Nematic Liquid-Crystal Colloids

materials

Igor Muševiˇc 1,2 1 J. Stefan Institute, Jamova 39, Ljubljana SI-1000, Slovenia; [email protected] 2 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana SI-1000, Slovenia

Received: 16 October 2017; Accepted: 14 December 2017; Published: 25 December 2017

Abstract: This article provides a concise review of a new state of colloidal matter called nematic liquid-crystal colloids. These colloids are obtained by dispersing microparticles of different shapes in a nematic liquid crystal that acts as a solvent for the dispersed particles. The microparticles induce a local deformation of the liquid crystal, which then generates topological defects and long-range forces between the neighboring particles. The colloidal forces in nematic colloids are much stronger than the forces in ordinary colloids in isotropic solvents, exceeding thousands of kBT per micrometer-sized particle. Of special interest are the topological defects in nematic colloids, which appear in many fascinating forms, such as singular points, closed loops, multitudes of interlinked and knotted loops or soliton-like structures. The richness of the topological phenomena and the possibility to design and control topological defects with laser tweezers make colloids in nematic liquid crystals an excellent playground for testing the basic theorems of topology.

Keywords: liquid crystal; colloids; experimental topology; handlebodies

1. Introduction In a nematic liquid crystal, rod-like organic molecules are on average spontaneously oriented along the direction called the director n, as shown in Figure1. The director can have an arbitrary direction in space, but in reality it always points along some preferred direction, because the liquid crystal is usually conﬁned to a measuring cell. The degree of order of the nematic liquid crystal is described by the order parameter S [1],

1 D E S = 3 cos2 θ − 1 2 Here, θ is the angle between the long axis of a selected molecule and the average direction of the orientation, the director n. The brackets hi denote the average over the angular distribution of molecules in the sample. The order parameter is temperature dependent and decreases when the temperature of the nematic liquid crystal is increased. If the nematic liquid crystal is heated above a certain temperature, the orientational order is suddenly and completely lost and there is a ﬁrst-order phase transition into the isotropic state of the nematic liquid crystal. In this isotropic phase, the nematic liquid crystal behaves as an ordinary liquid with complete orientational disorder, and therefore S = 0. It should be noted that the nematic state of matter is positionally disordered; hence, it is similar to positionally disordered ﬂuids. Because of the spontaneous orientational order of the nematic liquid crystal, the material properties of this state of matter are strongly anisotropic. For example, nematic liquid crystals have the largest optical birefringence ∆n observed in matter, which typically ranges from 0.1 to 0.4. The electric and magnetic properties of nematic liquid crystals are also anisotropic. The dielectric constant, for example, is a tensorial property with very different eigenvalues, typically between ε 10 and 1. The magnetic properties of liquid crystals are similar to those of anisotropic diamagnetic materials, and other physical properties, such as the electric conductivity and viscosity, are anisotropic as well.

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Figure 1. Molecular ordering and tensor material properties of a nematic liquid crystal: (a) Snapshot Figure 1. Molecular ordering and tensor material properties of a nematic liquid crystal: (a) Snapshot of of molecular order in a nematic liquid crystal. Molecules exhibit rapid diffusion and orientational molecular order in a nematic liquid crystal. Molecules exhibit rapid diffusion and orientational fluctuations; (b) Orientational order is described by the Q-tensor, where the order parameter S fluctuations; (b) Orientational order is described by the Q-tensor, where the order parameter S (a number) is the largest eigenvalue of this tensor. (a number) is the largest eigenvalue of this tensor. Because of the spontaneous orientational order of the nematic liquid crystal, the material propertiesThe appearance of this state of of spontaneous matter are strongly order is anis nototropic. only reflected For example, in the nema anisotropytic liquid of thecrystals physical have properties,the largest butoptical also inbirefringence the elasticity  ofn thisobserved state of in matter. matter, Nematic which liquidtypically crystals ranges are from fluids 0.1 because to 0.4. theyThe electric can flow, and but magnetic they also properties exhibit orientational of nematic liqu elasticityid crystals and areare ablealso toanisotropic. transmit staticThe dielectric torques. Theconstant, elastic for properties example, of is liquid a tensor crystalsial property are used with in displayvery differen applications,t eigenvalues, where typically the elasticity between restores  10 theand original 1. The statemagnetic after theproperties external of electric liquid field crystals is switched are similar off, therefore to those driving of anisotropic the pixel elementdiamagnetic into thematerials, off state. and other physical properties, such as the electric conductivity and viscosity, are anisotropic as well.Nematic liquid crystals can also be aligned microscopically by putting them in contact with certainThe surfaces. appearance For example, of spontaneous when a order nematic is liquidnot only crystal reflected is put in into the contact anisotropy with of a solidthe physical crystal, suchproperties, as graphite, but also the in molecules the elasticity are forced of this to state lie flat of matter. along the Nematic interface liquid throughout crystals theare contactfluids because region. Thistheytype can offlow, alignment but they is calledalso exhibit planar orientational alignment. Other elasticity types and of surfaces, are able for to example,transmit hairystatic polymertorques. surfaces,The elastic are properties able to induce of liquid either cr planarystals are or perpendicularused in display alignment applications, anchoring where of the the elasticity liquid crystal restores at theirthe original interface. state after the external electric field is switched off, therefore driving the pixel element into Thethe off combination state. of the long-range orientational order and the ability to align the director field alongNematic the surface liquid is thecrystals key characteristiccan also be aligned of nematic microscopically liquid-crystal by colloids,putting them also short-namedin contact with as nematiccertain surfaces. colloids [For2]. example, Here, micrometer-sized when a nematic particles, liquid crystal such asis put microspheres, into contact are with immersed a solid crystal, in the nematicsuch as liquidgraphite, crystal. the Becausemolecules of are the surfaceforced to alignment, lie flat along the director the interface field is throughout forced to align the alongcontact a closedregion. surface This type of theof alignment sphere, which is called results planar in thealig appearancenment. Other of types topological of surfaces, defects for ofexample, the nematic hairy orientationalpolymer surfaces, field [are3–6 ].able Topological to induce defectseither planar are regions or perpendicular where the order alignment cannot anchoring be defined of and, the asliquid a consequence, crystal at their the interface. orientational order is strongly depressed in the core of topological defects. This createsThe combination an interesting of the situation long-range where orientational the colloidal order inclusions and the ability are inevitably to align accompaniedthe director field by topologicalalong the surface defects is and the the key liquid characteristic crystal surrounding of nematic liquid-crystal the colloidal particlecolloids, is also strongly short-named deformed. as Thesenematic deformations colloids [2]. result Here, in micrometer-sized a strong elastic interaction particles, such between as microspheres, neighboring colloidalare immersed particles in the in thenematic nematics. liquid This crystal. interaction Because force of the triggers surface the alignment, spontaneous the assembly director offield nematic is forced colloids, to align where along the a processclosed surface of pair interactionof the sphere, is strongly which characterizedresults in the byappearance the topological of topological defects. Topologicaldefects of the defects nematic are thereforeorientational very field important [3–6]. Topological for the interaction defects ofare nematic regions colloidswhere the and order bring cannot in a particular be defined signature and, as a relatedconsequence, to topology. the orientational order is strongly depressed in the core of topological defects. This creates an interesting situation where the colloidal inclusions are inevitably accompanied by topological defects and the liquid crystal surrounding the colloidal particle is strongly deformed. These deformations result in a strong elastic interaction between neighboring colloidal particles in

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When looking through the literature covering a significant time period it is clear that studies of nematic colloids and topological defects appear in waves separated by tens of years. The first studies of defects in nematic liquid crystals were conducted in the 1970s using ordinary optical microscopy [7,8]. The second wave of exploration was triggered by the work of Poulin et al. in 1997 on water droplets in nematics [9], but was limited by the existing experimental methods of that time. During the next ten years, two important experimental techniques were developed and introduced to the study of nematic liquid-crystal colloids: Laser tweezers [10–27] and fluorescent confocal polarizing microscopy (FCPM) [28–32]. The first method made it possible to grab and manipulate individual micrometer-sized colloidal particles using laser light focused through the objective of an optical microscope. The forces between the colloidal particles could then be studied and the particles could be systematically assembled into 2D and 3D colloidal crystals. FCPM made it possible to study the director field around the colloidal particles and to explore the richness of the topological defects and the structures around the colloidal inclusions having various levels of complexity. The aim of this article is to describe the basic concepts and present some results from the field of nematic liquid-crystal colloids to scientists who are outside the field. There is also a short review of the fundamental topological properties of this system, which in my opinion are the most interesting phenomena that are observed in nematic colloids.

2. Forces between Spherical Microparticles in 2D Nematic Liquid Crystals The study of nematic colloids started nearly two decades ago with an experiment performed by Poulin et al. in 1997 [9]. They mixed water and a nematic liquid crystal and obtained a dispersion of micrometer-diameter water droplets floating in the nematic liquid crystal. Under a polarizing optical microscope they saw that the water droplets spontaneously formed chains, which were directed along the nematic director (the direction of the average orientation of liquid-crystal molecules). Although fluid, the water droplets did not coalesce into larger droplets, but remained separated from each other by a topological defect that was spontaneously created in the vicinity of each droplet. This experiment was clear evidence of a new type of force that acts between colloidal inclusions in nematic liquid crystals. When micrometer-diameter colloidal particles are dispersed in the nematic, the strength of this force is typically tens of pN at a typical surface-to-surface colloidal separation of a micrometer or so. The force has its origin in the ordering and elasticity of the nematic liquid crystal and, most importantly, in the alignment of the liquid-crystal molecules along the closed surface of the colloidal inclusion. When the colloidal inclusion is introduced into the nematic liquid crystal, the closed surface of the particle forces the liquid-crystal molecules to align all along the closed surface of the inclusion. We shall consider the homeotropic alignment of a liquid crystal on a colloidal surface where the molecules are locally perpendicular to that surface. As shown in Figure2, it is not possible to smoothly fill the space around such an inclusion with liquid-crystal molecules. There are regions where the liquid-crystal molecules “do not know” in which direction to orient themselves, which results in the local disordering of their long molecular axes. These deformed and disordered regions are called topological defects and are a consequence of the law of the conservation of topological charge. There are two important aspects of topological defects in liquid crystals. First, the topological defects are regions of reduced orientational order of the liquid-crystal molecules, which are also accompanied by strong elastic deformation of the liquid crystal. Because the liquid crystals are orientationally ordered fluids, their deformation costs energy. Similarly, reducing the degree of local orientational order also costs energy, because the nematic liquid crystal has to be nearly molten to achieve a lower order parameter. Second, in addition to free energy, topological defects are, mathematically speaking, points or loops where the ordering field is singular, i.e., the field is not defined in those regions. The defects of the orientational field are therefore of the same nature as the topological defects in any other physical field. For example, an electric field has its singularities in electric charges, which are the source of this field. In physics, charges always generate forces between themselves, which is also the case with the topological charge attributed to topological defects. Materials 2018, 11, 24 4 of 27

In addition, the total topological charge of the system has to be conserved at all times, which means that topological charges are always created in pairs of oppositely charged defects. The point defect accompanying the inserted spherical colloidal particle must be neutralized by another topological charge of opposite sign. In the case of a solid microsphere, this charge is a virtual one and is located in the center of the microsphere, as illustrated in Figure2. Materials 2018, 11, 24 4 of 27

