PHYSICAL REVIEW E 84, 051702 (2011)

Nematic disclinations as twisted ribbons

Simon Copar,1 Tine Porenta,1 and Slobodan Zumer1,2

1Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia2J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia

(Received 10 August 2011; revised manuscript received 10 October 2011; published 8 November 2011)

The disclination loops entangling colloidal structures in nematics are geometrically stabilized and restrictedaccording to topological rules. We focus on colloidal dimers and show how the writhe and twist, which areconstrained to sum to a constant value, are affected by changing the intercolloidal spacing, the twist angle of thecell, and the cholesteric pitch. We analyze the geometric properties of disclination loops using finite differencenumerical simulations of colloidal dimers. The observed trends are explained and correlated to the symmetryproperties and effects of liquid crystal elasticity.

DOI: 10.1103/PhysRevE.84.051702 PACS number(s): 61.30.Jf, 61.30.Dk, 82.70.Dd

I. INTRODUCTION

A nematic liquid crystal is an exemplary anisotropicmaterial with highly tunable physical characteristics and well-researched manipulation techniques [1]. Its switching capa-bilities and unique topological properties make it interestingfor use in both modern technology and academic research.Nematics can contain both point and line defects in theirordered structures [2], depending on the external fields andanchoring conditions enforced by the boundaries. Inclusion ofcolloidal particles in the nematic enables controlled creationof stable defects. In particular, colloidal particles that favorhomeotropic alignment of the nematic director on their surfacecatalyze the creation of −1/2 disclination loops [3].

Experiments with laser tweezers on samples containingspherical silica particles have shown these disclinations canbe cut and reconnected into entangled states [4]. Dependingon the choice of containment cell, particle size, and materialproperties, the disclinations can form nematic braids thatentangle particles into various structures, from linear chains [4]to small clusters and two-dimensional (2D) periodic lattices[5–7]. The rewiring of disclinations also controls the topology.Studies of particles in a π/2 twisted nematic cell haveeven shown that arbitrary knots and linked structures can beproduced by simple optical manipulation [8].

Compared to localized disclinations such as point de-fects and Saturn rings, entangled structures are topologicallycomplex and require a systematic approach to describetheir geometric properties. In the specific case of −1/2disclinations, it has been shown that different geometricconfigurations differ locally at tetrahedrally shaped rewiringsites and that the cumulative twisting of the disclinationline cross section can be described by an invariant calledthe self-linking number [9]. The theory behind the self-linking number of nematic disclinations predicts possibleentangled structures and provides restrictions and conservationlaws [8,9].

In this paper, we use numerical simulations based onfree-energy minimization to calculate length, writhe, and twistof the disclination lines around colloidal dimers contained in anematic cell. We investigate the dependence of these quantitieson parameters such as the cholesteric pitch, boundary condi-tions of the nematic cell, and intercolloidal separation. Wecompare the results with phenomenological expectations and

quantify the differences in shape using the geometric quantitiesobtained from the simulations.

II. THEORETICAL BACKGROUND

Nematic disclination lines have internal structure withnoncylindrical symmetry, so they must be treated as ribbons,not as simple curves. A ribbon is a geometric object thatholds the information about both the path the ribbon tracesin space and the orientation of its cross section in the planeperpendicular to the direction of propagation. The behavior of aribbon during its propagation in space can be separated into twofundamental contributions, twisting and writhing [Fig. 1(a)].Twisting is the rotation of the cross section around the ribbon’stangent. It is measured by the geometric twist,

Tw = 1

2π

∫t(s) ·

(u(s) × ∂u(s)

∂s

)ds, (1)

where t is the unit tangent to the curve and u is a unit vector,representing the orientation of the ribbon’s cross section [forexample, the gray sticks in Fig. 1(b)]. Writhing is the changingof the tangent direction along the curve and can be measuredby the quantity called writhe, defined by the Gauss integral

Wr = 1

4π

∫∫t(s) × t(s ′) · r(s) − r(s ′)

|r(s) − r(s ′)|3 ds ds ′. (2)

For a ribbon that forms a closed loop, the number of rotationsof the cross section during one round trip, the self-linkingnumber Sl, can be measured. The self-linking is related tothe geometric twist and writhe by the Calugareanu theorem,Sl = Wr + Tw [10]. It is a topological invariant, since it doesnot change under transformations that keep the loop closed. Aclosed ribbon must also satisfy continuity of its cross section,so the self-linking number is restricted to discrete values. Inthe case of −1/2 nematic disclinations, the director field ofthe cross section has a threefold symmetry, so its self-linkingnumber is restricted to third-integer values [9].

