An Introduction to Modeling Approaches of Active Matter
AAn Introduction to Modeling Approaches of Active Matter
L. Hecht, J. C. Ureña, and B. Liebchen ∗ Institut für Physik kondensierter Materie, Technische Universität Darmstadt,Hochschulstr. 8, 64289 Darmstadt, Germany (Dated: February 26, 2021)
CONTENTS
I. Introduction 1II. Wet and dry models: The role of thesolvent in active matter 1III. Dry active particles: The active Brownianparticle model and its alternatives 2IV. Continuum theories for dry active matter 7V. Hydrodynamics of microswimming: LowReynolds number and Stokes flow 10VI. Modeling hydrodynamics at themany-particle level 11VII. Continuum theories of microswimmerswith hydrodynamic interactions 13References 14
I. INTRODUCTION
This article is based on lecture notes for the MarieCurie Training school “Initial Training on Numer-ical Methods for Active Matter”. It provides anintroductory overview of modeling approaches foractive matter and is primarily targeted at PhDstudents (or other readers) who encounter someof these approaches for the first time. The aim ofthe article is to help put the described modelingapproaches into perspective.We begin with a brief discussion of the role of thesolvent in (soft) active matter, which is followed byan introduction to “dry particle-only models”, suchas the active Brownian particle model, before com-ing to models for wet active matter, which (explic-itly) include a solvent, and continuum descriptionsfor the collective behavior of many active particles. ∗ [email protected] II. WET AND DRY MODELS: THE ROLEOF THE SOLVENT IN ACTIVE MATTER
Models of active matter can be classified into “dry”and “wet” models. The former class of modelsinvolves only equations of motion for “particles”whereas the latter involves an explicit descriptionof a solvent in addition to the embedded activeparticles, which ensures momentum conservation,as will be discussed below.Dry models are naturally used to describe activesystems which do not involve a liquid solvent, suchas granular particles on vibrating plates [1–6], self-vibrating granular particles [7–9], bacteria glidingon a rigid surface [10], flocks of birds, animal herds,swarms of locusts [11–15], human crowds [16, 17],or flying drones [18]. However, they are also fre-quently used as simplified descriptions of activematter systems involving a solvent which is only ef-fectively represented and commonly acts as a ther-mal bath leading to fluctuations in the equationsof motion of the individual particles. In contrast,wet models are used to describe microswimmerssuch as synthetic active colloids [19–21], dropletswimmers [22–25], and biological microorganismslike bacteria [10, 26, 27], algae [28], or sperm cells[26], including their interaction with the surround-ing solvent and the corresponding cross interac-tions among different microswimmers. A particu-lar example of a wet system at larger scales, i.e.,beyond the soft matter realm, which microswim-mers belong to, can be found in schools of fish[13, 29]. To understand the applicability regimeof the various active matter models, it is instruc-tive to first discuss the impact of the solvent onactive systems:(i)
Fluctuations and dissipation:
Active particlesare typically orders of magnitude larger thanthe molecules of the surrounding solvent andare subject to collisions with the latter. Thisleads to fluctuations in their motion, analo-gously to Brownian motion of “passive” col-loids in equilibrium [cf. trajectory in Fig. 1(a)]. The trajectory of an isolated active par-ticle is then typically given by the combina-tion of ballistic motion due to self propul-sion and fluctuations due to collisions withthe solvent molecules. As can be seen in a r X i v : . [ c ond - m a t . s o f t ] F e b Fig. 1 (b), the motion of an active particle isnot straight because the collisions of the sol-vent molecules with the active particles fea-ture both a radial and a tangential compo-nent. The latter induces a stochastic turn ofthe particle orientation and, hence, reorien-tation of the self-propulsion direction, whichis called rotational Brownian motion or ro-tational diffusion. Following the fluctuation-dissipation theorem from statistical mechan-ics, these fluctuations are necessarily linkedto dissipation occurring, e.g., in the form ofStokes drag for spherical particles. For mi-croswimmers, i.e., for (active) particles atthe microscale, dissipation normally domi-nates over inertia. Hence, the motion is over-damped. These effects of the solvent, namelytranslational diffusion, rotational diffusion,and dissipation, are the only effects of the sol-vent which are typically taken into accountin dry models, such as the active Brownianparticle (ABP) model, which we will discussfurther below.(ii)
Momentum conservation:
Physically, in theabsence of external fields or boundaries, theoverall momentum of an active system hasto be conserved. For example, when amicroorganism or an active Janus colloidmoves forward, there is necessarily a counter-propagating solvent flow such that the over-all momentum of the active particle and thesurrounding solvent is conserved (swimmingin vacuum is impossible). Thus, the solventnot only acts as a bath providing fluctuationsand drag but also ensures momentum conser-vation.(iii)
Hydrodynamic interactions:
The solvent me-diates hydrodynamic interactions among dif-ferent active particles. These arise becausethe flow pattern induced by each active par-ticle as a consequence of its swimming actsonto all other particles in the system. Thesesolvent-mediated interactions are often long-ranged. In particular, in the absence of ex-ternal forces, they often decay as 1 /r forforce-dipole swimmers, such as various bac-teria or algae (explicit measurements of theflow field exist, e.g., for E. coli bacteria [30]),and as 1 /r for source-dipole swimmers, suchas Paramecium [31] or (idealized) Janus col-loids with a uniform surface mobility [32, 33].However, they can be weakened or decayfaster in the presence of a substrate or otherboundaries [34].(iv) Hydrodynamic boundary effects:
If the activeparticles are in contact with boundaries, suchas a glass substrate, which is frequently used
Figure 1. (Top panel) Experimental trajectories of aJanus colloid showing (a) passive and (b) active Brow-nian motion (kindly provided by J. R. Howse; see Ref.[19] for experimental details). (Bottom panel) Exem-plary trajectories obtained in the overdamped regimefrom simulations of (c) the passive Brownian particle(PBP) model, (d) the active Brownian particle (ABP)model, (e) the run-and-tumble particle (RTP) model,(f) the active Ornstein-Uhlenbeck particle (AOUP)model, and (g) the chiral active particle (CAP) model.Note that an isolated AOUP is equivalent to an under-damped PBP with m/γ = τ p (see Sec. III). in experiments with active colloids, or an-other liquid-solid or liquid-air interface, thesolvent can lead to additional interesting ef-fects. An example of these is constituted byosmotic flows at fluid-solid interfaces, such asthose induced by auto-phoretic particles [35]or by some modular swimmers involving ion-resins [36, 37]. At fluid-air interfaces, activeparticles can cause Marangoni flows [38–41],which act on all particles in the system andcan elicit interesting collective behaviors [42–45]. III. DRY ACTIVE PARTICLES: THEACTIVE BROWNIAN PARTICLE MODELAND ITS ALTERNATIVES
