Stable nematic droplets with handlesEkapop Pairama, Jayalakshmi Vallamkondua, Vinzenz Koningb, Benjamin C. van Zuidenb, Perry W. Ellisa,Martin A. Batesc, Vincenzo Vitellib, and Alberto Fernandez-Nievesa,1

aSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332; bInstituut-Lorentz for Theoretical Physics, Leiden University, 2333 CA, Leiden, TheNetherlands; and cDepartment of Chemistry, University of York, York YO10 5DD, United Kingdom

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 18, 2013 (received for review December 7, 2012)

We stabilize nematic droplets with handles against surfacetension-driven instabilities, using a yield-stress material as outerfluid, and study the complex nematic textures and defect structuresthat result from the competition between topological constraintsand the elasticity of the nematic liquid crystal. We uncovera surprisingly persistent twisted configuration of the nematicdirector inside the droplets when tangential anchoring is es-tablished at their boundaries, which we explain after consider-ing the influence of saddle splay on the elastic free energy. Fortoroidal droplets, we find that the saddle-splay energy screens thetwisting energy, resulting in a spontaneous breaking of mirrorsymmetry; the chiral twisted state persists for aspect ratios as largeas ∼20. For droplets with additional handles, we observe in experi-ments and computer simulations that there are two additional −1surface defects per handle; these are located in regions with localsaddle geometry to minimize the nematic distortions and hence thecorresponding elastic free energy.

geometric frustration | topology | torus | double twist | boojum

The liquid crystal in a common display is twisted due to theorientation of the molecules at the confining glass plates. By

manipulating this twist using electric fields, an image can begenerated. More exotic structures can emerge when the liquidcrystal is confined by curved rather than flat surfaces. The to-pology and geometry of the bounding surface can drive the systeminto structures that would not be achieved without the presenceof external fields. In this sense, the shape of the surface playsa role akin to that of an external field. Thus, under confinementby curved surfaces, the molecules can self-assemble into complexhierarchical structures with emergent macroscopic properties notobserved for flat liquid crystal cells. However, the design prin-ciples and properties of structures generated by this geometricroute are still largely unknown.The lowest energy state of an ordered material, such as a

liquid crystal or a simple crystal, is typically defect-free be-cause any disruption of the order will raise the elastic energy.However, the situation can be very different if the material isencapsulated within a confining volume and there is strongalignment of the molecules at the bounding surfaces. In thiscase, the preferred local order cannot be maintained throughoutspace. Such a material will be geometrically frustrated and itsground state could contain topological defects, which are spatialregions where the characteristic order of the material is lost.For nematic liquid crystals, the molecules tend to align alonga common director, n. The presence of defects at the bound-aries, which we characterize with their topological charge, s,giving the amount of n-rotation at the boundary as we encirclethe defect, raises the energy of the system. Thus, the formationof defects is normally disfavored due to this increase in energy.However, when an orientationally ordered material is confinedto a closed volume, the Poincaré–Hopf theorem establishes thatthe total topological charge on the bounding surface must beequal to its Euler characteristic, χ, a topological invariant givenby χ = 2ð1− gÞ, where g is the genus of the surface or its numberof handles (1). This theorem implies that the ground state of thesystem will, in many cases, incorporate topological defects. This

is indeed the case when the closed surface is spherical (2–4), be-cause χ = 2 for the sphere. Surfaces that are obtained by twisting,bending, stretching, or generally deforming the sphere withoutbreaking it are topologically equivalent. This is because none ofthese transformations introduce handles and thus they all haveχ = 2. In contrast, a toroidal surface is topologically different fromthe sphere because it has a handle and consequently χ = 0.Spherical nematics have been widely studied from experi-

mental, theoretical, and simulation points of view (5–14) andtheir intriguing technological potential for divalent nanoparticleassembly has been already demonstrated (15). In contrast, thereare virtually no controlled experiments with ordered media inconfined volumes with handles. A notable exception is the op-tically induced formation of cholesteric toroidal droplets insidea nematic host (16). This largely reflects the difficulties in gen-erating stable handled objects with imposed order. Althoughthe sphere is relatively easy to achieve in liquids due to surfacetension, the generation of stable droplets with handles remainsa formidable challenge.In this paper, we experimentally generate stable handled

