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Periodic orbits, pair nucleation, and unbinding of active nematic defects on cones
Farzan Vafa, David R. Nelson, Amin Doostmohammadi
TL;DR
The paper asks how confinement and a curvature singularity jointly affect active nematic defect dynamics on a cone. Using analytical arguments and numerical simulations under strong anchoring, it finds ordered one- and two-defect orbits and transitions involving unbinding, absorption, and pair nucleation.
Problem
Curved surfaces studied to date do not typically combine nontrivial Gaussian curvature with a boundary that confines the nematic.
Source: §I · p. 1Method
The authors combine Born-Oppenheimer analytical arguments, in which textures rapidly follow defect positions, with full numerical simulations of an active nematic on a strongly anchored cone.
Source: §I · p. 2, §VIII · p. 14Results
Strong anchoring yields circular orbits of one or two +1/2 flank defects, with transitions involving defect absorption, unbinding, and apex pair nucleation.
Source: §I · p. 1, §VIII · p. 14Takeaways & Limitations
The phase diagram identifies distinct mechanisms organizing transitions between ordered defect-orbit states on cones.
Source: §I · p. 1, B. Apex-mediated defect nucleation and… · p. 11, C. Stable cyclic apex-mediated defect… · p. 11
Abstract
from arXiv · showhide
Geometric confinement and topological constraints present promising means of controlling active materials. By combining analytical arguments derived from the Born-Oppenheimer approximation with numerical simulations, we investigate the simultaneous impact of confinement together with curvature singularity by characterizing the dynamics of an active nematic on a cone. Here, the Born-Oppenheimer approximation means that textures can follow defect positions rapidly on the time scales of interest. Upon imposing strong anchoring boundary conditions at the base of a cone, we find a a rich phase diagram of multi-defect dynamics including exotic periodic orbits of one or two $+1/2$ flank defects, depending on activity and non-quantized geometric charge at the cone apex. By characterizing the transitions between these ordered dynamical states, we can understand (i) defect unbinding, (ii) defect absorption and (iii) defect pair nucleation at the apex. Numerical simulations confirm theoretical predictions of not only the nature of the circular orbits but also defect unbinding from the apex.
I. INTRODUCTION
The paper studies how confinement and curvature jointly organize active nematic defects on a cone, where the apex supplies a singular geometric charge. It combines analytical and numerical approaches to characterize defect dynamics under strong anchoring.
I. INTRODUCTION
- Confinement has been studied extensively on flat geometries, while curved-surface studies have focused on surfaces that do not typically combine nontrivial Gaussian curvature with a confining boundary.The cone combines both features and concentrates Gaussian curvature at its apex.Source: §I · p. 1
- Strong anchoring on a cone produces periodic orbits of one or two +1/2 flank defects, with transitions mediated by absorption, unbinding, or apex pair nucleation.Zero-flank-defect states are also possible at high deficit angles.Source: §I · p. 1
- The authors analyze the cone problem by combining analytical predictions with full numerical simulations, including tests of defect-unbinding transitions.The Born-Oppenheimer approximation assumes textures rapidly adapt to defect motion; agreement is reported in the low-activity regime.Source: §I · p. 2
A. Metric and nematic tensor
- In two dimensions, isothermal coordinates express the metric using a position-dependent conformal factor, and the nematic order parameter has two nonzero components.The covariant derivatives of those components include derivatives of the conformal factor.Source: A. Metric and nematic tensor · p. 2
B. Conical geometry
The cone’s deficit angle is determined by its half-angle and describes the sector removed from a flat disk. The paper sets up the curved-surface free energy and its ordered-limit form for analyzing nematic textures.
