Active particles with polar alignment in ring-shaped confinement
aa r X i v : . [ c ond - m a t . s o f t ] S e p Active particles with polar alignment in ring-shaped confinement
Zahra Fazli ∗ and Ali Naji
1, 2, † School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran School of Nano Science, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran
We study steady-state properties of a suspension of active, nonchiral and chiral, Brownian particleswith polar alignment and steric interactions confined within a ring-shaped (annulus) confinementin two dimensions. Exploring possible interplays between polar interparticle alignment, geometricconfinement and the surface curvature, being incorporated here on minimal levels, we report asurface-population reversal effect, whereby active particles migrate from the outer concave boundaryof the annulus to accumulate on its inner convex boundary. This contrasts the conventional picture,implying stronger accumulation of active particles on concave boundaries relative to the convex ones.The population reversal is caused by both particle alignment and surface curvature, disappearingwhen either of these factors is absent. We explore the ensuing consequences for the chirality-inducedcurrent and swim pressure of active particles and analyze possible roles of system parameters, such asthe mean number density of particles and particle self-propulsion, chirality and alignment strengths.
I. INTRODUCTION
Active fluids constitute a fascinating class ofnonequilibrium systems and include numerous exam-ples such as bacterial suspensions and artificial nano-/microswimmers, where constituent particles, being sus-pended in a base fluid, display the ability to self-propel bymaking use of internal mechanisms and the ambient freeenergy [1–12]. Being of mounting theoretical and experi-mental interest in recent years, active fluids are found toexhibit many peculiarities, not paralleled with analogousproperties in thermodynamic equilibrium [13–18].While some of the key properties of active particles,such as their excessive nonequilibrium accumulation nearconfining boundaries [19–24] or their shear-induced be-havior in an imposed flow [25–32], can be captured withinnoninteracting models, their collective properties andphase behaviors are largely determined by particle in-teractions, including steric and hydrodynamic interac-tions [33–35]. Phenomenological models with polar align-ment interactions, which tend to align the directions ofself-propulsion of neighboring active particles, have alsoemerged as an important class of models, describing de-velopment of long-range orientational order in active sys-tems, as first proposed by Vicsek et al. [5, 36]. Coherentcollective motion, pattern formation and large-scale trav-eling structures, such as clusters and lanes, are among thenotable phenomena observed in Vicsek-type models [37–41]. In suspensions of active particles, steric interactions,near-field hydrodynamics and active stresses [42–45] playsignificant roles in engendering interparticle alignment.Geometric confinement has also emerged as an im-portant factor, determining various properties of activesystems, such as active flow patterns and stabilizationof dense suspensions into spiral vortices [46–49]. Whilemany interesting behaviors (e.g., capillary rise of active ∗ [email protected] (corresponding author) † [email protected] fluids in thin tubes, shape deformation of vesicles en-closing active particles, upstream swim in channel flow,etc; see Refs. [14, 50, 51] and references therein) havebeen reported to occur due to the near-surface behaviorof active particles, geometric constraints have also beenutilized in useful applications, such as steering living orartificial microswimmers in a desired direction [52–54].Active particles can lead to intriguing attractive andrepulsive effective interactions [55–59] between passivecolloidal objects and rigid surface boundaries as well,with interaction profiles that exhibit rather complex de-pendencies on system parameters, including the distancebetween the juxtaposed surfaces. The latter arises duepartly to the aforementioned boundary accumulation ofactive particles that can lead to pronounced near-surfaceparticle layering [57–61]. The surface accumulation ofactive particles has been associated with a number of dif-ferent factors, including hydrodynamic particle-wall cou-plings [19, 20, 23, 62, 63] and, on a more basic level, thepersistent motion of the particles [21, 64], causing pro-longed near-surface detention times. A remarkable man-ifestation of the foregoing effects is the so-called swimpressure produced by active particles on confining bound-aries, a notion of conceptual significance in describing thesteady-state properties of active fluids via possible analo-gies with equilibrium thermodynamics that has attractedmuch interest and debate in the recent past [16, 34, 65–79]. Swim pressure and effective coupling between ac-tive particles and curved boundaries are known to resultin a diverse range of phenomena, including negative in-terfacial tension [15, 80], bidirectional flows and helicalvortices in cylindrical capillaries [48], microphase separa-tion [35, 81] , and reverse/anomalous Ostwald ripening[24, 82], with the latter mechanism enabling stabilizedmesophases of monodispersed droplets (see also other re-lated works on active droplets in Refs. [83–86]).In this work, we study surface accumulation and swimpressure of active, nonchiral and chiral, Brownian parti-cles within a minimal two-dimensional model