Orientation-dependent propulsion of active Brownian spheres from self-advection to programmable cluster shapes Stephan Br oker1Jens Bickmann1Michael te Vrugt1Michael E. Cates2and Raphael Wittkowski1

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Orientation-dependent propulsion of active Brownian spheres: from self-advection to

programmable cluster shapes

Stephan Br¨oker,1Jens Bickmann,1Michael te Vrugt,1Michael E. Cates,2and Raphael Wittkowski1, ∗

1Institut f¨ur Theoretische Physik, Center for Soft Nanoscience,

Westf¨alische Wilhelms-Universit¨at M¨unster, 48149 M¨unster, Germany

2DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom

Applications of active particles require a method for controlling their dynamics. While this is

typically achieved via direct interventions, indirect interventions based, e.g., on an orientation-

dependent self-propulsion speed of the particles, become increasingly popular. In this work, we

investigate systems of interacting active Brownian spheres in two spatial dimensions with orientation-

dependent propulsion using analytical modeling and Brownian dynamics simulations. It is found

that the orientation-dependence leads to self-advection, circulating currents, and programmable

cluster shapes.

Active Brownian particles (ABPs) [1–4] combine Brow-

nian motion with directed self-propulsion, leading to an

inherently nonequilibrium dynamics. They are a prime

model system for active particles, which have great po-

tential for future applications including nanobots for

medical applications like microsurgery [5] or drug de-

livery [6–8] and programmable materials for industrial

applications [9–11]. Almost all applications have in com-

mon that general features of the dynamics of active par-

ticles, such as their collective dynamics, have to be con-

trolled. This is often achieved using direct interventions

[4, 12], where an external force or torque acts on the par-

ticles. Recently, methods based on indirect interventions,

where one instead changes the way the particles perceive

their environment, have become very popular. Previ-

ous work on such approaches focuses on motility maps,

where the particles’ propulsion speed becomes space-

dependent [13–26]. Such systems have already been re-

alized, e.g., via light-propelled particles in complex light

ﬁelds [14, 27]. Less well understood are indirect interven-

tions with respect to the particles’ orientations, as given,

e.g., by an orientation-dependent propulsion force. Such

forces arise, e.g., when particles are propelled by ultra-

sound [28] or light [29].

There exists theoretical as well as experimental work

on single particles with an orientation-dependent self-

propulsion [30, 31], but many-particle systems of inter-

acting ABPs with an orientation-dependent propulsion

have not been investigated so far. Of particular im-

portance in this context are the eﬀects of such an indi-

rect intervention on the collective dynamics of ABPs and

their intriguing nonequilibrium eﬀects, such as non-state-

function pressure [32, 33], reversed Ostwald ripening [34],

and motility-induced phase separation (MIPS) [35].

In this article, we address this issue by investi-

gating systems of interacting spherical ABPs with an

orientation-dependent propulsion velocity in two spatial

∗Corresponding author: raphael.wittkowski@uni-muenster.de

dimensions using analytical modeling and computer sim-

ulations. We derive a predictive ﬁeld-theoretical model

that describes the collective dynamics of such systems

and ﬁnd novel contributions that depend on the symme-

try properties of the orientation-dependent propulsion.

The model provides an analytical prediction for the spin-

odal corresponding to the onset of MIPS, which we com-

pare to state diagrams obtained by Brownian dynam-

ics simulations. Furthermore, we show that the orienta-

tion dependence of the propulsion gives rise to the self-

assembly of deformed MIPS clusters with, e.g., elliptical,

triangular, and rectangular shapes.

The considered system consists of Nspherical, in-

teracting ABPs in two spatial dimensions with center-

of-mass positions ri= (xi, yi)T, orientations ˆ

u(φi) =

(cos(φi),sin(φi))T, and polar orientation angles φi, where

i= 1, . . . , N. To model the microscopic dynamics of the

particles, we use the overdamped Langevin equations

ri=vA(φi)ˆ

u(φi) + vint,i({ri}) + p2DTΛT,i,(1)

φi=p2DRΛR,i,(2)

where an overdot denotes a derivative with respect to

time t. Equations (1) and (2) diﬀer from the standard

Langevin equations for ABPs [2, 3, 12, 33, 36, 37] by the

orientation-dependence of the propulsion speed vA(φ).

