Chiral crystals self-knead into whorls
Ephraim S. Bililign, Florencio Balboa Usabiaga, Yehuda A. Ganan, Vishal Soni, Sofia Magkiriadou, Michael J. Shelley, Denis Bartolo, William T. M. Irvine
CChiral crystals self-knead into whorls
Ephraim S. Bililign, Florencio Balboa Usabiaga, Yehuda A. Ganan, Vishal Soni, SofiaMagkiriadou, Michael J. Shelley,
2, 3, ∗ Denis Bartolo, † and William T. M. Irvine ‡ James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA Center for Computational Biology, Flatiron Institute, New York, NY 10010 Courant Institute, NYU, New York, NY 10012 University of Lyon, ENS de Lyon, University of Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France James Franck Institute, Enrico Fermi Institute and Department of Physics, University of Chicago, Chicago, IL 60637, USA
The competition between thermal fluctuations and potential forces is the foundation of our under-standing of phase transitions and matter in equilibrium. Driving matter out of equilibrium allowsfor a new class of interactions which are neither attractive nor repulsive but transverse. The ex-istence of such transverse forces immediately raises the question of how they interfere with basicprinciples of material self-organization. Despite a recent surge of interest, this question remainsopen [1–9]. Here, we show that activating transverse forces by homogeneous rotation of colloidalunits generically turns otherwise quiescent solids into a crystal whorl state dynamically shaped byself-propelled dislocations. Simulations of both a minimal model and fully resolved hydrodynamicsestablish the generic nature of the chaotic dynamics of these self-kneading polycrystals. Using acontinuum theory, we explain how odd and Hall stresses conspire to destabilize chiral crystals fromwithin. This chiral instability produces dislocations that are unbound by their self-propulsion. Theirproliferation eventually leads to a crystalline whorl state out of reach of equilibrium matter.
The celebrated interplay between configurational en-tropy and the energetics of topological defects in two-dimensional melting have provided a lens through whichto understand the phases of a plethora of condensed mat-ter systems [10], including superfluid films, colloids andliquid crystals [11–16]. While these material systemsspan a wide range of particle interactions, scales, and in-termediate phases, they are all unified in that the forcesbetween constituents are primarily longitudinal, their dy-namics are equilibrium, and their interactions are sym-metric under both time-reversal and parity. What hap-pens if the inter-particle interactions include transverseforces as well? This deceivingly minimal generalizationcan break these assumptions at a fundamental level.In equilibrium, Boltzmann statistics guarantee thattransverse forces cannot alter the phase behavior of con-densed matter. However, there is no such guarantee outof equilibrium and such transverse interactions generi-cally occur in collections of naturally and artificially spin-ning objects. Examples include planetary disks [17], spin-ning cell aggregates and membrane inclusions [18, 19], ac-tive colloids and grains [4, 8, 9, 20–24], atmospheric scaledynamics [25, 26], parity-breaking fluids [27–32] and sim-ple models of turbulence [33]. How does this more generaland ubiquitous form of matter generically self-organize,what are its stable phases, and how does it transitionbetween them?Figure 1a and Supplementary Movie 1 show a ∼ × µm region within a centimetre-scale monolayer ofmagnetic colloids. Each particle is uniformly spun byan externally applied magnetic field, resulting in theirself-organization into a dynamic and dense phase. Theactive rotation of the magnets gives rise to both longi- tudinal magnetic attraction and sustained chiral trans-verse hydrodynamic interactions. Crucially, the forcesare generic, separation-dependent, and can be tuned byvarying the rotation frequency, providing an ideal plat-form for exploring how transverse interactions shape thedense phases of chiral matter.Upon spinning our particles, we find that the sys-tem generically self-organizes into crystal ‘whorls’. Asnapshot, colored by the phase of the crystalline bond-orientational order parameter, ψ ( x ) (Figure 1b), re-veals a poly-crystalline arrangement of grains of trian-gular crystal order separated by topological defects orga-nized into grain boundaries. This picture is reminiscentof metallurgical crystalline phases with quenched disor-der; however, unlike their crystalline static counterpart,the structure is continually evolving, grain boundariesmove, collapse, and spontaneously emerge as crystallinedomains rotate like vortical whorls (see Figure 1c andSupplementary Movie 2). Segmenting the phase intocrystalline domains (see Supplementary Information) en-ables us to study its statistical properties. As shown inFigure 1d-e (and Supplementary Information), after ashort transient, the domains within this poly-crystal set-tle to a constant characteristic size and the cumulativelength of grain boundaries reaches a constant value.What powers this lively steady state? Careful inspec-tion reveals that the motion of topological defects in thecrystalline structure is unlike the familiar motion of dis-locations found in conventional passive materials. Con-ventional dislocations are either stationary or diffuse bi-directionally driven by thermal fluctuations; we observeinstead that in our chiral medium they move ballisticallylike self-propelled particles as seen in Figure 2a and Sup- a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1.