Figure 2. (a) A spherical colloidal particle with homeotropic surface anchoring forces the liquid Figure 2. (a) A spherical colloidal particle with homeotropic surface anchoring forces the liquid crystal crystal molecules to orient along its closed surface. The director field is represented by black lines. molecules to orient along its closed surface. The director field is represented by black lines. Far away Far away from the particle, the nematic liquid crystal is uniform. However, because of the alignment from the particle, the nematic liquid crystal is uniform. However, because of the alignment of the of the liquid crystal along the closed surface, it is not possible to have a smooth director field liquid crystal along the closed surface, it is not possible to have a smooth director field everywhere. everywhere. Instead, regions of disorder are created, where the molecules have no preferential Instead, regions of disorder are created, where the molecules have no preferential direction to align direction to align with. These regions are topological defects. There are two topological defects with. These regions are topological defects. There are two topological defects (blue and red dots), (blue and red dots), separated by rd. The first one is called a hyperbolic hedgehog (blue dot) and separated by rd. The first one is called a hyperbolic hedgehog (blue dot) and carries a topological carries a topological charge q  1. The second one is a virtual one and resides in the center of the charge q = −1. The second one is a virtual one and resides in the center of the particle. It is called a particle. It is called a radial hedgehog and carries the opposite topological charge q  1 . These two radial hedgehog and carries the opposite topological charge q = +1. These two topological defects topological defects form an elastic* dipole denoted as p . (b) Real image of a dipolar colloidal form an elastic dipole denoted as p el.(b) Real image of a dipolarel colloidal particle with a dark spot, whichparticle is awith hyperbolic a dark spot, point which defect. is Noa hyperbolic polarizers point were useddefect. when No polarizers taking the were photograph. used when taking the photograph. There are therefore two oppositely charged topological defects accompanying a microsphere with There are two important aspects of topological defects in liquid crystals. First, the topological a homeotropic surface anchoring in the nematic liquid crystal, as shown in Figure2. The first one defects are regions of reduced orientational order of the liquid-crystal molecules, which are also is the real topological defect, which has the hyperbolic internal structure of the director field and an accompanied by strong elastic deformation of the liquid crystal. Because the liquid crystals are axial symmetry around the axis connecting the defect and the center of the sphere. This is called a orientationally ordered fluids, their deformation costs energy. Similarly, reducing the degree of local hyperbolic hedgehog. The second, oppositely charged topological defect has a radial structure and orientational order also costs energy, because the nematic liquid crystal→ has to be nearly molten to is called a radial hedgehog. Together, they form an elastic dipole p , because of the strong elastic achieve a lower order parameter. Second, in addition to free energy,el topological defects are, deformation in the neighborhood of the hedgehogs. It is also called a topological dipole, formed of mathematically speaking, points or loops where the ordering field is singular, i.e., the field is not two oppositely charged point defects. defined in those regions. The defects of the orientational field are therefore of the same nature as the In topology, any smooth deformation of objects, such as the stretching of surfaces or the twisting topological defects in any other physical field. For example, an electric field has its singularities in of loops, is allowed. In our case, it is allowable to smoothly open a point into a loop, following the electric charges, which are the source of this field. In physics, charges always generate forces smooth process illustrated in Figure3[ 2,6,33]. This means that a hyperbolic hedgehog could be opened between themselves, which is also the case with the topological charge attributed to topological into a hyperbolic ring, and this ring could be smoothly pushed along the surface of the colloidal defects. In addition, the total topological charge of the system has to be conserved at all times, which particlemeans that to its topological equator. Incharges this way, are aalways new typecreated of colloidalin pairs of particle oppositely is formed charged that defects. has quadrupolar The point symmetry,defect accompanying as shown in the Figure inserted4b. The spherical hyperbolic coll ringoidal is alsoparticle called must the be Saturn neutralized ring [ 34 –by37 ]another because oftopological its similarity charge to the of planetopposite Saturn sign. and In the its rings.case of The a solid particle microsphere, in Figure4 thisa is acharge dipolar is colloida virtual with one a hyperbolicand is located point in defect.the center of the microsphere, as illustrated in Figure 2. There are therefore two oppositely charged topological defects accompanying a microsphere with a homeotropic surface anchoring in the nematic liquid crystal, as shown in Figure 2. The first one is the real topological defect, which has the hyperbolic internal structure of the director field and an axial symmetry around the axis connecting the defect and the center of the sphere. This is called a hyperbolic hedgehog. The second, oppositely charged topological defect has a radial structure and is  called a radial hedgehog. Together, they form an elastic dipole pel , because of the strong elastic deformation in the neighborhood of the hedgehogs. It is also called a topological dipole, formed of two oppositely charged point defects. In topology, any smooth deformation of objects, such as the stretching of surfaces or the twisting of loops, is allowed. In our case, it is allowable to smoothly open a point into a loop, following the smooth process illustrated in Figure 3 [2,6,33]. This means that a hyperbolic hedgehog

Materials 2018, 11, 24 5 of 27 Materials 2018, 11, 24 5 of 27 could be opened into a hyperbolic ring, and this ring could be smoothly pushed along the surface of couldthe colloidal be opened particle into toa hyperbolic its equator. ring, In this and way, this riang new could type be of smoothly colloidal pushed particle along is formed the surface that has of thequadrupolar colloidal symmetry,particle to asits shown equator. in FiIngure this 4b. way, The a hyperbolicnew type ringof colloidal is also called particle the isSaturn formed ring that [34–37] has quadrupolar symmetry, as shown in Figure 4b. The hyperbolic ring is also called the Saturn ring [34–37] Materialsbecause2018 of ,its11 ,similarity 24 to the planet Saturn and its rings. The particle in Figure 4a is a dipolar5 of 27 becausecolloid with of its a hyperbolicsimilarity topoint the defect.planet Saturn and its rings. The particle in Figure 4a is a dipolar colloid with a hyperbolic point defect.

Figure 3. A hyperbolic point can be smoothly opened into a hyperbolic ring, which encircles a real Figure 3. A hyperbolic point can be smoothly opened into a hyperbolic ring, which encircles a real Figuremicrosphere 3. A hyperbolic in a nematic point liquid can crystal. be smoothly Adapted opened from Ref.into [2].a hyperbolic ring, which encircles a real microsphere in a nematic liquid crystal. Adapted from Ref. [2]. microsphere in a nematic liquid crystal. Adapted from Ref. [2].

Figure 4. (a) A dipolar colloidal particle in a nematic liquid crystal. The dark dot on the right-hand Figure 4. (a) A dipolar colloidal particle in a nematic liquid crystal. The dark dot on the right-hand Figureside is 4.a hyperbolic(a) A dipolar point colloidal defect. particle (b) A quadrupolar in a nematic colloidal liquid crystal. particle The in darkthe nematic. dot on the The right-hand two dark side is a hyperbolic point defect. (b) A quadrupolar colloidal particle in the nematic. The two dark sidespots is on a hyperbolicthe top and point the bottom defect. are (b )a Across-section quadrupolar of colloidala hyperbolic particle ring inen thecircling nematic. the microsphere. The two dark spots on the top and the bottom are a cross-section of a hyperbolic ring encircling the microsphere. spots on the top and the bottom are a cross-section of a hyperbolic ring encircling the microsphere. Whether a colloidal particle is a dipole or a quadrupole depends on the strength of the orientationalWhether surfacea colloidal anchoring particle and is thea dipole diameter or aof quadrupole the colloidal depends particle. on If thethe particlestrength is of small the orientationalWhether asurface colloidal anchoring particle isand a dipole the diameter or a quadrupole of the dependscolloidal on particle. the strength If the of particle the orientational is small surface(less than anchoring a micrometer) and the and diameter the nematic of the director colloidal is particle.forced to Iffollow the particle a strongly is small curved (less spherical than a (lesssurface, than it willa micrometer) relax the elastic and theenergy nematic of distortion director by is formingforced to a largerfollow structure, a strongly i.e., curved the Saturn spherical ring. micrometer)surface, it will and relax the the nematic elastic directorenergy of is distortion forced to followby forming a strongly a larger curved structure, spherical i.e., the surface, Saturn it ring. will relaxFor larger the elastic particles energy and a of lower distortion Gaussian by forming curvature, a larger the elastic structure, energy i.e., can the be Saturn stored ring. in a hyperbolic For larger Forpoint larger hedgehog. particles The and dipolar a lower or Ga quussianadrupolar curvature, nature the of elasticthe particle energy also can dependsbe stored onin athe hyperbolic external particlespoint hedgehog. and a lower The dipolar Gaussian or curvature,quadrupolar the nature elastic of energy the particle can be also stored depends in a hyperbolic on the external point hedgehog.fields and the The confinement. dipolar or quadrupolar If the particles nature are confined of the particle to a very also thin depends nematic on layer, the external they will fields usually and fieldsobtain and a quadrupolar the confinement. structure If the with particles a Saturn are ring.confined to a very thin nematic layer, they will usually theobtain confinement. a quadrupolar If the structure particles with are confineda Saturn ring. to a very thin nematic layer, they will usually obtain a quadrupolarIf two dipolar structure or with quadrupolar a Saturn ring. particles are brought together, the regions of their elastic distortionIf two start dipolar to overlap or quadrupolar and the two particlesparticles areare eibroughtther attracted together, to or the repelled regions from of each their other, elastic as distortionIf two start dipolar to overlap or quadrupolar and the two particles particles are broughtare either together, attracted the to regions or repelled of their from elastic each distortion other, as startillustrated to overlap in Figure and the5 for two dipolar particles colloidal are either particles. attracted to or repelled from each other, as illustrated illustratedThe reason in Figure for this 5 for pair-interaction dipolar colloidal force particles. is in the overlapping of the elastically distorted regions in FigureThe 5reason for dipolar for this colloidal pair-interaction particles. force is in the overlapping of the elastically distorted regions and theThe consequent reason for this sharing pair-interaction of the elastic force energy. is in theIf both overlapping elastic fields of the are elastically in favor distorted of lowering regions the andtotal thefree consequent energy, the sharing particles of willthe elasticbe attracted energy. to Ifshare both as elastic much fields of that are energy in favor as ofpossible, lowering which the andtotalthe free consequent energy, the sharing particles of thewill elastic be attracted energy. to If bothshare elastic as much fields of are that in energy favor of as lowering possible, the which total freewill energy,then reduce the particles the total will energy be attracted of the pair. to share If the as sharing much ofis thatenergetically energy as unfavorable, possible, which the willparticles then will thenbe repelled reduce from the total each energy other. of the pair. If the sharing is energetically unfavorable, the particles reducewill be repelled the total from energy each of other. the pair. If the sharing is energetically unfavorable, the particles will be repelled from each other.

