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Hypercomplex Liquid
Crystals
Zvonimir Dogic,1,2, Prerna Sharma,1 and
Mark J. Zakhary1
1
Department of Physics, Brandeis University, Waltham, MA 02474;
email: zdogic@brandeis.edu, prerna@brandeis.edu, markz1@brandeis.edu
2
Institute for Advanced Study - Technische Universität München, Garching 85748,
Germany
Annu. Rev. Condens. Matter Phys. 2014. 5:137–57
Keywords
First published online as a Review in Advance on
December 18, 2013
soft matter, complex fluids, self-assembly, colloids, topological
defects, active matter, elasticity
The Annual Review of Condensed Matter Physics is
online at conmatphys.annualreviews.org
This article’s doi:
10.1146/annurev-conmatphys-031113-133827
Copyright © 2014 by Annual Reviews.
All rights reserved
Corresponding author
Abstract
Hypercomplex fluids are amalgamations of polymers, colloids, or
amphiphilic molecules that exhibit emergent properties not observed in elemental systems alone. Especially promising buildingblocks for assembly of hypercomplex materials are molecules with
anisotropic shape. Alone, these molecules form numerous liquid
crystalline phases with symmetries and properties that are fundamentally different from those of conventional liquids or solids.
When combined with other complex fluids, liquid crystals form
materials with diverse emergent properties. In equilibrium, the
interactions, dimensions, and shapes of these hypercomplex materials can be precisely controlled. When driven far from equilibrium,
these materials can deform and even spontaneously flow in the absence of external forces. Here we describe recent experimental
accomplishments in this rapidly developing research area. We emphasize how the common theme underlying these diverse efforts is
their reliance on the basic physics of molecular liquid crystals developed in the 1970s.
137
1. INTRODUCTION
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Upon reducing temperature, most molecular substances undergo a transition from a disordered
freely flowing liquid to an ordered solid crystal. Molecules with anisotropic shape that exhibit
a panoply of partially ordered liquid crystalline states between a high-temperature liquid and
a low-temperature solid are an important exception to this rule. Although the first evidence for
their existence dates back to the late 19th century, liquid crystals captured the attention and
imagination of the physics community in the 1970s, when many of the theoretical concepts that
describe their phase behavior, structure, and dynamics were developed (1, 2). These extensive
efforts have rapidly transformed the mysterious and poorly understood liquid crystalline materials
into highly controllable and designable substances with diverse applications, most notably including
liquid crystal displays. As a result of this success, one encounters the opinion that the fundamental
questions related to partial ordering of anisotropic molecules have been successfully answered. In our
opinion, this is not the case. Conventional molecular liquid crystals remain a vibrant research field
with numerous open questions. Furthermore, over the past two decades the ideas developed through
pioneering studies of molecular liquid crystals have been used to create increasingly complex
composite materials that remain at the forefront of soft-matter research. Here, we review a few
representative examples of these hypercomplex materials that illustrate the timelessness and importance 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. Other
notable systems belonging to the same class of materials are polymers, colloids, lipid bilayers, and
self-assembling amphiphiles. All of these classical systems have unique properties that are interesting from both the fundamental and application perspectives. Furthermore, extensive theoretical efforts have produced detailed understanding of how the macroscopic properties of
complex fluids are related to their microscopic structure and interactions, thus making these
materials tunable and designable. With this basic understanding in place, recent efforts have focused on developing next-generation materials that combine desirable features of different complex fluids. For instance, combining the amphiphilic nature of surfactant molecules with classical
polymers leads to block-copolymers, a highly versatile complex fluid with exceptional material
properties (3). Under specific conditions, such block-copolymers can further assemble into polymerosomes, which are analogs of classical lipid bilayer vesicles with greatly improved material
properties, such as durability and toughness (4).
In a similar vein, molecular liquid crystals have been merged with other complex fluids to
produce materials with emergent and unexpected properties. For example, combining molecular
liquid crystals with cross-linked polymer gels leads to nematic elastomers, which are materials that
exhibit soft elasticity, a phenomenon wherein a solid-like elastomer can behave like a liquid, with
zero energetic cost for certain shear deformations (5). Soft elasticity is not a property of molecular
liquid crystals or cross-linked rubber alone but only emerges when these two quintessential
complex fluids are combined in a single material. Such emergent materials have been called
hypercomplex fluids, a term first coined by Robert Meyer. As another example, combining
molecular liquid crystals with micron-sized colloidal particles leads to novel interparticle interactions 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 that
liquid crystal physics plays in their emergent behavior.
We first briefly recapitulate the basic physics of molecular liquid crystals. Next, we describe the
behavior of colloidal liquid crystals, which combine desirable features of colloidal suspensions
with those of classical liquid crystals. From here, we focus on how merging liquid crystals and
cross-linked rubber leads to solid networks with surprising elastic properties. Then, we describe
138
Dogic
Sharma
Zakhary
the behavior of liquid crystal–colloid mixtures, in which micron-sized particles suspended in
anisotropic fluids acquire novel interactions not observed in conventional isotropic solvents.
Finally, we review recent experimental advances in active liquid crystals, in which inanimate rods
are replaced by animate constituent objects that can coordinate their motion to produce inherently
far-from-equilibrium materials that spontaneously swim, flow, and crawl.