A crystal whorl state. a,
A dense and dynamic phase of spinning colloids is directly imaged through a microscopebetween crossed-polarizers. The rotation-averaged position of each particle appears as a bright spot and reveals intermittentcrystalline order. b, To illuminate the polycrystalline structure of this phase, we color particles by the angle of the local bond-orientational order parameter θ . The polycrystal can be segmented into domains, and boundaries are drawn between themaccording to the network formed across non-crystalline interstices (see Supplementary Information). The inset highlights theindividual dislocations underlying a grain boundary. c, The polycrystal is dynamical and displays intermittent vortical flowsas revealed when each region is colored by its vorticity (See also Supplementary Movie 3). While this phase evolves over time,it ultimately finds a steady-state in which the structural measures characterizing polycrystallinity hold constant, including the d, average grain size ξ θ and e, total length of grain boundary. f, When averaged over sufficiently long timescales, the bulkflows vanish revealing the absence of long lived excitations. plementary Movie 4. This motion must originate in thechiral drive, and consideration of the local structure sur-rounding a dislocation readily suggests a mechanism: thedipole in density that accompanies a dislocation (see e.g.[34]) naturally gives rise to an imbalance in transverseforces. The resulting shearing stress is naturally relievedby moving the dislocation in the direction of its Burgersvector, resulting in sustained self-propulsion.To isolate this phenomenon, akin to a self-generated Peach-Koehler force [34–37], we performed both full hy-drodynamic simulations of particles that closely approx-imate our experimental system as well as of a minimalmodel of the overdamped dynamics of spinners interact-ing via transverse, frequency-dependent forces and po-tential longitudinal forces. The transverse forces mimicviscous friction, while the longitudinal interactions re-flect the combination of short-range steric repulsion andlong-range magnetic attraction (see Supplementary In-
FIG. 2.
Self-propelled dislocations driven by odd stress.
Dislocations in the chiral crystalline phase behave like activeparticles. a, In the experiment, dislocations are observed to move ballistically in the direction of their Burgers vector. Samecolormap as in Fig. 1b. b, This behavior is reproduced in simulations of both the full hydrodynamic and minimal models.The motility can be precisely tuned by initializing a single dislocation and sweeping the particle rotation frequency. In acrystal of particles interacting via transverse forces, the dislocation’s direction of motion can be intuited by examining thedensity dipole. c, The precise dislocation motility depends weakly on the details of the interactions. In both models the speedgenerally increases with frequency. d, By tuning the frequency and initial separation, two defects that would otherwise attractand annihilate can be made to repel, overwhelming even elastic forces with transverse ones. Same colormap as in Fig. 1b. e, The collective dynamics of many defects arranged to form a grain boundary inherits this sensitivity to transverse forces.Such a grain boundary collapses when elastic forces dominate, and expands without bound when transverse forces dominate.Same colormap as in Fig. 1b. The addition of odd forces to compete with the familiar even forces enables the tuning of thepolycrystalline structure of the whorl state. f, A uniaxial gaussian density profile is superimposed onto an otherwise uniformcrystal (left), permitting an interplay between odd stresses and density gradients. This results in defect production (right), andsubsequent proliferation of additional defects, destabilizing the crystal from within. g, The self-kneading of crystalline patchesof material leads to the steady state illustrated in Figure 1, which is characterized by h, an exponential distribution of grainsizes. i, Accordingly, the size of grains in the steady-state tends to decrease with frequency. formation).We are able to observe the motion of dislocations iso-lated from interactions with other defects by initializingsimulations with a single dislocation in an otherwise per-fectly ordered triangular crystal. As shown in Figure 2b-c, we find confirmation of a rotation-frequency depen-dent velocity with defect motion parallel to the Burgersvector (i.e. transverse to the density gradient). In ourfull simulations (see Supplementary Information), a fre-quency threshold to this motility is also