by combin-ing the three main ingredients noted above, i.e., the stericand polar interparticle alignment interactions, geomet-ric confinement and boundary curvature. The two latterfactors are brought in by adopting a ring-shaped (annu-lus) confinement. This geometry is particularly usefulin that it allows one to examine the interplay betweenconvex and concave curvatures introduced by its circu-lar inner and outer boundaries, respectively. The systemis described in the steady state using a Smoluchowskiequation, governing the joint position-orientation prob-ability distribution function of active particles, which isnumerically analyzed to derive quantities such as particledensity and polarization profiles, chirality-induced cur-rent and swim pressure on the inner and outer circularboundaries of the annulus. We thus report a surface-population reversal, whereby active particles accumulatemore strongly near the convex inner boundary of theannulus rather than its concave outer boundary. Thiscontrasts the conventional picture implying preferentialaccumulation (due to longer detention times) of activeparticles near concave boundaries relative to the convexones, a behavior that is found for noninteracting particleswithin the present model as well. We thus show that thesaid population reversal is a direct consequence of bothalignment interactions and boundary curvature and willbe absent in the absence of either of them. The implica-tions of this effect for the chirality-induced current andswim pressure of active particles are then explored in de-tail. We describe our model and its governing equationsin Sections II and III, discuss our results in Section IV,and conclude the paper in Section V. II. MODEL
We consider a two-dimensional minimal model of ac-tive Brownian particles [87] confined between two imper-meable, concentric, circular boundaries with radii R and R > R , forming a ring-shaped (annulus) confinement;see Fig. 1a. Each particle has an intrinsic, constant, self-propulsion speed v > along its instantaneous directionof motion, determined by the unit vector ˆ u , and may alsopossess an intrinsic, constant, in-plane angular velocity(chirality) of signed magnitude ω ; see, e.g., Refs. [88–92]. Particles are assumed to interact via pair potentialsof two main types: (i) a phenomenological alignment in-teraction of strength U > to be modeled through thedot-product of particle orientations as − U ˆ u i · ˆ u j for thepair of particles labeled by i and j [93]; (ii) a local stericdelta-function potential U δ ( r i − r j ) for the given pair.The latter provides a formally straightforward route toincorporate a finite excluded area U for each particle inour later continuum formulation. Thus, the total pairinteraction energy U ij ≡ U ( r i , ˆ u i ; r j , ˆ u j ) , is U ij k B T = − U ˆ u i · ˆ u j + U δ ( r i − r j ) . (1)Particles interact with the circular boundaries of the confinement with the repulsive simple harmonic potential V i = k r i − R ) Θ( R − r i ) + k r i − R ) Θ( r i − R ) , (2)where V i ≡ V ( r i ) and r i is the radial distance of the i thparticle from the origin, see Fig. 1, Θ( · ) is the Heavisidestep function, and k = k are the respective harmonicconstants. These constants are fixed at a sufficiently largevalue to establish nearly hard boundaries, being also as-sumed to be torque-free (no dependence of V i on the par-ticle orientation); hence, the swim pressure on them isexpected to be a state function [66, 68]. It shows nodependence on the interfacial potential strengths. III. GOVERNING EQUATIONS
The translational and rotational dynamics of activeparticles are described by the Langevin equations ˙ r i = v ˆ u i − µ t ∇ i V i + X j = i U ij + p D t η i ( t ) , (3) ˙ ϕ i = ω − µ r ∂ ϕ X j = i U ij + p D r ζ i ( t ) , (4)with the shorthand notations ∇ i = ∂/∂ r i and ∂ ϕ = ∂/∂ϕ . Here, η i and ζ i are the translational and ro-tational white noises of zero mean and unit variances,with D t and D r being the single-particle translationaland rotational diffusivities, respectively. To ensure thatthe system behavior reduces to that of an analoguesequilibrium system, when the active sources are put tozero, v = ω = 0 , the diffusivities are assumed to ful-fill the Smoluchowski-Einstein-Sutherland relations D t = µ t k B T and D r = µ r k B T , where k B T is the ambient ther-mal energy scale and µ t and µ r are the translational androtational (Stokes) mobilities, respectively [94].Equations (3) and (4) can standardly be mapped toa time-evolution equation involving one- and two-pointPDFs, with the latter originating from the two-particleinteractions. Without delving into further details, weadopt a mean-field approximation as a closure scheme forsuch an equation by neglecting interparticle correlationsand assuming P ( r , ϕ, r ′ , ϕ ′ ; t ) = P ( r , ϕ ; t ) P ( r ′ , ϕ ′ ; t ) ,which is expected to hold in sufficiently dilute suspen-sions. This leads to an effective Smoluchowski equation,involving an effective translational flux velocity − µ t ( ∇U ) experienced on average by each particle due to its in-teractions with other particles, where the averaged two-particle interaction U = R d r ′ d ϕ ′ U P ( r ′ , ϕ ′ ) . Such anequation is nonlocal and nonlinear and can be expressedas ∂ t P = −∇ · (cid:2)(cid:0) v ˆ u − µ t ∇ ( V + U ) (cid:1) P − D t ∇P (cid:3) − ∂ ϕ (cid:2)(cid:0) ω − µ r ∂ ϕ U (cid:1) P − D r ∂ ϕ P (cid:3) . (5) - 0 5 15 25 35 45 00.10.20.30.40.50.60.70.8 (b) - (c) OO (a) Figure 1. (a) Left: Active particles are described by their position and orientation