It is easily shown that the integrand ω = t · (u × us) inthe expression for geometric twist Eq. (1) is simply the local“angular velocity” of the ribbon’s cross section around itstangent. Specifically, for −1/2 disclination lines, rotatingthe cross section profile at a rate ω results in a rotation ofthe director field n at a rate of 3/2ω, as demonstrated inFig. 1(b). The rate of the director field rotation is expressible in

051702-11539-3755/2011/84(5)/051702(7) ©2011 American Physical Society

SIMON COPAR, TINE PORENTA, AND SLOBODAN ZUMER PHYSICAL REVIEW E 84, 051702 (2011)

FIG. 1. (Color) (a) Illustration of the difference between writheand geometric twist. The writhe measures the amount of three-dimensional variation of the ribbon’s tangent, while the geometrictwist measures rotation of the director field around the tangent. (b)For −1/2 nematic disclinations, the geometric twist that representsthe rotation of its cross section (marked with gray sticks) is linearlyproportional to the local elastic twist that appears in the free energyexpansion (green ellipsoids).

differential form as 3/2ω = −n · (∇ × n), which is the twistcontribution to the free energy in Frank’s theory of elasticity[2]. The geometric twist described in the Calugareanu theoremis therefore naturally penalized by the elasticity of the liquidcrystal.

The theoretical derivation of the self-linking numbersinvolves setting geometric twist to zero [9]. However, thereis no reason for this condition to hold exactly in a real system.As the experiments cannot provide a three-dimensional pictureof the director field and disclination line profile, we usesimulations to approximate the physically realistic directorfield. For numerical simulations, we use a Landau–de Gennesapproach, based on the tensorial order parameter Qij and theone elastic constant approximation [2]. The method involvesminimizing the free energy functional

F = L

2

∫bulk

(∂Qij

∂xk

∂Qij

∂xk

+ 4q0εiklQij

∂Qlj

∂xk

)dV

+∫

bulk

(A

2QijQji + B

3QijQjkQki + C

4(QijQji)

2

)dV

+ W

2

∫surf

(Qij − Q0

ij

)(Qji − Q0

ji

)dS (3)

on a finite difference grid [4]. The grid resolution is 7.5 nm,similar to the nematic correlation length for our material.The material parameters are set to the one-elastic-constantapproximation of the 5CB (4-cyano-4′-pentylbiphenyl) at atemperature 10 K below the isotropic-nematic transition,L = 4 × 10−11 N, A = −0.172 × 106 J/m3, B = −2.12 ×106 J/m3, and C = 1.73 × 106 J/m3. The same parameterswere used by Ravnik in [4]. The inverse cholesteric pitch,q0 = 2π/p, is zero except when simulating chiral nematics[11]. Strong homeotropic anchoring on the particles is ensuredby setting W = 1 × 10−2 J/m2.

The disclination loops are found in the numerical simula-tions by searching for regions with low values of the scalarorder parameter. An initial point on the disclination is chosen,then a small sphere surrounding it is searched for the lowestvalue of the order parameter and a step is made in that direction.The process is repeated until the loop closes on itself. Theobtained polygonal approximation of the loop is then analyzedfor different properties, in particular their length, twist, andwrithe. As the twist and writhe are coupled by a knownself-linking number, only the writhe is calculated numerically,which is much easier than extracting the geometric twistfrom the director profile. The writhe can be calculated eitherdirectly from the Gauss integral Eq. (2) or by using the tangentindicatrix approach [12,13]. The tangent indicatrix is the curvetraced by the tip of the normalized tangent to the loop. Thiscurve lies on the surface of the unit sphere and the surfacearea A it bounds gives the writhe, Wr = A/2π − 1 mod 2.The modulo 2 reflects the fact that full sphere wraps can beadded to an area without changing the boundary curve. Thespherical area can be calculated from its boundary curve usingthe Gauss-Bonnet theorem,

