Spontaneous Segregation of Self-Propelled Particles with Dierent Motilities:
Supplemental Information
Samuel R. McCandlish, Aparna Baskaran, and Michael F. Hagan
Martin A. Fisher School of Physics, Brandeis University, Waltham, MA, 02454
(Dated: October 13, 2011)
In this supplemental material, we rst provide further
information about the clustering behavior of the rods,
(Fig. S3), the herds of passive rods are most evident.
4.
Relationship Between Clustering and Segregation:
including cluster size distributions and our method for
The enhanced local alignment due to rod propulsion gives
identifying clusters. We also explain the relationship be-
rise to collective motion of polar clusters.
tween the segregation parameter
σ̂
and the mean clus-
ter size discussed in the main text.
Then we present
in the same cluster if their representative line segments
snapshots from simulations of various parameters to dis-
are within
play the behavior of the mixed system as a function of
less than
motile fraction.
of clusters.
Finally, we present various pair corre-
We dene a
cluster as in [1, 2]. Two active rods are considered to be
3σ
π/6.
of each other, and their relative angle is
Traversing these connections yields a set
A single unconnected rod is counted as a
lation functions that illustrate the anatomies of clusters
cluster of size one. We do not consider passive rods when
and can be measured in experiments [1].
identifying clusters in this way.
1.
Clustering and swarming:
Various representative
cluster size distributions are shown in Fig. S1.
To measure the degree of segregation within clusters,
Due to
we also implemented an alternative algorithm that iden-
a lack of statistics at large cluster sizes, the cluster size
ties clusters in a manner independent of rod species. In
0.1
this algorithm, two rods are connected if their average
distributions are shown averaged over bins of width
log n.
in
For low
φ, L, fa ,
and Pe, the distributions
follow a power law with an exponential cuto. Increasing
φ, L, fa ,
and Pe yields distributions that `pile up'
relative velocity over a time
value
vmax .
τ
is smaller than a threshold
The size of clusters chosen by this method
was sensitive to the parameters
τ
and
vmax ,
but the cal-
at large cluster sizes, with the most extreme example
culation conrmed that clusters with a motile fraction
(L
between
= 21, φ = 0.8,
Pe
= 120)
exhibiting system-sized
clusters. The forms of these cluster size distributions are
qualitatively similar to those observed by [2].
We nd
0.2
and
0.8
are rare. Typically clusters contain
at least 99% active rods.
As noted in the main text, there is a close relationship
that segregation occurs even for parameters well below
between the mean cluster size
the threshold at which this plateau develops in the clus-
parameter
ter size distribution.
for which clusters are nearly pure and there is a one-
2. Determination of the Segregation Order Parameter:
σ̂ .
hnC i
and the segregation
This relationship holds under conditions
to-one mapping between the mean cluster size and the
The method for determining the segregation parameter
cluster size distribution.
σa (b)
as described in the main text is shown in Fig. S2
well within the interior of a cluster are almost entirely
as a function of the discretization box size
b. In order
σa (b) at zero Péclet
number, we subtract the same function σ0 (b), computed
for a system with the same parameter values except Pe =
0. This function is a nonmonotonic function of the box
size b. Then, we dene the segregation order parameter
σ̂ to be the maximum value of σa (b) − σ0 (b) over all b.
surrounded by other rods of the same species and are
to account for the nonzero value of
thus segregated, while rods at cluster interfaces have a
3.
Snapshots
from
Simulations:
We
display
in
Figs. S3,S4,S5,S6 snapshots from simulations at various
parameters. In these snapshots, active rods are colored
Under these conditions, rods
mixture of neighbors and thus tend to be integrated.
Therefore, the segregation parameter can be estimated
by calculating the fraction of rods
fint
which are found
in the interiors of clusters.
To estimate
fint
we represent a cluster of size
rectangle of width
w
2w,
and length
tirelyp
of of active rods with number density
w=
nC L/2.
nC
as a
which consists en-
L−1 ,
so that
This approximation is based on the typi-
red, and passive rods are colored blue. Figures S4,S5,S6
cal aspect ratio of clusters and the fact that clusters are
are arranged identically to gures 3(b-d) in the main text
close to 99% pure for conditions where the relationship
to facilitate comparison with the segregation order pa-
holds. The value of
rameter
σ̂ .
