Analytical Approach to Chiral Active Systems: Suppressed Phase

Analytical approach to chiral active systems: suppressed phase separation of interacting Brownian circle swimmers

Jens Bickmann a, Stephan Bröker a, Julian Jeggle a, and Raphael Wittkowski a∗ We consider chirality in active systems by exemplarily studying the phase behavior of planar systems of interacting Brownian circle swimmers with a spherical shape. Continuing previous work presented in [G.-J. Liao, S. H. L. Klapp, Soft Matter, 2018, 14, 7873-7882], we derive a predictive field theory that is able to describe the collective dynamics of circle swimmers. The theory yields a mapping between circle swimmers and noncircling active Brownian particles and predicts that the angular propulsion of the particles leads to a suppression of their motility-induced phase separation, being in line with previous simulation results. In addition, the theory provides analytical expressions for the spinodal corresponding to the onset of motility-induced phase separation and the associated critical point as well as for their dependence on the angular propulsion of the circle swimmers. We confirm our findings by Brownian dynamics simulations and an analysis of the collective dynamics using a weighted graph-theoretical network. The agreement between results from theory and simulation is found to be good.

1 Introduction bacteria 51 as well as spermatozoa 52. The latter’s particle inter- Active systems can show a variety of collective behaviors, includ- action at walls is known to play a crucial role in the fertilization 1,2 ing phenomena like shoaling and schooling of fish , flocking process 53. and swarming of birds 3–5, cell migration 6,7, swirling 8–10, lan- Individual circle swimmers were studied theoreti- ing 11–13, low-Reynolds-number turbulence 14–20, clustering 20–26, cally 39,47,48,50,54–58 as well as experimentally 35,41–45,47,51,53. crystallization 27,28, superfluidity 29, and even the emergence of There exist some works on alignment of circle swimmers via negative viscosities 30. The active agents can be macroscopic like anisotropic particle shapes 59,60 and an imposed alignment pedestrians 31,32 or animals 1–4 but also microscopic like bacteria mechanism as in the Vicsek model 5,61–64 as well as a descrip- or colloidal Janus particles 33–36. tion that models particle interaction via coupling to a field In the literature, usually particles with symmetric active propul- describing signalling molecules 65. The collective behavior of sion and thus no chirality or helicity are considered. However, circle swimmers, however, is not yet fully understood. A prime most systems are imperfect leading to, e.g., chirality in a two- example for the collective behavior of active colloidal particles is dimensional geometry 37,38. Since this generates a circular mo- motility-induced phase separation (MIPS) 26,66. MIPS is the effect tion, these particles are referred to as circle swimmers 39. They of forming a dense liquid-like phase of particles and a dilute are known already since 1901 from studies regarding microor- gas-like phase of particles out of a homogeneous distribution ganisms by Jennings 40. Chirality and chiral motion can origi- of repulsively interacting active particles. However, studies of nate from different types of asymmetry. A common example is a MIPS in systems of circle swimmers are rare. There exists a shape asymmetry leading to an effective propulsion torque, as it 67 arXiv:2010.05262v1 [cond-mat.soft] 11 Oct 2020 work of Liao and Klapp investigating MIPS in systems of circle is present, e.g., for an ‘L’-shaped particle 41,42. Further examples swimmers by computer simulations and an investigation of MIPS are Janus particles with an asymmetric coating 43, bacteria with in circle-swimmer systems at vanishing temperature on the basis unequally strong flagella 44,45, and structure formation of achiral of computer simulations and dynamic mean-field theory by Lei, particles 46. Even external fields (such as, e.g., gravitational 47, Ciamarra, and Ni 68. Another simulation study on the collective magnetic 48, or temperature 49 fields) and nearby surfaces 50 can dynamics of circle swimmers showed that chiral active particles bend trajectories. Especially the chiral behavior at walls plays can be rectified by transversal temperature differences 49. In an important role in nature and was studied for Escherichia coli addition to these mostly computer-simulation-based studies, a theory for rod-like active Brownian particles (ABPs) was recently published 69. It suggests a suppression of MIPS due to chirality. a Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms- The authors also discussed the idea of an effective rotational Universität Münster, D-48149 Münster, Germany ∗ Corresponding author: [email protected] diffusion. Their work, however, describes the dynamics by