Wr = − 1

2π

∑αi − 1

2π

∮kgds mod 1. (4)

The first term sums over the jump angles αi , the abrupt changesin direction of the tangent indicatrix curve, while the secondintegrates the local geodesic curvature, kg . For polygonalcurves, the tangent is piecewise constant. We assume thatthe abrupt changes of the tangent happen along the shortestpath on the unit sphere, which implies kg = 0. The modulo 1is a consequence of the 2π ambiguity in angles αi . We usethe Gauss integral method Eq. (2) to calculate the value ofthe writhe without the ambiguity and the tangent indicatrixmethod to verify the fractional part of the result and boundthe discretization error, which is discussed in [14]. Based onthe differences between results given by both methods and thediscretization errors, we estimate that the writhes in this paperhave an accuracy of ∼0.005.

III. VISUALIZATION TECHNIQUE

To examine the internal structure of −1/2 disclinationlines, their threefold profile has to be extracted from the orderparameter tensor field. The scalar splay-bend parameter SSP

is constructed from second derivatives of the order parametertensor Qij :

SSP = ∂2Qij

∂xi∂xj

. (5)

This expression equals the divergence of order- and flexo-electric polarization that arises from the deformation of thenematic field with the single flexoelectric constant set to 1 [15].Transforming from order parameter tensor into director fieldrepresentation n(r), and assuming there is no variation in thescalar order parameter S, allows the splay-bend parameter tobe rewritten as

3S

2∇(n(∇ · n) − n × (∇ × n)). (6)

051702-2

NEMATIC DISCLINATIONS AS TWISTED RIBBONS PHYSICAL REVIEW E 84, 051702 (2011)

FIG. 2. (Color) (a) Threefold profile of a −1/2 disclination line,visualized using the splay-bend parameter SSP. The positive valuesof this parameter highlight high-splay regions (blue), while thenegative values highlight high-bend regions (yellow). The core ofthe disclination exhibits low values of the order parameter S (red).(b–e) The well-known Saturn rings, theta, figure-eight, and omegadimer structures, respectively. The threefold ribbons formed by thehigh-bend and high-splay isosurfaces can be traced around the loopsto see if they end up with the same orientation. This reflects thatthe self-linking number is Sl = 0 for (b,c) and Sl = −2/3 for (d,e).Numerically integrated writhe Wr deviates from its ideal value of Slfor (d,e). Free energy differences �F from the ground state (b) areshown in arbitrary units. The interparticle separation is 0.116 nm.

This representation clearly shows the decomposition into splayand bend terms from the Frank expansion of the free energy[2,16]. Regions of strong splay deformation have large positiveSSP while negative values imply strong bend deformation.

For −1/2 disclinations, the divergence of the flexoelectricpolarization is known to have a threefold spatial profile [17],which is convenient for visualization of the disclination lines(Fig. 2). The self-linking number can be easily countedby following the ribbon, formed by the isosurfaces of SSP

[Figs. 2(b)–2(e)]. This technique is immune to moderatedeviations from the ideal director profile, as long as the splayand bend retain three extremes around the disclination. It canalso be used for defect lines with different winding numbers,excepting +1 defect lines, due to their cylindrical symmetry.

IV. DISCUSSION

Three separate numerical experiments have been devisedin order to explore the response of the disclinations to thevariation of material properties and confining geometry. Thetest case is the well-researched colloidal dimer state, whichconsists of two colloidal particles with strong homeotropicanchoring. In a parallel nematic cell, the disclinations can formsix different structures: the theta structure, the Saturn rings,and the omega and figure-eight structures of both chiralities[4,9]. The Saturn ring structure is generally the most stable[4], but the precise stability properties strongly depend onparticle size, cell thickness, and material properties. Full freeenergy minimization, including the dynamics of the colloidal

particles, shows that, in equilibrium, the colloidal particlesare not necessarily aligned with the rubbing direction [5].However, fixed particle simulations are used in our case, as ourgoal is to describe the behavior of the disclination loops, notthe dynamics and stability of entangled dimers. We simulateparticles of diameter 1 μm in a cell of thickness 1.5 μm.

