Quorum-sensing active particles with discontinuous motility
Andreas Fischer Friederike Schmid Thomas Speck





Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany
Graduate School Materials Science in Mainz, Staudinger Weg 9, 55128 Mainz, Germany

Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany

Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 7-9, 55128 Mainz, Germany


  We develop a dynamic mean-field theory for polar active particles that interact through a self-generated field, in particular one generated through emitting a chemical signal. While being a form of chemotactic response, it is different from conventional chemotaxis in that particles discontinuously change their motility when the local concentration surpasses a threshold. The resulting coupled equations for density and polarization are linear and can be solved analytically for simple geometries, yielding inhomogeneous density profiles. Specifically, here we consider a planar and circular interface. Our theory thus explains the observed coexistence of dense aggregates with an active gas. There are, however, differences to the more conventional picture of liquid-gas coexistence based on a free energy, most notably the absence of a critical point. We corroborate our analytical predictions by numerical simulations of active particles under confinement and interacting through volume exclusion. Excellent quantitative agreement is reached through an effective translational diffusion coefficient. We finally show that an additional response to the chemical gradient direction is sufficient to induce vortex clusters. Our results pave the way to engineer motility responses in order to achieve aggregation and collective behavior even at unfavorable conditions.


Quorum-sensing active particles with discontinuous motility
    Thomas Speck
    December 30, 2023
===========================================================







§ INTRODUCTION


Motility is a fundamental property of many biological systems, from swimming sperm <cit.> and crawling cells <cit.> to animal locomotion <cit.>. While animals coordinate motion in response to optical and mechanical sensory inputs, regulation of motility on the microscale requires very different mechanisms. A particular challenge is to overcome the physical limitations of sensing and gathering information about the environment on small scales <cit.>.

For example, sperm cells can detect and follow gradients of signaling molecules (a small peptide in the case of sea urchin) through changing the beat pattern of their flagellum <cit.>. An alternative strategy to gather information about the local environment is quorum sensing, in which members of a population exude signaling molecules <cit.>. The sensed concentration (or rather concentration change) influences gene expression and determines the phenotype. Quorum sensing facilitates the synchronized behavior of bacteria required, e.g., for biofilm formation <cit.>, virulence <cit.>, bioluminescence <cit.>, and motility control <cit.>. Moreover, there is evidence that quorum sensing also plays a role in the regulation of immune cell responses <cit.>. Beyond the understanding of large-scale collective behavior of biological entities, such communication mechanisms are interesting for synthetic microrobots with a limited set of responses. Even if individual robots can only follow simple rules, there have been studies recently reporting on the emerging collective behavior in ensembles of robots with the potential to fulfill complex tasks <cit.>. Another possible application is to guide the spontaneous assembly of active functional materials <cit.>.

Phoretic colloidal particles provide a well-characterized experimental model system in which we can study different aspects of emerging collective behavior <cit.>. Implementing responses beyond excluded volume and alignment requires control over particles motion <cit.>. For phoretic Janus particles triggered by light, individual control of motility has been demonstrated recently for two modes of perception: cohesive flocking through vision cones <cit.> and for quorum sensing <cit.>. For the later, a virtual concentration field is calculated from particle positions that determines the propulsion speed of each particle through a feedback mechanism. Switching between motile and passive according to a preset threshold, aggregation into a cluster is observed, the shape and density of which can be controlled through tailoring the response of particles.

Such aggregation of spherical active particles into dense clusters is often rationalized within the framework of motility-induced phase separation <cit.>, which is based on an effective propulsion speed v(ρ) that is reduced as the local density ρ increases. The system becomes inhomogeneous with coexisting dilute and dense domains <cit.>. Theory shows that this phase separation can be understood as a large-scale dynamic instability captured through non-linear evolution equations <cit.>. It requires the derivative of v(ρ) and thus a continuous function. While in principle a continuous response might be achieved through quorum sensing, in realistic applications as mentioned above the response is discontinuous.

Here we develop a comprehensive theoretical framework for this scenario, in which active particles interact through a self-generated chemical field. In particular, we consider discontinuous changes of particle motility (another possibility would be the diffusion coefficient <cit.>). Somewhat counter-intuitive, such discontinuous changes simplify the mathematical problem since the resulting evolution equations are now linear, which allows us to obtain analytical results for two simple geometries: a planar interface and circular clusters. Our theory is a mean-field theory neglecting fluctuations. However, we argue that the dominating effect of fluctuations is to increase the diffusive current out of clusters. Taking this effect into account, we achieve excellent quantitative agreement of our theoretical predictions with simulations of interacting active particles. In the final part we consider vortex clusters, which in active matter are typically induced by a combination of aligning inter-particle interactions and confinement <cit.>, or are related to “active turbulence” <cit.>. Here we show that similar patterns can be achieved through an additional response of particle orientations to the chemical field gradient.





§ MODEL




 §.§ Basic equations


We consider a suspension of polar active particles in two dimensions governed by the stochastic equations of motion

    = ve⃗ + ,     = e⃗·χs⃗ + η.

Each particle is described by a unit orientation e⃗≡(cos,sin)^T (with  the angle enclosed between the orientation and the x-axis) along which it is propelled with speed v. Here,  and η are translational and orientational Gaussian noise with zero mean and diffusion coefficients D_0 and , respectively. With these parameters, we define the typical length and speed scale,

    ℓ≡√(D_0/),      v_∗≡ 4√(D_0) = 4D_0/ℓ.