B. Conical geometry
- The cone deficit angle is 2πχ, with χ = 1 − sin β; χ = 0 corresponds to a disk with no sector removed.Physical coordinates represent the cone as an unrolled disk with a missing sector.Source: B. Conical geometry · p. 2, Figure 2 · p. 3
C. Free energy
- In regions of zero Gaussian curvature, the two Frank-elastic terms are equivalent by integration by parts, while the final free-energy term governs the isotropic-nematic transition.The coherence length is controlled by ϵ, and S0 sets the equilibrium nematic-order magnitude.Source: C. Free energy · p. 3
- Deep in the ordered limit, the free energy simplifies with J = K + K′ and Q = e^−φe^iα.The accompanying sketches show single- and two-defect nematic textures and their polarization angle.Source: C. Free energy · p. 3, Figure 3 · p. 3
D. Multi-defect solution
The multi-defect ansatz gives a texture minimizing the free energy for fixed defect positions and charges. The resulting free energy includes elastic, self-energy, and geometry-mediated interactions, with image defects implementing strong anchoring.
D. Multi-defect solution
- The multi-defect ansatz minimizes the free energy for fixed defect positions and charges, with image charges available to impose boundary conditions.The global phase ψ sets the overall nematic orientation.Source: D. Multi-defect solution · p. 3
- Defect polarizations are determined jointly by defect positions, charges, and ψ rather than being independent.For neighboring defects, the polarizations are antiparallel; for a +1/2 pair with ψ = π/2, they are perpendicular to the joining line.Source: D. Multi-defect solution · p. 3
E. Computation of the free energy
- Strong anchoring is imposed by fixing the director angle relative to the cone base and adding an image defect for each defect.The total defect charge is one, and the cone apex acts as an effective charge −χ.Source: E. Computation of the free energy · p. 4
- The free energy separates elastic interactions, defect self-energy, and a Coulombic interaction between defects and cone geometry.The self-energy must be supplemented by a microscopic defect core energy.Source: E. Computation of the free energy · p. 4
F. Forces
The force on a defect is obtained from the free energy and consists of Coulombic interactions with defects, image charges, and the cone apex.
F. Forces
- The defect force is computed from the free energy as its z̄-component.The force expression for defect 1 is then specialized to the two-defect case.Source: F. Forces · p. 4
- The force includes Coulombic interactions with the other defects and with their image charges.These are the first two contributions identified in the force expression.Source: F. Forces · p. 4
- A third Coulombic force contribution comes from interaction with the cone apex.The passage identifies this term alongside defect and image-charge interactions.Source: F. Forces · p. 4
III. STATIONARY SOLUTIONS IN PASSIVE CASE ON A CONE
For sufficiently small cone deficit, two classically stable passive defect configurations coexist, although the apex-plus-flank state has lower free energy. The section then introduces active nematic dynamics through relaxational, advective, and overdamped force-balance ingredients.
A. Stability for two defects
- For sufficiently small χ, cones admit two classically stable configurations—one apex-plus-flank pair and one two-flank pair—with the former minimizing F globally.The two-flank state is a local but not global minimum.Source: A. Stability for two defects · p. 4
- At χ = 0.15, the two-flank potential has a local minimum at x = xc = 0.59R, but its global minimum lies as x → 0.The plot marks the defect positions at the potential extrema.Source: Figure 4 · p. 5
IV. DYNAMICAL EQUATION
- The nematic order parameter’s dynamics combine relaxational evolution and flow advection, with the generalized advective derivative also accounting for flow-gradient reorientation.The molecular field appears in the stated dynamical setup.Source: §IV · p. 5
- In the overdamped limit, active-force and frictional-damping balance determines the flow; µ is friction and ζQ is the scalar activity coefficient.Source: §IV · p. 5
A. Born-Oppenheimer approximation
The Born-Oppenheimer approximation treats defect motion as slow enough for the nematic texture to readjust rapidly. For a defect pair, the framework retains Coulombic forces and imposes polarization constraints, alongside a simulation-informed phase diagram.
