Hypercomplex LiquidCrystalsZvonimir Dogic,1,2,� Prerna Sharma,1 andMark J. Zakhary1

1Department of Physics, Brandeis University, Waltham, MA 02474;email: zdogic@brandeis.edu, prerna@brandeis.edu, markz1@brandeis.edu2Institute for Advanced Study - Technische Universität München, Garching 85748,Germany

Annu. Rev. Condens. Matter Phys. 2014. 5:137–57

First published online as a Review in Advance onDecember 18, 2013

TheAnnual Review of Condensed Matter Physics isonline at conmatphys.annualreviews.org

This article’s doi:10.1146/annurev-conmatphys-031113-133827

Copyright © 2014 by Annual Reviews.All rights reserved

�Corresponding author

Keywords

soft matter, complex fluids, self-assembly, colloids, topologicaldefects, active matter, elasticity

Abstract

Hypercomplex fluids are amalgamations of polymers, colloids, oramphiphilic molecules that exhibit emergent properties not ob-served in elemental systems alone. Especially promising building-blocks for assembly of hypercomplex materials are molecules withanisotropic shape. Alone, these molecules form numerous liquidcrystalline phases with symmetries and properties that are funda-mentally different from those of conventional liquids or solids.When combined with other complex fluids, liquid crystals formmaterials with diverse emergent properties. In equilibrium, theinteractions, dimensions, and shapes of these hypercomplex materi-als can be precisely controlled. When driven far from equilibrium,these materials can deform and even spontaneously flow in the ab-sence of external forces. Here we describe recent experimentalaccomplishments in this rapidly developing research area. We em-phasize how the common theme underlying these diverse efforts istheir reliance on the basic physics of molecular liquid crystals devel-oped in the 1970s.

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1. INTRODUCTION

Upon reducing temperature, most molecular substances undergo a transition from a disorderedfreely flowing liquid to an ordered solid crystal. Molecules with anisotropic shape that exhibita panoply of partially ordered liquid crystalline states between a high-temperature liquid anda low-temperature solid are an important exception to this rule. Although the first evidence fortheir existence dates back to the late 19th century, liquid crystals captured the attention andimagination of the physics community in the 1970s, when many of the theoretical concepts thatdescribe their phase behavior, structure, and dynamics were developed (1, 2). These extensiveefforts have rapidly transformed the mysterious and poorly understood liquid crystalline materialsinto highly controllable anddesignable substanceswith diverse applications,most notably includingliquid crystal displays. As a result of this success, one encounters the opinion that the fundamentalquestions related topartial orderingof anisotropicmolecules havebeen successfullyanswered. Inouropinion, this is not the case. Conventional molecular liquid crystals remain a vibrant research fieldwith numerous open questions. Furthermore, over the past twodecades the ideas developed throughpioneering studies of molecular liquid crystals have been used to create increasingly complexcomposite materials that remain at the forefront of soft-matter research. Here, we review a fewrepresentative examples of these hypercomplex materials that illustrate the timelessness and im-portance of the fundamental concepts related to the ordering of anisotropic fluids.

Liquid crystals belong to a broader category of soft materials known as complex fluids. Othernotable systems belonging to the same class of materials are polymers, colloids, lipid bilayers, andself-assembling amphiphiles. All of these classical systems have unique properties that are in-teresting from both the fundamental and application perspectives. Furthermore, extensive theo-retical efforts have produced detailed understanding of how the macroscopic properties ofcomplex fluids are related to their microscopic structure and interactions, thus making thesematerials tunable and designable. With this basic understanding in place, recent efforts have fo-cused on developing next-generation materials that combine desirable features of different com-plex fluids. For instance, combining the amphiphilic nature of surfactant molecules with classicalpolymers leads to block-copolymers, a highly versatile complex fluid with exceptional materialproperties (3). Under specific conditions, such block-copolymers can further assemble into poly-merosomes, which are analogs of classical lipid bilayer vesicles with greatly improved materialproperties, such as durability and toughness (4).

In a similar vein, molecular liquid crystals have been merged with other complex fluids toproduce materials with emergent and unexpected properties. For example, combining molecularliquid crystals with cross-linked polymer gels leads to nematic elastomers, which arematerials thatexhibit soft elasticity, a phenomenon wherein a solid-like elastomer can behave like a liquid, withzero energetic cost for certain shear deformations (5). Soft elasticity is not a property of molecularliquid crystals or cross-linked rubber alone but only emerges when these two quintessentialcomplex fluids are combined in a single material. Such emergent materials have been calledhypercomplex fluids, a term first coined by Robert Meyer. As another example, combiningmolecular liquid crystals with micron-sized colloidal particles leads to novel interparticle inter-actions that drive the assembly of colloidal matter that is not achievable with other methods (6).Here, we review some examples of these hypercomplex materials while emphasizing the role thatliquid crystal physics plays in their emergent behavior.

We first briefly recapitulate the basic physics ofmolecular liquid crystals. Next, we describe thebehavior of colloidal liquid crystals, which combine desirable features of colloidal suspensionswith those of classical liquid crystals. From here, we focus on how merging liquid crystals andcross-linked rubber leads to solid networks with surprising elastic properties. Then, we describe

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the behavior of liquid crystal–colloid mixtures, in which micron-sized particles suspended inanisotropic fluids acquire novel interactions not observed in conventional isotropic solvents.Finally, we review recent experimental advances in active liquid crystals, in which inanimate rodsare replaced by animate constituent objects that can coordinate theirmotion to produce inherentlyfar-from-equilibrium materials that spontaneously swim, flow, and crawl.

2. PHYSICS OF MOLECULAR LIQUID CRYSTALS

Spherically shaped molecules can form either a liquid state, which is characterized by short-rangepositional order, or a solid crystal, which is characterized by long-range positional order (7). Inaddition to these phases, anisotropic molecules form numerous liquid crystalline phases that haveorientational and/or partial positional order (Figure 1). Transitions between these liquid crys-talline phases are induced by varying either temperature (for thermotropic liquid crystals) orsample concentration (for lyotropic liquid crystals). The nematic phase is the most common liquidcrystalline phase. Upon decreasing temperature or increasing particle concentration, a disorderedisotropic liquid composed of anisotropic molecules undergoes a first-order phase transition intoanematic liquid crystal inwhich all themolecules are, on average, aligned along the samedirection,which is described by the nematic director (Figure 1b). The resulting nematic phase has the long-range orientational order characteristic to solids. However, the center of mass of each rod retainsthe positional disorder inherent to liquids and the time-averaged density is spatially uniform. It isthis confluence of both liquid and solid properties that leads to the name liquid crystals and theemergence of their highly desirable materials properties. In many substances, decreasing thetemperature of the nematic phase leads to a smectic phase (Figure 1c). In addition to long-rangeorientational order, smectic liquid crystals have a quasi-long-range 1D positional order. In otherwords, smectic liquid crystals are composed of liquid monolayers of aligned molecules stacked ontop of each other. Along with nematic and smectic phases, there are dozens of other liquidcrystalline phases with more exotic symmetries. Furthermore, many different types of anisotropic

a b c

Figure 1

Schematic illustration of isotropic, nematic, and smectic liquid crystalline phases formed in a suspensionof rod-like molecules. (a) The isotropic phase has short-range orientational and positional order. (b) Thenematic phase is characterized by long-range orientational order, although the center-of-mass of each rod-likemolecule retains the short-range order characteristics of liquids. (c) In the smectic phase, rods formtwo-dimensional one-rod-long liquid-like layers. Entropic-excludedvolume interactionsbetweenhard rods arearguably the simplest interactions that drive the formation of both nematic and smectic phases.