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2. PHYSICS OF MOLECULAR LIQUID CRYSTALS
Spherically shaped molecules can form either a liquid state, which is characterized by short-range
positional order, or a solid crystal, which is characterized by long-range positional order (7). In
addition to these phases, anisotropic molecules form numerous liquid crystalline phases that have
orientational and/or partial positional order (Figure 1). Transitions between these liquid crystalline phases are induced by varying either temperature (for thermotropic liquid crystals) or
sample concentration (for lyotropic liquid crystals). The nematic phase is the most common liquid
crystalline phase. Upon decreasing temperature or increasing particle concentration, a disordered
isotropic liquid composed of anisotropic molecules undergoes a first-order phase transition into
a nematic liquid crystal in which all the molecules are, on average, aligned along the same direction,
which is described by the nematic director (Figure 1b). The resulting nematic phase has the longrange orientational order characteristic to solids. However, the center of mass of each rod retains
the positional disorder inherent to liquids and the time-averaged density is spatially uniform. It is
this confluence of both liquid and solid properties that leads to the name liquid crystals and the
emergence of their highly desirable materials properties. In many substances, decreasing the
temperature of the nematic phase leads to a smectic phase (Figure 1c). In addition to long-range
orientational order, smectic liquid crystals have a quasi-long-range 1D positional order. In other
words, smectic liquid crystals are composed of liquid monolayers of aligned molecules stacked on
top of each other. Along with nematic and smectic phases, there are dozens of other liquid
crystalline 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 suspension
of rod-like molecules. (a) The isotropic phase has short-range orientational and positional order. (b) The
nematic phase is characterized by long-range orientational order, although the center-of-mass of each rod-like
molecule retains the short-range order characteristics of liquids. (c) In the smectic phase, rods form
two-dimensional one-rod-long liquid-like layers. Entropic-excluded volume interactions between hard rods are
arguably 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-core molecules. In this
report, 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 than
a directional vector. This observation has important consequences for the nature of liquid
crystalline 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 in
which all rods point in the same direction. As with most other complex fluids, nematic liquid
crystals are very soft materials, i.e., thermal fluctuations are sufficiently strong to induce significant distortions away from the minimum energy state of uniform alignment. For nematic liquid
crystals, 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 the
Frank elastic free energy (8). A consequence of such low-energy modes is that distortions of the
nematic director scatter light much more strongly compared with ordinary liquids, endowing
nematic liquid crystals with their characteristic turbidity (1).
Along with slowly varying elastic distortions, another pervasive property of nematic liquid
crystals is the presence of topological defects: localized imperfections in the liquid crystalline order
that cannot be removed by continuous deformations of the order parameter. The charge of a topological defect is related to the director rotation as one traverses a loop enclosing the defect. By
convention, topological defects that involve director rotations of 360° have a charge of 61. The
sign is determined by the sense of director rotation when traversing a closed path in an anticlockwise direction. In a conventional nematic characterized by quadrupolar symmetry, the lowest
energy defects, called disclinations, have a topological charge of 61/2 (Figure 2a,b). In comparison, a liquid crystal with polar order cannot form such defects because rotating a vector by
180° results in a physically distinct state. In such materials, the lowest energy defects would have
a charge of 61 (Figure 2c–e). Therefore, a simple inspection of topological defects can determine
the 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 nanometers and a few microns, thus ensuring that thermal fluctuations suppress particle sedimentation (9).
a
b
c
d
e
m = +½
m = –½
m = +1
m = +1
m = –1
Figure 2
Schematic illustrations of a nematic director enclosing topological defects found in quadrupolar (nematic) and
polar liquid crystals. (a,b) Disclination defects of charge þ1/2 and 1/2. Moving along a loop that encircles
such defects results in net director rotations of 180°. Because polar vectors need to rotate 360° to return
to their original state, they cannot form defects with a charge of 1/2. (c–e) Topological defects of charge 1 can be
formed by liquid crystals with both polar and nematic order. Abbreviation: m, topological charge of the defect.
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Two principal features make colloidal suspensions a highly desirable system to study. First, it is possible
to systematically engineer colloidal interactions, enabling experiments that directly probe the relationship between microscopic interactions and macroscopic phase behavior (10–12). Second,
micron-sized colloids can be directly tracked with video microscopy and manipulated with optical
tweezers (13–17).
The classic model system exemplifying these features is the suspension of micron-sized
spherical latex or silica beads (18). By tuning relevant parameters, it is possible to engineer
these 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 an
experimental realization of a theoretical model that has played a crucial role in developing
a modern theory of the liquid state (19, 20). With increasing concentration, a suspension of hardsphere-like colloids undergoes a liquid-to-solid phase transition (21). The ability to simultaneously
track all the particles in real time within a colloidal liquid or crystal has yielded essential insight into
important 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 of
colloidal suspensions are universal. Thus, studying colloidal spheres increases our understanding
not only of complex fluids but also of materials on the molecular length scale. It is helpful to think
of colloids as giant atoms that can be visualized, tracked, and manipulated, and whose interactions
can 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 subject
that 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), an
anisotropic rod-like colloid (25). Soon thereafter, John Bernal and coworkers prepared TMV
suspensions 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 the
I-N phase transition, a scientific accomplishment that was many years ahead of its time and has
laid 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 overlap
between any two rods leads to infinitely large interaction energy, so that such states are never
realized. Consequently, the only contribution to the internal energy of the ensemble, U, is the
kinetic energy U } NkB T, where N is the number of particles. This argument holds for any fluid
with hard-core interactions. It follows that the free energy of hard-particle fluid scales linearly with
temperature, F } U TS } TðkB N SÞ, and the free energy attains a minimum for a phase that
maximizes the overall system entropy. It might seem that entropy-maximizing fluids will always
remain in a disordered state. However, as explained below, at surprisingly low densities an aligned
hard-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 rods
form an ordered nematic phase at volume fractions as low as 1%. The remarkably low density of
the I-N phase transition makes it possible to expand the free energy of a hard-rod fluid in powers of
density while still being able to quantitatively describe the I-N phase transition (28–30). Because of
the much higher density at which hard spheres crystallize, a similar approach cannot be used to
describe a liquid-to-crystal transition. In this sense, the fluid of hard rods is simpler than that of
hard spheres.
