Trapping flocking particles with asymmetric obstacles

Raul Martinez,1, 2 Francisco Alarcon,2, 3 Juan Luis Aragones,1, ∗ and Chantal Valeriani2, †

1Departamento de F́ısica Teórica de la Materia Condensada,
Instituto Nicolás Cabrera and Condensed Matter Physics Center (IFIMAC),

Universidad Autónoma de Madrid, E-28049 Madrid, Spain
2Departamento de Estructura de la Materia, F́ısica Térmica y Electrónica,

Facultad de Ciencias F́ısicas, Grupo Interdisciplinar de Sistemas Complejos,
Universidad Complutense de Madrid, 28040 Madrid, Spain.

3Departamento de Ingenieŕıa F́ısica, División de Ciencias e Ingenieŕıas,
Universidad de Guanajuato, Loma del Bosque 103, 37150 León, Mexico

(Dated: December 2, 2019)

Asymmetric obstacles can be exploited to direct the motion and induce sorting of run-and-
tumbling particles. In this work, we show that flocking particles which follow the Vicsek model
aligning rules experience a collective trapping in the presence of a wall of funnels made of chevrons,
concentrating at the opposite side of a wall of funnels than run-and-tumbling particles. Flocking
particles can be completely trapped or exhibit a dynamical trapping behaviour; these two regimes
open the door to the design of a system with two perpendicular flows of active particles. This
systematic study broaden our understanding about the emergence of collective motion of microor-
ganisms in confined environments and direct the design of new microfluidics devices able to control
these collective behaviours.

PACS numbers: Valid PACS appear here

I. INTRODUCTION

Self-generated motion is ubiquitous in nature; microor-
ganisms and cells are required to move to execute vital
biological processes. The dynamical behaviours emerging
at the microscopic scale from this non-equilibrium nature
are fascinating [1]. Recently, our attention has shifted
towards not only the effect of solid interfaces on the dy-
namical behaviour of these active systems [2], but also
the effect of their dynamics on their surroundings [3, 4].

One of the most paradigmatic active matter systems
are bacteria, whose locomotion mechanism [5] and collec-
tive behaviour [6] are highly influenced by the properties
and structure of the supporting media. Some bacteria
such as E. Coli move following a run-and-tumble proto-
col, which alternates periods of swimming in a straight
line at constant speed with tumblings in which the move-
ment orientation is randomly chosen, resulting into a dif-
fusive dynamics, in the absence of external fields. How-
ever, the presence of asymmetric walls, such as V-shaped
obstacles, referred to as chevrons, produce rectification
of bacterial motion, resulting into directed motion [7, 8].
It has been recently shown that this ratchet effect can be
utilised to extract work from a bacterial suspension by
means of the motion of anisotropic mobile obstacles such
as gears [9–11] or chevrons [12–15]. Similarly, anisotropic
fixed obstacles such as a wall of chevrons, with funnels
between neighbouring obstacles, induce an increase of
bacteria concentration on the concave side of the fun-
nelled wall [16–18]. This is a consequence of the persis-

∗ Co-corresponding Author. E-mail: juan.aragones@uam.es
† Corresponding Author. E-mail: cvaleriani@ucm.es

tence length of the bacteria movement, which increases
the probability of crossing the funnel from the convex to
the concave side of the funnelled wall, effectively concen-
trating bacteria on the latter [19, 20].

Anisotropic obstacles are not only capable to induce
ratchet effects on suspensions of run-and-tumbling par-
ticles, but also affect the collective behaviour of active
aligning particles, such as Vicsek particles [21]. This
model, despite the simplicity of the inter-particle inter-
action rule, provides a successful platform to study self-
organisation of active particles [22–24], describing flock
formation and collective behaviour [25, 26]. In the pres-
ence of anisotropic obstacles such as a funnelled walls,
Vicsek particles concentrate on one side of the wall [27].
In this system, the rectification’s direction can be re-
versed by changing inter-particle steric-repulsion radius.
In addition, it has been suggested that the alignment in-
teractions between active particles interacting via align-
ment rules and obstacles play a key role in rectifying
active particles’ motion in the presence of a wall of fun-
nels in an infinite periodic channel in which particles dy-
namics obeys an over-damped Langevin rule with finite
angular time correlation [28, 29].