The orientation of particles aligns perpendicular to the direction s⃗ with strength χ. Both the speed v(c) and the coupling χ(c) are functions of the local concentration c(,t) of some chemical that is produced or consumed by the particles, see Fig. <ref>. This model focuses on the interactions mediated by the chemical field and neglects direct interactions (e.g. through volume exclusion) but also phoretic and hydrodynamic interactions.



Previous studies of related models <cit.> have focused on chemotactic behavior in which alignment occurs parallel to ∇ c, describing either chemo-attractive or chemo-repellent particles depending on the sign of χ. Setting ve⃗∝∇ c has been used to model the dynamics of chemically active uniform particles <cit.>. In contrast, here we will consider the case of perpendicular alignment to the concentration gradient (s⃗∝∇ c). More importantly, we will consider discontinuous functions v(c) and χ(c) that switch between two constant values depending on a threshold c̅ (as depicted in Fig. <ref>).

An equivalent but for our purposes more convenient representation of Eq. (<ref>) is through the evolution of the joint probability ψ(,;t) of position and orientation, which reads

    ψt = -∇·[ve⃗-D_0∇]ψ - χ[e⃗·s⃗ψ] + ^2ψ^2.

Decomposing ψ(,;t) into Fourier modes leads to a hierarchy of coupled dynamic equations <cit.>. Since we are interested in the large-scale collective behavior, we close the hierarchy at the second level and treat the system in terms of the first two moments of the joint probability, the particle density ρ(,t) and the polarization p⃗ (,t). The derivation of their evolution equations can be found in appendix <ref>. The equation for the density is the continuity equation

    ρt = -∇·j⃗

with particle current

    j⃗ = vp⃗ - D_0∇ρ.

The total current is the sum of an active current due to a non-zero polarization and a diffusive current due to density gradients.

The general expression for the polarization is a bit lengthy and reads

    p⃗t = ∇·M⃗ + 1/2χρR⃗·s⃗ - (+χ^2/8|s⃗|^2)p⃗
    
      + v/16[(∇·χs⃗)R⃗·p⃗-(∇·R⃗·χs⃗)p⃗],

where the divergence involves the matrix

    M⃗≡ -1/2vρ𝕀 + D_0∇p⃗ + D_0v/v_∗^2[(∇ vp⃗)+R⃗·(∇ vp⃗)·R⃗] 
    
      + χ v/16[3(s⃗p⃗)·R⃗-R⃗·(s⃗p⃗)]

akin to the stress in conventional hydrodynamics. It is convenient to employ the asymmetric matrix

    R⃗≡([  0 -1;  1  0 ])

describing a rotation by π/2.

We assume that the chemicals are produced uniformly by every particle with rate  and diffuse independently with diffusion coefficient D_c in the semi-space above the plane in which the active particles are moving. Moreover, assuming a separation between the time scale on which active particles move and the time scale for the chemical profile to relax, one has

    c() = /4π D_c^2'ρ(-')e^-r'//r'

with decay length  due to the degradation of chemicals and r'=|'|. For an unbounded system with uniform density ρ_0, we find the uniform concentration

    c_0 = /2D_cρ_0.




 §.§ Active Brownian particles


For χ=0 and a linear relationship v(c)=v_0-ζ c between speed and local concentration, our model reduces to that of conventional active Brownian particles. For densities that vary only weakly on the interaction range , we can expand ρ(-')≈ρ()-'·∇ρ+1/2'^2∇^2ρ, leading to

    v(,t) = v_0 - ζ/2D_cρ(,t) - ζ^3/2D_c∇^2ρ(,t)

so that the effective speed indeed depends on the local density. In contrast, here we go beyond such a simple analytical relation and study situations where the speed v and the coupling χ change discontinuously depending on the chemical field c reaching a threshold c̅ [Fig. <ref>(a)].



 §.§ Continuity condition


Drawing an analogy with electrostatics, the speed (and coupling) is akin to a dielectric constant that jumps at the interface between different media. Consequently, the discontinuity of v and χ imposes jump conditions for p⃗ () at ∂𝒞 via Eq. (<ref>), which we derive in analogy to the jump conditions for the electromagnetic fields. To this end, we assume that a closed domain 𝒞 has formed with speed v^< and coupling χ^< inside, and v^> and χ^> outside [Fig. <ref>(a)]. The domain is bounded by the curve ∂𝒞 with normal vector n⃗ pointing outwards. In the steady state (∂_tp⃗=0), integrating Eq. (<ref>) over an infinitesimal area δ A with one half inside and the other half outside 𝒞 [cf. Fig. <ref>(b)] yields

    ^2δ A∇·M⃗ + ^2δ Av/16(∇·χs⃗)R⃗·p⃗ = 0.

The first integral can be evaluated using the divergence theorem. The second integral involves the gradient of χ, which jumps at =_c (with _c a parametrization of ∂𝒞) and is given by

    ∇χ = (χ^>-χ^<)n⃗δ(-_c).

The speed v also jumps at _c and we follow Ref.  to evaluate δ(-_c), by which we arrive at the jump condition

    1/16(s⃗·n⃗)(χ^>-χ^<)1/2(v^>R⃗·p⃗^>-v^<R⃗·p⃗^<) 
     +n⃗· (M⃗^>-M⃗^<) = 0

with (M⃗^< and M⃗^> are defined analogously)

    p⃗^< = lim_|| → 0p⃗(_c-),     p⃗^> =lim_|| → 0p⃗(_c+).