The behavior of disclination lines is observed understretching of the dimer, variation of the angle between therubbing directions of the top and the bottom wall of thecell, and, finally, changing from ordinary to chiral nematic.In simulations where interparticle distance is fixed, it is setto 116 nm, which is larger than in equilibrium but leaves thedisclinations more space to move around and pronounces thedifferences in writhe and twist.

A. Parallel nematic cell

We first focus on dimers in a parallel cell and a nonchiralnematic. Simulations have been performed for different inter-particle distances and the writhes and disclination line lengthsmeasured. For structures with two loops (Saturn rings and thetastructures), the values of individual loops are added together.The two components of the Saturn ring structure always haveequal properties anyway, because they are symmetric underthe π rotations around the rubbing direction.

The length of the disclinations increases with interparticledistance because it must span a larger gap between them. Anotable exception is the central ring of the theta structure,which shrinks when the particles are not close enough toact as a sterical constraint [Fig. 3(b)]. The disclination linesmostly contribute to the free energy cost because of theirisotropic core. This cost increases linearly with the disclinationlength, which results in approximately constant interactionforce under stretching, as established in existing work [4]. Weonly present the results for the disclination lengths, whichare comparable across simulations with different materialparameters and boundary conditions.

The theory predicts that the theta structure and the Saturnrings should have a zero self-linking number, while thefigure-eight and omega structures should have Sl = ±2/3,where the sign depends on the chirality [9]. The visualizationof the threefold profile of the simulated data confirms this fact(Fig. 2). Ideally, the geometric twist of the disclination loopshould be zero and the writhe should equal the self-linkingnumber. In the parallel cell, numerical calculation confirmsthis: the writhe for the theta structure and Saturn ringsstructures is exactly zero, as expected from the symmetry[Figs. 2(b) and 2(c)]. For the figure-eight and omega structures,the deviation of writhe from the ideal value is almost zerowhen the particles touch and increases approximately linearlywith the interparticle distance [Fig. 3(a)]. Larger deviationmeans more geometric twist, which is energetically unfavor-able due to elasticity, as demonstrated in the previous section.At greater distances, the tendency to shorten the disclinationscompetes with the tendency to reduce the geometric twist.Increasing the interparticle separation produces opposite ef-fects for the omega and figure-eight structures [Fig. 3(a)],which has to do with the fact that the omega structure tends toplanarity if it is stretched, while the figure-eight crosses overitself [Figs. 2(c) and 2(d)].

051702-3

SIMON COPAR, TINE PORENTA, AND SLOBODAN ZUMER PHYSICAL REVIEW E 84, 051702 (2011)

7.0

7.1

7.2

7.3

0.50.70.91.11.31.51.7

0 50 100 150 200 250

disc

linat

ion

leng

th[μ

m]

distance [nm]

-2/3

-0.8

-0.7

-0.6

-0.5

writ

he

omegafigure-eighttheta (total)

theta (ring only)

FIG. 3. (Color) Influence of the interparticle separation on thewrithe and total disclination length. (a) With increasing separation,the writhe of the omega structure increases and the writhe of thefigure-eight structure decreases. Linear extrapolation suggests thatthe writhe converges to the ideal value of −2/3 when the particles aretouching. (b) The disclination length increases monotonously with theinterparticle distance, except for the theta structure, where the centralring shrinks with the increasing gap (bottom data set). The data forthe omega structure stops at 200 nm separation when it collapses intotwo Saturn rings.

B. Twisted nematic cell

The symmetry of the parallel cell preserved the zero writheof the theta structure and made the left- and right-handedomega and figure-eight structures exact mirror images. Intwisted nematic cells, this symmetry is broken. Particularlyrelevant to experiments is the π/2 twisted cell, which has beenshown to promote 2D crystal self-assembly and linking of thestructures [8]. The gradual transition from parallel cell to ±π/2twisted cell has been simulated, without introducing chiralterms into the free energy minimization (Fig. 4). Using bothdirections of cell twisting allows observation of the symmetrybreaking for the omega and figure-eight structures, which haveinherent chirality. Similar simulations have been performedpreviously, but only on the figure-eight structure and withoutcalculation of the geometric properties [18].

