|  Effect of Confinement and Topology: 2-TIPS vs MIPS†
---|---
|  Nayana Venkatareddy,a Jaydeep Mandal,a and Prabal K. Maiti∗a
|  2-TIPS (Two Temperature induced phase separation) refers to the phase
separation phenomenon observed in mixtures of active and passive particles
which are modelled using scalar activity. The active particles are connected
to a thermostat at high temperature while the passive particles are connected
to the thermostat at low temperature and the relative temperature difference
between "hot" and "cold" particles is taken as the measure of the activity
$\chi$ of the non-equilibrium system. The study of such binary mixtures of hot
and cold particles under various kinds of confinement is an important problem
in many physical and biological processes. The nature and extent of phase
separation are heavily influenced by the geometry of confinement, activity,
and density of the non-equilibrium binary mixture. Investigating such 3D
binary mixtures confined by parallel walls, we observe that, the active and
passive particles phase separate, but the extent of phase separation is
reduced compared to bulk phase separation at high densities and enhanced at
low densities. However, when the binary mixture of active and passive
particles is confined inside a spherical cavity, the phase separation is
radial for small radii of the confining sphere and the extent of phase
separation is higher compared to their bulk counterparts. Confinement leads to
interesting properties in the passive(cold) region like enhanced layering and
high compression in the direction parallel to the confining wall. In 2D, both
the bulk and confined systems of the binary mixture show a significant
decrement in the extent of phase separation at higher densities. This
observation is attributed to the trapping of active particles inside the
passive cluster, which increases with density. Thus the 2D systems show
structures more akin to dense-dilute phase co-existence, which is observed in
Motility Induced Phase Separation in 2D active systems. The binary mixture
constrained on the spherical surface also shows similar phase co-existence.
Our analyses reveal that the coexistent densities observed in 2-TIPS on the
spherical surface agree with the findings of previous studies on MIPS in
active systems on a sphere.

##

††footnotetext: aCentre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India ; E-mail:
<EMAIL_ADDRESS>† Electronic Supplementary Information (ESI)
available:

##  1 Introduction:

Active matter systems are composed of self-driven particles that can convert
internal free energy to some mechanical work 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11. Collective motion and high mechanical stress are some of the
characteristics of such systems. These systems are ubiquitous in nature and
range from a very small to large length scales, e.g. from a colony of bacteria
to a flock of birds. The behaviour of such systems is studied using a variety
of theoretical and computational models, the majority of which are vectorial
in nature. Among many remarkable features shown by such systems, Motility
Induced Phase Separation (MIPS) is one of the most important aspects.

For a non-equilibrium system with particles having different spatially varying
speeds, the local density is inversely proportional to the local speed 12.
Cates and Tailleur 13 theorised the density beyond which, the homogeneous
distribution of a system of active particles gives rise to a spinodal
decomposition into a dense and a dilute phase, the phenomena termed as
Motility Induced Phase Separation (MIPS). In 2012, Fily $et.al$ 14 confirmed
that a system of active Brownian particles (ABPs) in the 2D suspension phase
separates in a dense and dilute region from a homogeneous structure at high
densities and swimming speeds. Later, Redner $et.al.$ 15 showed that, with
increasing density of the system, the phase separation occurs at a lesser
activity. Further experimental confirmations on MIPS were also observed 16,
17.

Besides purely active matter systems, studying a binary mixture of active and
passive particles has significant biological and industrial applications. Such
systems show various features such as phase segregation 18, 19, 20, 21 and
turbulent behaviour 22. To model such binary mixtures without directionality,
Weber $et.al.$ 23 used two different diffusivities to simulate the behaviour
of active and passive particles and observed phase separation between them,
where ’cold’ particles (particles with smaller diffusivities) form dense
clusters. Ganai $et.al$, in 2014 24 explained the chromatin separation inside
the nucleus using the two-temperature model. Theoretical models 25, 26, 27 and
simulations 28, 29, 30, 31, 32, 33, 34 on various soft matter systems show
phase separation between active (hot) and passive (cold) particles under the
two-temperature picture.

What is the effect of different confinements on such binary mixtures? What is
the effect of geometry and topology? Before trying to answer these questions
in this work, we emphasize that the surface curvature is an important aspect
that plays a key role in different biological phenomena such as collective
cell motility in embryogenesis 35 or development of corneal epithelium 36 or
alignment direction for particle aggregation in fluid vesicles 37. Surface
curvature can also induce various topological defects on systems of passive
nematogens 38, 39, 40, 41, 42, which can be controlled systematically 43. An
intricate interplay between the elasticity and active interaction leads to
dynamics of defect structure for active nematic shells 44. In the case of
colloids, structural, thermodynamical and various other properties are also
dependent on the surface curvature 45, 46. A rich array of spatiotemporal
characteristics are observed in a system of active Brownian particles on
spherical surface 47. Self-organisation of active systems on constrained
geometries has also been studied 48. On the other hand, the study of various
condensed matter systems inside confined geometry is also of physical and
biological importance 49, 50, 51, 52, 53, 54. The various phenomena that
follow are consequences of the interaction between the confining wall and the
system, as well as the geometry of the confinement, which results in
aggregation of the particles near the walls 55, 56, 57, 58, 59. The presence
of wall leads to the formation of new phases which can be found only at
extremely high pressure in bulk systems60. Confined geometry affects the
interaction of colloidal particles as well 61, 62. Therefore, we see that
there is a great influence of confinement, in general, on various kinds of
soft matter systems. The geometry of the wall can also alter the ordering in
various active matter properties such as MIPS63, 64, 65, 66, 67, 68. The
active Brownian particles under confinement aggregate near the walls at low
rotational diffusion rates63. Binary mixtures of microscopic algae and passive
colloids in microfluidic channel also show accumulation of passive particles
near the boundaries64.

In this paper, we have addressed the question of how confinement and topology
can play an important role in the phase separation of the active and passive
particles in a binary mixture, using the two-temperature model, the details of
which are discussed in section 2. We first introduce confining parallel walls
in a 3D periodic system and observe that, although the active and passive
particles do tend to phase separate in the confined system, the extent of
phase separation is enhanced at low densities and reduced at high densities
compared to the free periodic boundary/bulk cases. We also observe that for
confined systems with parallel walls, the degree of phase separation reduces
with increase in density. Then we confine the binary mixture in a spherical
cavity, where we observe, the phase separation between active and passive
particles, but the nature of phase separation is radial for small radii of the
cavity. With the increase in the radius of the spherical cavity, the bulk
effect begins to set in. Interestingly, the extent of phase separation between
active and passive particles in the spherical cavity is observed to be
enhanced compared to the bulk periodic systems at all densities. Next, we
study the effect of confinement in 2 dimensions. Interestingly, as opposed to
3D bulk systems, the 2D periodic/bulk systems show a decrement in phase
separation as density is increased.

Similar effects are observed for the 2D confined cases as well. We also
observe that for higher densities, the phase separation between active and
passive particles in confined systems is reduced compared to 2D periodic
systems. These phenomena are the results of increased trapping of hot
particles in cold cluster. Finally, we constrain the system on the surface of
the sphere and observe that phase separation between active and passive
particles is again very small for high packing fractions and the structure
observed on the spherical surface is compared with the motility induced phase
separation observed in active systems in 2 dimensions. The details of the
above-mentioned results are given in section 3. Finally, we draw our
conclusion and shed some light on some of the future directions of the work in
section 4

Fig. 1: Initial configurations of a binary mixture of hot and cold LJ particle
under various geometric confinements. A) Initial configuration of the 3D
binary mixture with parallel walls perpendicular to $\hat{x}$ axis at
equilibrium when $\rho^{*}=0.8$ and $T_{h}^{*}=T_{c}^{*}=2$. (B)Initial
configuration of the 3D binary mixture under spherical confinement at
equilibrium when $\rho^{*}=0.8$ and $T_{h}^{*}=T_{c}^{*}=2$ and the radius of
the spherical wall is $10\sigma$. C)Initial configuration of the 2D binary
mixture with walls perpendicular to $\hat{x}$ axis at equilibrium when
$\rho^{*}=0.8$ and $T_{h}^{*}=T_{c}^{*}=2$. D) Initial configuration of the
binary mixture constrained on the surface of sphere with radius $20\sigma$ at
equilibrium when $T_{h}^{*}=T_{c}^{*}=1.0$ and $\eta=N/(16R^{2})=0.6$. Fig. 2:
(A) and (B) Instantaneous configurations of the phase-separated system of hot
(red) and cold (blue) particles when $\rho^{*}=0.8$ and
$T_{h}^{*}=80,T_{c}^{*}=2$ in the presence of walls which are perpendicular to
$\hat{x}$ and $\hat{y}$ directions respectively. C) Instantaneous
configuration of the phase-separated system with walls perpendicular to
$\hat{x}$ when $\rho^{*}=0.2$ and $T_{h}^{*}=80,T_{c}^{*}=2$.

