Colloidal entanglement in highly twisted chiral nematic colloids: Twisted loops, Hopf links, and trefoil knots

PHYSICAL REVIEW E 84, 031703 (2011)

Colloidal entanglement in highly twisted chiral nematic colloids: Twisted loops,Hopf links, and trefoil knots

V. S. R. Jampani,1 M. Skarabot,1,2 M. Ravnik,2,3 S. Copar,2 S. Zumer,2,1 and I. Musevic1,2

1J. Stefan Institute, Jamova 39, SLO-1000 Ljubljana, Slovenia2Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SLO-1000 Ljubljana, Slovenia

3Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OXI 3NP, United Kingdom(Received 25 May 2011; revised manuscript received 22 July 2011; published 16 September 2011)

The topology and geometry of closed defect loops is studied in chiral nematic colloids with variable chirality.The colloidal particles with perpendicular surface anchoring of liquid crystalline molecules are inserted in atwisted nematic cell with the thickness that is only slightly larger than the diameter of the colloidal particle.The total twist of the chiral nematic structure in cells with parallel boundary conditions is set to 0, π , 2π , and3π , respectively. We use the laser tweezers to discern the number and the topology of the −1/2 defect loopsentangling colloidal particles. For a single colloidal particle, we observe that a single defect loop is windingaround the particle, with the winding pattern being more complex in cells with higher total twist. We observe thatcolloidal dimers and colloidal clusters are always entangled by one or several −1/2 defect loops. For colloidalpairs in π -twisted cells, we identify at least 17 different entangled structures, some of them exhibiting linkeddefect loops-Hopf link. Colloidal entanglement is even richer with a higher number of colloidal particles, wherewe observe not only linked, but also colloidal clusters knotted into the trefoil knot. The experiments are in goodagreement with numerical modeling using Landau-de Gennes theory coupled with geometrical and topologicalconsiderations using the method of tetrahedral rotation.

DOI: 10.1103/PhysRevE.84.031703 PACS number(s): 61.30.Jf, 64.70.pv

I. INTRODUCTION

Defects in nematic liquid crystals (NLCs) have attractedscientists because of their intriguing structure and conceptualsimilarities with defects in nature, such as the cosmic stringsin the universe and vortices in magnetic systems. Originallyconsidered as the line or point singularities of the molecularorientation [1], their internal structure is much more complexand involves an interplay between the decrease of the degreeof uniaxial molecular ordering in the core of defects, and theonset of the biaxial order. While defects are usually not desiredin optical applications because they strongly scatter light,they play a prominent role in the crystallization of nematiccolloids. When foreign particles, favoring, for example, normalorientation of the NLC at the surface, are introduced intothe homogeneous bulk NLC, defects in the form of points[2] and small defect loops [3–5] are spontaneously createdat each particle. They are responsible for huge long-rangedistortions of the NLC in the vicinity of particles, whichtogether with defects form elastic (topological) dipoles orelastic quadrupoles [6]. As the free energy of two nearbyparticles depends on their separation, structural forces betweenthe colloidal particles appear. These forces [7,8], originatingfrom the nematic topological defects and elastic distortionof the liquid crystal (LC), are responsible for the assemblyof colloidal particles into interesting geometrical forms, suchas chains of particles [2] and two-dimensional (2D) colloidalcrystals [9]. Another type of colloidal interaction has beenobserved in nematic colloids, which results in topologicallyentangled colloidal states, such as the figure of eight, figureof omega, and figure of theta [10–12]. Here, a defect loopis encircling two or more colloidal particles, binding themtogether as an elastic strip.

The entanglement of nematic colloids into one-dimensionalcolloidal structures—colloidal chains and wires—was ob-

served in planar nematic cells [12], but 2D entangled nematiccolloids were never found, although predicted theoretically[13]. The possible reason is the distinct symmetry and thecharacteristic director profile of the −1/2 defect loops; ifsimply circular, the loops are also called the Saturn rings [3–5].The director field surrounding these loops lies generally in aplane, perpendicular to the direction of the defect line. There-fore, in planar cells, the −1/2 loops can propagate easily (i.e.,energetically favorably) only in an “easy plane,” i.e., a planeperpendicular to the overall director, which strongly limits thepossible conformations of the loops. Indeed, such directionallylimited propagation of the defect loops can be observed inentangled colloidal chains and wires in planar cells, wherethe defect loops deviated from the characteristic easy planesonly slightly, by a small twist or circulation. The attemptsto create 2D entangled colloidal crystals in planar cells withlaser tweezers were unsuccessful because they require linepropagation along the director, which is not possible in planarcells. However, this limitation is avoided in twisted nematiccells, where the direction of the easy plane varies with location.Here, the loops can propagate generally in any direction bysimply adjusting its position within the globally twisted di-rector. The experiments show that already moderately twistedπ/2 cells allow for advanced manipulation of such entangleddefect loop structures [14]. All of these arguments indicatethat we should expect the interaction of colloidal particles inthe chiral nematics to be much richer in its topology comparedto the simple homogeneous nematic colloids.