1 effective quantities and provides no analytical prediction or mann constant kB and absolute temperature T, DT is the trans- result for the effective rotational diffusion or for the spinodal lational diffusion coefficient of the particles, ~Fint,i({~ri}) describes ~ corresponding to the onset of MIPS. There exists also a dynamical the interaction force acting on the i-th particle, and ξT,i(t) and 70 71 density functional theory (DDFT) for circle swimmers , but it ξR,i(t) are statistically independent Gaussian white noises with cannot predict the spinodal for MIPS, since the DDFT approach zero mean for the translational and rotational degrees of free- is limited to low particle densities and weak propulsion, where dom of the i-th particle, respectively. The correlations of the 70 ~ ~ MIPS does not occur . However, there is recent work on a pre- noise terms are given by hξT,i(t1) ⊗ ξT, j(t2)i = 2DT12δi jδ(t1 − t2) 63 dictive hydrodynamic theory for mixtures of circle swimmers and hξR,i(t1)ξR, j(t2)i = 2DRδi jδ(t1 − t2) with the ensemble aver- that is based on the Vicsek model. This hydrodynamic theory age h·i, dyadic product ⊗, 2 × 2-dimensional identity matrix 12, 5 can describe effects like flocking , but it is unable to describe Kronecker delta δi j, Dirac delta function δ(t), and rotational dif- clustering and MIPS, since no spatial interactions are taken into fusion constant DR. The angular velocity ω constitutes the dif- account. This problem will be addressed in the present work. ference between ABPs in two spatial dimensions 72,74–81 and the We study ABPs as an important class of active colloidal parti- circle swimmers considered here and in Ref. [67]*. As usual, we cles, where hydrodynamic interactions are not present but the consider an additive pairwise interaction so that the interaction ~F − N ~ U kr − r k hydrodynamic resistance exerted on the particles is taken into force can be written as int,i = ∑ j=1, j=6 i ∇~ri 2( ~i ~ j ), where account, and consider a translational and rotational propulsion ~ T ∇~ri = (∂x1,i ,∂x2,i ) is the del operator in two spatial dimensions of these particles leading to circle swimming. Extending previ- with respect to ~ri and U2(k~ri −~r jk) is the pair-interaction poten- ous predictive field-theoretical models for ABPs 72,73 towards cir- tial depending on the distance of the i-th and j-th particles. In cle swimmers, we obtain a 2nd-order-derivatives model that de- the following, we derive a predictive local field-theoretical model scribes the collective dynamics of circle swimmers. Interestingly, for the collective dynamics of the circle swimmers based on the we find that the diffusive dynamics of circle swimmers can be Langevin equations (1) and (2). mapped onto that of ABPs via an effective rotational diffusion coefficient. Furthermore, this model predicts a spinodal that we The statistically equivalent Smoluchowski equation corre- compare for different values of the particles’ angular propulsion sponding to the Langevin equations (1)and (2) is given by velocity to data we obtained from Brownian dynamics simula- N P˙ D 4 D ∂ 2 − ω∂ P tions. The critical point associated with the spinodal is found = ∑( T ~ri + R φi φi ) to have a remarkable dependence on the angular propulsion ve- i=1 (3) locity. While the critical density is independent of the angular ~ − ∇r · (βDT~Fint,i({~ri}) + v0uˆ(φi))P . propulsion velocity, the critical Péclet number describing the ac- ~i tivity of the particles at the critical point increases with the angu- Here, P denotes the many-particle probability density lar propulsion velocity. P({~ri},{φi},t) and 4~ri is the Laplacian with respect to ~ri. This article is structured as follows. In section 2, we derive the At this point in the derivation, the angular velocity gives rise to analytical model and present the details of our computer simula- the term −ω∂φi P, which is of first order in the angular derivative. tions. We discuss our results in section 3. Finally, we conclude in Remarkably, there is no such term in corresponding models for section 4. ordinary ABPs in two 72 and three 73 spatial dimensions. As we will show further below, this term has important consequences. 2 Methods We derive our model for the collective dynamics of circle swim- 2.1 Analytical model mers via the interaction-expansion method 72,73,76. As a first step, We consider a suspension of N spherical Brownian circle swim- we calculate the one-particle density ψ by integrating both sides mers in two spatial dimensions. Each particle has a constant of the Smoluchowski equation over the degrees of freedom of all translational propulsion parallel to its instantaneous orientation but one particles and renaming its position vector and orientation and an additional constant angular propulsion. We denote the angle as~r and φ, respectively: translational speed of a free particle by v0 and the angular ve- N Z Z 2π locity by ω. The center of mass of the i-th particle is denoted 2 ψ(~r,φ,t) = N ∏ d r j dφ j P . (4) T 2 ~ri=~r, by the vector ~r = (x ,x ) and its orientation by the unit vector j=1 R 0 i 1 2 φi=φ T j=6 i uˆ(φi) = (cos(φi),sin(φi)) that is parametrized by the polar angle φi. ~ri(t) and φi(t) depend on time t. The dynamics of the parti- Following the procedure described in Refs. [ 72,73], we cles can be described via the overdamped Langevin equations for perform a Fourier expansion, a gradient expansion 82–84, a 37,49,55–57,67 circle swimmers Cartesian orientational expansion 85, and a quasi-stationary approximation 72,73,76. Moreover, the pair-distribution ~r˙ = v uˆ(φ ) + βD ~F ({~r }) +~ξ , (1) i 0 i T int,i i T,i function g(~r,~r0,φ,φ 0,t) that relates the two-particle density ψ(2)(~r,~r0,φ,φ 0,t) to the one-particle density ψ(~r,φ,t) is intro- φ˙i = ω + ξR,i, (2) where a dot over a variable denotes a derivative with respect to time, β = 1/(kBT) is the thermodynamic beta with the Boltz- * The angular velocity ω introduced here is equivalent to ω0 in Ref. [67].