##  2 Simulation details:

We start with a system of an equal mixture of $N=8000$ hot and cold particles
in a cubic periodic box as has been used in our earlier works 28. The
particles interact via the Lennard-Jones(LJ) potential

$\displaystyle
U=\begin{cases}4\varepsilon\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right],&r<3.0\sigma\\\
0,&r\geq 3.0\sigma,\end{cases}$ (1)

where $\epsilon$ carries the unit of energy, $\sigma$ is the diameter of the
particles and $r$ is the distance between the particles. We have used the
reduced units throughout this work, where the various thermodynamic quantities
are defined as follows: temperature $T^{*}=k_{B}T/\varepsilon$, pressure
$P^{*}=P\sigma^{3}/\varepsilon$. In our calculations, we take mass $m^{*}=1$,
measure time in units of $(\varepsilon/m\sigma^{2})^{1/2}$ and distances in
units of $\sigma$.
We simulate the binary mixture of hot and cold particles under four different
geometrical confinements as shown in Fig:1 :
A) The 3D binary mixture of hot and cold LJ particles is confined by placing
two parallel walls at the boundaries of simulation volume(Fig:1A),
perpendicular to any one of the Cartesian directions (say along $\hat{x}$
axis). The interactions between the wall and particles inside the simulation
volume is repulsive in nature and described by the following potential.

$\displaystyle
U_{wall}=\begin{cases}\varepsilon\left[\frac{2}{15}\left(\frac{\sigma}{r_{w}}\right)^{9}-\left(\frac{\sigma}{r_{w}}\right)^{3}\right]+\varepsilon,&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color<EMAIL_ADDRESS>0,&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}r_{w}\geq
0.4^{1/6}\sigma},\end{cases}$ (2)

where $r_{w}$ is the distance between the wall and a particle.
B) The 3D binary mixture of hot and cold particles is confined inside
spherical cavities of different radii(Fig:1B). The LJ particles interact with
the wall by Weeks-Chandler-Andersen (WCA) potential.

$\displaystyle
U=\begin{cases}4\varepsilon\left[\left(\frac{\sigma}{r_{w}}\right)^{12}-\left(\frac{\sigma}{r_{w}}\right)^{6}\right]+\varepsilon,&r_{w}<2^{1/6}\sigma\\\
0,&r_{w}\geq 2^{1/6}\sigma,\end{cases}$ (3)

where $r_{w}$ is the distance between the particles and the wall.
C) We perform simulations of 2D binary mixtures of hot and cold particles in
the presence of parallel walls at boundaries(Fig:1C). The interaction of the
particles with the wall is given by eqn 2.
D) Finally, to study the effect of curvature, we constrain the mixture of hot
and cold particles on the surface of a sphere (Fig:1D). At each timestep, all
the particles obey two constraint equations:

$\displaystyle|\vec{r_{i}}|=R,$ (4)
$\displaystyle\vec{r_{i}}\cdot\vec{v_{i}}=0$ (5)

where $\vec{r_{i}},\vec{v_{i}}$ indicate the position and velocity of the
$i$-th particle on the surface of the sphere of radius $R$. The origin of the
coordinate system is taken at the center of the sphere.

The simulations were carried out via the LAMMPS software 69 in NVT ensemble.
We have used the Nosé-Hoover 70 thermostat for both 3D and 2D simulations,
whereas the Berendsen thermostat 71 was used to control the temperature for
the system on the spherical surface. The RATTLE algorithm 72 was used to
constrain the particles on the surface of the sphere. The timestep of
integration was chosen to be $\delta t=0.0005$ and the time constant for the
thermostats was taken as $\tau_{T}=100*\delta t$.
Initially, all the particles are assigned the same temperature and we allow
the system to equilibrate. The choice of the initial temperature is made such
that the system remains in a fluid state in equilibrium for all the simulated
densities both for 2D and 3D 73, 74. Accordingly, the initial temperature of
the system is set at T*=2 for 3D and 2D and at T*=1 for spherical confinement
(for reasons discussed in the corresponding subsection). After the
equilibration, we introduce two-temperature model: connect half of the
particles to a cold thermostat maintained at $T_{c}^{*}=2$ ($T_{c}^{*}=1.0$
for confinement on spherical surface) and the rest of the particles to a hot
thermostat initially maintained at temperature $T_{h}^{*}=2$ ($T_{h}^{*}=1.0$
for confinement on spherical surface). Hence, $N_{hot},N_{cold}=N/2$ = the
total number of hot and cold particles in the system. Then we increase the
temperature of hot particles $T_{h}^{*}$ from 2 to 5 to 10 …..to 80 (1 to 5 to
10….to 100 in spherical confinement). At each of the increased temperatures of
hot particles $T_{h}^{*}$, we allow the system to reach a non-equilibrium
steady state for 1 million(M) time steps. Finally, we perform a production run
of another 1M time steps to obtain data for further analysis.
The activity of the non-equilibrium system is measured using the relative
temperature difference between the hot and cold particles,
$\chi_{imp}=\frac{T_{h}^{*}-T_{c}^{*}}{T_{c}^{*}}$. However, due to heat
exchange between hot and cold particles from collisions, the effective
temperature of the cold(hot) particles obtained from equipartition theorem
$T_{c}^{eff*}$($T_{h}^{eff*}$) is higher(lower) than the temperature imposed
on cold(hot) thermostat $T_{c}^{*}$($T_{h}^{*}$). Therefore the effective
activity is defined as the relative effective temperature difference between
the two types of particles.

$\chi=\chi_{eff}=\frac{T_{h}^{eff*}-T_{c}^{eff*}}{T_{c}^{eff*}}$ (6)

We observe phase separation between hot and cold particles at high activities
in the presence of any kind of confining walls in both 3D and 2D systems and
on the surface of the sphere. To quantify the extent of phase separation in
the presence of parallel walls, we calculated the order parameter for the
phase separation in the following way. We divide the simulation box into sub-
cells and calculate the order parameter as 28, 30

$\phi(T_{h}^{*})=\frac{1}{N_{cell}}\bigg{<}\sum_{i=1}^{N\textsubscript{cell}}\frac{|(n^{h}(i)-n^{c}(i))|}{(n^{h}(i)+n^{c}(i))}\bigg{>}$
(7)

where $n_{h},n_{c}$ are the number of hot and cold particles in each of the
sub-cells, $N_{cell}$ is the total number of subcells in the simulation volume
and $<...>$ means the average over all steady state configurations. For the
cases of particles on a spherical surface, to calculate order parameter
$\phi$, the surface is divided such that each sub-section takes almost equal
areas. The polar angle $\theta\in[0,\pi]$ was divided into larger intervals
near the poles of the sphere and in smaller intervals near the equator,
whereas the azimuthal angle ($\phi\in[0,2\pi]$) was divided into equal parts
throughout.

However, for hot and cold particles confined inside spherical walls in 3D,
dividing the simulation volume into sub-cells with equal volume is
challenging. So we perform cluster analysis28, 29 on phase separated cold
particles. We define two particles belong to the same cluster if the distance
between them is less than cut-off distance $r_{c}$. The cut-off distance
$r_{c}$ is obtained from the first peak of the radial distribution
function(RDF) of the cold particles. Using the above criteria of a cluster, we
have calculated the normalized number of clusters of cold particles
$K_{cl}^{cold}$ and the fraction of cold particles in the largest cold cluster
$f_{cl}^{cold}$.