In the present work, we investigate the behaviour of ac-
tive aligning particles interacting with arrays of chevron-
like obstacles that form parallel walls of funnels with al-
ternating orientations, as shown in Fig. 1A. Vicsek par-
ticles spontaneously accumulate on the convex side of
the funnelled walls. We study the dynamics of this col-
lective self-trapping behaviour and identify two regimes,
one static and other dynamical in which particles are ef-
fectively trapped by constantly escaping and getting into
the channels. Exploiting these two regimes, we engineer
a system showing two perpendicular flows of Vicsek par-
ticles.

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2

II. METHODS

To model active aligning particles we use the two-
dimensional Vicsek model [22, 23], in which point-like
particles align with their first neighbours’ average orien-
tation. Vicsek particles move at a constant speed of v =
0.2, while their direction of motion, θi, determines their
velocity ~vi = v(cos(θi)x̂+sin(θi)ŷ). The position of each
Vicsek particle, ~xi(t), is updated at regular time intervals
∆t according to:

θi(t+ ∆t) = arg

 ∑
j∈Ni(t)

eiθj(t)

+ 2πηζ (1)

~xi(t+ ∆t) = ~xi(t) + ~vi(t+ ∆t)∆t (2)

where η ∈ [0, 1] is the noise amplitude, ζ ∈ [−0.5, 0.5] is
a delta-correlated white noise and Ni is the list of first
neighbours of particle i, which are defined as the parti-
cles inside a cut-off radius, R, from particle i. We refer
to this scheme as metric Vicsek model [22, 23]. Alter-
natively, we consider Vicsek particles that only interact
with their first Voronoi neighbours independently of the
distance between them, referred to as topological Vicsek
model [30]. In our simulations, we set R = 1 and ∆t =
1 and normalise all quantities by R and ∆t.

We consider N = 2000 Vicsek particles within a two-
dimensional simulation box of length L = 32 at constant
density ρ = N/L2 = 1.953. We apply periodic boundary
conditions in both directions. For this system, the order-
disorder transition occurs at noises below η = 0.5 [22, 31];
thus, we only consider noises ranging between 0 and 0.5,
where flocking is observed. Besides Vicsek, we also study
a system of run-and-tumble particles without aligning in-
teractions. When considering run-and-tumble particles,
we neglect the first term in Eq. 1 and set η = 1. The
speed is set to 1 and so the persistence length. For a
direct comparison with the Vicsek particles simulations,
we consider a system with the same density (L = 32,
N = 2000).

In our system, active particles interact with fixed
anisotropic obstacles with a chevron shape (V-shape).
Details about their implementation can be found in the
SM. Briefly, when an active particle encounters a chevron
it bounces, thereby reorienting its direction of motion.
We consider three different bouncing rules: i) repul-
sive, ii) elastic and iii) aligning, schematically shown in
Fig. 1B. The repulsive bouncing rule is similar to that of
Ref. [27], under which an active particle bouncing with
a chevron changes its orientation to be perpendicular to
the wall. According to the elastic bouncing rule, active
particles are scattered from the chevrons according to
the Snell’s law; the approaching angle towards the wall
and the outgoing angle from the wall are the same. The
aligning bouncing rules accounts for the tendency of the
active particles to align with solid interfaces [16]. Active
particles colliding with a solid wall change their orienta-
tions to align with it independently of the approaching

FIG. 1. Scheme of the system. A) Geometry of the
channel made of walls of funnels formed by pairs of chevrons
of angle α with an opening δ and separated by a distance dc.
B) Scheme of the bouncing rules of Vicsek particles with the
considered obstacles. Snapshots of the steady state reached
by run-and-tumbling particles (C) at v = 1, δ = 0.1 and by
Vicsek particles (D) at v=0.2 and η = 0.1 swimming between
two parallel walls of funnels with δ = 1, dc = 2 and α =
30 degrees. Schemes showing the possible collisions between
active particles and a funnel: E) for run-and-tumbling parti-
cles the probability to pass from the wide towards the narrow
opening side is high, while F) for a flock of Vicsek particles
aligned with the wall, collisions with the narrow opening side
of the funnels force particles towards the wide opening side
and collision on the wide opening side only bounce them back.

angle, plus a small angular random noise in the interval
[0, ηπ], being η the noise for Vicsek particles or η = 1/32
for run-and-tumble particles [16]. To compute the time
evolution of the particles that bounce with solid walls
we use an event-driven dynamics, whereas Vicsek update
rules are implemented at regular intervals. Details about
the implementation of the models, interactions with ob-
stacles and dynamics can be found in the supplementary
material (SM).