The lowest-order term in the virial (density) expansion of the hard-rod free energy, associated
with 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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a
c
L
Vexc
L
d
Concentration (mg/mL)
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2D
b
L
2D
Vexc
30
20
10
5
Theory
Isotropic
Nematic
1
10
100
1,000
Ionic strength (mM)
Figure 3
The Onsager theory of the I-N phase transition can be understood as a competition between mixing entropy, which favors the isotropic
phase, 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, and
Vexc scales as L2 D. (b) In the nematic phase, rods are, on average, aligned with each other, and Vexc scales as LD2 . (c) A suspension of rodlike Tobacco mosaic viruses at high-enough concentrations exhibits coexistence between the isotropic and nematic phases. The nematic
phase 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 above
the nematic phase, is not birefringent and appears dark. (d) Coexisting isotropic and nematic data measured in a suspension of the rod-like
bacteriophage fd at different ionic strengths, which determines the effective rod diameter. Lines indicate predictions of the Onsager
theory, which have been extended to include effects of semiflexibility and the polyelectrolyte nature of filamentous viruses, and the open
and closed dots are concentrations of coexisting isotropic and nematic phases, respectively (49). Data provided by Seth Fraden.
state and dominates at higher densities. The packing entropy is directly related to the excluded
volume between two rods, which is defined as the volume inaccessible to the center of mass of
a colloid owing to the presence of other impenetrable colloids. Minimizing the excluded volume
increases the packing entropy of the system. In an isotropic phase, rods are on average perpendicular to each other, resulting in a large excluded volume (Figure 3a). In contrast, two parallel
rods in the nematic phase have a much smaller excluded volume and thus higher packing entropy
(Figure 3b). The competition between the entropy of mixing and the entropy of packing determines
the exact location of the I-N phase transition (Figure 3c,d). The Onsager argument becomes
quantitative for rods with an aspect ratio larger than 100, as the higher-order virial terms become
negligible at densities in which the I-N phase transition occurs. The basic Onsager argument
outlined here can be extended to account for important properties of experimental systems, such as
surface charge, rod bidispersity and polydispersity, and flexibility (31–34).
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Following the development of the Onsager theory in 1949, there was a period of reduced activity in the field over the next few decades. However, the emergence of computer simulations and
the success of the hard-sphere theory of the liquid state led to resurgent interest in studying other
hard particle fluids in the 1980s, which has continued unabated ever since. Computer simulations
have 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 drive
hard rods to form a stable smectic phase (37).
Any colloid that significantly deviates from a spherical shape undergoes an I-N phase transition. Consequently, numerous experimental systems exhibit a nematic phase, including biological
polymers 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 colloids
that 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 tests
of the Onsager theory. TMV is an important system in which the I-N phase transition has been
studied in detail (Figure 3c). However, the aspect ratio of TMV rods is not sufficiently large to
satisfy the Onsager criterion, leading to significant quantitative discrepancies between experiments and theory (48). Another useful experimental system is the fd virus, which has an aspect
ratio of 130. However, fd wt (wild type) is semiflexible, which considerably influences the nature of
the I-N phase transition and again leads to quantitative differences from the original Onsager rigidrod theory (Figure 3d) (49, 50). Recently, a mutant fd Y21M virus has been identified that has the
same aspect ratio as fd wt but is significantly stiffer (51). Preliminary results indicate that the
behavior of such rods is quantitatively described by the original Onsager theory. However, further
studies 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 wide
I-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 is
filamentous viruses, which are naturally identical and thus serve as an ideal model system to study
smectic colloidal liquid crystals. Indeed, the smectic-A (Sm) phase was first conclusively demonstrated in suspensions of rigid TMV particles (54). Subsequently, studies on fd wt and more rigid
fd Y21M rods have demonstrated the important effect of flexibility on the nature of the N-Sm
phase transition (55, 56). Flexibility destabilizes the smectic phase, drives the transition to higher
densities, and changes the transition from second order to strongly first order. The length of fd wt
viruses, which is in the micron range, also makes it possible to directly visualize smectic layers, thus
exploiting the second appealing feature of colloidal suspensions (Figure 4a). Visualization of
smectic layers opened a path toward imaging real-space structure and dynamics of partially
ordered colloidal liquid crystals (56, 57). Another recently developed system that holds promise is
that of chemically synthesized silica-like bullet colloids with a narrow length distribution. Dense
sediments of these particles display beautiful smectic layers that can be directly visualized with
optical microscopy (Figure 4b) (58, 59).
From a fundamental perspective, studies of colloidal liquid crystals have established the
quantitative relationship between the microscopic properties of anisotropic particles and their
macroscopic phase behavior. Furthermore, by using an extension of the Onsager theory, it
becomes possible to predict how elastic constants that describe the long-wavelength, low-energy
distortions of the nematic director depend on the microscopic parameters of constituent rods (60,
61). From a practical perspective, the foundational principles developed through work on colloidal
liquid crystals are being applied to an ever-wider range of systems. For example, recent advances in
chemical synthesis have yielded highly monodisperse nanorod suspensions with tunable aspect
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Hypercomplex Liquid Crystals
143
a
b
2.5 μm
5 μm
Figure 4
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The length scale of rod-like colloids enables direct visualization of smectic layers and the dynamics of
constituent rods. (a) Smectic layers of micron-long fd wt viruses observed with differential interference contrast
(DIC) microscopy (55). (b) Confocal microscopy image of a smectic phase formed with fluorescently
labeled 1.4-mm colloidal silica bullets. Adapted with permission from Reference 59.
ratios. The next challenge is to self-assemble such particles into well-defined 3D structures that can
be 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 extensions
provide a valuable guiding principle for this endeavor. Finally, there is a recent renewed emphasis
on studying particles with shapes that are far more complex than those of simple rods and
cylinders, 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 molecular liquid crystals to shed insight into the fundamental interactions that drive the assembly of
all anisotropic fluids. Colloids also allow for the possibility of systematically engineering more
complex interactions, making it possible to observe structures that are not feasible in molecular
liquids. Here, we describe how introducing attraction between rod-like colloids considerably
expands the complexity of the phase diagram. These systems merge three different types of
complex fluids—colloids, liquid crystals, and polymers—to produce materials with surprising
emergent behavior.