##  3 Results:

###  3.1 Confinement of 3D binary mixture by parallel walls

Fig. 3: Plot of order parameter $\phi$ versus activity $\chi$ at various
densities $\rho^{*}$. The solid lines are the results of the binary mixture
under confinement by parallel walls while the dotted lines are the results
from bulk simulations. For high densities $\rho^{*}$=0.8 and 0.5, the order
parameter $\phi$ has a lower value under confinement compared to bulk phase
separation. At low density $\rho^{*}$=0.2, the order parameter $\phi$ has
higher value under confinement compared to bulk phase separation.

We have observed that in the presence of confining parallel walls, the hot and
cold particles phase separate at high activities. The extent of phase
separation under confinement is density dependent. The degree of phase
separation is reduced compared to bulk phase separation at high densities and
enhanced at low densities.
Fig:1A shows the initial configuration of the binary mixture with parallel
walls perpendicular to $\hat{x}$ axis when $\rho^{*}=N\sigma^{3}/V=0.8$ ($V$
is the volume of the simulation box) and $T_{h}^{*}=T_{c}^{*}=2$. The initial
system of hot and cold particles is well mixed. Fig:2A and B show the
instantaneous snapshot of phase separated system at $\rho^{*}=0.8$ when the
temperature of hot particles is raised to $T_{h}^{*}=80$ in the presence of
parallel walls perpendicular to $\hat{x}$ and $\hat{y}$ axis respectively.
Fig:2C shows the snapshot of phase separated system at $\rho^{*}=0.2$ when the
temperature of hot particles is raised to $T_{h}^{*}=80$ and confined by walls
parallel to $\hat{x}$ axis.
One of the distinguishing features of phase separation under parallel
confinement is that the interface between phase-separated hot and cold
particles is always parallel to the confining walls(See Fig 2A and B) and the
phase-separated cold particles are always present in the vicinity of walls. To
quantify the phase separation of the binary mixture under parallel wall
confinement, we have calculated the order parameter $\phi$ defined by equation
7 for three simulated densities $\rho^{*}$=0.8, 0.5 and 0.2 and compared those
with the results from bulk unconfined simulations. Fig 3 shows the plot of
order parameter $\phi$ versus activity $\chi$ at various densities $\rho^{*}$.
The solid lines are the results of binary mixture under confinement by
parallel walls while the dotted lines are the results of bulk simulation. For
all densities, we see that the order parameter $\phi$ increases with activity
$\chi$ indicating that the hot and cold particles phase separate at high
activity. We observe that for high density of $\rho^{*}$=0.8 and 0.5, the
magnitude of order parameter $\phi$ is lower under confinement compared to
bulk implying a reduction in the extent of phase separation. However, at a low
density of $\rho^{*}$=0.2, the value of $\phi$ under confinement is much
greater than the value of $\phi$ from bulk simulations indicating a
significant enhancement in extent of phase separation.
We can qualitatively understand the reason for the density-dependent reduction
or enhancement of phase separation under parallel confinement by analyzing the
composition of the phase-separated cold dense region. From Fig:2A, we can see
that in the phase-separated cold region, a significant number of hot particles
are trapped (greater than the number of hot particles trapped in the bulk
simulations without walls) at a high density of $\rho^{*}$=0.8. Under periodic
boundary conditions, the cold region has two interfaces between hot and cold
zones that facilitate the diffusion of hot particles trapped in the cold
region. However, in the case of parallel confinement, the cold region is
formed near the wall leading to only one interface between cold and hot
particles. The wall prevents the escape or diffusion of the trapped hot
particles resulting in significant number of hot particles in the cold region
which leads to the reduction of order parameter $\phi$ compared to bulk. This
can be confirmed from the radial distribution function (RDF) of hot and cold
particles for $T_{h}^{*}$=80 as shown in Fig:4. From the RDF of bulk
simulations(Fig:4A) at $\rho^{*}$=0.8 and $T_{h}^{*}$=80, we can see that the
cold particles are in the crystalline state but hot particles are in the
gaseous state indicating that a negligible number of hot particles are trapped
in the cold region. However, under confinement(Fig:4B) at $\rho^{*}$=0.8 and
$T_{h}^{*}$=80, the hot particles also show crystalline structure because a
significant fraction of hot particles are trapped in the cold zone. The number
of hot particles trapped in the phase-separated cold region decreases with
density. Hence, at a low density of $\rho^{*}$=0.2, we observe an enhancement
in phase separation compared with bulk. We note that our order parameter
values from the simulations (Fig:3) are dependent on the initial configuration
of the binary mixture. To obtain the results which are independent of the
initial configuration, the heating rate must be very slow or the simulations
must be run for very long time. The presence of wall in the system prevents
the escape of trapped hot particles and hence would require extremely long
simulation run time to reach its true steady state.

Fig. 4: A) The radial distribution function of the particles for a 3D periodic
system at $\rho^{*}=0.8,T_{h}^{*}=80.0$. The peaks in the cold cluster
indicate crystalline ordering whereas, the hot particles are forming a gaseous
phase. B) The radial distribution functions for the case of binary system
confined by walls perpendicular to $\hat{x}$ direction, at
$\rho^{*}=0.8,T_{h}^{*}=80.0$. In this case, we see some peaks in the plot of
hot particles as well, which indicates that a significant amount of the hot
particles are trapped inside the cold cluster.

###  3.2 Effect of spherical confinement:

Fig. 5: Instantaneous configurations of the phase separated hot and cold
particles inside spherical cavities of different radii when
$T_{h}^{*}=80,T_{c}^{*}=2.0$ and $\rho^{*}=0.8$. A) and B) Final configuration
of spherical cavity of $R=4\sigma$ and its cross-section respectively . The
phase separation is radial with cold particles near the periphery and hot
particles in the interior of the sphere. C) Cross-section of final
configuration of spherical cavity of $R=10\sigma$ also shows radial phase
separation similar to $R=4\sigma$. D)Cross-section of final configuration of
spherical cavity of $R=13.36\sigma$. For $R=13.36\sigma$, the phase separation
is not radial but takes place along a cartesian direction similar to bulk
phase separation.

In the previous section, we studied the effects of confinement of hot and cold
particles by parallel walls placed at the boundary of the simulation volume.
Here, we simulate the mixture of hot and cold LJ particles inside spherical
cavities of different radii to study the influence of isotropic confinement on
phase separation. We have systematically studied the effect of confining
radius $R$, density $\rho^{*}$ and activity $\chi$ on the phase separation of
hot and cold LJ particles.

####  3.2.1 Effect of confining radius

We have simulated equal mixture of hot and cold LJ particles inside spherical
cavities of radii $R=4\sigma(N=214)$, $6\sigma(N=724)$, $8\sigma(N=1716)$,
$10\sigma(N=3352)$ and $13.36\sigma(N=8000)$, while maintaining the constant
density of $\rho^{*}=3N/(4\pi R^{3})=0.8$. Using the same procedure mentioned
in section 2, starting from an equilibrium configuration, we maintain the
temperature of cold particles at $T_{c}^{*}=2$ and increase the temperature of
hot particles from $T_{h}^{*}=2$ to $T_{h}^{*}=5,10,15,...80$ in steps of
five. We observe that the phase separation between hot and cold LJ particles
is heavily influenced by the geometry of confinement. Instantaneous snapshots
of the phase-separated non-equilibrium system at different radii are shown in
Fig 5. When both hot and cold particles are at the same temperature
$T_{c}^{*}=T_{h}^{*}=2$, the particles are well mixed (Fig:1B). For
$R=4\sigma$, when the temperature of hot particles is $T_{h}^{*}=80$, the
particles undergo radial phase separation with hot particles in the interior
and cold particles at the periphery of the sphere (Fig:5A and B). This radial
phase separation is also observed for $R=6,8,10\sigma$ at high activities
(Fig:5C). However, for a larger radius of $R=13.36\sigma$, when
$T_{h}^{*}=80$, the phase separation is not radial but takes place along a
Cartesian direction similar to the bulk phase separation(Fig:5D). Here we
would like to point out that for radii $R=4$ to $10\sigma$ (Fig:5A-C), the
radially phase separated cold "phase" is just 1-2 layer thick as the volume of
cold region is limited by the number of particles in the finite-sized system.
So it is difficult to identify the co-existing phases and their interface in
the phase separated system (in contrast with phase separation under parallel
walls in section 3.1 where hot and cold phases co-exist with a well defined
interface). Hence we refer to the above-mentioned radial phase separation as
micro-phase separation75, 76. As the radius of confining sphere is increased,
the phase separation becomes more similar to bulk phase separation.