The ﬁrst studies of the pair-interaction forces between the spherical inclusions in nematic liquid crystals were theoretical and are closely related to studies of the director structure around a spherical inclusion, ﬁrst analyzed by Terentjev et al. [35,36]. Kuksenok et al. [38] analyzed the Frank free-energy Materials 2018, 11, 24 6 of 27

The first studies of the pair-interaction forces between the spherical inclusions in nematic liquid crystals were theoretical and are closely related to studies of the director structure around a Materials 2018, 11, 24 6 of 27 spherical inclusion, first analyzed by Terentjev et al. [35,36]. Kuksenok et al. [38] analyzed the Frank free-energy functional provided by the surface-anchoring energy and used multipole expansion for functionalthe director provided field. byRamaswamy the surface-anchoring et al. [39] predic energyted and the used elastic multipole interaction expansion force between for the director colloidal field.particles Ramaswamy that was et calculated al. [39] predicted within thean elasticelectrostatic interaction analogy force following between Brochard colloidal particlesand de Gennes that wasusing calculated multipole within expansion an electrostatic [40]. For analogy two followingquadrupolar Brochard particles, and dethey Gennes predicted using a multipole power-law expansiondependence [40]. of For the two elastic quadrupolar attractive particles, force. A they very predicted comprehensive a power-law mean-field dependence analysis of thefor elasticthe pair attractiveinteraction force. of A elastic very comprehensivemultipoles andmean-field in particular analysis topological for the dipoles pair interaction was performed of elastic by multipoles Lubensky et andal. in [5,9], particular stressing topological the importance dipoles wasof the performed topology. by Further Lubensky mean-field et al. [5,9 approaches], stressing the to importancenematic colloids of theinclude topology. investigations Further mean-field of the stability approaches of colloidal to nematic clusters colloids by Lev include and Tomchuk investigations [41] and of the the stability effects of of colloidalconfining clusters walls on by the Lev colloidal-pair and Tomchuk interaction [41] and the by effectsFukuda of et confining al. [42,43]. walls Fukuda on the et colloidal-pair al. used a fully interactiontensorial by Landau-de Fukuda et Gennes al. [42,43 approach]. Fukuda to et the al. usedanalysis a fully of tensorialcolloidal-pair Landau-de interactions Gennes in approach a nematic to theliquid analysis crystal. of Pergamentschik colloidal-pair interactions and Uzunova in aused nematic a refined liquid approach crystal. Pergamentschikto colloidal nematostatics and Uzunova [44–46]. usedThey a refined observed approach that, toin colloidalspite of nematostaticsthe analogy [with44–46 electrostatics,]. They observed the that, three-dimensional in spite of the analogy colloidal withnematostatics electrostatics, is thesubstantially three-dimensional different colloidal in both nematostatics its mathematical is substantially structure different and its in physical both itsimplications. mathematical A structure review of and this its approach physical can implications. be found in A Ref. review [47]. of Chernyshuk this approach et canal. [48] be foundconsidered in Ref.the [47 theory]. Chernyshuk of colloidal et al. elastic [48] considered interactions the in theory the pr ofesence colloidal of an elastic external interactions electric inor themagnetic presence field of anusing external the Green’s electric function or magnetic method. field using the Green’s function method.

FigureFigure 5. ( a5.) ( Twoa) Two dipolar dipolar particles particles with with the the same same orientation orientation of theirof their elastic elastic dipoles dipoles will will attract attract each each other,other, if placed if placed collinearly. collinearly. (b) ( Ifb) placedIf placed side side by by side, side, they they will will ﬁrst first repel repel each each other other and and ﬁnd find their their minimumminimum energy energy through through a side-by-sidea side-by-side attraction. attraction. The The black black trail trail is is the the trajectory trajectory ofof eacheach particle,particle, as as determineddetermined by thethe particle-trackingparticle-tracking method. ( (cc)) Pair Pair potential potential for for collinear collinear attraction attraction shown shown in in (a) (a)(squares) and side-by-side attractionattraction inin ((b)) (circles).(circles). Scale bar, 55 µµm.m. AdaptedAdapted fromfrom Ref.Ref. [[2].2].

TheThe first first experimental experimental evidence evidence for for the the existence existence of elasticof elastic forces forces between between spherical spherical inclusions inclusions in ain nematic a nematic liquid liquid crystal crystal was was given given for for small-angle small-angle neutron neutron scattering scattering (SANS) (SANS) experiments experiments on on a a lyotropiclyotropic nematic nematic crystal crystal by by Raghunathan Raghunathan et al.et al. [49 [49].]. The The first first experiments experiments on on the the pair-interaction pair-interaction forcesforces between between inclusions inclusions in the in the nematic nematic liquid liquid crystal crystal were were performed performed in the in seminal the seminal experiments experiments of Poulinof Poulin et al. [et9]. al. They [9]. performedThey performed an experiment an experime on waternt on droplets water droplets dispersed dispersed in a 5CB in nematic a 5CB liquidnematic crystal.liquid They crystal. observed They thatobserved each waterthat each droplet water was droplet accompanied was accompanied by a point defect,by a point and defect, water droplets and water spontaneouslydroplets spontaneously assembled into assembled chains with into topological chains with defects topological separating defects two waterseparating droplets. two They water explaineddroplets. this They chaining explained of liquid this chaining droplets of in liquid another droplets liquid in by another a topological liquid (elastic)by a topological dipole-dipole (elastic) interactiondipole-dipole force, interaction which is expected force, which to follow is expected a power to law follow [5,9]. a There power are law many [5,9]. different There studiesare many of thedifferent elastic studies forces betweenof the elastic spherical forces colloidal between particles spherical in nematics colloidal [ 50particles–61]. The in observednematics forces[50–61]. areThe relatively observed strong forces and are long relatively range, andstrong show and a long power-law range, and dependence show a power-law on the separation dependence between on the theseparation two particles, between together the two with particles, a strong together angular wi dependence.th a strong Theyangular can dependence. be well described They can with be an well electrostaticdescribed analogy with an and electrostatic behave like analogy electric and dipoles behave in an lik externale electric electric dipoles field, in an which external is analogous electric field, to thewhich far-field is analogous nematic director. to the far-field nematic director.

Materials 2018, 11, 24 7 of 27 Materials 2018, 11, 24 7 of 27

The elasticelastic colloidal colloidal forces forces can can be expressedbe expressed in terms in terms of multipole of multipole moments. moments. Two dipolar Two particlesdipolar 4 willparticles therefore will therefore show a 1/ showr interaction a 1 r 4 interaction force with force a strong with angular a strong dependence, angular dependence, similar to the similar electric to forcethe electric between force two between electric dipoles.two electric It is dipoles. also possible It is also to measure possible the to pair-interactionmeasure the pair-interaction energy of the nematicenergy of colloids, the nematic and colloids, it turns outand thatit turns dipole-dipole out that dipole-dipole binding energies binding are energies of the orderare of ofthe 3000 orderkB ofT for 1-micrometer-diameter microspheres. Figures5c and6 show the pair-interaction energy for two 3000 kBT for 1-micrometer-diameter microspheres. Figures 5c and 6 show the pair-interaction energy dipolarfor two colloidaldipolar colloidal particles, particles, which iswhich obtained is obtained by integrating by integrating the work theof work the attractiveof the attractive force alongforce thealong trajectory the trajectory of one of ofone the of particles.the particles. The The interaction interaction energies energies are are huge, huge, even even for for micrometer-sizemicrometer-size particles, and are due to the energy of the elasticallyelastically distorted nematic director between and around the colloidal pair.

Figure 6. Dipole-dipole pair-interaction energy in nematic liquid crystal for two parallel and antiparallel Figure 6. Dipole-dipole pair-interaction energy in nematic liquid crystal for two parallel and antiparallel dipoles. Adapted from Ref. [2]. dipoles. Adapted from Ref. [2].

Two quadrupoles show a 1 r 6 interaction force, which is much weaker than the dipole-dipole Two quadrupoles show a 1/r6 interaction force, which is much weaker than the dipole-dipole force. An example of the measured pair-interaction energy for two quadrupolar colloidal particles is force. An example of the measured pair-interaction energy for two quadrupolar colloidal particles is shown in Figure 7. Typically, the quadrupolar colloidal interaction is several times weaker than the shown in Figure7. Typically, the quadrupolar colloidal interaction is several times weaker than the dipolar one for the same size of particles. dipolar one for the same size of particles. Finally, the dipole-quadrupole interactions show a 1 r5 power-law dependence, similar to the Finally, the dipole-quadrupole interactions show a 1/r5 power-law dependence, similar to the interaction of an electricelectric dipoledipole withwith anan electricelectric quadrupole.quadrupole. Because of thethe power-lawpower-law interactioninteraction nature ofof this this interaction interaction force, force, two two micrometer-diameter micrometer-diameter colloidal colloidal particles particles in the nematicin the nematic will interact will acrossinteract a across very large a very separation large sepa ofration tens of of micrometers. tens of micrometers.

Materials 2018, 11, 24 8 of 27 Materials 2018, 11, 24 8 of 27

Materials 2018, 11, 24 8 of 27

Figure 7. (a) Series of snapshots showing the attraction between two quadrupolar nematic colloids. Figure 7. (a) Series of snapshots showing the attraction between two quadrupolar nematic colloids. (b) Quadrupole-quadrupole pair-interaction energy in a nematic liquid crystal. Scale bar, 5 µm. (b) Quadrupole-quadrupole pair-interaction energy in a nematic liquid crystal. Scale bar, 5 µm. Adapted from Ref. [2]. AdaptedFigure from 7. (a Ref.) Series [2]. of snapshots showing the attraction between two quadrupolar nematic colloids. (b) Quadrupole-quadrupole pair-interaction energy in a nematic liquid crystal. Scale bar, 5 µm. 3. Microrods in Nematic Liquid Crystals 3. MicrorodsAdapted in Nematic from Ref. [2]. Liquid Crystals The force of the interaction in nematic liquid-crystal colloids has a strong topological signature, becauseThe3. Microrods force of the of in thepresence Nematic interaction of Liquid topological in Crystals nematic defects. liquid-crystal The interaction colloids force has is strongly a strong anisotropic, topological because signature, becausethe appearance ofThe the force presence ofof thethe interaction oftopological topological in defect nematic defects. breaks liquid-cry The the interaction continuousstal colloidsforce rotationalhas a is strong strongly symmetry topological anisotropic, of signature,a spherical because thecolloidal appearancebecause particle.of the of presence the However, topological of topological it is allowed defect defects. breaks in topology The the interaction continuous to smoothly force rotational deformis strongly objects symmetry anisotropic, and this of because asmooth spherical colloidal“morphing”the appearance particle. of However, theof theobject topological itby is stretching allowed defect inbreaksand topology de theforming continuous tosmoothly the surfacerotational deform will symmetry keep objects their of and atopological spherical this smooth “morphing”propertiescolloidal of particle.unchanged. the object However, by This stretching strikingit is allowed andproperty deforming in topology is reflected the to surfacesmoothly and observed will deform keep inobjects their real topological experimentsand this smooth properties with unchanged.colloidal“morphing” Thisparticles strikingof the of objectdifferent property by stretchingshapes is reﬂected in andnematic and de observedforming liquid the incrystalsreal surface experiments [62–70]. will keep For with theirexample, colloidal topological we particles can of differentsmoothlyproperties shapestransform unchanged. in nematica sphere This striking liquidinto a crystalsshortproperty cylinder [ 62is –reflected70 by]. stretching For and example, observed and morphing we in can real smoothly experiments the surface transform withof the a sphere. In terms of topology, the cylinder will have the same sort of topological defects as the sphere, spherecolloidal into a shortparticles cylinder of different by stretching shapes andin nematic morphing liquid the surfacecrystals of[62–70]. the sphere. For example, In terms we of topology,can whichsmoothly is indeed transform observed a sphere in the into experiments a short cylinder [63], as by shown stretching in Figure and 8.morphing the surface of the the cylinder will have the same sort of topological defects as the sphere, which is indeed observed in sphere. In terms of topology, the cylinder will have the same sort of topological defects as the sphere, the experimentswhich is indeed [63 ],observed as shown in the in Figureexperiments8. [63], as shown in Figure 8.