The colloidal interactions in homogeneous nematic liquidcrystal are now well understood, but the LC topology-mediatedinteractions in chiral nematic LCs are much less studiedand understood. It has been observed in the early work ofZapotocky et al. [15] that micrometer-diameter silica particlesimmersed in the cholesteric liquid crystal, with the helicalperiod larger than the particle diameter, formed a dense

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V. S. R. JAMPANI et al. PHYSICAL REVIEW E 84, 031703 (2011)

network of disclination lines. The resulting colloidal structurebehaved as a particle-stabilized defect gel with solidlikeelasticity due to a connected network of defect lines acting assprings between clusters of colloidal particles. The experimentof Hijnen et al. [16] on colloidal particles, with planar surfaceanchoring of the cholesteric LC and the helical pitch smallerthan the particle diameter, revealed not only the formationof colloidal chains, but also particle segregation in individualplanes and the formation of 2D aggregates.

While the macroscopic elasticity is an interesting effectin the particle-stabilized defect gel of a cholesteric liquidcrystal, the fine structure of defects around individual colloidalparticles has remained unexplored. The recent work of Tkalecet al. [17] is a systematic step toward understanding chiralnematic colloidal interactions, and demonstrates that thechirality of the nematic solvent indeed strongly influencesdefect formation and results in the formation of nonsingular,escaped hyperbolic loops of topological charge −2, mediatingthe attractive interaction between colloidal particles in a 90◦twisted nematic cell. This is a clear indication that the topologyof chiral nematic colloids might be much richer comparedto homogeneous, nonchiral nematics. It also indicates thatcolloidal structures with escaped hyperbolic defect loops arelikely to form in cholesteric twisted cells, in addition to theentangled colloidal structures based on −1/2 defect loops, aspresented in this paper.

Recent numerical calculations by Lintuvuori et al. [18,19]represent a significant step forward in understanding the defectstructure in chiral nematic colloids, as it is predicted thatcolloidal particles with planar surface anchoring in the bulkcholesteric liquid crystal are decorated by −1/2 defect loops,which are winding around the particle. This was, in fact,observed by Senyuk et al. [20] some time ago for planarcolloidal particles in moderately twisted chiral nematic LCs.The natural helical modulation of the orientational order inthe cholesteric liquid crystal, therefore, appears to impose akind of confinement to the inserted colloidal particles becausethe point defect is stretched into a Saturn ring [21], which inturn twists around the particle when increasing the chirality.The “strength” of this “chiral confinement” is expressed viathe ratio R/p of the particle diameter R to the cholestericpitch p. When this ratio is small, R � p, the colloidal particleexperiences practically homogeneous nematic order, and thedefect, which carries the topological charge [6], is either asmall singular point of the director field, characteristic foran elastic dipole [22], or a circular defect ring (Saturn ring)surrounding the particle well around its equator, producing anelastic quadrupole [3,23]. However, by increasing the strengthof chiral confinement, i.e., by shortening the helical pitch, weenter into a regime where R ∼ p, and the original point defector the circular defect loop transforms into a strongly deformeddefect loop. This strongly deformed defect loop carries thetopological charge and winds around the colloidal particle inthe form of a twisted −1/2 Saturn ring [4,5]. This situationis most interesting, as the winding of the defect line in spacecould provide the means for a much richer binding in a systemof a larger number of colloidal particles in 2D [9].

In this work, we present an experimental and theoreticalstudy of the nature of defect loops winding around colloidalparticles that impose a perpendicular (homeotropic) surface

anchoring of the surrounding chiral nematic liquid crystal. Weexplore the role of increasing chirality of the liquid crystal bysystematically shortening the helical pitch of the cholestericstructure, while keeping the colloidal diameter fixed. We thusstudy defects around colloidal particles with the diameter2R in cholesteric planar cells of fixed thickness h ∼ 2R andthe twisting angle of π , 2π , and 3π , respectively. We alsostudy the colloidal aggregation and find that already the singlecolloidal pair in the cholesteric liquid crystal can be boundin at least 17 topologically different structures. In the case ofentangled topological defects, we observe that loops can bespontaneously linked and knotted, thus forming the Hopf linkand the trefoil knot. The experiments are well supported withthe numerical and theoretical results of the tensorial Landau-deGennes modeling and topological analysis of single colloidsand colloidal pairs.