Particle interactions are incorporated using the term

vint,i({ri}) = −βDTPN

j=1,j6=i∇riU2(kri−rjk). Here,

β= 1/(kBT) is the thermodynamic beta with Boltzmann

constant kBand temperature T,DTthe translational

diﬀusion coeﬃcient, ∇ri= (∂xi, ∂yi)Tthe del operator

with respect to ri,U2a two-particle interaction poten-

tial, k·k the Euclidean norm, DR= 3DT/a2the rota-

tional diﬀusion coeﬃcient, and athe particle diameter.

Thermal ﬂuctuations are modeled via zero-mean, unit-

variance statistical white noises ΛT,i(t) and ΛR,i(t).

Using the interaction-expansion method [36–41], we

derived from Eqs. (1) and (2) an advection-diﬀusion

model that describes the time evolution of the number

density ρ(r, t) of the particles, depending on position

r= (x, y)Tand time. The derivation (see Ref. [42])

arXiv:2210.13357v1 [cond-mat.soft] 24 Oct 2022

assumes short-ranged interactions and a dependence of

vA(φ) on φthat can be well approximated with few

Fourier modes. The resulting model reads

˙ρ=−∇·µ(1)ρ+∇·D(ρ)∇ρ,(3)

where D(ρ) is a density-dependent diﬀusion tensor with

elements

Dij (ρ)=(DT+c1ρ+c2ρ2)δij +c3ρµ(2)

ij +µ(2)

ik µ(2)

2DR

.(4)

(We sum over repeated lower indices from here on.) The

coeﬃcients ciare given in Ref. [42]. Moreover, δij de-

notes the Kronecker delta, µ(1) =R2π

0dφ vA(φ)ˆ

u(φ)/(2π)

the particles’ orientation-averaged propulsion velocity,

and µ(2)

ij the elements of the symmetric velocity tensor

µ(2) =R2π

0dφ vA(φ)ˆ

u(φ)⊗ˆ

u(φ)/π with dyadic product

⊗. Formally, µ(1) and µ(2) are the zeroth- and ﬁrst-order

contributions, respectively, of the orientational expansion

of the propulsion velocity v(φ) = vA(φ)ˆ

u(φ) into Carte-

sian tensors [43, 44]: v(φ) = µ(1) +ˆ

u(φ)·µ(2) +O(ˆ

u2).

Including the O(ˆ

u2) contributions would not lead to ad-

ditional terms in Eq. (3) unless we also include higher

orders in derivatives. For the special case of an isotropic

propulsion speed, we recover the purely diﬀusive model

from Ref. [36]. Equation (3) is our ﬁrst main result.

The orientational contributions µ(1) and µ(2) are the

novel features of this model compared to the model de-

rived in Ref. [36] for isotropic propulsion. A nonvanishing

µ(1) corresponds to an internal polarization of the propul-

sion velocity that, similar to an external ﬁeld [12], gives

rise to (self-)advection. This is easily seen from the fact

that the ﬁrst term on the right-hand side of Eq. (3) can be

eliminated using the Galilei transformation r→r−µ(1)t.

The self-advection results, like the motility of individual

active particles [45–47], from an (r↔ −r)-symmetry

breaking. In contrast, µ(2) breaks the (x↔y)-symmetry

of the diﬀusion tensor (4). This also occurs in systems

with chirality such as circle swimmer systems [37, 48].