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molecules form liquid crystalline phases, such as rods, discs, and arced bent-coremolecules. In thisreport, we focus our attention mainly on rod-shaped constituents.

Theoretical models of liquid crystals require identification of their underlying symmetries.Most liquid crystal–forming molecules are polar and have a distinguishable head and tail.However, in a liquid crystalline phase these rods are equally likely to point in either direction.Therefore, the local orientational order is more accurately described by an arrowless line thana directional vector. This observation has important consequences for the nature of liquidcrystalline phase transitions as well as for the topology of the associated defects.

The lowest energy state of a nematic liquid crystal is one of a uniform spatial alignment inwhich all rods point in the same direction. As with most other complex fluids, nematic liquidcrystals are very soft materials, i.e., thermal fluctuations are sufficiently strong to induce signifi-cant distortions away from the minimum energy state of uniform alignment. For nematic liquidcrystals, such low-energy deformations can be accounted for by a director that slowly twists,bends, and splays throughout space (1, 7). Quantitatively, these distortions are described by theFrank elastic free energy (8). A consequence of such low-energy modes is that distortions of thenematic director scatter light much more strongly compared with ordinary liquids, endowingnematic liquid crystals with their characteristic turbidity (1).

Along with slowly varying elastic distortions, another pervasive property of nematic liquidcrystals is the presence of topological defects: localized imperfections in the liquid crystalline orderthat cannot be removed by continuous deformations of the order parameter. The charge of a to-pological defect is related to the director rotation as one traverses a loop enclosing the defect. Byconvention, topological defects that involve director rotations of 360� have a charge of 61. Thesign is determined by the sense of director rotation when traversing a closed path in an anti-clockwise direction. In a conventional nematic characterized byquadrupolar symmetry, the lowestenergy defects, called disclinations, have a topological charge of 61/2 (Figure 2a,b). In com-parison, a liquid crystal with polar order cannot form such defects because rotating a vector by180� results in a physically distinct state. In such materials, the lowest energy defects would havea charge of 61 (Figure 2c–e). Therefore, a simple inspection of topological defects can determinethe fundamental symmetry of an underlying liquid crystalline material.

3. LIQUID CRYSTALS FROM ANISOTROPIC COLLOIDS

Colloids are solid particles suspended in a background fluid. Their sizes range between a few nano-meters and a fewmicrons, thus ensuring that thermal fluctuations suppress particle sedimentation (9).

m = +½ m = –½ m = +1 m = +1 m = –1

a b c d e

Figure 2

Schematic illustrations of a nematic director enclosing topological defects found in quadrupolar (nematic) andpolar liquid crystals. (a,b) Disclination defects of charge þ1/2 and �1/2. Moving along a loop that encirclessuch defects results in net director rotations of 180�. Because polar vectors need to rotate 360� to returnto their original state, they cannot formdefectswith a charge of 1/2. (c–e) Topological defects of charge 1 can beformed by liquid crystals with both polar and nematic order. Abbreviation:m, topological charge of the defect.

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Twoprincipal featuresmakecolloidal suspensionsahighlydesirable systemto study.First, it is possibleto systematically engineer colloidal interactions, enabling experiments that directly probe the re-lationship between microscopic interactions and macroscopic phase behavior (10–12). Second,micron-sized colloids can be directly tracked with video microscopy and manipulated with opticaltweezers (13–17).

The classic model system exemplifying these features is the suspension of micron-sizedspherical latex or silica beads (18). By tuning relevant parameters, it is possible to engineerthese colloids to have either repulsive or attractive interactions of predetermined shapes, ranges,and strengths. Of particular interest are colloids with hard-sphere-like interactions. These are anexperimental realization of a theoretical model that has played a crucial role in developinga modern theory of the liquid state (19, 20). With increasing concentration, a suspension of hard-sphere-like colloids undergoes a liquid-to-solid phase transition (21). The ability to simultaneouslytrack all the particles in real timewithin a colloidal liquidor crystal has yielded essential insight intoimportant yet hard to study phenomena, such as crystal nucleation and glass formation (22–24).Because hard-core repulsive potentials arise at all length scales, many aspects of the physics ofcolloidal suspensions are universal. Thus, studying colloidal spheres increases our understandingnot only of complex fluids but also of materials on the molecular length scale. It is helpful to thinkof colloids as giant atoms that can be visualized, tracked, andmanipulated, andwhose interactionscan be tuned and engineered.

Colloidal liquid crystals combine the unique features of colloids and molecular liquid crystals,giving rise to myriad complex materials. In some ways, colloidal liquid crystals are an old subjectthat has experienced a renaissance over the past two decades. In a pioneering work in 1935,Wendell Meredith Stanley performed the first purification of Tobacco mosaic virus (TMV), ananisotropic rod-like colloid (25). Soon thereafter, John Bernal and coworkers prepared TMVsuspensions at a high-enough concentration to observe an isotropic-nematic (I-N) phase transition(26). Among others, these experiments motivated Onsager to formulate his seminal theory of theI-N phase transition, a scientific accomplishment that was many years ahead of its time and haslaid the intellectual foundation for the entire field of colloidal liquid crystals (27).

To describe the I-N phase transition, Onsager considered a fluid of hard rods. In such systems,the only accessible configurations are those without any particle overlaps (Figure 3). An overlapbetween any two rods leads to infinitely large interaction energy, so that such states are neverrealized. Consequently, the only contribution to the internal energy of the ensemble, U, is thekinetic energy U}NkBT, where N is the number of particles. This argument holds for any fluidwith hard-core interactions. It follows that the free energy of hard-particle fluid scales linearlywithtemperature, F}U � TS}TðkBN � SÞ, and the free energy attains a minimum for a phase thatmaximizes the overall system entropy. It might seem that entropy-maximizing fluids will alwaysremain in a disordered state.However, as explained below, at surprisingly lowdensities an alignedhard-rod nematic phase has higher entropy then a disordered isotropic phase.