3

III. RESULTS AND DISCUSSION

To start with, we study the behaviour of run-and-
tumble and metric Vicsek particles in the presence of
two parallel funnelled walls made of chevrons, where
the walls are oppositely oriented, as shown in Fig. 1A.
This geometry results in two regions, one on the concave
and the other on the convex side of the funnels. Once
reached the steady-state, both types of particles expe-
rience a trapping behaviour induced by the anisotropy
of the obstacles’ arrays. However, while run-and-tumble
particles concentrate on the concave side of the funnels
(Fig. 1C), Vicsek particles are trapped on the convex
side (Fig. 1D). These different trapping behaviours result
from their differences in the interactions with the obsta-
cle arrays. Run-and-tumble particles tend to go from the
convex to the concave side of the funnelled wall due to
their alignment with the chevrons’s edges, as schemat-
ically shown in Fig.1E. Thus, to induce trapping, the
opening between chevrons must be small compared to
the run’s persistence length [19]. On the contrary, Vicsek
particles are confined in the region between the convex
sides of the funnelled walls. Differently from the trapping
experience by run-and-tumbling particles, the trapping
of Vicsek particles is of collective nature. Initially, Vic-
sek particles align with each other forming flocks. Next,
flocks align their direction of motion with the funnelled
walls, independently of the chosen bouncing rules (see
SM). When flocks are on the concave side of the walls of
funnels, the particle’s collisions with the chevrons push
them towards the convex side. Alternatively, the parti-
cles of flocks on the convex side of a funnelled wall are
bounced back, as represented in Fig. 1F. Therefore, for a
flock of particles aligned with a funnelled wall, the prob-
ability of passing from the concave to the convex side is
much higher than the other way around (see SM). Since
this trapping of Vicsek particles on the convex side of the
funnelled walls is due to the interaction with the chevrons
of previously formed and wall-aligned flocks, the aligning
interactions between Vicsek particles on the concave side
and those on the convex side of the wall are not driving
this trapping behaviour. We observe the same steady-
state in systems where Vicsek particles are not allowed
to interact through funnelled walls (data not shown).

This collective trapping behaviour on the convex side
of the funnelled walls clearly differs from the one de-
scribed in Ref. [27], where point-like Vicsek particles are
preferentially located on the concave side of the wall.
Both systems differs in the chevrons’ geometry, particles’
swimming speed and that we consider periodic boundary
conditions instead of a finite simulation box bounded by
repulsive walls. A detailed comparison can be found in
the SM. However, the different trapping behaviours ob-
served are consequence of the fact that both rectification
mechanisms are of different nature. In our system the
rectification is of collective nature; it is the flock aligning
with the funnelled walls what enables the rectification,
which results into chevrons directing the particles to-

wards the convex side of the wall, as depicted in Fig. 1F.
Whereas in the system of Ref. [27], in which the flocks do
not align with the funnelled walls due to the chevron’s
angle and the presence of repulsive boundaries of the sim-
ulation box, the rectification of Vicsek particles is similar
to the one experienced by run-and-tumbling particles on
the concave side of the wall of funnels. Therefore, one
can switch between rectification mechanism, and thus
trapping localisation, by promoting or neglecting flock’s
alignment with the funnelled walls. For example, one can
increase the angle of the chevrons to hinder flock’s align-
ment with the walls of funnels, or increase the size of the
chevrons to promote the alignment.

