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LIQUID CRYSTALS
Three-dimensional crystals ofadaptive knotsJung-Shen B. Tai1 and
Ivan I. Smalyukh1,2,3*
Starting with Gauss and Kelvin, knots in fields were postulated
to behave likeparticles, but experimentally they were found only as
transient features orrequired complex boundary conditions to exist
and could not self-assemble intothree-dimensional crystals. We
introduce energetically stable, micrometer-sizedknots in helical
fields of chiral liquid crystals. While spatially localized and
freelydiffusing in all directions, they resemble colloidal
particles and atoms, self-assemblinginto crystalline lattices with
open and closed structures. These knots are robustand topologically
distinct from the host medium, though they can be morphed
andreconfigured by weak stimuli under conditions such as those in
displays. A combinationof energy-minimizing numerical modeling and
optical imaging uncovers the internalstructure and topology of
individual helical field knots and the various
hierarchicalcrystalline organizations that they form.
Topological order and phases represent anexciting frontier of
modern research (1), buttopology-related ideas have a long history
inphysics (2). Gauss postulated that knots infields could behave
like particles, whereas
Kelvin, Tait, and Maxwell believed that matter,including
crystals, could be made of real-space,free-standing knots of
vortices (2–4). These earlyphysics models, introduced long before
the veryexistence of atoms was widely accepted, gaveorigins to
modern mathematical knot theory(2–4). Expanding this topological
paradigm,Skyrme and others modeled subatomic par-ticles with
different baryon numbers as non-singular topological solitons and
their clusters(3–5). Knotted fields emerged in classical andquantum
field theories (3–7) and in scientificbranches ranging from fluid
mechanics to par-ticle physics and cosmology (2–11). In
condensedmatter, arrays of singular vortex lines and
low-dimensional analogs of Skyrme solitons werefound as
topologically nontrivial building blocksof exotic thermodynamic
phases in superconduc-tors, magnets, and liquid crystals (LCs)
(12–14).Could they be knotted, and could these knotsself-organize
into three-dimensional (3D) crys-tals? Knotted fields in condensed
matter foundmany experimental and theoretical embodiments,including
both nonsingular solitons and knottedvortices (7–9, 15–23).
However, they were meta-stable and decayed with time (7–9, 15–17)
orcould not be stabilized without colloidal inclu-sions (18, 19) or
confinement and boundary con-ditions (20–22), and could not
self-organize into
3D lattices (22, 23). We introduce energeticallystable,
micrometer-sized adaptive knots in chiralLCs that, unexpectedly,
materialize the knottedvortices and nonsingular solitonic knots at
thesame timeand indeedbehave like particles, under-going 3D
Brownian motion and self-assemblinginto 3D crystals.Helical fields,
as in the familiar example of
circularly polarized light with electric and mag-netic fields
periodically rotating around the Poyn-ting vector, are ubiquitous
in chiral materialssuch as magnets and LCs. These helical
fieldscomprise a triad of orthonormal fields (Fig. 1A),namely
thematerial alignment fieldnðrÞ (of rod-like molecules in LCs or
spins in magnets) andthe immaterial line fields along the helical
axiscðrÞ, analogous to the Poynting vector, and tðrÞ⊥nðrÞ⊥cðrÞ. For
LCs, nðrÞ is nonpolar but can bedecorated with unit vector fields
(14, 24). Thedistance over whichnðrÞ and tðrÞ rotate aroundcðrÞ by
2p within the helical structure is thehelical pitchp (Fig. 1A). We
demonstrate knottedfields that in nðrÞ are topological solitons
withinterlinked, closed-loop preimages resemblingHopf fibration
(Fig. 1B). At the same time, thenonpolar nature of cðrÞ and tðrÞ
permits thehalf-integer singular vortex lines to form varioustorus
knots (Fig. 1C) while retaining the fully non-singular nature
ofnðrÞ. Therefore, our topologicalsoliton in the helical field is a
hybrid embodimentof both interlinked preimages and knotted
vortexlines, which can be realized to have this
solitonicnonsingular nature in systems with either polaror nonpolar
nðrÞ (12–14). We find these knotsolitons, which we call
“heliknotons,” embeddedin a helical background and forming
spontane-ously after transition from the isotropic to LCphase when
a weak electric fieldE is applied to
apositive-dielectric-anisotropy chiral LC along thefar-field
helical axis c0 . The materials used areprepared as mixtures LC-1
through LC-3 (24) ofcommercially available, room-temperature
ne-matics and chiral dopants. In bulk LC samples oftypical
thickness within d = 10 to 100 mm (24),
heliknotons display 3D particle–like propertiesand form a dilute
gas at low number densities(Fig. 1D), with orientations of
shape-anisotropicsolitonic structures correlated with their
posi-tions along c0 (Fig. 1, D and E). Depending onmaterials and
applied voltageU, heliknotons canadopt different shapes (Fig. 1, D
to G), which arereproduced by numerical modeling (insets ofFig. 1,
F and G) based on minimization of thefree energy (24):
F ¼
∫d3rK
2ð∇nÞ2 þ 2pK
pn � ð∇� nÞ � e0De
2ðn � EÞ2
� �
ð1Þwhere K is the average elastic constant, De isthe LC’s
dielectric anisotropy, and e0 is the vac-uum permittivity. The
integrand comprises energyterms originating from elastic
deformation, chiral-ity, and dielectric coupling, respectively.