A fluid of hard spheres crystallizes at 50%volume fraction. In contrast, highly anisotropic rodsform an ordered nematic phase at volume fractions as low as 1%. The remarkably low density ofthe I-N phase transitionmakes it possible to expand the free energy of a hard-rod fluid in powers ofdensitywhile still being able to quantitatively describe the I-N phase transition (28–30). Because ofthe much higher density at which hard spheres crystallize, a similar approach cannot be used todescribe a liquid-to-crystal transition. In this sense, the fluid of hard rods is simpler than that ofhard spheres.

The lowest-order term in the virial (density) expansion of the hard-rod free energy, associatedwith the entropy of mixing, favors a rotationally disordered phase and dominates at low densities.The second-lowest-order term, associated with the entropy of packing, favors a strongly aligned

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state and dominates at higher densities. The packing entropy is directly related to the excludedvolume between two rods, which is defined as the volume inaccessible to the center of mass ofa colloid owing to the presence of other impenetrable colloids. Minimizing the excluded volumeincreases the packing entropy of the system. In an isotropic phase, rods are on average perpen-dicular to each other, resulting in a large excluded volume (Figure 3a). In contrast, two parallelrods in the nematic phase have a much smaller excluded volume and thus higher packing entropy(Figure 3b). The competition between the entropyofmixing and the entropyof packingdeterminesthe exact location of the I-N phase transition (Figure 3c,d). The Onsager argument becomesquantitative for rods with an aspect ratio larger than 100, as the higher-order virial terms becomenegligible at densities in which the I-N phase transition occurs. The basic Onsager argumentoutlined here can be extended to account for important properties of experimental systems, such assurface charge, rod bidispersity and polydispersity, and flexibility (31–34).

L

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Ionic strength (mM)

30

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1 10 100 1,000

Theory

NematicIsotropic

L

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Vexc

Vexc

Figure 3

The Onsager theory of the I-N phase transition can be understood as a competition between mixing entropy, which favors the isotropicphase, and packing entropy, which favors the aligned nematic state. (a) Packing entropy is closely associated with excluded volume Vexc

(indicated by gray shading) between two rods. In the isotropic phase, the rods are, on average, perpendicular to each other, andVexc scales asL2D. (b) In the nematic phase, rods are, on average, alignedwith eachother, andVexc scales asLD2. (c) A suspensionof rod-like Tobacco mosaic viruses at high-enough concentrations exhibits coexistence between the isotropic and nematic phases. The nematicphase sediments to the bottom. When viewed between cross-polarizers (right image), the nematic phase exhibits optical anisotropy(birefringence), which is a consequence of the structural anisotropy of the liquid crystalline phase. The isotropic phase, located abovethe nematic phase, is not birefringent and appears dark. (d) Coexisting isotropic and nematic datameasured in a suspension of the rod-likebacteriophage fd at different ionic strengths, which determines the effective rod diameter. Lines indicate predictions of the Onsagertheory, which have been extended to include effects of semiflexibility and the polyelectrolyte nature of filamentous viruses, and the openand closed dots are concentrations of coexisting isotropic and nematic phases, respectively (49). Data provided by Seth Fraden.

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Following the development of the Onsager theory in 1949, there was a period of reduced ac-tivity in the field over the next few decades. However, the emergence of computer simulations andthe success of the hard-sphere theory of the liquid state led to resurgent interest in studying otherhard particle fluids in the 1980s, which has continued unabated ever since. Computer simulationshave mapped a detailed phase diagram of hard spherocylinders as a function of their aspect ratio(35, 36). An important finding that has emerged from these efforts is that entropic forces also drivehard rods to form a stable smectic phase (37).

Any colloid that significantly deviates from a spherical shape undergoes an I-N phase transi-tion. Consequently, numerous experimental systems exhibit a nematic phase, including biologicalpolymers such as DNA, actin, and various filamentous viruses as well as cellulose and collagen(38–42). Nematic phases are also seen in a wide variety of chemically synthesized rod-like colloids(43–46).Moreover, suspensions of plate-like particles are another well-developed class of colloidsthat form nematic phases, a case that was also considered by Onsager himself (47).

However, only a few of these systems are sufficiently monodisperse to enable quantitative testsof the Onsager theory. TMV is an important system in which the I-N phase transition has beenstudied in detail (Figure 3c). However, the aspect ratio of TMV rods is not sufficiently large tosatisfy the Onsager criterion, leading to significant quantitative discrepancies between experi-ments and theory (48). Another useful experimental system is the fd virus, which has an aspectratio of 130.However, fdwt (wild type) is semiflexible,which considerably influences thenature ofthe I-Nphase transition andagain leads to quantitative differences from the originalOnsager rigid-rod theory (Figure 3d) (49, 50). Recently, a mutant fd Y21M virus has been identified that has thesame aspect ratio as fd wt but is significantly stiffer (51). Preliminary results indicate that thebehavior of such rods is quantitatively described by the originalOnsager theory.However, furtherstudies including detailed analysis of the order parameter are needed to confirm these findings.

Rods with a broad length distribution still form a nematic phase but exhibit a relatively wideI-N coexistence region (52). In contrast, to form a smectic phase, rod monodispersity is essential(53). Thus, there are very few colloidal liquid crystals that form a smectic phase. The exception isfilamentous viruses, which are naturally identical and thus serve as an ideal model system to studysmectic colloidal liquid crystals. Indeed, the smectic-A (Sm) phase was first conclusively dem-onstrated in suspensions of rigidTMVparticles (54). Subsequently, studies on fdwt andmore rigidfd Y21M rods have demonstrated the important effect of flexibility on the nature of the N-Smphase transition (55, 56). Flexibility destabilizes the smectic phase, drives the transition to higherdensities, and changes the transition from second order to strongly first order. The length of fd wtviruses, which is in themicron range, alsomakes it possible to directly visualize smectic layers, thusexploiting the second appealing feature of colloidal suspensions (Figure 4a). Visualization ofsmectic layers opened a path toward imaging real-space structure and dynamics of partiallyordered colloidal liquid crystals (56, 57). Another recently developed system that holds promise isthat of chemically synthesized silica-like bullet colloids with a narrow length distribution. Densesediments of these particles display beautiful smectic layers that can be directly visualized withoptical microscopy (Figure 4b) (58, 59).

From a fundamental perspective, studies of colloidal liquid crystals have established thequantitative relationship between the microscopic properties of anisotropic particles and theirmacroscopic phase behavior. Furthermore, by using an extension of the Onsager theory, itbecomes possible to predict how elastic constants that describe the long-wavelength, low-energydistortions of the nematic director depend on the microscopic parameters of constituent rods (60,61). Fromapractical perspective, the foundational principles developed throughworkon colloidalliquid crystals are being applied to an ever-wider range of systems. For example, recent advances inchemical synthesis have yielded highly monodisperse nanorod suspensions with tunable aspect

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ratios. The next challenge is to self-assemble such particles intowell-defined 3D structures that canbe used for various applications, ranging from light-emitting diodes to photovoltaic devices (62,63). With its emphasis on universal particle shape, the Onsager theory and its various extensionsprovide a valuable guiding principle for this endeavor. Finally, there is a recent renewed emphasison studying particles with shapes that are far more complex than those of simple rods andcylinders, and perhaps it is possible to consider Onsager as a forerunner of these efforts (64).

