactions allows defects inthe bulk nematic phase far below the interface todirect assembly at the interface. Other recent workon producing ordered arrangements of particles atliquid crystal interfaces beyond simple triangularlattices – such as chains [5], stripes [6], and densequasihexagonal lattices [7] – have focused on con-fining the nematic in thin film or droplet geometriesand varying the surface coverage fraction. Our sen-sitive control over substrate topography providesthe ability to tune the defects’ positions and theirinfluence on the interface, offering a route to tun-able non-trivial colloidal assemblies.

To understand the bulk’s influence on theboundary, it is essential for us to first know what ison the inside. To that end, we have carefully char-acterized, simulated, and analyzed the response ofthe nematic liquid crystal, 5CB, to the micropost-patterned substrate. We begin with a descriptionof the sample preparation, our microscopy results,and an analytic approximation to characterize thedefect complexion. We corroborate this with nu-merical modeling and find, to our surprise, that theconnection between the boundary conditions andthe substrate topography is not always entirely ge-ometric. Finally, we present our main finding: themigration and arrangement of colloids at the sur-face in response to disclinations in the bulk.

Results and Discussion

We prepare a square array of cylindrical micro-posts and impose homeotropic anchoring at allsurfaces, resulting in the appearance of a discli-nation ring around each micropost. The defectlines encircle the microposts and appear as brightlines in an otherwise black polarized optical mi-crograph shown in Fig. 1A. Under sample rota-tion, the dark regions remain black, confirminghomeotropic anchoring, while bright field opticalmicroscopy corroborates this observation (inset ofFig. 1A). When the sample is heated above theclearing point (TNI = 34◦C), the defect line is ab-sorbed into the isotropic phase, resulting in a com-pletely black image under crossed polarizers. Uponcooling back to the nematic phase (either slowly at0.1 ◦C/min or in contact with the room environ-

ment), the disclination reappears, confirming thatthe texture is due to the equilibrium elasticity ofthe NLC and the frustrated director field near thesurface of the micropost. We determine the verticalposition of the disclination line using fluorescenceconfocal polarizing microscopy (FCPM) and findthat, for a wide range of post heights, the disclina-tion occurs near the mid-height position (Fig. 1B).In addition, we deduce through FCPM that thedisclination sits within 5µm of the micropost sur-face. Though both bright field and polarized mi-croscopy suggest a larger gap between the postsand defects, we believe this is an optical effect dueto the curved interface and the birefringence of theLC. Finally, by varying the cross section of the mi-croposts we demonstrate that the disclination linebears the signature of the micropost shape as shownin Fig. 1C.

What does the observed disclination ring im-ply about the bulk director field and the bound-ary conditions associated with confinement? To ad-dress this theoretically, we study a Q-tensor based,Landau-de Gennes (LdG) model of the nematicconfined in micropost geometries. Using a finitedifference scheme to minimize the LdG free energy[8] we determine the director field and defect lo-cations. We use the unequal elastic constants mea-sured for the elastic anisotropy of 5CB. We numeri-cally model the micropost as a right cylinder bridg-ing two planar interfaces and find four distinct lo-cal minima of the free energy: two in which thereis no defect in the bulk (Fig. 2B and its reflectionthrough the horizontal), one with a disclination ringaround the micropost with +1/2 winding geometry(Fig. 2C), and one in which the disclination ringhas instead a −1/2 geometry (Fig. 2D). All theminima exhibit axial symmetry and we find thatthe director field has no axial component as in Fig.2B-D. It follows that we can discuss the topologyof the texture in terms of two-dimensional nematicscorresponding to each radial slice, promoting the±1/2 geometry into a pseudo-topological charge.The disclination ring is topologically charged in thethree-dimensional sense, as the director field pro-file is essentially constant along its contour [9–11];we will henceforth focus exclusively on the two-dimensional pseudo-charge.