The continuity equation (<ref>) determines whether the density is continuous (D_0>0) or also becomes discontinuous (D_0=0, see next Sec. <ref>). Note that in any case, the concentration profile Eq. (<ref>) remains a continuous function everywhere.







§ NO ALIGNMENT




 §.§ Polarization



We now study the conditions under which particles aggregate and form a stationary inhomogeneous system. We first simplify the general problem and assume that there is no alignment of orientations with the field gradient through setting χ=0, cf. Fig. <ref>(a). The speed v jumps across the interface but is constant within each region with

    M⃗_0 = -1/2vρ𝕀+D_0(1+v^2/v_∗^2)∇p⃗

and the evolution of the polarization is thus governed by

    p⃗t = -v/2∇ρ + D_0(1+v^2/v_∗^2)∇^2p⃗ - p⃗.

In the following, we exploit that in the steady state (with no-flux boundary conditions) not only the divergence but the particle current vanishes, j⃗=0. We then substitute ∇ρ=(v/D_0)p⃗ [Eq. (<ref>)] to obtain a differential equation for the polarization p⃗ alone. Clearly, p⃗=0 is a solution, which corresponds to a homogeneous system.

Now let us assume that a cluster has formed due to a reduced speed v^<. The continuity condition Eq. (<ref>) simplifies to

    n⃗· (M⃗_0^>-M⃗_0^<) = 0.

If we set D_0=0 then from Eq. (<ref>) we immediately find p⃗=0 everywhere and thus vρ=const. This relationship has been exploited to create complex density patterns through spatially modulating the speed <cit.>. If v^<=0 then ρ^>=0, i.e., all particles condensate into the cluster and stop moving. In the following, we consider the coexistence of motile and passive particles with non-vanishing diffusion coefficient D_0>0.



 §.§ Planar interface



We first assume a geometry in which the interface is planar with normal vector n⃗=e⃗_x. The system is translationally invariant along the transversal direction so that ρ(x) only depends on x and the polarization is p⃗=p(x)e⃗_x. For x<0 we set v^<=0 implying a constant density ρ_c and p=0. For x>0 with v^>=v_0 we find the differential equation p”-p/ξ_0^2=0 (the prime denotes the derivative) with decay length

    ξ_0 ≡ℓ[1+(v_0/v_∗)^2/1+8(v_0/v_∗)^2]^1/2,

which equals ℓ for v_0=0 and drops to ℓ/√(8) for v_0→∞. The solution is p(x)=ae^-x/ξ_0 with integration constant a. The polarization thus has a discontinuity at x=0, where it jumps from a<0 to zero. The density follows through integrating Eq. (<ref>) as

    ρ(x) = ρ_c - v_0ξ_0/D_0 a (e^-x/ξ_0-1)

for x⩾ 0 and ρ=ρ_c for x<0. For x≫ξ_0 the active gas reaches ρ_g=ρ_c+v_0ξ_0/D_0 a. The continuity condition (<ref>) yields

    0 = 1/2v_0ρ_c + D_0(1+v_0^2/v_∗^2)a/ξ_0,

which relates a to the density ρ_c. Equation (<ref>) can be rewritten into

    ρ_g/ρ_c = (1+8v_0^2/v_∗^2)^-1

showing that the ratio of gas and cluster density drops quadratically with the propulsion speed.



The final condition reads c(x=0)=c̅ with concentration

    c(x=0) = 
    /4π D_cr0∞ e^-r/[πρ_c + θ0πρ(rsinθ)]

at the interface. The integrals can be performed analytically, which yields the closed expression

    c̅ = /2D_c[ρ_c+v_0ξ_0/2D_0a(1-2/πcos^-1(/ξ_0)/√(1-(/ξ_0)^2))].

For a given threshold c̅ and decay length , Eqs. (<ref>), (<ref>), and (<ref>) can be solved for the passive and active gas densities ρ_c and ρ_g, which are plotted Fig. <ref>(a) as a function of the reduced outer speed v_0/v_∗. The coexistence of a passive and an active domain is possible for speeds v_0 higher than a minimal speed v_min which is set by the condition ρ_g=ρ_0. For smaller v_0, the system remains a homogeneous active gas with ρ(x)=ρ_0. With rising speed, the cluster density ρ_c saturates at

    ρ_c(v_0→∞) = ρ̅[1/2+cos^-1(√(8)/ℓ)/π√(1-8(/ℓ)^2)]^-1

with ρ̅≡2D_c/c̅, while the gas density ρ_g converges to zero due to the enhanced active current into the passive region.



 §.§ Coexistence without a critical point


We now gather our results to produce the phase diagram shown in Fig. <ref>(b) varying speed v_0 and threshold c̅. There are two homogeneous phases in which all particles are either motile (forming an active gas with concentration c_0<c̅) or passive (c_0>c̅). Between these homogeneous phases, the system becomes spatially inhomogeneous and develops an interface between a dense passive and a dilute active region. At greater thresholds c̅, a higher speed v_min is required to enable phase coexistence.