II. EXPERIMENT

In our experiments we used silica particles with a di-ameter of d = 20 μm (Bangs Laboratories, Inc.). Theirsurfaces were treated with DMOAP [octadecyldimethyl (3-trimethoxysilylpropyl)ammonium chloride, ABCR GmbH],as described in Ref. [23], to obtain strong perpendicularsurface anchoring of LC molecules on a chemisorbed DMOAPmonolayer. Silica colloids were dispersed in 4-cyano-4′-pentylbiphenyl (5CB) (Nematel), which was previously dopedwith various concentrations of the right-handed chiral dopant4-(2-methylbutyl)-4′-cyanobiphenyl (CB15) (Merck), to in-duce the right-handed chiral nematic phase of variable helicalpitch. The concentration of the chiral dopant and, therefore, thelength of the helical pitch of the mixture was adjusted to obtainπ -, 2π -, and 3π -twisted nematic structures in glass cells witha gap of 22 μm, respectively. Glass cells were assembled fromtwo indium-tin-oxide (ITO) coated, 0.7-mm thick glass slidescovered with a thin layer of a polyimide (PI-2555, NISSANChemicals). The polyimide layer was rubbed unidirectionallyto obtain an excellent planar alignment of doped 5CB, withdirections of the alignment on both surfaces set parallelduring the assembly of the cell. The gap thickness of thecell was controlled with mylar spacers and was determinedby measuring the transmission spectra of empty cells. Thechiral nematic dispersion was introduced into the glass cell bycapillary action at room temperature.

To observe and manipulate the colloidal particles in thechiral nematic environment, we used a laser tweezers [8,24–29] setup that was built around an inverted microscope (NikonEclipse, TE2000-U) with an infrared fiber laser operating at1064 nm as a light source and a pair of acousto-optic deflectorsdriven by a computerized system (Aresis, TWEEZ 70) for trapmanipulation [30].

III. THEORY AND MODELING

Cholesteric liquid crystal colloids are materials governed byan interplay between the liquid crystalline elasticity, materialchirality, formation of distorted defect regions, and surfaceanchoring at particle surfaces. A strong approach to describeall these mechanisms in cholesteric colloids with micrometer-sized particles is to use chirality-extended Landau-de Gennes

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(LdG) free energy [31], and then minimize it using numericalmethods. The Landau-de Gennes free energy F is based onthe order parameter tensor Qij , which incorporates all basicliquid crystalline orientational degrees of freedom. The freeenergy F reads

F =∫

LC

(L

2

∂Qij

∂xk

∂Qij

∂xk

+ 2q0Lεikl Qij

∂Qlj

∂xk

)dV

+∫

LC

[A

2QijQij + B

3QijQjkQki + C

4(QijQij )2

]dV

+∫

S

[W

2

(Qij − Q0

ij

)(Qij − Q0

ij

)]dS. (1)

Here, L is a single elastic constant, q0 = 2π/p is theinverse pitch, εijk is the totally asymmetric tensor, A, B, andC are material constants, W is the particle surface anchoringstrength, Q0 is the order parameter tensor imposed by thehomeotropic surface, and the summation over repeated indicesis assumed. In Eq. (1), the first two terms—the elastic andthe degree-of-order term—are bulk free energies and areintegrated over all volumes of the liquid crystal, whereas thethird term is the surface anchoring free energy integratedover the surfaces of all particles. The total free energy isminimized numerically by using the finite difference explicitEuler relaxation algorithm on a cubic mesh; modeling detailsare explained in Ref. [32]. In the calculations, the followingvalues of the material parameters are used, which map theexperiments qualitatively: L = 4 × 10−11N; A = −0.172 ×106J/m3; B = −2.12 × 106J/m3; C = 1.73 × 106J/m3; andW = 10−2J/m2. For single colloidal particles in Fig. 4, theparticle radius is set to R = 1.5μm and the cell thickness ish = 3.2μm. For colloidal dimers in Fig. 5, these parameterswere R = 1μm and h = 2.2μm. The pitch of the material ischosen for all Nπ -twisted cells to fit with the cell thickness:p = 2h/N . In the calculations, the size of the particles 2R

and the cell thickness h are smaller compared to those inreal experiments; however, it is the appropriately matchedratio 2R/h that proves sufficient to achieve good qualitativeagreement between the calculations and the experiments.