The total pseudo-charge can be calculated viathe winding of the director around the boundary ofthe sample. However, as our numerics demonstrate

3

in Fig. 2, the sharp corners at the cylinder-planarjunctions require careful consideration of the topol-ogy in each slice. Each corner can be resolved viaa splay or bend texture leading to a director rota-tion of ±π/2, respectively. Thus the defect pseudo-charge in each radial slice is determined by the de-tails of each junction. When the winding sense ispositive at corner X and negative at corner Y (Fig.2B) or vice versa, there is no disclination ring inthe bulk. On the other hand, a disclination ringof winding number +1/2 (Fig. 2B) is observed atmid-height around the micropost when the two cor-ners both have positive winding. Likewise, a discli-nation ring of winding number −1/2 (Fig. 2C) isobserved at mid-height around the micropost whenthe two corners both have negative winding. Thus,the experimental observation of a disclination inthe bulk implies that the director winding has thesame sign at both corners. The nematic degree oforder is diminished locally at the corners due tothe incompatible boundary conditions. It should benoted that the computed free energies of the stateswith the bulk disclination ring (Figs. 2C and 2D)are slightly higher than that of the states with nobulk defect (Fig. 2B), whereas the bulk disclina-tion is stable in experiment. A similar multista-bility due to sharp-cornered boundaries has beenreported for nematics in square wells with planaranchoring [12, 13]. There, the choice of windingsense at each corner gives rise to two optically dis-tinct states, a useful situation for bistable displays.

In the numerical model, we can induce a localpreference for a given winding sense by altering thegeometry of either corner. For example, Fig. 3Ashows a 5CB-air interface curving upward to meetthe micropost at a nonzero pinning angle as in theexperiment. As a result, bend is favored over splayand the winding is negative. Conversely, curva-ture at the bottom of the micropost favors positivewinding, as shown in Fig. 3B. With this resolu-tion of the sharp boundaries, the bulk disclinationring is not even metastable in the numerics. Toprobe this experimentally, we replaced the sharpcorner at X with a curved base to favor a posi-tive winding sense (+1/4) as in Fig. 3B; the sur-face pinning maintains the preference for negativewinding at corner X. In contrast with the numer-ics, we find that when the base of the micropostis slightly curved (Fig. 3C) the bulk disclinationpersists but is pushed away from the base, closer

to the 5CB-air interface. We verify the defect posi-tion using FCPM as shown in Fig. 3D and verify thestability of the defect by heating into the isotropicphase and re-cooling several times. Each time,upon cooling into the nematic phase, the defectre-forms. Further, the bulk disclination remainseven when microposts are completely submergedin a thick nematic film. However, we also find thatwhen the microposts are tapered all the way to thetop (Fig. 3E-G), the bulk disclination ring fails toform (Fig. 3F and 3G), as verified by FCPM, whichis in agreement with the numerical results. Thus,the geometry of the boundary alone is not suffi-cient to predict the qualitative features of the bulkdirector field. We suspect that variations in sur-face chemistry, roughness, or relative magnitudesof anchoring strength are important for the stabi-lization of bulk defects. On the other hand, thesefindings demonstrate that sensitive control over theboundary shape offers the ability to tune the stableposition of the disclination ring. Lateral confine-ment of disclination rings by neighboring microp-osts holds promise as a way to further manipulatethe defects, as shown by studies of nematic-filledcapillaries [14, 15]. However, simply decreasing themicropost center-to-center spacing (even down to1.25 times the micropost diameter) does not affectthe existence or circular shape of the bulk disclina-tions.

Armed with our careful characterization of thecomplexion, we now turn to our main interest: theinteraction of the liquid crystal with colloidal par-ticles. Prior work has established that colloids areattracted to disclination lines, an effect that hasbeen exploited in the bulk to form wire-like chainsof colloidal particles along the defects [17–20]. Herehowever, we use the distortion field generated bythe defect to remotely steer particles trapped atthe 5CB-air interface. When a 5µm diameter silicamicrosphere with homeotropic anchoring is placedon the interface, it migrates radially toward the mi-cropost until contact (Fig. 4A). Though capillaryinteractions are known to induce particle migra-tion along curved surfaces [21], capillarity effectsare absent in this system since the interfacial dis-tortions induced by the microspheres are negligi-ble [5, 22]. Further, when the system is heatedabove the nematic-isotropic transition to annihi-late the disclination lines we no longer observe mi-gration and thus conclude that the observed mi-