vectors parametrized in polar coordinatesas r = r (cos θ, sin θ ) and ˆ u = (cos ϕ, sin ϕ ) ; Right: They are confined within a ring-shaped confinement, or annulus, withimpermeable circular boundaries of radii R and R from the origin O. (b) and (c) Steady-state PDF, ˜ P (˜ r, ψ ) of activeparticles within the shown confined geometry is plotted in the ψ − ˜ r plane for nonaligning (panel b, U = 0 ) and interaligningparticles (panel c, U = 10 ) with fixed parameter values are ˜ R = 10 , ˜ R = 40 , Pe = 10 , ˜ ρ = 0 . and Γ = 1 . Our focus will be on the steady-state solution ˜ P ( r , ϕ ) .Because of the rotational symmetry in the present geom-etry, the angular dependence of the PDF can be assumedto occur only through the relative angle ψ = ϕ − θ , al-lowing the relations ∂ ϕ P = − ∂ θ P = ∂ ψ P and R d r ′ d ϕ ′ ≡ R r ′ d r ′ d ψ ′ to be used, where necessary. In this case, themean-field interaction energy reads U k B T = − U Z r ′ d r ′ d ψ ′ cos( ψ ′ − ψ ) P ( r ′ , ψ ′ )+ U Z d ψ ′ P ( r, ψ ′ ) . (6)Equation (5) is supplemented by no-flux boundary con-ditions in the radial direction on the two circular bound-aries at r = R and R and periodic boundary conditionover the angular coordinate ψ for r ∈ [ R , R ] . Since wehave assumed a closed system with a fixed number N of active particles within the annulus, the normalizationcondition for the PDF can be expressed using the meannumber density ρ = N/ [ π ( R − R )] of particles as Z R R Z π − π r d r d ψ P ( r, ψ ) = πρ (cid:0) R − R (cid:1) . (7) A. Dimensionless representation
We proceed by rescaling units of length/time with thecharacteristic length/time a = ( D t /D r ) / and /D r , re- spectively, and the units of energy with k B T ; thus, e.g.,the boundary interaction energy is rescaled as ˜ V (˜ r ) = V ( a ˜ r ) / ( k B T ) . The PDF is suitably rescaled as ˜ P (˜ r, ψ ) = P ( a ˜ r, ψ ) /ρ . The parameter space is thus spanned bythe set of dimensionless parameters defined by the (swim)Péclet number and the chirality strength Pe = vaD r , Γ = ωD r , (8)respectively, the rescaled interaction parameters U , ˜ U = U /a , and ˜ k , = k , a / ( k B T ) , the rescaled radii ˜ R , = R , /a , and the rescaled mean particle density ˜ ρ = ρ a . The steady-state Smoluchowski equation canthen be expressed in dimensionless units as r ∂ ˜ r (cid:16) ˜ r ˜ J ˜ r (cid:17) + 1˜ r ∂ ψ ˜ J ψ = 0 , (9)where we have defined the rescaled spatial and angularprobability flux densities ˜ J ˜ r and ˜ J ψ , respectively, as ˜ J ˜ r = − ∂ ˜ r ˜ P + (cid:16) Pe cos ψ − ∂ ˜ r ˜ V (cid:17) ˜ P − ˜ U ˜ ρ (cid:18) ∂ ˜ r Z d ψ ′ ˜ P (˜ r, ψ ′ ) (cid:19) ˜ P , (10) ˜ J ψ = − r (cid:0) r (cid:1) ∂ ψ ˜ P + ( − Pe sin ψ + Γ˜ r ) ˜ P + U ˜ ρ ˜ r (cid:0) r (cid:1) (cid:18)Z ˜ r ′ d˜ r ′ d ψ ′ sin( ψ ′ − ψ ) ˜ P (˜ r ′ , ψ ′ ) (cid:19) ˜ P , (11)where R ˜ R ˜ R R π − π ˜ r d˜ r d ψ ˜ P (˜ r, ψ ) = 1 . Equation (9) is an integrodifferential PDE, which is solved using finite- (a) (b) Figure 2. (a) Rescaled number density profile of active particles, ˜ ρ (˜ r ) , as a function of the radial distance inside the annulusin the absence ( U = 0 ) and presence ( U = 10 ) of alignment interactions. Inset shows the corresponding polarization profiles, ˜ m (˜ r ) . Here, we have fixed Γ = 1 and other parameters are as in Fig. 1. (b) Same as (a) but plotted for different values ofchirality strength, as indicated on the plot both without (inset, U = 0 ) and with (main set, U = 10 ) alignment interactions. element methods within the annulus subject to the afore-mentioned boundary conditions. Other rescaled steady-state quantities such as the particle density profileswithin the annulus and swim pressure on the boundariesfollow directly from the numerically obtained PDF [24].The choice of dimensionless parameter values and theirrelevance to realistic systems is discussed in Appendix A.Since Eq. (9) remains invariant under chirality and orien-tation angle reversal, Γ → − Γ and ψ → − ψ , we restrictour discussion to only positive (counterclockwise) valuesof Γ , bearing in mind that chirality-induced currents willbe reversed for negative values of Γ . IV. RESULTSA. Particle distribution and radial density profile
In Figs. 1b and c, the steady-state PDF of active par-ticles inside the annulus is plotted in the ψ − ˜ r planein the absence (panel b, U = 0 ) and presence (panel c, U = 10 ) of alignment interactions between the particlesfor a representative set of parameter values. As seen,most of active particles accumulate near the boundaries(represented by larger probability densities appearing inyellow and red colors in the plots), a well-known conse-quence of the persistent motion of particles [62]. Also, inboth the absence and the presence of alignment interac-tions, the typical orientations of particles on the outer(inner) boundaries are expectedly in the outward (in-ward) directions pointing away (toward) the origin of theannulus, respectively. The PDF of nonchiral active par-ticles [24] would be peaked exactly at ψ = 0 and ψ = ± π on the outer and inner boundaries, respectively, reflect-ing the normal-to-boundary orientations of the particles.In the plots, however, we have taken a relatively small,counterclockwise, particle chirality ( Γ = 1 ) that producesa shift of the probability-density peaks from the noted angular orientations to the first and third angular quad-rants near the outer and inner boundaries, respectively.Also, as seen in Fig. 1b, nonaligning particles are morestrongly accumulated near the outer boundary than theinner one, in accord with the standard