Since the seminal theoretical study by Asakura & Oosawa (AO) (12), it has been widely appreciated that adding nonadsorbing polymers to colloids induces tunable attractive interactions
that can significantly affect the colloidal phase behavior (Figure 5a) (65–68). The AO model treats
polymers as ghost spheres whose diameter is equal to the polymer radius of gyration (Rg). Polymers
pass through each other without any energetic penalty and can thus be described by an ideal gas
equation 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
D2
4p
D3
pleting polymers is reduced by p Rg þ
Lþ
, where L and D are the length
Rg þ
2
3
2
and diameter of the spherocylinder, respectively. As two colloids approach within two Rg of each
other, the excluded volume shells overlap, effectively increasing the free volume available to the
depleting molecules. Because the number density of depletants is much greater than that of
colloids, the entropic cost associated with clustering a few large colloids is more than offset by the
increase in free volume (entropy) accessible to depletant molecules. The depletion interaction
provides a versatile mechanism for controlling colloidal interactions, as the range and strength of
attractive interactions can be tuned by changing the polymer size and concentration, respectively.
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a
b
c
d
5 μm
5 μm
Figure 5
Attractive depletion interactions assemble rod-like viruses into 2D liquid monolayers, which are colloidal
analogs of lipid bilayers. (a) Schematic illustration of the depletion effect. The volume excluded to the center-ofmass of the depleting polymer because of the presence of each spherocylinder is indicated with light shadowing.
Bringing two spherocylinders together results in an overlap of excluded volume shells (indicated with dark
shadowing), increases the accessible volume (entropy) of small depleting polymers, and leads to effective and
tunable attractive interactions. (b) Schematic illustration of a colloidal membrane, a one-rod-length-thick
liquid-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 fluorescently
labeled rods (green) reveals the dynamics of constituent rods within a membrane whose structure is
simultaneously visualized with phase contrast microscopy (red). Adapted with permission from Reference 81.
The mesoscopic size of colloids makes it possible to directly measure the depletion-induced
effective interactions (10, 11, 69).
Adding nonadsorbing polymers to colloidal rods significantly alters the liquid crystalline phase
diagram (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 no
analogs in pure suspensions of hard rods (75–77). For example, a depleting polymer whose size is
approximately one tenth of the colloidal rod length destabilizes the nematic phase and induces the
formation 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 in
mixtures of hard rods and polymers are determined by geometric (entropic) considerations alone.
An entirely different hypercomplex structure emerges when depletant molecules are added to
a dilute isotropic phase of rod-like molecules. For colloidal spheres, the strength of the depletion
interaction depends only on the radial separation between the particles. In comparison, the
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depletion interaction between two rod-like molecules depends on both their center-of-mass separation 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 of
aligned rods, which coalesce laterally to form mesoscopic one-rod-length liquid-like monolayer
disks that contain thousands of rods. This assembly constitutes a mesoscopic phase separation into
polymer-rich and rod-rich phases, as the disks contain no polymer. At high depletant concentration, small micrometer-sized disks stack on top of each other to form smectic filaments that are
tens of micrometers in length (80). For large-enough disks, the stacking interactions reduce excluded volume more effectively than lateral association. Surprisingly, below a well-defined critical
depletant concentration the face-on interactions between disks become repulsive, and stacking
ceases. However, mesoscopic disks are still free to coalesce laterally, leading to the formation of
smectic monolayers that can reach macroscopic size (Figure 5b–d) (81).
The transition from face-on stacking to lateral coalescence can be understood by considering
competition between depletion-induced attractive interactions and entropic repulsion that is associated 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 entropy
of closely spaced disks and leading to effective repulsive interactions. On one hand, the range of
disk-disk attractive interactions is determined by the depletant size and is independent of its
concentration. On the other hand, the range of repulsive interactions arising because of protrusion
fluctuations increases with decreasing depletant concentration. Therefore, at sufficiently low
depletant concentration the longer-ranged repulsion overwhelms the attraction, leading to
thermodynamically 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 membranes (81). Furthermore, the repulsive protrusion interactions that lead to colloidal monolayer
stability 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 been
named colloidal membranes. However, from a microscopic perspective the forces driving assembly of
colloidal membranes and lipid bilayers are quite distinct. Colloidal membranes are assembled from
chemically homogeneous rod-like molecules, whereas lipid bilayers are assembled from chemically
heterogeneous amphiphilic lipids. Nonetheless, colloidal membranes represent a unique opportunity
to investigate membrane biophysics from an entirely new perspective on a length scale where it is
possible to visualize constituent building blocks. Along these lines, recent work has demonstrated that
chirality of the constituent rods controls the edge line tension and can be used for assembly of various
complex structures (87). From a practical perspective, colloidal membranes offer an easily scalable
method for assembling sheets of semiconducting nanorods, which might be useful as photovoltaic
devices (88).