observed (Fig-ure 2c), which is consistent with the notion that for mo-tion to occur the self-propulsion forces must exceed thelocal Peierls barriers [38, 39].This self propulsion provides a significant twist on thecollective dynamics of defects. For example two defectsthat would normally attract and annihilate in responseto elastic forces, can instead unbind when propelled bytransverse forces as shown in Figure 2d.The collective dynamics of several defects is similarlyaffected. Figure 2e shows snapshots from simulations inwhich we initialized a finite size grain in an otherwiseperfect crystal and varied the rotation frequency. Al-tering the frequency affects transverse interactions moststrongly and thus enables us to tune the balance betweenstabilizing elastic interactions and defect motility. Asshown in Figure 2e the grain size is set by the compe-tition between motility and defect interactions. At lowrotation frequencies the grains are stable, and their sizebecomes larger when the defect’s propulsion direction isoutwards. Similarly, they shrink and collapse for inwardsdefect motility. When the system is driven sufficientlystrongly, the defect motility overpowers the elastic inter-actions resulting in unstable grain boundaries.In this regime, the coupling between density variationsand transverse forces further conspires with gradients inthe density to make the crystalline phase unstable to dis-location unbinding. Figure 2f and Supplementary Movie5, show how an initial density gradient relaxes by pro-ducing defect pairs which go on to both self-propel andreact with each other. For strong rotational drives thedensity variations that surround and propel dislocationsfurther act as nucleation sites for additional defect pairproduction, powering a cascade to a steady state thatrapidly forgets the initial fluctuations that gave rise to it.Nonetheless, even in the chaotic steady state we observein both experiment and simulations, this balance betweeneven and odd forces remains the controlling parameterthat determines the characteristic size of the grain distri-bution as in Figure 2g-h (see also Supplementary Infor-mation). This sustained proliferation and annihilationof motile dislocations is reminiscent of active nematicswhere self-propelled disclinations power spatiotemporalchaos [40]. Here they give rise to self kneading crystalwhorls.Ever since the iconic footage by Bragg and Nye ofdislocations moving through bubble rafts [41], disloca- tions have been understood as the primary mechanismof plastic deformation in a crystal subject to externalloads. Here plastic flows are driven from within by theintrinsic motility of the dislocations. The resulting flowfield v ( x ) can be readily measured from individual par-ticle trajectories and its corresponding vorticity ω ( x i )is shown in Figure 1c. Averaging over sufficiently longtimescales, the dynamics of this chaotic internal flow av-erage to a quiet bulk (Figure 1f) and a lively edge [7].However our measurements reveal instantaneous dynam-ics shaped by unsteady vortical flows as illustrated inFigure 1c. The dynamical and structural pictures of thischiral whorl state are aligned. As seen in Figure 1b-c, thegrain boundaries support strongly localized flows havinga vorticity opposite to Ω. By contrast, the grains corre-spond to low positive vorticity, intermittently interruptedby isolated dislocations zipping through.This self-kneading of the crystal phase results in en-hanced mixing which can be qualitatively captured by ar-tificially tagging the colloids and watching them spread;see Figure 3a and supplementary Movie 6. As shown inFigure 3b, an artificially dyed blob spreads anisotropi-cally before disintegrating into separate blobs, hinting ata mechanism reminiscent of Richardson diffusion in tur-bulence. We investigate this quantitatively by trackingthe mean squared separation between pairs of particles inthe chiral phase. Figure 3c shows that pair separation issuper-diffusive above a separation that corresponds tothe characteristic grain size due to a punctuated mixof conventional diffusion within crystalline whorls andRichardson-like diffusion between them.Is the crystal whorl state generic? To approach thisquestion we perform a linear stability analysis of a proto-typical chiral material and find that chiral elastic solidsare intrinsically unstable. We consider the continuumdescription of a visco-elastic solid with a local displace-ment u ( r , t ) and local velocity v ( r , t ), that is allowed toexperience all stresses consistent with broken parity andtime-reversal [7, 42].The symmetric stress includes even and