droplets of a nematic liquid crystal, using a continuous hostwith a yield stress. This approach allows us to perform uniqueexperiments that probe nematic materials confined withindroplets that are topologically different from the sphere. Weobserve that the toroidal nematic droplets formed are defect-free. However, they exhibit an intriguing twisted structureirrespective of the aspect ratio of the torus. The stability ofthis configuration, which is in contrast to existent theoreticalexpectations (17), results from the often-neglected saddle-splay contribution to the elastic free energy. Upon switchingfrom one to multiply handled droplets, we observe both inexperiments and in simulations the presence of two defects,each with topological surface charge −1, per additional han-dle. These defects are nucleated in regions with local saddlegeometry to minimize the nematic distortions and hence thecorresponding elastic free energy.

Toroidal DropletsTo make nematic toroidal droplets, we inject a liquid crystal [4-n-pentyl-4′-cyanobiphenyl (5CB)] through a needle into a rotat-ing bath containing a yield-stress material consisting of (i) 1.5wt% polyacrylamide microgels (carbopol ETD 2020), (ii) 3 wt%glycerin, (iii) 30 wt% ethanol, (iv) 1 wt% polyvinyl alcohol(PVA), and (v) 64.5 wt% ultrapure water. The presence of PVAguarantees degenerate tangential (or planar) anchoring for theliquid crystal at the surface of the droplets; we confirmed this by

Author contributions: M.A.B., V.V., and A.F.-N. designed research; E.P., J.V., V.K., B.C.v.Z.,P.W.E., M.A.B., V.V., and A.F.-N. performed research; E.P., J.V., V.K., B.C.v.Z., P.W.E., M.A.B.,V.V., and A.F.-N. analyzed data; and E.P., J.V., V.K., M.A.B., V.V., and A.F.-N. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: alberto.fernandez@physics.gatech.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1221380110/-/DCSupplemental.

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making spherical droplets and checking their bipolar character.We also note that the continuous phase is neutralized to pH 7,where the sample transmission is more than 90% (18). However,the most relevant property of this phase is its yield stress, σy.During formation of the torus, the stresses involved are larger thanσy and hence the continuous phase essentially behaves as if it werea liquid. The combination of the viscous drag exerted by the outerphase over the extruded liquid crystal and its rotational motioncauses the liquid crystal to form a curved jet, as shown in Fig. 1A,which eventually closes onto itself, resulting in a toroidal nematicdroplet, such as that shown in Fig. 1B in bright field and in Fig. 1Cbetween cross-polarizers. Once the torus has been formed, theelasticity of the continuous phase provides the required force toovercome the surface tension force that would naturally tend totransform the toroidal droplet into a spherical droplet. There aretwo ways this transformation can happen: either through a dropletbreakup mechanism reminiscent of the Rayleigh–Plateau breakupof a jet into smaller spherical droplets or through the shrinkage ofthe droplet toward its center to form a single spherical droplet(19). The relevant length scale that changes in the breakup is thetube radius, a, whereas for shrinking it is the inner radius, R, de-fined in Fig. 1B. The minimum yield stress required to stabilize thetoroidal droplet against either transformation is σy = γ=ac orσy = γ=Rc, where γ is the interfacial tension between the two liq-uids, and ac and Rc are the critical tube and inner radii of the torusbelow which either breakup or shrinking occurs. Using this tech-nique we can successfully generate stable nematic toroids with anaspect ratio or slenderness ξ= ½R+ a�=a.Remarkably, when these droplets are observed along their side

view under cross-polarizers, their central region remains brightirrespective of the orientation of the droplet with respect to theincident polarization direction, as shown in Fig. 1 D–F; the