Figure S3 displays the segregation behavior
w is obtained from the distribution of
cluster sizes for a given parameter set. The area of the in-
at high area fraction. These snapshots display the vari-
terior of a cluster
ety of emergent behavior that occurs at various system
that lies a distance of at least
parameters.
rectangle, where
At low motile fraction
fa ,
we nd active clusters in a
passive gas, and the simple dependence of segregation
on the cluster size distribution is clear.
At higher
fa ,
the passive rods form herds which also increase the degree of segregation. At high
fa
and high area fraction
φ
Aint
b
is dened as the area of the region
b/2
from the edge of the
is the box side length used to calcu-
w < b then there is no
Aint = (w − b)(2w − b).
late the segregation parameter. If
interior,
Aint = 0,
and otherwise
Then, the interior fraction at a given set of parameters
is dened from the cluster size distribution
Π(n) for that
parameter set as the total interior of all clusters divided
2
species can be either P (passive) or A (active). Similarly,
by the total area of all clusters:
fint
the polar and nematic orientation correlation functions
P∞
n=1 Π(n)Aint
.
= P
∞
n=1 Π(n)A
are dened by
CP,XY (x, y) = h(ŷi · ŷj )δ[xx̂i + y ŷi − (~ri − ~rj )]iij
As shown in Fig. S7, comparing this parameter with the
segregation parameter shows a strong correlation, which
corresponds to the correlation between the mean cluster size and segregation parameter presented in the main
text.
5.
Pair Correlation Functions:
correlation functions
G(x, y)
We compute pair
between the dierent rod
species as follows. For a given rod i, we dene a local coordinate system according to the reference frame of rod i,
with
x̂i
as the unit vector in the direction perpendicular
to the axis of rod
i and ŷi as the unit vector parallel to the
rod's axis. Then, the two dimensional pair interspecies
correlation function is dened as
GXY (x, y) =
1
ρ
δ[xx̂i + y ŷi − (~ri − ~rj )]
j6=i
correlation functions
GAA
L = 21, φ = 0.2, Pe= 120.The
and GPP indicate the cluster-
ing of active and passive rods, and show the approximate
shape and size of the clusters. Also, the region less than
1 near the origin of
GAP
and
GPA
indicates a deciency
in the density of passive rods near active ones and vice
verse, which is a consequence of segregation. The empty
where the average is taken only over rods
i
GAP (x, y) as a deciency at y < 0.
As a consequence, passive rods behind active clusters are
i
j
These pair correlation functions are shown in Figs.
S8, S9, S10 for parameters
rods are seen clearly in
+
and the sum is only over rods
CN,XY (x, y) = (2(ŷi · ŷj )2 − 1)δ[xx̂i + y ŷi − (~ri − ~rj )] ij
regions behind active clusters which are devoid of passive
*
X
and
of species X,
of species Y, and the
[1] H. P. Zhang, A. Be'er, E.-L. Florin, and H. L. Swinney,
Proc. Natl. Acad. Sci. U. S. A. 107, 13626 (2010).
aligned in the same direction as the cluster, as seen in
CN,AP .
The regions of aligned rods in
CN,PA
are due to
the eect of clusters colliding with passive rods, pushing
them into ordered regions.
[2] Y. Yang, V. Marceau, and G. Gompper, Phys. Rev. E 82,
1 (2010).