4

gration must be orchestrated solely by the elasticdirector field. In other words, the elastic distor-tion created by the spherical particle is interactingwith the bulk nematic texture. As noted in [5],however, the distortions made by the particles arequadrupolar in nature and die off rapidly away fromthe surface. Thus we expect that the colloids willonly interact with large director deformations. Asshown in Fig. 2B-D, we can imagine three potentialsources of large director distortions which include aregion of splay gradient along the interface due toits meniscus; a defect line near the interface at theupper corner Y (as in Fig. 2A); and the defect ringin the bulk. We can eliminate the first potentialsource by submerging the micropost in the liquidcrystal so that it no longer intersects the nematic-air interface. Yet, even without distortion of theinterface, migration occurs. When several particlesare present at the interface, they form a ring abovethe immersed micropost mimicking the bulk discli-nation (Fig. 4B). The second potential source isshown to be negligible by changes to the micropostgeometry discussed below, which alter the speed ofcolloidal migration without changing the defect atcorner Y. These observations confirm that the as-sembly of particles at the free interface is, in fact,driven by the third source, the defect ring withinthe bulk liquid crystal.

As we increase the surface density of the micro-spheres, a single ring of colloids around each mi-cropost (Fig. 4B and 4C) gives way to complexstructures that form owing to the balance of at-traction to the micropost and the long range inter-particle elastic repulsion. This competition resultsin highly ordered structures as shown in Fig. 4Dand Fig. 4E that nucleate from the post and evolvewith distance away from the post into a hexagonallattice.

Since the particles migrate in creeping flow, wecan estimate the interaction energy by balancingthe elastic and viscous forces. We track particletrajectories and use drag coefficients implied by theBrownian fluctuation of particles at the interfaceand the Stokes-Einstein relation. We infer elasticinteraction energies Ep ≈ 104 kT (Fig. 5A) andfind that the energy falls off as the inverse squareof the radial position from the post center. Addi-tionally, we test the elastic potential numerically bymoving a small spherical colloid with a quadrupo-lar defect at fixed height near the top surface to-

ward a micropost with a defect ring. As shownin Fig. 5B, the elastic energy is well described byEp = E0−α(r−rd)−2, where rd is the radius of thedisclination ring around the micropost and E0 andα are fitted coefficients. The dependence of thispotential on distance from the disclination, ratherthan from the center or edge of the micropost, forexample, further underscores the role of the discli-nation as the principal cause of the elastic attrac-tion.

Finally, we find that as the height increases therate of migration decreases (inset of Fig. 5A). Sincethe disclination line is always at or below the mi-cropost mid-height for un-tapered microposts, thereduced strength of interaction follows from the in-creased separation between the microspheres at thesurface and the defect ring in the bulk. We confirmthe latter result by measuring the variation of theelastic force magnitude fp for micropost heights of33 µm, 60 µm, and 96 µm at fixed radial position ofthe particle r = 62 µm. We find that fp decreaseswith values of 1.39 pN, 0.633 pN, and 0.406 pN, re-spectively. Similarly, when the micropost base ge-ometry is changed from a “sharp” base to a curvedbase, the defect rings are closer to the interface, andthe migration rates increase. Together these obser-vations offer additional confirmation of the directinteraction of the bulk disclination and the colloids.

To summarize, we have demonstrated that bulkdefects can be generated in confined liquid crystalsand guide the assembly of remote colloidal parti-cles into ordered structures through nematic elas-ticity. The colloids assemble to mimic the defectstructure in the bulk, in this case, a ring aroundthe micropost. Even when the microposts are sub-merged under liquid crystal, the elastic effect of thetrapped topological defects remotely organizes thecolloids on the surface, allowing us to probe theinterior through “colloid-aided tomography.” Athigh surface coverage, attraction to the defect ringis coupled with long range interparticle repulsionand leads to highly ordered structures that nucleateradially outward from microposts. Coupled withthe noted ability to manipulate the nematic direc-tor through flow, fields, and functionalization, ourwork paves the way to dynamically tunable assem-blies at fluid interfaces.