corresponding bright-field images are shown in Fig. 1 G–I. Notethat for an axial torus with its director field along the tube, thecross-polarized image should appear black for an orientationof 0° and 90° with respect to the incident polarization direction.Hence our result is suggestive of a twisted structure. In fact,twisted bipolar droplets also have a central bright region, whenviewed between cross-polarizers, irrespective of their orienta-tion (10, 20–22). In addition, theoretical studies of DNA intoroidal geometries have also shown that the DNA condensatecan be twisted as, in this case, some of the bending energy ofthe untwisted axial structure is released at the price of a smallamount of twist energy (17). Interestingly, the theory predictsthere is a critical value of slenderness, ξc = 1.4 for 5CB, beyondwhich the trade-off between bend and twist energies is un-favorable and the toroidal DNA condensate remains axial.To explore this possibility, we generate toroidal droplets with

different ξ and observe them between crossed polarizers alongtheir side view, for different orientations with respect to the in-cident polarization direction. We find that the central region ofall droplets remains bright for all orientations, as shown in Fig.2 A and D for a drop with ξ = 18.5 and orientations of 0° and45° with respect to the incident polarization direction. Thus,our observations suggest that we do not observe the transitionfrom the twisted to the axial configuration predicted for to-roidal DNA condensates.To explain the lack of axial structure in our experiments, we

consider the full Frank free energy

F =12

ZdV

�K1ð∇ · nÞ2 + K2ðn ·∇ × nÞ2 + K3ðn×∇ × nÞ2�

−K24

ZdS · ðn∇ · n+ n×∇ × nÞ; [1]

which, besides the well-known bulk terms representing splay,twist, and bend deformations weighted with elastic constants K1,K2, and K3, respectively, also contains the less familiar surfaceterm representing saddle-splay deformations with elastic con-stant K24. Our calculations use an ansatz for the unit directorfield, n= nrer + nϑeϑ + nϕeϕ, with er , eϑ, and eϕ the orthonormalbasis vectors in the r, ϑ, ϕ direction, respectively, and with nr = 0,

nϑ =ω r=a1− γ r=ðR+ aÞcosϑ, and nϕ =

ffiffiffiffiffiffiffiffiffiffiffiffi1− n2ϑ

q. In these expressions, ϑ

and r are the polar angle and the radial distance in the circularcross section of the torus, and ϕ is the angle in the plane per-pendicular to the symmetry axis of the torus, as shown in Fig. 3A.

A LC

a

R

CB

D E F

G H I

y

Fig. 1. Toroidal droplets. (A) Formation of a toroidal liquid crystal dropletinside a material with yield stress σy . (B and C) The top view of a typicalstable toroidal droplet of nematic liquid crystal, having tube and inner radiia and R, is shown in B when viewed in bright field and in C when viewedunder cross-polarizers. (D–F) Side view of a typical toroidal droplet with ξ =1.8 when viewed under cross-polarizers for orientations of 0°, 45°, and 90°with respect to the incident polarization direction. Note that the center partof the toroid remains bright irrespective of its orientation. (G–I) Corre-sponding bright-field images. The dark regions of the toroid in these imagesare due to light refraction. (Scale bar: 100 μm.)

D E F

A B C

Fig. 2. Persistence of the doubly twisted configuration. (A and D) Side viewof the central part of a torus with ξ = 18.5 when viewed under cross-polarizers for orientations of 0° and 45° with respect to the incident polar-ization direction. (Scale bar: 200 μm.) (B, C, E, and F) Computer simulation ofthe nematic texture of a torus with ξ = 2 when viewed along its side andbetween cross-polarizers and an orientation of 0° and 45° with respect tothe incident polarization direction. (B and E) ω = 0.4; (C and F) ω = 0.1.

9296 | www.pnas.org/cgi/doi/10.1073/pnas.1221380110 Pairam et al.

The variational parameter ω determines the nematic organiza-tion, which continuously evolves from the axial structure, whereω= 0 and hence n= eϕ, to a twisted configuration, where ω≠ 0.For simplicity, we first set γ = 1. The resulting nematic field isthen free of splay distortions and automatically obeys the tan-gential boundary conditions because nr = 0. Moreover, detailedinspection of the nematic arrangement inside the torus revealsthat the configuration is doubly twisted, as shown in Fig. 3A, wherewe use nails to represent the out-of-plane tilt of the director. Thestable nematic organization is obtained from minimization of theelastic free energy with respect to ω. After volume integration, weobtain, to leading order in ω,

The physical implications of this equation are better seen in the limitof large ξ, where the Frank free energy to quartic order in ω reads