Single homeotropic particles in Nπ cholesteric cells (N �1) each stabilize one deformation pattern of the director.However, as seen in Fig. 4 of the next section, already a pair ofcolloidal particles in the Nπ cell can stabilize multiple defectloop structures. Energetically, these colloidal dimer structuresare stable and metastable, with their relative metastabilitydepending sensitively on the specific material parameters. Toapproach these various structures with numerical modeling,appropriate Ansatze (initial conditions) for Qij are constructed,which are then relaxed to the equilibrium. In a specific Ansatz,the profile of Qij is split into a near-field and far-fieldregion, where the near field (far field) corresponds to theinside (outside) of an ellipsoidal surface that encapsulatesboth particles. The director projection of Qij in the far-fieldregion n turns out to be similar to the field of elastic dipoles inuniform cells [30,31], but now appropriately twisted to complywith the overall twisted director profile of the cholesteric. Thefollowing Ansatz is used:

n = e + cr − rcol

|r − rcol|3 − cr + rcol

|r + rcol|3 , (2)

where r is the position vector, e = (cos φ, sin φ,0) is thefar-field twisted director (φ is the twist angle along thethickness of the cell), c is a constant, and rcol and −rcolare the positions of the two particles. The profile of Qij inthe near-field region—the region close and in between thetwo particles—is less affected by the overall twist in thecholesteric cell, and interestingly turns out to be locally similarto the profile of entangled structures in planar (nontwisted)cells. Therefore, for the inner region, we can use previousnumerically calculated equilibrium tensor profiles of theentangled dimers in planar cells (figure of eight, figure ofomega) [33].

IV. RESULTS AND DISCUSSION

A. Colloids in a π -twisted cell

When chiral nematic dispersion of colloidal particles isintroduced into the twisted nematic cell, the structure of thedisclination lines around the individual colloidal particle is noteasy to discern because of the closed packing of the defect linesin a small volume around the particle. We have, therefore, usedthe laser tweezers to grab and stretch the defect lines windingaround the particle in order to easily visualize and discerntheir structure. We see, in all cases (π , 2π , and 3π cells),a single and closed −1/2 disclination loop that winds in amore or less complicated manner around the colloidal particle,which is the topological source of the defect loop. It can,therefore, be considered as a single Saturn ring that is observedin quadrupolar nematic colloids in a spatially homogeneousnematic liquid crystal, which is now twisted because of thetwisting of the chiral nematic surroundings.

Figure 1 illustrates our general strategy of how to analyzeand determine the spatial distribution of defect lines in atwisted nematic environment. The colloidal particle of thediameter slightly smaller than the thickness of the π -twistedcell (h ≈ 2R) is shown on the left side of the left panel ofFig. 1(a). In this top view, one can see a blurry contour of thedefect line, which is similar to the “figure-of-eight” pattern onthe area occupied by the particle. Now, as this is the top view, itis difficult to discern how the twisted line propagates along thez direction. Using the fluorescent confocal polarization spec-troscopy does not help much because of the faint fluorescentcontrast between the defect line and the bulk liquid crystal. Wehave, therefore, used a rather simple and robust principle ofanalysis by grabbing and stretching the twisted loop with thehigh-power, focused beam of the laser tweezers, which locallymelts the liquid crystal due to the absorption of light. This isclearly visible as a microscopic island of the isotropic phaseon the right side of the left panel of Fig. 1(a). The position ofthe beam waist is indicated with a small cross. This isotropicisland acts as an effective particle, which strongly captures the−1/2 defect line, as visible on the right panel of Fig. 1(a),thus providing pulling and manipulation of the defect line. Itis clear already from this image that the defect loop is in aform of the figure of eight, which is embracing the sphericalparticle. This is further supported by capturing and stretchingthe twisted figure-of-eight defect loop in the orthogonal (i.e.,rubbing) direction by using a pair of laser tweezers, as one cansee from the panels in Fig. 1(b). Finally, the microscope view

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(a)

(b)

10 m 10 m

10 mStart Step 1 Step 2

noitceri

Dg

nib

bu

R

xz

y

(c)

z=8 mµ z= -8 mµ

(d)