Active Brownian particle model:
One of the simplest and most popular modelsto describe active particles is the active Brown-ian particle (ABP) model [46–52], originally intro-duced to describe the motion of colloidal particleswhich smoothly change their self-propulsion direc-tion due to rotational diffusion. It treats the sol-vent as a bath providing only fluctuations and dragwithout ensuring momentum conservation and, atleast in its most commonly used form, withoutaccounting for hydrodynamic interactions amongparticles. The ABP model does not explicitly de-scribe the mechanism leading to self propulsion ei-ther, which arises through the interactions of theactive particles with the surrounding solvent (orwith a substrate), but simply replaces it with aneffective force that drives the particle forward. Mi-croscopically, this is not correct because, as dis-cussed above, microswimmers are force free, but itleads to a simple generic model for the dynamicsof active particles, which stays agnostic on manydetails of the specific underlying realization. In itssimplest form, the ABP model in two-dimensionalspace is defined by the overdamped Langevin equa-tionsd ~r i ( t )d t = v ~p i ( t ) − γ ∇ ~r i U + √ D~ξ i ( t ) , (1)d φ i ( t )d t = p D R η i ( t ) , (2)where ~r i = ( x i , y i ) and φ i are the position andthe orientation angle of the i -th spherical ABP,respectively, v is the self-propulsion speed, γ isthe Stokes drag coefficient, U = P i 6, Pé-clet number Pe = 200 / √ 2, and time step ∆ t/τ p =5 × − . The ABPs interact via the purely repulsiveWeeks-Chandler-Anderson (WCA) potential [62] withstrength (cid:15) = 10 k B T . (e) Time evolution of the meancluster size L ( t ) for the ABP model (adapted from Ref.[53] - Published by The Royal Society of Chemistry).(f) Schematic of the mechanism which leads to MIPS(with permission from Ref. [21] - © 2013 by the Amer-ican Physical Society). cles with a sufficiently large Péclet number andpurely repulsive interactions arising because, e.g.,the individual particles cannot overlap, the ABPmodel predicts a spectacular phenomenon knownas motility-induced phase separation (MIPS) [21,50, 58–61]. A sequence of snapshots of the state ofan ensemble of ABPs which interact via the purelyrepulsive Weeks-Chandler-Anderson (WCA) po-tential [62] and for which MIPS occurs is shown inFig. 3 (a)–(d). Initially, the ensemble is uniformlydistributed. For suitable parameters (large Pécletnumber and high density), the uniform state losesstability and the particles aggregate in small clus-ters. These clusters grow following the coarseninglaw shown in Fig. 3 (e) until a single macroclus-ter, which coexists with a low-density active gas, iseventually formed. Overall, while phase separationin equilibrium generally requires inter-particle at-tractions, active systems can phase separate evenin their complete absence [60, 63]. The mechanismunderlying MIPS is shown in Fig. 3 (f): When par-ticles collide, they block each other until their ori-entations are randomized and they can separatefrom each other. Broadly, MIPS occurs if the ac-tive particles are fast and numerous enough for col-lisions with existing clusters to occur more oftenthan particles in these clusters leave them due torotational diffusion.The ABP model can also be generalized to accountfor inertial effects [50, 64–67], which is used, e.g., tomodel active granular particles on vibrating plates[1, 2]. Alternatives to the ABP model: Several alternative models have been designed thathave a similar scope to that of the ABP model inthe sense that they also treat the solvent as a bathwhich only provides fluctuations and drag ratherthan accounting for momentum conservation andhydrodynamic interactions. Run-and-tumble model: The run-and-tumble par-ticle (RTP) model [68–71] was originally intro-duced to describe the characteristic motion pat-terns of certain bacteria such as E. coli [72–74],but it has now advanced to a standard model forthe description of active particles. (In fact, thefirst theory for MIPS was formulated for RTPs[68] and MIPS has been observed in simulationsof this model as well [69].) In contrast to ABPs,RTPs alternate running periods, during which theself-propulsion direction remains unchanged, withidealized tumbling events, in which the orientationof the particles is randomized [cf. Fig. 1 (e)]. Theequations of motion for the i -th RTP readd ~r i ( t )d t = v ~p i ( t ) − γ ∇ ~r i U, (5)d φ i ( t )d t = X n ∆ φ n δ ( t − T n ) , (6)where the parameters of Eq. (5) are defined as inEq. (1). The values of ∆ φ n are uniformly dis-tributed between 0 and 2 π , with tumbling eventstaking place at discrete times T n [71]. In prac-tice, the times T n are chosen either randomly with h T n+1 − T n i = λ − (and, e.g., tumbling eventsfollowing a Poisson distribution, which leads toexponentially distributed times between tumblingevents, as originally found for E. coli [75]) orequally spaced. In any case, the (mean) tum-bling rate λ t is fixed, yielding a persistence time τ p = 1 /λ t , which plays the role of the (mean) timebetween tumbling events.Remarkably, the many-particle dynamics followingfrom the RTP and the ABP models turn out to beequivalent at coarse-grained scales if ( d − D R = λ t , where d > t (cid:28) τ p , since tumbling events are statisticallyunlikely on this timescale and translational diffu-sion is not considered. The latter can also be takeninto account, resulting in the emergence of a dif-fusive regime for t (cid:28) D/v , as in the ABP model,where D is the translational diffusion coefficient. Active Ornstein-Uhlenbeck model: Another alter-native to the ABP model is the active Ornstein-Uhlenbeck particle (AOUP) model [76–79], whichhas certain advantages compared with the ABPmodel regarding the theoretical description of themany-body dynamics of dry active particles. Thisis due to the fact that the AOUP model avoidsthe strongly nonlinear dependence of the center-of-mass motion on the particle orientation, whichis present in the ABP model [cf. Eqs. (1) and (2)],by using colored noise to generate self propulsion.The equation of motion for particle i in the AOUPmodel (in the overdamped regime) readsd ~r i ( t )d t = ~v ,i ( t ) − γ ∇ ~r i U, (7)where γ is the Stokes drag coefficient and U isthe total interaction potential. Whereas the self-propulsion speed v remains constant