Minimiza-tion ofF at differentU andDe (table S1) reveals
thatheliknotons can be stable, metastable, or unstablewith respect
to the helical background (Fig. 1H),comprising localized regions
(depicted in gray inFig. 1, B and C) of perturbed helical fields
andtwisting rate.Heliknotons undergo Brownian motions (Fig.
1I and movie S1) and exhibit anisotropic inter-actions while
moving along c0 and in the lateraldirections (Fig. 1, E and J, and
movie S2) (24).The inter-heliknoton pair-interaction potential
isanisotropic and highly tunable, from attractive torepulsive and
from tens to thousandskBT (wherekB is Boltzmann constant and T is
temperature),depending on the choice of LC, U, and samplethickness
(Fig. 1J). Similar to nematic colloids(25, 26), interactions
between localized helikno-tons arise from sharing long-range
perturbationsof the helical fields around them andminimizingthe
overall free energy for different relative spa-tial positions of
these solitons. These interactionslead to a plethora of crystals,
including low-symmetry and open lattices that were recentlyachieved
in colloids (26–28) (Fig. 2). In thin cellsof thickness d≲4p ,
heliknotons localize aroundthe sample’s horizontal midplane, making
theiranisotropic interactions quasi-2D. Heliknotonsself-assemble
(movie S3) into a 2D rhombic latticeboth when the attractive
potential is ~1000 kBT(Fig. 2, A and B) and when it is ~10 kBT
(Fig. 2,C and D). From initial positions defined by lasertweezers
(24), heliknotons self-assemble into astretched kagome lattice with
anisotropic bind-ing energies ~100 kBT (Fig. 2E). Such open
lat-tices have interesting topological properties (28),potentially
bringing about an interplay betweentopologies of the crystal’s
basis and lattice. Crys-tallographic symmetries and lattice
parameterscan be controlled through tuning
reconfigurableinteractions, such as switching reversibly
betweensynclinic and anticlinic tilting of heliknotons viachanging
U by 4p,
when anisotropic interactions yield triclinic lat-tices (Fig. 2,
H to N). One can watch initially
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1Department of Physics, University of Colorado, Boulder,CO
80309, USA. 2Materials Science and EngineeringProgram, Soft
Materials Research Center, and Departmentof Electrical, Computer
& Energy Engineering, Universityof Colorado, Boulder, CO 80309,
USA. 3Renewable andSustainable Energy Institute, National Renewable
EnergyLaboratory and University of Colorado, Boulder, CO80309,
USA.*Corresponding author. Email: [email protected]
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quasi-2D pre-self-assembled crystallites interact-ing with each
other while moving in lateral andaxial directions (Fig. 2, J to M,
and movie S5),forming different crystallographic planes of the3D
triclinic lattice. The helical background LCs,individual
heliknotons, and the ensuing latticesare all chiral. The
lowest-symmetry triclinic pediallattices can have primitive cells
comprising two(Fig. 2, H and J toM, andmovie S5) or three (Fig.2, I
andN, andmovie S6) crystallographic planes,depending on relative
orientations of helikno-tons within these planes. The two lattices
withparallel (Fig. 2M) and orthogonal (Fig. 2N) rela-tive
orientations of heliknotons in consecutiveheliknoton layers are
just examples as the anglebetween heliknotons within
crystallographic pla-nes along c0 can be tuned (Fig. 1E) byU,
material,and geometric parameters. Because the helikno-tons have
anisometric shape (like LC molecules)and can exhibit spatial
twists, hierarchical topo-logical solitons comprising heliknotons
couldpotentially emerge. Heliknoton crystals exhibitgiant
anisotropic electrostriction (Fig. 2O andmovies S7 and S8). For
example, upon changingfrom U = 3.0 to 4.4 V, one lattice parameter
inthe insets of Fig. 2O extends by ∼44%, whereasthe other only by
∼4%. This electrostriction isconsistent with the free-energy