paradigm that ac-tive particles spend longer ‘detention’ times at concaverather than convex boundaries [95–97]. Remarkably, wefind a reversed situation in the presence of interparti-cle alignment (panel c) with larger particle probabilitiesappearing near the inner boundary. This appears to sug-gest an alignment-induced crossover between two differ-ent configurational states with active particles primarilyattracted to only one of the boundaries. Such a behav-ior, which shall examine more closely later, can also bediscerned from the local density of active particles insidethe annulus defined as ˜ ρ (˜ r ) = Z π − π d ψ ˜ P (˜ r, ψ ) . (12)This quantity being plotted in Fig. 2a indicates a re-versal in the relative accumulation of active particlesnear the two boundaries, when the alignment interac-tion strength U is changed from U = 0 (dotted curve)to U = 10 (solid curve). The figure also shows that thewide plateau present in the density profile of nonaligningparticles changes to a linear stretch of particle densitybetween the inner and outer peaks, when the alignmentinteraction is switched on. The active particles withinthis linear region of the density profiles turn out to showno specific orientational order, as it can be verified usingthe first angular moment of the PDF defined as ˜ m (˜ r ) = Z π − π d ψ cos( ψ ) ˜ P (˜ r, ψ ) . (13)Being shown in the inset of Fig. 2a, ˜ m (˜ r ) reflects theexpected outward- (inward-) pointing mean polarizationof the particles near the outer (inner) boundaries but no Figure 3. Mean fraction of active particles found within theinner and outer semi-annuli, Q ( ˜ R c ) and − Q ( ˜ R c ) , respec-tively (see the text for definitions), as functions of the align-ment interaction strength for fixed parameter values ˜ R = 10 , ˜ R = 40 , Pe = 10 , ˜ ρ = 0 . , Γ = 1 . Symbols are numericaldata and curves are guides to the eye. mean orientational order ( ˜ m ≃ ) elsewhere across thecentral regions, neither for nonaligning nor for interalign-ing particles.When the chirality strength Γ , is increased, as shown inFig. 2b, the near-boundary accumulation of active parti-cles is suppressed in both the absence (inset, U = 0 ) andthe presence (main set, U = 10 ) of alignment interac-tions. One eventually finds a homogeneous density pro-file at elevated Γ (see the red dotted curve for Γ = 100 ),consistent with known results [98, 99], indicating that thebehavior of chiral active particles reduces to that of pas-sive particles in the limit of infinite chirality (see Ref. [24]for a systematic derivation). At intermediate Γ , however,our results reveal nontrivial variations in the density pro-file, with the density peak at the inner (outer) boundariesfollowed (preceded) by a shallow dip, indicating partialdepletion of active particles (green dot-dashed curve for Γ = 3 ). This is a consequence of the fact that chiral-ity effects in suppressing the persistent motion of activeparticles are more apparent near the boundaries and thedensity peaks are suppressed more strongly than the lin-ear bridge connecting them, as Γ is increased. B. Surface-population reversal
Further insight into the alignment-induced effects canbe obtained by conventionally defining the inner andouter fractions Q ( ˜ R c ) and − Q ( ˜ R c ) of active particlesas those found in radial distances ˜ r < ˜ R c and ˜ r > ˜ R c ,respectively, where ˜ R c = ( ˜ R + ˜ R ) / is the mean radiusof the annulus. We thus have Q ( ˜ R c ) = R ˜ R c ˜ R ˜ r d˜ r ˜ ρ (˜ r ) R ˜ R ˜ R ˜ r d˜ r ˜ ρ (˜ r ) . (14)These fractions are shown as functions of the alignmentinteraction strength in Fig. 3. In the nonaligning case (a)(b) Figure 4. (a) Dependence of U c , the alignment strength giv-ing the maximum population reversal, on the mean numberdensity (main set) and chirality strength (inset) of active par-ticles for fixed ˜ R = 10 , ˜ R = 40 , Pe = 10 , and with
Γ = 1 (main set) and ˜ ρ = 0 . (inset). (b) Dependence of U c onthe radii of the inner/outer boundaries of the annulus for fixed ˜ ρ = 0 . , Pe = 10 , Γ = 1 , and with ˜ R = 100 (main set) and ˜ R = 10 (inset). Symbols are numerical data and curves areguides to the eye. ( U = 0 ), a larger fraction of particles is found in theouter semi-annulus, which, as noted before, is becauseself-propelling particles accumulate more strongly nearconcave than convex boundaries due to their lingered de-tention times. As seen in the figure, the inner fraction Q ( ˜ R c ) increases with U and − Q ( ˜ R c ) decreases withit until they are equalized at a certain value of U ∗ (here, U ∗ ≃ ). This particular value corresponds to the onsetof a counterintuitive surface-population reversal , beyondwhich active particles are more strongly accumulated bythe convex inner boundary with the smaller radius ofcurvature rather than the concave outer boundary withthe larger radius of curvature. The maximum popula-tion reversal is achieved at a slightly larger value of U c (here, U c ≃ ), where Q ( ˜ R c ) displays a global maxi-mum and − Q ( ˜ R c ) a global minimum. Beyond thispoint, both quantities level off and gradually tend to-ward 1/2, indicating an even distribution of particles de-veloping in the infinite U limit within the annulus (notshown). The designated values U ∗ and U c are typi-cally found to be close and, to examine the dependenceof the population reversal phenomenon on other systemparameters, we concentrate on the latter quantity.Figure 4a (main set) shows the dependence of the max-imum population reversal plotted as a function of themean number density of particles inside