FIG. 1. (Color online) (a) Silica microsphere of diameter d =20 μm with homeotropic surface anchoring in a right-handed π -twisted nematic cell with thickness h = 22 μm. Note the defect linewrapping around the colloidal particle. The circular object on the rightside of the left panel is the isotropic island of 5CB, produced by localheating with high-power laser tweezers, indicated with a small cross.The laser power was 100 mW for each trap. Right panel shows howthe molten island of 5CB captures the defect line and stretches it in thex direction. (b) Two equal beams of the laser tweezers were used tostretch the defect line in the y direction. The rubbing directions set theorientation of LC molecules at the top and bottom glass, respectively.(c) The defect line is stretched by two beams in the y direction, andthe images between crossed polarizers are shown for two different zpositions of the focal plane of the microscope, separated by 16 μm.Note that only a part of the defect line is in focus on both panels. (d)Schematic view of the laser-tweezers-stretched defect line, wrappingthe colloidal particle in the π -twisted nematic cell.

of the laser-stretched defect loop under crossed polarizers inFig. 1(c) confirms our conjecture: the −1/2 defect loop in theπ -twisted cell is in the form of the figure of eight, embracingthe colloidal particle, as schematically presented in Fig. 1(d).It is important to note that this figure-of-eight defect loop istopologically different (has different topological charge) fromthe figure-of-eight structure found in colloidal dimers in planarnematic cells [12].

B. Single colloidal particle in a 2π - and 3π -twisted cell

A colloidal particle with homeotropic surface anchoring ina 2π -twisted nematic cell is shown in Fig. 2(a). At first sight,the defect line, wrapping around the particle, looks similarto that in a π -twisted nematic cell in Fig. 1(a). However,a more detailed inspection reveals an additional thin whitering, encircling the particle shown in Fig. 2(a), indicating amore complicated winding of the closed defect loop aroundthe colloidal particle in the case of the 2π -twisted nematiccell. This ring is definitely not visible in Fig. 1(a). In the

(c)

(b)

10 m

noitceriD

gnibbuR

z=8 mµ z=-8 mµz=0 mµ

Step 1Start

(a)

10 m

Step 2

(d)

xz

y

10 m

FIG. 2. (Color online) (a) Silica microsphere of diameter d =20 μm with homeotropic surface anchoring in a right-handed 2π -twisted nematic cell with thickness h = 22 μm. Note the defect linewrapping around the colloidal particle. The circular object on the rightside of the left panel is the isotropic island of 5CB, produced by localheating with high-power laser tweezers, indicated with a small cross.The laser power was 170 mW for each trap. The right panel showshow the molten island of 5CB attracts the defect line and stretchesit in the x direction. (b) Two equal beams of the laser tweezers wereused to stretch the defect line in the y direction. (c) The defect lineis stretched by two beams in the y direction, and the images betweencrossed polarizers are shown for two different z positions of the focalplane of the microscope, separated by 16 μm. (d) Schematic viewof the laser-tweezers-stretched defect line, wrapping the colloidalparticle in the 2π -twisted nematic cell.

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xz

y

(a)

(b)

noitceriD

gni bb uR

Start Step 1

Step 2

(c)

10 mµ 10 mµ

10 mµ

FIG. 3. (Color online) (a) Silica microsphere of diameter d =20 μm with homeotropic surface anchoring in a right-handed 3π -twisted nematic cell with thickness h = 22 μm. The three circularobjects are the isotropic islands of 5CB, produced by local heatingwith high-power laser tweezers, indicated with a small cross. Thelaser power was set to 200 mW for each trap. The right panel showshow the molten islands of 5CB attract the defect line and stretchit. (b) The same situation as in (a), stretching the defect loop withthree independent traps and sequentially observed between crossedpolarizers. (c) Schematic view of the laser-tweezers-stretched singledefect line, wrapping the colloidal particle in the 3π -twisted nematiccell. Each part of the single defect line between two isotropic islandsis marked differently.

2π -twisted nematic cells, when grabbing and stretching thedefect loop by the laser tweezers in the x or y directions,the ring is deformed in a quite different manner comparedto Figs. 1(a) and 1(b). Finally, the images between crossedpolarizers, with the defect loop stretched by the two lasertweezers in the y directions, and taken at two different heightsof the focal plane of the microscope, as presented in Fig. 2(c),clearly reveal the winding pattern of the defect loop, which isillustrated schematically in Fig. 2(d).