2 duced: coefﬁcients

(2) 0 0 0 0 0 0 R 2π dt R 2π dt g(r,t ,t )cos(k t + k t ) ψ (~r,~r ,φ,φ ,t) = g(~r,~r ,φ,φ ,t)ψ(~r,φ,t)ψ(~r ,φ ,t). (5) g r 0 1 0 2 1 2 1 1 2 2 (9) k1,k2 ( ) = 2 . π (1 + δk1,0)(1 + δk2,0) By assuming translational and rotational invariance as well as time invariance of the system, which correspond to a homoge- Remarkably, the obtained diffusion equation has a scalar diffusion neous stationary state, the dependence of the pair-distribution coefficient and, albeit the chirality of the particles, no mixing of ‡ 0 spatial derivatives occurs . function reduces to g(r,φr − φ,φ − φ) with the parametrization 0 0 The 2nd-order-derivatives model (6) for circle swimmers can ~r −~r = ruˆ(φr) of the relative position ~r −~r. Since ω leads to a breaking of rotational symmetries, the assumption of rotational be compared with the model for ABPs in two spatial dimensions invariance is well justified in the limit ω = 0 but becomes increas- presented in Ref. [72]. There, the achiral case ω = 0 is consid- ingly inaccurate for, according to amount, larger angular veloci- ered and a 2nd-order-derivatives model was derived using anal- ties. A sketch of the setup illustrating the used absolute and rela- ogous approximations. The comparison of that model, given by tive coordinates of two particles is shown in Fig. 1. Eqs. (21) and (22) in Ref. [72], with ours yields an interesting and important result: The collective dynamics of circle swimmers are, up to the considered orders, qualitatively identical to the collective dynamics of ordinary ABPs, but the former have a larger rotational diffusion. It is therefore possible to map the dynamics of circle swimmers onto that of ordinary ABPs with an effective rotational diffusion coefficient

∗2 DR,eff = DR(1 + ω ). (10)