Next, we study the effect of both chevrons and fun-
nels’ geometry on the collective self-trapping of Vicsek
particles. To characterise this trapping behaviour, we re-
duce the area of the region in between the convex sides
of the funnelled walls to one tenth of the total area, by
decreasing the distance between the two funnelled walls,
thus forming a narrower channel where particles tend to
get trapped, as seen in Fig. 2A. Then, we establish the
dependence of trapping on the opening length of the fun-
nels δ, the chevron angle α, and the distance between
neighbouring chevrons dc (Fig. 1A).

A flock of Vicsek particles swimming along the concave
side of the funnels is eventually pushed towards the con-
vex side as particles pass through the funnels openings
δ. Therefore, it is expected that δ plays a major role in
the collective self-trapping behaviour of Vicsek particles.
For small funnels’ openings, trapping within the region
in between the convex sides of the funnelled walls is ef-
fective up to high values of Vicsek noise η (Fig. 2B). As δ
increases, the probability of particles traveling from the
convex towards the concave side increases, reason why
self-trapping is effective up to lower values of η. We also
explore the effect of the chevrons angle α, while keeping
δ and dc constant (Fig. 2C). For small angles there is a
small difference between the probability of particles pass-
ing towards the convex side from the concave one and
viceversa, resulting into collective self-trapping only at
small noises where particles form compact flocks aligned
with the funnelled walls. Similarly, chevrons with large
angles only produce trapping at small noises since these
funnels are not very effective in keeping the flock aligned
parallel to the wall. On the contrary, angles larger than
60 degrees promote the aligning of the flocks perpendic-
ular to the funnelled walls. Thus, the collective self-
trapping of Vicsek particles occurs at an optimal trap-
ping of about 40 degrees. Finally, we study the effect of
the distance between chevrons dc, while keeping δ and α
constant. As shown in Fig. 2D, trapping is not affected
by dc, and in all cases is not effective for noises greater
than 0.25.

We also consider the effect of the alignment rules be-
tween Vicsek particles and swimming velocity on this
collective trapping behaviour. We observe that elastic
bouncing rules slightly enhance trapping with respect
to the aligning rule, but the overall behaviour remains



4

FIG. 2. Funnels geometry. A (metric) E (topological):
Characteristic snapshot of the trapped steady state is shown
for η = 0.1, dc= 2, α= 30 degrees, δ = 0.5 and Atrapping/Atotal

= 1/10. The colormaps represent the averaged number of
trapped Vicsek particles over ten independent simulations as
a function of η and: δ with dc = 2 and α = 30 degrees for
metric (B) and topological (F) Vicsek particles, α with δ = 1
and dc = 2 for metric (C) and topological (G) Vicsek particles,
and dc with δ = 0.5 and α = 30 degrees for metric (D) and
topological (H) Vicsek particles. The red solid lines represent
the threshold above which dynamical trapping occurs and the
grey dashed region corresponds to particles trapped inside the
concave part of chevrons.

unchanged (see Fig. S3 in SM). However, the repulsive

bouncing rule makes trapping a lot more erratic, and
we no longer find a clear dependence with the opening
length of the chevrons, δ. Reducing the swimming speed,
v, diminishes the importance of the bouncing rule in the
trapping behaviour (see Fig. S4 in SM). At a swimming
speed of v = 0.004 [27], the trapping of Vicsek particles
is slightly more effective, but the dependence with the
opening length of the chevrons disappears.

Topological Vicsek particles exhibit the same type of
collective self-trapping behaviour described for metric
Vicsek particles. However, due to their lower tendency
to form flocks [30], this collective trapping mechanism is
less effective, resulting into a smaller number of particles
trapped within the channel. Therefore, independently on
the chosen parameter of the funnelled wall geometry, low
noise levels are needed to enhance the number of trapped
particles as shown in Figs. 2F, G and H. The differences
of the collective self-trapping behaviour between metric
and topological Vicsek particles is more evident when
considering more localised traps such as circular traps
instead of infinite channels. These traps are formed by
circular arrays of funnels of radius Rt, whose convex side
point towards the center of the trap, as shown in Fig. 3.
Since we apply periodic boundary conditions, this sys-
tem is a square lattice of circular traps. Independently
on the Vicsek model, either metric (Fig. 3A) or topolog-
ical (Fig. 3B), Vicsek particles at low noise are preferen-
tially located inside the trap, while performing circular
motion along the boundary. Metric and topological Vic-
sek differ in the organisation of particles inside the trap,
while metric Vicsek particles form flocks, topological ones
are placed uniformly along the circular perimeter. Since
most of the particles are located at the circle’s boundary,
either inside or outside of the funnelled wall, to quan-
tify the trapping in this geometry instead of counting
the number of particles inside the circle, we calculate the
number of particles performing a circular motion around
the center of the trap by means of an order parameter
λ ∈ [0, 1] averaged over iterations in the steady state,