Such a coexistence of different densities is a hallmark of passive liquid-gas phase separation (at constant volume), which requires attractive interactions between particles. Coexistence is then related to the equality of the chemical potential in both phases. It has been shown that the mean-field behavior of active Brownian particles can be mapped onto an effective free energy <cit.> and thus follows the same scenario as passive liquid-gas coexistence, although the non-equilibrium nature manifests itself in phenomena like a negative interfacial tension <cit.> and “bubbly” phase separation <cit.>. Nevertheless, the generic relation with passive liquid-gas coexistence has been corroborated by detailed numerical investigations of the phase behavior <cit.>. The resulting phase diagram in the v_0–ρ plane is sketched in Fig. <ref>(a). Quenching the system inside the two-phase region, the stable steady state is inhomogeneous with the coexisting densities given by the points on the binodal. Changing the global density changes the relative size of the regions covered by either phase as expressed by the lever rule, but not the coexisting densities. Phase coexistence is terminated in a critical point characterized by diverging fluctuations below which the system remains homogeneous at any density.

For the discontinuous motility studied here, we also reach a state of coexistence. As for conventional liquid-gas coexistence, varying the global density ρ_0 yields the same coexisting densities. There are, however, important differences [Fig. <ref>(b) for fixed threshold c̅]. First, for global densities ρ_0>ρ̅ the full system becomes homogeneous and passive since c_0>c̅. For coexistence, we need to cross the gas density, v_0>v_min, with ρ_c>ρ̅. Second, there is no critical point. The two-phase region becomes more narrow as the speed v_0 is decreased and ends in a single point at v_0=0 with density ρ̅ [cf. Eq. (<ref>)]. This feature enables stable aggregation even at very low speeds in a region of parameter space that is mainly controlled by the threshold c̅.



 §.§ Circular clusters



We now assume a circular cluster of radius  with normal vector n⃗=e⃗_r of the bounding curve ∂𝒞. In steady state, the current j⃗=0 still vanishes and we thus anticipate a radially symmetric density profile ρ(r) so that the polarization p⃗(r)=p(r)e⃗_r only has a radial component. Going to polar coordinates (with r=0 the center of mass), we obtain the modified Bessel differential equation

    0 = p” + p'/r - p/r^2 - p/ξ_0^2

with the length scale ξ_0 defined in Eq. (<ref>). The general solution of Eq. (<ref>) reads

    p(r) = a_1I_1(r/ξ_0) + a_2K_1(r/ξ_0),

where I_n(x) and K_n(x) are modified Bessel functions of order n of the first and second kind, respectively, and a_1 and a_2 are integration constants to be determined. In the outer region, p→ 0 as r→∞ and thus a_1^>=0. The density profile is obtained through integrating ρ'=(v_0/D_0)p [Eq. (<ref>)],

    ρ(r) = v_0ξ_0/D_0[a_1I_0(r/ξ_0)-a_2K_0(r/ξ_0)] + b

with another integration constant b. For the integral of the density (viz. number of particles) to vanish as →0 requires to set a_2^<=0. Moreover, b^>=ρ_g since the density is that of the active gas for r→∞.

We now consider the formation of clusters through turning off the self-propulsion when triggering the threshold (v^<=0), which is the situation realized experimentally in Ref. . As for the planar interface, this implies p=0 and a constant density ρ_c inside the cluster. The density profile is thus fixed by ρ_c, ρ_g, , and a_2^>, which we need to determine.

To this end, the first condition follows from the continuity of the density at ,

    ρ_c = -v_0ξ_0/D_0a_2^>K_0(/ξ_0) + ρ_g.

To fulfill the continuity condition (<ref>), we need the expression

    e⃗_r·M⃗_0 = [-1/2vρ + D_0(1+v^2/v_∗^2)p']e⃗_r,

where the derivative of the polarization can be expressed as (inside the cluster with b^<=ρ_c)

    p' = D_0/vξ_0^2(ρ-b) - p/r.

Our second condition then follows through the continuity condition, which can be recast into

    ρ_c = (1+8v_0^2/v_∗^2)ρ_g + v_0/(1+v_0^2/v_∗^2)p_c/.

The polarization jumps from p=0 inside the cluster to 

    p_c = a_2^>K_1(/ξ_0) < 0

at . As expected, for a cluster of infinite size (→∞), Eq. (<ref>) becomes equal to the result for the planar interface, Eq. (<ref>).

The concentration inside the cluster needs to exceed the threshold c̅ with c()=c̅ at the boundary, which constitutes our third condition. Plugging the density profile Eq. (<ref>) into Eq. (<ref>), the condition becomes

    c̅/c_0 = r'0∞ r'ρ(r')/ρ_0k0∞k/√(1+(k)^2) J_0(k)J_0(kr')

with J_0(x) the Bessel function of first kind and order zero. No closed analytical expression is available for these integrals but they can be solved easily numerically.

For the remainder of this paper, we adopt as specific geometry a circular confinement of radius R containing a single cluster with the center of mass at the center of the confinement. This geometry has been studied previously in Ref.  but for run-and-tumble dynamics without translational diffusion. Our fourth and final condition then follows through the conservation of particle number,

    ^2ρ() = ρ_0 A,

where A=π R^2 is the area of the system. We thus have four conditions for four unknowns, which we solve iteratively for given threshold c̅ and decay length . The minimal threshold above which a passive cluster can coexist with an active gas is given by c̅_min=c(R) and can be calculated analytically from Eq. (<ref>) as

    c̅_min/c_0=1/2[1-I_0(2R/)+L_0(2R/)]

with L_0(x) the modified Struve-function of order zero.