Fig. 6:  Density variation of hot, cold and all particles along the radial
direction. A)For $R=4\sigma$, $T_{c}^{*}=T_{h}^{*}=2$ and $\rho^{*}=0.8$, the
density of both hot and cold particles oscillate as we move along in the
radial direction showing layering phenomena. B) For $R=4\sigma$,
$T_{h}^{*}=80$ and $\rho^{*}=0.8$, the density of cold particles are
concentrated near the wall while the density of hot particles is uniform in
the interior. C) For $R=13.36\sigma$, $T_{c}^{*}=T_{h}^{*}=2$ and
$\rho^{*}=0.8$, the density oscillations are present only near the wall and
the density is constant in the interior due to the large radius of the sphere.
D) For $R=13.36\sigma$, $T_{h}^{*}=80$ and $\rho^{*}=0.8$, we see that cold
particles don’t show complete radial phase separation and are present in
radius range $6<r<13.36$, indicating phase separation takes place along the
cartesian direction.  Fig. 7:  A) Normalized number of clusters of cold
particles $K_{cl}^{cold}$ versus activity $\chi$ for different values of $R$
when $\rho^{*}=0.8$ . As the activity $\chi$ increases, the number of clusters
of cold particles $K_{cl}^{cold}$ for all radii decreases. B) Fraction of cold
particles in the largest cluster $f_{cl}^{cold}$ versus activity $\chi$ for
different values of $R$ when $\rho^{*}=0.8$. The size of the largest cluster
increases with activity $\chi$ for all radii. The size of the largest cluster
increases as we increase the radius upto $R=10\sigma$ and then decreases
slightly for $R=13.36\sigma$.

To understand the arrangement of hot and cold particles inside the sphere, we
have calculated the density variation $\rho^{*}(r)$ in the radial direction
$r$ from the center of the sphere to the wall (Fig:6). The sphere is divided
into spherical shells along the radial direction and the density of hot and
cold particles is calculated in each shell. The density of both hot and cold
particles oscillate along the radial direction for $R=4\sigma$ when
$T_{c}^{*}=T_{h}^{*}=2$ and $\rho^{*}=0.8$ (Fig:6A) and indicates layering of
the particles. Such layering of particles leading to density oscillation is
well studied in equilibrium systems77, 78, 79. The magnitude of density
oscillations increases with density and decreases with increase in confining
volume. Fig:6B shows density variation for $R=4\sigma$ when $T_{h}^{*}=80$
where we can clearly see that cold particles are present near the wall and hot
particles are present in the interior. Due to the high temperature of hot
particles layering is absent in the interior. Also, we can see that the
density of cold particles near the periphery is very high (higher than the
density of phase-separated cold region in bulk as reported in our previous
work28) and the density of the hot particles is lower than the average
density. Fig:6C and D show the density variation for $R=13.36\sigma$ and
$\rho^{*}=0.8$ when $T_{h}^{*}=2$ and $T_{h}^{*}=80$ respectively. For a large
radius of $R=13.36\sigma$ in equilibrium, the density of both hot and cold
particles is constant in the interior and oscillates only near the wall. When
$T_{h}^{*}=80$, the phase separation is not radial as we can see that cold
particles are present in the range $6\sigma<r<13.36\sigma$. Also the region
with phase separated cold particles shows layering due to the high density of
phase separated cold particles.
The phase separated cold particles in the binary mixture always seem to
aggregate near the wall as if there was an effective attraction between the
cold particles and the wall. This attraction between the cold particles and
the wall can be quantified by using the mean radial position of cold
particles(Refer Fig:S1 and Section I in Supplementary Information(SI)) which
increases as the activity $\chi$ of the non-equilibrium system is increased.
For radius $R=4\sigma$ to $10\sigma$, the 3D phase separation of hot and cold
particles leads to a 2D spherical shell of cold particles where the particles
are arranged in 2D hexagonal lattice 80. For radius $R=13.36\sigma$, the
phase-separated cold cluster has particles arranged in HCP and FCC lattice
similar to the bulk phase separation. Further details on the structure of
phase separated cold particles are given in Section II of the SI.
The hot and cold particles inside a spherical cavity phase separate at high
activities for all radii even though the nature of phase separation depends on
the radius. The hot particles due to their high temperature, force the cold
particles to aggregate and form clusters. To study the number and size of the
cold clusters, we perform cluster analysis of cold particles as defined in
section II. Fig:7A and B give the plot of the normalized number of clusters of
cold particles $K_{cl}^{cold}$ (normalized by the total number of cold
particles $N_{cold}$) versus activity $\chi$ and plot of the fraction of cold
particles in the largest cluster $f_{cl}^{cold}$, defined as
$f_{cl}^{cold}=n_{cl}^{cold}/N_{cold}$ (where $n_{cl}^{cold}$ is the number of
cold particles in the largest cluster), versus activity $\chi$ respectively
for different values of $R$. Initially for low activity, the number of
clusters of cold particles $K_{cl}^{cold}$ is large indicating that the cold
particles are spread out in the simulation volume showing no phase separation.
As the activity $\chi$ is increased, the number of clusters of cold particles
$K_{cl}^{cold}$ decreases with activity $\chi$ as small clusters join to form
larger clusters for all radii $R$. We see that $K_{cl}^{cold}$ decreases with
an increase in radii from $R=4\sigma$ to $10\sigma$ even below the value for
bulk phase separation. So, the spherical confinement of hot and cold particles
enhances phase separation compared to bulk phase separation. The plot of
$K_{cl}^{cold}$ for $R=13.36\sigma$ closely follows the plot of bulk phase
separation. The fraction of cold particles in the largest cluster
$f_{cl}^{cold}$ is proportional to the size of the largest cold cluster. The
fraction of cold particles in the largest cluster $f_{cl}^{cold}$ increases
with activity $\chi$ for all radii which implies that the size of the cold
cluster grows with activity. The size of the cold cluster also increases as we
increase the radius from $R=4\sigma$ to $10\sigma$ even surpassing the bulk
phase separation. For $R=13.36\sigma$, the size of the cluster is reduced
compared to its preceding radius($10\sigma$) but closely follows the value of
bulk phase separation.

Fig. 8: A)Instantaneous snapshot of the final configuration of the system of
N=8000 hot and cold particles inside spherical cavity when $T_{h}^{*}=80$ at
density $\rho^{*}=0.2$. We can clearly see that the hot and cold particles
phase separate. B)Plot of normalized number of clusters of cold particles
$K_{cl}^{cold}$ versus activity $\chi$ for different densities $\rho^{*}$. The
solid lines are plots for spherically confined systems and the dashed lines
are plots for bulk systems. As the activity $\chi$ increases, the number of
clusters of cold particles $K_{cl}^{cold}$ decreases for all densities. The
number of clusters of cold particles $K_{cl}^{cold}$ in spherically confined
systems is always lower than the bulk systems indicating that the extent of
phase separation is enhanced in spherically confined systems at all densities
compared to their bulk counterparts. C)Plot of fraction of cold particles in
the largest cluster $f_{cl}^{cold}$ versus activity $\chi$ for different
densities $\rho^{*}$. We can clearly see that the size of the cluster is
higher in spherically confined systems at all densities compared to their bulk
counterparts.