λ =
1

N

∣∣∣∣∣
N∑
i=1

((rxi − rx0)vyi − (ryi − ry0)vxi
|~r − ~r0| |~v|

∣∣∣∣∣ (3)

where ~r0 = (rx0, ry0) is the center of the circular trap,
~ri = (rxi, ryi) and ~vi = (vxi, vyi) are the ith-particle’s
position and velocity, respectively. When particles are
trapped, thus performing a circular trajectory around its
center, λ is close to 1, while λ will be lower if particles
are not trapped.

In Figs. 3C and D, the values of λ as a function of the
noise level and distance between chevrons, dc, are shown
for both metric and topological Vicsek particles, respec-
tively. In this case, the distance between chevrons, dc, is
measured along the circumference arc. As it can be seen,
the closer the chevrons, the collective trapping within the
circular trap becomes more effective. Although this trend
is the same for both types of Vicsek particles, metric Vic-
sek particles can be partially trapped, while topological



5

FIG. 3. Circular traps. A) metric Vicsek particles get
trapped in a circular trap made of funnels (dc = 2.356). B)
Steady-state reached by topological Vicsek particles in the
presence of a circular trap (dc = 1.508). In both cases the
conditions where the same: v = 0.2, δ = 0.8, α = 30 deg,
radius of trap Rt = 12 and C and D) Colormaps representing
λ as a function of η and dc.

Vicsek particles are either trapped as seen in the snap-
shot of Fig. 3B or particles are not trapped at all. Thus,
metric Vicsek particles exhibit a wider range of trapping
inside the circles, but at the boundaries of this colormap
we observe flocks that temporarily leave a trap and even-
tually are confined by a neighbouring trap.

We now study the dynamics of the collective trapping
of Vicsek particles in channels made of funnels(Fig.4).
For a given chevron geometry we compute the time evo-
lution of particles transiting through the funnelled walls
in (dark symbols) and out of the channel (light symbols).
As shown in Fig.4A, at low Vicsek noises (up triangles),
flocks form and more particles enter into the channel than
particles leave it, although the system does not reach
steady state during the simulated time. As the noise in-
creases, η = 0.1 (squares), the rate of particles entering
the channel is higher, reaching a trapping steady state
at shorter times. For η around 0.2 (down triangles) the
system reaches a different steady state in which parti-
cles are constantly passing through the funnelled walls
in both directions, while keeping most of the particles
inside the channel. We refer to this behaviour as dynam-
ical trapping. At higher noises, above η = 0.4 (circles),
Vicsek particles do not experience funnels’ anisotropies,
resulting in a homogeneous distribution of particles.

The number of particles crossing from the convex side

FIG. 4. Trapping dynamics. A) Time evolution of the cu-
mulative sum of particles transiting into the channel, repre-
sented by black symbols, and out of the channel, represented
by light symbols at different Vicsek noises η: 0.02 (up tri-
angles), 0.1 (squares), 0.2 (down triangles) and 0.4 (circles).
The chevron geometry is kept constant at δ = 0.3, dc = 2.0
and α = 30 degrees. B) Normalized flux of particles through
the top walls of the channel, J̄⊥+ , as a function of the opening
between chevrons, δ, for several values of Vicsek noises, 0.08
(black pentagons), 0.10 (brown diamonds), 0.12 (blue down
triangles), 0.14 (green up triangles), 0.16 (orange circles), 0.18
(light blues squares) and 0.20 (grey crosses). The horizontal
red line corresponds to the threshold value above which we
consider a dynamical trapping of Vicsek particles.