Colloids in a 3π -twisted nematic cell exhibit an even morecomplicated winding pattern of the defect loop, which ispresented in Figs. 3(a)–3(c). Three different laser traps hadto be used in this case to discern the winding pattern of thedefect loop, which is schematically shown in Fig. 3(c).

The winding patterns of defect loops in π -, 2π -, and3π -twisted nematic cells are summarized in Figs. 4(a)–4(d),together with the results of the LdG numerical analysis

noitceriD

gnibbuR

(a)

10 mµ 10 mµ

10 mµ

10 mµ

(b)

(c)

xz

y

xz

y

xz

y

x

z

yxz

y

x

z

y

No Twist

Increasing Twist

(d)

x

z

y

xz

y

10 mµ

10 mµ

xz

y

xz

y

(e)

FIG. 4. (Color) (a) Unpolarized and polarized images of 20 μmsilica colloidal particle with homeotropic surface anchoring in a h =22 μm thick and right-handed π -twisted cell of 5CB. The right panelshows the result of LdG numerical analysis. The red line presents theregions of the CLC with the order parameter S = 0.51. The size ofthe colloidal particle is 3 μm and the cell thickness is 3.2 μm. (b) Thesame as in (a), but the twist of the cell is now 2π . (c) The same as in(a), but the twist of the cell is now 3π . (d) Numerical LdG calculationof winding of the defect loop in planar, π -, 2π -, and 3π -twisted CLCcells. Defects are shown in red as isosurfaces of S = 0.51. (e) Topand side view of the winding of the defect line in a 10π -twisted CLCcell.

displayed on the far-right panels. A reasonably good visualagreement can be seen by comparing the microscope imageson the left and middle panels with the theoretically predicteddefect structures in the right panels in Figs. 4(a)–4(c).Figure 4(d) is a schematic presentation, using numericallycalculated images, of what happens to the defect loop whenthe twist of the surrounding cholesteric liquid crystal (CLC)medium is increasing from planar to π -, 2π -, and 3π -twistednematic structures. The first panel in Fig. 4(d) shows a colloidalparticle in a planar, nontwisted cell, which is only slightlylarger than the colloidal diameter. The two glass surfaces areabove and below the particle. Because of the confinement,there is the well-known Saturn ring, which encircles theparticle and lies in the plane, perpendicular to the director.

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Now, one can easily imagine that the π -twisted cell is obtainedby rotating the upper glass plate around the z axis by π . Thistwisting results not only in the π -twisted nematic cell, but alsoin the π -twisted Saturn ring, which is shown in the secondpanel from the left in Fig. 4(d). It is easy to guess what happenswith this twisted ring when the nematic structure is twisted byNπ : the ring is transformed into a N -half-turn-twisted ring, ascan be seen by following the subsequent panels in Fig. 4(d).

As we have now a rather simple explanation of the windingof the defect loop around the homeotropic colloidal particlein a twisted nematic cell, one can predict the structure of adefect line for arbitrary twist. For example, the defect loop ina 10π -twisted nematic cell should show nine visible parts ofthe defect line, winding around the particle, plus an additionaltwo lines at the upper and bottom “cap,” when observed fromthe side along the x direction. This is indeed in agreementwith numerically calculated images of the twisted defect loopshown in Fig. 4(e).

C. Entangled colloidal dimers and clusters in π -twisted cells

The winding of the defect loop around the colloidal particlesin the cholesteric liquid crystal should have a pronouncedeffect on the particle pair interaction, and, in particular, on thenumber and structure of the equilibrium states of topologicallyentangled colloidal dimers and clusters. In planar nematiccells, three different colloidal entangled states have beenreported so far [12], in which two of them could appear eitherin right- or left-handed form. In a twisted environment, it isexpected that more entangled colloidal structures are stableand can be more easy formed because the defect lines havemore freedom of propagation through the twisted structure ofthe cholesteric liquid crystals.

We have performed the experiments where a pair ofcolloidal particles was positioned at a close proximity using thelaser tweezers. Then, the power of the tweezers was increasedso that the CLC was locally molten into the isotropic phasebecause of the light absorption in a thin ITO layer on the innerglass interface. The tweezer power was adjusted so that bothcolloidal particles were in the isotropic melt and then the lightwas shut down. This resulted in a rapid temperature quench ofthe isotropic island, which was accompanied by the formationof a dense tangle of topological defect loops. In the courseof time, most of them annihilated, while one or two of themremained in a form of twisted loops, entangling both colloidalparticles.