Figure 8. (a) The Saturn ring is encircling a microsphere with homeotropic surface anchoring in a nematic liquid crystal. (b) The Saturn ring is also encircling a microrod with homeotropic surface Figure 8. (a) The Saturn ring is encircling a microsphere with homeotropic surface anchoring in a Figureanchoring. 8. (a) TheScale Saturn bars, 10 ring µm. is encircling a microsphere with homeotropic surface anchoring in a nematic liquid crystal. (b) The Saturn ring is also encircling a microrod with homeotropic surface nematicanchoring. liquid Scale crystal. bars, ( b10) Theµm. Saturn ring is also encircling a microrod with homeotropic surface anchoring. Scale bars, 10 µm.

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Figure8 shows a microsphere and a microrod, both being encircled by a Saturn ring. This is a clear manifestation of the strength of the topology in nematic colloids, suggesting that a microsphere and a microrod are topologically equivalent. Indeed, the nature of the topological defects that are accompanying the objects inserted in a nematic liquid crystal is governed by the Euler characteristics of the object or, alternatively, the genus g of that object [2]. Loosely speaking, genus denotes the number of handles of the object. For example, a sphere has no handles and the genus of the sphere is g = 0. By attaching a single handle to the sphere we obtain an object that is homotopic to a toroid. The toroid therefore has genus g = 1. By increasing the number of handles, the genus of the object increases in proportion to the number of handles. The appearance of topological defects in a nematic liquid crystal around an immersed particle is governed by the basic laws of topology and will be explained in more detail in Section6. Briefly, a topological charge is attributed to the topological defects and similar to Gauss’s law in classical electrostatics this charge is measured by embracing the defect with a closed surface and measuring the number of streamlines of the director field through that surface. If a vector field is pinned to orient perpendicular to a closed surface, the Gauss-Bonnet theorem states that the topological charge of the surface is connected to the Euler characteristic of that surface. In our case, this surface is the surface of the object that is immersed in the liquid crystal. Because the overall topological charge of the inserted object and the liquid crystal around that object must be conserved at all times, the total topological charge is zero. The topological charge in this case consists of the topological charge of the surface and the topological charge of defects accompanying that surface. This means that the topological charge of the defects in a liquid crystal is directly related to the Euler characteristics of the surface of the inserted object. For a microsphere with genus g = 0, the Euler characteristics is 1 − g and there should be a unit topological charge accompanying the inserted spherical surface. In our case this unit topological charge is a hyperbolic hedgehog or a hyperbolic Saturn ring. Because the rod is homotopic to a sphere and has genus g = 0, it should also be accompanied by a unit topological charge, i.e., a hyperbolic hedgehog or a hyperbolic Saturn ring, as observed in the experiments. The force between two microrods in nematic liquid crystals is therefore expected to be similar to the force between two microspheres in nematic liquid crystals, with possible variations due to the different geometry (shape) of a rod compared to a sphere. Experimental studies of the pair-interaction forces between microrods in a nematic liquid crystal indeed confirm the predicted separation dependence of the interaction forces [63]. Whereas topology determines the unit charge of a defect accompanying a microrod, the number of defects on such a rod is not limited, as long as the total topological charge of all the defects adds up to −1. This could be realized on a very long fiber that is immersed in a nematic liquid crystal, where the separation between the defects should be large enough to prevent the spontaneous attraction and annihilation of the oppositely charged defects. The question of how to create additional defects on a fiber (or any other object) was solved by Nikkhou et al. [71–74], who used laser tweezers to heat and quench the micrometer-diameter area of a nematic liquid crystal surrounding the microfiber. By using a relatively strong laser light, the liquid crystal is locally heated into the isotropic phase, which creates a circular island of molten liquid crystal surrounded by a nematic liquid crystal. If such an experiment is performed using a thin layer of nematic liquid crystal without a fiber, after the light is switched off, we observe from a sequence of images shown in Figure9a dense tangle of topological defects. At the beginning of the quench, the structure of this quenched region is finely grained, with the grains representing individual domains of the nematic liquid crystal, which are randomly oriented. These domains are growing with time and they meet each other, forming topological defect structures at their interfaces. Materials 2018, 11, 24 10 of 27 Materials 2018, 11, 24 10 of 27

FigureFigure 9.9.( a(a)) Laser-tweezers-induced Laser-tweezers-induced quench quench of aof nematic a nematic liquid liquid crystal crystal without without a fiber a createsfiber creates a tangle a oftangle topological of topological defects, defects, which all which annihilate all annihilate back to the back vacuum. to the (bvacuum.) If a long (b fiber) If a islong present, fiber most is present, of the defectsmost of annihilate, the defects except annihilate, for a pairexcept of for rings a pair encircling of rings the encircling fiber. These the twofiber. rings These are two like rings a particle are like and a itsparticle anti-particle and its carryinganti-particle opposite carrying topological opposite charges. topological Scale charges. bar, 10 µ m.Scale Adapted bar, 10 fromµm. Ref.Adapted [71]. from Ref. [71]. This phenomenon, well known in cosmology as the Kibble mechanism of monopole This phenomenon, well known in cosmology as the Kibble mechanism of monopole production production [75–78], involves a field being rapidly quenched across a phase transition. This results in a [75–78], involves a field being rapidly quenched across a phase transition. This results in a number of number of topological defects, which are regions where various domains of nucleated low-temperature topological defects, which are regions where various domains of nucleated low-temperature phase phase meet. In condensed matter this mechanism was translated by Zurek and is one of the current meet. In condensed matter this mechanism was translated by Zurek and is one of the current hot hot topics in the physics of phase transitions. At later stages of the Kibble-Zurek mechanism the topics in the physics of phase transitions. At later stages of the Kibble-Zurek mechanism the topological defects annihilate, resulting in a coarsening of the structure [77,78]. In our case, the final topological defects annihilate, resulting in a coarsening of the structure [77,78]. In our case, the final state is a vacuum if there is no fiber present in the nematic liquid crystal. state is a vacuum if there is no fiber present in the nematic liquid crystal. The presence of the fiber alters the outcome of the coarsening dynamics after the thermal quench, The presence of the fiber alters the outcome of the coarsening dynamics after the thermal as seen from the sequence of images in Figure9b. Instead of a complete annihilation of all the quench, as seen from the sequence of images in Figure 9b. Instead of a complete annihilation of all topological defects, a pair of topological defects remains after the coarsening dynamics. If they are far the topological defects, a pair of topological defects remains after the coarsening dynamics. If they apart, these two rings are stable objects; if they are brought close to each other using the laser tweezers, are far apart, these two rings are stable objects; if they are brought close to each other using the laser they start attracting each other and rapidly annihilate to vacuum. This is clear evidence that these tweezers, they start attracting each other and rapidly annihilate to vacuum. This is clear evidence two rings are analogous to a particle and its anti-particle. They both carry a unit topological charge that these two rings are analogous to a particle and its anti-particle. They both carry a unit of opposite polarity and their total topological charge adds up to zero. In experiments with stronger topological charge of opposite polarity and their total topological charge adds up to zero. In laser light and a larger starting isotropic area, several pairs of topological defects could be created. experiments with stronger laser light and a larger starting isotropic area, several pairs of topological Topological defects always appear in an even number, so that their total topological charge is kept defects could be created. Topological defects always appear in an even number, so that their total to zero in all experiments. There are many different ways of producing oppositely charged pairs of topological charge is kept to zero in all experiments. There are many different ways of producing topological charges on a fiber, depending on the orientation of the fiber with respect to the nematic oppositely charged pairs of topological charges on a fiber, depending on the orientation of the fiber director, the thickness of the cell and the chirality of the liquid crystal. A nematic liquid crystal with a with respect to the nematic director, the thickness of the cell and the chirality of the liquid crystal. A fiber that is a strongly elongated colloidal particle is in fact an ideal experimental setting for studying nematic liquid crystal with a fiber that is a strongly elongated colloidal particle is in fact an ideal the topology of the director field with inclusions. experimental setting for studying the topology of the director field with inclusions. 4. Assembly of 2D and 3D Nematic Colloidal Crystals 4. Assembly of 2D and 3D Nematic Colloidal Crystals The pair-wise attraction of colloidal particles in nematics is very strong and long range, and can The pair-wise attraction of colloidal particles in nematics is very strong and long range, and can therefore be used to assemble 2D and even 3D colloidal crystals. In 2D the colloidal crystals of dipolar, therefore be used to assemble 2D and even 3D colloidal crystals. In 2D the colloidal crystals of quadrupolar and mixed (dipole-quadrupole) structures were demonstrated. An example of building dipolar, quadrupolar and mixed (dipole-quadrupole) structures were demonstrated. An example of up a dipolar 2D colloidal crystal is shown in Figure 10[ 55,79]. Isolated dipolar colloidal particles are building up a dipolar 2D colloidal crystal is shown in Figure 10 [55,79]. Isolated dipolar colloidal first brought together to assemble into dipolar colloidal chains, as shown in Figure 10a. The orientation particles are first brought together to assemble into dipolar colloidal chains, as shown in Figure 10a. of the dipolar chain could be controlled by choosing the proper direction of the individual dipolar The orientation of the dipolar chain could be controlled by choosing the proper direction of the individual dipolar particles forming the chain. The direction of the topological dipole could also be

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Materials 2018, 11, 24 11 of 27 particles forming the chain. The direction of the topological dipole could also be changed by using laserchanged tweezers by using [21]. laser After tweezers a chain [21]. is formed, After a anotherchain is formed, chain is another assembled chain in is which assembled the dipoles in which are pointingthe dipoles in the are opposite pointingdirection in the opposite with respect direction to the with ﬁrst respect chain to (Figure the first 10b). chain When (Figure these 10b). two chainsWhen withthese anti-parallel two chains dipoleswith anti-parallel are placed dipoles close to are each plac other,ed close they to attract each andother, form they a smallattract 2D and crystallite. form a Thissmall procedure 2D crystallite. is then This repeated procedure by addingis then repeated dipolarchains by adding with dipolar opposite chains directions with opposite of dipoles, directions which resultsof dipoles, in rather which large results 2D dipolarin rather colloidal large 2D crystals, dipolar ascolloidal shown crystals, in Figure as 10 shownc. in Figure 10c.

