Dynamical Crystallites of Active Chiral Particles

One of the intrinsic characteristics of far-from-equilibrium systems is the nonrelaxational nature of the system dynamics, which leads to novel properties that cannot be understood and described by conventional pathways based on thermodynamic potentials. Of particular interest are the formation and evolution of ordered patterns composed of active particles that exhibit collective behavior. Here we examine such a type of nonpotential active system, focusing on effects of coupling and competition between chiral particle self-propulsion and self-spinning. It leads to the transition between three bulk dynamical regimes dominated by collective translative motion, spinning-induced structural arrest, and dynamical frustration. In addition, a persistently dynamical state of self-rotating crystallites is identified as a result of a localized-delocalized transition induced by the crystal-melt interface. The mechanism for the breaking of localized bulk states can also be utilized to achieve self-shearing or self-flow of active crystalline layers.

Although much effort has been devoted to investigating either of these two mechanisms of self-propulsion and self-spinning, effects of their mutual coupling are much less explored.Also less understood is the corresponding crystallization process, for which a statistical continuum description that can access large length and time scales much beyond the restrictions encountered in discrete particle-based approaches, has still been lacking.Here, by introducing a continuum density-field description that is nonpotential and nonvariational, we show that the coupling and competition between self-propulsion and selfspinning result in a surprisingly rich behavior of nonrelaxational dynamical crystallized states.They feature both translational and rotational collective motion, governed by persistent dynamics.Two types of transition for active crystalline patterns are identified, i.e., bulk traveling-localization and interfacial localized-delocalized transitions, each mediated by a crossover regime showing dynamical frustration of active chiral particles.Of particular interest is the effect of the controlled crystal-melt interfaces, leading to an emergent state of self-rotating crystallites embedded in a homogeneous active melt, or self-shearing or wriggling flow of crystalline layers.
The active system here is described by a local particle density variation field ψ and a local polar particle orientation field P, the dynamics of which are governed by where , together with the average density ψ 0 , controls the transition between homogeneous (liquid) and crystalline phases, v 0 is the self-propulsion strength, D r is the rotational diffusion constant, and M = M ẑ represents the strength of self-spinning caused by an active torque.When M = 0 we recover the previous active phase field crystal (PFC) description [19,20].
Here we focus on C 1 > 0, counteracting any spontaneous polarization by e.g., rotational diffusion, to preclude any explicit alignment interactions and be consistent with related experiments.Our description can be derived from a microscopic particle-based formulation and dynamical density functional theory (DDFT) [49].All the model parameters are rescaled and dimensionless, giving a diffusion time scale and a spatial scale set by the periodicity of the ordered phase with q 0 = 1.
It is noteworthy that Eqs. ( 1) and ( 2) could be reduced to a gradient, relaxational form only when v 0 = M = 0 for passive particles, giving ∂ t ψ = ∇ 2 δF/δψ and 4}, a combination of the PFC free energy [84][85][86][87] and a Landau expansion of the polarization field.Thus for any active system with nonzero v 0 , the corresponding system dynamics is nonrelaxational, i.e., does not follow the minimization of F. We examine this nonpotential system through a series of numerical simulations.Each starts either from a homogeneous state with random initial conditions or from initially small crystalline nuclei, with system sizes ranging from 128 × 128 to 2048 × 2048 grid points (around 230 to 60, 000 density peaks) subjected to periodic boundary conditions.The system parameters are chosen as ( , g, D r , C 1 , C 4 ) = (−0.98,0, 0.5, 0.2, 0), while values of M , v 0 , and ψ 0 are varied to control the competition between self-propulsion and self-spinning.