5. LIQUID CRYSTALS IN CROSS-LINKED GELS
One of the characteristics that distinguishes solids from liquids is that the former deform only in
response to an applied stress, whereas the latter change shape even in the absence of external
forces. In this respect, a weakly cross-linked polymer, such as conventional rubber, behaves as
a solid, albeit a very soft one with a modulus that is orders of magnitude smaller than that of
conventional molecular solids. The origin of rubber elasticity can be understood by considering
the dynamics of the constituent polymer coils. An isolated polymer coil in suspension assumes, on
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average, an entropy-maximizing spherical shape. Pulling on such a chain deforms its equilibrium shape,
reduces the number of accessible states, and decreases the polymer entropy, which leads to a measurable
force that tends to restore the polymer to a spherical shape (89, 90). In rubber, a large fraction of
polymer segments behave similarly to those of a free chain. However, the chains are also tethered at
a few isolated points to a solid-like background matrix composed of other chains. Applying external
stress on a rubber deforms the tethering points, which in turn stretches individual chains, decreasing
their entropy and increasing the free energy of the rubber. Thus, the origin of macroscopic rubber
elasticity is intimately connected with the microscopic states available to the constituent polymers
chains.
Polymer liquid crystals are polymers with liquid crystalline mesogens incorporated into the
monomer subunits; the mesogens are located either within the polymer backbone or on a side
chain. 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 is
the phenomena of soft elasticity, wherein a nematic elastomer, a solid-like material, can behave
like a liquid in that it undergoes energy-free deformations (91, 92).
By applying appropriate boundary conditions, it is possible to prepare conventional nematic
liquid crystals in a uniformly aligned state. Creating such defect-free samples is essential for elucidating the basic properties of liquid crystals. It has proven significantly more difficult to create
monodomain nematic elastomers. Simple one-step cross-linking of polymer liquid crystals yields
polydomain samples that obscure much of the interesting behavior of nematic elastomers. In an
important breakthrough in 1991, the first monodomain samples were prepared via a two-step
process (5). In the first step, a nematic polymer melt is sparsely cross-linked to form a weak
gel. Subsequently, the gel is stretched to align the nematic domains and then cross-linked a second
time to reinforce the uniformly aligned state. This important advance has enabled a series of
foundational experiments that have uncovered the surprising properties of nematic elastomers.
The tight coupling between the nematic order parameter and polymer chain conformations
endows liquid crystalline elastomers with their unique properties. In uncross-linked isotropic
samples, 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 crystalline background, the polymer backbone preferentially aligns along the nematic director, leading
it to assume the shape of a prolate ellipsoid elongated along the average liquid crystalline orientation (Figure 6b). Thus, the I-N phase transition of the background liquid crystals is intimately
coupled to the shape change of the polymer coils. Cross-linking polymer chains couples such
microscopic 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 of
the elastomer structure. The effect is dramatic: Deformations as large as 350% are not hard to
achieve (Figure 6c) (93). Detailed analysis confirms that the temperature dependence of the nematic 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 an
isotropic state induces anisotropy of polymer chains that, in turn, induce the formation of
a paranematic liquid crystalline state (96, 97). As an alternative to temperature-induced shape
deformations using thermotropic liquid crystals, it is also possible to induce such transformations
with 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 nematic
director is the phenomena known as soft or semisoft elasticity (Figure 6d) (91). A nematic elastomer can respond to an applied stress by either distorting polymer chains from their elongated
equilibrium distribution or rotating them while preserving their ellipsoidal shape, both of which
align the nematic director along the direction of the applied stress (98, 99). However, the latter is the
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c
Figure 6
Macroscopic shape changes and soft elasticity of nematic elastomers. (a) Schematic illustration of a cross-linked
elastomer in an isotropic phase in which the constituent polymers assume a free energy–minimizing
spherical shape. A single polymer (orange) is shown. (b) The background nematic phase induces the polymer
chains to assume an elongated shape aligned along the nematic director. The microscopic shape change of
constituent polymers is intimately coupled to the macroscopic shape change of the entire elastomer. (c)
A sequence of images illustrating the large macroscopic shape change of a nematic elastomer as it undergoes an
I-N phase transition. Images courtesy of E. Terentjev (93). (d) The elastic stress as function of uniaxial extension
applied in the direction perpendicular to the nematic director demonstrates that for certain conditions
nematic elastomers can undergo macroscopic deformations that have essentially no energetic cost. This
phenomenon is known as soft or semisoft elasticity. Adapted with permission from Reference 100.
favored response, as rotation preserves the preferred microscopic conformation of the constituent
polymers. The evidence for this can be gained by analyzing the behavior of a nematic director in an
elastomer under external stress. Experiments show that applying an external stress perpendicular to
the direction of the initial alignment results in a rotation of the nematic director (98, 99). The director
can rotate either continuously or discontinuously until it eventually aligns along the direction of
applied stress. Throughout this operation, the shape and, therefore, the entropy of the constituent
polymers 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 to
various environmental cues has led to numerous applications. An elastomer based on a cholesteric
liquid crystal makes it possible to assemble a tunable and mirrorless laser (101). Also, a dye-sensitized
nematic elastomer behaves as an autonomously motile object that swims away from a light source
(102). The fact that nematic elastomers can behave as molecular actuators has led to a proposal to use
them as artificial muscles (103).
6. COLLOIDS IN LIQUID CRYSTALS
Interactions between colloidal particles suspended in an isotropic liquid have been studied in great
detail. By tuning the solvent properties, introducing nonadsorbing polymers, or coating colloids
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with a polymeric brush, it is possible to engineer microscopic colloidal interactions that can drive
the assembly of well-defined macroscopic materials. We discuss how anisotropic liquid crystalline
solvents lead to an even greater diversity of colloidal interactions that can be used to assemble
structures of unprecedented complexity.