####  3.2.2 Effect of density on phase separation inside spherical cavity.

We find that for all densities under spherical confinement, the phase
separation is enhanced under confinement compared to bulk simulations. We
simulate a fixed number of hot and cold particles (N=8000) inside spherical
cavities at densities $\rho*=0.8(R=13.36\sigma)$, $0.5(R=15.63\sigma)$,
$0.2(R=21.21\sigma)$ and $0.1(R=26.72\sigma)$. Since the number of particles
is fixed, the radius of the sphere is varied to obtain desired density. We
observe phase separation at high activities for all densities (Fig:8A).
However, due to the large radii of confining spheres, the phase separation is
not radial but takes place along a cartesian direction similar to the bulk
phase separation. Again, to study the number and size of the cold clusters, we
perform cluster analysis on cold particles as defined in section II. Fig:8B
and C show plot of normalized number of clusters of cold particles
$K_{cl}^{cold}$ versus activity $\chi$ and plot of fraction of cold particles
in the largest cluster $f_{cl}^{cold}$ versus activity $\chi$ respectively for
different densities. The results of spherically confined systems (solid lines)
are compared with bulk systems (dashed lines) at all densities. As the
activity $\chi$ increases, the number of clusters of cold particles
$K_{cl}^{cold}$ for all densities decreases. The number of clusters of cold
particles $K_{cl}^{cold}$ in spherically confined systems is always lower than
the bulk systems. We conclude that confining a non-equilibrium system of hot
and cold particles inside a sphere enhances the extent of phase separation
compared to their bulk counterparts. The fraction of cold particles in the
largest cluster $f_{cl}^{cold}$ is also higher in spherically confined systems
compared to their bulk counterparts. Under spherical confinement, phase
separation is enhanced even at high densities where we observe a reduction in
the extent of phase separation under parallel walls. Due to the geometry of
spherical confinement, the phase-separated hot and cold particles have a
larger interface(See Fig:5B, C, D and 8A) relative to the interface under
parallel confinement. So, the number of trapped hot particles in the cold
region is lower in spherically confined systems compared to parallel
confinement, which leads to an increase in the extent of phase separation.

Fig. 9:  A) Variation of normal pressure $P_{N}$=$P_{xx}^{*}$ and tangential
pressure $P_{T}$=$P_{yy}^{*}$=$P_{zz}^{*}$ along $\hat{x}$ direction
perpendicular to parallel walls when $\rho^{*}=0.8$ and
$T_{c}^{*}=T_{h}^{*}=2$. B)Variation of normal pressure $P_{N}$=$P_{xx}^{*}$
and tangential pressure $P_{T}$=$P_{yy}^{*}$=$P_{zz}^{*}$ along $\hat{x}$
direction perpendicular to parallel walls when $\rho^{*}=0.8$ and
$T_{h}^{*}=80$. We can see that in the phase-separated cold region the
tangential pressures $P_{yy}^{*}$ and $P_{zz}^{*}$ are significantly higher
than the normal pressure $P_{xx}^{*}$ indicating that the cold particles are
under compression.

###  3.3 Pressure anisotropy in confined systems

The pressure variation across the interface of phase-separated hot and cold
particles has been studied in bulk systems 31, 30, 29, 34 which reveal that
the high kinetic pressure of the phase-separated hot particles is balanced by
the high virial pressure of the dense cold clusters. Also, pressure anisotropy
has been studied in bulk31, 34 systems which reveal that in the cold region of
the phase-separated system, the value of tangential pressure $P_{T}$ is always
lower than the normal pressure $P_{N}$ (See Fig:4 in SI for bulk results of
binary mixture of LJ particles). In case of confined systems with either
parallel or spherical walls, despite the repulsive interaction of the wall
with the particles, the cold particles at high activity aggregate near the
walls as if there is an effective attraction between the cold particles and
the wall. This attraction is also evident from the fact that the cold
particles under confinement reach densities greater than their bulk
counterparts. So, we investigate the effect of confinement on the tangential
and normal pressures of the binary mixture in the presence of parallel walls
at the boundaries of the simulation volume. The simulation volume is divided
into sub-volumes along the direction perpendicular to the wall which is along
the $\hat{x}$ axis. The component of pressure tensor $P_{\alpha\beta}$ 81, 82,
83, 84in the $i^{th}$ sub-volume is given by

$P_{\alpha\beta}(i)=P_{\alpha\beta}^{k}(i)+P_{\alpha\beta}^{c}(i)$ (8)

where $P_{\alpha\beta}^{k}$ is the kinetic part and $P_{\alpha\beta}^{c}$ is
the configurational or virial part of the pressure tensor.

$P_{\alpha\beta}^{k}(i)=\frac{1}{V(i)}\sum_{j\in
V(i)}mv_{\alpha}^{j}v_{\beta}^{j}$ (9)
$P_{\alpha\beta}^{c}(i)=\frac{1}{V(i)}\int_{C_{jk}\in
V(i)}f_{\alpha}^{jk}dl_{\beta}$ (10)

where $j\in V(i)$ implies summation over all the particles in sub-volume
$V(i)$ and $v_{\alpha}^{j}$ is the velocity of $j^{th}$ particle along
$\alpha$ direction. $f_{\alpha}^{jk}$ is the force between $j^{th}$ and
$k^{th}$ particles along $\alpha$ direction, $l$ is a point on contour
$C_{jk}$ connecting particles $j$ and $k$ and $C_{jk}\in V(i)$ implies that
only particles $j$ and $k$ contribute to the integral if a part of contour
$C_{jk}$ lies in volume $V(i)$. Fig:9 A) and B) shows the variation of normal
pressure $P_{N}$=$P_{xx}^{*}$ and tangential pressure
$P_{T}$=$P_{yy}^{*}$=$P_{zz}^{*}$ along $\hat{x}$ direction perpendicular to
parallel walls at $\rho^{*}=0.8$ when $T_{c}^{*}=T_{h}^{*}=2$ and
$T_{h}^{*}=80$ respectively. The condition of mechanical equilibrium
$\nabla\cdot P=0$ requires the normal component of pressure $P_{N}$ to be
constant. In equilibrium, when $T_{c}^{*}=T_{h}^{*}=2$, we see that normal
component $P_{xx}^{*}$ is constant along $\hat{x}$ axis but the tangential
pressures $P_{yy}^{*}$ and $P_{zz}^{*}$ oscillate near the walls due to
layering effect. When $T_{h}^{*}=80$, normal component $P_{xx}^{*}$ is
constant as dictated by the condition of mechanical equilibrium. The
tangential pressures $P_{yy}^{*}$ and $P_{zz}^{*}$ in the cold region show
oscillations of high amplitude(due to the high density of cold region layering
effect is very much enhanced ) which exceeds the normal pressure $P_{xx}^{*}$
by nearly order of magnitude. As mentioned before, the strong effective
attraction of cold particles with the wall is also reflected in the pressure
calculation, where in contrast to bulk, tangential pressure in the cold region
is much greater than the normal pressure inducing strong compression in the
cold regions perpendicular to the direction of the wall.

###  3.4 Effect of confinement in 2 dimensional plane

Fig. 10: Snapshots of the non equilibrium steady states at
$T_{h}^{*}=80.0,T_{c}^{*}=2.0$ at $\rho^{*}=N/(L^{2})=0.2$ for A) 2D bulk and
B) x-confined system, where $L=$ length of the 2D plane. Formation of a single
large cold cluster with hot particles trapped inside can be observed in the
configurations. The red and blue particles are the hot and cold ones
respectively.

We also study the effect of dimension and confinement on the phase separation
of such binary mixtures. In 3D systems, we observed that for low density
($\rho^{*}=0.2$), the phase separation increases with parallel confinement
(wall normal to the edges along $\hat{x}$ direction), whereas the phase
separation reduces with confinement for the high density ($\rho^{*}=0.8$). To
examine the effect of dimensions with similar parallel confinements, we have
simulated a binary mixture of $N=8000$ particles on a 2D plane (square) with
different densities $\rho^{*}=N/(L^{2})$, where $L$ is the length of the
plane. We have studied three cases: i) Periodic plane (2D bulk) ii) Plane with
parallel walls in x direction (normal to $\hat{x}$ direction at the edges of
the simulation region, implemented by a repulsive potential described by eqn
2) (x-confined) and iii) Plane with parallel walls in both directions
($\hat{x}$ and $\hat{y}$) (xy-confined). Each of these three confined systems
was simulated at three different densities a) $\rho^{*}=0.2,$ b)
$\rho^{*}=0.5$, and c) $\rho^{*}=0.8$. The densities and other parameters for
the simulations were chosen such that the initial configuration assumes a
fluid phase.

Fig. 11: The effect of density on phase separation at specific confinement in
2D- Plot of order parameter $\phi$ as a function of activity $\chi$ at
different densities, $\rho^{*}=0.2,0.5$, and $0.8$: for A) periodic plane (2D
bulk), and B) x-confined system. The extent of phase separation reduces with
an increase in the density of the system for both the cases.