4. COLLOIDAL LIQUID CRYSTALS IN NONADSORBING POLYMERS

Colloidal liquid crystals merge desirable properties of colloidal suspensions with those of mo-lecular liquid crystals to shed insight into the fundamental interactions that drive the assembly ofall anisotropic fluids. Colloids also allow for the possibility of systematically engineering morecomplex interactions, making it possible to observe structures that are not feasible in molecularliquids. Here, we describe how introducing attraction between rod-like colloids considerablyexpands the complexity of the phase diagram. These systems merge three different types ofcomplex fluids—colloids, liquid crystals, and polymers—to produce materials with surprisingemergent behavior.

Since the seminal theoretical study by Asakura & Oosawa (AO) (12), it has been widely ap-preciated that adding nonadsorbing polymers to colloids induces tunable attractive interactionsthat can significantly affect the colloidal phase behavior (Figure 5a) (65–68). The AOmodel treatspolymers as ghost sphereswhose diameter is equal to the polymer radius of gyration (Rg). Polymerspass through each other without any energetic penalty and can thus be described by an ideal gasequation of state. However, polymer coils interact with colloids via excluded volume interactions.For each spherocylindrical colloid added to a polymer suspension, the volume available to de-

pleting polymers is reduced by p

�Rg þ D

2

�2Lþ 4p

3

�Rg þ D

2

�3, where L and D are the length

and diameter of the spherocylinder, respectively. As two colloids approach within two Rg of eachother, the excluded volume shells overlap, effectively increasing the free volume available to thedepleting molecules. Because the number density of depletants is much greater than that ofcolloids, the entropic cost associated with clustering a few large colloids is more than offset by theincrease in free volume (entropy) accessible to depletant molecules. The depletion interactionprovides a versatile mechanism for controlling colloidal interactions, as the range and strength ofattractive interactions can be tuned by changing the polymer size and concentration, respectively.

a b

2.5 μm2.5 μm2.5 μm 5 μm5 μm

Figure 4

The length scale of rod-like colloids enables direct visualization of smectic layers and the dynamics ofconstituent rods. (a) Smectic layers ofmicron-long fdwt viruses observedwith differential interference contrast(DIC) microscopy (55). (b) Confocal microscopy image of a smectic phase formed with fluorescentlylabeled 1.4-mm colloidal silica bullets. Adapted with permission from Reference 59.

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The mesoscopic size of colloids makes it possible to directly measure the depletion-inducedeffective interactions (10, 11, 69).

Adding nonadsorbing polymers to colloidal rods significantly alters the liquid crystalline phasediagram (70–72). For example, introducing depleting molecules widens the I-N phase coexistence(73, 74). Furthermore, the presence of depletants also leads to new emergent phases that have noanalogs in pure suspensions of hard rods (75–77). For example, a depleting polymer whose size isapproximately one tenth of the colloidal rod length destabilizes the nematic phase and induces theformation of a lamellar phase, which consists of intercalating liquid-like layers of rods and polymers(78, 79). Such phases are similar to those observed in conventional block-copolymers. However,whereas the latter assembly is driven by chemically heterogeneous molecules, lamellar phases inmixtures of hard rods and polymers are determined by geometric (entropic) considerations alone.

An entirely different hypercomplex structure emerges when depletant molecules are added toa dilute isotropic phase of rod-like molecules. For colloidal spheres, the strength of the depletioninteraction depends only on the radial separation between the particles. In comparison, the

a b

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5 μm5 μm5 μm5 μm

Figure 5

Attractive depletion interactions assemble rod-like viruses into 2D liquid monolayers, which are colloidalanalogs of lipid bilayers. (a) Schematic illustrationof the depletion effect. The volume excluded to the center-of-mass of the depleting polymer because of the presence of each spherocylinder is indicatedwith light shadowing.Bringing two spherocylinders together results in an overlap of excluded volume shells (indicated with darkshadowing), increases the accessible volume (entropy) of small depleting polymers, and leads to effective andtunable attractive interactions. (b) Schematic illustration of a colloidal membrane, a one-rod-length-thickliquid-like monolayer of aligned rods. The rod aspect ratio is reduced by a factor of 10 for visual clarity. (c)Differential interference contrast image of a colloidal membrane. (d) A low volume fraction of fluorescentlylabeled rods (green) reveals the dynamics of constituent rods within a membrane whose structure issimultaneously visualized with phase contrast microscopy (red). Adapted with permission from Reference 81.

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depletion interaction between two rod-like molecules depends on both their center-of-mass sep-aration and mutual orientation. The free energy–minimizing (excluded volume–maximizing)configuration is the one in which two rods are mutually aligned along their long axes (Figure 5a).For this reason, adding depletant to dilute isotropic rods induces the formation of clusters ofaligned rods, which coalesce laterally to form mesoscopic one-rod-length liquid-like monolayerdisks that contain thousands of rods. This assembly constitutes amesoscopic phase separation intopolymer-rich and rod-rich phases, as the disks contain no polymer. At high depletant concen-tration, small micrometer-sized disks stack on top of each other to form smectic filaments that aretens of micrometers in length (80). For large-enough disks, the stacking interactions reduce ex-cluded volumemore effectively than lateral association. Surprisingly, below awell-defined criticaldepletant concentration the face-on interactions between disks become repulsive, and stackingceases. However, mesoscopic disks are still free to coalesce laterally, leading to the formation ofsmectic monolayers that can reach macroscopic size (Figure 5b–d) (81).

The transition from face-on stacking to lateral coalescence can be understood by consideringcompetition between depletion-induced attractive interactions and entropic repulsion that is as-sociated with rod protrusion fluctuations along the direction of the average rod alignment (82).Bringing two disks together face-on restricts such protrusion fluctuations, decreasing the entropyof closely spaced disks and leading to effective repulsive interactions. On one hand, the range ofdisk-disk attractive interactions is determined by the depletant size and is independent of itsconcentration.On the other hand, the range of repulsive interactions arising because of protrusionfluctuations increases with decreasing depletant concentration. Therefore, at sufficiently lowdepletant concentration the longer-ranged repulsion overwhelms the attraction, leading tothermodynamically stable colloidal monolayer membranes.