This mapping is similar to the mapping of ABP systems onto 86–90 detailed-balance systems via an effective temperature Teff . For two spatial dimensions, the effective temperature is given by 91 Pe2 T = T 1 + (11) eff 6 with the Péclet number Pe = v σ/D , where σ is the diameter Fig. 1 Absolute and relative positions and orientations of two circle swim- 0 T of the particles. Indeed, one can define an effective rotational mers with angular velocity ω, where eˆx1 denotes the unit vector in x1 di- rection. temperature ∗2 TR,eff = T(1 + ω ) (12) that corresponds to the orientational motion of the circle swim- For the dynamics of the local density field ρ(~r,t) = mers, where ω∗ can be considered as a rotational Péclet number. R 2π dφ ψ(~r,φ,t), one then obtains, up to the 2nd order in deriva- 0 Taking into account that, via the Einstein relation, D is propor- tives, the model† R tional to the temperature T so that DR,eff should be proportional ρ˙(~r,t) = ∂i(D(ρ)∂iρ) (6) ∗ to TR,eff, one can justify the similar scaling of DR,eff with ω and of ~ with the compact notation ∂i = (∇)i, the density-dependent diffu- Teff with Pe. We stress that the validity of the mapping we found is sion coefficient not restricted to low-density and near-equilibrium systems, which A(1,0,0) is the case for the mapping of ABPs onto passive Brownian par- D(ρ) = D + ρ T π ticles. Reasons for this are the fact that angular interactions are not present in the case of spherical particles as considered here 1 2 and the fact that our derivation is applicable even for arbitrarily + ∗2 v0 − (A(0,1,0) + A(0,1,−1))ρ (7) 2DR(1 + ω ) π large Pe. However, this mapping does not account for chirality 4 effects and is only moderately applicable for higher-order mod- v − A(0,1,0)ρ , 0 π els. It should therefore be applied predominantly for low values of ω∗. ∗ 67 and the dimensionless rescaled angular velocity ω = ω/DR . If one wants to study the dynamics of the circle swimmers ex- The coefficients A(m,k1,k2) are given by Z ∞ 2 m+1 0 A(m,k1,k2) = −π βDT dr r U2(r)gk1,k2 (r) (8) 0 ‡ Actually, a matrix-like diffusion coefficient and a mixing of spatial derivatives were obtained during the derivation. However, the nondiagonal terms cancel each other 0 with the shorthand notation U2(r) = dU2(r)/dr and the expansion out, since the diffusion matrix was built upon two base matrices: the Kronecker delta δi j and the later introduced antisymmetric two-dimensional Levi-Civita symbol

εi j given by Eq. (19). For the special case of a 2nd-order-derivatives model, no contributions from εi j can occur due to its properties. For higher-order models, † From here onwards, summation over indices appearing twice in a term is implied. there exist a tensorial diffusion coefﬁcient and mixing of spatial derivatives.