minimization(Fig. 2P) for 3D crystals of heliknotons withtunable
lattice parameters at different U. The ex-perimentally observed
soliton crystals correspondto minima of free energy within a broad
range
of applied voltages (24), consistent with theirfacile
self-organization into triclinic pedial crys-tals and other
reconfigurable 3D and 2D lattices(Fig. 2). As the applied field is
increased evenfurther, the heliknoton crystals become meta-stable
and then unstable with respect to theunwound state with nðrÞ||E
(24).Numerical modeling and experiments reveal
detailed structures of the fields within a heli-knoton (24)
(Fig. 3, A to I, and fig. S1). InnðrÞ, thecontinuous localized
configuration of a heliknotonis embedded in a helical background
(Fig. 3, Ato C) and has all closed-loop preimages linkedwith each
other once, with the linking number +1(Fig. 3J and fig. S1). Up to
numerical precision,this matches the Hopf index calculated
throughnumerical integration (22):
Q ¼ 164p2
∫d3rDijkAiBjk ð2Þ
where Bij ¼ Dabcna@inb@jnc , D is the totally anti-symmetric
tensor, Ai is defined as Bij ¼ ð@iAj�@jAiÞ=2, and the summation
convention isassumed. Spatial structures of cðrÞ and tðrÞ
arederived from the energy-minimizing nðrÞ usingthe eigenvector of
the chirality tensor (24, 29, 30)(Fig. 3, D to I). They exhibit
torus knots of spa-tially colocated singular half-integer
vortices,withinwhichcðrÞandtðrÞnonpolar fields rotateby 180° around
the vortex line in the plane lo-cally orthogonal to it (Fig. 3, D
to I). The closedloop of the vortex line is the righthanded
T(2,3)
trefoil torus knot, also labeled as the 31 knot inthe
Alexander–Briggs notation (Fig. 3, K and L).The singular vortex
knots in cðrÞ and tðrÞ alsocorrespond to a colocated knot of a
meron (topo-logically nontrivial structure of a fractional
2Dskyrmion tube) in nðrÞ (fig. S2). Handedness ofthe knots and
links matches that of chiral nðrÞ,implying that the sign of Hopf
indices of suchenergy-minimizing solitons is dictated by
LC’schirality. Simulated and experimental depth-resolved nonlinear
optical images of helikno-tons for different polarizations of
excitationlight closely agree (Fig. 3, M to O),
confirmingexperimental reconstruction of the field (24).Unlike the
Shankar solitons (11), which exem-plify condensed matter models
with topologyof a triad of orthonormal fields similar to thatof
Skyrme solitons in nuclear physics, helikno-tons exhibit
nonsingular structure only in oneof the three fields, though they
are still overallnonsingular in the material field. Differing
fromtransient textures of linked loops of nonsingulardisclinations
(15, 16) and metastable loops ofsingular vortices (17) in
cholesteric LCs, ourheliknotons are stable torus knots of
colocatedmerons innðrÞand vortices incðrÞandtðrÞ thatenable
ground-state 3D crystals of knots (Fig. 2and fig S2).In addition to
the Q ¼ 1 heliknotons with
equilibrium dimensions between p and 2p,we also find larger Q ¼
2 topological solitons(Fig. 4A), for which experimental
polarizing
Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 2 of
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Fig. 1. Knots in helices. (A) Helical field compris-ing a triad
of orthonormal fields nðrÞ, cðrÞ,and tðrÞ, with nðrÞ being either
polar (left) ornonpolar (right). (B) Preimages in nðrÞ of
aheliknoton colored according to their orientations
on S2 (top right inset). (C) Knotted colocatedhalf-integer
vortex lines in cðrÞ and tðrÞ. In (B)and (C), the gray isosurfaces
show regionsof distorted helical background. (D) A gas
ofheliknotons in LC-1 sample of thickness 30 mmat U = 4.5 V. (E)
Two heliknotons interact in 3Dwhile forming a dimer in LC-2 sample
of thickness30 mm at U = 11.0 V. (F and G) Polarizing
opticalmicrographs of metastable and stable heliknotonsat U = 4.3
and 4.5 V, respectively, in a samplewith d ¼ 10 mm, with
computer-simulated counter-parts shown in the bottom right insets.