Self-assembled hard-rod monolayers have the same symmetry as lipid bilayers. Consequently,the elastic free energy for lipid bilayers written down by Helfrich also applies to colloidal mem-branes (81). Furthermore, the repulsive protrusion interactions that lead to colloidal monolayerstability are reminiscent of the coarse-grained Helfrich entropic repulsions between lipid bilayers;however, in the former case the fluctuations occur on molecular rather than continuum length scales(83–86). For these reasons the monolayer assemblages observed in rod-polymer mixtures have beennamed colloidal membranes. However, from amicroscopic perspective the forces driving assembly ofcolloidal membranes and lipid bilayers are quite distinct. Colloidal membranes are assembled fromchemically homogeneous rod-like molecules, whereas lipid bilayers are assembled from chemicallyheterogeneous amphiphilic lipids. Nonetheless, colloidal membranes represent a unique opportunityto investigate membrane biophysics from an entirely new perspective on a length scale where it ispossible to visualize constituent building blocks. Along these lines, recent work has demonstrated thatchirality of the constituent rods controls the edge line tension and can be used for assembly of variouscomplex structures (87). From a practical perspective, colloidal membranes offer an easily scalablemethod for assembling sheets of semiconducting nanorods, which might be useful as photovoltaicdevices (88).

5. LIQUID CRYSTALS IN CROSS-LINKED GELS

One of the characteristics that distinguishes solids from liquids is that the former deform only inresponse to an applied stress, whereas the latter change shape even in the absence of externalforces. In this respect, a weakly cross-linked polymer, such as conventional rubber, behaves asa solid, albeit a very soft one with a modulus that is orders of magnitude smaller than that ofconventional molecular solids. The origin of rubber elasticity can be understood by consideringthe dynamics of the constituent polymer coils. An isolated polymer coil in suspension assumes, on

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average, anentropy-maximizing spherical shape. Pullingonsuchachaindeforms its equilibriumshape,reduces thenumberofaccessible states, anddecreases thepolymerentropy,which leads toameasurableforce that tends to restore the polymer to a spherical shape (89, 90). In rubber, a large fraction ofpolymer segments behave similarly to those of a free chain. However, the chains are also tethered ata few isolated points to a solid-like background matrix composed of other chains. Applying externalstress on a rubber deforms the tethering points, which in turn stretches individual chains, decreasingtheir entropy and increasing the free energy of the rubber. Thus, the origin of macroscopic rubberelasticity is intimately connected with the microscopic states available to the constituent polymerschains.

Polymer liquid crystals are polymers with liquid crystalline mesogens incorporated into themonomer subunits; the mesogens are located either within the polymer backbone or on a sidechain. Light cross-linking of a polymer liquid crystal solution yields a liquid crystalline elastomer,a material with emergent properties not found in any other system. Perhaps the most surprising isthe phenomena of soft elasticity, wherein a nematic elastomer, a solid-like material, can behavelike a liquid in that it undergoes energy-free deformations (91, 92).

By applying appropriate boundary conditions, it is possible to prepare conventional nematicliquid crystals in a uniformly aligned state. Creating such defect-free samples is essential for elu-cidating the basic properties of liquid crystals. It has proven significantly more difficult to createmonodomain nematic elastomers. Simple one-step cross-linking of polymer liquid crystals yieldspolydomain samples that obscure much of the interesting behavior of nematic elastomers. In animportant breakthrough in 1991, the first monodomain samples were prepared via a two-stepprocess (5). In the first step, a nematic polymer melt is sparsely cross-linked to form a weakgel. Subsequently, the gel is stretched to align the nematic domains and then cross-linked a secondtime to reinforce the uniformly aligned state. This important advance has enabled a series offoundational experiments that have uncovered the surprising properties of nematic elastomers.

The tight coupling between the nematic order parameter and polymer chain conformationsendows liquid crystalline elastomers with their unique properties. In uncross-linked isotropicsamples, liquid crystal mesogens are isotropically oriented and polymer coils assume, on average,an entropy-maximizing spherical shape (Figure 6a). By contrast, in an anisotropic liquid crys-talline background, the polymer backbone preferentially aligns along the nematic director, leadingit to assume the shape of a prolate ellipsoid elongated along the average liquid crystalline ori-entation (Figure 6b). Thus, the I-N phase transition of the background liquid crystals is intimatelycoupled to the shape change of the polymer coils. Cross-linking polymer chains couples suchmicroscopic shape changes of individual coils to the macroscopic shape change of the entire gel.The I-N phase transition of constituent mesogens is reflected in the stretching and contracting ofthe elastomer structure. The effect is dramatic: Deformations as large as 350% are not hard toachieve (Figure 6c) (93). Detailed analysis confirms that the temperature dependence of the ne-matic order parameter is quantitatively related to the overall anisotropy of the nematic elastomer(94). The reverse holds true as well: Applying an external stress to a liquid crystal elastomer in anisotropic state induces anisotropy of polymer chains that, in turn, induce the formation ofa paranematic liquid crystalline state (96, 97). As an alternative to temperature-induced shapedeformations using thermotropic liquid crystals, it is also possible to induce such transformationswith optical signals by utilizing light-sensitive liquid crystals based on azo dyes (95).

Another more subtle consequence of the coupling between the polymer shape and nematicdirector is the phenomena known as soft or semisoft elasticity (Figure 6d) (91). A nematic elas-tomer can respond to an applied stress by either distorting polymer chains from their elongatedequilibrium distribution or rotating them while preserving their ellipsoidal shape, both of whichalign the nematic director along the direction of the applied stress (98, 99).However, the latter is the

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favored response, as rotation preserves the preferred microscopic conformation of the constituentpolymers. The evidence for this can be gained by analyzing the behavior of a nematic director in anelastomer under external stress. Experiments show that applying an external stress perpendicular tothe directionof the initial alignment results in a rotation of thenematic director (98, 99). The directorcan rotate either continuously or discontinuously until it eventually aligns along the direction ofapplied stress. Throughout this operation, the shape and, therefore, the entropy of the constituentpolymers do not change despite the macroscopic deformation of the material. As a consequence,essentially no energy is needed to drive the deformation of the nematic elastomer in this geometry,and one obtains a macroscopic shape change without any energetic cost (100).

The remarkable property of liquid crystalline elastomers to smoothly change shape in response tovarious environmental cues has led to numerous applications. An elastomer based on a cholestericliquid crystal makes it possible to assemble a tunable andmirrorless laser (101). Also, a dye-sensitizednematic elastomer behaves as an autonomously motile object that swims away from a light source(102). The fact that nematic elastomers can behave asmolecular actuators has led to a proposal to usethem as artificial muscles (103).

6. COLLOIDS IN LIQUID CRYSTALS

Interactions between colloidal particles suspended in an isotropic liquid have been studied in greatdetail. By tuning the solvent properties, introducing nonadsorbing polymers, or coating colloids

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Macroscopic shape changes and soft elasticity of nematic elastomers. (a) Schematic illustrationof a cross-linkedelastomer in an isotropic phase in which the constituent polymers assume a free energy–minimizingspherical shape. A single polymer (orange) is shown. (b) The background nematic phase induces the polymerchains to assume an elongated shape aligned along the nematic director. The microscopic shape change ofconstituent polymers is intimately coupled to the macroscopic shape change of the entire elastomer. (c)A sequence of images illustrating the largemacroscopic shape change of a nematic elastomer as it undergoes anI-Nphase transition. Images courtesyof E.Terentjev (93). (d) The elastic stress as functionof uniaxial extensionapplied in the direction perpendicular to the nematic director demonstrates that for certain conditionsnematic elastomers can undergo macroscopic deformations that have essentially no energetic cost. Thisphenomenon is known as soft or semisoft elasticity. Adapted with permission from Reference 100.