3
L=4
fa = 1.0
0.1
0.001
PHnL
10 -5
10 -5
10 -7
10 -7
1
10
100
1000
1
10
0.1
1000
1
10
L = 10
fa = 0.5
10 -5
10 -7
100
1000
L = 21
fa = 0.5
Ê
f=0.05 Pe=10
‡
f=0.05 Pe=120
Ï
f=0.8 Pe=10
Ú
f=0.8 Pe=120
0.001
PHnL
PHnL
10 -7
1000
0.1
0.001
10 -5
100
n
0.1
0.001
PHnL
100
n
L=4
fa = 0.5
10
10 -5
10 -7
n
1
L = 21
fa = 1.0
0.1
0.001
PHnL
0.001
PHnL
L = 10
fa = 1.0
0.1
10 -5
10 -7
1
10
n
100
n
1000
1
10
100
1000
n
Figure S1. Cluster size distributions Π(n) compared across aspect ratio L, motile fraction f , area fraction φ, and Péclet
number Pe. The top row corresponds to pure systems (f = 1) and the bottom row corresponds to mixed systems (f = 0.5).
a
a
Segregation
1.0
0.8
Ê
0.6
‡
0.4
Ï
0
‡ s0
Ê
Ê
‡
0.2
0.0
Ê sa
Ï sa -s0
Ê
Ê
Ê
ÊÊ
ÊÊ
ÊÊ
ÏÏÏÏ Ê Ê Ê Ê Ê Ê
ÏÏÏÏÏ Ê Ê Ê Ê Ê
‡
ÏÏÏÏÏÏ Ê Ê Ê
‡ ‡
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‡ ‡ ‡
‡ ‡ ‡ ‡ ‡ ‡ ‡
‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
ÏÏÏÏ
ÏÏ
Ï
‡
50
100
150
Box Edge Length
Figure S2. Determination of the segregation parameter σ̂ for
L = 10, φ = 0.2, Pe = 120. The parameters are calculated
for varying box sizes in the mixed case (σa ) and the zero
force case (σ0 ). The segregation parameter σ̂ is dened as
the maximum dierence between σa and σ0 .
a
4
Péclet Number (Pe)
120
80
40
10
0.1
0.3
0.5
0.7
Motile Fraction (fa)
Figure S3. Snapshots from simulations of varying Pe, f with xed L = 10, φ = 0.8.
a
0.9
5
Péclet Number (Pe)
120
80
40
10
0.1
0.3
0.5
0.7
Motile Fraction (fa)
Figure S4. Snapshots from simulations of varying Pe, f with xed L = 10, φ = 0.4.
a
0.9
6
Péclet Number (Pe)
120
80
40
10
0.02
0.2
0.4
Area Fraction (𝜙)
0.6
0.8
Figure S5. Snapshots from simulations of varying Pe, φ with xed L = 10, f = 0.5.
a
7
Péclet Number (Pe)
120
80
40
10
0.02
0.2
0.4
Area Fraction (𝜙)
0.6
0.8
Figure S6. Snapshots from simulations of varying Pe, φ with xed L = 21, f = 0.5. The frames at Pe = 10, φ = 0.4, 0.6, 0.8
display nematic lanes did not reach a steady segregation state within our simulation time, and the frames shown are from the
end of the trajectory.
a
8
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‡
Segregation Hs̀L
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Pe = 0.25
Pe = 0.5
Pe = 1
Pe = 3
Pe = 10
Pe = 20
Pe = 30
Pe = 40
Pe = 60
Pe = 80
Pe = 100
Pe = 120
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Interior Fraction
Figure S7. Comparison between the segregation parameter
σ̂ and the interior fraction f
for L = 10, f = 0.5 and
b = 5.0. These parameters are highly correlated when clusters
are suciently pure; the relationship does not hold for very
low Pe, where the clusters identied are due only to number
uctuations.
int
a
9
(a)
(b)
(c)
(d)
Figure S8. Pair correlation functions: (a) G
φ=0.2, Pe=120.
AA
(x, y), (b) GPP (x, y), (c) GAP (x, y), (d) GPA (x, y). Parameters are L=21,
(a)
Figure S9. Polar orientation correlation functions: (a) C
(b)
P,AA
(x, y), (a) CP,PA (x, y). Parameters are L=21, φ=0.2, Pe=120.
10
(a)
(b)
(c)
(d)
Figure S10. Nematic orientation correlation functions: (a) C
Parameters are L=21, φ=0.2, Pe=120.
N,AA
(x, y), (b) CN,PP (x, y), (c) CN,AP (x, y), (d) CN,PA (x, y).