the annulus. Asseen, a larger (smaller) mean density, ˜ ρ , necessitates aweaker (stronger) alignment interaction strength, U c , toachieve the maximum population reversal. The mono-tonically decreasing trend fits closely with a functionaldependence of the form U c ∼ ˜ ρ − α with α ≃ , a re-markable scaling-like behavior that remains to be under-stood. Figure 4a (inset) shows that U c increases almostlinearly with chirality strength, which is plausible as, forlarger chirality strengths, particle directions change morerapidly, requiring a larger value of alignment interactionstrength to establish surface-population reversal.Figure 4b, on the other hand, shows that population re-versal is more easily established as the difference betweenthe radii of the inner and outer boundaries increases. At afixed value of the outer boundary radius (here R = 100 ,main set) or at a fixed value of the inner boundary radius(here R = 10 , inset), U c increases as the radial widthof the annulus is reduced and vice versa. This is indica-tive of the fact that in a narrower ring-shaped confine-ment, active particles tend distribute more evenly withinthe confinement and easily interchange between the twocircular boundaries, necessitating a stronger alignmentinteraction to produce population reversal. Our numeri-cal results (not shown) indicate that U c does not sig-nificantly vary with Pe > for which self-propulsiondominates particle diffusion (for Pe < , particles aremostly dispersed nearly uniformly within the confine-ment and varying the alignment strength does not ef-fectively change particle distribution). C. Chirality-induced current
Chiral active particles can generate net rotational cur-rents near circular boundaries. This effect emerges as aresult of the deviation of the most probable orientation ofchiral particles from the normal-to-surface direction (seeSection IV A), creating a tangential velocity componenton the surface. To examine the steady-state chirality-induced current in the present context with alignmentand steric interactions between particles (see Ref. [24]for the special case of noninteracting active particles),we begin by integrating the Smoluchowski equation overthe rotational degree of freedom, ϕ , which gives a rela-tion as ∇ · ˜ J = 0 , where the current density, ˜ J = ( ˜ J ˜ r , ˜ J θ ) ,has the following two components ˜ J ˜ r = Z π − π d ψ ˜ A ˜ r (˜ r, ψ ) , ˜ J θ = Z π − π d ψ ˜ A θ (˜ r, ψ ) , (15)where ˜ A ˜ r = h Pe cos ψ − ∂ ˜ r ˜ V i ˜ P − ∂ ˜ r ˜ P − ˜ U ˜ ρ ˜ P ∂ ˜ r Z d ψ ′ ˜ P , (16) -1.2-0.8-0.400.40.8 10 20 30 40-0.500.5 10 15 20 25 Figure 5. Main set: Rescaled chirality-induced current, ˜ J θ , asa function of radial distance for active particles with ( U = 40 )and without ( U = 0 ) alignment interactions. Inset: ˜ J θ for aselected set of U values (here, U c = 8 ). Other parametersfixed as ˜ R = 10 , ˜ R = 40 , Pe = 10 , ˜ ρ = 0 . and Γ = 1 . ˜ A θ = Pe sin ψ ˜ P − r ∂ ψ ˜ P (17) + U ˜ ρ ˜ r ˜ P Z ˜ r ′ d˜ r ′ d ψ ′ ˜ P sin( ψ − ψ ′ ) . The radial current density component, ˜ J ˜ r , is zero forall parameter values. The angular component, ˜ J θ , repre-sents the chirality-induced current and is always nonzeroon the boundaries, see Fig. 5, while it falls off to zeroas one moves away from the boundaries, where the netcurrent density of particles passing through a particu-lar point vanishes [24]. Figure 5 (main set) also showsthis quantity for different values of the alignment interac-tion strength for a representative set of parameter valuesfor which the maximum population reversal occurs at U c = 8 (being nearly equal to its onset).The results rep-resented in Fig. 5 are typical behaviors obtained for thechirality strength Γ = 1 that are shown for the sake ofillustration and we can observe similar results for otherrepresentative values of Γ near their corresponding pop-ulation reversal. Thus, while in the absence of align-ment interactions ( U = 0 , red dashed curve), the innerpopulation shows clockwise (negative current) of smallermagnitude relative to the outer population that showscounterclockwise (positive current) of larger magnitude,the situation is reversed for active particles with strongalignment interactions ( U = 40 , blue solid curve).For the rotational current on the outer boundary, in-creasing U only results in a smaller current, one thatnever changes sign on this boundary as it diminishes withstrengthening the alignment interaction. For the currenton the inner boundary, we find a more complex behav-ior. In the inset of Fig. 5 we show ˜ J θ near the innerboundary for a few different values of alignment strengthchosen around the onset of population reversal. As seen,by increasing U from U = 4 (purple dashed curve) to U = U c = 8 (green dotted curve), the shallow dip withnegative rotational current close to the inner boundary (a) (b) s w i m p r e ss u r e s w i m p r e ss u r e Figure 6. (a) Rescaled swim pressures, ˜ P and ˜ P , on the inner and outer boundaries, respectively, as functions of Pe with( U = 10 ) and without ( U = 0 ) alignment interactions for fixed ˜ R = 10 , ˜ R = 40 , ˜ ρ = 0 . and Γ = 1 . (b) Same as (a) butplotted for swim pressures as functions of alignment interaction strength, U , for fixed ˜ R = 10 , ˜ R = 40 , Pe = 10 , ˜ ρ = 0 . and Γ = 1 . Symbols are numerical data and curves are guides to the eye. turns to a region of positive current with a nonmono-tonic profile with a pronounced peak. On increasing U further to U = 9 (orange dot-dashed curve) and then U = 10 (pink dashed curve), the positive current pro-file is suppressed, even though it can now exhibit regionswith both positive and negative currents ( U = 9 ) and apronounced dip with a strong negative current ( U = 10 )near the inner boundary. On increasing U further, oneonly finds a negative current on this boundary, whosemagnitude is further enhanced.The generation of rotational current profile of varyingsign near the population reversal are reminiscent of par-ticle layering that develops at the onset of flocking tran-sition in Vicsek-type models [38, 100], even though thering-shaped confinement is expected to have a dominantrole in the present context. D. Swim pressure