For what follows, we specify the orientation-dependent

propulsion speed vA(φ) as

vA,n(φ) = ¯v1−ν+ 2νsin2(nφ/2),(5)

which involves an n-fold symmetry and is parametrized

by the orientation-averaged propulsion speed ¯vand the

dimensionless angular modulation amplitude ν. A sim-

ilar form was used in Ref. [31]. We focus on the cases

n= 1,...,4. Using Eq. (5), we obtain

µ(1) =−¯vδn,1

2(ν, 0)T,(6)

µ(2) = ¯v1−σ3δn,2

2,(7)

where 1is the identity matrix and σ3is the third Pauli

matrix. Only for n= 1, µ(1) does not vanish. The

tensor µ(2) is diagonal and nonzero for all n. For n= 2,

its diagonal elements are not identical.

To investigate the system further, we performed Brow-

nian dynamics simulations [49] based on the Langevin

equations (1) and (2). For the interactions, we

chose the Weeks-Chandler-Andersen potential U2(r) =

(4ε[(a/r)12 −(a/r)6] + ε)Θ(21/6a−r) [50] with interac-

tion strength ε, particle distance r, and Heaviside step

function Θ. The particle diameter a, Lennard-Jones time

scale τLJ =a2/(εβDT), and interaction strength εare

chosen as units of length, time, and energy, respectively.

Nondimensionalization of Eqs. (1) and (2) leads to the

P´eclet number Pe = ¯va/DT, which is a measure for the

activity of the particles, and the overall packing density

Φ0=π¯ρa2/4, where ¯ρis the spatially-averaged number

density of the particles. We ﬁxed the average propulsion

speed to ¯v= 24a/τLJ and changed Pe via the tempera-

ture T. If not stated otherwise, we chose Pe = 150 and

Φ0= 0.4. Additional details on the computer simulations

can be found in Ref. [42].

Equation (6) predicts the self-advection velocity µ(1) of

MIPS clusters for n= 1. We conﬁrmed this by comparing

Eq. (6) with the velocity vc= (vc,x, vc,y )Tof macroscopic

MIPS clusters of ABPs that we observed in Brownian

dynamics simulations. Figure 1 shows that our analytical

prediction and the velocity of the clusters are in excellent

agreement, demonstrating that our theory is applicable

also to self-organized structures. The x-component of vc

decreases linearly for increasing νand the y-component is

zero since vA,1(φ) = vA,1(−φ). Our results suggest that,

0 0.2 0.4 0.6 0.8 1

-0.1

-0.2

-0.3

-0.4

-0.5

vc,y/¯v

vc,x/¯v

vc,x

vc,y

µ(1)

x=−¯vν/2

µ(1)

y= 0

FIG. 1. Comparison of our analytical results for the self-

advection velocity µ(1) = (µ(1)

x, µ(1)

y)T(see Eq. (6)) with the

time-averaged velocity of particle clusters vc= (vc,x, vc,y )T

that we obtained from six Brownian dynamics simulations for

each considered value of the angular modulation amplitude ν.

by changing the ﬁrst Fourier mode of vA(φ), one can steer

many-particle structures of ABPs in arbitrary directions.

This eﬀect can be useful for applications where one wants,

e.g., to steer clusters of ABPs through a maze or into or

out of a trap, including active carrier particles that need

to swim to certain regions and release their cargo there

[6–8]. The ability to steer ABPs collectively is relevant,

e.g., for drug delivery [51] and active microstructures [15].

For n= 2, no self-advection arises, but the diﬀu-

sion tensor has no longer identical diagonal elements:

Dij ∝f(ρ)δij −¯vν(c3ρ+ ¯v/DR)σ3,ij with a scalar func-

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Orientation-dependentpropulsionofactiveBrownianspheres:fromself-advectiontoprogrammableclustershapesStephanBroker,1JensBickmann,1MichaelteVrugt,1MichaelE.Cates,2andRaphaelWittkowski1,1InstitutfurTheoretischePhysik,CenterforSoftNanoscience,WestfalischeWilhelms-UniversitatMunster,48149Munster,G...

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Orientation-dependent propulsion of active Brownian spheres from self-advection to programmable cluster shapes Stephan Br oker1Jens Bickmann1Michael te Vrugt1Michael E. Cates2and Raphael Wittkowski1

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