Figure 10. The 2D dipolar colloidal crystal is assembled in several steps by using the laser tweezers to Figure 10. The 2D dipolar colloidal crystal is assembled in several steps by using the laser tweezers to grab and manipulate the colloidal particles. (a) The dipolar colloidal particles are pairwise assembled grab and manipulate the colloidal particles. (a) The dipolar colloidal particles are pairwise assembled into dipolar colloidal chains. (b) Individual dipolar chains are assembled into small 2D colloidal into dipolar colloidal chains. (b) Individual dipolar chains are assembled into small 2D colloidal crystallites. (c) Additional dipolar chains are brought close to the already assembled structures so crystallites. (c) Additional dipolar chains are brought close to the already assembled structures so that that they spontaneously assemble into large 2D colloidal crystals. Scale bar, 5 µm. Adapted from Ref. [2]. they spontaneously assemble into large 2D colloidal crystals. Scale bar, 5 µm. Adapted from Ref. [2]. A 2D colloidal crystal consists of micrometer-diameter colloidal particles that are very strongly boundA 2Dtogether, colloidal which crystal makes consists them of very micrometer-diameter robust, but still relatively colloidal elastic. particles The that space are verybetween strongly the boundmicrosphere together, is filled which with makes an elastically them very distorted robust, but liquid still crystal, relatively which elastic. acts Theas a spacecontinuum between elastic the microspheremedium binding is filled the withmicrospheres an elastically together. distorted The latti liquidce spacing crystal, of which a 2D nematic acts as acolloidal continuum crystal elastic can mediumbe controlled binding by the an microspheres external electric together. field. The Du latticee to the spacing dielectric of a 2D anisotropy, nematic colloidal the liquid-crystal crystal can bemolecules controlled in by between an external the electricmicrospher field.es Due tend to to the align dielectric either anisotropy, along or perpendicular the liquid-crystal to moleculesthe field, inwhich between forces the the microspheres microsphere tend to move to align apart either or closer along ortogether. perpendicular In this way, to the the field, lattice which constant forces of the a microspheredipolar 2D colloidal to move crystal apart or could closer be together. changed In by this up way,to 10%. the lattice constant of a dipolar 2D colloidal crystalLike could with be changeddipolar bycolloids, up to 10%.quadrupolar colloids also form 2D colloidal crystals [60,79]. The Likedifference with is dipolar in the much colloids, weaker quadrupolar binding of th colloidse quadrupolar also form colloidal 2D colloidalcrystal with crystals respect [ 60to ,the79]. Thedipolar difference colloidal is crystal, in the much as the weakerbinding binding energies of are the an quadrupolar order of magnitude colloidal smaller. crystal It withis also respect possible to theto assemble dipolar colloidal 2D colloidal crystal, structures as the from binding dipolar energies and quadrupolar are an order colloidal of magnitude particles smaller. [61,80,81]. It is This also possiblemixed interaction to assemble is 2D very colloidal interesting structures [61], from because dipolar it andis strongly quadrupolar anisotropic colloidal and particles shows [61 several,80,81]. Thispossible mixed stable interaction constellations is very of interesting a colloidal [61 pair.], because As a result, it is strongly the variety anisotropic of dipolar-quadrupolar and shows several 2D possiblecolloidal stable crystals constellations is very large, of with a colloidal a huge pair.number As aof result, different the motifs variety [81] of dipolar-quadrupolarthat can be achieved 2Dby colloidallaser-tweezers-assisted crystals is very assembly. large, with a huge number of different motifs [81] that can be achieved by laser-tweezers-assistedWhereas the assembly assembly. of nematic colloids in 2D thin layers is relatively straightforward, it is veryWhereas difficult theto control assembly a colloidal of nematic assembly colloids in 3D. in 2DThis thin is because layers is of relatively the difficulty straightforward, involved in a it3D is veryobservation difficult of to colloidal control astructures colloidal assemblyin optically in birefringent 3D. This is becauseliquid crystals of the difficultyand the difficulty involved of in amanipulating 3D observation the ofcolloidal colloidal particles structures in 3D in using optically laser birefringenttweezers. In spite liquid of crystals these difficulties, and the difficulty Nych et ofal. manipulating[82] demonstrated the colloidal the successful particles assembly in 3D usingof dipolar laser nematic tweezers. colloids In spite in 3D of nematic these difficulties, colloidal crystal, as shown in Figure 11. The structure is tetragonal with anti-parallel dipolar colloidal chains regularly distributed in space. Similar to 2D nematic colloidal crystals, the 3D nematic colloidal crystals respond strongly to an external electric field. Using positive-dielectric-anisotropy liquid

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Nych et al. [82] demonstrated the successful assembly of dipolar nematic colloids in 3D nematic colloidal crystal, as shown in Figure 11. The structure is tetragonal with anti-parallel dipolar colloidal chainsMaterials regularly 2018, 11, 24 distributed in space. Similar to 2D nematic colloidal crystals, the 3D nematic colloidal12 of 27 crystals respond strongly to an external electric ﬁeld. Using positive-dielectric-anisotropy liquid crystal,crystal, the the crystal crystal can can shrink shrink by upby toup 30%, to 30%, whereas whereas in a negative-dielectric-anisotropy in a negative-dielectric-anisotropy liquid crystal, liquid thecrystal, 3D crystal the 3D can crystal be rotated can be by rotated tens of by degrees. tens of degrees.

FigureFigure 11.11. ((aa)) SnapshotsSnapshots ofof thethe assemblyassembly processprocess forfor aa 3D3D nematicnematic colloidalcolloidal crystalcrystal andand (b(b)) 3D3D reconstructionreconstruction using using ﬂuorescentfluorescent confocal confocal microscopy.microscopy. P P and and A A indicate indicate the the direction direction of of the the polarizer polarizer andand analyzer, analyzer, respectively. respectively. Adapted Adapted from from Ref. Ref. [82 [82].].

5.5. Entanglement,Entanglement, KnottingKnotting andand LinkingLinking ofof NematicNematic ColloidsColloids in in 2D 2D SingleSingle colloidal colloidal particles particles with with a a simple simple shape, shape, such such as as a spherea sphere or aor short a short rod, rod, are are accompanied accompanied by aby single a single topological topological defect defect in the in form the ofform a point of a or point a small or loop.a small The loop. numerical The numerical simulations simulations of different of groupsdifferent in 2003groups and in 2006 2003 predicted and 2006 the predicted existence the of aex newistence colloidal of a state,new colloidal where two state, or many where colloidal two or particlesmany colloidal are entangled particles together are entangled by a single together disclination by a single loop disclination [83–86]. Figure loop 12 [83–86]. shows theFigure prediction 12 shows of the prediction of Landau-de Gennes numerical modeling [86] for the rapid cooling of a nematic liquid crystal around two colloidal particles. At the beginning, a dense tangle of topological defects is formed around both colloidal particles, which gradually annihilate until a single loop is left, as shown in the final panel of Figure 12.

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Landau-de Gennes numerical modeling [86] for the rapid cooling of a nematic liquid crystal around two colloidal particles. At the beginning, a dense tangle of topological defects is formed around both colloidal particles, which gradually annihilate until a single loop is left, as shown in the ﬁnal panel ofMaterials Figure 2018 12,. 11, 24 13 of 27

Figure 12. (a–c) Numerical experiment where the liquid crystal is rapidly cooled across the Figure 12. (a–c) Numerical experiment where the liquid crystal is rapidly cooled across the isotropic-nematicisotropic-nematic phasephase transition,transition, leavingleaving behindbehind aa singlesingle defectdefect looploop entanglingentangling aa pairpair ofof colloidalcolloidal particles.particles. Scale bar 1 µµm.m. (d–f) InIn realreal experiments,experiments, rapidrapid quenchingquenching acrossacross thethe isotropic-nematicisotropic-nematic phasephase transitiontransition is performed by by using using laser laser tweezer tweezers.s. Scale Scale bar, bar, 20 20 µm.µm. Numerical Numerical panels panels courtesy courtesy of ofM. M. Ravnik. Ravnik.

In real experiments [86] the rapid quenching of the nematic liquid crystal with colloidal In real experiments [86] the rapid quenching of the nematic liquid crystal with colloidal particles particles is obtained by using high-power laser tweezers. The tweezers are focused to a particular is obtained by using high-power laser tweezers. The tweezers are focused to a particular place in place in the sample close to the dispersed particles and the optical power is increased to the desired the sample close to the dispersed particles and the optical power is increased to the desired level. level. Because of the light absorption in a liquid crystal or in glass slides covered with a thin Because of the light absorption in a liquid crystal or in glass slides covered with a thin transparent transparent layer of Indium Tin Oxide (ITO), the liquid crystal is locally molten into the isotropic layer of Indium Tin Oxide (ITO), the liquid crystal is locally molten into the isotropic phase, as shown phase, as shown in Figure 12d. Similar to a numerical experiment, the real experiment shows the in Figure 12d. Similar to a numerical experiment, the real experiment shows the rapid annihilation of rapid annihilation of topological defects until a single loop is left, which entangles both the colloidal topological defects until a single loop is left, which entangles both the colloidal particles. particles. There are three different configurations of the defect loop entangling two colloidal particles, which There are three different configurations of the defect loop entangling two colloidal particles, appear in real experiments with different probabilities of their formation, as shown in Figure 13[ 86]. which appear in real experiments with different probabilities of their formation, as shown in The first configuration is the figure-of-eight entanglement, shown in Figure 13a, whereas the simulation Figure 13 [86]. The first configuration is the figure-of-eight entanglement, shown in Figure 13a, is shown in Figure 13d. Here, the −1/2 defect loop is entangling both colloidal particles in a form whereas the simulation is shown in Figure 13d. Here, the −1/2 defect loop is entangling both resembling the number 8. This configuration is most probable and is formed in approximately 36% colloidal particles in a form resembling the number 8. This configuration is most probable and is of experiments [86]. The second most probable configuration is the figure of Omega, shown in formed in approximately 36% of experiments [86]. The second most probable configuration is the Figure 13b,e. It is a single loop that does not cross itself, as in case of the figure of eight, but makes figure of Omega, shown in Figure 13b,e. It is a single loop that does not cross itself, as in case of the a small loop in between the two colloidal particles. The least probable and also thermodynamically figure of eight, but makes a small loop in between the two colloidal particles. The least probable and unstable configuration is the entangled point defect, where an extra point defect is located in between also thermodynamically unstable configuration is the entangled point defect, where an extra point two colloidal particles and the −1/2 loop is entangling them, as shown in Figure 13c,f. defect is located in between two colloidal particles and the −1/2 loop is entangling them, as shown in Figure 13c,f.

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Figure 13. EntangledEntangled colloidal colloidal pairs pairs appear appear in in three three different different configurations. conﬁgurations. ( a)) Figure Figure of of 8, 8, observed observed inin the experiment, and and ( d) LdG simulation. ( (bb)) Figure Figure of of Omega Omega in in the the experiment experiment and and ( (ee)) LdG simulations.simulations. ( (cc)) Entangled Entangled point point defect defect in in the the experiments experiments and and ( (ff)) theory. theory. Scale Scale bar bar in in ( (aa––cc),), 20 20 µm.µm. Scale bar in ( d–f), 1 µm.µm. Adapted from Ref. [86]. [86].