Our simulations indicate that in the bulk state of active crystals three characteristic regimes of system dynamics can be identified.As shown in Fig. 1(a), the propelling-spinning competition leads to a sharp transition between a unidirectionally traveling ordered state driven by particle self-propulsion at large enough v 0 and small M [Fig.1(b)] and a localized or arrested crystalline state with vanishing velocity v of each density peak at large enough M [Fig.1(d)].The effect of active torque causes the self-spinning or localized self-circling of individual chiral particles and hence the localization of each density peak, which is consistent with previous DDFT results for noncrystallized states [88].It thus induces the arrest of the whole pattern as observed here.This traveling-localization transition can be accompanied by phase transformations between ordered structures as induced by particle self-spinning when M increases.Examples include transformations from a traveling rhombic or distorted-hexagonal structure to a localized hexagonal phase [Figs.1(b)-1(d)], or from a traveling-square to traveling-rhombic to localized-hexagonal structures at large active drive v 0 (Fig. 2).
In a narrow crossover regime near the transition threshold (M ∼ M c ) the incompatibility between two dynamical effects dominated by self-propelled translation and spinning-induced localization becomes explicit.When these two competing dynamics are of similar degree, none of the corresponding optimal collective behaviors can be achieved, leading to the local dynamic frustration of active particles [see Fig. 1(c)] or a wavy, swinging translative motion of the whole crystalline pattern characterized by alternative regions of density peaks traveling at varying directions (see Supplemental Movie S1).The balancing of translation and localization also leads to a phenomenon of migrating crystallites.As shown in Fig. 1(e) and Movie S2, during the evolution and coarsening of faceted crystallites or grains, the density peaks are localized within each grain while the whole crystallites travel within the coexisting medium of homogeneous melt, impinging and coalescing with each other.
The above results can be further understood by rewriting Eq. ( 2) in terms of a local divergence field S = ∇ • P and the self-spinning field Ω = (∇ × P) z /2; for C 4 = 0, Equation ( 4) indicates that S serves as an effective source for nonzero Ω in the steady state, generating particle selfspinning locally, as confirmed numerically in the inset of Fig. 1(a).Nonzero S also drives the propagation of density patterns when entering Eq. ( 1), while its own source is in turn provided by the variation of the density field [see v 0 ∇ 2 ψ in Eq. ( 3)].Simultaneously, the term M Ω in Eq. ( 3) causes the damping of S and plays the role of an inhibitor that hinders the particle migration, leading to the effect of localization observed in the simulations.
Given the linear form of Eqs. ( 3) and ( 4), it is straightforward to express the Fourier components of S and Ω in terms of those of density ψ in the nonequilibrium steady state with a constant pattern migration speed v m .For the hexagonal phase, in one-mode approximation ; otherwise v m = 0.This analytic result indicates that there exists a critical threshold M c (or v 0c ); when |M | > M c (or v 0 < v 0c ) the active crystal is localized with v m = 0. q0 is the selected wave number of the ordered pattern, difficult to be determined analytically for a nonpotential system [1][2][3].Our simulations indicate that q0 is in the vicinity of q 0 , an approximation used in evaluating Eq. ( 5) as presented in Fig. 1(a) without any parameter fitting.We find a reasonably good agreement between analytic and numerical results for v m , with the deviations attributed to the employed one-mode approximation.
Remarkably, the effect of chiral particle self-spinning, which leads to the bulk structure localization identified above, can be utilized to generate a further localizeddelocalized transition in the interfacial state and, consequently, a phenomenon of crystallite self-rotation or selfshearing.The crystal-melt interface is set up through a sufficiently steep gradient of the "temperature"-type parameters and C 1 (corresponding to the spatial variation of, e.g., chemicals or heat sources controlling homogeneous vs. crystalline phases in experiments).Our obtained results are robust against specific realizations of the setup, as verified in simulations.Below, we present results for a kink-type two-phase profile with the same form for C 1 .This represents a circular cavity of radius r 0 , enclosing a crystallite embedded in an outside coexisting homogeneous medium of active melt.Similar kink-type profiles can be set up for other interfacial geometries.Here we set ( in , out , C in 1 , C out 1 ) = (−0.98,0, 0.2, 1) and ∆ around 1 to 5 grid spacings.