Insight into liquid crystal–mediated colloidal interactions can be gained by considering the
topology of the liquid crystalline ordering around a spherical colloid (104–106). For such
experiments, it is necessary to control the liquid crystalline anchoring conditions, which can be
either tangential or homeotropic (perpendicular) at the colloid surface. Placing a spherical colloid
with well-defined anchoring conditions into a uniformly aligned nematic phase leads to topological frustration. It is not possible to deform a nematic director so that it continuously transitions
from 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 colloidal
particle. Two types of defects are possible: either a point defect called hyperbolic hedgehog (Figure
7a) 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 and
multiple liquid crystal emulsions, in which a small aqueous droplet suspended in a nematic liquid
crystal plays the role of the perturbing colloid (Figure 7c) (6). The multiple emulsion system allows
for simultaneous observation and precise control of the liquid crystal anchoring in the far field as
well as locally around an embedded droplet. For example, it is possible to observe large nematic
droplets (∼50 mm) suspended in an aqueous solvent, each encapsulating a single or multiple
a
c
b
d
5 μm
e
5 μm
f
g
20 μm
Figure 7
Interactions and self-assembly of colloidal particles in anisotropic (nematic) solvents. (a) A point defect called a hyperbolic hedgehog enables
a nematic director to transition between uniform alignment at the boundaries and homeotropic anchoring at a colloid surface. The red
lines 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 large
liquid crystalline droplet interact through dipolar interactions and assemble into elongated chains. Adapted with permission from
Reference 6. (d) Chains of colloidal spheres formed under similar conditions to panel c. (e) Colloids with Saturn ring defects interact laterally
to form kinked chains aligned perpendicularly to the nematic director (110). (f) Colloids with quadrupolar Saturn ring defects assemble
into crystals with the shape of a parallelogram. (g) Colloids with dipolar coupling form chains that can be further assembled into
a 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 liquid
crystalline droplet to minimize the elastic energy. For a one-droplet configuration, it is possible to
continuously deform the nematic director from the inner surface of the large nematic droplet to the
outer surface of the smaller encapsulated aqueous droplet, unlike the case discussed above of
a colloid in a uniformly aligned nematic field. However, introducing an additional droplet into such
a 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 crystalline
droplets that contain n encapsulated water droplets contain n 1 defects. This geometry of multiple
emulsions allows one to explore the interactions between the droplets and their associated defects. It
is observed that multiple aqueous droplets assemble into elongated chains, wherein individual
droplets are not in contact but are separated by a well-defined distance (Figure 7c). High-resolution
images reveal the presence of point defects separating the droplets and thus suppressing their
coalescence.
Theoretically, the colloid-induced elastic deformation of the liquid crystalline director in the
far-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 known
that dipoles minimize their interaction energy by forming elongated chains, which explains experimental observations of droplet (colloids) chaining (Figure 7c,d). By applying an external
magnetic field, it is possible to vary the separation of a pair of liquid crystal–bound droplets doped
with a ferrofluid (108). Measuring the force between droplets at different separations directly
confirms the dipolar nature of these liquid crystal-mediated colloidal interactions.
Along with colloids with bound point defects, theoretical modeling also predicts another
possible 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 external
field, decreasing the particle size, or confining the background nematic liquid crystal to a thin
monodomain layer; experimentally, Saturn rings have been primarily observed in confined samples.
The electrostatic analogy predicts that colloids with Saturn defects interact with quadrupolar
symmetry. Therefore, whereas defects with hyperbolic hedgehogs line up to form elongated chains
aligned along the nematic director, colloids with Saturn rings bind laterally at a well-defined angle
with respect to the director field to form kinked chains (Figure 7e) (110).
Recent advances have demonstrated another liquid crystal–mediated mechanism that leads to
very different binding of spherical colloids (111, 112). Quenching a bulk isotropic phase into
a nematic is accompanied by the formation of an entangled network of disclination loops, which
anneal over time to form a lower energy state characterized by a uniform director field. The same
procedure can be performed locally around colloidal particles by heating the liquid crystal with
a laser. A subsequent quench generates disclination loops that cannot easily anneal due to the
presence of the colloids. These disclination loops entangle colloid particles to create strongly bound
pairs. Pulling entangled colloids apart with optical tweezers requires extension of the disclination
lines and leads to an interaction energy that increases with increasing particle separation and can
easily reach thousands of kBT.
The basic understanding of liquid crystal–mediated colloidal interactions at a pair level enables
assembly of macroscopic colloidal materials that are of both fundamental and applied interest and
are difficult to achieve with other methods of colloidal assembly (110). In particular, it has proven
possible to bring together hundreds of colloids with quadrupolar symmetry to assemble a 2D crystal
in the shape of a parallelogram (Figure 7f ). Increasing the thickness of the confining nematic induces
a switch from Saturn rings to hyperbolic point defects. As a result, colloids align along the nematic
field to form elongated chains. Dipolar chains formed from colloids with hedgehog defects either
repel 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 organized into higher-order 2D crystals with hexagonal symmetry (Figure 7g).
Macroscopic structures have also been obtained by quenching a liquid crystal–silicone oil mixture
from a uniform nematic phase into a biphasic I-N region (113). In conventional systems, a phaseseparating 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 background
nematic liquid crystal but coarsen only to an upper-critical size. At this point, they are stabilized by the
presence of defects and associated liquid crystal–mediated interactions. Such uniform droplets organize into macroscopic 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 as
diverse as photonics, materials science, and basic topology (114, 115). Here we have only described liquid crystal–mediated interactions between spherical colloids and their assembly into
bulk materials. Shifting to more complex shapes that also include topologically distinct structures
significantly affects the nature of the surrounding liquid crystalline field and considerably increases
the diversity of colloidal interactions that can potentially be used to assemble even more exotic
colloidal structures (116, 117).