Fπ2K3a

≈1ξ+�4K2 −K24

K3ξ−

54ξ

�ω2 +

12ξω4: [3]

Note that the saddle-splay modulus acts as an external field thattends to align n along the ϑ direction at the surface of the torus.Similar to the Landau theory of magnetism (23), the switching ofthe sign of the quadratic term in Eq. 3 from positive to negativeestablishes a spontaneous symmetry-breaking transition from theaxial to the doubly twisted configuration. Because the relevantquadratic term is zero when K2 −K24

K3= 5

16ξ2c, the relative magnitude

of twist and saddle splay determines whether the axial or thedoubly twisted structure is the preferred nematic arrangement.When K2 −K24

K3> 0, the quadratic term can be either negative or

positive, depending on the slenderness. Hence there is a criticalξc above which the lowest energy state corresponds to the axial

torus; this state is shown by the dashed line in Fig. 3B. Note,however, that ξc can be pushed to much higher values comparedwith the saddle-splay free case. Below ξc, the lowest free-energystate has nonzero ω, corresponding to the doubly twisted torus.In this case, there are two minima of equal depth correspondingto the two possible configurations in which the handedness of thetwisted nematic director is either positive or negative, as shownby the solid line in Fig. 3B. Remarkably, when K2 −K24

K3< 0, the

quadratic term is always negative and the only possible structureis the doubly twisted configuration. This result holds irrespectiveof ξ, as shown in Fig. 3C, where we plot the phase boundary,

obtained from Eq. 2, separating the axial from the doubly twistedregions, in a ξ vs. K24/K2 diagram. For 5CB, K24 ≈K2 (24–28) andhence the axial to doubly twisted transition is either pushed toextremely slender tori or completely lost, consistent with ourexperimental observations.We confirm our interpretation of the experimental results by

first performing computer simulations of the nematic texturesbased on Jones calculus (29, 30) and on the ansatz above for thedirector field inside the torus; these quantify how the polariza-tion state of the incident light changes as it travels through thesample and analyzer. Consistent with the experimental results,we find that indeed the central region of the torus remainsbright, when viewed along its side between crossed polarizers,irrespective of its orientation with respect to the incident po-larization direction. This is shown for a nematic torus with ξ = 2and ω = 0.4 in Fig. 2 B and E. Interestingly, for this aspect ratio,the center is brighter for an orientation of 0° than it is for anorientation of 45°. This is also seen experimentally (Fig. 1 D and

Z r

A C

K24/K2

0.3 0.6 0.9 1.2 1.5 1.81

10

100

-0.2 -0.1 0.0 0.1 0.2

B

1.990

1.995

2.000

F/[K

3a]

2.005

Fig. 3. Spontaneous chiral symmetry breaking and the significance of saddle-splay distortions. (A) Circular cross section of the torus illustrating the relevantcoordinates: ϑ is the polar angle, r is the radial distance from the center of the cross section, and ϕ is the azimuthal angle. The nails indicate the tilt direction ofthe director; it is tilted outward at the top, where r = a and ϑ= 90°, and inward at the bottom, where r = a and ϑ= 270°. The presented configurationcorresponds to a twisting strength ω = 0.49, for a torus with aspect ratio ξ = 2. Note the structure is doubly twisted. The director configuration inside thewhole torus is obtained by rotating the director field in this cross section around the Z axis. (B) Normalized elastic free energy, F=ðK3aÞ, vs. the variationalparameter ω, for ξ = 5 and two different values of ðK2 −K24Þ=K3. For ðK2 −K24Þ=K3 = 0:02 (red dashed line), there is only one energy minimum at ω= 0corresponding to the axial structure shown schematically on the left in C. For ðK2 −K24Þ=K3 = 0:01 (blue solid line), there are two minima located at ω≈ ± 0:1corresponding to the two possible handednesses of the doubly twisted structure shown schematically on the right in C. The ratio K24=K2 determines whetherthere is a transition between the axial and the doubly twisted structure and if so what the critical value of ξ is or whether the doubly twisted structure remainsirrespective of ξ. This is shown in the structural phase diagram of C, where we have used that K2 = 0:3K3 for 5CB (35). Because for 5CB, K24 ≈K2 (24–28), theaxial to double-twist transition is either completely lost or shifted to very high values of ξ, consistent with our experimental observations.