Simulation results for a circular cavity are presented in Fig. 3, showing three regimes of crystallite self-motion: (i) self-translation dominated, (ii) self-rotation dominated, and (iii) the transition between them, as a result of the competition between self-propelled translative particle motion, interface-induced tangential motion of density peaks, and localization through particle selfspinning.In regime (i) for small M , high-density blocks constantly crystallize from the active melt at one side of the cavity, propagate across it, and remelt into the homogeneous medium at the other side, as seen in Fig. 3(b) and Movie S3.At the same time the whole crystallite still rotates slowly, with small but nonzero averaged an-gular velocity |ω| [estimated as the orientation change rate of v , given the dominance of translative motion, with the center of rotation located outside the crystallite; see Fig. 3(a)].The maximum magnitude of the average translational velocity | v | (among realizations of different ψ 0 ) decreases with increasing M , due to stronger localization through particle self-spinning.At large enough M [regime (ii)], persistent self-rotation of the faceted crystallite about the cavity center is observed, as illustrated in Fig. 3(c) and Movie S4.The direction of crystallite rotation (clockwise) is opposite to that of individual particle self-spinning (counterclockwise).The maximum averaged rotation rate |ω| (here |ω| = |v − v |/r) is reduced at larger M with enhanced bulk localization.
This phenomenon of self-rotating crystallites can be understood by examining the spatial variation of the polarization field P. At the cavity boundary the average polarization of a boundary density peak at the melt side is weaker than at the inner crystalline side, generating a local spatial gradient of P (and Ω) that is suppressed asymmetrically by its vicinity to the interface.Thus, the active drive at the inner side of each boundary peak dominates over that of its outer side, causing the corresponding self-flow of the interfacial crystalline layer.It subsequently overcomes the localization of (i.e., delocalizes) interior particles via collective dynamics and drives the overall self-rotation of the crystallite.The direction of self-rotation follows the orientation of the net polarization at the inner side of interfacial peaks and thus, interestingly, is opposite to that determined by the chirality of individual self-spinning.When the sign of M is reversed, both directions of self-spinning and self-rotation are reversed.The mechanism here is different from that underlying the rotation or edge flows of active spinners found in previous studies of no-flux rigid boundary walls; there the chirality of the boundary/edge flow is the same as that of the individual spinners or rotors [24][25][26][27][28][29].
In the transitional regime the crystallite shows doubledegenerate behaviors, one dominated by self-translation and the other by self-rotation, as depicted in Figs.3(d) and 3(e), respectively.In the latter, although the overall crystallite self-rotates as driven by the interfacial layer, some inner layers exhibit local frustration and even intermittent inverse rotation [leading to regions of local selfshearing; see Fig. 3(e) and Movie S5].This reflects the competition among self-propulsion, self-spinning-induced localization, and interface-induced delocalization.
The dynamical regimes identified for both bulk and circularly interfacial systems are summarized in Fig. 4, in terms of M -v 0 state diagrams obtained from simulations across different values of ψ 0 that lead to hexagonal crystallized states.Major parts of the M -v 0 space are occupied by the self-translation-dominated state and the states of self-spinning-induced bulk localization [Fig.4 namical frustration in both cases, is broader in Fig. 4(b) that involves crystal-melt interfaces.
The interface-induced driving mechanism of crystalline layers should apply to any geometry of nonrigid (soft) crystal-liquid boundaries or edges.In the example of a slab crystallite [Fig.5(a)], the top and bottom interfacial layers are expected to be driven towards opposite directions due to their inverse crystal-to-liquid interface orientations, leading to self-shearing of the crystallite as verified in our simulations [see Fig. 5(a) and Movie S6].Conversely, in the transitional regime local frustration of density peaks occurs as a consequence of the comparable strengths of self-propulsion and self-spinning, as seen in Fig. 5(b).This leads to either fluctuating modes of traveling crystalline layers (Movie S7) or even a snaking/worming type of layer flow (Movie S8).
Finally, similar consequences of opposing interfacial orientations can be manifested in a ring-like configuration, as shown in Fig. 5(c) and Movie S9.In many cases, although the whole ring-shaped crystallite still rotates collectively as driven by the outermost circular layer with longest perimeter [Fig.5(c)], its angular velocity is significantly reduced due to the hindrance by the counteracting drive of the innermost annulus.The scenario of self-shearing occurs when the two interfacial drives are of comparable strength, for which the outer and inner crystalline layers rotate to opposite directions (Movie S9).