odd elastic andviscous contributions σ ij = pδ ij + K ijkl ∂ k u (cid:96) + η ijkl ∂ k v (cid:96) .In addition, the inner drive imposed by the spinners isencoded by an antisymmetric stress σ spin ij = 2 η R (cid:15) ij Ω. Ig-noring inertia, the dynamics is given by the balance be-tween viscoelastic stresses and substrate drag defined bya friction coefficient Γ:Γ v = ∇ · σ + ∇ · σ spin , (1)and the continuity equation ∂ t ρ + ∇ · ( ρ v ) = 0.Linearizing about a homogeneous quiescent base state u = v = 0 and ρ = ρ , making the ansatz u , v , ρ ∝ exp ( − iωt + i k · r ), we readily obtain an expression forthe dispersion ν ≡ Re( ω ) and damping α ≡ Im( ω ) ofdisplacement and density waves as detailed in the Sup-plementary Information. We find a generic exponential FIG. 3.
Transport in a self-kneading crystal. a,
The constant structural kneading of the chiral whorl state by topo-logical defects introduces novel mixing properties which can be imaged by artificially dying stratified layers in a crystal thatsubsequently bleed into each other. b, By contrast to conventional diffusive processes, the smearing of the fluid over time is astrongly anisotropic process, wherein an example blob of fluid is pulled apart by the flow between two chiral whorls. c, Thepairwise separation ( δx ) plotted versus time t for particles initially in close proximity suggests that this abnormal spreadinggives rise to superdiffusive behavior, which itself is a function of rotational frequency. amplification of density fluctuations. A number of dif-ferent combinations of off-diagonal material parametersin η and K result in different instabilities. However,they all reflect the same mechanistic picture sketched inFigure 4a: A density fluctuation is converted to a local-ized rotation. The resulting net shear across the densityfluctuation is in turn converted into an outward forceamplifying the initial perturbation and so on to insta-bility. Competition with the stabilizing influence of theconventional bulk and shear moduli of the elastic soliddetermine the shape of the dispersion curve. Crucially,this generic mechanism relies on the coupling betweenstresses and strains having different spatial symmetries,which is only allowed when time reversal, and, or paritysymmetries are broken at the microscopic level [42, 43].To test this simplified model, we measure the flows v ( x ) that occur in the bulk of a large crystallite driven atfinite rotation frequency, and measure the Fourier spec-trum of all scalar measures of deformation, including vor-ticity, dilation and shears. Figure 4b shows that follow-ing the onset of rotations the azimuthally averaged spec-tra all change before eventually settling into a steadystate. By comparing the spectra (see Supplementary In-formation) at different times, we extract the mode growthcurves shown in Figure 4d-e. They reveal the presenceof an instability at finite wavelength for all modes withrelatively constant growth above a characteristic scale.As shown in Figure 4c, a spatial map of the integratedgrowth rate reveals that this instability is at the originof the destabilization of our chiral crystal.The predicted growth rate, plotted in Figure 4f for es-timated experimental hydrodynamic and elastic parame-ters, demonstrates excellent agreement with the exper- imental growth rate measurements for these alternatedynamical and structural variables, demonstrating theinterplay between the onset of crystalline disorder andgradients in the resulting flow. Further, we achieve qual-itative agreement with the analogous growth rates mea-sured from both hydrodynamic and minimal model sim-ulations, which are initialized as uniform crystals withvarying levels of noise (see Supplementary Information),demonstrating the universality of our continuum theoryto this generic class of materials.Unlike in equilibrium matter where crystals are stableuntil entropic forces overcome the cohesive elastic forcesbetween defects, chiral crystals are generically unstable.Their parity-odd response and stresses act together toamplify density fluctuations, unbind topological defects,and sustain their proliferation. This mechanism, rootedin chiral transverse forces at the microscopic scale, resultsin a new state of matter: a self-kneading crystal whorlstate. We have demonstrated the existence of genericprinciples that govern the stability and structure of two-dimensional isotropic chiral solids. Our work naturallyraises questions about the existence of fundamental prin-ciples that govern the crystallography and metallurgy ofnonequilibrium chiral matter in two and three dimen-sions. Acknowledgments.