for a singleparticle in the ABP and RTP model, it evolveswith time in the AOUP model according to τ p d ~v ,i ( t )d t = − ~v ,i ( t ) + √ D~ξ i ( t ) , (8)where τ p is the persistence time and ~ξ i ( t ) is Gaus-sian white noise with unit variance and zero mean.As a result, the velocity components of an iso-lated AOUP are represented by colored Gaussiannoise with correlation function D v ( α )0 ,i (0) v ( β )0 ,j ( t ) E = δ ij δ αβ ( D/τ p )e − t/τ p between components α and β of particles i and j . Note that a single particle inthe AOUP model, i.e., U = 0, is formally identicalto an underdamped passive Brownian particle with m/γ = τ p [cf. Fig. 1 (f)]. Hence, a single AOUPshows a ballistic regime for t (cid:28) τ p followed by adiffusive regime for t (cid:29) τ p , as shown in Fig. 2.Since it involves colored noise, the AOUP modeldoes not permit formulation of an exact Fokker-Planck equation for the corresponding probabilitydistribution. However, it is still possible to de-rive an approximate Fokker-Planck equation forthe many-body dynamics, which does not dependon the particle orientation but only on the par-ticle positions [76, 78–80]. MIPS has also beenreported for the AOUP model [77, 79] suggestingthat it provides a useful alternative for the descrip-tion the many-body dynamics of active particlesalthough the single-particle properties significantlydiffer from those of the ABP and RTP model. Chiral particle model: A further class of modelsdescribes chiral active particles (CAPs) [50, 56, Whereas white noise is delta-correlated in time, the cor-relation function of colored noise takes finite values forfinite time differences. Figure 4. Simulation snapshots of active particleswith alignment interactions [cf. Eqs. (10) and (11)] for D = 0 and (a) ω = 0 (smooth variant of the Vic-sek model) and (b), (c) ω > 57, 81–87], which experience an additional effectivetorque arising from an anisotropy in their shapeor propulsion mechanism. For an isolated CAP,this leads to circular trajectories in the limit ofzero noise, whereas the orientation angle of the i -th CAP in the presence of noise evolves accordingto d φ i ( t )d t = ω + p D R η i ( t ) , (9)where ω is a constant angular velocity. As inthe ABP model, the position ~r i of the i -th CAPgenerally evolves with time according to Eq. (1).An exemplary trajectory of a CAP and the timeevolution of its MSD are shown in Figs. 1 (g)and 2, respectively. Examples of circle swimmersinclude E. coli bacteria near surfaces and inter-faces [88, 89], sperm cells [90, 91] and artificialmicroswimmers such as L-shaped particles [92],“spherical-cap particles” near a substrate [93], andasymmetric Quincke rollers [94].An overview of the models introduced thus far isprovided in Tab. I. This table is intended as aguide to numerically implementing the previouslydescribed models on a single-particle level. To thisend, the equations of motion are presented in di-mensionless form. Monte Carlo simulations: A final example to de-scribe isotropic dry active particles is based on ki-netic Monte Carlo simulations [95–97], where thedisplacements of the particles are correlated intime. Namely, the displacement during a certaintime step is drawn from a Gaussian distributionwhose mean equals the displacement in the previ-ous time step. Models with explicit alignment interactions: Thus far, we have focused our discussion onisotropic active particles, i.e., on particles withoutexplicit alignment interactions. The most popu-lar model for describing (dry) active particles with Table I. Dimensionless equations of motion and parameters of a single active particle in the ABP, RTP, AOUPand CAP models. The variables ~ r , ~ v , and t shown in the equations of the second column have been non-dimensionalized by rescaling the original dimensional variables with respect to the natural time and length scalesshown in the fourth column: ~ r = ~r/l , ~ v = ~v τ p /l , t = t/τ p . The dot over these variables denotes the derivativewith respect to the dimensionless time t . Pe is the Péclet number [49]. DRY ACTIVE PARTICLE MODELS FOR A SINGLE PARTICLEModel Equations of motion Parameters Natural units ABP ˙ ~ r ( t ) = ~p ( t ) + Pe − ~ξ ( t ) Pe = v √ DD R Time scale: τ p = D − ˙ φ ( t ) = √ η ( t ) Length scale: l = l p = v D − RTP ˙ ~ r ( t ) = ~p ( t ) None a Time scale: τ p = λ − ˙ φ ( t ) = P n ∆ φ n δ ( t − ˜ T n ) Length scale: l = l p = v λ − AOUP ˙ ~ r ( t ) = ~ v ( t ) None Time scale: τ p ˙ ~ v ( t ) = − ~ v ( t ) + √ ~ξ ( t ) Length scale: l = p Dτ p CAP ˙ ~ r ( t ) = ~p ( t ) + Pe − ~ξ ( t ) Pe = v √ DD R Time scale: τ p = D − ˙ φ ( t ) = ˜ ω + √ η ( t ) ˜ ω = ωτ p Length scale: l = l p = v D − for equally spaced T n or Poisson-distributed tumbling events as found in E. coli [75] and without translational diffusion;with the latter, the equation of motion for the position reads ˙ ~ r ( t ) = ~p ( t ) + Pe − ~ξ ( t ) with Pe = v / √ Dλ t . (polar) alignment interactions is the Vicsek model[98, 99], which accounts for self-propelled parti-cles (“birds”) that align their orientation with thatof their neighbors. A generalized continuous-timevariant of the Vicsek model comprising CAPs withalignment interactions can be defined byd ~r i ( t )d t = v ~p i ( t ) + √ D~ξ i ( t ) , (10)d φ i ( t )d t = ω + KπR X j ∈ S ( i ) R sin( φ j − φ i )+ p D R η i ( t ) , (11)where ω is an angular velocity, K is the strength ofthe alignment interactions (for K=0 this model re-duces to the CAP model) and the sum is calculatedover all particles within a circle S ( i ) R of radius R centered at the position of particle i [81, 100–102].The hallmark of this model is that particles tendto follow the orientation of their neighbors, whichcan induce polar order, e.g., in the form of the trav-eling bands shown in Fig. 4 (a) for ω = 0. Whenconsidering CAPs with polar interactions (Eq. (11)with ω > Applicability regime of dry active particlemodels: The ABP model and its alternatives are commonlyused to perform particle-based simulations of ac-tive particles and also as a starting point for theformulation of continuum theories, as we shall dis-cuss hereunder. These models have proven usefulwhen applied to, e.g., the following problems con-cerning active particles:(i) When we are concerned with single activeparticle flow fields, the ABP model has beenvery successful, e.g., to predict correlationfunctions in close agreement with experi-ments of Janus colloids [108].(ii) When hydrodynamic interactions play a mi-nor role such as for certain active colloids,their many-body behavior is reasonably welldescribed by ABPs [21]. Similarly, when hy-drodynamic interactions are dominated byother interactions such as, e.g., phoretic inter-actions of autophoretic colloids with a near-uniform surface mobility, the ABP modelserves as a useful starting point for the deriva-tion of simple models with effective phoreticpair interactions [33].