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with a polymeric brush, it is possible to engineer microscopic colloidal interactions that can drivethe assembly of well-definedmacroscopic materials.We discuss how anisotropic liquid crystallinesolvents lead to an even greater diversity of colloidal interactions that can be used to assemblestructures of unprecedented complexity.

Insight into liquid crystal–mediated colloidal interactions can be gained by considering thetopology of the liquid crystalline ordering around a spherical colloid (104–106). For suchexperiments, it is necessary to control the liquid crystalline anchoring conditions, which can beeither tangential or homeotropic (perpendicular) at the colloid surface. Placing a spherical colloidwith well-defined anchoring conditions into a uniformly aligned nematic phase leads to topo-logical frustration. It is not possible to deform anematic director so that it continuously transitionsfrom uniform alignment at large distances to being locally perpendicular at the colloid surface.This frustration is resolved by forming a topological defect that remains bound to the colloidalparticle. Two types of defects are possible: either a point defect called hyperbolic hedgehog (Figure7a) or a disclination loop that envelops the equator of the particle, called a Saturn ring (Figure 7b).

An important experimental advance in this field was the successful assembly of inverted andmultiple liquid crystal emulsions, in which a small aqueous droplet suspended in a nematic liquidcrystal plays the role of the perturbing colloid (Figure 7c) (6). The multiple emulsion system allowsfor simultaneous observation and precise control of the liquid crystal anchoring in the far field aswell as locally around an embedded droplet. For example, it is possible to observe large nematicdroplets (∼50 mm) suspended in an aqueous solvent, each encapsulating a single or multiple

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Interactions and self-assembly of colloidal particles in anisotropic (nematic) solvents. (a) A point defect called a hyperbolic hedgehog enablesa nematic director to transition between uniform alignment at the boundaries and homeotropic anchoring at a colloid surface. The redlines indicate the orientation of the local nematic director. (b) The same boundary conditions can be fulfilled by forming a disclination loop(also called a Saturn ring), which envelops the colloid equator. (c) Multiple micron-sized aqueous droplets that are confined within a largeliquid crystalline droplet interact through dipolar interactions and assemble into elongated chains. Adapted with permission fromReference 6. (d) Chains of colloidal spheres formed under similar conditions to panel c. (e) Colloids with Saturn ring defects interact laterallyto form kinked chains aligned perpendicularly to the nematic director (110). (f) Colloids with quadrupolar Saturn ring defects assembleinto crystals with the shape of a parallelogram. (g) Colloids with dipolar coupling form chains that can be further assembled intoa 2D hexagonal crystal. Adjacent dipolar chains point in opposite directions. Images d–g adapted with permission from Reference 110.

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micron-sized aqueous droplets. Small aqueous droplets migrate toward the center of the liquidcrystalline droplet to minimize the elastic energy. For a one-droplet configuration, it is possible tocontinuously deform the nematic director from the inner surface of the large nematic droplet to theouter surface of the smaller encapsulated aqueous droplet, unlike the case discussed above ofa colloid in a uniformly aligned nematic field. However, introducing an additional droplet into sucha geometry leads to topological frustration that requires the formation of an accompanying defect.Each additional droplet leads to the generation of another defect so that large liquid crystallinedroplets that contain n encapsulated water droplets contain n� 1 defects. This geometry ofmultipleemulsions allows one to explore the interactions between the droplets and their associated defects. Itis observed that multiple aqueous droplets assemble into elongated chains, wherein individualdroplets are not in contact but are separated by a well-defined distance (Figure 7c). High-resolutionimages reveal the presence of point defects separating the droplets and thus suppressing theircoalescence.

Theoretically, the colloid-induced elastic deformation of the liquid crystalline director in thefar-field-limit can be mapped onto an equivalent problem in electrostatics (107). In this analogy,a sphere with a bound hyperbolic hedgehog defect behaves as an electrostatic dipole. It is knownthat dipoles minimize their interaction energy by forming elongated chains, which explains ex-perimental observations of droplet (colloids) chaining (Figure 7c,d). By applying an externalmagnetic field, it is possible to vary the separation of a pair of liquid crystal–bound droplets dopedwith a ferrofluid (108). Measuring the force between droplets at different separations directlyconfirms the dipolar nature of these liquid crystal-mediated colloidal interactions.

Along with colloids with bound point defects, theoretical modeling also predicts anotherpossible defect, the Saturn ring, which consists of a disclination loop encircling the sphere equator(109). A hyperbolic hedgehog can be transitioned into a Saturn ring by either applying an externalfield, decreasing the particle size, or confining the background nematic liquid crystal to a thinmonodomain layer; experimentally, Saturn rings have been primarily observed in confined samples.The electrostatic analogy predicts that colloids with Saturn defects interact with quadrupolarsymmetry. Therefore, whereas defects with hyperbolic hedgehogs line up to form elongated chainsaligned along the nematic director, colloids with Saturn rings bind laterally at a well-defined anglewith respect to the director field to form kinked chains (Figure 7e) (110).

Recent advances have demonstrated another liquid crystal–mediated mechanism that leads tovery different binding of spherical colloids (111, 112). Quenching a bulk isotropic phase intoa nematic is accompanied by the formation of an entangled network of disclination loops, whichanneal over time to form a lower energy state characterized by a uniform director field. The sameprocedure can be performed locally around colloidal particles by heating the liquid crystal witha laser. A subsequent quench generates disclination loops that cannot easily anneal due to thepresence of the colloids. These disclination loops entangle colloid particles to create strongly boundpairs. Pulling entangled colloids apart with optical tweezers requires extension of the disclinationlines and leads to an interaction energy that increases with increasing particle separation and caneasily reach thousands of kBT.

The basic understanding of liquid crystal–mediated colloidal interactions at a pair level enablesassembly of macroscopic colloidal materials that are of both fundamental and applied interest andare difficult to achieve with other methods of colloidal assembly (110). In particular, it has provenpossible to bring together hundreds of colloids with quadrupolar symmetry to assemble a 2D crystalin the shape of a parallelogram (Figure 7f ). Increasing the thickness of the confining nematic inducesa switch from Saturn rings to hyperbolic point defects. As a result, colloids align along the nematicfield to form elongated chains. Dipolar chains formed from colloids with hedgehog defects eitherrepel or attract each other depending on whether they are in parallel or antiparallel alignment,

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respectively. Therefore, two antiparallel chains form a tightly bound pair that can be further or-ganized into higher-order 2D crystals with hexagonal symmetry (Figure 7g).