(H) Freeenergy of individual heliknotons versus E, whereenergy of
the helical state equals zero; the helical
state and heliknotons are unstable atffiffiffiffiffiffiDe
pE ≳ 1V/mm
when the field tends to align nðrÞ||E (24).(I) Displacement
histograms Dx and Dy showingdiffusion of the heliknoton in (F) in
orthogonallateral directions perpendicular to c0. Experimentaland
numerical data were obtained for LC-1 in (D)and LC-2 in (E) to (I).
(J) Pair interaction ofheliknotons. Data shown in red (at voltages
○,4.2 V; △, 2.9 V) were obtained for LC-2 andthose in blue (○, 1.4
V; △, 1.0 V; ◇, 1.7 V)for LC-1 at d ¼ 10 mm (24). Scale bars
indicate10 mm and p ¼ 5 mm.
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micrographs also match their numerical coun-terparts. A Q ¼ 2
heliknoton contains a largerregion of distorted helical background
in both thelateral and axial directions (Fig. 4, A to D, andfigs.
S3 and S4). Preimages for two antiparal-lel vertical orientations
of nðrÞ form a pair ofHopf links (Fig. 4H), linked twice, like all
otherpreimage pairs. Singular vortex lines incðrÞ andtðrÞ form
closed cinquefoil T(2,5) torus knots(also labeled as 51 knots)
colocated with a sim-ilar knot of a meron tube in nðrÞ. A Q =
3heliknoton contains three Hopf links of preim-ages with a net
linking number of 3 for eachpreimage pair (Fig. 4I and figs. S5 and
S6). Thesingular vortex lines incðrÞandtðrÞ form a T(2,7)
torus knot (the 71 knot), colocated with the sameknot of a meron
tube in nðrÞ (Fig. 4, E, F, andI). Figure 4, G to I, shows both
preimages of theantiparallel vertical orientations of nðrÞ
andvortex lines in cðrÞ and tðrÞ , as well as theReidemeister moves
simplifying their struc-tures. Topologically disinct heliknotons
havedifferent numbers of crossings in the free-standing knots of
vortex lines and differentlinking of preimages, which were key
topo-logical invariants in early models of atoms andsubatomic
particles (2–6). For different heli-knotons, Q is related to
crossing number N ofthe vortex knots: N ¼ 2Q þ 1. The closed loopof
preimages in nðrÞ are interlinked with the
torus knots of vortices incðrÞandtðrÞ, as shownin Fig. 4, G to
I.Differing from transient vortex lines, which
shrink with time owing to energetically costlycores and
distorted order around them, vortex-meron knots in heliknotons are
energeticallyfavorable because they are nonsingular in thematerial
nðrÞ field and comprise twisted struc-tures with handedness
matching that of the LC.The stability of our 3D topological
solitons asspatially localized structures is assisted by thechiral
term in Eq. 1, which introduces their fi-nite dimensions and plays
a role analogous tothat of high-order nonlinear terms in
solitonicmodels of subatomic particles (3–6) and the
Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 3 of
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Fig. 2. Crystals of heliknotons. (A and B) Snapshots showing
self-assembly of a 2D crystal (d ¼ 10 mm, U = 3.5 V). (C to E) 2D
closedrhombic [(C) and (D)] and open (E) lattices of heliknotons at
U = 1.9 and1.7 V, respectively (d ¼ 15 mm). (F and G) Crystallites
with aligned (F) andanticlinically tilted (G) heliknotons at U =
1.8 and 2.3 V, respectively(d ¼ 17:5 mm). (H and I) Primitive cells
of 3D heliknoton crystals wherethe solitons in neighboring
horizontal layers have relative parallel (H)or perpendicular (I)
orientations. Isosurfaces (gray) show the localized
3D regions of heliknotons with distorted helical background
whencolocated with both vortex knots (light red) and preimages
ofantiparallel vertical orientations in nðrÞ (black and white). (J
to L) 3Dinteractions and self-assembly of heliknoton crystallites
(d ≈ 30 mm and
U = 2.8 V). (M and N) 3D heliknoton lattices comprising
crystalliteswith parallel (M) or perpendicular (N) orientations,
where (N) and itsinset are polarizing micrographs obtained when
focusing at differentcrystalline planes ∼10 mm apart (d ≈ 30 mm in
both cases and U = 2.8 and3.4 V, respectively). (O)
Electrostriction of a heliknoton crystal. Insetsshow lattices at
different U, with the lattice parameters shown inblue and red (d ¼
10 mm). (P) Free energy of heliknoton crystals perprimitive cell
for two LCs at different E. The heliknotons become
metastable with respect to the unwound state
atffiffiffiffiffiffiDe
pE ≳ 0.8 V/mm
for LC-1 and atffiffiffiffiffiffiDe
pE ≳ 1.2 V/mm for LC-2. Data were obtained using LC-2
in (A), (B), and (O) and using LC-1 in (C) to (N) (24). Scale
barsindicate 10 mm and p ¼ 5 mm.