to the concave side of the channel through the top fun-
nelled wall, N⊥+ , and bottom wall, N⊥− , results into a per-
pendicular flux of particles escaping the channel through

the top wall at each time step, J⊥+ =
N⊥+
L∆t or the bot-

tom wall, J⊥− =
N⊥−
L∆t , respectively. The normalised flows

through the top wall, J̄⊥+ = J⊥+
1
ρv , as a function of the

opening between chevrons for different Vicsek noises are
shown in Fig. 4B. For small values of δ, Vicsek particles
are trapped within the channel and thus, there is not
flux of particles through the walls. As δ increases, parti-
cles are able to escape from the channel and the flux of
particles through the walls increases. The value of δ at
which particles start to escape from the channel decreases
with the Vicsek noise. To define the boundary between
trapping and dynamical trapping regimes, we arbitrarily
choose a minimum flux of particles through the top wall,
J̄⊥+ = 0.004 (horizontal red line in Fig. 4B). The bound-
ary between these two trapping regimes is shown by the
solid red lines in Fig. 2 as a function of the funnel’s geom-
etry (δ, dc and α) and Vicsek noise (η). We observe the
same dynamical regimes in the topological Vicsek model,
but at smaller Vicsek noises (see Fig. S6 in the SM).

Combining the dynamical trapping regime with differ-
ent geometries of the top and bottom funnelled walls of
a channel, we can obtain two perpendicular flows of Vic-
sek particles. Instead of a single channel, we simulate
four parallel channels to minimise possible finite size ef-
fects (Fig 5A). The bottom wall of the channels has an
opening of δbottom = 0.3, while the top wall has wider
openings ranging from δtop = 0.3 to δtop = 1.6. Thus,
the bottom wall works in the trapping regime, while the
top wall works in the dynamical trapping regime except



6

FIG. 5. Cross flow of metric Vicsek particles. A) Snap-
shot of the simulation box containing four parallel channels
formed by pairs of oppositely oriented funnelled walls, with
the bottom wall with opening of δbottom = 0.3 and the top wall
has an opening of δtop = 1.2 (dc = 2, α = 30 degrees, η =
0.12, L = 128, N = 32000 and v = 0.2). The light blue arrow
shows the swimming direction of each flock along the channels
and the orange the perpendicular direction. B) Normalised
flows of Vicsek particles: net perpendicular flow J̄⊥net (orange
squares), top perpendicular flow J̄⊥+ (blue top triangles) and
down perpendicular flow J̄⊥− (red down triangles).

in the symmetric case, δtop = δbottom, where the system is
in the trapping regime and Vicsek particles are trapped
within the channels: J̄⊥+ = J̄⊥− ≈ 0 (Fig 5B). As soon as
we consider the top wall working in the dynamical trap-
ping regime, δtop/δbottom = 3, the Vicsek particles have a
higher probability to pass through the funnels with wider
openings, a fraction of which travels out of the channel
towards the neighbouring channel. This results into a net
flux of Vicsek particles along the perpendicular direction
of the channels towards the top region, J̄⊥net = J̄⊥+ − J̄⊥− .
This flux perpendicular to the channel increases as the
opening of the funnels of the top wall increases, δtop,
as shown in Fig. 5. Even though each channel is also
characterised by a net flux of particles along the channel
(J||), there is not net flux for the entire system due to
the absence of a preferential swimming direction parallel
to the channels. However, a net flux of particles parallel
to the channels could be obtained by breaking the sym-
metry along the channels either by tilting the chevrons
or adding a chevron inside of the channel.

The same behaviour is observed for topological Vic-
sek particles within channels formed by a funnelled wall
working on the trapping regime combined with a wall
working on the dynamical trapping regime. However,
while for metric Vicsek particles the direction of the
flocks swimming along the channel does not have any
effect on the magnitude of the perpendicular flows of
Vicsek particles, in the case of topological Vicsek par-
ticles the relative direction of the swimming direction of
the flocks between two adjacent channels strongly affects
the magnitude of the perpendicular flows of Vicsek par-
ticles, J̄⊥. In Fig. 6A we show a snapshot of the simula-
tion box when all four channels have their Vicsek flows
aligned. The corresponding cross flows measured as a
function of the wider δ value are shown in Fig. 6B, which
are similar to those obtained for metric Vicsek particles.