Fπ2K3a

≈ 2�ξ−

ffiffiffiffiffiffiffiffiffiffiffiffiξ2 − 1

q �+

�1− 9ξ2 + 6ξ4 + 6ξ

ffiffiffiffiffiffiffiffiffiffiffiffiξ2 − 1

p− 6ξ3

ffiffiffiffiffiffiffiffiffiffiffiffiξ2 − 1

p �ξ2 + 4ðK2 −K24Þξ4=K3

ξ2 − 13=2 ω2: [2]

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E). In addition, the regions to both sides of the center, encircledwith a dashed line in Fig. 2 B and E, are darker for an orientationof 0° than they are for an orientation of 45°. This is also seenexperimentally (Fig. 1 D and E). However, the situation reversesfor a torus with smaller twist distortions. In this case, the center ofthe torus is darker for an orientation of 0° than it is for an ori-entation of 45°, as shown in Fig. 2 C and F for ω = 0.1. This is inagreement with the experimental results as well (Fig. 2 A and D).We then quantify our results by measuring the twist angle in

our toroidal droplets along the Z direction, from (r = a, ϑ = 90°)to (r = a, ϑ = 270°) (Fig. 3A). The method relies on the fact thatlinearly polarized light follows the twist of a nematic liquidcrystal if the polarization direction is either parallel or per-pendicular to the nematic director at the entrance of thesample, provided the Mauguin limit is fulfilled (30); the cor-responding mode of propagation is referred to as extraordinaryor ordinary waveguiding, respectively. We then image the torusfrom above (Fig. 4A), rotate the polarizer to ensure that theincident polarization direction is parallel or perpendicular tothe nematic director at (r = a, ϑ = 90°), and then rotate theanalyzer an angle ϕexit with respect to the polarizer whilemonitoring the transmitted intensity, T. The minimum in T,shown in Fig. 4B, reflects the lack of light propagation throughthe analyzer, indicating that the incident polarization directionhas rotated an amount τ such that it is perpendicular to theanalyzer after exiting the torus at (r = a, ϑ = 270°). The imageof the torus in this situation exhibits four black regions whereextinction occurs, as shown in Fig. 4C; these correspond towaveguiding of ordinary and extraordinary waves. It is alongthese regions that we measure T. The counterclockwise rota-tion of the incident polarization direction by an angle of ∼56°exactly corresponds to the twist angle of the nematic along theZ direction through the center of the circular cross section.However, to increase the precision of our estimate, we fit theT vs. ϕexit results to the theoretically expected transmission(30), leaving τ as a free parameter. We find τ = (52.9 ± 0.4)o forξ = 3.5. Moreover, within the experimentally accessed ξ-range,we find that the twist is nonzero and that it monotonouslydecreases with increasing aspect ratio, as shown in Fig. 4D.Remarkably, these features are captured by our theoreticalcalculations for large ξ, as shown by the dashed line in Fig. 4D.We note that the deviations of the experiment and the theoryfor small ξ result from the inadequacy of the ansatz in de-scribing the highly twisted structures observed experimentallyat these low values of ξ. This can be partially resolved by lifting

the constraint that γ = 1. This introduces a second variationalparameter in the ansatz, which allows the nematic field to splay.The result qualitatively captures the experimental trend for allaspect ratios, as shown by the solid line in Fig. 4D. By furthercomparing the experiment to the theory in the high ξ-region,we obtain a value for the saddle-splay elastic constant of K24 =1.02K2, which is slightly larger than the twist elastic constant,confirming our previous conclusions and supporting our in-terpretation on the relevance of saddle-splay distortions. How-ever, our analysis cannot exclude the possibility of a slightlysmaller value of K24 and hence of a twisted-to-axial transitionfor extremely large ξ.