In equilibrium, at $T_{c}^{*}=2.0$, the particles are uniformly distributed
(Fig:1(C)). When two temperature scalar activity is introduced, the hot and
cold particles phase separate for both the unconfined and confined systems
(see Fig:10A-B). We find that the phase separation phenomenon is indeed
affected by the dimension of the space. In contrast to 3D bulk results, in the
2D periodic case (2D bulk), we observe that the extent of phase separation
reduces with density, as is evident from the saturation values of the order
parameter, as shown in Fig:11(A). Similar findings are made for x-confined
systems as well (Fig:11(B) Similar to the 3D confined systems, the 2D
x-confined and xy-confined systems also reveal that the particles accumulate
near the wall and form the dense cluster close to the wall. Therefore we
conclude that the accumulation of particles near the walls is a phenomena
independent of the spatial dimensions.

Fig. 12: Effect of confinement on phase separation at a specific density in 2D
- Plot of order parameter $\phi$ vs activity $\chi$ for three different
confining cases: 2D bulk, x-confined and xy-confined case. A) For density
$\rho^{*}=0.2$, there is no observed effect of confinement. The order
parameter $\phi$ almost takes the same value for bulk systems and confined
systems (x-confined and xy-confined). B) For a density of $\rho^{*}=0.5$, the
effect of confinement is observed to reduce the phase separation.

We also explore the effect of applying different confinements on the phase
separation at fixed packing densities. We observe that at low densities such
as $\rho^{*}=0.2$, (Fig:12(A)), the effect of confinement is not prominent in
the phase separation, but at higher densities $\rho^{*}=0.5$, the phase
separation again reduces with the insertion of wall in the system (Fig:12(B)),
compared to 2D bulk systems.

As the spatial dimensions decrease from 3D to 2D, the reduced effective
diffusivity of particles leads to the reduced thermal escape of the hot
particles from the cold cluster. In addition, the interface between hot and
cold clusters turns into a line in 2D systems from a surface in 3D systems.
Also, the 2D systems can be viewed as a 3D confined system having an
infinitesimal extent in one of the directions. These characteristics result in
enhanced trapping of hot particles in cold clusters in two-dimensional
systems. This phenomenon offers an explanation for the features observed in
Fig:11(A) and (B). To quantify the trapping of hot particles, we define:
$p_{cl}^{hot},p_{cl}^{cold}=(n_{cl}^{hot}/n_{cl}),(n_{cl}^{cold}/n_{cl})$,
which denotes the fraction of hot and cold particles in the largest cluster
with respect to the total particles in the cluster:
$n_{cl}=n_{cl}^{hot}+n_{cl}^{cold}$. Also $f_{cl}=n_{cl}/N$ denotes the
fraction of particles taking part in the largest cluster and
$f_{cl}^{hot}=(n_{cl}^{hot}/N_{hot})$ denotes the fraction of hot particles in
the largest cluster, with respect to the corresponding hot system size.

Fig. 13: A) Composition of the largest cold cluster plotted against activity
for the different densities considered: $\rho^{*}=0.2,0.5$ and $0.8$ for the
2D bulk systems. The solid lines and dashed lines indicate the total number of
particles $f_{cl}$ and the trapped hot particles inside the cluster
$q_{cl}^{hot}(=f_{cl}^{hot}/2)$ respectively. The rise of the saturation
values of $q_{cl}^{hot}$ with density $\rho^{*}$ indicates the enhanced
trapping of hot particles in cold cluster with density. For convenience in
comparison, we took the value of $1.1\sigma$ to be the cut-off distance for
each density. B)The local packing fractions $\eta(x)$ for the non-equilibrium
steady state for the x-confined system, plotted against the distance along the
wall $(x)$, for different initial equilibrium densities $\rho^{*}=0.2,0.5$ and
$0.8$. The phase-separated cold cluster gives rise to a high value of the
local packing fraction, whereas the low values are due to the isotropic hot
zone. The maximum value of the local density in cold cluster increases with
the overall densities $\rho^{*}$.

In fig. 13A) we show the behaviour of $f_{cl}$ and
$q_{cl}^{hot}(=f_{cl}^{hot}/2)$ with $\chi$ for 2D bulk systems. We observe
that for a specific density $\rho^{*}$, there is a rapid rise in the values of
both the quantities with activity. This indicates that the size of the largest
cluster as well as the number of trapped hot particles in the cluster rises
with activity and then saturates to a certain value. We also note that the
saturation value of $q_{cl}^{hot}$ also grows with density, which is evidence
of the enhanced trapping of hot particles as density is increased. This, in
turn, explains why even for 2D bulk cases, we observe reduction in the phase
separation with density (fig 11 A).

We also showed in Fig:11B that for confined systems there is a reduction in
the phase separation with density. In addition to the enhanced trapping as
discussed earlier, we also find that the variation of the local packing
fractions in the simulation region also plays a role. We divide the x-confined
system into a number of slabs along x-axis and plot the local packing fraction
$\eta(x)$ in each slab as a function $x$ (Fig:13). The local packing fraction
$\eta(x)$ is defined as $\eta(x)=\frac{N(x)\pi}{4L\Delta x}$, where $N(x)$
denotes the number of particles in a slab of width $\Delta x$ at a distance
$x$. Due to the ordered structures in the cold domain near the wall, the local
packing fraction attains a high value, whereas the dilute hot zone gives rise
to the lower values of $\eta(x)$. We notice that the maximum local packing
fraction for the system increases with the initial density $\rho^{*}$. To give
a quantitative idea, we note that for $\rho^{*}=0.2$, the maximum local
packing fraction $\eta_{max}(x)\approx 0.8$ (observed near $x\approx 0$),
whereas for $\rho^{*}=0.8$, $\eta_{max}(x)\approx 1.3$ (observed near
$x\approx 10.0$). This means that as the total density $\rho^{*}$ increases,
the virial pressure acting on the hot particles that are trapped inside the
cold cluster also rises considerably. This prohibits their diffusion to the
hot zone and keeps them trapped inside the cold cluster. This sustained
trapping of hot particles in the cold cluster in turn reduces the phase
separation between the active and passive particles for confined systems as
the density increases.

Periodicity |  % of total particles in largest cluster ($f_{cl}\times 100$) |  % of hot particles in cluster w.r.to cluster size ($p_{cl}^{hot}\times 100$) |  % of cold particles in cluster w.r.to cluster size ($p_{cl}^{cold}\times 100$)
---|---|---|---
2D bulk |  76.59 $\pm$ 0.78 |  38.18 $\pm$ 1.27 |  61.82 $\pm$ 0.85
x-confined |  89.97 $\pm$ 0.33 |  46.54 $\pm$ 0.43 |  53.46 $\pm$ 0.35
xy-confined |  87.44 $\pm$ 0.37 |  45.13 $\pm$ 0.51 |  54.87 $\pm$ 0.37
Table 1: Table illustrating the enhanced trapping of hot particles in the cold
cluster for confined systems in 2D at a specific density and activity
$\rho^{*}=0.5,T_{h}^{*}=80.0$: The first column shows the different states of
confinement. The second column presents the percentage of total particles
(both hot and cold) $f_{cl}$ in the largest cluster. The 3rd column
$p_{cl}^{hot}$ in the table shows that with confinement, more hot particles
are trapped in the largest cluster.

In general, for 2D bulk and confined systems, as the activity is increased,
the size of the largest cluster ($f_{cl}$ or $n_{cl}$) increases. In Fig 12B),
we observe that for a specific density of $\rho^{*}=0.5$, there is a reduction
in the extent of phase separation for confined systems compared to their bulk
counterparts. This can be explained by analyzing $p_{cl}^{hot}$ values for the
bulk and confined systems at density $\rho^{*}=0.5$. For the confined system,
at $\rho^{*}=0.5$, the fraction of hot particles trapped in the largest
cluster also is greater compared to the bulk systems
($(p_{cl}^{hot})_{x-confined}>(p_{cl}^{hot})_{bulk}$ at
$\rho^{*}=0.5,T_{h}^{*}=80.0$, see table 1). As a result, when $\rho^{*}=0.5$,
because the fraction of trapped hot particles is higher in confined cases than
in periodic conditions, the order parameter is lower in confined systems than
in bulk systems. In this way, the rise in hot particle trapping in cold
clusters for confined systems explains the observed drop in phase separation
order parameter for such confined systems in 2D at $\rho^{*}=0.5$ (Fig:12B)

We conclude this section by briefly summarising the main results of this
section. We observe that when the state of confinement is fixed and the
density is varied, both the 2D bulk and confined systems show a reduction of
phase separation between active and passive particles with density. This
phenomenon is explained by analysing the trapping of hot particles in the
largest cluster in the system and also using the spatial variation of the
local packing fractions of the confined systems (fig 13). Next, we observe
that fixing the density and varying the state of confinement, at
$\rho^{*}=0.5$, the saturation value of $\phi$ is reduced for x-confined and
xy-confined states compared to bulk systems ( Fig:12). We justify this
observation via the analysis of enhanced trapping of hot particles in the cold
cluster (table 1). A general finding about 2D systems is the significant
amount of trapping of hot particles in the cold cluster, which rises with
density in both the bulk and confined scenarios. As a result, 2D binary
mixtures at high densities can be thought of as the coexistence of a dense and
dilute phase. This phenomenon is typical of 2D systems, and we address it in
greater depth for the case of confinement on spherical surface in the
following section.