In summary, we have analyzed and predicted the collective behavior of spatially ordered structures featuring both active propulsion and active rotation.The interplay between individual particle self-propulsion and selfspinning during crystallization results in various novel states of collective and persistent dynamics that are enabled by the nonrelaxational nature of the active system.The competition leads to a traveling-frustrationlocalization transition in active crystals with increasing strength of self-spinning, which also induces a transformation between ordered phases as a result of pattern selection.A breaking of the localization and structural arrest occurs for interfacial states, revealing persistently dynamical states of self-rotating crystallites and self-shearing or self-flowing crystalline layers.The direction of crystallite self-rotation or layer propagation is opposite to that given by the chirality of the individual self-spinning particles, an effect caused by the crystalmelt interface-induced spatial variation of local polarization and the subsequent edge-originated delocalization and collective motion of active particles.
These predictions open new possibilities to explore the emergence of novel dynamical phenomena and unveil complex mechanisms underlying a wide variety of nonequilibrium active systems governed by persistent, nonrelaxational dynamics.Although here we focused on a dry environment, the results and mechanisms identified above will in many cases still apply to leading order in additional fluid surroundings [49].It should even be possible to disentangle the effects of both environments in an experiment when surrounding granular or colloidal spinners by a viscous fluid with varying viscosity for the tuning of hydrodynamic couplings.Our results can be realized and verified in various experimental setups, such as a collective of granular rotors [25,28,89], light-controlled anisotropic colloidal Janus particles [90] and colloidal molecules [91], or self-propelled particles equipped with magnetic dipole moments [92,93] to perform active spinning under a magnetic field.

Dynamical Crystallites of Active Chiral Particles -Supplemental Material
Zhi-Feng Huang, 1 Andreas M. Menzel, 2 and Hartmut Löwen  In this supplemental material, we provide a more microscopic basis of the active phase field crystal (PFC) description introduced in Eqs. ( 1) and (2) of the main text.Frequently, these equations are directly used as an input to corresponding evaluations.However, a derivation and motivation from a microscopic picture is possible, as we summarize below.We start from a particle-based picture of active microscopic objects that besides pure translational self-propulsion feature self-spinning as well.It is demonstrated how Eqs. ( 1) and ( 2) of the main text follow from the corresponding statistical continuum representation of the discrete microscopic picture under suitable rescaling.In addition, effects of hydrodynamic interactions on the active PFC description are briefly discussed.

Details on microscopic derivation of the active PFC description
We start from a system of N microscopic spherical active particles located at positions r 1 (t), ..., r N (t) and featuring marked orientations given by unit vectors û1 (t), ..., ûN (t), where t denotes time.Considering overdamped dynamics of the Langevin type, the time evolution of the positions and orientations of the particles is determined by their deterministic velocities v det,i and angular velocities ω det,i that can be calculated from the deterministic forces and torques acting on them, plus the influence of stochastic translational noise ξ tr,i and stochastic orientational noise ξ rot,i acting on each particle (i = 1, ..., N ).The stochastic noises are typically viewed as thermal in origin, delta-correlated in time, and Gaussian in distribution.Originating from thermal equilibrium fluctuations, the fluctuation-dissipation relation sets its strength [1].During an infinitesimal time step dt, the positional and orientational changes are governed by Because of the dynamics of the individual particles, the probability density P(r 1 , ..., r N , û1 , ..., ûN , t) for finding the system at a time t in a certain configuration r 1 , ..., r N , û1 , ..., ûN evolves with time.Through textbook procedures [2,3], by averaging over the noise, Eqs. ( 1) and ( 2) can be transformed into a statistical conservation equation for the time evolution of P. Since we consider overdamped dynamics, the time evolution of P is solely driven by the velocities v i and angular velocities ω i of the particles (with i = 1, ..., N ).Along these lines, the dynamics of P thus follows the conservation equation [4] Both the stochastic noise acting on the particles as well as the deterministic forces and torques control the particle velocities v i and angular velocities ω i (i = 1, ..., N ).Here we consider deterministic potential interactions between the particles, deriving from the overall potential U (r 1 , ..., r N , û1 , ..., ûN ).Since we are working towards the probabilistic conservation equation for P in Eq. ( 3), instead of explicitly working out the influence of stochastic noises in Eqs. ( 1) and ( 2), an equally valid approach at this stage is to supplement U by an entropic potential k B T ln P [4], with k B denoting the Boltzmann constant and T the absolute temperature.This entropic potential implies a diffusional behavior as obtained by explicitly averaging the noise terms in Eqs. ( 1) and (2).Considering active particles self-propelling and self-spinning on a substrate, the resulting forces and torques acting on the particles are given by with i = 1, ..., N .Here F a is the strength of the active self-propulsion force driving particle i along its orientation ûi , and T a i is the active torque driving the selfspinning of each particle i.Both F a and T a are assumed to be constant and identical for all the particles in the present investigation.