3 2 plicitly, the values of the coefficients A(n,k1,k2) have to be known. τLJ = σ /(εβDT), and ε are used as units for length, time, and 0 This requires information about the product U2(r)g(r,t1,t2). For a energy, respectively. The Péclet number was varied only by vary- 92 2 Weeks-Chandler-Andersen (WCA) interaction potential ing T and we fixed FA = 24ε/σ as well as DT/(kBT) = σ /(ετLJ)  analogous to Refs. [74,76,81,93]. Following the Stokes-Einstein- σ 12 σ 6 1 4ε r − r + ε, if r < 2 6 σ, Debye relation for spheres, the rotational diffusion coefficient is U2(r) = (13) 2 0, else, given by DR = 3DT/σ . As simulation domain, a quadratic area with side length ` = 128σ and periodic boundary conditions was where r = k~rk is the center-of-mass distance of two particles and chosen. For the initial condition, we used a random distribution ε is the interaction energy, this product is analytically known of the particles. Particle trajectories were integrated for a total 81 93 −5 in two and three spatial dimensions. The WCA interac- simulation period of 250τLJ with a time-step size ∆t = 5 · 10 τLJ. tion potential is purely repulsive and often used for ABP simu- The number of particles we simulated ranged from 4172 to 18775, lations 67,74,79,81,93, since it suits well the interaction behavior of based on the overall packing density Φ of the system. The latter active particles used in experiments 21. Using the available an- ranged as Φ ∈ [0.2,0.9]. alytical results for the pair-distribution function, the coefficients To study the collective behavior of circle swimmers includ- occurring in the density-dependent diffusion coefficient (7) are ing the emergence of MIPS, we calculated state diagrams that approximately given by 72,81 show the state of a system as a function of the Péclet number and the packing density. However, this requires a good measure A(1,0,0) = 38.2 + 18.4e2.87Φ, (14) that allows to distinguish MIPS from other states of the system. In the following, we propose a method that is based on graph- A(0,1,0) = 36.9, (15) theoretical networks. We discarded the initial period of duration A(0,1,−1) = −0.232 − 13.6Φ, (16) 125τLJ from the simulation data to give all considered systems enough time to relax to a stationary state. Afterwards, we ex- 2 where Φ = ρπσ /4 denotes the overall packing density. tracted the particles’ positions every 2.5τLJ. From these positions, Additionally, for potential further applications, we derive a we constructed a graph-theoretical network: The nodes are the phase-field model describing the collective dynamics of circle different particles and they are connected by an edge to another swimmers up to 4th order in derivatives. Since the model would node if these two particles interact. This is straightforward to de- be rather lengthy and complicated, we make a further approxi- termine, since the WCA potential has a sharp interaction length. mation by neglecting terms of second or higher order in ω. The In addition, we assigned to each edge of the graph a weight ac- model can be written in conservative form cording to a weight function ζ(r). We found ζ(r) = U2(r)/ε (i.e., an edge weight corresponding to the energy of the represented (chiral) (achiral) ρ˙ = −∂i(Ji + Ji ) (17) interaction) to be a robust choice for outlining the region of clustering in Pe-Φ space. Summing the edge weights for this choice with the model-specific current of ζ then gives a dimensionless measure for the total potential (chiral) 2 3 4 energy Epot stored in the system. In the end, we averaged over Ji = ωεi j(µ1ρ + µ2ρ + µ3ρ + µ4ρ )∂ j4ρ (18) all extracted data sets so that every point in the state diagram is 2 2 + ωεi j(µ5 + µ6ρ + µ7ρ )(∂ jρ)(∂kρ) based on 50 graphs and therefore 50 measurements of Epot. For the state diagrams, we considered the mean potential energy per (achiral) and the current Ji for the achiral case ω = 0, which is given particle, since this measure is independent of the size of the sys- by Eq. (25) in Ref. [72]. Expressions for the coefficients µi are tem. given in the Appendix. Considering this order in derivatives, a Using this procedure, one gets a clear and consistent measure mixing of spatial derivatives occurs. In two spatial dimensions, for the state of a nonequilibrium system as an alternative to other this mixing is characterized by the two-dimensional Levi-Civita common measures used for determining state diagrams of ABPs, matrix such as visual inspection 74,76, the size of the biggest cluster 67, ! 0 1 and the characteristic length 74,81,93,95,96. Our method has ad- ε = , (19) −1 0 vantages compared to these other methods. Visual inspection has the drawback of not being objective and the size of the biggest which obeys the relation ∂φ uˆ(φ) = −εuˆ(φ). In the case of the cluster is a measure that depends on the size of the system. The phase-field model (17), terms proportional to εi j are responsible for the mixing of spatial derivatives. characteristic length, albeit being dependent on the size of the system, works very well for ABPs where the clusters are rather static and will merge eventually into one big cluster. However, we 2.2 Computer simulations observed that the clusters of circle swimmers behave much more We carried out Brownian dynamics simulations by integrating the dynamically: They rotate, break, and form again§. Furthermore, Langevin equations (1) and (2) using a modified version of the molecular dynamics simulation package LAMMPS 94. In these simulations, we expressed physical quantities in terms of dimen- § In the supplementary material, we provide videos of MIPS occurring in systems of sionless Lennard-Jones units, where σ, the Lennard-Jones time circle swimmers.

