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Analytical Approach to Chiral Active Systems: Suppressed Phase Analytical approach to chiral active systems: sup- pressed phase separation of interacting Brownian cir- cle 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 sys- tems 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 sep- aration, 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(f~rig) describes ~ corresponding to the onset of MIPS. There exists also a dynamical the interaction force acting on the i-th particle, and xT;i(t) and 70 71 density functional theory (DDFT) for circle swimmers , but it xR;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 hxT;i(t1) ⊗ xT; j(t2)i = 2DT12di jd(t1 − t2) 63 dictive hydrodynamic theory for mixtures of circle swimmers and hxR;i(t1)xR; j(t2)i = 2DRdi jd(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 di j, Dirac delta function d(t), and rotational dif- clustering and MIPS, since no spatial interactions are taken into fusion constant DR. The angular velocity w 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 co- efficient. 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 − w¶ P tions. The critical point associated with the spinodal is found = ∑( T ~ri + R fi fi ) to have a remarkable dependence on the angular propulsion ve- i=1 (3) locity. While the critical density is independent of the angular ~ − ∇r · (bDT~Fint;i(f~rig) + v0uˆ(fi))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(f~rig;ffig;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 −w¶fi 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 y 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 f, respectively: translational speed of a free particle by v0 and the angular ve- N Z Z 2p locity by w.
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