###  3.5 Effect of topology

Fig. 14: Snapshots of the non-equilibrium steady state of binary mixture
confined on the surface of the sphere. The configuration of the system at
$T_{h}^{*}=100.0,T_{c}^{*}=1.0,R=20.0\sigma$, A) $\eta=N/(16R^{2})=0.2$ and B)
$\eta=0.6$. The configuration at $T_{h}^{*}=100.0$ shows a co-existence of
dense and dilute regions on the spherical surface. The dilute region is hot-
dominated, and the dense one is made up of both kinds of particles. The
enhanced trapping with density can be visualized from the snapshots. Fig. 15:
A) Variation of the observed activity $\chi$ with the activity imposed by the
thermostats $\chi_{imp}$, for the xy-confined system and particles confined on
spherical surface case. For the case of particles confined on spherical
surface, due to the constraint equation imposed on the position and velocities
of the particles, it is more difficult to numerically maintain the desired
activity. In contrast, for the xy-confined planar case, though the observed
activity is less than the desired activity due to the collision between the
particles, still this maintains the activity between the particles better than
the spherical confinement. B) Comparison of order parameter $\phi$ as a
function of activity $\chi$ between the xy-confined and the case of particles
confined on spherical surface. There is not much effect of the topological
constraints on the phase separation phenomena, as is evident from the order
parameter plots.

The xy-confined state of confinement describes a system where the particles
are confined from all directions. A spherical surface too corresponds to a
similar finite-sized confined system. Such spherical systems will also reveal
the effect of curvature on the various aspect of phase separation. Thus, to
study the effect of topology and curvature, we simulate a binary system of LJ
particles where particles are constrained to move on the surface of the
sphere. At every step, the positions and force vectors are projected back onto
the local tangent plane to restrict the particles to the curved surface (eqn
(4) and (5)). The equation of motion was integrated using the RATTLE algorithm
72 in LAMMPS. This algorithm yields results that, for suitably small
integration timestep $\delta t$, are identical to those of actual motion on a
spherical surface 85. In line with the prevailing practice seen in many recent
studies 86, 47, 87, 88, 42, we have chosen to adhere to the convention of
taking packing fraction as the measure of the concentration of the particles
on the spherical surface. The packing fraction of particles on the spherical
surface is defined as $\eta=N/(16R^{2})$. Also, we restrain from studying the
effect of the radius of curvature of the constraining sphere, and hence the
simulations were carried out for a specific radius of the sphere,
$R=20.0\sigma$. Hence to change the packing fractions, the number of particles
($N$) is changed, keeping the radius $R$ fixed. Also, the challenge of
sustaining high activity for such spherical confinement, as elaborated further
in subsequent discussions, prompts us to adopt an equilibrium temperature of
$T_{c}^{*}=T_{h}^{*}=1.0$. The instantaneous equilibrium configuration for
such a system at packing fraction $\eta=0.6$, at $T_{c}^{*}=1.0$ is shown in
fig 1D. The particles are in a liquid state and are homogeneously dispersed
over the surface of the sphere. Implementing the two-temperature model for the
simulation of non-equilibrium behaviour, the observed steady-state
configuration at $T_{h}^{*}=80.0$ is shown in Fig:14B ( the corresponding
snapshot for $\eta=0.2$ is shown in Fig:14A).

The collision between the hot and cold particles helps in the exchange of
energy between the particles, due to which we observe $\chi<\chi_{imp}$. In
addition to that, as the mechanical constraints eq. 4 and 5 imposed on the
particle position and velocity at each step of the simulation, it is harder to
maintain the desired activity using the equilibrium thermostats for such
spherical confinements, and therefore, we obtained a significantly lesser
value of the observed activity $\chi$ compared to the desired activity
$\chi_{imp}$ (fig 15A). The definitions of $\chi,\chi_{imp}$ are given in
section 2.

At equilibrium ($T_{h}^{*}=T_{c}^{*}=1.0,\eta=0.6,R=20.0\sigma$) the LJ
particles form a liquid phase and they are homogeneously distributed over the
surface of the sphere. When we increase the temperature of the hot particles
to a high value ($T_{h}^{*}=100.0,T_{c}^{*}=1.0$), we observe the formation of
a dilute fluidlike phase, rich with hot particles, in co-existence with a
solid-like dense region, which is made up of many hot particles trapped inside
cold clusters (fig 14 A, B). There is a circular band of cold particles (with
finite width) forming at the interface between the two regions, which is
necessary for the stability of the two co-existing phases. Since the particles
are constrained on the surface of the sphere, defects are inevitable in the
dense solid-like region. Therefore we calculated the defect structures for the
solid-like phase and found that, apart from particles having co-ordination
number 6, there are disclination defects where the number of nearest
neighbours of a particle can be either 7 or 5 (see SI, fig 4). These two types
of defect points often stay together forming dislocations, which are also
"grain boundary scars", which separate the different domains within the
crystalline structure. We note that earlier simulation works 88 of a binary
mixture of self-propelled active and passive particles on spherical surface
exhibit various other organisations and structures depending on the interplay
between the polar alignment and rotational diffusion of the active particles.

Next, We studied the effect of curved surfaces on the phase separation between
the active and passive particles. In order to quantify the phase separation
order parameter for such spherical surface confinement, we divide the whole
surface of the sphere into equal areas. For that, the azimuthal angle and the
polar angle were divided accordingly. The details are discussed in section 2.

The results of order parameter calculations are shown in Fig:15B. We plotted
the phase separation order parameter $\phi$ vs activity $\chi$ for two
different packing fractions for spherical confinement, namely $\eta=0.2$ and
$\eta=0.5$. To compare the results with a xy-confined system, we also plotted
the order parameter for the same at similar packing
fractions($\eta=\frac{N\pi}{4L^{2}},L=$length of the plane) (Fig:15B). We
observe that: i) for the spherical surface confinement of the LJ binary
mixture, the degree of phase separation between active and passive particles
decreases as the packing fraction is increased. This is the same effect as
observed in any other 2D system (bulk, x-confined or xy-confined) as discussed
in detail in subsection 3.4. Therefore we conclude that the reduction in the
extent of phase separation between active and passive particles as the packing
fraction of the particles is increased- is a general phenomenon observed in 2D
systems, independent of curvature or topology of the confining surface. ii)
When we compare the order parameter values between the xy-confined and
spherically confined systems for a given packing fraction at different
activities($\chi$), we find that the values are nearly identical. Hence, we
conclude that the curvature has no substantial influence on the phase
separation order parameter. Note that these comparisons are carried out by
simulating the xy-confined and spherically confined systems with equal surface
areas of the confining geometry, to bring out solely the effect of constant
curvature of the surface.

Although we showed that the order parameters are similar for the xy-confined
and spherically confined systems, there are certain aspects where these two
systems differ. As discussed, one such aspect is the difference between the
imposed and obtained activity ($\chi_{imp}-\chi$), which is larger for the
spherically confined system (as shown in Fig:15(A)), the reason for which is
discussed earlier.

A general observation for all considered 2D systems: bulk, planar confinement,
and spherical confinement, is that as density increases, the phase separation
reduces, which is the result of the increased trapping of hot particles in
cold cluster. Hence, the systems can be also analysed in view of a dense-
dilute phase co-existence, which is evocative of MIPS in 2D ABPs. The details
are discussed in the next subsection.