(iii) When a solvent is absent but fluctuations arestill relevant as, e.g., for granular particles onvibrating plates, where quasi-deterministicchaos arises and leads to effective random-ness, which can be described as Browniannoise, the ABP model can be used as a nu-merical model [4, 6].(iv) The ABP model is also useful for fundamen-tal theoretical explorations, e.g., when we aremore interested in the fundamental conse-quences of activity on the collective behaviorof active particles rather than in the specificlink to experimental realizations. Advantages and limitations: Compared with most “wet” models, a key advan-tage of the ABP model and its alternatives istheir simplicity from both a conceptual and a com-putational viewpoint. In particular, these mod-els allow one to simulate very large ensembles ofactive particles (state-of-the-art simulations oftenuse 10 − particles [53, 63, 64, 109–111]). Onekey limitation of these models regarding the de-scription of soft active matter systems is that theydo not account for momentum conservation and of-ten not for hydrodynamic interactions either. Thiscan be particularly relevant for the description ofthe collective behavior or for describing single mi-croswimmers near walls. The ABP model is popu-lar when simulating the collective behavior of au-tophoretic active colloids as well. Here, beside hy-drodynamic interactions, also phoretic interactionscan play a crucial role and are also neglected by thestandard ABP model [33], which can however beextended to take them into account [112–118]. IV. CONTINUUM THEORIES FOR DRYACTIVE MATTER To understand the collective behavior of (dry) ac-tive particles, one often uses continuum models,which can be used for a purely theoretical analysisor a numerical analysis based on continuum simu-lations. In general, one can distinguish between (i)phenomenological and (ii) microscopic theories.(i) Phenomenological theories: This class isoften based on an identification of the rele-vant “slow variables” (e.g., the density field ρ ( ~r, t ) in the case of isotropic active systemswith particle number conservation or the density field and polarization density for po-lar active systems with polar alignment inter-actions) and on writing down all terms whichare allowed by symmetry and conservationlaws up to a certain order. Accordingly, thesetheories are sometimes called Landau theo-ries. A key advantage of phenomenologicaltheories is that they predict the structure ofthe field equations essentially based on sym-metry, conservation laws, and dimensional-ity of the system without requiring any ref-erence to the details of the underlying par-ticle system (such as the precise form ofthe interactions). Thus, these field theoriesare sometimes called “generic” in this senseand can even be formulated (and numeri-cally solved) if no underlying particle-basedmodel is known. However, phenomenologi-cal field theories do not provide informationabout the values of the coefficients. Thus,one often treats all occurring coefficients asindependent parameters and studies the phe-nomenology of the field equations as a func-tion of all these parameters. A related im-portant drawback of this approach is thatit then remains unclear if there is an un-derlying particle-based model or realizationwhich leads to the corresponding parametervalues. A specific example of a phenomeno-logical theory is discussed further below.(ii) Microscopic theories: In contrast to phe-nomenological theories, microscopic theo-ries involve a systematic derivation of thefield equations typically from the underly-ing equations of motion for the individualactive particles. This approach yields equa-tions of motion for the relevant fields, whichdirectly follow from the underlying particle-based model. Thus, in contrast to the formerclass of theories, one advantage of this secondapproach is that one obtains, in addition tothe structure of equations, an explicit linkbetween the coefficients of the particle-basedmodel and the continuum theory. This typi-cally leads to a (much) smaller number of in-dependent parameters than one would obtainfrom phenomenological approaches. Anotheradvantage is that, following the microscopicapproach, terms which are allowed by sym-metry cannot be missed, which has happenedfor various standard models of active matterin the past when following the phenomeno-logical approach.In the following, we will illustrate both approachesbased on specific examples for isotropic and polaractive systems. Example: Phenomenological theory forisotropic active matter Collective phenomena of isotropic active matter,such as phase separation, can be described, e.g.,by the phenomenological active model B+, whichis based on the common model B that describesphase separation in equilibrium systems [119].Here, the density field ρ ( ~r, t ) is assumed to be theonly slow variable of the system and the order pa-rameter φ is related to it by the linear transforma-tion φ = (2 ρ − ρ H − ρ L ) / ( ρ H − ρ L ), where ρ H and ρ L denote the density at the low-density and thehigh-density critical point, respectively [120, 121].The active model B+ is given by the equations ∂φ∂t = − ∇ · (cid:20) − M ∇ (cid:18) δ F δφ + λ |∇ φ | (cid:19) + ζM (cid:0) ∇ φ (cid:1) ∇ φ + √ D~ Λ i , (12) F [ φ ] = Z d r (cid:20) a φ + b φ + K |∇ φ | (cid:21) . (13)Here, the free-energy functional F is approximatedup to the order φ and up to square-gradient terms[122]. Equation (12) has the form of a continu-ity equation, and hence it ensures particle num-ber conservation, whereas reaction terms are notallowed in Eq. (12). The order parameter φ is sub-ject to a Gaussian white noise field ~ Λ( ~r, t ) withzero mean and unit variance. The diffusion coef-ficient is denoted by D and M is the mobility ofthe active particles. For active particles, the time-reversal symmetry (TRS) is broken locally. Thisfact is included in the active model B+ by the ad-ditional terms proportional to λ and ζ . The activemodel B+ describes the phase separation behaviorof isotropic active matter and predicts two typesof patterns: The first one is characterized by phaseseparation into a dense and a dilute phase and theadditional occurrence of vapor bubbles inside thedense phase, which are continuously created andmove to the surface of the dense phase [cf. Fig. 5(a)]. The second pattern is characterized by theemergence of dense clusters that do not grow be-yond a certain characteristic size [cf. Fig. 5 (b)].The coefficients a, b, K, λ, ζ are not known in thisphenomenological approach and are treated as pa-rameters of the model. Thus, there is no obviousconnection to particle-based models such as theABP model, whereas in microscopic theories allparameters are directly related to the underlyingparticle-based model, as will be discussed next. Example: Microscopic theories for isotropicactive