A. Born-Oppenheimer approximation
- At small activity, the passive multi-defect ansatz remains suitable if defects move slowly and the texture rapidly readjusts to their positions.Source: A. Born-Oppenheimer approximation · p. 5
B. Forces and polarization for a pair of +1/2 defects
- Within the Born-Oppenheimer approximation, Coulombic forces on a defect pair remain valid during motion, and each +1/2 defect has a motile force.The paper compares theoretical predictions with full numerical simulations to assess the approximation.Source: B. Forces and polarization for a pair… · pp. 5–6
- The phase diagram combines simulation data with colored curves that guide the eye, except the red curve, which follows Eq. (30) with c′ = 0.9.Source: Figure 5 · p. 6
- For neighboring defects, polarizations are antiparallel and their orientation relative to the line joining them does not change.Source: B. Forces and polarization for a pair… · p. 6
C. Simulation details
The simulations impose strong anchoring at the cone base and numerically evolve the active nematic texture. The study proceeds from a disk toward increasing cone deficit angle and compares two defect arrangements.
C. Simulation details
- Simulations set K′ = 0 because the cone has zero Gaussian curvature except at its apex, and impose strong anchoring that fixes the director angle relative to the base boundary.Source: C. Simulation details · p. 6
- The authors solve Eq. (16) by the method of lines, using predictor-corrector time evolution and five-point central differences for spatial derivatives.Source: C. Simulation details · p. 6
- The analysis starts with small activity on a disk, increases the deficit angle, and studies two-flank defects before an apex-plus-flank configuration.Source: C. Simulation details · p. 6
A. Two defect orbits on a disk (χ = 0)
On a disk, radial force balance sets the defects’ distance from the center, while tangential motility produces circular orbits. Simulations support the predicted orbit behavior and speed.
A. Two defect orbits on a disk (χ = 0)
- Generically, the defects orbit at constant speed with zero net radial force, rather than being driven by a centripetal force.The radial position balances radial motility against net Coulomb force; tangential motility remains nonzero.Source: A. Two defect orbits on a disk (χ = 0) · pp. 6–7
- For ψ = π/4 and π/2, two defects form stable circular orbits; ψ = 0 produces no orbit because motility has no tangential component.The orbital radius adjusts so radial motile and net Coulomb forces balance, while the tangential motile component drives motion.Source: A. Two defect orbits on a disk (χ = 0) · pp. 6–7
- Simulations confirm the theoretical speed prediction, with an approximately 1.4 predicted ratio close to the numerical ratio of approximately 1.5.The authors describe this agreement as surprisingly good despite the ansatz not being exact.Source: A. Two defect orbits on a disk (χ = 0) · p. 7
B. Two defect orbits on cone with 0 < χ < .31
For cone deficit angles χ < 0.31, small activity supports stable circular orbits of two flank defects. Radial force balance sets a χ-dependent radius, while the qualitative orbital behavior is independent of χ.
B. Two defect orbits on cone with 0 < χ < .31
- For χ < 0.31, small activity supports stable circular orbits of two flank defects on a cone.This extends the stable two-flank configuration from the disk to cones with small deficit angles.Source: B. Two defect orbits on cone with 0 <… · p. 7
- The orbit radius depends on χ, but the defects generically move at constant speed because radial motility balances net Coulomb force while tangential motility remains constant.The authors characterize this behavior as qualitatively independent of χ.Source: B. Two defect orbits on cone with 0 <… · p. 7
- The theory–simulation comparison covers distances for χ = 1/10 and ζ/(Jγ−1) = .01 at ψ = 0, π/4, and π/2.Table II reports theoretical predictions and simulation results for these conditions.Source: B. Two defect orbits on cone with 0 <… · p. 7, Table II · p. 8
VI. SINGLE DEFECT ORBITS AND DEFECT UNBINDING ON A CONE
A cone’s passive ground state can place one +1/2 defect on its flank, providing a setting for studying isolated-defect dynamics. With small activity, the flank defect orbits the apex; the ansatz becomes less accurate as χ and activity increase.