5

Materials and Methods

Liquid Crystal 5CB. We use 4-cyano-4’-

pentylbiphenyl (5CB, Kingston Chemicals Limited),

a thermotropic liquid crystal with a nematic phase

between 18◦C ≤ T ≤ 34◦C.

Supporting 5CB-Air Interfaces in Microp-

ost Arrays with Homeotropic Anchoring. KMPR

negative photoresist (Microchem Corp.) is used to

fabricate micropost arrays on transparent quartz sub-

strates. The cylindrical microposts have nominal radii

R = 50µm, heights ranging from 33µm ≤ h ≤ 96µm,

and are positioned in square arrays with a pitch

p = 400µm. The micropost height is tuned by

changing spin speed when depositing the photoresist

while different micropost cross-sections are created

by varying the photomask design. Microposts with

curved bases are achieved by spin coating SU-8 2002

on a micropost array at speeds between 1500 and 6000

rpm and then developing the epoxy using standard

SU-8 lithographic procedures. All substrate surfaces

are treated to induce strong homeotropic anchoring by

sputtering a 30 nm film of chromium using a home built

sputtering system and then incubating the samples

in a 3% weight solution of 3-(Trimethoxysilyl)propyl-

octadecyldimethylammonium chloride, (DMOAP,

Sigma Aldrich) in a 9:1 mixture of ethanol:water.

Samples are rinsed and heated at 110◦C for 1 hour

to complete the substrate treatment [24]. Untreated

substrate surfaces have degenerate planar anchoring.

To confine the LC film, the micropost array substrate is

heated to 60◦C and 5CB is applied by spin-coating. At

the top edge of each micropost, the nematic-air interface

is pinned with a typical angle 15◦ ≤ ψ ≤ 20◦ measured

with respect to the horizontal using a goniometer.

Preparing and Depositing Microspheres. Dry

silica microspheres of nominal diameter 5µm (Poly-

sciences, Inc.) are treated to induce strong homeotropic

anchoring using the same chemistry described above

for the microposts. Particles are then dried in a

vacuum oven overnight at 110◦C and subsequently

aerosolized in a closed container with compressed air.

Once aerosolized, the substrate supporting the 5CB-air

interface is introduced and particles are effectively

deposited with minimal aggregation.

Optical Characterization. We characterize our

system using an upright microscope in transmission

mode equipped with crossed polarizers (Zeiss AxioIm-

ager M1m) and a high resolution color camera (Zeiss

AxioCam HRc). Fluorescence confocal polarization

microscopy (FCPM) images are obtained using an

inverted IX81 microscope equipped with an FV300

confocal scanbox and a half wave plate between the

objective and filter cubes to rotate the polarization of

the scanning laser. FCPM allows us to determine di-

rector orientation of LC molecules doped with a 0.01%

wt. dye N,N’-Bis(2,5-di-tert-butylphenyl)-3,4,9,10-

preylenedicarboximide (BTBP, Sigma Aldrich). At

these low concentrations, dye molecules align along the

axis of the LC molecules while preserving the properties

of 5CB and fluoresce when aligned parallel to the

excitation light polarization vector.

Numerical modeling. Our numerical modeling

follows the Landau-de Gennes numerical approach

reviewed in [8]. Our finite difference scheme employs

a regular cubic mesh on which is defined a traceless,

symmetric, rank-2 tensor Q, from which the nematic

director is recovered as the eigenvector corresponding to

the leading eigenvalue S. The LdG free energy density

is a sum of two terms,

fphase = 12AQijQji + 1

3BQijQjkQki + 14C (QijQji)

2,

fgrad =1

2L1∂Qij

∂xk

∂Qij

∂xk+

1

2L2∂Qij

∂xj

∂Qik

∂xk

+1

2L3Qij

∂Qkl

∂xi

∂Qkl

∂xj

with summation over repeated indices. In fgrad, we

use L1 = 3.8 × 10−12 N, L2 = 5.3 × 10−12 N, and

L3 = 5.3 × 10−12 N to model 5CB with elastic con-

stants K1 = 0.64 × 10−11 N, K2 = 0.3 × 10−11 N,

K3 = 1×10−11 N [23]. It is necessary to include a third-

order term in fgrad in order to have K1 6= K3 [8]. From

Ref. [8], we take typical values for the material constants

of 5CB: A = −0.172×106 J/m3, B = −2.12×106 J/m3,

C = 1.73 × 106 J/m3. From a random or specified ini-

tial configuration, we minimize the LdG free energy over

Q(~x) using a conjugate gradient algorithm from the AL-

GLIB package [25]. The mesh spacing corresponds to 4.4

nm. The microposts modeled have diameter 440 nm and

height 264 nm. The simulation box has length 1320 nm

in both horizontal dimensions.