FIG. 6. Cross flow of topological Vicsek particles. Top
row, a snapshot of a situation where all flows have the same
direction and the results for J⊥ and J|| in that case. Bottom
row, the same for the opposite flows situation. All parameters
have been taken to be the same than in figure 4 of the main
text, but using topological Vicsek model instead of metric.

In Fig.6C and D, we show a system where there are Vic-
sek flows along the channels in different directions and
the dependence of the cross flows with δ using only cases
with opposite flows, respectively. As it can be seen, when
the Vicsek flows between two neighbouring channels have
opposite directions, the cross flows are considerably en-
hanced. Since topological Vicsek particles interact with
their first Voronoi neighbours, particles that scape from
one bottom channel gradually change their velocities to
align it with that of the flock in the top channel. This
effectively push particles towards the top channel, reduc-
ing the density of particles in the region in between two
channels with flocks swimming in opposite directions, as
shown in Fig. 6C. Given that the definition of each par-
ticle neighbours does not depend on the distance, this
interaction is independent of the distance between chan-
nels and thus, the cross flow in this asymmetric case only
depends on the phase between the channels, which is con-
stant at 180 degrees.

IV. CONCLUSIONS

To summarise, we have studied the trapping behaviour
of active particles by means of anisotropic obstacles. V-
shaped obstacles induce the rectification of active parti-
cles, resulting into the concentration of run-and-tumbling
particles within the region at the concave side of channels



7

formed by oppositely oriented walls of funnels. On the
contrary, particles interacting via aligning interactions,
as described by the Vicsek model, concentrate at the
convex side of the funnelled walls. Differently from run-
and-tumbling particles, the trapping behaviour of Vicsek
particles is driven by collective effects. We have shown
that this collective trapping is independent of the align-
ing rules with the obstacles, either repulsive, elastic or
aligning. In addition, this effect is not dependent on the
definition of particles’s neighbours, either metric or topo-
logical Vicsek particles. The key element to observe this
collective self-trapping behaviour is the alignment of the
flocks of particles with the funnelled walls; systems in
which flocks are not allowed to align with the walls of
funnels tend to concentrate on the concave side of the
walls, similar to run-and-tumbling particles. We care-
fully considered how the geometry of the wall of funnels
condition the collective trapping behaviour exhibited by
Vicsek particles. We observe that trapping is more ef-
fective for narrow openings of the funnels and short dis-
tances between chevrons when considering an optimal
angle of the chevrons of about 40 degrees. Moreover, we
observed three well-defined dynamical regimes for this
collective trapping effect as a function of the geometry of
the funnels and Vicsek noise: i) particles pass through a
wall of funnels from the convex side towards the concave
side of the wall, resulting into their trapping within this
region of the channel between the funnel’s walls, ii) par-

ticles pass from both sides of a wall of funnels at equal
rates and do not get trapped and iii) particles exhibit dif-
ferent rates passing through a wall funnels from one side
or the other, reaching a steady state in which particles
are effectively trapped within the channel, but are con-
stantly transiting through the walls of funnels. Exploit-
ing the latter dynamical trapping regime we engineered
a system that exhibits two perpendicular flows of Vicsek
particles, whose magnitude and direction can be tuned
by modifying the geometry of the walls of funnels.

V. ACNOWLEDMENNTS

The authors acknowledge funding from Grant FIS2016-
78847-P of the MICINN and the UCM/Santander
PR26/16- 10B-2. F.A. acknowledges support from
the Juan de la Cierva program (FJCI-2017-33580).
The project that gave rise to these results received
the support of a fellowship from ”la Caixa” Foun-
dation (ID 100010434). The fellowship code is
LCF/BQ/LI18/11630021. R.M. thanks MICINN by the
award of a FPI pre-doctoral grant (BES-2017-081108)
and J.L.A acknowledges the support of the IFIMAC
(MDM-2014-0377) and NVIDIA corporation with the do-
nation of the Titan Xp GPU used for this research.

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	Trapping flocking particles with asymmetric obstacles
	Abstract
	I Introduction
	II Methods
	III Results and Discussion
	IV Conclusions
	V Acnowledmennts
	 References