Double and Triple ToriNoteworthy, nematic toroids have no defects in their groundstate. However, this should not be the case if we add handlesbecause the Euler characteristic and hence the total topologicalcharge decrease by −2 with every additional handle. However,the Poincaré–Hopf theorem provides only a conservation lawthat prescribes the total topological surface charge. It tells usnothing about the individual defect charges, whether they arepoint defects (called boojums) or singular lines (31) inside thedroplet terminating at the boundaries, the number of defects, ortheir locations. To understand defect formation in higher-genusnematic droplets, we use computer simulations of a simplenematogenic lattice model (32). For this model, the elasticconstants are equivalent, K1 = K2 = K3, and no special consid-eration of the saddle-splay contributions to the elastic energy istaken. Hence, we do not expect to observe any twist in theresulting structures. For the double torus, we find two types ofdefect configurations with comparable energy. Both of thesehave two defects on the surface of the double torus, each withtopological charge −1. The defects are located either at the in-nermost regions of the inner ring of each torus, as shown in Fig.5A, or in the outermost regions where the individual tori meet, asshown in Fig. 5B. In both cases, the defects are located in regionsof local saddle geometry where the Gaussian curvature, G, de-fined as the product of the two principal curvatures, is negative.This finding shows similarity to the theoretical insight that neg-atively charged defects in a 2D curved nematic liquid crystal areattracted to regions with negative curvature (33–35).To investigate handled droplets experimentally, we exploit the

elastic character of the continuous phase below the yield stressand generate two nearby single tori that are merged together bythe addition of liquid crystal in the region between them. The top

180120600.00

0.25

0.75

0.50

1.00

exit (deg)

T

60

(deg

)

30

90

015105

A CB D

P

A

PA

Fig. 4. Determination of the twist angle and its dependence with slenderness. (A) A torus with ξ = 3.5 when viewed from the top and between cross-polarizers. (B) Transmission, T, as a function of the angle between the incident polarization direction and the analyzer, ϕexit. The line is a fit to the theoreticalexpectation in the Mauguin limit (30) with the twist angle, τ, as the only free parameter. We obtain τ = (52.9 ± 0.4)o. (C) Top view of the same torus at theminimum of the transmission curve. We measure T along the four black regions that are observed, which are darkest for the indicated direction of thepolarizer and analyzer. The sense of rotation of the analyzer indicates the nematic arrangement is right-handed; this likely results from the way the torus isgenerated, as all tori generated in the same way have the same handedness. (D) Twist angle as a function of ξ. The dashed line represents the theoreticalprediction based on Eq. 2, for K24 = 1.02K2. The solid line represents the theoretical prediction based on the improved ansatz including the second variationalparameter γ for the same value of K24, where we have used that K1 = 0.64K3 for 5CB (36). (Scale bar: 200 μm.)

9298 | www.pnas.org/cgi/doi/10.1073/pnas.1221380110 Pairam et al.

view of a typical droplet is shown in Fig. 5C; the correspondingcross-polarized image is shown in Fig. 5D. Interestingly, whenthis droplet is viewed along its side and between cross-polarizers,we observe that there is a defect in the very center of the droplet,as shown in Fig. 5E. We also observe the four-brush texturetypical of jsj = 1 defects. To determine the sign of this charge, werotate the double torus and observe that the black brushes alsorotate in the same direction (Movie S1), indicating that the

defect has a topological charge of -1 (36). This defect is in theback of the droplet when looked along its side, with an identicaldefect on its front. Hence there are two defects of charge -1 onthe surface of the double torus, consistent with the constraintsimposed by the Poincaré-Hopf theorem. Furthermore, they arelocated in regions with G < 0, consistent with the findings of ourcomputer simulations. We note, however, that the structure istwisted; we know this from realizing that the central region ofeach of the two tori forming our droplet remains bright irre-spective of its side view orientation with respect to the incidentpolarization direction. We also note that the location of the de-fects obtained experimentally is consistent with just one of thetwo configurations obtained in the computer simulations. Theother configuration has not been observed in our experiments,presumably because the way the double torus is made biasesthe director toward the structure in Fig. 5B.We can also generate more complex droplets with, for exam-