The lengths of the disclination lines we measured have aquadratic dependence on the twist angle of the cell, with theresiduals within the confidence interval of the measurements.The quadratic coefficients are similar for all four structures

[Fig. 4(b)]. The undisturbed nematic in a twisted cell has thefree energy proportional to the square of the twist angle due tothe elastic twist contribution [2]. We subtracted this valuefrom free energies obtained from our simulations to extractthe free energy contribution of disclinations and distortedparts of director fields around the particles. The dependenceof these differences is also quadratic and qualitatively matchesthe disclination length dependence [Fig. 4(a)], so the twoproperties are roughly interchangeable for the purpose ofstability estimation. This correlation suggests that the effect oftwisting on the disclinations can be understood as equilibrationof the free energy cost between the unfavorable bulk elastictwist and the length of the isotropic disclination core.

The disclination lengths of the theta and Saturn rings havea symmetric dependence, whereas the figure-eight and omegastructures exhibit an offset, with shorter disclinations andtherefore lower energies at a nonzero twist. The minimumdisclination length is around the twist angle of 11◦ for theomega structure and −23◦ for the figure-eight structure. Notethat changing the gap between the colloidal particle and thecell walls would greatly influence this result. Left-handed(Sl = −2/3) varieties of these two structures were simulated,but their optimal configurations occur at twists of oppositesign, which means the different shapes of the loops can havedifferent global chiral effects even if their writhes are the same.The Sl = +2/3 structures should produce similar results, butmirrored and multiplied by −1.

All three entangled structures survive the gradual twist to±π/2 without breaking apart. However, the theta and omegastructures, which are already less stable in the parallel cell,have significantly longer disclinations than the figure-eightstructure of the correct chirality for the chosen twist directionof the cell. Figure 4(a) shows that the figure-eight structure’sfree energy is even lower than could be estimated from thedisclination lengths alone and is the most stable structure intwisted cells.

The writhe of all four structures changes with twisting ofthe cell. The symmetric structures develop nonzero writhe withan extended range of linear proportionality to the twist angle.The figure-eight structure has an even better match in twistedcells, regardless of the direction of the twist, while the omegastructure behaves differently [Fig. 4(c)]. The qualitativelydifferent behavior is caused by the fact that a twisting of thecell acts globally by changing the boundary conditions and itseffect depends strongly on the relative position of disclinationsto the preferred director field.

C. Chiral nematic in a parallel cell

In the Calugareanu decomposition, the global shape isdescribed by the writhe, while the local behavior is describedby the geometric twist. To influence the geometric twistdirectly, a local effect must be introduced. As demonstratedin Fig. 1, the geometric twist is closely related to the elastictwist in Frank elastic theory, so the natural way to affect it isto use a chiral nematic, which has an intrinsic bias toward onedirection of twisting.

The variable chiral nematic was simulated in the parallelnematic cell, with the same setup as in previous simulations.The input variable was the inverse cholesteric pitch q0 [Eq. (3)],

051702-4

NEMATIC DISCLINATIONS AS TWISTED RIBBONS PHYSICAL REVIEW E 84, 051702 (2011)en

ergy

[arb

.uni

t]

7

7.1

7.2

7.3

7.4

7.5

7.6

7.7

-100o -80o -60o -40o -20o 0o 20o 40o 60o 80o 100o

twist angle

disc

linat

ion

leng

th[ µ

m]

omegafigure-eight

thetasaturn rings

-2/3

-0.1

0.0

0.1

-0.7

-0.6

-90o -75o -60o -45o -30o -15o 0o 15o 30o 45o 60o 75o 90o

writ

he

twist angle

omegafigure-eight

thetasaturn rings

FIG. 4. (Color) Dependence of the free energy, total disclination length, and writhe on the twist angle for all four dimer structures. Thehorizontal axis measures the angle between the top and and bottom rubbing directions. The interparticle separation is fixed at 116 nm. (a) Thefree energy contribution of the structures is quadratic with the twist angle. The Saturn rings structure is the ground state in a parallel cell andtwisted cells with very small twist, but at largest twist angles, the figure-eight structure is the most stable. The free energy scale is in arbitraryunits. (b) A similar dependence is observed for disclination lengths. The chiral structures (figure-eight and omega) have the shortest disclinationsat a nonzero angle, while the other two structures exhibit a symmetric dependence. For twist angles greater than 30◦, the figure-eight structurehas shorter disclinations than the Saturn rings. The quadratic terms of the fitted curves are all in the interval (0.12 ± 0.005) μm rad−2. (c)Changing the nematic cell from parallel to twisted causes changes in writhe. Even the symmetric structures like theta and Saturn rings exhibitnonzero writhe in a chiral environment. For the theta structure, writhe of the central ring is negligible.