We model the anchoring strength on all the

substrate surfaces as infinite by imposing a fixed,

uniaxial Q at the boundaries. The infinite anchoring

strength approximation is reasonable because 5CB near

DMOAP treated surfaces experiences an anchoring

strength W ≈ 10−2 N/m [17], so the anchoring extrap-

olation length ξW = K/W is a few nanometers, close to

the nematic coherence length ξN ∼√L1/A. We have

6

verified that finite anchoring strength in the strong

anchoring regime does not significantly change the

results using a Rapini-Papoular-type surface potential

as used in Ref. [8]. Periodic boundary conditions in

the horizontal directions are used to simulate a square

array of microposts.

Acknowledgments

The authors acknowledge support from NationalScience Foundation (NSF) through MRSEC Grant

DMR11-20901. This work was supported in partby a gift from The Mark Howard Shapiro andAnita Rae Shapiro Charitable Fund and by NSFGrant PHY11-25915 under the 2012 Kavli Insti-tute for Theoretical Physics miniprogram “KnottedFields”. DAB was supported by the NSF througha Graduate Research Fellowship under Grant No.DGE-1321851. This work was partially supportedby a Simons Investigator award from the SimonsFoundation to RDK. DAB and RDK thank theIsaac Newton Institute for their hospitality whilethis work was completed.

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[3] Aharony O, Gubser SS, Maldacena J, Ooguri H, OzY (2000) Large N field theories, string theory andgravity. Phys Rept 323:183-386.

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[8] Ravnik M, Zumer S (2009) Landau-de Gennes mod-elling of nematic liquid crystal colloids. LiquidCrystals 36:1201-1214.

[9] [1] Janich K (1987) Topological properties of ordi-nary nematics in 3-space. Acta Applicandae Math-ematica, 8:6574.

[10] Copar S, Zumer S (2011) Nematic braids: topolog-ical invariants and rewiring of disclinations. Physi-cal review letters, 106:177801.

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liquid crystal filled wells. Applied physics letters,90:111913111913.

[13] Luo C, Majumdar A, Erban R (2012) Multistabilityin planar liquid crystal wells. Physical Review E,85:061702.

[14] De Luca G, Rey AD (2007) Ringlike cores of cylin-drically confined nematic point defects. The Jour-nal of chemical physics, 126:094907.

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[16] Phillips PM, Rey AD (2011) Texture formationmechanisms in faceted particles embedded in a ne-matic liquid crystal matrix. Soft Matter 7:2052-2063.

[17] Skarabot M, et al. (2008) Hierarchical self-assemblyof nematic colloidal superstructures. Phys Rev E77:061706.

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[20] Senyuk B, et al. (2012) Shape-dependent orientedtrapping and scaffolding of plasmonic nanoparticlesby topological defects for self-assembly of colloidaldimers in liquid crystals. Nano Lett 12:955-963.

[21] Cavallaro Jr. M, Botto L, Lewandowski EP, WangM, Stebe KJ (2011) Curvature-driven capillary mi-gration and assembly of rod-like particles. ProsNatl Acad Sci USA 108:20923-20928.

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[25] http://www.alglib.net.

A B

~ 28h mP

~ 60h mP

PA

C

FIG. 1: Micropost induced bulk defect rings. (A) Po-larized optical microscopy (POM) image of a micropostarray where all surfaces have homeotropic anchoring re-sulting in defect rings that circumscribe each micropost.INSET: Bright field image of a single micropost wherethe bright line indicates the approximate lateral positionof the defect loop. (B) Fluorescence confocal polariza-tion microscopy (FCPM) image indicating the locationof defects in an otherwise uniform director field. Top: Atop view of the micropost. Bottom: A z-stack of FCPMimages in which the maximum intensity represents thelocation of the defect core that occurs at approximatelythe post mid-height. (C) Disclination lines are dictatedby the shape of the micropost as shown around trian-gular, square and pentagonal microposts. All scale barsare 50 µm.