Other interesting quantities we can explore in this sys-tem are swim pressures on the inner and outer bound-aries, denoted by ˜ P and ˜ P , respectively, which can becalculated by integrating the force density exerted by ac-tive particles on the boundaries as ˜ P = Z d˜ r ˜ ρ (˜ r ) ∂ ˜ r ˜ V , ˜ P = Z ∞ Λ d˜ r ˜ ρ (˜ r ) ∂ ˜ r ˜ V . (18)where Λ is a radial distance away from the two bound-aries and inside the annulus (where ∂ ˜ r ˜ V vanishes), whichwe arbitrarily set equal to the mean radius of the annulus.The swim pressures ˜ P and ˜ P both turn out to bemonotonically increasing functions of Péclet number inthe presence and absence of alignment interactions, asseen in Fig. 6a (for the given set of fixed parametervalues in the figure, the maximum population reversaloccurs around U c ≃ . ; hence, the two cases shownoccur are far from the onset of the reversal). For the case with aligning particles, ˜ P is larger than its cor-responding value in the nonaligning case (compare bluesquares and triangle-downs), but ˜ P shows the reverseproperty (compare red circles and triangle-ups). Thus,alignment interactions increase the swim pressure on theinner boundary and decrease it on the outer one. At small Pe , ˜ P and ˜ P converge and tend to zero, as expected. ˜ P and ˜ P diverge as soon as Pe increases beyond Pe = 1 .Also, for all nonvanishing Pe , the difference between ˜ P and ˜ P is larger in the aligning case, which is a direct con-sequence of population reversal and particles departingfrom outer to the inner boundary.Figure 6b shows ˜ P and ˜ P as functions of the align-ment strength (for the given set of fixed parameter values, U c ≃ ). As seen, ˜ P first increases rapidly to a max-imum value and then smoothly drops to finite values atlarge U . ˜ P behaves in an opposite way, as it first dropsto almost zero and then slowly approaches a small andfinite value at large U . For U = 0 , the swim pressure onthe inner boundary is slightly smaller than the outer one.Upon switching on the alignment interaction, particlesstart to migrate to the inner boundary and this causes ˜ P to rapidly increase and ˜ P to decrease with U untilthey reach their maximum and minimum, respectively,at U c , with the overall trends occurring in accord withFig. 3. In Appendix B, we derive an analytical expres-sion for the swim pressure in the presence of alignmentinteractions, corroborating the foregoing discussions.In Fig. 7, we show ˜ P and ˜ P as functions of radiiof the circular boundaries (for all values of ˜ R and ˜ R shown in the figure, U = 10 is larger than U c ). In theabsence of alignment interactions, ˜ P and ˜ P vary weaklywith both ˜ R and ˜ R and the difference between ˜ P and ˜ P is small. In the presence of alignment interactions, wefind a different behavior. Figure 7a shows that the swimpressure on the inner boundary (blue squares) decreasesmonotonically as a function of ˜ R , attaining its maximumvalue at the smallest inner radii ˜ R = 10 , corresponding s w i m p r e ss u r e (a) (b) s w i m p r e ss u r e Figure 7. (a) Rescaled swim pressures, ˜ P and ˜ P , on the inner and outer boundaries, respectively, as functions of the innerradius, ˜ R , with ( U = 10 ) and without ( U = 0 ) alignment interactions for fixed ˜ R = 100 . (b) Same as (a) but plotted forswim pressures as functions of the outer radius, ˜ R , for fixed ˜ R = 10 . Other parameters are fixed in the plots as Pe = 10 , ˜ ρ = 0 . and Γ = 1 . Symbols are numerical data and curves are guides to the eye. to the largest confinement, given that ˜ R = 100 is fixed.This is reflective of stronger active-particle accumulationon the inner boundary as the system is deep inside thepopulation-reversal regime. In this case, the swim pres-sure on the outer boundary (red triangle-ups) varies onlyweakly with ˜ R . The aforementioned behaviors are cor-roborated by those in Fig. 7b, depicting ˜ P and ˜ P asfunctions of ˜ R . Also, as seen in Fig. 7a, all four curvesare found to converge to a common value as the ring-shaped confinement becomes narrower, with ˜ R tendingtoward ˜ R . This is due to the fact that in extremelynarrow annuli, active particles tend to distribute almostevenly within the confinement with equal probabilities ofinteracting with either of the boundaries, regardless oftheir alignment interactions. V. SUMMARY