matter There are several approaches to developing micro-scopic theories. To exemplify one of them, weconsider a system of N active particles without Figure 5. Numerical results of the active model B+:(a) Coexistence of a liquid phase (yellow to red colors)comprising continuously created vapor bubbles and agas phase (black and purple). (b) Phase separationinto a dense (yellow to red colors) and a dilute phase(black and purple). Dense clusters stabilize at a certaincluster size in the steady state (taken from Ref. [122]). alignment interactions. Then, we write down theSmoluchowski equation for the N -particle proba-bility density and integrate out variables to ob-tain the one-particle density field. This approachhas been used, for instance, to formulate a mi-croscopic theory of MIPS in overdamped ABPswith positions ~r i and orientations ϕ i [123]. Let X = { ~r , ..., ~r N , ϕ , ..., ϕ N } denote the state ofthe N -particle system. The corresponding Smolu-chowski equation [124] for the joint probability dis-tribution ψ N ( X , t ) reads ∂ψ N ∂t = N X k =1 ∇ ~r k · (cid:20) ( ∇ ~r k U ) γ − v ~p k + D ∇ ~r k (cid:21) ψ N + D R N X k =1 ∂ ψ N ∂ϕ k , (14)with U = P k Figure 6. Interpretation of the Smoluchowski equation[cf. Eq. (14)] for an ABP as probability conservationlaw. The contribution of each particle to the proba-bility current can be decomposed into a translationaland a rotational current. The former includes trans-lational diffusion and a drift term due to the interac-tion potential as well as the self-propulsion velocity,whereas the latter considers rotational diffusion. TheSmoluchowski equation can then be interpreted as acontinuity equation ensuring probability conservation. second hierarchy of equations, which has again tobe closed using a suitable closure scheme. To studyphase separation, one possible approximation toavoid the first type of hierarchy is to assume thatthe density varies slowly in space such that thelocal density is constant within the range of theinteraction potential resulting in an effective self-propulsion speed v ( ρ ) = v − ζρ with constant ζ .This density-dependent self-propulsion speed effec-tively accounts, to some extend, for the net effectof the repulsive interactions, namely the slowdownof active particles in regions of high density. Theresult of this microscopic approach fits well to com-puter simulations of ABPs and predicts MIPS inoverdamped ABPs [123].An alternative approach, sometimes called the“Dean approach” [127], is based on an explicitcoarse-graining of the Langevin equations for theindividual particles. This approach has been ap-plied in several works, e.g., to describe MIPSin systems of RTPs [68], pattern formation inself-propelled particles with alignment interactions[101], collective phenomena in systems of CAPs[81], pattern formation in systems of phoreticallyinteracting active colloids [112], or active systemsshowing nematic order [128]. Here, one uses Itôcalculus [127, 129, 130] to deduce a stochastic dif-ferential equation, which involves multiplicativenoise, for the (fluctuating) combined probability f ( ~r, ϕ, t ) = P Ni =1 δ ( ~r i ( t ) − ~r ) δ ( ϕ i − ϕ ) to find oneparticle with orientation ϕ at position ~r at time t . To derive the one-particle density field ρ ( ~r, t ),one can then, for example, choose to neglect themultiplicative noise term (mean field) and derivea hierarchy of equations in a similar way to theSmoluchowski approach. Example: Microscopic theories for polar ac-tive matter The aforementioned continuum theories for dry ac-tive matter were focused on isotropic active matterthat can be described by only considering the den-sity field. However, if the particles feature align-ment interactions such as in the Vicsek model, po-lar order can arise. Thus, describing these sys-tems additionally requires the consideration of themean local orientation of the particles by means ofa polarization density ~p ( ~r, t ). Corresponding theo-ries for the density field and the polarization den-sity can be derived based on the Smoluchowski ap-proach or the Dean approach. Another approach,which is aimed at describing the collective behav-ior of the Vicsek model (which is discrete in timein its original formulation) and is given by Ref.[131], is based on the Liouville equation for the N -particle probability density ψ N ( X , t ) and appliedto the well-known Vicsek model [98]. Within thismodel, the particles only interact during a collisionevent by aligning their orientation to that of theirnext neighbors and the orientation is subject toGaussian white noise. Under the assumption thatthe particles are uncorrelated prior to a collision,the N -particle density is written as a product ofone-particle densities, which is a good approxima-tion if the noise strength is large and if the mean-free path between two collisions is larger than theinteraction radius. Then, the one-particle proba-bility distribution is obtained by integration. How-ever, the solution contains complicated collision in-tegrals that are approximated using the Chapman-Enskog expansion [132], which takes the stationarystate as a reference and expands around it in pow-ers of the gradients. Finally, this leads to a setof two coupled differential equations for ρ and ~p .This set of equations is similar to that of the phe-nomenological Toner-Tu model [11] except for ad-ditional gradient terms, which occur only in themicroscopic approach.Independently of whether a theory is phenomeno-logical or microscopic, the relevant field equationscan then be studied based on various analyticaland numerical techniques ranging from perturba-tion theories, linear stability analyses, or dynami-cal renormalization group calculations in the pres-ence of additional noise terms to explicit numericalsolutions based on, e.g., finite difference, finite vol-ume, or finite element methods.0 V. HYDRODYNAMICS OFMICROSWIMMING: LOW REYNOLDSNUMBER AND STOKES FLOW The ABP model and its alternatives do not re-solve the self-propulsion mechanism, but insteadinvolve an effective force to phenomenologicallymodel the resulting directed motion. To under-stand and describe the self-propulsion mechanismof a microswimmer, one has to explicitly model theflow field produced by the microswimmer and itsinteraction with the body of the swimmer. Microhydrodynamics: Let us now briefly dis-cuss the basic equations which are involved in themodeling of a single microswimmer. While swim-ming at the macroscale involves inertia and leadsto flow fields which are described by the Navier-Stokes equation, microswimmers have to employswimming mechanisms which work even in the ab-sence of inertia since, at the microscale, viscouseffects dominate over inertial effects. This is quan-tified by the Reynolds number, which measures therelative importance of inertial and viscous forcesand is given by Re = ( ρLv ) /η , where the numer-ator represents the product of the fluid density,a characteristic length scale, and a typical flowspeed, whereas the denominator contains the sol-vent viscosity. For microswimmers, Re (cid:28) 1: For E. coli bacteria in water, for example, we have L ∼ µ m , v ∼ µ m/s , η = 0 . 