We would like to acknowledge dis-cussions with P. Wiegmann, A. Abanov, C. Scheibner,M. Han, V. Vitelli, M. Fruchart. This work was pri-marily supported by the University of Chicago Materi-als Research Science and Engineering Center, which isfunded by the National Science Foundation under awardnumber DMR2011854. Additional support was providedby NSF DMR1905974, NSF EFRI NewLAW 1741685
FIG. 4.
Measuring an elasto-hydrodynamic instability. a,
The chiral phase can respond to a perturbation in spinnerdensity in two ways: (1) the crystal can be stabilized by elasticity to restore order; (2) density gradients can be converted torotations, producing flow gradients that are transformed into Hall stresses that destabilize the crystal. b, As the chiral phaseapproaches the steady-state of Figure 1, we can measure the spectrum of its flow to see decay at large scales and growth atsmall scales. c, The total amount of growth measured at these smaller scales can be mapped over time and compared to thegrain boundaries to reveal enhanced growth within initially crystalline patches and vanishing growth at later times. d, Theestimated spectrum of growth ˆ α ( q ) reveals stabilization at scales λ (cid:29) (cid:104) ξ θ (cid:105) and constant destabilization at scales λ (cid:28) (cid:104) ξ θ (cid:105) ,which is in excellent agreement with the result of a linear stability calculation of this system. e, Computing the growth ratefor alternative scalar decompositions of the strain rate tensor reveals uniform growth across dilation, rotation, and shear andqualitative agreement with theory in all cases. f, By estimating the hydrodynamic and elastic parameters of the experimentalsystem, we predict the growth rate and demonstrate stability in the absence of either an odd response or a coupling betweendilation and rotation. and the Packard Foundation. M.J.S. acknowledges thesupport from NSF grants DMR-1420073 (NYU-MRSEC)and DMS-1463962. D. B. acknowledges the support fromARN grant WTF and IdexLyon Tore. E.S.B. was sup-ported by the National Science Foundation GraduateResearch Fellowship under Grant No. 1746045. D.B.and W.T.M.I. gratefully acknowledge the Chicago-FranceFACCTS programme. The Chicago MRSEC (US NSFgrant DMR2011854) is also gratefully acknowledged foraccess to its shared experimental facilities. F.B.U. ac-knowledges support from ”la Caixa” Foundation (ID100010434), fellowship LCF/BQ/PI20/11760014, andfrom the European Union’s Horizon 2020 research and in-novation programme under the Marie Sk(cid:32)lodowska-Curiegrant agreement No 847648. M.J.S. acknowledges the support from NSF grant DMR-1420073 (NYU-MRSEC). ∗ [email protected] † [email protected] ‡ [email protected][1] Cafiero, R., Luding, S. & Herrmann, H. J. Rotationallydriven gas of inelastic rough spheres. EPL (EurophysicsLetters) , 854 (2002).[2] Grzybowski, B. A., Stone, H. A. & Whitesides, G. M.Dynamic self-assembly of magnetized, millimetre-sizedobjects rotating at a liquid-air interface. Nature ,1033–1036 (2000).[3] Yan, J., Bae, S. C. & Granick, S. Rotating crystals ofmagnetic Janus colloids.
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