7. ACTIVE LIQUID CRYSTALS
So far we have described a few representative examples of hypercomplex liquid crystals that emerge
when conventional anisotropic fluids are merged with other complex fluids, such as colloids and
polymers. All of these systems are assembled from inanimate, passively diffusing building blocks. As
such, their behavior can be explained with equilibrium statistical mechanics, which entails minimizing the overall free energy of the system to determine stable phases. However, the laws of
equilibrium statistical mechanics impose severe constraints on the properties of such materials; for
example, these materials cannot exhibit spontaneous motion or perform macroscopic work. Recently, an entirely new research direction has emerged that is focused on developing materials with
orientational order that are assembled from animate, energy-consuming building blocks. Being
actively driven, the emergent properties of such far-from-equilibrium materials are not constrained
by the laws of equilibrium statistical mechanics, and they can become motile or spontaneously flow
(118–122). Among research topics within active matter, the study of active liquid crystals has received a great deal of attention. Since its inception, this field has been largely driven by theoretical
efforts. However, the past few years have witnessed the emergence of well-defined experimental
systems that can be used to test existing theoretical models and guide their further development. We
briefly review these experiments while emphasizing similarities as well as differences between active
liquid crystals and their equilibrium analogs. For a more thorough discussion, the reader is referred to
extensive review articles on active matter (123–125).
In an equilibrium nematic, all molecules are equally likely to point up or down, with the local
orientation of molecules described by an arrowless bar. In contrast, active liquid crystals can have
either polar or nematic (quadrupolar) symmetry, and both have been experimentally realized. The
polar active liquid crystals were beautifully demonstrated in a biological motility assay based on
an actin-myosin system (126–128). F-actin is a semiflexible filament that plays an essential role in
a variety of biological processes, ranging from cellular division and motility to muscle contraction.
Myosin is a motor protein that utilizes energy from ATP hydrolysis to take nanometer-sized steps
along an actin filament. Myosin motors move in a polar fashion toward one end of an intrinsically
polar actin filament. In a motility assay, one affixes a high density of myosin motors onto a solid
microscope slide (Figure 8a,b) (129). In the presence of ATP, actin filaments continuously glide on
a bed of myosin motors in a ballistic fashion.
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100
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T=0s
–100
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–100
0
x (μm)
T = 185 s
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200
Figure 8
Active polar nematic liquid crystals observed in a gliding actin-myosin motility assay. (a) Schematic of a motility assay in which surfacebound motors propel the gliding of isolated filaments. The light-blue arrow indicates the direction of filament motion. (b) Image of a
low density (isotropic phase) of fluorescently labeled actin filaments gliding on the coverslip surface. (c) Coherent filament motion
in a high-density liquid crystalline phase exhibits structures that are reminiscent of defects with topological charge equal to unity. To
illustrate filament dynamics, ten consecutive images are overlaid, starting with the single image depicted in the inset. (d) The velocity
profile associated with the coherently moving filaments in panel c. (e) The trajectory of the center of a liquid crystalline defect that
remains stable for ∼400 s before falling apart. Adapted with permission from Reference 126.
In a traditional motility assay, one places a few dilute filaments to ensure absence of any filament interactions. The Bausch group has explored the behavior in the opposite limit, in which the
microscope slide is covered with a dense ensemble of gliding filaments (126). When two gliding
filaments encounter each other, there is a small but finite probability of them reorienting and
moving in the same direction because of steric interactions. A number of surprising collective
effects emerge in such dense monolayers of actively gliding filaments. With increasing filament
concentration, one observes the spontaneous formation of dense flocks in which thousands of
motile filaments move in a coordinated fashion. The dense flocks of aligned filaments coexist with
a 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 is
possible to assign direction to each filament by simply pointing the arrow in the direction of
filament motion. Because in a dense flock all the filaments point in the same direction, driven actin
filaments form a far-from-equilibrium liquid crystal with polar symmetry. The evidence for a polar
liquid crystal can also be extracted by examining the nature of liquid crystalline defects. As
discussed previously, disclination defects of charge 61/2 observed in apolar nematic liquid crystals
cannot 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, and
local orientation is described by an arrowless line. The first experimental example of an active
nematic consisted of a dense monolayer of elongated melanocyte cells (130, 131). Observations of
disclination defects of charge 61/2 confirmed the underlying apolar nematic symmetry (Figure
9a). However, in a living system one cannot control relevant microscopic parameters, such as cell
motility, which precludes detailed comparison with theory. A distinct system consisting of shaken
granular rod monolayers also exhibits properties of active nematics (Figure 9b) (132). In this case,
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c
Time
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Time
b
Figure 9
Disclination defects govern dynamics of active liquid crystals with nematic symmetry. (a) Observation of 1/2
disclination 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 in
a vibrated monolayer of granular rods. An asymmetric þ1/2 defect (green circles) is highly mobile, whereas
a more symmetric 1/2 defect (red circles) is mostly stationary. Adapted with permission from Reference 132.
(c) Unbinding of oppositely charged defects in active liquid crystals based on microtubules and the molecular
motor kinesin. The active nematic is unstable toward bend fluctuations. Once initiated, the bend deformation
grows in amplitude, eventually leading to the fracture of the nematic director, which is terminated at a pair of
oppositely charged defects (red and green arrows). The fracture line spontaneously heals, leaving behind a pair
of unbound defects. Steady-state is reached when the rate of defect unbinding is balanced by the rate of defect
annihilation. Adapted from Reference 133.