001 Pa s, and ρ = 1 g/cm [49]. Thus, Re ∼ − − − (cid:28) ∼ − [133].At low Reynolds number, the Navier-Stokes equa-tion reduces to the Stokes equation, which de-scribes “creeping flow” and reads η ∇ ~u − ∇ p + ~f = 0 , (15)where ~u ( ~r, t ) and p ( ~r, t ) are the solvent velocityfield and the pressure field, respectively, and ~f ( ~r, t )is the force density representing the forces ex-erted by the microswimmers on the solvent. TheStokes equation is typically complemented by theincompressibility condition ∇ · ~u = 0 leading to acomplete set of equations to determine ~u ( ~r, t ) and p ( ~r, t ) for a given ~f ( ~r, t ) and given boundary condi-tions. Notably, the Stokes equation does not con-tain any time derivatives, and therefore, the sol-vent responds instantaneously to the applied forces(no motion would take place once the forcing termis switched off), which reflects the absence of iner-tia. Accordingly, the swimming mechanism of scal-lops, which move by periodically opening and clos-ing their shells, would not work at low Reynolds (a)(b) (c) Figure 7. (a) Motion of a scallop. By quickly clos-ing and slowly opening the two shells, the scallop pro-duces a net flow and starts moving. At low Reynoldsnumber, the net displacement is zero for this recip-rocal motion (taken from Ref. [135]). (b) Schematicof the non-reciprocal motion of an E. coli bacterium(taken from Ref. [136]). (c) Electron microscope imageof Chlamydomonas reinhardtii algae showing the flag-ella producing self propulsion by non-reciprocal motion(taken from Ref. [137]). number [cf. Fig. 7 (a)]. Likewise, any other mecha-nism based on reciprocal motions would not lead todirected motion. This is Purcell’s scallop theorem[134].The general procedure to model microswim-mers which move by body-shape deformations(or squirmers) at low Reynolds numbers consistsin solving the Stokes equation with appropriateboundary conditions for the solvent velocity field ~u on the surface of the microswimmers. Thisyields the solvent velocity field ~u , from which thestress tensor σ = η (cid:0) ∇ ⊗ ~u + ( ∇ ⊗ ~u ) T (cid:1) can be ob-tained. The latter then allows one to calculatethe total force ~F = R S d S σ ( ~r, t )ˆ n and the torque ~T = R S d S ~r × ( σ ( ~r, t )ˆ n ) which act on the mi-croswimmer, where S and d S denote the surface ofthe microswimmer and a differential element of it,respectively. Then, for a solid particle, the rigiditycondition ~u ( ~r ) = ~v + ~ω × ~r, ~r ∈ S (16)is typically assumed to apply at the surface S ofthe particle and links the particle velocity ~v andangular velocity ~ω to ~F and ~T . Finally, the torque-free ( ~T = 0) and force-free ( ~F = 0) conditionsallow one to solve for ~v and ~ω [49, 138]. Since mi-croswimmers often deform in a cyclic way, the netdisplacement during one cycle of period T is givenby R T d t ~v ( t ), which is zero for reciprocal move-ment in the regime of low Reynolds numbers [34].Thus, non-reciprocal body-shape deformations arerequired to produce directed motion. Two exam-ples of biological microswimmers that self propelby non-reciprocal motion are demonstrated in Fig.7 (b) and (c). A minimal microswimmer modelcan be constructed, e.g., based on three spheres1connected by two arms, which periodically changetheir length (three-sphere swimmer) [139–143] orbased on two spheres which can contract or ex-pand radially and are connected by an elastic arm[140, 144, 145]. VI. MODELING HYDRODYNAMICS ATTHE MANY-PARTICLE LEVEL In ensembles of microswimmers, each of them gen-erates a specific flow pattern which typically de-cays slowly in space and leads to long-ranged hy-drodynamic cross interactions among different mi-croswimmers as well as to hydrodynamic (self) in-teractions with walls and interfaces. These hydro-dynamic interactions are typically not included inmodels of dry active matter such as the ABP modeland its alternatives. One way of simulating severalinteracting microswimmers is to explicitly modelthe detailed self-propulsion mechanism of each mi-croswimmer, i.e., to alternately solve the Stokesequation with the microswimmer-solvent boundaryconditions for all swimmers simultaneously and topropagate the swimmers based on the force- andtorque-free conditions. While such an approach isconceptually relatively simple and accurate in prin-ciple, it creates a huge numerical effort and typi-cally becomes unfeasible even for moderately largemicroswimmer ensembles. In the following, webriefly discuss some alternative approaches, whichallow for more efficient numerical descriptions ofmicroswimmer ensembles. Minimal models and hydrodynamic far-fieldinteractions: To model the dynamics of large microswimmer en-sembles, an explicit modeling of the solvent flowincluding the detailed particle-solvent boundaryconditions occurring in real microswimmers is of-ten numerically so demanding that very large sys-tem sizes remain unreachable. Therefore, one oftenlooks for a compromise between the ABP model,which neglects hydrodynamic interactions and mo-mentum conservation altogether, and an explicitmodeling of the self-propulsion mechanism of allinteracting microswimmers in a given ensemble.One common approach involves formulating hydro-dynamically consistent minimal models for the col-lective behavior of microswimmers, where one doesnot explicitly describe the self-propulsion mech-anism of each microswimmer but replaces eachmicroswimmer with a simpler representative thatcreates a similar (far-field) flow pattern. To thisend, one uses a multipole expansion of the flowfield (similar to that used, e.g., in electrodynam-ics) [146–149] and only considers the leading-orderterms. In the simplest case, these are the so-called (a) Point force (b) Force dipole (c) Source dipole Figure 8. Illustration of the velocity field ~u ( ~r ) of (a) apoint force, (b) a force dipole, and (c) a source dipole. “singularity solutions” of the Stokes equation (e.g.,the flow field of a force dipole), which are thenused to replace the flow field created by each mi-croswimmer and are equivalent to the far-field flowpattern generated by the actual microswimmer tobe modeled. For example, it is well known that E. coli bacteria produce essentially the same far-field flow pattern as a force dipole (pusher) [30]and Chlamydomonas algae produce a far-field flowpattern which can be represented by the flow fieldproduced by an oscillatory force dipole [150]. Letus briefly discuss three common singularity solu-tions of the Stokes equation:(i) Point force (“Stokeslet”): The flow gener-ated by a point force ~f p = f ˆ eδ ( ~r − ~r ) placedat position ~r and pointing along the direc-tion ˆ e is similar to the far-field flow of aparticle that is driven by an external force[138, 147]. By setting ~f = ~f p in the Stokesequation [cf. Eq. (15)], the resulting velocityfield reads ~u PF ( ~r ) = f πηr [ˆ e + (ˆ r · ˆ e ) ˆ r ] , (17)where r = | ~r | , ˆ r = ~r/r , and η denotes theviscosity of the solvent. The velocity field isshown in Fig. 8 (a). Since microswimmersare force free (momentum conservation), theStokeslet solution alone is unsuitable to rep-resent them.