4 the characteristic length becomes increasingly problematic at very ber and growing with ω∗2 and that the simulation results pre- high densities, since it cannot distinguish well between a random- cisely confirm the analytical prediction (10). The independence close-packed system and an active cluster that has a similar den- of DR,eff/DR from Pe is reasonable, since Pe characterizes the ac- sity but an increased pressure in its interior. This comes from the tivity corresponding to the translational propulsion of the parti- fact that the interaction potential is very steep at low distances so cles, which does not affect their orientations, and the quadratic that the mean potential energy per particle inside a MIPS cluster dependence on ω∗ can be understood from the analogous scaling can be much larger than outside of such a cluster, while the parti- of the effective rotational temperature (12). Since we analyze the cle densities inside and outside of a MIPS cluster are very similar. rotational diffusion for noninteracting particles, the concept of an The graph-theoretical method can handle dynamical clusters and effective temperature is applicable both for the translational and distinguish between those states, which makes it a perfect choice for the rotational motion of the particles. In fact, in Eq. (21) a for the systems considered in the present work. quadratic dependence on both the Péclet number Pe corresponding to the particles’ translational motion and the rescaled angu- 3 Results lar propulsion velocity ω∗ that constitutes a Péclet number cor- Before we investigate the collective dynamics of circle swimmers, responding to their rotational motion is observed. While the ef- we address their effective rotational diffusion DR,eff and test the fective translational temperature (11) can no longer be defined analytical result (10). For this purpose, we study the long-time when the interactions of the circle swimmers are taken into ac- diffusion constant of a circle swimmer 39,54 count, the effective rotational temperature (12) and the effective rotational diffusion coefficient (10) are still applicable in this 1 2 Dcs = lim (~r(t) −~r(0)) . (20) t→∞ 2t case, since the considered spherical particles have only positional but not angular interactions. If also angular interactions come For a noninteracting circle swimmer, we obtain from Eq. (6) the into play (e.g., by investigating anisotropic particles like ellip- exact analytical result soids), we suspect DR,eff to become density-dependent. Pe2 D = D 1 + (21) Furthermore, we investigated the collective dynamics of in- cs T 6γ teracting circle swimmers. For this purpose, we first applied a with the rescaled effective rotational diffusion γ = DR,eff/DR = linear-stability analysis to our 2nd-order-derivatives model (6). ∗2 1 + ω . We therefore calculated Dcs using Eq. (20) based on This analysis yields the spinodal condition describing the onset of Brownian dynamics simulations with 10,000 noninteracting cir- MIPS cle swimmers and determined the corresponding result for the D(ρ) = 0. (22) rescaled effective rotational diffusion D D by using Eq. (21). R,eff/ R Figure 3 presents our simulation results for the state of a sys- This procedure was repeated for different values of ∗ and Pe. ω tem of circle swimmers as a function of the Péclet number Pe and In Fig. 2, the simulation results for D D are compared to R,eff/ R packing density Φ in the form of state diagrams for various values the analytical result (10). This figure shows that the rescaled of the rescaled angular velocity ω∗ as well as our predictions for the spinodal given by Eq. (22). For comparison, also the earlier 2 Analytical prediction: simulation results for the spinodal, which are presented in Ref. 2 [67] for ω∗ = 0 and ω∗ = 1, are plotted (see Figs. 3(a) and 3(e)). DR,eff /DR = 1 + ω∗ These results are consistent with ours, which cover a larger pa- 1.8 Simulation data (ρ 0): → rameter space and have a finer resolution. Our simulation results Pe = 10 show three different states. They are a homogeneous state occur- R 1.6 Pe = 50 ring at low densities, a solid state at high densities and low Péclet /D Pe = 250 numbers, and a MIPS state at high densities and high Péclet num- eff , ∗

R bers. Interestingly, for increasing rescaled angular velocity ω , 1.4 D the MIPS region is shifted to higher Péclet numbers and packing densities. However, the maximum value for the mean potential 1.2 energy per particle does not change significantly in the cluster state when ω∗ is increased. The solid region is similar to the findings of Refs. [69,77] and seems to be only weakly affected by the 1 additional torque ω∗. Our analytical prediction (22) for the spin- 0 0.25 0.5 0.75 1 odal corresponding to the onset of MIPS is in good agreement ω∗ with our simulation results for the state diagram and those of Ref. [67]. It is important to take into account that the simulation Fig. 2 The dimensionless effective rotational diffusion DR,eff/DR of circle swimmers as a function of the dimensionless angular propulsion velocity data can show MIPS slightly outside the predicted spinodal, since ω∗ for different Péclet numbers Pe. The solid line represents the analyti- fluctuations in the simulations can lead to a transition into the cal prediction (10). cluster state if the system is in the binodal region. Furthermore, the results shown in Fig. 3 are in line with the predictions of Ref. effective rotational diffusion is independent of the Péclet num- [24], where they studied ABPs with different rotational diffusion