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 the
density fluctuations are significantly stronger than in equilibrium systems, are long lived, and
decay logarithmically in time.
Until recently, the majority of experimental model systems of animate liquid crystals described,
including gliding filaments, crawling cells, and shaken rods, required the presence of a solid 2D
substrate. Another recent advance has demonstrated the assembly of an active nematic in
microtubule-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 microtubule in discrete 8-nm steps (134). Using depletion interactions, short microtubules are assembled 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 multimotor
clusters. Such clusters simultaneously bind multiple microtubules within a nematic layer, inducing interfilament sliding and driving bundle extension. This extension lacks directionality;
consequently, active microtubule liquid crystals have nematic (quadrupolar) symmetry. The
existence of disclination defects of charge 1/2, similar to those observed in shaken rods or
crawling cells, supports this hypothesis (Figure 9c).
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In conventional liquid crystals, defects are largely static inanimate structures and their existence and location are determined by either internal frustrations or externally imposed
boundary conditions or forces. The behavior and role of defects in active liquid crystals is
dramatically different. Most notably, defects embedded in active liquid crystals acquire motility,
and their exact dynamics depends on their topological charge. The motility of þ1/2 defects is
powered by their asymmetric structure, whereas the more symmetric 1/2 defects are largely
static. 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. An
extensile active nematic in a uniform state is inherently unstable against long-wavelength bend
fluctuations (121). Once initiated, a bend deformation grows in amplitude and eventually
buckles, resulting in the spontaneous fracture of the nematic director accompanied by an unbinding of a pair of oppositely charged disclination defects (Figure 9c) (135, 136). Upon
generation, the motile þ1/2 moves away from the largely static 1/2 defect. It follows that
a steady state of active nematics is characterized by a dense suspension of highly interacting
motile defects of fleeting existence. Defect motility drives the streaming flow dynamics of the
entire liquid crystal. Microtubule-based active nematics are easily assembled on flat interfaces as
well as on curved liquid-liquid interfaces, such as emulsion droplets. When such emulsions are
confined, the internally powered streaming liquid crystalline flows drive droplet motility,
resulting in objects that spontaneously roll around on the surface. These studies illustrate the
essential 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 materials
with emergent properties. A number of notable recent advances in soft-matter physics have involved development of hypercomplex materials based on liquid crystals and anisotropic fluids. We
have described a few representative examples that illustrate the diversity of these materials as well
as their emergent properties. These examples show how the foundational ideas developed through
original research on molecular liquid crystals are being applied to an ever-widening range of
phenomena. Finally, hypercomplex liquid crystals also demonstrate how assembly of advanced
materials does not necessarily require engineering of more complex building blocks through sophisticated chemistry but can also be accomplished through knowledgeable merger of relatively
simple systems with competing interactions.
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings that
might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
Support from National Science Foundation (MRSEC-0820492, CAREER-0955776, CMMI1068566) and W.M. Keck are gratefully acknowledged. Z.D. also acknowledges support from
the Technische Universität München–Institute for Advanced Study, which is funded by the
German Excellence Initiative. To our knowledge, the term hypercomplex fluid was first coined by
R.B. Meyer, and we thank him for many useful discussions.
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Condensed Matter
Physics
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Contents
Volume 5, 2014
Whatever Happened to Solid State Physics?
John J. Hopfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Noncentrosymmetric Superconductors
Sungkit Yip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Challenges and Opportunities for Applications of Unconventional
Superconductors
Alex Gurevich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Correlated Quantum Phenomena in the Strong Spin-Orbit Regime
William Witczak-Krempa, Gang Chen, Yong Baek Kim,
and Leon Balents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Dirac Fermions in Solids: From High-Tc Cuprates and Graphene to
Topological Insulators and Weyl Semimetals
Oskar Vafek and Ashvin Vishwanath . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A Quantum Critical Point Lying Beneath the Superconducting Dome
in Iron Pnictides
T. Shibauchi, A. Carrington, and Y. Matsuda . . . . . . . . . . . . . . . . . . . . . 113
Hypercomplex Liquid Crystals
Zvonimir Dogic, Prerna Sharma, and Mark J. Zakhary . . . . . . . . . . . . . . 137
Exciton Condensation in Bilayer Quantum Hall Systems
J.P. Eisenstein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Bird Flocks as Condensed Matter
Andrea Cavagna and Irene Giardina . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Crossover from Bardeen-Cooper-Schrieffer to Bose-Einstein Condensation
and the Unitary Fermi Gas
Mohit Randeria and Edward Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
v
Crackling Noise in Disordered Materials
Ekhard K.H. Salje and Karin A. Dahmen . . . . . . . . . . . . . . . . . . . . . . . . 233
Growing Length Scales and Their Relation to Timescales in Glass-Forming
Liquids
Smarajit Karmakar, Chandan Dasgupta, and Srikanth Sastry . . . . . . . . . . 255
Multicarrier Interactions in Semiconductor Nanocrystals in Relation to
the Phenomena of Auger Recombination and Carrier Multiplication
Victor I. Klimov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
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Polycrystal Plasticity: Comparison Between Grain-Scale Observations of
Deformation and Simulations
Reeju 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/GaAs
Heterostructures: Enabling Physics in Low-Dimensional
Electronic Systems
Michael J. Manfra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Simulations of Dislocation Structure and Response
Richard LeSar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Errata
An online log of corrections to Annual Review of Condensed Matter Physics articles
may be found at http://www.annualreviews.org/errata/conmatphys
vi
Contents