Denoting by µ t and µ r the translational and rotational mobilities, respectively, the velocity and angular velocity of particle i are given by When substituting Eqs. ( 4)-( 7) into Eq.( 3), we find arXiv:2010.01252v1[cond-mat.soft]3 Oct 2020 Following the standard procedure and assuming identical particles, from the above equation we can then identify the dynamic equations for the n-particle densities ρ In the following we confine ourselves to the twodimensional (2D) subspace that is defined by the planar substrate on which the active particles are self-propelling and self-spinning.In principle, most of the steps described below follow analogously in three dimensions, but the terms involving the rotational operator ûi × ∇ ûi can be substantially simplified in two dimensions.Denoting by ẑ the normal to the 2D plane and within this plane using the angle ϕ i to represent the orientation ûi of the ith particle measured from a fixed axis, we have ûi × ∇ ûi = ẑ ∂ ϕi .Moreover, for particles self-spinning within the plane we set T a = T a ẑ, with T a the constant magnitude of the active torque.
Next, applying the operation defined by Eq. ( 9) to Eq. ( 8) and denoting ρ (1) (r 1 , û1 , t) as ρ(r, û, t), we find the dynamic equation for the one-particle density ρ as where we have abbreviated and split U into the potential V for mutual particle interactions and the possible influence of an external potential Φ ext .If V only includes pairwise interactions, U reads and P∇V = dr dû ρ (2) (r, r , û, û )∇V (r, r , û, û ), ( 14) As a central step, we need to find a closure for these expressions in terms of ρ.One strategy is provided by dynamical density functional theory (DDFT) [5][6][7], which has been successfully developed and evaluated to describe, amongst many others, the dynamics of selfpropelled particles [8][9][10][11][12].It applies exact equilibrium relations for the density to the instant state of the system density at each moment of time t during the nonequilibrium dynamics.This adiabatic approximation is supported by the overdamped dynamics of the system.Here we provide a short summary of the procedure.
Classical density functional theory exploits that the grand potential Ω in equilibrium shows an extremum as a functional of the density [13,14], Covering the effect of a possible chemical potential by the external potential, this implies where F id = k B T dr dû ρ ln(λ 2 ρ) − 1 is the free energy associated with noninteracting particles (i.e., the contribution from entropy) and λ denotes the thermal de Broglie wavelength.F exc accounts for the additional contributions to the free energy resulting from interparticle interactions, while F ext = dr dû ρ Φ ext includes into the free energy the effect of the external potential.Thus, we get from Eq. ( 17) Simultaneously, in equilibrium ∂ρ/∂t = 0, F a = 0, and T a = 0 in Eq. (10).Thus, the remaining translational and rotational parts in the first and second lines on the right-hand side of Eq. ( 10) need to vanish (separately).Together with Eq. ( 18), we obtain We remark that a similar functional form can be found if we apply the mean-field approximation ρ (2) (r, r , û, û ) = ρ(r, û)ρ(r , û ) in Eqs. ( 14) and (15).Then, the role of the expression δF exc /δρ in Eqs.(19) and ( 20) is taken by the mean-field potential dr dû ρ(r , û )V (r, r , û, û ).(21) Using Eqs. ( 19) and (20) in Eq. ( 10), we have derived the conservation equation for the one-particle density in terms of DDFT.Without taking into account the role of an external potential as in the main text, i.e., setting Φ ext = 0 from now on, which leads to F = F id + F exc , this fundamental DDFT equation reads We now perform an expansion of the density ρ with respect to its orientational dependence [15][16][17], ρ(r, û, t) = ρ + ρ φ1 (r, t) + ρ û • P(r, t) + ..., (23) and use this expansion to express Eq. ( 22) in terms of φ1 and P, where we apply the approximation of constant mobility.