ple, three handles aligned along a common axis, as shown by thetop-view image in Fig. 5F, or arranged in a triangle, as shown inFig. 5G. In the first case, there are four defects, each of topo-logical surface charge −1, located in the regions where the in-dividual tori meet, as shown when the droplet is viewed along itsside between cross-polarizers in Fig. 5H. In addition, the directoris twisted, as expected on the basis of the results for the singleand double tori. In contrast, when the handles are arranged ina triangle, there are two −1 defects that cluster together in one ofthe three regions where the single tori meet, as indicated inFig. 5G. In this situation, in addition to the natural frustrationimposed by the bounding surface, there is an additional frus-tration arising from the lack of a sufficient number of negative-curvature regions between the single tori to position the defects.There are only three natural regions for the defects to be locatedand four defects. We find that a possible solution to this problemis to cluster two of the four defects together in one of the threenatural regions where they could be located.

ConclusionsWe have generated stable nematic droplets with handles, usinga material with a yield stress as continuous phase to stabilize theseotherwise unstable droplets. Nematic toroids have no defects andexhibit a doubly twisted configuration, similar to that observed inblue phases (37), irrespective of aspect ratio, which in our ex-periments ranges from ∼2 to ∼20; this results from importantsaddle-splay contributions to the elastic free energy. Interestingly,the comparison of the experimental measurement of the twistangle with our theoretical predictions provides a robust and simpleway to measure K24; this is important given the difficulty in de-termining the value of this elastic constant with current methods(24–28). For droplets with additional handles, we observe thatthere are two −1 surface defects per handle located in regions ofG < 0 where elastic distortions are minimized.Our work highlights the role of nematic confinement as a reliable

route to induce field configurations with unique geometrical andtopological properties. The chiral nematic texture observed in ourtoroidal droplets closely resembles a Seifert fibration of the 3-sphere,a slightly more general configuration than the celebrated Hopffibration (38, 39). Intense experimental effort has been recently di-rected toward constructing soft structures with nontrivial topologicalproperties, using external fields or unique sample preparation.Examples include fluid knots (40), optically created nematic torons(16), hybrid systems composed of nematic dispersions of colloidalparticles with various shapes (41, 42), or densely packed filamentousassemblies (17, 43, 44). Our experiments open up a versatile ap-proach to generate topological soft materials that exploits nematicself-assembly within macroscopic droplets with handles, stabilizedusing a yield-stress material as the outer fluid.

DC

-1

-1

H

F

E

-1

-1

-1

-1-1

-2

G

-1

A B

Fig. 5. Double and triple toroids. (A and B) The two textures found bycomputer simulations of a typical double torus. They both have two defects(solid spots) on the surface of the double torus, each with topological charge−1, located in regions of negative Gaussian curvature, either (A) at the in-nermost regions of the inner ring of each torus or (B) at the outermostregions where the individual tori meet. (C–H) Experimental double and tripletoroids. (C) Top view of a double toroid in bright field. Solid dark circles in-dicate the location of the two −1 surface defects. (D) The same image undercross-polarizers. (E) Side view of the double toroid under cross-polarizerswhen focused at its back. The four black brushes in the region where the twosingle toroids meet indicate the presence of a topological defect with chargejsj = 1. The sign of this charge is determined by rotating the double torus.Because the brushes rotate in the same sense as the rotation, we conclude thedefect has charge s = −1 (Movie S1). By changing the focal plane, we confirmthere is another s = −1 defect at the front of the double toroid. (F) Top-viewimage of a triple toroid with a side-by-side arrangement of the handles. (G)Top-view image of a triple toroid with a triangular arrangement of thehandles. Solid circles show the defect locations found by looking at thedroplets between cross-polarizers along different viewing directions. (H) Sideview of a triple toroid with a side-by-side arrangement of the handles viewedunder cross-polarizers. The defects are located in the outer regions where theindividual toroids meet. (Scale bar: 100 μm.)

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ACKNOWLEDGMENTS. We thank Randall Kamien, Hiroshi Yokoyama, BryanChen, and Nitin Upadhyaya for illuminating discussions. We also thank PabloLaguna and the Center for Relativistic Astrophysics for the use of their

Cygnus cluster. We acknowledge funding from the National Science Foun-dation (DMR-0847304), Stichting voor Fundamenteel Onderzoek der Mate-rie, and The Netherlands Organization for Scientific Research.

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