 §.§ Simulations
 




  §.§.§ Simulation details


To test our analytical predictions we now turn to numerical simulations. We numerically solve Eq. (<ref>) with χ=0 for N=800 particles additionally interacting via the repulsive Weeks-Chandler-Anderson (WCA) potential <cit.>

    u(r_ij) =
      
        4[(/r_ij)^12 - (/r_ij)^6 
          + 1/4]     (r_ij/ < 2^1/6) 
    
        0     (r_ij/≥ 2^1/6),

with particle separation r_ij=|_j-_i| and _i the position of the ith particle. We use a potential strength =100 modeling hard discs with an effective diameter _eff=1.10688 <cit.>. The rotational diffusion coefficient is set to =3D_0/_eff^2. The same set of parameters has been used in previous studies of active Brownian particles <cit.>. From now on, we employ , ^2/D_0, and k_B T as units of length, time, and energy, respectively. The particles move in a circular confinement with radius R=42.4 at packing fraction ϕ=N (_eff/2)^2/R^2=0.136 (corresponding to particle number density ρ_0=0.141). To prevent the accumulation of particles at the boundary (due to their persistent motion <cit.>), particle orientations are flipped instantaneously and redirected towards the center of the confinement when reaching the border.

Every particle acts as a point source of the chemical, thus the instantaneous concentration sensed by particle i is

    c_i ≡ c(_i) = /4π D_c∑_j ≠ i^N e^-r_ij//r_ij.

When c_i is larger than the threshold value c̅, the particle's propulsion speed is set to zero. 

For different decay lengths , we perform simulations varying thresholds c̅ and propulsion speeds v_0 at which the system separates into a dense cluster of passive particles surrounded by a dilute gas of active particles, see snapshot in Fig. <ref>(a). After the system has reached the steady state, we measure the radial density profile ρ(r) with respect to the center of mass of all particles. Density profiles can be well fitted by

    ρ(r) = ρ_c+ρ_g/2+ρ_c-ρ_g/2tanh(r-/2ω),

from which we obtain the coexisting densities ρ_c and ρ_g, the cluster size , as well as the width of the interface ω. Both ρ_c and ρ_g increase with rising threshold c̅, while the cluster radius decreases, cf. Fig. <ref>(b,c), i.e., the cluster compactifies.

For a quantitative comparison of the density profiles measured in the simulations to our theoretical prediction (<ref>), we normalize concentrations c with respect to the average concentration c_0 in an (unconfined) uniform system at density ρ_0. In the mean-field theory (which neglects the repulsive interactions) c_0 is given by Eq. (<ref>). In the simulations c_0 is reduced due to the finite size of the particles. It can be approximately calculated via Eq. (<ref>) by assuming ρ(-')≈ρ_0 Θ(|-'|-_eff) (with Θ(x) the Heaviside step-function) in the uniform system. In this way, we obtain

    c_0=/2D_cρ_0 e^-_eff/

as corrected normalization constant in the simulations.



  §.§.§ Role of the interface


For =10, speed v_0=20, and threshold c̅=1.4c_0, Fig. <ref>(a) compares the density profile of circular clusters obtained in the numerical simulations with the theoretical prediction. We observe that the theoretical density profile decays very sharply at the interface between the regions of velocity zero and v_0 (the decay length ξ_0 is small), which is in contrast to the broad interface the simulated system exhibits. In Fig. <ref>(a) we plot the interfacial width ω against the propulsion speed v_0 in the gas for different decay lengths of the chemical λ. We observe a notable increase of ω for smaller v_0, which we attribute to enhanced fluctuations. Additionally, the interface becomes broader with decreasing decay length, as already observed in Ref. . This can be explained by fluctuations: For smaller λ, a particle receives chemical signals from fewer particles and so the total concentration it measures fluctuates stronger. Because of this, a small density fluctuation can trigger a particle at the cluster surface to switch its motility from passive to active (or reverse), which leads to a rougher and thus broader interface. Due to the excluded volume of the particles, additional fluctuations arise in the simulations.





Second, the theory notably overestimates the cluster density and underestimates the gas density. This suggests that in the theory the particle current from the cluster into the gas is significantly smaller than in the simulations. In the mean-field picture, a particle leaves the cluster (it turn from passive to active) when it moves from r< to r>, which can only happen  via translational diffusion of the particle itself (the current out of the cluster is -D_0 ∂_r ρ). But in the simulations the concentration a particle senses fluctuates (even if the particle would be held fixed) due to the motion of the surrounding particles. Consequently, particles at the cluster boundary constantly change between sensing super- and sub-threshold concentrations (and thus switch between passive and active). When – due to a concentration fluctuation – a particle at the surface turns active and its orientation points towards the gas, it will leave the cluster (if it moves far enough away from the surface before it turns passive again due to another fluctuation). Due to this additional fluctuation-induced particle current, the disagreement between theory and simulations is not surprising.

