Activity Driven Phase Separation and Ordering Kinetics Of Passive Particles

Shambhavi Dikshit1∗ and Shradha Mishra1†1Department of Physics, Indian Institute of Technology (BHU), Varanasi, India 221005

(Dated: August 23, 2021)

The steady state and phase ordering kinetics in a pure active Borwnian particle system are studiedin recent years. In binary mixture of active and passive Brownian particles passive particles are usedas probe to understand the properties of active medium. In our present study we study the mixtureof passive and active Brownian particles. Here we aim to understand the steady state and kinetics ofsmall passive particles in the mixture. In our system,the passive particles are small in size and largein number, whereas ABPs are large in size and small in number. The system is studied on a two-dimensional substrate using overdamp Langevin dynamic simulation. The steady state and kineticsof passive particles are studied for various size and activity of active particles. Passive particles arepurely athermal in nature and have dynamics only due to bigger ABPs. For small size ratio andactivity the passive particles remain homogeneous in the system, whereas on increasing size ratioand activity they form periodic hexagonal close pack (HCP) spanning clusters in the system. Wehave also studied the kinetics of growing passive particle clusters. The mass of the largest clustershows a much slower growth kinetics in contrast to conserved growth kinetics in ABP system. Ourstudy provides an understanding of steady state and kinetics of passive particles in the presence ofbigger active particles. The mixture can be thought of as effect of big microorganism moving inpassive medium.

I. INTRODUCTION

Collection of active Brownian particles (ABPs) under-goes a motility induced phase separation without anycohesion at packing density much lower than the densityfor phase separation in corresponding passive system[1–5]. The mechanism of phase separation is due to theenhanced persistent motion of active particles [6, 7].

Recent researches have focussed the kinetics and steadystate of pure ABPs or in the mixture of passive particles[8–12]. In a recent work of [13], a monodisperse mixtureof active passive particles is studied for varying activityand packing fraction of ABPs. Whereas in other studiesa field theoretic approach is used to understand thepropagation of active passive interface in the mixtureof passive and active particles [14]. In the study of[15], a mixture of active passive particle is studied, anddifferent phases and dynamics of system is studied.A variety of interesting properties and phases havebeen found when ABPs are placed in the mixture withpassive particles. Asymmetric passive particles lead todirectional transport and trapping when placed in thesea of ABPs [16–19]. Such systems can be useful inindustrial and pharmaceutical applications. Symmetricpassive particles in the mixture of ABPs has also beenused as a probe to characterise the properties of ABPs[20–22]. Experiments on the dynamics of large passivebeads in active bacterial fluid show the persistencemotion of bead in bacterial solution [21]. In the binarymixture of ABPs with passive bead, large passiveparticles experience an effective attractive interaction

∗ shambhavidikshit.rs.phy18@itbhu.ac.in† smishra.phy@iitbhu.ac.in

analogous to depletion induced attraction in asymmetricequilibrium binary mixture [5, 22, 23].Most of the above studies of binary active-passivemixture are studied where passive particles are bigger insize and treated as a probe to characterise the propertiesof active medium [24–26] or monodisperse mixture ofactive passive particles and dynamics of different phasesare studied [13–15]. Here we ask the question, howthe ABPs which are bigger in size can influence thecharacteristics of athermal passive particles ? Thesystem resembles big microorganisms moving in passivefluid. In general thermal and hydrodynamic effects areimportant in normal passive fluid, but we ignore it hereto make the model simple and only study the effect ofactivity on the properties of passive particles.We consider a minimal model of mixture of small passiveand big active Brownian particles on a two-dimensionalsubstrate. The both types of particles interact througha short range soft repulsive interaction. The dynamicsof active particles is driven by the active self-propulsionforce and interaction with the particles in its surround-ings whereas passive particles can move only due to theinteraction with other particles. The packing fractionof both particles are same and total packing fractionis kept fixed at 0.6. The dynamics and steady state ofpassive particles are studied for various size ratios ofactive and passive particles and dimensionless activityof ABPs.Our main results are as follows: Starting from therandom homogeneous mixture of active and passiveparticles, passive particles start to phase separate withtime. The phase separation order parameter of passiveparticles grows with time and saturates to value ∼ 1for large size ratio and activity and remains much lowerthan 1 for small size ratio and activity. Hence a phasediagram is found in the plane of size ratio and activity.For moderate size ratio and activity the clustered

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2passive particles form hexagonal close packed structuresand start to overlap for large size ratio and activity.The cluster size distribution changes from exponentialto power law and power converges to −2 for large sizeratio and activity. Hence passive particles form thespanning clusters [27] for larger size ratio and activity.We have further studied the kinetics of the phaseseparation. The mass of the largest cluster grows asas a power law with time, with a much slower growthkinetics in contrast to equilibirum conserved passivesystem [28]

Rest of the article is divided in the following manner. Innext section II, we describe the model in detail. SectionIII discusses the results of our study and finally in sectionVI we conclude our results.