ρ is the average density.Moreover, we introduce the translational diffusion constant D = k B T µ t , the rotational diffusion constant Dr = k B T µ r , and the effective velocity of self-propulsion v = µ t F a .Additionally, we determine via integrations dû and dû û the zeroth-and first-order orientational moments of Eq. ( 22) to find the dynamic equations for φ1 and P. Defining β = 1/k B T as usual, this guides us to In the last term, for compact notation we implicitly involve the direction ẑ normal to the plane confining the particles.More precisely, here we have defined the effective angular velocity of self-spinning M = µ r T a π ẑ.At this stage, our theory would be complete.However, exact expressions for F exc that contributes to F = F id + F exc in Eqs. ( 24) and ( 25) are hardly known, except for some special model cases.Therefore, approximations for F are necessary.We split the free energy functional into a purely φ-dependent part Fpfc (where φ = φ 0 + ρ φ1 relates to the spatial density variations) and a purely Pdependent part F P, i.e., F = Fpfc + F P. First, for the φ-dependent part we choose the well-established phase field crystal (PFC) description [18][19][20] where a, λ, g, and u set the magnitudes of the corresponding energetic contributions, the expressions of which have been derived from either classical DFT [20] or DDFT [21,22] in terms of microscopic direct correlation functions.∆T in general is a measure for the temperature difference from the melting point, while 1/q 0 is related to the length scale of the crystalline-type structures that emerge, the properties of which have been studied within the PFC description.The PFC framework has previously been supported by comparison with DDFT solutions [21] and molecular dynamics simulations [23].We remark that in principle the cubic term gφ 3 /3 in Eq. ( 26) can be effectively removed by choosing a specific reference point for φ, while we still keep this term in the present formulation for completeness.(In our calculations in the main text, the coefficient g is set to zero.) Second, for the P-dependent part we include as analyzed in theories of flocking [24,25].Again, C1 and C4 quantify the magnitudes of the corresponding energetic contributions, for which likewise expressions in terms of microscopic correlation functions are available [15][16][17]26].A combination of C1 < 0 and C4 > 0 implies spontaneous polarization supporting collective motion.It could arise on the microscopic level, for instance, from a pairwise alignment potential V or (û i , ûj ) [27,28], if in Eq. ( 13) we split the pairwise interaction potential V into a purely positional and a purely orientational part, i.e., V (r i , r j , ûi , ûj ) = V pos (r i , r j ) + V or (û i , ûj ).This type of splitting is reasonable, for example, for spherical particles.In the present work, we do not consider predominant alignment interactions and instead focus on the regime C1 > 0. If only concentrating on the lowest-order contribution in Eq. ( 27), we may therefore set C4 = 0 as adopted in the simulations described in the main text.
The orientational decorrelation associated with C1 > 0 results on the microscopic particle level, for example, from basic rotational diffusion.
Finally, the expressions in Eqs. ( 26) and ( 27) are inserted via F = Fpfc + F P into Eqs.( 24) and (25).Next, the functional derivatives are evaluated, followed by two sequences of rescaling.In the first step, the rescalings set in Ref. [19] are applied, i.e., together with a rescaling of length scale by 1/q 0 .In the second step, we apply together with a rescaling of time by 2π/β D ρ λq 6 0 .As a result, we finally obtain Eqs. ( 1) and ( 2) of the main text governing the rescaled density variation field ψ and the rescaled polarization field P. We remark that in fact the inverse length scale q0 is scaled out by the above procedure.Nevertheless, to be able to refer to it, we maintain it as q 0 = 1 in Eq. ( 1) of the main text.