Macroscopic structures have also been obtained by quenching a liquid crystal–silicone oil mixturefrom a uniform nematic phase into a biphasic I-N region (113). In conventional systems, a phase-separating mixture continues to coarsen until reaching macroscopic dimensions. For liquid crystals,upon a temperature quench, silicon oil-rich droplets start to phase separate from the backgroundnematic liquid crystal but coarsen only to anupper-critical size. At this point, they are stabilized by thepresence of defects and associated liquid crystal–mediated interactions. Such uniform droplets or-ganize intomacroscopic chains that repel each other and permeate the entire liquid crystalline sample.

Colloids in liquid crystals remain a remarkably vibrant research area that merges fields asdiverse as photonics, materials science, and basic topology (114, 115). Here we have only de-scribed liquid crystal–mediated interactions between spherical colloids and their assembly intobulk materials. Shifting to more complex shapes that also include topologically distinct structuressignificantly affects the nature of the surrounding liquid crystalline field and considerably increasesthe diversity of colloidal interactions that can potentially be used to assemble even more exoticcolloidal structures (116, 117).

7. ACTIVE LIQUID CRYSTALS

So far we have described a few representative examples of hypercomplex liquid crystals that emergewhen conventional anisotropic fluids are merged with other complex fluids, such as colloids andpolymers. All of these systems are assembled from inanimate, passively diffusing building blocks. Assuch, their behavior can be explained with equilibrium statistical mechanics, which entails mini-mizing the overall free energy of the system to determine stable phases. However, the laws ofequilibrium statistical mechanics impose severe constraints on the properties of such materials; forexample, these materials cannot exhibit spontaneous motion or perform macroscopic work. Re-cently, an entirely new research direction has emerged that is focused on developing materials withorientational order that are assembled from animate, energy-consuming building blocks. Beingactively driven, the emergent properties of such far-from-equilibrium materials are not constrainedby the laws of equilibrium statistical mechanics, and they can becomemotile or spontaneously flow(118–122). Among research topics within active matter, the study of active liquid crystals has re-ceived a great deal of attention. Since its inception, this field has been largely driven by theoreticalefforts. However, the past few years have witnessed the emergence of well-defined experimentalsystems that can be used to test existing theoretical models and guide their further development.Webriefly review these experiments while emphasizing similarities as well as differences between activeliquidcrystals and their equilibriumanalogs. For amore thoroughdiscussion, the reader is referred toextensive review articles on active matter (123–125).

In an equilibrium nematic, all molecules are equally likely to point up or down, with the localorientation of molecules described by an arrowless bar. In contrast, active liquid crystals can haveeither polar or nematic (quadrupolar) symmetry, and both have been experimentally realized. Thepolar active liquid crystals were beautifully demonstrated in a biological motility assay based onan actin-myosin system (126–128). F-actin is a semiflexible filament that plays an essential role ina variety of biological processes, ranging from cellular division andmotility tomuscle contraction.Myosin is a motor protein that utilizes energy from ATP hydrolysis to take nanometer-sized stepsalong an actin filament.Myosin motors move in a polar fashion toward one end of an intrinsicallypolar actin filament. In a motility assay, one affixes a high density of myosin motors onto a solidmicroscope slide (Figure 8a,b) (129). In the presence of ATP, actin filaments continuously glide ona bed of myosin motors in a ballistic fashion.

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In a traditional motility assay, one places a few dilute filaments to ensure absence of any fil-ament interactions. The Bausch group has explored the behavior in the opposite limit, inwhich themicroscope slide is covered with a dense ensemble of gliding filaments (126). When two glidingfilaments encounter each other, there is a small but finite probability of them reorienting andmoving in the same direction because of steric interactions. A number of surprising collectiveeffects emerge in such dense monolayers of actively gliding filaments. With increasing filamentconcentration, one observes the spontaneous formation of dense flocks in which thousands ofmotile filaments move in a coordinated fashion. The dense flocks of aligned filaments coexist witha dilute background phase of isolated filaments that have an isotropic orientation distribution.Such behavior is reminiscent of the equilibrium I-N phase transition. In the motility assay, it ispossible to assign direction to each filament by simply pointing the arrow in the direction offilament motion. Because in a dense flock all the filaments point in the same direction, driven actinfilaments form a far-from-equilibrium liquid crystalwith polar symmetry. The evidence for a polarliquid crystal can also be extracted by examining the nature of liquid crystalline defects. Asdiscussed previously, disclination defects of charge61/2 observed in apolar nematic liquid crystalscannot exist in polar systems. Indeed, such defects are never observed with driven actin filaments;instead, one observes vortex defects of charge 1,which form in polar liquid crystals (Figure 8c,d,e).

It is also possible to assemble active liquid crystals with conventional nematic (quadrupolar)symmetry. In such systems, the constituent subunits are equally likely to point up or down, andlocal orientation is described by an arrowless line. The first experimental example of an activenematic consisted of a densemonolayer of elongatedmelanocyte cells (130, 131). Observations ofdisclination defects of charge 61/2 confirmed the underlying apolar nematic symmetry (Figure9a). However, in a living system one cannot control relevant microscopic parameters, such as cellmotility, which precludes detailed comparison with theory. A distinct system consisting of shakengranular rodmonolayers also exhibits properties of active nematics (Figure 9b) (132). In this case,

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Active polar nematic liquid crystals observed in a gliding actin-myosin motility assay. (a) Schematic of a motility assay in which surface-bound motors propel the gliding of isolated filaments. The light-blue arrow indicates the direction of filament motion. (b) Image of alow density (isotropic phase) of fluorescently labeled actin filaments gliding on the coverslip surface. (c) Coherent filament motionin a high-density liquid crystalline phase exhibits structures that are reminiscent of defects with topological charge equal to unity. Toillustrate filament dynamics, ten consecutive images are overlaid, starting with the single image depicted in the inset. (d) The velocityprofile associated with the coherently moving filaments in panel c. (e) The trajectory of the center of a liquid crystalline defect thatremains stable for ∼400 s before falling apart. Adapted with permission from Reference 126.

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the random energy that propels rods is fed into the system from the vibrating external boundaries.Using this system, it is possible to test theoretical predictions for active materials, namely that thedensity fluctuations are significantly stronger than in equilibrium systems, are long lived, anddecay logarithmically in time.