5 (a) (b) (c) 0.9 ω∗ = 0 0.9 ω∗ = 0.25 0.9 ω∗ = 0.5 solid solid solid 0.8 MIPS 0.8 MIPS 0.8 MIPS 0.7 0.7 0.7 8 0.6 critical 0.6 critical 0.6 critical c point c point c point predicted Φ predicted Φ predicted Φ Φ Φ Φ 7 0.5 spinodal 0.5 spinodal 0.5 spinodal for MIPS for MIPS for MIPS 0.4 0.4 0.4 6 0.3 Liao & Klapp 0.3 0.3 (2018) 0.2 homogeneous 0.2 homogeneous 0.2 homogeneous 5

10 50 100 150 200 250 10 50 100 150 200 250 10 50 100 150 200 250 )

Pec Pec Pec εN

Pe Pe Pe (

4 / (d) (e) (f)

100 pot

0.9 ω∗ = 0.75 0.9 ω∗ = 1 0.9 2 E solid solid Pec = 20.4(1 + ω∗ ) 2 4 3 MIPS MIPS +√446+863ω∗ +416ω∗ 0.8 0.8 0.8 80 0.7 0.7 0.7 2 critical critical 60

0.6 0.6 0.6 c c point predicted c point predicted c Φ Φ Φ Φ Φ spinodal Φc = 0.588 Pe 0.5 spinodal 0.5 0.5 for MIPS for MIPS 40 1 0.4 0.4 0.4 Liao & Klapp (2018) 0.3 0.3 0.3 20 0 homogeneous homogeneous 0.2 0.2 0.2 0 10 50 100 150 200 250 10 50 100 150 200 250 0 0.25 0.5 0.75 1 Pec Pec Pe Pe ω∗

Fig. 3 State diagram for (a) ABPs and (b)-(e) circle swimmers with various rescaled angular velocities ω∗. The color bar shows the rescaled mean potential energy per particle Epot/(εN) that reveals the different states of the system. These include a homogeneous state at low packing densities Φ, a solid state at very high Φ and low Péclet numbers Pe, and a state corresponding to MIPS at high Φ and Pe. In subfigures (a) and (e), the computer- simulation results of Ref. [67] for the spinodal are shown for comparison. The associated curves in these subfigures correspond to the criterion that the fraction of the largest cluster passes a value of about 0.3. Our analytical prediction for the spinodal describing the onset of MIPS (see Eq. (22)) is indicated in each state diagram. Also our analytical predictions for the coordinates of the critical point, being the critical packing density Φc and the ∗ critical Péclet number Pec, are marked in the state diagrams. Subfigure (f) shows the dependence of Φc and Pec on ω . constants. They only observed MIPS in systems of small angular 4 Conclusions reorientation but not in systems corresponding to a high angular diffusion (see Fig. 8 in Ref. [24]). We have investigated the dynamics of interacting spherical Brow- nian circle swimmers in two spatial dimensions based on de- riving a predictive field theory describing their collective be- With the equation (22) for the spinodal, we were also able to havior and performing Brownian dynamics simulations. The derive an analytical prediction for the associated critical point, derivation yielded a 2nd-order-derivatives model and a 4th-order- which is given by the critical Péclet number Pec and the critical derivatives model that are applicable for small values of the par- packing density Φc. Our predictions for Pec and Φc as functions ticles’ rescaled angular velocity ω∗. Using the 2nd-order model, ∗ of ω are given by we showed that the dynamics of circle swimmers can be mapped

∗2 p ∗2 ∗4 onto that of common ABPs via introducing an effective rotational Pec = 20.4(1 + ω ) + 446 + 863ω + 416ω , (23) diffusion coefﬁcient DR,eff. We also obtained analytical expressions for the spinodal corresponding to the onset of MIPS and Φc = 0.588. (24) for the associated critical point, which allow to assess how the These expressions are visualised in Fig. 3(f). Interestingly, for spinodal and the critical point depend on ω∗. All these analyt- ∗ increasing ω , the critical point shifts to higher Pec but remains ical predictions were found to be in good agreement with our 67 at a constant Φc. To the best of our knowledge, the critical point simulation results and available results from the literature . To for systems of circle swimmers has not yet been located in the determine state diagrams where MIPS can clearly be identiﬁed literature. from our simulation data, we computed the potential energy of