(ii) Force dipole: The far-field solution of theStokes equation in the presence of two pointforces ~f + = f ˆ eδ ( ~r − ~r − ( l/ e ) and ~f − = − f ˆ eδ ( ~r − ~r + ( l/ e ), which are separatedby a distance l , reads ~u FD ( ~r ) = f l πηr h e · ˆ r ) − i ˆ r (18)in the limit l → r (cid:29) l and it is represented in Fig. 8 (b) for f > f < 0, where2all flow field lines are reverted, correspondsto a “puller”.(iii) Source dipole: The point-force and force-dipole solutions are obtained by solving theStokes equation together with the incom-pressibility condition ∇ · ~u = 0. In thepresence of sources of solvent molecules, theStokes equation is unchanged, but the incom-pressibility condition changes to ∇ · ~u ( ~r ) = s ( ~r ), where s ( ~r ) denotes the source density[147]. While a point source is of limited rel-evance (it would lead to a net flow of sol-vent molecules entering or leaving the do-main), the source dipole is an important sin-gular solution to the Stokes equation. Itssource density consists of two point sources s + ( ~r ) = Qδ ( ~r − ~r − ( l/ e ) (source of solventmolecules) and s − ( ~r ) = − Qδ ( ~r − ~r + ( l/ e )(sink of solvent molecules) that are separatedby a distance l , where Q > l → ~u SD ( ~r ) = Ql πr [3 (ˆ e · ˆ r ) ˆ r − ˆ e ] (19)and its velocity field is demonstrated in Fig.8 (c).Since self-propelled particles are force free, thesimplest representation of active particles by sin-gularity solutions of the Stokes equation is givenby source and force dipoles. Examples of simu-lations of microswimmer models based on thesesingularity solutions comprise, e.g., studies ofmotile suspensions of active rod-like particles [151],of the dynamics of a single molecule composedof microswimmers [152], of RTPs with hydrody-namic interactions [153], or of microswimmers nearboundaries [154].To simulate microswimmers based on singularitysolutions of the Stokes equation, one often modelsthe external fluid velocity field ~u as a sum of all mi-croswimmer singularity solutions and applies cer-tain boundary conditions on the surface of each mi-croswimmer. The velocity ~v of each microswimmeris then calculated using the force-free and torque-free conditions based on the stress tensor, as pre-viously discussed for a single microswimmer, vianumerical integration. To obtain self propulsion,one shifts the force or source dipole away fromthe center of the particles [151, 152]. Moreover,one can also combine the singularity solutions withnumerical solvers such as the Lattice-Boltzmannmethod [155] discussed below. Beside simulations,the force and source dipole models are used to de-velop continuum theories for active matter with hydrodynamic interactions, which we will discussin the last section of this article. Squirmer models: An alternative (not necessarily unrelated) ap-proach to formulate hydrodynamically consistentmodels of microswimmers is to consider squirmers,i.e., spherical particles with a prescribed solventflow along the surface (without explicitly model-ing the origin of the latter) [146, 156–165]. Onthe surface of the squirmer particle, the verti-cal fluid velocity is set to zero and the tangen-tial surface velocity is prescribed by a series offirst derivatives of Legendre polynomials, whichcan be used, e.g., to model the net effect of au-tophoresis, which leads to a slip velocity acrossthe surface of Janus particles [166]. The squirmermodel has been used in several works, e.g., incombination with the lattice-Boltzmann method[163, 167, 168] or multi-particle collision dynam-ics simulations [156, 157, 159–162, 164, 165].In contrast to the ABP model and its alternatives,microswimmer models based on combinations ofsingularity solutions of the Stokes equation or onsquirmers are momentum conserving and can cor-rectly describe hydrodynamic interactions at largeinter-particle distances for a given active system.However, they do not necessarily account for thecorrect hydrodynamic near-field interactions andare therefore mainly useful to model active systemsat low density (squirmer models, when used to rep-resent Janus particles, may serve as an exception,which is expected to correctly describe hydrody-namic interactions down to distances on the orderof the slip length [166]). These effective models areoften used also as a starting point for continuumtheories as briefly discussed further below. Explicit simulations of the solvent: In the following, we briefly introduce several nu-merical methods which are frequently used in ac-tive matter physics to explicitly determine the flowfield and to simulate hydrodynamic interactions,often beyond the far-field approximation. Lattice-Boltzmann method: One popular methodto solve fluid dynamics problems is the lattice-Boltzmann method (LBM), where one solves theBoltzmann equation instead of the (Navier-)Stokesequation and exploits the fact that the latter equa-tion can be derived from the former [163, 169–175].Interestingly, the Boltzmann equation is numeri-cally often more convenient when combined withsuitable approximations. It describes the particledistribution function f ( ~r, ~v, t ), which is the densityof particles with velocity ~v at position ~r and time t . With the so-called collision operator Ω( f ), the3Boltzmann equation reads [170, 176] ∂f∂t + ~v · ∇ ~r f + ~Fm · ∇ ~v f = Ω( f ) , (20)where m denotes the mass of the particles and ~F is the external force field acting on them. The sec-ond term on the left-hand side describes advectionof the particles with velocity ~v , whereas the thirdterm describes external forces acting on the solventparticles and affecting their velocity. The sourceterm on the right-hand side of Eq. (20) describesthe local redistribution of the solvent particles dueto collisions. This collision operator is often ap-proximated by Ω( f ) = − ( f − f eq ) /τ , which de-scribes the relaxation of the distribution f towardsthe equilibrium distribution f eq on the time scale τ and is known as the Bhatnagar–Gross–Krook(BGK) collision operator [177]. In the LBM, thecontinuous Boltzmann equation [i.e., Eq. (20)] isdiscretized in position, velocity, and time and nu-merically solved on a lattice with spacing ∆ x atdiscrete times with time step ∆ t . The velocity ~v can only take discrete values ~c i , which are givenby a discrete set { ~c i , w i } with weights w i . Thediscretized Boltzmann equation is