Remarks on hydrodynamic interactions
Recently, we have considered systems of microscopic particles suspended in a surrounding fluid under low-Reynolds-number conditions.In such a situation, hydrodynamic interactions [29,30] mediated between the parti-cles through the induced hydrodynamic fluid flows make additional contributions to the particle dynamics.Especially, the thermal noises ξ tr,i and ξ rot,i become coupled through the hydrodynamic background as well.Their strengths now depend on the particle configurations [31][32][33] and need to be evaluated at each time step, which is a tedious procedure when evaluating the corresponding Langevin equations.It thus becomes more favorable to target at a statistical equation for P, similar to the alternative procedure mentioned above, by including an entropic potential k B T ln P instead of directly working with the stochastic noise [4].
When including hydrodynamic interactions, the translational velocity v i and angular velocity ω i of the ith particle now read instead of Eqs. ( 6) and (7).Here the hydrodynamic mobility matrices µ tt ij , µ tr ij , µ rt ij , and µ rr ij quantify the translation-translation, translation-rotation, rotationtranslation, and rotation-rotation hydrodynamic couplings between particles i and j [29,30].In our case, the force F j and torque T j on the jth particle are still determined by Eqs. ( 4) and (5).Substituting Eqs. ( 30 which includes the hydrodynamic interactions via the mobility matrices. In the bulk fluid, denoting the distance between particles i and j as r ij = |r i − r j |, the hydrodynamic mobility matrices in the far-field approximation scale as µ tt i=j , µ rr i=j ∼ (1/r ij ) 0 = 1, µ tt i =j ∼ 1/r ij , µ rt ij , µ tr ij ∼ (1/r ij ) 2 , and µ rr i =j ∼ (1/r ij ) 3 for spherical particles [29,30].Thus, inserting µ tt i=j = µ t 1 and µ rr i=j = µ r 1 [29,30], with 1 representing the unit matrix, and focusing only on the zeroth-order effects, Eq. ( 32) is reduced to Eq. ( 8) and the above results for the active PFC description analyzed in the main text can be recovered.In principle, such a situation could be realized experimentally in a bulk fluid, for instance, by tracking the positions of colloidal particles in real time [34,35] and driving accord-ingly the particles by optical tweezers [36,37].In such a setup, the deterministic trajectories resulting from both self-propulsion and self-spinning as well as from interparticle interactions would need to be programmed into the optical gearing of the colloidal particles, plus the effect of positional and orientational noises.Instead, in this work we mainly focus on the study of self-propelling and self-spinning active particles confined on a 2D substrate.Generally, if such a system is embedded in a surrounding fluid, the substrate can significantly limit the influence of the above-mentioned hydrodynamic interactions, particularly for hydrodynamic no-slip surfaces.It is thus expected that the results presented in the main text will in many cases still apply in a fluid environment.

FIG. 1 .
FIG. 1.(a) Magnitude of average density-peak velocity | v | as a function of M , for ψ0 = −0.3 and v0 = 0.35, 0.4, and 0.5.Dashed curves represent analytic results of Eq. (5) assuming q0 q0 = 1.Inset: |Ω| measuring particle self-spinning.(b)-(d) Snapshots of parts of the ψ profiles in three characteristic regimes, with the density peaks shown in red, their velocities v indicated by large arrows, and the polarization field P by fine arrows.(e) Snapshot of migrating crystallites in an active melt (see Movie S2).Inset: Enlarged portion of a two-grain impinged corner, with a penta-hepta dislocation.

FIG. 3 .
FIG. 3. (a) Averaged translational and rotational velocities (| v | and |ω| at v0 = 0.5) as a function of M , containing translation-and rotation-dominated regimes for crystallites in a circular cavity and a transitional regime (shaded symbols) enlarged in the inset.(b)-(e) Snapshots of crystallite patterns simulated [as marked in (a)], with v vectors indicated.
FIG. 4. Dynamical state diagrams in the M -v0 space for three dynamical regimes in (a) bulk and (b) interfacial systems.