The dominant effect of this process is to induce an additional diffusive current j⃗_fluc=-D_fluc∇ρ through the interface. Since the density gradient is small away from the interface, we replace D_0 by the increased effective diffusion coefficient D_eff=D_0+D_fluc. Interestingly, we find that it is possible to choose a single D_eff such that the coexisting densities ρ_c and ρ_g as well as the cluster size  show excellent agreement between theory and simulations, see Fig. <ref>(a). At these parameters, the best agreement (minimum of the sum of squared errors for ρ_c and ρ_g) is found for D_eff=10.6D_0, which shows the relevance of the fluctuation-induced current. Moreover, for fixed v_0 and  this agreement is independent of the threshold value c̅ (in the regime of stable cluster formation), cf. Fig. <ref>(b,c). We therefore treat D_eff(v_0,) as an effective parameter depending on decay length  and propulsion speed v_0, by which we simultaneously fit the theory curves for ρ_c(c̅) and ρ_g(c̅) to the simulation data.

The quantity ω/v_0 can be regarded as a measure for the time a particle (which is escaping from the cluster at propulsion speed v_0) needs to travel through the interface of width ω to enter the active gas. When a particle has to spend a longer time in the interfacial region in order to escape, the probability that it turns passive again by a concentration fluctuation (and thus reenters the cluster before reaching the gas) increases. This leads to a diminishing number of particles that successfully leave the cluster and thus to decreasing D_eff. Combining the fitted values D_eff(v_0,) with the measured interfacial widths ω(v_0,), we plot D_eff versus ω/v_0 in Fig. <ref>(b). Interestingly, we observe a collapse of the data for

    D_eff/D_0 = ^/3f(^ω/v_0)

with exponent ≃0.57 and scaling function f(x). This function is non-monotonous and decays as f(x)∼ x^-1/3 for large x. In this regime the effective diffusion coefficient becomes independent of the interaction range  and decreases, indicating that fluctuations are less important. In the opposite limit of small x corresponding to intrinsically sharp interfaces [with v_0 large, cf. Fig. <ref>(a)] the fluctuations captured by the effective diffusion coefficient also become less important.



  §.§.§ Phase Diagrams


Employing the obtained D_eff values, we now construct the theoretical v_0–ρ and v_0–c̅ phase diagrams for circular clusters, which we quantitatively compare to the simulation results.

We first note that the theory curve c() [Eq. (<ref>)] for the concentration at the cluster boundary exhibits a maximum c̅_max and thus, in principle, allows for two solutions (r_c,ρ_c,ρ_g) for given c̅, v_0 and , cf. Fig. <ref>(a,b). However, the solution with the smaller  is unstable: Increasing , the rim concentration becomes larger than the threshold and thus particles at the rim turn passive and the cluster grows. At the larger solution the reverse holds, increasing the cluster size the rim concentration is below the threshold and rim particle turn active, thus stabilizing the cluster. Consequently, in the simulations always the solution with larger cluster size  is observed. For threshold values higher than c̅_max no solution exists, which implies that the system is an active gas. This upper limit diminishes with decreasing speed v_0, see Fig. <ref>(a). Hence (as for the planar interface in Sec. <ref>), for given c̅ a phase-separated state exists only above a minimal speed v_min [determined by c̅_max(v_0)=c̅]. This is confirmed plotting the coexisting densities as a function of propulsion speed in Fig. <ref>(c), which again shows very good agreement with the theoretical predictions.

In Fig. <ref>(d), we finally plot the phase diagram for circular clusters [cf. Fig. <ref>(b)]. In the inhomogeneous regime for small thresholds c̅, in the simulations we observe an instantaneous formation of a dense cluster starting from a disordered initial configuration [green region in Fig. <ref>(d), Supplemental Video 1]. Increasing c̅, we observe a transition to a nucleation-like regime, where clusters only form after a waiting time when a sufficiently strong density fluctuation has occurred [orange region in Fig. <ref>(d), Supplemental Video 2]. In order to access the steady state within reasonable time, in this (metastable) regime we initialize the system in a crystalline configuration. The upper threshold up to which we observe a stable phase-separated state in the simulations is very close to the theoretical prediction c̅_max(v_0), cf. Fig. <ref>(d). Figure <ref>(c) shows that also at thresholds in the metastable regime the coexisting densities still agree well with the theory. The remaining discrepancies probably arise due to fluctuations and the influence of the excluded volume, which are both not included in the theory. As there is no upper limit for the cluster density in the theory, at very high speeds it can rise above that of dense packing (corresponding to ρ_c≈6.66ρ_0). At the highest speeds and thresholds we have probed in the simulations, clusters reach densities of ∼ 90% of dense packing. Furthermore, also the prediction for the lower limit c̅_min below which the system is homogeneous and passive [Eq. (<ref>)] is in good agreement with the simulations.







§ VORTEX CLUSTERS



We now include the coupling of the particle orientations to the instantaneous direction of the concentration gradient, s⃗_i=∇ c/|∇ c| evaluated at the particle position _i [cf. Fig. <ref>(b)]. In particular, we set χ^<=χ≠ 0, χ^>=0. Thus, according to Eq. (<ref>), particles with c_i>c̅ experience a torque aligning them perpendicular to s⃗_i. Moreover, we need to consider finite speeds v^<>0 inside the cluster to induce an angular particle current.