VI. SINGLE DEFECT ORBITS AND DEFECT UNBINDING ON A CONE
- The passive global-minimum configuration has one +1/2 defect on the cone flank, providing a setting to study isolated-defect dynamics.Source: §VI · p. 9
A. Single defect orbits
- For small activity, the isolated flank defect circularly orbits the apex at constant speed, with vanishing net radial force rather than a centripetal force.The analysis starts from the single-flank-defect ground state valid for χ < 2/3.Source: A. Single defect orbits · p. 9
- The multi-defect ansatz works better at small χ and ζ; at larger values, space-dependent global phase is not captured and the prediction becomes less accurate.The simulations show outward-rotated defect polarizations and a numerical distance larger than the prediction in the χ = 1/3, ζ Jγ−1 = 0.25 case.Source: A. Single defect orbits · p. 9
B. 1 →2 defect transition via defect unbinding
Simulations show that sufficiently strong activity can unbind an apex defect, while increased activity and cone deficit also support apex-mediated pair nucleation and transitions among one- and two-defect orbits. These transitions complement the stabilization of two-flank-defect orbits at larger deficit angles.
B. 1 →2 defect transition via defect unbinding
- At sufficiently large activity, a +1/2 defect can escape the apex, and the critical activity predicted by Eq. (30) agrees well with simulations.The threshold is calibrated with c′ ≈ 0.9; simulations show the escaping defect can join circular flank motion.Source: B. 1 →2 defect transition via defect… · p. 9, Figure 11 · p. 10, Figure 10 · p. 10
A. Stable two defect configurations for χ > 0.31
- For χ > 0.31, simulations show that sufficiently large activity can stabilize a two-defect orbit even though low-activity two-flank configurations are not stable.For ψ = π/2, outward-rotated defect polarizations may counteract attraction toward the apex; the multi-defect ansatz becomes less accurate as χ and activity rise.Source: A. Stable two defect configurations… · p. 11
- For the simulated χ = 1/3, ζ/(Jγ−1) = 0.5 cases, the authors lack an explanation for the ψ = 0 orbit and propose polarization rotation for ψ = π/2.The larger numerical defect distance at ψ = π/2 is consistent with outward-pointing polarizations and a space-dependent global phase not captured by the ansatz.Source: A. Stable two defect configurations… · p. 11
B. Apex-mediated defect nucleation and emission
- At sufficiently large cone deficit and activity, a neutral pair nucleates at the apex; its −1/2 defect annihilates one orbiting +1/2 defect, leaving one on the flanks.The force-balance estimate gives χc = 2/5 as a necessary but not sufficient condition; simulations show the threshold also depends on activity.Source: B. Apex-mediated defect nucleation and… · p. 11
C. Stable cyclic apex-mediated defect pair nucleation, emission, and unbinding
Simulations reveal cyclic apex-mediated defect pair nucleation, emission, annihilation, and unbinding at sufficiently high activity and cone deficit. The discussion connects these dynamics to related disk systems and identifies possible experimental tests and further geometries to study.
C. Stable cyclic apex-mediated defect pair nucleation, emission, and unbinding
- At sufficiently high activity and cone deficit, the simulated cycle combines pair nucleation and −1/2 emission that reduce two orbiting +1/2 defects to one, followed by unbinding back to two.The authors interpret the cycle as successive 2 →1 and 1 →2 transitions.Source: C. Stable cyclic apex-mediated defect… · p. 11, Figure 12 · p. 12, Figure 13 · p. 13
VIII. DISCUSSION
- The simulations suggest curvature can act as a lightning rod, with activity catalyzing neutral-pair nucleation and mobile −1/2 defect emission despite local three-fold symmetry.This mechanism goes beyond what the simple Born-Oppenheimer approximation derives.Source: §VIII · p. 14
- Related disk studies show similar pair-nucleation and annihilation transitions, while the cited experiments nucleate pairs at the boundary rather than at a cone apex.The authors also report agreement with a disk/annulus phase diagram at small cone deficit in the stated small-inner-radius regime.Source: §VIII · p. 14
- The authors identify hyperbolic cones and inverted truncated cones as further geometries for studying curvature-dependent active-defect dynamics.For hyperbolic cones, they note preliminary passive ground states with apex-nucleated neutral pairs; large-activity behavior is left for future study.Source: §VIII · p. 14
- Conical droplets, cell monolayers, and living liquid crystals are proposed as experimental settings for probing the predicted defect phase diagram and dynamics.The discussion suggests varying cone sharpness and activity to examine defect emission, nucleation, and unbinding.Source: §VIII · pp. 14–15