The −1/2 defect loop, which appears in all three configurations of entanglement, is an object that The −1/2 defect loop, which appears in all three configurations of entanglement, is an object that requires energy in order to form. In the first-order approximation, this energy is simply proportional requires energy in order to form. In the first-order approximation, this energy is simply proportional to to the physical length of the loop. This means that the loop acts as a kind of string that provides the physical length of the loop. This means that the loop acts as a kind of string that provides constant constant force. If the two colloidal particles are forced to move apart by an external force, a constant force. If the two colloidal particles are forced to move apart by an external force, a constant force is force is needed to separate the colloids. The string-like force is therefore similar to the force of the needed to separate the colloids. The string-like force is therefore similar to the force of the surface surface tension. It is possible to measure this force by using two optical traps that grab each of the tension. It is possible to measure this force by using two optical traps that grab each of the particles particles and separate them. If the light is switched off, the two colloidal particles will be driven back and separate them. If the light is switched off, the two colloidal particles will be driven back by the by the force of the entanglement. By using a standard technique of particle tracking with video force of the entanglement. By using a standard technique of particle tracking with video microscopy, microscopy, it is possible to determine the force between the particles and the work of this force it is possible to determine the force between the particles and the work of this force during the motion during the motion of the colloids. It turns out that the force of entanglement is even larger than the of the colloids. It turns out that the force of entanglement is even larger than the colloidal forces due to colloidal forces due to the sharing of defects and the elastic distortion. Typically, the two colloidal the sharing of defects and the elastic distortion. Typically, the two colloidal particles are entangled by particles are entangled by an energy close to 10,000 kBT. When the experiments with the colloidal an energy close to 10,000 k T. When the experiments with the colloidal entanglement are performed in entanglement are performedB in planar nematic cells where the liquid crystal is homogeneous, it is planar nematic cells where the liquid crystal is homogeneous, it is possible to entangle many colloidal possible to entangle many colloidal particles into chains or “colloidal wires”. However, it is not particles into chains or “colloidal wires”. However, it is not possible to entangle colloidal particles possible to entangle colloidal particles in 2D to form an entangled colloidal crystal, although such in 2D to form an entangled colloidal crystal, although such structures were predicted by numerical structures were predicted by numerical simulations. The reason for this is the difficulty of simulations. The reason for this is the difficulty of propagating a −1/2 defect line along an arbitrary propagating a −1/2 defect line along an arbitrary direction in a homogeneous nematic liquid crystal. direction in a homogeneous nematic liquid crystal. When the entanglement experiments are performed in a chiral nematic liquid crystal, another When the entanglement experiments are performed in a chiral nematic liquid crystal, another class of entanglement is immediately observed, which is illustrated in Figure 14 [87–89]. class of entanglement is immediately observed, which is illustrated in Figure 14[87–89]. When the number of colloidal particles is small (up to 4), we observe a trivial topological loop When the number of colloidal particles is small (up to 4), we observe a trivial topological conformation, consisting of a single loop encircling a single colloidal particle (Figure 14a) or entangling loop conformation, consisting of a single loop encircling a single colloidal particle (Figure 14a) or a colloidal dimer (Figure 14b), trimer (Figure 14c) or tetramer (Figure 14d,e). Here, the chiral entangling a colloidal dimer (Figure 14b), trimer (Figure 14c) or tetramer (Figure 14d,e). Here, the chiral environment is simply a 90° left-twisted nematic cell, with the molecular arrangement represented environment is simply a 90◦ left-twisted nematic cell, with the molecular arrangement represented in in the upper-right corner of Figure 14a. All the images are taken between crossed-polarizers and the the upper-right corner of Figure 14a. All the images are taken between crossed-polarizers and the field field of view is bright because of the waveguiding of the polarization in a twisted nematic structure. of view is bright because of the waveguiding of the polarization in a twisted nematic structure.

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Figure 14. Defect loops tie links and knots in chiral nematic colloids. (a) A twisted Saturn ring is an Figure 14. Defect loops tie links and knots in chiral nematic colloids. (a) A twisted Saturn ring is unknot and appears spontaneously around a microsphere in a chiral nematic. The molecular an unknot and appears spontaneously around a microsphere in a chiral nematic. The molecular arrangement in all the panels is illustrated in the top-right corner. (b–e) The defect loops of the arrangement in all the panels is illustrated in the top-right corner. (b–e) The defect loops of the colloidal colloidaldimer, trimer dimer, and trimer tetramers and te aretramers also unknots. are also (unknots.f) The ﬁrst (f) non-trivial The first non-trivial topological topological binding of binding a colloidal of acluster colloidal is thecluster Hopf is the link. Hopf Two link. separate Two separate defect loopsdefect are loops interlinked. are interlinked. In the In lower the lower panels panels (a–f )(a the–f) theloop loop conformations conformations were were calculated calculated using using numerical numerical Landau-de Landau-de Gennes Gennes modeling. modeling. (g– (jg)– Aj) seriesA series of ofalternating alternating torus torus knots knots and and links links on on colloidal colloidal particle particle arrays arrays of of m m× × nn particlesparticles areare knittedknitted by the laser. The The actual actual defect defect loops loops are are schematically redrawn to show the relaxation mapping from the experimental planar projection to the ﬁnalfinal knot diagram. The knots are labeled using the standard notation C,C, i,i, N,N, where where C C is is the the minimum minimum number number of crossings,of crossings, i distinguishes i distinguishes the different the different knot typesknot typesand N and counts N counts the number the number of loops. of loops. Scale bars,Scale 5bars,µm. 5 Adapted µm. Adapted from Ref.from [87 Ref.]. [87].

The first topologically nontrivial object is observed in Figure 14f, where a colloidal tetramer is The ﬁrst topologically nontrivial object is observed in Figure 14f, where a colloidal tetramer is entangled by two separate, mutually interlinked loops. The number of loops and their configuration entangled by two separate, mutually interlinked loops. The number of loops and their conﬁguration in space is determined experimentally by changing the focus of the microscope and following the in space is determined experimentally by changing the focus of the microscope and following the trajectories of the loops, which always appear dark because of the scattered light. In this way it is trajectories of the loops, which always appear dark because of the scattered light. In this way it is possible to reconstruct a planar projection (i.e., the projection on a microscope plane) for practically possible to reconstruct a planar projection (i.e., the projection on a microscope plane) for practically any kind of entangled loop. The colloidal clusters in Figure 14a–f were built using laser tweezers that enable grabbing and then manipulating the individual colloidal particles into a preferred colloidal

Materials 2018, 11, 24 16 of 27 any kind of entangled loop. The colloidal clusters in Figure 14a–f were built using laser tweezers that enable grabbing and then manipulating the individual colloidal particles into a preferred colloidal constellation [89,90]. Once a selected number of particles is brought together, the laser tweezers are used to cut and rewire the individual defect loops in many possible configurations. In the continuation, the colloidal array was enlarged by systematically adding rows of particles to the pre-existing p × q particle array, as shown in Figure 14g–j. At the same time, the laser tweezers were used to systematically cut and rewire the defect loops, as shown in the sequence of microscope images in Figure 14g–j. For example, Figure 14g shows nine colloidal particles arranged in a 3 × 3 particle array. There are many possible methods of loop formation and entanglement in this colloidal array. However, for the sake of a systematic study, the individual loops of colloidal particles were manipulated in a way that results in a single defect loop running all around the colloidal particles and entangling them in a firmly bound structure. The question that poses itself is: What is the topology of this loop? To answer this question we graphically extract the loop and form the projection of the loop on the imaging plane, which is shown with the first red line labeled “knot projection” in Figure 14g. It is immediately clear that there is a single defect loop and its topology is elucidated by applying Reidemeister moves [90]. These moves make the knot projection smoother and do not change the topology of the object. Finally, we obtain the knot diagram of this loop, which turns out to be a Trefoil knot. It has three crossings and a single loop. By adding an additional three colloidal particles to the 3 × 3 particle array and manipulating the defect loops to continue the entanglement in a logical way, we obtain an entangled colloidal array of 3 × 4 colloidal particles. It turns out that this is the Solomon link, consisting of two loops that are mutually interlinked to form four crossings. Note the similarity of the Solomon link with four crossings and the Hopf link with two crossings in Figure 14f. Using the procedure of adding particles in a logical way, we obtain 3 × 5 particle array that is entangled with a Pentafoil knot, as shown in Figure 14i. By adding an additional three particles we end up with a colloidal array that is entangled by a Star of David. We see from Figure 14 that the systematic addition and building of colloidal arrays results in a series of alternating torus knots and links. This result indicates that a generically knotted series of knots and links is possible and that knots and links of arbitrary complexity can be formed by taking a sufficiently large colloidal array. The knots and links of topological defect loops entangling colloids in chiral nematics can also be reversibly retied. This involves using the strong light of the laser tweezers to locally influence the defect loops in order to rewire them into another configuration. Certain topological rules apply when such a rewiring is performed, as illustrated in Figure 15. The first panel in Figure 15a shows an array of 4 × 3 colloidal particles, and we are concentrated on the possible rewiring of the tangled crossing indicated by the white dashed circle in Figure 15a. The local constellation of the two defect lines that are crossing each other in this area is shown in the upper-left corner of this panel. If we enlarge this panel, as in Figure 15b, we see a graphical visualization of a small volume in between the four colloidal particles where the line crossing takes place. An interesting feature is observed: It is possible to construct a small tetrahedron with the corners illustrated by red dots in Figure 15b. The corresponding illustration of the director field inside this tetrahedron shows that it has a certain symmetry. Namely, the tetrahedron could be rotated by steps of 120◦ around a symmetry axis of the tetrahedron that is running through its corner. By performing a 120◦ rotation (anti-clockwise green arrow in Figure 15b) the director field in the tetrahedron is preserved, but the two defect lines are now rewired, as shown in the next panel in Figure 15c. This imaginary rotation of the tetrahedron, which is allowed by local symmetry, results in the rewiring of the crossings and changes the global topology of the colloidal cluster. We can make exactly three possible tangles by reversibly transforming one into another. These local transformations change the topology and the handedness of the knots and links [91]. Materials 2018, 11, 24 17 of 27 Materials 2018, 11, 24 17 of 27

Figure 15. Rewiring of knots and links using laser tweezers. (a) A right-handled Trefoil knot is Figure 15. Rewiring of knots and links using laser tweezers. (a) A right-handled Trefoil knot is realized realized on a 4 × 3 colloidal array. The dashed circles indicate a unit tangle that can be rewired with × on athe 4 laser3 colloidal beam. array.The tangle The dashedconsists circles of two indicate perpendicular a unit tangle line segments that can beand rewired the surrounding with thelaser beam.molecular The tangle field. consists (b) By rewiring of two perpendicularthe unit tangle that line corresponds segments and to a the 2π/3 surrounding rotation of the molecular encircledfield. (b) Bytetrahedron, rewiring thea new unit composite tangle that knot, corresponds shown in (c), to is a knitted. 2π/3 rotation The sequence of the of encircled tangle rewirings tetrahedron, in (b–f a) new compositeresults knot,in switching shown between in (c), is the knitted. knots and The links, sequence demonstrated of tangle in rewirings (a,c,e). The in procedure (b–f) results for rewiring in switching betweenthe topological the knots anddefect links, loops demonstrated is not an exact in (anda,c,e repe). Theatable procedure procedure, for rewiringbut is rather the topologicala skill that is defect loopslearned is not after an exact trying and and repeatable repeating a procedure, good number but of is experiments. rather a skill It thatis ne iscessary learned to learn after how trying to and repeatingadjust athe good applied number power, of the experiments. position of the It is tweez necessaryers’ focus to learnand the how direction to adjust of the the movement applied power,of the positionthe trap that of the is cutting tweezers’ the defect focus loop. and Scale the direction bars, 5 µm. of Adapted the movement from Ref. of [87]. the trap that is cutting the defect loop. Scale bars, 5 µm. Adapted from Ref. [87]. In real experiments this colloidal cluster is then systematically knotted and linked with the laser tweezers into the desired configurations by a localized laser-induced micro-quench. This is achieved byIn realadjusting experiments the laser thispower colloidal to the clusterlevel that is theninfluences systematically locally the knotteddefect loops and and linked enables with their the laser tweezersrewiring. into In the the desired example configurations given in Figure by 15 a localizedwe start with laser-induced the right-handed micro-quench. Trefoil knot This (Figure is achieved 15a), by adjustingwhich the is transformed laser power in to the levelleft-handed that influences composite locally knot in the Figure defect 15c loops and, and finally, enables into theirthe third rewiring. In thepossible example configuration, given in Figure which is15 the we two-component start with the lin right-handedk in Figure 15e. Trefoil It turns knot out (Figure that practically 15a), which is transformedany kind of knot in the or left-handedlink can be realized composite by taking knot a insufficiently Figure 15 largec and, colloidal finally, array. into the third possible configuration, which is the two-component link in Figure 15e. It turns out that practically any kind of 6. Exotic Nematic Colloids: Handlebodies and Koch-Star Particles knot or link can be realized by taking a sufficiently large colloidal array. So far we have presented the rich topology of a nematic liquid crystal around a spherical or 6. Exoticfiber-like Nematic inclusion, Colloids: i.e., all Handlebodies topologically trivial and Koch-Starobjects. These Particles experiments clearly demonstrate that the main parameter that enhances the richness of the observed topological structures is the chirality So far we have presented the rich topology of a nematic liquid crystal around a spherical or of the nematic liquid crystal. For achiral liquid crystals the topological defects around colloidal fiber-likeparticles inclusion, are rather i.e., simple, all topologically with the most trivial complex objects. being These the experiments colloidal entanglement clearly demonstrate of a pair thator the mainseveral parameter particles. that enhancesHowever, thewhen richness the nematic of the observedliquid crystal topological is made structureschiral, the istopology the chirality of the of the nematicnematic liquid director crystal. becomes For achiral extremely liquid rich, crystals as demo the topologicalnstrated by defects the knotting around and colloidal linking particles of the are rathernematic simple, orientational with the most field complex around beingmultiple the colloidalspherical entanglementcolloids. Nevertheless, of a pair orthe several topological particles. However,properties when of the these nematic inclusions liquid are crystal rather is trivial made since chiral, microspheres the topology are of all the topological nematicdirector objects with becomes extremelygenus rich,g = 0, asi.e., demonstrated having no handles. by the knotting and linking of the nematic orientational field around multipleModern spherical micro-fabrication colloids. Nevertheless, techniques the allow topological us to fabricate properties colloidal of theseparticles inclusions with complex are rather trivialshapes, since such microspheres as tori and are other all handlebodies topological objects [70], spiraling with genus microrods g = 0, [92], i.e., Koch-star having no microparticles handles. [93],Modern and micro-fabricationeven knotted and techniqueslinked polymer allow colloidal us to fabricate particles colloidal [94]. It is particles possiblewith to produce complex such shapes, particles by using 2D lithography or 3D two-photon photo-polymerization techniques. The ability to such as tori and other handlebodies [70], spiraling microrods [92], Koch-star microparticles [93], and produce topologically nontrivial colloidal particles raises the question of the resulting topology of eventhe knotted director and field linked that is polymer forced to colloidalalign along particles the closed [94 surfaces]. It is possibleof these colloidal to produce particles. such particles by using 2D lithography or 3D two-photon photo-polymerization techniques. The ability to produce topologically nontrivial colloidal particles raises the question of the resulting topology of the director field that is forced to align along the closed surfaces of these colloidal particles. Materials 2018, 11, 24 18 of 27