In this paper, we study a two-dimensional systemof nonchiral and chiral active Brownian particles con-strained to move in a circular ring-shaped confinement(annulus) with impermeable confining boundaries. Theactive particles are assumed to have constant intrinsic,linear (self-propulsion) and angular, velocities and in-teract through alignment as well as steric pair poten-tials. The alignment interaction between particles is as-sumed to have a dot-product form in a way that it tendsto align the self-propulsion directions of a particle pair.The steady-state properties of active particles are ana-lyzed using a probabilistic Smoluchowski equation, whichis solved numerically. This equation takes the form ofan integrodifferential equation due to nonlocal couplingterms generated by the particle interactions that are thentreated using a mean-field approximation.While active particles are typically known to accu-mulate more strongly at concave rather than convexboundaries due to their longer near-boundary detentiontimes in the former case [24, 95–97], we show that the presence of alignment interactions causes a reverse phe-nomena to take place in the present setting; hence, alarger fraction of active particles are found to accumu-late at the inner boundary of the annulus, which is convexand has a smaller radius of curvature. Such a surface-population reversal is both an alignment-induced and acurvature-induced effect and will be absent in the absenceof alignment interactions and/or in a planar confinement.The effect is quite robust in the sense that it emergesover a wide range of moderately large alignment interac-tion strengths, U , and moderately large Péclet numbers(with the latter required to be only so large as to fa-cilitate surface accumulation of active particles againstparticle diffusion into the bulk). The population rever-sal is typically maximized at an alignment strength U c that is only slightly larger than the onset of the rever-sal, indicating a rapid crossover at the onset, followed bya nonmonotonic behavior, i.e., a relatively sharp humpand then decay of the surface populations down to cer-tain saturation values, as U is increased. We also findthat the wider the annulus (the larger the difference be-tween the outer and inner radii) the weaker will be thealignment strength required to cause the reversal, andvice versa. Similar results are found for the dependenceof the population reversal on the mean particle numberdensity, as in a more dilute (denser) system the reversalis realized for a stronger (weaker) alignment strength.We study the implications of the aforementioned effectfor the chirality-induced current at and swim pressure onthe inner and outer boundaries of the annulus. As gen-erally expected, particle chirality leads to suppression ofactivity-induced effects such as boundary-accumulationof active particles and, as such, chiral active particlesrequire a larger alignment strength to establish popu-lation reversal relative to nonchiral active particles. Aremarkable effect emerges in the case of near-boundarychirality-induced currents. While for alignment strengthssufficiently far from the onset of population reversal, wefind rotational current of a single well-defined sign (clock-wise or counterclockwise) forming near each of the innerand outer boundaries (albeit with opposing signs on theinner relative to the outer boundary), the situation turnsout to be more complex near the onset of the populationreversal; hence, the currents near the two boundaries cantake similar signs and thus rotate in the same direction,and multiple ‘layers’ of rotating currents can be seen,especially for weak chirality strengths.As the surface-population reversal is directly caused bythe presence of interparticle alignment, it may be tempt-ing to compare it with the bulk flocking transition inVicsek-type models [5]. Our results indicate that theconfinement and boundary curvature play key roles inregulating the population reversal, making it possible tocompare it also with surface-induced transitions, e.g., inwetting [101, 102] and capillary [50, 103] systems, and incounterion condensation phenomena [104, 105].Our model treats the problem at hand on the levelof a minimal model of active Brownian particles [87].Hence, it neglects several other important factors thatcould be considered for a more comprehensive analysisin the future. These include the roles of hydrodynamicinteractions between particles and between particles andthe boundaries (see, e.g., Refs. [62, 106]). In thesecontexts, an interesting problem would be that of theso-called pusher and puller microswimmers with dipo-lar flow fields (see, e.g., Refs. [10, 107, 108]) and howthe ensuing active stress due to these markedly differenttypes of active particles might influence the behaviorspredicted with the current setting, especially at elevatedarea fractions. Since the far-field hydrodynamic inter-actions may be screened by confinement effects [42], thenear-field hydrodynamics will be of particular interestin the analysis of active particle distributions within anannulus. The interparticle alignment due to such hy-drodynamic effects may thus compete or cooperate withthe dot-product alignment model considered here, pavingthe way for more intriguing possible scenarios. When therodlike nature of active particles is accounted for, particleinteractions with the boundaries will not be torque-freeanymore and the swim pressure can vary depending onthe type of surface potentials [68, 109]. Active rods incircular geometries [110, 111] can also be subjected toimposed shear flows, constituting another potential di-rection of research that can be explored in the future. VI. CONFLICTS OF INTEREST
There are no conflicts of interest to declare.
VII. ACKNOWLEDGEMENTS
Z.F. thanks A. Partovifard and M. R. Shabanniyafor useful discussions and comments. A.N. acknowl-edges partial support from the Associateship Scheme ofThe Abdus Salam International Centre for TheoreticalPhysics (Trieste, Italy). We thank the High PerformanceComputing Center of the Institute for Research in Fun-damental Sciences (IPM) for computational resources.