 §.§ Simulations


We perform simulations as described in Sec. <ref> with additional orientational coupling. Throughout this section, we set =20, c̅=c_0, and v^>=30. We employ a slightly smaller confinement radius R=36.7, implying a packing fraction ϕ=0.182 (ρ_0=0.189). Again, we observe the formation of a single cluster with reduced speed v^< surrounded by a gas with higher speed v^>. Additionally, the orientational coupling induces a angular particle current inside the cluster [see Fig. <ref>(a) for χ=-32.7 and v^<=1/8v^>].

The emergence of this vortex structure can be understood as follows: On average ∇ c points towards the center of the cluster, cf. Fig. <ref>(b,c). Due to the aligning torque particles orient perpendicular to the gradient and thus swim along regions of constant c, which correspond to circles around the cluster's center, see Fig. <ref>(b). Depending on the sign of χ, particles swim clockwise (χ>0) or counter-clockwise (χ<0).

In order to quantify the orientational order in the system, we change to polar coordinates with the particles' center of mass at the origin and unit vectors e⃗_r and e⃗_θ. We calculate the average radial and angular component of the particle orientations

    e_r=e⃗·e⃗_r,   e_θ=e⃗·e⃗_θ

as a function of the distance r to the center of mass. Since rotational diffusion is counteracting the aligning torque, e_θ increases with rising torque strength χ [see Fig. <ref>(e)]. At the inner side of the cluster-gas interface, e_θ has a maximum and decays both into the gas (where χ=0) and towards the cluster's center. The latter arises from fluctuations of the concentration field: as the magnitude of ∂_r c diminishes towards the center, the variance of fluctuations of s⃗_i around its mean s⃗_i=-e⃗_r increases, cf. Fig. <ref>(c). In turn, the direction along which particles are aligned by the torque is fluctuating stronger, leading to the drop of e_θ. Regions of increased angular alignment have a lower outgoing particle current in radial direction. Therefore, the cluster density is not homogeneous anymore but exhibits a peak close to the position of maximal e_θ, cf. Fig. <ref>(d).

The e_r-profile exhibits two peaks: a strong negative peak at the interface between the regions of small and high v (particle orientations on average point into the cluster) and a weaker positive peak on the inner side of the density maximum [see Fig. <ref>(f)]. The negative peak diminishes with rising torque strength, which leads to a stronger particle current out of the cluster and thus to an increased gas density, cf. Fig. <ref>(d).



 §.§ Theory


For completeness, we generalize the analysis of circular clusters to orientational coupling and to provide the explicit expressions. We set χ^<=χ and χ^>=0 and use the mean direction s⃗=-e⃗_r pointing towards the origin. The polarization p⃗(r)=p_r e⃗_r+p_θ(r)e⃗_θ now has a radial and angular component. In the steady state, the radial component j_r of the current still vanishes, j_r=0, implying ∂_rρ=(v/D_0)p_r. In contrast, the angular current is now non-zero and given by j_θ=v p_θ.

For the polarization equation (<ref>), in the inner region we need the matrix M⃗^<=M⃗_0+M⃗_χ with M⃗_0 from Eq. (<ref>) plus the matrix

    M⃗_χ = χ v/16(-3p_θe⃗_re⃗_r + p_re⃗_re⃗_θ + 3p_re⃗_θe⃗_r + p_θe⃗_θe⃗_θ)

due to the coupling of the orientation. We then arrive at the two coupled inhomogeneous modified Bessel equations

    0 =    p_r” + p_r'/r - p_r/r^2 - p_r/ξ_χ^2 - 3χ v/v_∗^2+v^2(p_θ'+p_θ/r), 
    
     0 =    p_θ”+ p_θ'/r -p_θ/r^2 -p_θ/ξ̃_χ^2 + 3χ v/v_∗^2+v^2( p_r'+p_r/r) 
                          -χ/2D_0[1+(v/v_∗)^2]ρ,

with radial

    ξ_χ≡ℓ[1+(v/v_∗)^2/1+8(v/v_∗)^2+18(χ/)^2]^1/2

and angular decay lengths

    ξ̃_χ≡ℓ[1+(v/v_∗)^2/1+18(χ/)^2]^1/2,

which generalize the decay length ξ_0 [Eq. (<ref>)] obtained in the absence of orientational coupling.



Obtaining a closed analytical solution of Eqs. (<ref>) and (<ref>) appears to be a formidable task. Nevertheless, one can draw some qualitative insights into the behavior of the system through further simplifications. Here we just give one example for the maximum e_θ_max of the angular orientation profiles. As simplification, we consider the limit of vanishing translational diffusion. In that case, the density jumps between the constant values ρ_c and ρ_g at r_c and p_r=0 everywhere. Then, Eq. (<ref>) implies p_θ=0 for r>r_c and 

    p_θ=-1/2χ//1+χ^2/(8^2)ρ_c

in the cluster, resulting from the competition of alignment and rotational diffusion. Equation (<ref>) implies an average angular orientation e_θ=p_θ/ρ_c. Comparing this prediction to the maximum e_θ_max of the angular orientation profiles from the simulations, we find good agreement for small v^< and χ, cf. Fig. <ref>. With increasing v^<, (density and concentration) fluctuations become more important, leading to stronger deviations from the mean-field prediction. For χ≳ 3, e_θ_max saturates in the simulations, but the theory prediction decays to zero for χ→∞ (data not shown). In order to capture the high χ-behavior, we probably would need to take into account higher moments of the probability density.