The topology of the director field around the colloidal particles is governed by the law of the conservation of topological charge [2]. This law is similar to the law of the conservation of electric charge, but instead of the electric field, we have the director field, which describes the ordering of the nematic liquid crystal. Topological charges in nematic liquid crystals are therefore associated with singularities of the director field, which appear in the core of point defects, also called hedgehogs, and the defect lines or loops, also called disclination lines. The law of the conservation of topological charge is similar to Gauss’s law for the electrostatic field, and the topological charge is measured in a similar way. To measure the topological charge, a closed surface is constructed around the field’s singularity and the topological flux is measured by integrating it along the closed surface. The resulting topological charge q is formally calculated as an integral of the topological flux (i.e., the number of director “streamlines”) along the closed surface embracing this singularity:

* * ! 1 I * ∂ n ∂ n q = εijk n · × dSi (1) 8π ∂xj ∂xk σ

Here, εijk is the Levi-Civita totally asymmetric tensor and xi are the Cartesian coordinates, and dSi is the surface differential. This expression for the topological charge is odd in the director * field n and, as a consequence, the sign of the topological charge for our director field is not well defined. The nematic director is a “headless” vectorial field where +n and −n are formally equivalent. The expression “headless” actually tells us that we are using an inappropriate vectorial field for the description of the tensorial field. The director field is a tensorial field and the direction (and the director) defines the direction of the long axis of the molecular probability distribution. As a consequence, the sign of the topological charge in the nematic liquid crystals has no meaning, the only strict condition is that there are two kinds of charges and they are different in signs. This difference in signs helps us to count the total charge, which in many cases mutually compensates. Whereas in vectorial fields the combined topological charge for two hedgehogs is simply the algebraic sum of their charges, in nematics, where the director field is tensorial, the combined charge of two hedgehogs is either the sum or the difference of their charges. In nematic colloids, solid, liquid or gas particles are inserted into the nematic liquid crystal and the nematic liquid-crystal molecules are forced to align along the closed surface of the inclusions. There are two characteristic types of local alignment for the liquid-crystal molecules at the interface: Parallel and perpendicular. In rare cases, the surface alignment is oblique. Because the molecules are forced to align all along the closed surface, topological defects have to be created in the surrounding nematic liquid crystal, as in general it is not possible to fill the surrounding orientational field without orientational singularities, which are our topological defects. It turns out that the topological charge of the surrounding defects created by the immersion of the colloidal particles into the nematic liquid crystal is determined by a topological invariant called the Euler characteristic χ of the inserted particle. The Euler characteristic is in a one-to-one correspondence with the topology of the immersed particle and does not change under any smooth transformation of the surface of the particle. The Euler characteristic for a closed surface depends on the type of the closed surface and is directly related to the number of handles and holes that the object possesses. For example, a sphere has a closed surface with no handles and holes and its Euler characteristic is equal to χ = 2. If we make one hole through the sphere, we obtain a torus with a different Euler characteristic of χ = 0. More precisely, the Gauss-Bonnet theorem claims that the Euler characteristic χ of any closed surface is related to the total Gaussian curvature of that surface:

" * * # 1 I I * ∂ ν ∂ ν χ = KdS = dθ dΦ ν · × (2) 2π σ σ ∂θ ∂Φ

Here, K = κ1 · κ2 is the local Gaussian curvature and κi are the two principal curvatures at a given * point on the surface. The vector ν is the local normal to the surface σ at the point of consideration, Materials 2018, 11, 24 19 of 27 deﬁned by the two angles θ and Φ. Furthermore, it can be shown that the Euler characteristic is related to the genus g of the surface of the particle:

χ = 2(1 − g) (3)

Here, the genus g is the number of handles attached to the sphere that results in the formation of handle-bodies. For example, we obtain a torus by adding a single handle to the sphere and then smoothly transforming this closed surface into a torus. If we now consider any closed surface inserted into a vector field when this field is perpendicular at all the points of the inserted surface, the topological charge q of this surface is equal to the integral given by Equation (1). This surface integral reduces to the total Gaussian curvature of the closed surface divided by 4π. Because the Gauss-Bonnet theorem states that the total Gaussian curvature of any closed surface without a boundary is quantized in units of 2π (Equation (2)), the resulting hedgehog charge mc of the vectorial field aligned along the surface S is:

2π · χ m = ± = ±(1 − g) (4) c 4π Far away from the surface, the ﬁeld is considered homogeneous and the topological charge of this larger volume is zero. On the other hand, there must be topological defects inside this volume, carrying the topological charge md, which compensates for the topological charge of the surface:

mc + md = 0 (5)

For a vector ﬁeld the Euler characteristic of the immersed surface of the particle therefore uniquely deﬁnes the total topological charge md of all the defects accompanying the surface:

md = 1 − g (6)

For a sphere the genus g = 0, which means that the net charge of all topological defects accompanying the sphere should add up to md = 1. In principle it is, therefore, allowed to have many different topological defects accompanying the sphere, which mutually compensate for their charge, except for a single hedgehog charge md = 1. However, all these additional charges increase the elastic energy and in reality in most cases we observe a single hedgehog unit charge accompanying the sphere. This is in perfect agreement with the observation of a point hedgehog or a Saturn ring encircling the microspheres in the nematic liquid crystals. Moving from spheres with g = 0 to tori with g = 1, the theorems require that the total topological charge of the hedgehogs accompanying the torus should be zero, md = 0. There are two possibilities in the case of a torus: (i) there are no topological defects accompanying the torus or, (ii) there is an even number of topological defects that are oppositely charged and mutually compensate. Both scenarios were observed for a torus [95–98]. The first case with no topological singularities was observed in the experiment where nematic toroidal dispersions were created [95,96]. In this case, the torus is filled with the nematic that creates a planar boundary condition that requires no topological defects inside the torus. The second scenario of mutually compensating pairs of topological defects floating outside the solid micro-torus and handlebodies was observed by Senyuk et al. [70], Campbell et al. [97] and Tasinkevych et al. [98]. By using fluorescent confocal microscopy they were able to reconstruct a director field around the particle and they observed that the topological defects were oppositely charged, thus preserving the total topological charge of zero. Figure 16a,b show a toroidal colloidal particle made of photo-polymerizable material with a perpendicular surface anchoring immersed in a planar nematic liquid-crystal cell [70]. It is clear that there are two topological defects: One centered inside the toroidal particle and the other resting on its outer perimeter. In Figure 16b the point defect has opened into a small loop. The panels in Materials 2018, 11, 24 20 of 27

Figure 16d,e show the schematics of the director ﬁeld around a toroidal particle with perpendicular surfaceMaterials anchoring 2018, 11,of 24 the liquid-crystal molecules. 20 of 27