8

?m

icro

post

LC-air interface

substrate

X

Y

14

§ ·�¨ ¸© ¹

¨ ¸14

§ ·�© ¹

12

�12

�

B

DC

A

FIG. 2: Numerical and topological evaluation of thedirector field. (A) The system has two corners – X andY – where the director field can choose between twowinding senses. (B-D) Director fields corresponding tothe relative minima of the Landau-de Gennes free en-ergy for a cylindrical micropost and planar interfaces,found numerically. Isosurfaces of S = 0.48 are shown inred, where S is the leading eigenvalue of Q. Blue ellip-soids indicate the director field. The nematic is locallymelted at the sharp corners. (B) Opposite winding atthe two corners precludes the possibility of a disclinationin the bulk. (C) A bulk disclination with +1/2 (i.e.,anticlockwise) winding number requires positive wind-ing at both corners. (D) A bulk disclination with −1/2(i.e., clockwise) winding number requires negative wind-ing at both corners. Since the numerics show that the(meta)stable states have azimuthal symmetry and thatthe director has no azimuthal component, we may thinkof these winding numbers as pseudo-charges.

9

A

~ 56h mP

~ 96h mP

DB

E F GPA

PA

C

FIG. 3: The effect of surface curvature. (A) Where the5CB-air interface curves upward to meet the micropost,LdG numerical modeling predicts that negative windingis favored. (B) Likewise, curvature at the bottom of themicropost favors positive winding; the surface defect inthis case can be viewed as virtual, inside the micropost.(C) SEM image of microposts with a curved base anda corresponding (D) FCPM image detecting the pres-ence of a disclination loop when the micropost array isfilled with LC. Typically this defect will sit towards theupper half of the micropost. (E-G) Curved micropoststapered along their entire lengths, SEM image shown in(E), do not induce the formation of a bulk disclinationring (verified with FCPM), as is evident when viewedbetween crossed polarizers. The relaxation of the di-rector to the vertical direction with increasing distancefrom the micropost is much more gradual in (F) thanin Fig. 1A. Axial symmetry is lost at smaller micropostspacing (G). All scale bars are 50 µm.

10

A B E

C D't = 1 s

FIG. 4: Elastic migration of colloidal particles induced by bulk topological defects. (A) Time-lapsed images ofthe migration of spherical colloids towards a micropost (∆t = 1 s). (B) A ring of colloids forms above a submergedmicropost. (C) At moderate surface coverage, ordered rings assemble around the micropost due to attraction bythe bulk defect and repel one another via long range interparticle repulsion. (D) As particle density continues toincrease, radial assemblies evolve into hexagonal ordering until (E) highly ordered structures form at a very highsurface coverage. All scale bars are 50 µm.

11

A

B

33 mP60 mP60 mP96 mP96 mP

FIG. 5: Elastic potential. (A) Migration of colloidsimplies bulk-colloid interaction energies on the order of1 × 104 kT and follow Ep ∝ r−2 (solid curves), whereEp is the inferred elastic potential and r is the radialdistance in the horizontal plane from the center of themicropost. As the height of a micropost is increased(different curves), attractions become weaker due to anincreased separation between particles and the bulk de-fect. INSET: Migration rates are faster for micropostswith base curvature compared to those of comparableheight that have a sharp corner at the base (corner Y inFig. 2A). This is due to the tendency of the bulk discli-nation to position itself closer to the free interface formicroposts with curved bases. Closed symbols: microp-osts with curved base. (B) Numerically modeled energyof a colloidal sphere approaching a micropost captur-ing the Ep ∝ (r − rd)−2 dependence of the quadrupo-lar colloid interacting with a disclination ring. The reddashed line represents the asymptotic value for Ep atlarge distances and rd is the radius of the micropost’sdisclination ring. INSET: Representative image of thenumerical modeling.