The results of the quenching experiments are surprising, aswe found that a single colloidal pair in a π -twisted CLC cellcould be entangled in at least 17 different ways. Some selectedexamples of entangled colloidal dimers in a π -twisted LC cellare shown in Figs. 5(a)–5(f), together with the probability oftheir formation in a quenching experiment. A closer inspectionof the entangled colloidal dimers reveals that they are allvariations of the same structure. These variations can beunderstood by applying the method of localized tetrahedralrotation to the orthogonal −1/2 defect lines crossings (ortangles) [33]. The differences between the dimer structurescan be viewed as localized rewirings at four tetrahedrallyshaped rewiring sites, as schematically illustrated in Fig. 6.Each rewiring site (labeled A, B, C, and D in Fig. 6) can be

(e)

no itc eri

Dg

nib

bu

R

10 mµ

(f)

(c)

(b)

(a)

(d)

10 mµ

10 mµ

10 mµ

10 mµ

10 mµ

kni

Lf

po

Hst

onk

nU

kn il

nU

FIG. 5. (Color online) Entangled colloidal dimers in the right-handed π -twisted CLC cell of h = 22 μm thickness. All structureswere obtained by quenching a colloidal pair from the isotropic phase,created by local heating with the laser tweezers. Left panels are takenin unpolarized light; right panels present either numerically calculatedcolloidal dimers (S = 0.51) or just a schematic presentation ofdefect loop topology, as depicted from left panels. In a total of 176experiments performed, the various structures [(a)–(f)] appear in (a)6.8%, (b) 7.3%, (c) 6.2%, (d) 26.1%, (e) 5.1%, and (f) 10.8% of theexperiments.

in one of three distinct states, depending on which verticesof the tetrahedron are connected with a disclination line[33]. This gives 34 = 81 possible structures, but not all ofthem are different, since the defect loop conformations andthe surrounding director are symmetric under rotations forπ around the x, y, and z axes. The distinct structures canbe discriminated (from all 81 structures) by applying thejoint π rotations to the four rewiring sites, which actuallycorresponds to permutations in the sequence of the rewiringsites (A,B,C,D). For example, a symmetric π rotation along thez axis changes the sequence (A,B,C,D) into (C,B,A,D). Sinceeach rewiring site is in one of three possible tangle states (cross,bypass along x, or bypass along y), all 34 = 81 structuresare generated and tested under the permutations, substitutingeach state sequence with its first equivalent permutation inlexicographical order and removing the duplicates.

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(a) (b)

(d)(c)

A

B

C

D D

B

D

x

yz

FIG. 6. (Color online) Schematic depiction of entangled colloidaldimers with four rewiring sites labeled A–D. The rewiring sitesare local tetrahedrons, where disclinations can take different paths,forming different entangled structures. The chosen structures (a)–(d)only differ in the top (B) and the bottom (D) rewiring sites, i.e.,tetrahedrons. Rewiring of the tetrahedron D transforms the structure(a) into (b). Subsequent transformations into (d) and (c) are performedin a similar manner, as shown in the figure. (a) is the most symmetricentangled dimer that is invariant to rotations for π around any of theprincipal axes. (b)–(d) are the same structures as Figs. 5(a), 5(b), and5(f).

We find 36 topologically different dimers, which includesuncoupled Saturn ring defects and the six structures, asdepicted in Fig. 5. This classification only accounts fortopology and symmetry of the configurations. The tendency ofa physical system to minimize the free energy and shorten thedisclination lines leads to energy differences and variations indisclination geometry, which is evident from a comparison ofthe experiments and the simulations [Figs. 5(a), 5(b), and 5(f)]with corresponding schematic structures [Figs. 6(b), 6(c), and6(d)]. The chirality of the medium also breaks the symmetrybetween different tangles [compare the B and D tangle inFig. 6(c)], which energetically distinguishes structures ofdifferent handedness. Depending on the specific materialproperties and particle size, the opposite handed structurescan be energetically stable, metastable, or even unstable withrespect to one another. Finally, it is important to note that thisclassification of possible dimers (total of 36 distinct structures)incorporates structures with explicitly four presented rewiringsites of −1/2 defect loops. For example, dimers based onescaped −1 defect loops and the elastic dipoles could also beenvisaged.