§ CONCLUSIONS


Taking into account a discontinuous motility leads to a novel paradigm for aggregation in scalar active matter different from chemotaxis and the motility-induced phase separation of active Brownian particles <cit.>. Particles aggregate into dense domains through a reduction of their motility in response to an event, here passing a predefined threshold of a self-generated field. This constitutes a simple communication mechanism at the microscale through which collective behavior can be initiated. It is exploited in nature through quorum-sensing of signaling molecules exuded by all members of a population and has recently been demonstrated in synthetic active matter <cit.>. Particles switch their motility depending on external cues, which sets them apart from mixtures of particles with different but fixed motility <cit.> (or diffusivity <cit.>). Implementing such communication beyond direct physical (steric, hydrodynamic, etc.) forces allows to “program” responses and thus to tailor collective behavior. In particular, the aggregation of building blocks is a crucial step in the autonomous assembly of ordered, hierarchical structures. Controlling the self-assembly of active matter is an important step towards microscopic “machines of machines” <cit.> and will become important for the emerging field of microrobots performing tasks collectively.

We have derived and investigated the coupled linear dynamic equations for density and polarization, whereby the propulsion speed v(c) is now a function of the concentration c and the concentration is a linear functional of the density [Eq. (<ref>)]. Specifically, we have studied a piece-wise constant function v(c) that drops to zero above a threshold c̅. The resulting spatially inhomogeneous steady state is reminiscent of the phase coexistence of an active gas with a dense liquid separated by an interfacial region. As for active Brownian particles, coexistence is determined by the balance of diffusive and active particle current. The later is due to a non-zero polarization induced by the interface, which decays into the bulk on a length scale ξ_0 [Eq. (<ref>)]. Most notable, however, is the absence of a critical point for discontinuous motilities. Introducing as single fit parameter an effective diffusion coefficient accounting for the fluctuations of the concentration field within the interface, excellent agreement with particle-based numerical simulations has been demonstrated. The effective diffusion coefficient is independent of the threshold and obeys a scaling form [cf. Eq. (<ref>) and Fig. <ref>]. This success of a mean-field theory might indeed be attributed to the absence of a critical point, which is characterized by diverging fluctuations and, consequently, the break-down of mean-field approximations in its vicinity.


The theoretical framework presented here is not restricted to the exponential kernel in Eq. (<ref>) for the chemical concentration field and other forms, e.g. generated through light, might be considered. Moreover, while for clarity we have focused on a single circular cluster in confinement, the aggregation is not an artifact of the confinement but can also be observed in simulations employing periodic boundary conditions (Supplemental Video 3). We have also discussed a novel route to the formation of vortex clusters, in which particles respond to the local field gradient through a torque. Such torques can be implemented experimentally for colloidal Janus particles <cit.>. Again, vortex formation is not linked to confinement but occurs in extended systems using periodic boundary conditions (Supplemental Video 4), which has been observed experimentally for magnetic rollers <cit.>.

In this work we have treated only steady state profiles. It would be interesting to investigate in more detail the nature of density fluctuations and the temporal evolution of interfaces, which connects active matter to the classical Stefan problem <cit.>.




  We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through the Graduate School of Excellence “Materials Science in Mainz” (GSC 266) and the priority program SPP 1726 (grant no. 254473714). Numerical computations were carried out on the MOGON2 Cluster at ZDV Mainz.






§ DERIVATION OF EFFECTIVE HYDRODYNAMIC EQUATION



In order to derive the dynamic equations (<ref>) and (<ref>) for the density and polarization, respectively, we write Eq. (<ref>) in cartesian coordinates (x,y) and measure the particle orientation  with respect to the x-axis. We follow Ref.  and calculate the evolution equations for the moments

    ψ_n(x,y)=02πcos(n)ψ(x,y,), 
    ϕ_n(x,y)=02πsin(n)ψ(x,y,)

yielding a hierarchy of equations. For n=0 we obtain

    ψ_0t=-x(vψ_1)-y(vϕ_1)+D_0∇^2ψ_0.

For n=1 we have

    ψ_1t=-1/2[x(vψ_0)+x(vψ_2)+y(vϕ_2)] 
     + D_0∇^2ψ_1 
      + 1/2χ[s_y(ψ_2-ψ_0)-s_xψ_2]-ψ_1

and

    ϕ_1t=-1/2[y(vψ_0)+x(vϕ_2)-y(vψ_2)] 
     + D_0∇^2ϕ_1 
      + 1/2χ[s_yϕ_2+s_x(ψ_2+ψ_0)]-ϕ_1.

By setting ψ_3=ϕ_3=0 and neglecting time and spatial derivatives of ψ_2 and ϕ_2, at the next order n=2 we obtain

    ψ_2=1/4[-1/2x(vψ_1)+1/2y(vϕ_1)-χ(s_xϕ_1+s_yψ_1)]

and

    ϕ_2=1/4[-1/2x(vϕ_1)-1/2y(vψ_1)-χ(s_yϕ_1-s_xψ_1)],

which now only couple to the n=1 modes. Finally, plugging Eqs. (<ref>,<ref>) into Eqs. (<ref>,<ref>) and identifying density ρ≡ψ_0 and polarization p⃗≡(ψ_1,ϕ_1) we arrive at Eqs. (<ref>) and (<ref>) from the main text. Note that for the divergence of a tensor, we use the convention (∇·M⃗)_j=∂_i M_ij. 











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