VIII. AUTHOR CONTRIBUTIONS
Z.F. performed the theoretical derivations and numer-ical coding, generated the output data and produced thefigures. Both authors analyzed the results, contributedto the discussions and wrote the manuscript. A.N. con-ceived the study and supervised the research.
Appendix A: Choice of parameter values
We fix the interparticle and particle-wall steric inter-action strengths at representative values of ˜ U = 10 and ˜ k = ˜ k = 10 and vary other system parameterswith a range representative values (i.e., Pe = 1 − , ˜ R , ˜ R = 10 − , U = 0 − , ˜ ρ = 0 . − . , Γ = 0 − )to explore different regions of the parameter space. Themain advantage of using dimensionless values is that theycan be mapped to a wider range of actual parameter val-ues relevant to realistic cases of synthetic and biologi-cal active particles [2, 112–116]. For instance, choosing a ≃ µ m and D r in the range D r ≃ . − (thermal dif-fusivity) to − (active tumbling), the range of Pécletnumbers used here to discuss the representative behaviorof the system can be mapped to self-propulsion speedsof up to v ≃ µ m / s . Artificial active particles can takea wide range of chirality strengths as well, with exam-ples furnished by curved self-propelled rods ( | Γ | ≃ − )[117, 118], self-assembled rotors ( | Γ | ≃ − ) [119], andJanus doublets ( | Γ | ≃ − ) [120]. Our investigatedrange of rescaled mean densities ( ˜ ρ = 0 . − . ) canalso be compared with those in Refs. [110, 111]. Appendix B: Analytic expressions for swim pressure
Here, we derive useful analytical expressions for theswim pressure for the system of confined interacting par-ticles described in the text. Swim pressure on each of theinner/outer circular boundaries can be calculated by inte-grating the force exerted per unit length of that bound-ary, which was achieved by numerically calculating thePDF as discussed in the text; see Section IV D. The saidanalytical expression can be established as follows.Starting from the Smoluchowski equation (5) and writ-ing it in polar coordinates ( r, ψ ), we have0 ∂ t P = − r ( ∂ r r " (cid:0) v cos ψ − µ t ∂ r V − µ t ∂ r U (cid:1) P − D t ∂ r P + ∂ ψ (cid:20)(cid:18) − v sin ψ + rω − r (cid:0) µ t + r µ r (cid:1) ∂ ψ U (cid:19) P − r (cid:0) D t + r D r (cid:1) ∂ ψ P (cid:21) ) , (B1)The steady-state local density of particles follows as thezeroth angular moment of the PDF with ∂ t ρ = 0 . Hence,integrating the above equation over the angular coordi-nate and substituting the definition of the swim pressuresfrom Eq. (18), we find these latter quantities in termsof the mean particle density and the local orientationalorder parameter (first moment) defined through Eq (13).Thus, for the inner boundary (with the same procedure being applicable to P , not to be repeated here), we have P = ρ µ t (cid:20) D t (cid:18) ρ U (cid:19)(cid:21) + vµ t Z d r m ( r ) . (B2)Multiplying both sides of Eq. (B1) by cos ψ and thenintegrating over the angular coordinate gives the steady-state equation, governing the first moment of equation(B1) in the steady state and in the regime of sufficientlylarge radii of curvature as D r m ≃ r ( − vm + µ t m ∂ r V + D t ∂ r m + U D t m ∂ r ρ ) − ∂ r (cid:16) v ρ + m ) − µ t m ∂ r V − D t ∂ r m − U D t m ∂ r ρ (cid:17) − ω Z d ψ sin ψ P ( r, ψ ) − U D r Z d r ′ Z d ψ ′ r ′ P ( r ′ , ψ ′ ) Z d ψ sin ψ sin( ψ ′ − ψ ) P ( r, ψ ) , (B3)where m ( r ) is the second angular moment of the PDF.Using the above relation in Eq. (B2), we arrive at the approximate decomposition for the swim pressure as P ≃ P f + P ch + P ex + P int , with the definitions P f = ρ µ t (cid:18) D t + U ρ D t v D r (cid:19) , (B4) P ch = − vωµ t D r Z π − π Z d ψ d r sin ψ P ( r, ψ ) , (B5) P ex = vµ t D r Z d r r ( − vm + µ t m ∂ r V + D t ∂ r m + U D t m ∂ r ρ ) , (B6) P int = − vU µ t Z d r ′ Z π − π d ψ ′ r ′ P ( r ′ , ψ ′ ) Z d r Z π − π d ψ sin ψ sin( ψ ′ − ψ ) P ( r, ψ ) . (B7)The term P f gives the standard swim pressure of nonchi-ral active particles on a flat boundary and P ex is the ex-cess pressure due to the curvature of the circular bound-aries [24]. The first term in the expression for P f on ther.h.s. of Eq. (B4) is the ideal gas pressure, the secondterm is the second virial term due to steric interactionsbetween the particles that scale with U , while the lastterm is the corresponding nonequilibrium swim pressure. P ex also involves an additional term explicitly dependenton the steric particle interactions, but neither of the threecontributions P f , P ch and P ex vary explicitly with the alignment interactions, which enter implicitly in nonidealterms through the solution of the PDF. It is only the lastterm P int that explicitly scales with the alignment inter-action strength, U . P ch is also the only term that ex-plicitly scales with the chirality strength. 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