The experimentally observed entangled colloidal dimerscan also be classified into two categories, depending onthe number of closed defect loops entangling both particles.Figures 5(a)–5(d) present entangled colloidal dimers, whichare bound by a single closed defect loop. The interesting formsof the defect loops raise the question of whether some of thesesingle loops are knotted. Now, in topology, this question couldbe answered by performing the Reidemeister moves [34,36],which involve the arbitrary manipulation of the closed defectloop, except for cutting it and rewiring it. After performingthe Reidemeister moves, one observes the projection of the

FIG. 7. (Color) Entangled colloidal clusters in the right-handedπ -twisted CLC cell. The images on the left are taken under crossedpolarizers. The panels in the middle column represent drawings ofthe defect loops, as deduced from images on the left. The rightpanels show the topologically minimized defect loop structure, afterperforming Reidemeister moves. These panels were drawn usingKNOTPLOT 1.0. (a)–(e) Colloidal particles of diameter d = 20 μmin h = 22 μm thick cell. (f) Colloidal cluster of h = 10 μm silicamicrospheres in a h = 12 μm thick cell.

loop onto a plane and looks for any crossing of the loop.If, by any means, one cannot get a simple circle (i.e., withoutloop crossings) after performing the Reidemeister moves, thenwe have a knotted loop. It turns out that no structure inFigs. 5(a)–5(d) is a knotted one, and neither are any of theother structures predicted by the formalism of the tetrahedralrotations.

Among structures entangled by two defect loops inFigs. 5(e) and 5(f), we find another interesting topologicalobject, shown in Fig. 5(f). The two defect loops are linked, i.e.,one loop is “passing” through the other, similar to the linked

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structure of a chain. In mathematics and physics, this topolog-ical structure is known as the Hopf link. The linking of closedloops is, in fact, quite common in supramolecular chemistry,where the linked molecular rings are called “catenanes” [35].The linking of defect loops was reported in liquid crystalsby Bouligand [36]. The linking and knotting of topologicaldefect loops in chiral nematic colloids has quite recently beenanalyzed and reported [14].

By increasing the number of colloidal particles, we obtaincolloidal clusters with even richer topological defect struc-tures, as shown in Figs. 7(a)–7(f). The colloidal trimer inFig. 7(a) is entangled by a single defect loop, which is anunknot. Similar is the single defect loop, entangling a colloidaltetramer in Fig. 7(b). However, a colloidal trimer in Fig. 7(c)is entangled by two closed defect loops that are linked, thusforming the Hopf link. The colloidal tetramer in Fig. 7(d)is entangled by a pair of interlinked defect loops, thus alsoforming the Hopf link. There is another topologically distinctstructure in a colloidal tetramer, which is presented in Fig. 7(e).Here, three defect loops are mutually interlinked, thus forminga short topological chain of loops. And finally, the colloidalcluster of a regular shape, shown in Fig. 7(f), is entangled by asingle defect loop, which is knotted. It has three crossings andis known in topology as a trefoil knot [34].

V. CONCLUSION

In conclusion, we have experimentally demonstrated thatthe chirality of the surrounding medium has a prominent effecton the geometry and topology of the closed defect loopsaround colloidal particles. We find that for a single colloidalparticle in the chiral nematic liquid crystal, the −1/2 defect

loop winds around the particle, which is in agreement withrecent theoretical predictions. Even more surprising is theinfluence of the chirality on the structure and topology ofentangled colloidal dimers, trimers, and clusters. These arealways entangled by twisted loops and the number of loopsand their topology are surprisingly rich, as a single colloidalpair in a π -twisted cell can theoretically be entangled in 36ways, 17 of which have been observed experimentally. Therichness of colloidal interactions is even more pronounced incolloidal trimers and tetramers, where we observe that severaldefect loops can be interlocked, forming the Hopf link, andthey can also be knotted into the trefoil knot in larger colloidalclusters. It is interesting that links can be formed in colloidaldimers, while the formation of the simplest knot, i.e., thetrefoil, requires at least seven colloidal particles, as observedin the experiments. This study demonstrates that the chiralityof the medium providing the colloidal interaction is essentialfor the spontaneous formation of novel colloidal topology,where the colloidal particles are entangled, linked, and knottedinto a new type of colloidal matter.

ACKNOWLEDGMENTS

This work was supported by the European Commission(EC) 7OP Marie Curie ITN project HIERARCHY Grant No.PITN-GA-2008-215851 (V.S.R.J.), the Slovenian ResearchAgency (ARRS) Contracts No. P1-0099 and No. J1-3612,and in part by the Center of Excellence NAMASTE. M.R.acknowledges the support of the European Commission (EC)under the Marie Curie Program Active Liquid Crystal Colloids(ACTOIDS).

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