###  3.6 2-TIPS vs MIPS

Fig. 16: Distribution of the Voronoi packing fraction $\eta_{V}$ of the
particles confined on the spherical surface at packing fraction $\eta=0.6$.
(A) In equilibrium, $T_{h}^{*}=1.0,T_{c}^{*}=1.0$, the histogram plot of the
probability distribution of the Voronoi packing fraction associated with each
of the particles shows a truncated Gaussian distribution, the centre of which
is around $\eta_{V}=0.6$. This denotes the equilibrium packing fraction
$\eta=0.6$. (B) As the temperature of the hot particles is increased to
$T_{h}^{*}=100$, we observe the distribution develops a very tiny peak around
$\eta_{V}\approx 0.2$, and a larger peak around $\eta_{V}\approx 0.9$. This
indicates that there is a co-existence between these two packing fractions at
the steady state, indicating a MIPS-like behaviour. Although the curve does
not show bimodality with equal strengths on the two modes. Fig. 17: Average
velocity of individual particles plotted against the average Voronoi packing
fraction associated with each particle on the surface of the sphere. We
observe that the velocity decreases with packing fraction (similar to MIPS),
for each activity. Calculations are done at initial equilibrium packing
fraction $\eta=0.6,R=20.0\sigma.$

From the earlier sections we observe that, for 2D systems, the phase
separation between the active and passive particles decreases at high
densities. In fact, for the cases of spherical confinement, at high packing
fractions ($\eta=0.6$, fig 14B), we observe that the system phase separates
between dilute and dense phases where a significant portion of the hot
particles are trapped inside the domain of cold ones. For example, we found
that $f_{cl}=0.862\pm 0.007$, $p_{cl}^{hot}=0.434\pm 0.010$ and
$p_{cl}^{cold}=0.566\pm 0.006$, which means that $86.2\%$ of the total number
of particles forms the largest cluster and $43.4\%$ of the particles in the
largest cluster are hot particles (at $T_{h}^{*}=100.0$).

Therefore, for the binary mixture of active and passive particles on the
spherical surface, instead of further examining the phase separation between
the two types of particles, an alternative approach is to consider the system
as exhibiting the coexistence between a dense crystalline solid-like region
and a dilute gaseous region. The dense region is composed of particles having
small velocities and in the dilute region, the particle velocities are high.
This effect is reminiscent of the Motility Induced Phase separation (MIPS)
that has been observed for the purely active systems, which shows a co-
existence between a dense and a dilute phase. We, therefore, have studied some
of the aspects of MIPS in our cases, to have the comparison between 2-TIPS and
MIPS phenomena.

In fig 16, we calculate the Voronoi packing fraction (which is defined as
$\eta_{V}=1/A_{V}$, where $A_{V}$ is the Voronoi area associated with each of
the particles) for each of the particles at different activities. Note that,
due to the intrinsic topology of the confinement, the local packing fractions
are either undefined or carry significant statistical error due to a lesser
number of data points. Hence, to quantify the packing fractions of the co-
existing phases, we resort to the calculation of Voronoi packing fractions
associated with each particle, and not the calculation of local packing
fractions. Fig:16A) shows that at equilibrium (at $T_{h}^{*}=1.0$ at
$\eta=0.6$), the distribution of $\eta_{V}$ is a Gaussian one (with finite
extent), centered around the value $0.6$. At $T_{h}^{*}=100.0$, the
distribution of $\eta_{V}$ (Fig:16B) shows a prominent peak around the value
$0.9$, whereas another small peak is observed at around $\eta_{V}=0.2$. This
shows a co-existence of a dense and dilute phase on the spherical surface, at
corresponding packing fractions $\eta_{1},\eta_{2}\approx 0.2,0.9$. Similar
values of co-existent densities were obtained by Iyer $et.al$ 89 in a system
of ABPs on a spherical surface. Simulating a system of ABPs on a 2D plane,
Redner $et.al.$ 15 showed, starting with an initial packing fraction $0.65$,
the system demonstrates co-existence of two phases at packing fractions
$\eta_{1},\eta_{2}\approx 0.2,1.2$. In addition to these observations, in
Fig:17, we also show the average velocity of the particles and their
dependence on their Voronoi packing fractions. We see a clear decreasing
behaviour of the velocities with the Voronoi packing fractions, for different
activities, which is a signature characteristic of MIPS.

Note that although the co-existence of a dense and dilute region of the
particles on spherical surface, and the decreasing velocity of the particles
with packing fraction are some of the similarities of the 2-TIPS on the
spherical surface with MIPS, there are some distinct dissimilarities too. MIPS
is observed in purely active systems, whereas we are observing it in a binary
mixture of active and passive systems. Also, the phase co-existence is
spontaneous for purely active systems and all the particles start with equal
self-propelled velocity, but in 2-TIPS at high activity, there is an inherent
velocity difference in the two types of particles, manifested by the two
different temperatures.

##  4 Conclusion:

In this work, we examined the effect of confinement, geometry and topology on
the phase separation phenomena in a system of a mixture of active and passive
particles using the two-temperature model. We observe that:

i) For parallel wall confinement in 3D and for all 2D systems with or without
confinement, the phase separation order parameter (although increasing with
activity always) decreases as density is increased, which is completely
opposite to the 3D bulk behaviour.

ii) For a specific density, the parallel confinement can either enhance (3D,
$\rho^{*}=0.2$) or reduce (3D, $\rho^{*}=0.8$;2D, $\rho^{*}=0.5$), or almost
unalter (2D, $\rho^{*}=0.2$) the extent of phase separation between the active
and passive particles compared to unconfined systems. The interface between
the phase-separated dense cold cluster and the dilute gaseous hot domain is
always observed to form parallel to the confining wall. Enhancement of
clustering due to confinement 65 and accumulation of particles near the wall
surface 90, with experimental evidence 64 has been reported earlier as well.

iii) When the particles are confined inside of a spherical cavity, we observe
the active and passive particles clearly phase separate and the type of phase
separation changes with increasing the radius of the spherical cavity. When
the radius of the cavity is small, the phase separation is observed in the
radial direction, where the cold particles occupy the periphery of the
confining sphere and the hot dilute zone is formed in the interior. As the
radius of the spherical cavity increases, the bulk effects set in, and the
phase separation is of Cartesian type. Spherical cavity confinement is always
observed to enhance the phase separation compared to bulk cases at all
observed densities. The confined steady states show a compression along the
direction parallel to the wall in the cold zone, another feature completely
opposite to the bulk (unconfined) cases.

iv) 2D systems in general shows enhanced trapping of hot particles in the cold
cluster due to the reduced diffusivity of the particles. Thus, in the case of
the binary mixture constrained on the surface of a sphere, our investigation
encompassed not only the analysis of phase separation between active and
passive particles but also the examination of a phase coexistence scenario
involving a dilute gaseous phase and a densely packed crystalline domain.

The dense crystalline phase on the surface of the sphere shows various defect
structures which exist solely due to the topological constraint. These
structures were compared with MIPS in 2D and various similarities as well as
dissimilarities were figured out.

The future directions of this work may include the simulation of mixed systems
on other topological surfaces and see whether they play any significant role
in the phase separation phenomena or not. It will also be interesting to study
the interplay between shape anisotropy and confinement by simulating mixtures
of anisotropic particles under different geometric confinement. By changing
the ratio between the active and passive particles, some critical activity can
be determined for the different topological constraint cases, which would lead
to some phase diagrams, which we can experimentally verify. These kinds of
phase diagrams can be helpful in the experiments such as the behaviour of
bacterial microswimmers on spherical liquid droplets and so on. Further
analysis is required to study the dynamics of cold cluster formation both in
the presence and absence of walls. The study of finite size effects and
cluster dynamics along with stability analysis of the clusters will lead to a
better understanding of the physics behind the phase separation processes in
such binary mixtures in confined geometry and curved surfaces. We would like
to publish further works along these lines in the upcoming days.

## Author contributions

Nayana Venkatareddy and Jaydeep Mandal: conceptualization, methodology,
investigation, visualization, formal analysis, writing. Prabal K. Maiti:
conceptualization, visualization, supervision, writing-review and editing. The
manuscript was written through contribution from all authors.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

NV and JM thank MHRD, India for the fellowship. PKM thanks DST, India for
financial support and SERB, India for funding and computational support. We
also thank Profs. Chandan Dasgupta and Sriram Ramaswamy for the insightful
discussions.

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