Figure 16. (a,b) Cross-polarized images of a toroidal particle with perpendicular surface anchoring of Figure 16. (a,b) Cross-polarized images of a toroidal particle with perpendicular surface anchoring of the liquid-crystal molecules immersed in a planar nematic cell. Two point defects can be seen in the liquid-crystalpanel (a). In panel molecules (b) one immersed point defect in ahas planar opened nematic into a small cell. Twoloop. point (c) A defectshandlebody can with be seen genus in panel = (a). Ing panel 2 (b is) accompanied one point defect by three has point opened defects. into (d a–f small) Schematic loop. drawing (c) A handlebody of the director with field genus for theg 2 is accompaniedthree images by three in (a– pointc). The defects. scale bar ( dis– 5f )µm. Schematic Adapted drawingfrom Ref. of[70]. the director field for the three images in (a–c). The scale bar is 5 µm. Adapted from Ref. [70]. A smooth torus can be morphed into a Koch-star colloidal particle that is produced mathematically Afollowing smooth torusthe prescribed can be morphed procedure into for a constructi Koch-starng colloidal the particles. particle Bending that isand produced shaping mathematicallythe torus keeps its Euler characteristic unchanged, only the geometry and shape are changing. This means that following the prescribed procedure for constructing the particles. Bending and shaping the torus keeps the Koch-star particles, shown in Figure 17, are topologically equivalent to a torus and the total its Eulercharge characteristic of all defects unchanged, of a Koch-star only particle the should geometry always and be shapezero. are changing. This means that the Koch-star particles,The topological shown equivalence in Figure of 17 Koch-star, are topologically particles and equivalent toroidal particles to a torus is demonstrated and the total in chargethe of all defectsexamination of a Koch-star of the number particle of should topological always defects be zero. around such particles with polarized optical Themicroscopy topological and laser equivalence tweezers93. of Figure Koch-star 18 show particless some andexamples toroidal of Koch-star particles particles is demonstrated having in the examinationdifferent iterations of the as number viewed under of topological an optical micr defectsoscope. around Panel suchI shows particles the particle with in polarizedthe isotropic optical microscopyphase of and the laserliquid tweezerscrystal, where [93]. the Figure optical 18 irregu showslarities some due examples to production of Koch-star can be detected. particles Panel having differentII shows iterations the crossed-polarized as viewed under images an optical of Koch-star microscope. particles Panel in a uniformly I shows thealigned particle planar in nematic the isotropic cell, where these particles are levitating in the middle of the nematic layer. If there was no surface phase of the liquid crystal, where the optical irregularities due to production can be detected. Panel II alignment of the liquid crystal on these particles, the crossed-polarized images would be completely showsblack. the crossed-polarizedThere is a strong surface images alignment of Koch-star in the direction particles perpendicular in a uniformly to the alignedlocal surface, planar which nematic cell, wheremeans thesethat the particles orientation are of levitating the liquid in crystal the middle is strongly of thedistorted nematic in the layer. vicinity If there of the was polymer no surface alignmentwalls ofof the Koch-star liquid crystal particles. on theseBecause particles, of this rather the crossed-polarized large distortion, extending images micrometers would be completelyfrom black.the There surfaces, is a strongthe direction surface of the alignment local optical in the axis direction is no longer perpendicular parallel to the to far-field the local liquid-crystal surface,which meansalignment. that the orientationThis local rotation of the of liquidthe optical crystal axis isresults strongly in strong distorted birefringent in the colors vicinity accompanying of the polymer walls ofthe the walls Koch-star of the Koch-star particles. particles. Because These of colors this rather are not large uniform distortion, along all extendingthe sides of micrometersthe Koch-star from particles. For example, for the first iteration Koch-star particle (a triangle), shown in Figure 18 IIa, we the surfaces, the direction of the local optical axis is no longer parallel to the far-field liquid-crystal can clearly see uneven coloring of the liquid crystal along the side parallel to the far-field director. alignment.This indicated This local the rotation presence of of the a opticaltopological axis defect results in inthe strong middle birefringent of this side colors of the accompanying triangle. the wallsAdditional of the Koch-staroptical inhomogeneities particles. These are colorsclearly areobservable not uniform in the along corner all of the the sides triangle, of the where Koch-star particles.topological For example, defects are for also the located. first iteration Similar color Koch-star inhomogeneities particle are (a triangle),observed in shown panels III, in Figurewhere a 18 IIa, we canspecial clearly optical see unevenpolarizing coloring techniqu ofe is the used liquid (the crystalred plate along or the theλ plate), side which parallel helps to theus to far-field determine director. This indicatedthe local alignment the presence of the of aoptical topological axis of defectthe liqu inidthe crystal. middle In combination of this side with of the optical triangle. tweezers, Additional opticalthese inhomogeneities optical polarization areclearly techniques observable allow us into count the corner the number of the of triangle, topological where defects. topological defects are also located. Similar color inhomogeneities are observed in panels III, where a special optical polarizing technique is used (the red plate or the λ plate), which helps us to determine the local alignment of the optical axis of the liquid crystal. In combination with optical tweezers, these optical polarization techniques allow us to count the number of topological defects. Materials 2018, 11, 24 21 of 27 MaterialsMaterials 2018 2018, 11,, 1124, 24 21 21of of27 27

Figure 17. Scanning electron images of Koch-star colloidal particles produced by a two-photon Figure 17. Scanning electron images of Koch-star colloidal particles produced by a two-photon Figurepolymerization 17. Scanning technique electron using images the Photonicof Koch-star Profes cosionallloidal System particles by Nanoscrproducedibe. byNote a two-photonthat these polymerization technique using the Photonic Professionalg 1 System by Nanoscribe. Note that these polymerizationparticles are topologically technique using equivalent the Photonic to torii with Profes sional . Adapted System fromby Nanoscr Ref. [93].ibe. Note that these particles are topologically equivalent to torii with g = 1 . Adapted from Ref. [93]. particles are topologically equivalent to torii with g  1 . Adapted from Ref. [93].

Figure 18. Koch-star colloidal particles of iterations 0 (a,b), 1 (c,d), 2 (e) and 3 (f) with a fractal pattern of the director field reveal nematic topological states. Panels I are taken with no polarizers; panels II are taken between crossed polarizers; panels (III) are taken between crossed polarizers and the red Figureplate; 18. panels Koch-star IV are colloidal numerical particles Landau of de iterations Gennes simulations. 0 (a,b), 1 (c ,Adaptedd), 2 (e) and from 3 Ref.(f) with [93]. a fractal pattern Figure 18. Koch-star colloidal particles of iterations 0 (a,b), 1 (c,d), 2 (e) and 3 (f) with a fractal pattern of the director field reveal nematic topological states. Panels I are taken with no polarizers; panels II of the director ﬁeld reveal nematic topological states. Panels I are taken with no polarizers; panels II are taken between crossed polarizers; panels (III) are taken between crossed polarizers and the red are taken between crossed polarizers; panels (III) are taken between crossed polarizers and the red plate; panels IV are numerical Landau de Gennes simulations. Adapted from Ref. [93]. plate; panels IV are numerical Landau de Gennes simulations. Adapted from Ref. [93].

Materials 2018, 11, 24 22 of 27

When we consider the next iterations of the Koch-particles, the optical pattern is even more complicated, as shown for the Star of David particles in Figure 18c. Nevertheless, the number of topological defects can be counted, and it turns out that there is always an even number of topological defects that mutually compensate and result in the overall zero topological charge. The experimental observations were complemented by numerical simulation using the finite-element method, and the resulting director fields together with topological defects are shown in panel IV. The number of topological defect pairs increases exponentially with fractal iterations and equals 108 at the third fractal iteration. There are other examples of colloidal particles of exotic shapes in liquid crystals, such as spiraling rods, prisms, platelets, knots and links, etc. They all show topological defects that obey the fundamental law of the conservation of topological charges, which makes liquid crystals a unique playground for the topology of fields.

7. Conclusions This short review presents aspects of the rapidly developing subﬁeld of liquid-crystal colloids, which has generated many attractive and fascinating results over the past 10–15 years. Of course, it is not possible to present all the important achievements in this ﬁeld; this would be a task comparable to writing a book. Instead I have presented a rather personal perspective with an emphasis on experimental topology. Indeed, the experimental topology involving studies of topological defects, their structure, emergence and dynamics is in my opinion one of the most appealing phenomena in nematic colloids. Defects in liquid-crystal colloids are relatively easy to observe, study and manipulate, thus giving us an insight into the foundations of condensed-matter topology. This fundamental aspect of nematic colloidal science is complemented by emerging applications of topological defects in future optical and photonic devices. One of them is generating optical wave fronts with complex structures, such as Laguerre Gaussian beams, where the topological charge of the condensed matter is imprinted into the topological charge of the electromagnetic waves.

8. Materials and Methods

8.1. Preparation of Nematic Colloids The simplest way to prepare nematic colloids is by mixing the microspheres of different materials in a nematic liquid crystal. Typically, the nematic liquid crystal 5CB is used in all experiments because it is commercially available, stable and its properties are well known. The typical material for the microspheres is silica, because of its excellent surface smoothness, ease of surface functionalization and commercial availability over a large range of diameters. One of the most important aspects of colloidal preparation is the chemical treatment of silica microspheres to induce excellent surface alignment. A good way to obtain the surface anchoring is to treat the silica microspheres with silanes such as DMOAP. This surface modification creates a well-defined and stable monolayer of DMOAP molecules, which aligns practically all liquid crystals perpendicular to the surface. The method of surface preparation is described in detail in Refs. [2,21,25]. The next step is the introduction of dried and surface-functionalized silica microspheres into a nematic liquid crystal, typically with a weight ratio of 1%. After mixing the colloidal mixture, the micrometer colloidal particles are usually well dispersed. However, smaller nematic colloids tend to preserve the clusters and aggregates of particles, and have to be treated with strong ultrasound to break up these clusters. After the mixture is prepared it is introduced by capillary action into a measuring cell. This is a standard cell made of two parallel glass plates separated by a spacer, which defines the thickness of the gap to be filled with the nematic colloidal dispersion. The cell is pre-treated in order to fabricate an alignment layer, which then aligns the director of the nematic liquid crystal uniformly throughout the cell surfaces. This alignment can be either parallel (planar cell) or perpendicular (homeotropic cell) to the surface of the cell walls. Materials 2018, 11, 24 23 of 27

8.2. Fabrication of Colloidal Particles Using 2-Photon Polymerization Whereas particles with a spherical or rod-like shape are commercially available, micrometer-sized particles of practically arbitrary shape can be produced using the 2-photon polymerization of a light-sensitive polymer. Typically, commercially available instruments, such as Photonic professional (Nanoscribe GmbH, Eggenstein-Leopoldshafen, Germany), are able to produce tens of micrometer-sized particles with complex shapes using different photosensitive polymers. These particles are produced on a supporting glass surface and have to be surface functionalized, detached from the glass surface and introduced into the liquid crystal. DMOAP is used to ensure a good homeotropic surface anchoring of the liquid crystal on a typical 2-photon polymer. However, it turns out that not all photosensitive polymers can be functionalized with DMOAP.

8.3. Manipulation of Nematic Colloids by Laser Tweezers Once the nematic colloidal dispersion is introduced into the cell, the particles can only be moved and manipulated with a non-contact method, such as laser tweezers. These optical tweezers are a rather simple instrument, consisting of a strong laser source, a microscope and the system for moving the focus of the laser beam in the microscope. Typically, a 100-mW laser is used (such as CW Nd:YAG), which is sent to one of the entrance ports of an inverted optical microscope. The laser light is then focused in the focal plane of the objective. When the measuring cell with the nematic colloidal dispersion is observed in such a microscope, the laser focus is in the same imaging plane as the observer (or camera). At a low laser power (several mW) the optical trap has a relatively small effect on the colloidal particles. In this regime, the refractive index of the colloidal particles plays an important role. It is known from optical trapping that colloidal particles can be trapped by the tweezers only if their refractive index is larger than the index of the surroundings. This is difficult to achieve in liquid crystals, since the refractive indices of typical liquid crystals are between 1.5 and 1.7. This is close to or even higher than the refractive index of glass (1.5), which means that glass particles in a nematic liquid crystal cannot be trapped and manipulated by the tweezers. However, it was shown in 2004 that the optical trapping of particles in liquid crystals is much more complex than the optical trapping of particles in ordinary liquids. There are two reasons for this: (i) optical trapping in a liquid crystal is polarization sensitive, which means that the electric field of the laser light couples to the dielectric anisotropy of the liquid crystal. Close to the colloidal particles, the liquid crystal is aligned along the normal to the surface of the particle, which creates an elastically distorted region around the particle. The polarized light of the tweezers couples to that region and by moving the trap, we also move the region. The particle is forced to follow the trap, because it has to follow the moving, elastically distorted region around itself. (ii) If the power of the laser tweezers is strong (100 mW), the liquid crystal is locally heated by the absorption of light. This creates a small volume of liquid crystal where the order parameter and the elastic properties are different from the surroundings. Such a volume acts as an artificial particle, which attracts other colloidal particles. By using either of the two mechanisms we can trap and manipulate colloidal particles of an arbitrary material and shape in the nematic liquid crystal.

Acknowledgments: The author wishes to acknowledge contribution of Miha Škarabot, Slobodan Žumer, Miha Ravnik, Andriy Nych, Ulyana Ognysta, Simon Coparˇ and many others with whom the author collaborated during the last 15 years in this field. The support of Slovenian Research Agency (ARRS) through contracts P1-0099 and J1-6723 is acknowledged. Conflicts of Interest: The author declares no conflict of interest.

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