which goes from zero for an ordinary nematic, to 2π/p fora chiral nematic with preferred pitch p. The forced parallelboundary condition ensures that we observe only the localeffects of pitch change, without mixing with the twisted-celleffect, which we measured separately. The shortest pitch thatwe simulated should have 1.5 windings per cell thickness inthe lowest energy state, so the observed state with paralleldirector field is actually only a metastable state.

The results show that the effect of the inverse cholestericpitch q0 on the geometric twist has the same sign andapproximate slope for all three entangled structures, except forlarge q0 where sterical restrictions begin to play a role (Fig. 5).This reflects the proportionality between geometric twist of thedisclination line and elastic twist of the director field. Sincethe geometric twist and the writhe are constrained to add up tothe self-linking number, the twist change manifests as writhe,which has an observable effect. For the left-handed figure-eightstructure, a positive q0 causes the loop to open, while a negativeq0 tightens it (Fig. 5). The visual appearance of the resultingdisclination loops is similar to that from the twisting of theboundary conditions (Fig. 4), but only the central portion ofthe structure is affected.

V. CONCLUSIONS

In nematic colloidal dispersions with homeotropic particles,the equilibrium disclinations tend to wind tightly around the

particles and make the bridges between the particles as short aspossible. This confinement enables the local rewiring processand stabilizes the resulting structures. The confining effecton the disclinations is observable through their geometricproperties, which also simplifies formulation of the topologicaltheory behind the rewiring [9].

Prediction of disclination structures using the formalismof tetrahedral rotations assumes the differences between themare completely localized at the tetrahedral rewiring sites. Inthis ideal case, the geometric twist of all related structuresis equal and equals zero if the model system has sufficientsymmetry. Deviation of the twist from the zero value thereforemeasures the amount of relaxation that causes the structuresto slightly differ at the parts that are not directly affectedby the rewiring. For large deviations, the recognition ofthe differences as local may be ambiguous, so the clas-sification of the structures with a unified model becomesinappropriate.

Our results confirm that the theoretical assumption of zerotwist is well obeyed in practice. The numerical experimentswith cell twisting and stretching of the entangled colloidalstructures show that variation of the boundary conditionsdoes not change the geometric twist for more than 0.15,so the self-linking number can be correctly extracted fromthe writhe by rounding to the nearest multiple of 2/3.This fact can be used to justify estimation of the self-linking number using writhe, which is much easier to extract

051702-5

SIMON COPAR, TINE PORENTA, AND SLOBODAN ZUMER PHYSICAL REVIEW E 84, 051702 (2011)

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0

.

1

-1.5 -1 -0.5 0 0.5 1 1.5

twis

t(T

w=

Sl-W

r)

inverse pitch [π/d]

omegafigure-eighttheta (total)

theta (ring only)

FIG. 5. (Color) The geometric twist of the entangled dimer statesin a parallel nematic cell, calculated from the theoretical valueof the self-linking number and the numerically obtained writhe.The geometric twist has a universal tendency to decrease with theinverse cholesteric pitch, a consequence of the relationship betweenthe geometric and elastic twist. For the theta structure, the twistcontribution of the central ring is shown separately. The x axis is theinverse cholesteric pitch in the units of π turns per cell thickness. Theinterparticle separation is 116 nm.

from the numerical data or experiments than the geometrictwist.

The inherent symmetries of the structures are reflected inthe behavior of the writhe and twist in response to the changingparameters. We have demonstrated that the geometric twistis correlated to the pitch in chiral nematics and confirmedthis fact with a numerical experiment. As the introduction ofchirality adds a similar amount of geometric twist to all thestructures, it is still almost preserved under rewiring. As long

as the geometry of the cell stays the same, the chirality of thesystem may be ignored for the purposes of classification.