Until recently, themajority of experimental model systems of animate liquid crystals described,including gliding filaments, crawling cells, and shaken rods, required the presence of a solid 2Dsubstrate. Another recent advance has demonstrated the assembly of an active nematic inmicrotubule-kinesin mixtures that is not coupled to a 2D solid surface (133). Similar to myosin,kinesin is a motor protein that uses chemical energy from ATP hydrolysis to walk along a mi-crotubule in discrete 8-nm steps (134). Using depletion interactions, short microtubules are as-sembled into bundles that are subsequently depleted at a high density onto a liquid-liquid interface,where they form a dense 2D nematic. Simultaneously, kinesins are bound into multimotorclusters. Such clusters simultaneously bind multiple microtubules within a nematic layer, in-ducing interfilament sliding and driving bundle extension. This extension lacks directionality;consequently, active microtubule liquid crystals have nematic (quadrupolar) symmetry. Theexistence of disclination defects of charge 1/2, similar to those observed in shaken rods orcrawling cells, supports this hypothesis (Figure 9c).

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Disclination defects govern dynamics of active liquid crystals with nematic symmetry. (a) Observation of�1/2disclination defects reveals that a dense monolayer of anisotropic melanocyte cells has nematic symmetry.Adapted with permission from Reference 130. (b) Dynamics of a pair of disclination defects observed ina vibrated monolayer of granular rods. An asymmetric þ1/2 defect (green circles) is highly mobile, whereasa more symmetric�1/2 defect (red circles) is mostly stationary. Adapted with permission fromReference 132.(c) Unbinding of oppositely charged defects in active liquid crystals based on microtubules and the molecularmotor kinesin. The active nematic is unstable toward bend fluctuations. Once initiated, the bend deformationgrows in amplitude, eventually leading to the fracture of the nematic director, which is terminated at a pair ofoppositely charged defects (red and green arrows). The fracture line spontaneously heals, leaving behind a pairof unbound defects. Steady-state is reached when the rate of defect unbinding is balanced by the rate of defectannihilation. Adapted from Reference 133.

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In conventional liquid crystals, defects are largely static inanimate structures and their ex-istence and location are determined by either internal frustrations or externally imposedboundary conditions or forces. The behavior and role of defects in active liquid crystals isdramatically different.Most notably, defects embedded in active liquid crystals acquiremotility,and their exact dynamics depends on their topological charge. The motility of þ1/2 defects ispowered by their asymmetric structure, whereas the more symmetric �1/2 defects are largelystatic. The first evidence of defect motility was obtained in shaken granular rods (Figure 9b)(132). The role of defects is even more pronounced in microtubule-based active matter. Anextensile active nematic in a uniform state is inherently unstable against long-wavelength bendfluctuations (121). Once initiated, a bend deformation grows in amplitude and eventuallybuckles, resulting in the spontaneous fracture of the nematic director accompanied by an un-binding of a pair of oppositely charged disclination defects (Figure 9c) (135, 136). Upongeneration, the motile þ1/2 moves away from the largely static �1/2 defect. It follows thata steady state of active nematics is characterized by a dense suspension of highly interactingmotile defects of fleeting existence. Defect motility drives the streaming flow dynamics of theentire liquid crystal.Microtubule-based active nematics are easily assembled on flat interfaces aswell as on curved liquid-liquid interfaces, such as emulsion droplets. When such emulsions areconfined, the internally powered streaming liquid crystalline flows drive droplet motility,resulting in objects that spontaneously roll around on the surface. These studies illustrate theessential role that defects play in the behavior of active liquid crystals.

8. CONCLUSIONS AND OUTLOOK

Hypercomplex fluids are combinations of two or more complex fluids that yield novel materialswith emergent properties. A number of notable recent advances in soft-matter physics have in-volved development of hypercomplexmaterials based on liquid crystals and anisotropic fluids.Wehave described a few representative examples that illustrate the diversity of these materials as wellas their emergent properties. These examples show how the foundational ideas developed throughoriginal research on molecular liquid crystals are being applied to an ever-widening range ofphenomena. Finally, hypercomplex liquid crystals also demonstrate how assembly of advancedmaterials does not necessarily require engineering of more complex building blocks through so-phisticated chemistry but can also be accomplished through knowledgeable merger of relativelysimple systems with competing interactions.

DISCLOSURE STATEMENT

The authors are not aware of any affiliations, memberships, funding, or financial holdings thatmight be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

Support from National Science Foundation (MRSEC-0820492, CAREER-0955776, CMMI-1068566) and W.M. Keck are gratefully acknowledged. Z.D. also acknowledges support fromthe Technische Universität München–Institute for Advanced Study, which is funded by theGerman Excellence Initiative. To our knowledge, the term hypercomplex fluid was first coined byR.B. Meyer, and we thank him for many useful discussions.

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Annual Review of

Condensed Matter

Physics

Volume 5, 2014Contents

Whatever Happened to Solid State Physics?John J. Hopfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Noncentrosymmetric SuperconductorsSungkit Yip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Challenges and Opportunities for Applications of UnconventionalSuperconductorsAlex Gurevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Correlated Quantum Phenomena in the Strong Spin-Orbit RegimeWilliam Witczak-Krempa, Gang Chen, Yong Baek Kim,and Leon Balents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Dirac Fermions in Solids: From High-Tc Cuprates and Graphene toTopological Insulators and Weyl SemimetalsOskar Vafek and Ashvin Vishwanath . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Quantum Critical Point Lying Beneath the Superconducting Domein Iron PnictidesT. Shibauchi, A. Carrington, and Y. Matsuda . . . . . . . . . . . . . . . . . . . . . 113

Hypercomplex Liquid CrystalsZvonimir Dogic, Prerna Sharma, and Mark J. Zakhary . . . . . . . . . . . . . . 137

Exciton Condensation in Bilayer Quantum Hall SystemsJ.P. Eisenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Bird Flocks as Condensed MatterAndrea Cavagna and Irene Giardina . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein Condensationand the Unitary Fermi GasMohit Randeria and Edward Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

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Crackling Noise in Disordered MaterialsEkhard K.H. Salje and Karin A. Dahmen . . . . . . . . . . . . . . . . . . . . . . . . 233

Growing Length Scales and Their Relation to Timescales in Glass-FormingLiquidsSmarajit Karmakar, Chandan Dasgupta, and Srikanth Sastry . . . . . . . . . . 255

Multicarrier Interactions in Semiconductor Nanocrystals in Relation tothe Phenomena of Auger Recombination and Carrier MultiplicationVictor I. Klimov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

Polycrystal Plasticity: Comparison Between Grain-Scale Observations ofDeformation and SimulationsReeju Pokharel, Jonathan Lind, Anand K. Kanjarla,Ricardo A. Lebensohn, Shiu Fai Li, Peter Kenesei, Robert M. Suter,and Anthony D. Rollett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Molecular Beam Epitaxy of Ultra-High-Quality AlGaAs/GaAsHeterostructures: Enabling Physics in Low-DimensionalElectronic SystemsMichael J. Manfra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Simulations of Dislocation Structure and ResponseRichard LeSar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375

Errata

An online log of corrections to Annual Review of Condensed Matter Physics articlesmay be found at http://www.annualreviews.org/errata/conmatphys

vi Contents

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