6 the system by means of a weighted graph-theoretical network. τ2 µ2 = (128DTτA(0,1,0)(A(0,1,−1) + A(0,1,0)) Using these analytical and numerical methods, we also investi- 32π2 gated the suppression of MIPS by circle swimming, which was + τ2v2(−48A(0,1,−1)2 + 5(A(0,1,1) − 42A(0,1,0)) previously reported in Ref. [67], in more detail. We found that 0 the spinodal shifts to larger Péclet numbers Pe and packing densi- A(0,1,−1) + 60A(0,1,0)2 + A(0,1,1)2 ties Φ when ω∗ is increased, whereas the associated critical point shifts only to larger Pe but remains at a constant Φ. A solid state, + 20A(0,1,0)A(0,1,1)) − τv0(A(0,1,1)(3A(1,2,−2) which was also observed in the state diagrams, showed no significant dependence on ω∗. These results show that the occurrence + 2A(1,2,0)) + A(0,1,−1)(32A(1,0,−1) − 32A(1,0,0) (26) of MIPS and thus the collective dynamics of circle swimmers can be tuned via their angular velocity. Since circle swimming can be + 32A(1,0,1) + 3A(1,2,0)) + A(0,1,0)(64A(1,0,−1) induced and easily tuned, e.g., by a rotating external magnetic − 96A(1,0,0) + 64A(1,0,1) + 9A(1,2,−2) ﬁeld in combination with magnetic ABPs, it offers a simpler way to suppress MIPS than alignment 64,69. This makes suspensions of − 64A(1,2,−1) + 18A(1,2,0))) + A(1,2,−2)A(1,2,0) circle-swimming active particles to a particularly interesting class of active materials. + 8A(0,1,0)A(2,1,−1) + 20A(0,1,−1)A(2,1,0)

+ A(1,2,0)2 − 4A(0,1,0)A(2,1,0)),

The 4th-order-derivatives model we derived can be used for τ3 µ = − A(0,1,0)(τv (−192A(0,1,−1)2 + (25A(0,1,1) further investigations of the collective behavior of interacting cir- 3 24π3 0 cle swimmers. With this model, one could address, e.g., the occurring finite cluster sizes or interfacial profiles. For example, − 329A(0,1,0))A(0,1,−1) + 75A(0,1,0)2 + 4A(0,1,1)2 the model could be used to study at which value of the particles’ rescaled angular velocity ω∗ the complete phase separation oc- + 53A(0,1,0)A(0,1,1)) − 64A(0,1,0)A(1,0,−1) curring in systems of ordinary ABPs is replaced by the arrested + 64A(0,1,0)A(1,0,0) − 64A(0,1,0)A(1,0,1) phase separation that can be observed for circle swimmers. Fur- (27) thermore, we believe that our method for computing the potential − 6A(0,1,0)A(1,2,−2) − 6A(0,1,1)A(1,2,−2) energy of the system via a graph-theoretical network will prove useful also for other dynamical many-particle systems. Using a + 64A(0,1,0)A(1,2,−1) − 15A(0,1,0)A(1,2,0) weighted graph-theoretical network can have significant benefits, since such networks are mathematically well established and nu- − 4A(0,1,1)A(1,2,0) − A(0,1,−1)(64A(1,0,−1) merous further parameters that are associated to a graph might turn out to be interesting properties for characterizing a physical − 64A(1,0,0) + 64A(1,0,1) + 9A(1,2,0))), system. In the future, this study could be extended towards circle 4 swimmers with other shapes, which will very likely lead to the τ 2 2 µ4 = 4 A(0,1,0) (−96A(0,1,−1) + (15A(0,1,1) observation of fascinating new effects. 8π (28) − 81A(0,1,0))A(0,1,−1) + 15A(0,1,0)2 + 2A(0,1,1)2

+ 17A(0,1,0)A(0,1,1)), A Appendix

We introduce the angular relaxation time τ = 1/DR for abbrevia- tion. The coefﬁcients for the ﬁrst-order corrections in ω occurring in Eq. (18) are then given by the following expressions:

τ2v µ = − 0 (64D τ(A(0,1,−1) + A(0,1,0)) 1 32π T

2 2 + τ v0(−43A(0,1,−1) + 15A(0,1,0) + A(0,1,1)) (25) − τv0(16A(1,0,−1) − 32A(1,0,0) + 16A(1,0,1)

+ 3A(1,2,−2) − 16A(1,2,−1) + 3A(1,2,0))

+ 4(A(2,1,−1) − A(2,1,0))),

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