then solved nu-merically as discussed, e.g., in Ref. [170]. To simu-late microswimmers that, e.g., create directed mo-tion through body-shape deformations, one oftendescribes the microswimmer surface as a set ofboundary links that define a closed surface andsolves the discretized Boltzmann equation togetherwith suitable boundary conditions [170]. Multi-particle collision dynamics: Another popular approach to simulate the dynam-ics of microswimmers is based on multi-particlecollision dynamics (MPCD), where, in contrastto the LBM, the solvent is represented by point-like particles which have continuous positions andvelocities [178–184]. To model active particles,one usually combines the MPCD method for thesolvent molecules with molecular dynamics (MD)simulations of the active particles, which are cou-pled to the solvent and are represented either asa single particle or by a quasi-continuous distribu-tion of particles which are connected with (time-dependent) springs and represent the surface ofa (deformable) microswimmer [185]. The MPCDmethod has been used in several works to investi-gate, e.g., chemotactic Janus colloids [186], activeparticles with phoretic interactions [187], dynam-ics of active particles in chemically active media[188], the motion of squirmers [157, 162, 164], theinfluence of hydrodynamic interactions on phaseseparation in systems of microswimmers [160], col-lective behavior of sperm cells [189], and activeparticles in filament networks [190]. Dissipative-particle dynamics: Another coarse-grained approach to modeling thesolvent is given by dissipative-particle dynamics(DPD) simulations. Here, each DPD particle rep-resents a small solvent region and, similar to theMPCD simulations, the positions and velocities ofthe DPD particles take continuous values. TheDPD particles interact via three types of effec-tive forces: A weak conservative force models thesoft repulsion of the solvent molecules, a dissipa-tive force models the friction, and a random forceaccounts for thermal fluctuations. Knowing theseforces, Newton’s equation of motion is solved forthe DPD particles to obtain the hydrodynamicsof the solvent [191–193]. This model has beenadapted, e.g., to active suspensions [194] and tomodel the self-propulsion of Janus colloids [195]. Microscopic solvent simulations: Finally, beside the previously discussed mesoscale-simulation methods, particle-based simulations ofthe solvent molecules based on direct MD simula-tions, which allow one to resolve very small spatialand temporal scales, are possible. Nevertheless,these simulations are computationally very intense,which makes it impossible to study systems of themicroscale over time scales of seconds, which arerelevant to most active matter systems. Still, thisexplicit modeling of the solvent has been success-fully used to model a self-propelled particle in aLennard-Jones solvent [196].Overall, the LBM, MPCD, and the DPD methodsare mesoscale simulation methods, which can beapplied to many hydrodynamic problems in softand active matter physics and beyond. Since theDPD method is based on particles moving in con-tinous space, it avoids lattice artifacts and allowssimulations capturing much larger length and timescales than typically possible in MD simulations.However, DPD simulations include a large num-ber of parameters (in order to model the differentforces), which have to be chosen carefully. TheMPCD method, on its part, which models the neteffect of individual collisions rather than account-ing for every collision event, is computationallyvery efficient and can be efficiently parallelized.This applies also to the LBM, which numericallysolves the Boltzmann equation and is well suited,e.g., for implementing complex (moving) bound-aries [170]. VII. CONTINUUM THEORIES OFMICROSWIMMERS WITHHYDRODYNAMIC INTERACTIONS Based on the previously discussed effective mi-croswimmer models, continuum theories for large4ensembles of particles can be formulated which ex-plicitly account for hydrodynamic interactions atleast at low density. These theories describe wetactive matter and can be formulated, e.g., basedon the puller and pusher solutions of the Stokesequation. One popular approach to account for hy-drodynamic far-field interactions is to write downthe (overdamped) equations of motion for the posi-tion and orientation of each microswimmer, whichcouple with the overall fluid velocity field. Thecontribution of each microswimmer to the overallvelocity field is modeled by singularity solutions ofthe Stokes equation such as force or source dipoles(which can be superimposed due to the linearity ofthe Stokes equation). One then derives a continu-ity equation for the N -particle probability density,which typically takes the form of a Fokker-Planckequation [197–200]. From here, one can proceedin a similar way to that of microscopic theories fordry active matter in order to derive an equation ofmotion for the one-particle density. Since the de-scribed approach to formulate continuum theoriesfor wet active matter is based on the singularitysolutions of the Stokes equation, which only de-scribe the far-field flow pattern of active particles,near-field hydrodynamic effects are not included inthis approach. However, although complicated inpractice, one can go beyond the far-field regime inprinciple, e.g., by using superimposed singularitysolutions to represent the flow field contributiondue to each swimmer or by starting with squirmer models.Let us finally mention that one can alternativelyformulate phenomenological minimal models ofwet active matter. Following a similar spirit tothe case of dry active matter, these models aregeneric in the sense that they are largely basedon considerations of symmetry, conservation laws,and dimensionality and do not refer to details suchas the specific self-propulsion mechanism, whichis employed by the microswimmers. One exam-ple of such a minimal model for wet active mat-ter is given by the phenomenological active modelH [201], which accounts for momentum conserva-tion. It is based on the active model B [121] andis closely related to the model H for equilibriumsystems [119]. The active model H addresses thephase separation behavior of wet active matter andcouples the generalized density field φ ( ~r, t ) to thevelocity field ~v ( ~r, t ) of the solvent. The general ideais that diffusive dynamics of the active particlestake place in the moving frame of the solvent andthe velocity field of the solvent is given by the cor-responding Navier-Stokes equation. There are alsophenomenological models for specific phenomenasuch as bacterial turbulence, which are based onphenomenological equations to describe the fluidvelocity field [202].More generally, there is a large number of alterna-tive approaches to formulating continuum theoriesfor microswimmers. 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