The examined disclination loops match the well-knownlinear growth of the free energy under stretching [4] andshow quadratic growth under twisting. Using only disclinationlengths instead of an expression for the entire free energyextracts the part of the free energy that is significantlyinfluenced by the geometry and leaves out the bulk part thatcontains strong dependence on the material parameters andsample size.

For visualization, we have used the splay-bend parameter,related to the flexoelectric theory. It serves as a powerfulvisualization tool, since it shows the cross section of thedisclinations without the need for visualization of the directorfield in three dimensions. It can be used to distinguish −1/2disclinations from +1/2 and twist disclinations and to detectif the cross section changes along the loop, which is morecommon in chiral liquid crystals [19]. We used this techniqueto verify that none of our experiments destroys the threefoldsymmetry of the disclinations so the use of the self-linkingnumber as an invariant remains valid.

In experiments, many parameters can be varied to produce adesired effect in the medium. Knowledge about the geometricresponse of the disclinations is an important step forwardtoward the systematic choice of optimal parameters. Otherfields, such as the theory of DNA loops [20] and polymers [21],take advantage of geometric properties of curves to formulatethe free energy of the system in terms of writhe and twist[22]. Our work shows that the same path can be taken fornematics where geometrical confinement stabilizes nematicbraids. This could open a complementary perspective to thestandard Landau–de Gennes approach.

ACKNOWLEDGMENTS

We acknowledge the contribution of M. Ravnik in designingthe numerical experiments, and support from the SlovenianResearch Agency (research program P1-0099 and Project No.J1-2335) and Centre of Excellence NAMASTE.

[1] I. I. Smalyukh, A. N. Kuzmin, A. V. Kachynski, P. N.Prasad, and O. D. Lavrentovich, Appl. Phys. Lett. 86, 021913(2005).

[2] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals(Oxford University Press, Oxford, 1993).

[3] M. Kleman and O. D. Lavrentovich, Philos. Mag. 86, 4117(2006).

[4] M. Ravnik, M. Skarabot, S. Zumer, U. Tkalec, I. Poberaj,D. Babic, N. Osterman, and I. Musevic, Phys. Rev. Lett. 99,247801 (2007).

[5] M. Ravnik and S. Zumer, Soft Matter 5, 269 (2009).[6] I. Musevic, M. Skarabot, U. Tkalec, M. Ravnik, and S. Zumer,

Science 313, 954 (2006).[7] U. Ognysta, A. Nych, V. Nazarenko, M. Skarabot, and I. Musevı,

Langmuir 25, 12092 (2009).

[8] U. Tkalec, M. Ravnik, S. Copar, S. Zumer, and I. Musevı,Science 333, 64 (2011).

[9] S. Copar and S. Zumer, Phys. Rev. Lett. 106, 177801(2011).

[10] R. D. Kamien, Rev. Mod. Phys. 74 (2002).[11] M. Ravnik, G. P. Alexander, J. M. Yeomans, and S. Zumer,

Faraday Discuss. 144, 159 (2010).[12] F. B. Fuller, Proc. Natl. Acad. Sci. USA 86, 815 (1971).[13] M. R. Dennis and J. H. Hannay, Proc. R. Soc. A 461, 3245

(2005).[14] J. Cantarella, Numer. Anal. 42, 1846 (2005).[15] T. Porenta, M. Ravnik, and S. Zumer, Soft Matter 7, 132

(2011).[16] T. C. Lubensky, D. Pettey, N. Currier, and H. Stark, Phys. Rev.

E 57, 610 (1998).

051702-6

NEMATIC DISCLINATIONS AS TWISTED RIBBONS PHYSICAL REVIEW E 84, 051702 (2011)

[17] E. Dubois-Violette and O. Parodi, J. Phys. (Paris) Colloq. 30,C4-57 (1969).

[18] M. Ravnik, Ph.D. thesis, University of Ljubljana, 2009(unpublished).

[19] J.-i. Fukuda and S. Zumer, Phys. Rev. Lett. 106, 097801 (2011).

[20] M. Sayar, B. Avsaroglu, and A. Kabakcioglu, Phys. Rev. E 81,041916 (2010).

[21] S. Kutter, Ph.D. thesis, University of Cambridge, 2002(unpublished).

[22] J. Samuel and S. Sinha, J. Stat. Phys. 143, 399 (2011).

051702-7