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Tai et al., Science 365, 1449–1453 (2019) 27 September 2019 4 of
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Fig. 3. Structure of an elementary heliknoton. (A to I)
Computer-simulated cross-sections of nðrÞ [(A) to (C)], cðrÞ [(D)
to (F)], and tðrÞ [(G)to (I)] of a heliknoton. Vertical
cross-sections and the viewing directionsare marked in (A), (D),
and (G), respectively. nðrÞ is shown with arrowscolored according
to S2 [(A), inset], and cðrÞ and tðrÞ are shown with
ellipsoidscolored according to their orientations on the doubly
colored S2=ℤ2 [(D) and(G), insets].The vortex lines in cðrÞ and
tðrÞ are marked by red circles in (D) to(I). (J) Preimages of
vertical orientations of nðrÞ forming a Hopf link (bottom
right inset) and the cross-section of nðrÞ. (K and L) The
singular vortex linein cðrÞ and tðrÞ forming a trefoil knot (bottom
right insets) visualized bylight-red tubes and the cross-sections
of cðrÞ and tðrÞ, respectively.(M to O) Computer-simulated and
experimental nonlinear optical images ofnðrÞ in the cross-sections
of a heliknoton obtained with marked linearpolarizations [(M) and
(N)] and circular polarization [(O)]. Images on the leftare
numerical and those on the right are experimental, all obtained for
LC-3at d ¼ 10 mm and U = 1.7 V (24). Scale bars indicate 5 mm and p
¼ 5 mm.
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Dzyaloshinskii–Moriya term in models of mag-netic skyrmions (13,
14). Applied field along c0tends to reorient nðrÞ along E as
compared tothe helical state withnðrÞ⊥E. Consequently, theknots
emerge as local or global energy minimawithin a certain range of
voltages (24) by reduc-ing the dielectric term inEq. 1 comparedwith
thehelical state (Figs. 1H and 2P). At low E, elasticenergetic
costs are high and heliknotons collapseinto the helical background
through spontane-ous creation and annihilation of singular
defectsin nðrÞ. The strong dielectric coupling betweennðrÞ and E
aligns nðrÞ||E at high applied fields,eventually making both the
helical structure andsolitons unstable, but heliknotons are the
glob-al free-energy minima within broad, material-dependent ranges
of E (24).We have demonstrated 3D topological solitons
in helical fields of chiral LCs that can be ground-state and
metastable configurations, forming 3Dcrystalline lattices. Unlike
the atomic, molecular,and colloidal crystals, heliknoton crystals
ex-hibit giant electrostriction andmarked symmetrytransformations
under voltage changes
-
Three-dimensional crystals of adaptive knotsJung-Shen B. Tai and
Ivan I. Smalyukh
DOI: 10.1126/science.aay1638 (6460), 1449-1453.365Science
, this issue p. 1449; see also p.
1377Sciencearrangements.topologically distinct from the host medium
and diffuse and organize like colloidal particles, forming regular
crystallinestructures in cholesteric liquid crystals using electric
fields (see the Perspective by Alexander). The knots are polymers,
proteins, DNA, and even chemical molecules. Tai and Smalyukh
describe the creation of localized knottedsystems spans many
disciplines, including fluid and optical vortices, Skyrmion states,
liquid crystals, excitable media,
Although for some, knots are merely a frustration caused by
poorly tied shoelaces, interest in knots in physicalOptical
micrograph of a self-assembled lattice of knots in a chiral liquid
crystal
Knot all that it seems
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