II. MODEL

We consider a binary mixture of small athermal passiveparticles in the presence of large active Brownian parti-cles (ABPs) on a two-dimensional substrate. The activeand passive particles are modeled as discs of radius raand rp respectively. We choose ra > rp, active parti-cles are larger in size compare to passive particles. Thesize ratio S = ra

rpis one of the control parameters in the

model. The radius of the passive particles is kept fixedand radius of active particles is tuned to vary the size ra-tio. We keep the packing fractions of both types of parti-

cles φa =πr2aNa

L2 = φp =πr2pNp

L2 = 0.3, hence the numberof active particle Na are less in comparison to that ofpassive particles Np. Both active and passive particlesare defined by their position ri(t) and active particlesare also having their orientation θi(t), which determinestheir direction of self-propulsion. They self-propel alongtheir direction of orientation θi(t) with a constant self-propulsion speed v0. The dynamical Langevin’s equa-tions of motion for position and orientation of activeparticles are

drai (t)

dt= v0ni(t) + µFi(t) (1)

where ni(t) = (cos(θi(t)), sin(θi(t)) is the unit directionof self-propulsion of ABP. The change in the orientationof the active particle is given by:

dθidt

=√νrηi(t) (2)

here ηi(t) is the random Gaussian white noise with mean

zero and variance, < ηri (t)ηrj (t

′) >= δijδ(t − t

′), where

νr is the rotational diffusion constant of active particles.The equation of motion for the passive particle is givenas:

drpi (t)

dt= µFi(t) (3)

Here, the mobility, µ is chosen to be the same for bothtypes of particles. Fi(t) is the force acting on the ith

particles, due to all other particles interacting with it

Fi(t) =∑j 6=i

Fij(t) (4)

The force is obtained from the soft-repulsive pair poten-tial Fij = −∇U(rij), where U(rij) = K(rij − (rαi +rα′j))

2 if rij ≤ (rαi + rα′j) and 0 otherwise. rij=|ri -rj | and K is the force constant and rα, is the radius ofactive or passive particles for α and α′= a or p respec-tively. ν−1r is the time scale over which the orientationof an active particle changes. Hence, lp = v0ν

−1r , is the

persistence length or run length, is the typical distancetravelled by an active particle before it changes its di-rection. In our study, lp = (100rp to 600rp) is tuned bytuning SPPs v0. The (µK)−1 = 0.7 defines the elastictime scale in the system. We define the dimensionless ac-

tivity v =lprp

as the ratio of persistent length to the size

of passive particles. The size ratio S and v are the twotuning parameters in the model. A schematic cartoon ofsystem is shown in Fig. 1, where red big particles areABPs and small gray particles are passive. The whitedots on red particles represent their instantaneous direc-tion of orientation θ.We start with random non-overlapping arrangement ofactive and passive particles on a two dimensional squaresubstrate of linear dimension L = 250rp with periodicboundary condition. The equations 1 − 3, are updatedand one simulation step is counted after updation of allthe particles once. The time step τ = 5×10−4 and total6× 104 simulation steps are used to get the results. Wefirst characterise the effect of size ratio and activity onthe steady state of the athermal passive particles in themixture. Then we study the kinetics to the steady state.

FIG. 1. (color online) Schematic diagram of the model. Itshows the initial homogeneous state of the system. Here thecolours and sizes represent two types of particles. Biggerparticles (red) are active particles and smaller particles (grey)are passive particles. Small white dot on red particles denotethe instantaneous orientation direction θ of ABPs.

3III. RESULTS

We start with the random homogeneous distribution ofactive and passive particles and the equations 1− 3 areintegrated to update the position and orientation of ac-tive and position of passive particles. In Fig. 2 we plotthe time evaluation snapshot of local density of passiveparticles at different times = 0.05, 0.5 and 30.0 and fortwo different size ratios S = 10 and 6 and for v =100, 300 and 600. The bright and dark regions showthe lower and higher local density of passive particlesrespectively. Local density ρp is defined as: the num-ber of passive particles in the small coarse-grained area(a = 5rp × 5rp). In Fig. 2(h-i) we plot the probabilitydistribution function (PDF) of local density P (ρp) forthe same parameters as in Fig. 2(a-f). The tail of thedistribution is larger for large size ratio S = 10 and ac-tivity v = 600. As time progresses the passive particlestarts to come close to each other. Hence local density ρpof passive particle rich region grows or passive particlesphase separate. To characterise the phase separation, wecalculate the phase separation order parameter (PSOP),φ(t) =< (np(t)− na(t))/(np(t) + na(t)) >, where np(t)and na(t) are calculated as the number of active andpassive neighboring particles around a passive particle.< .. > means average over all passive particles and 20independent realisations. With time φ(t) grows and ap-proaches a steady state. We calculate the steady stateφ =< φ(t) >t, where < .. >t is the average of φ(t) overa time interval in the steady state. In Fig. 2(g) weplot the phase diagram in the plane of activity and sizeratio (S, v). The color shows the magnitude of PSOP,φ(t). For large activity and size ratio, phase separationincreases in the system.

IV. CHARACTERISTICS OF STEADY STATE

A. Radial distribution function (RDF) gpp(r)

As discussed in previous section, system shows the phaseseparation for larger size ratio and activity. Hence clus-tering increases with increasing S and v. We furthercharacterise the structure of the clusters by calculatingthe radial distribution function (RDF) of passive-passiveparticles for different size ratio S and v. The RDF, gpp(r)gives the probability of finding a passive particle at a ra-dial distance r from the center of the given passive par-ticle. In Figure 3 we plot the gpp(r) vs. scaled distancer/rp, for passive-passive particles for three different ac-tivities v = 100, 300 and 600 and varying the size ratioS = 6, 8 and 10. For small activity and S = 6 as shownin Fig. 3(a), the first peak appears at r/rp ≈ 2 and sec-ond peak is at 4 and few more higher order peaks arepresent. But as we increase S > 8, the location of firstpeak remains almost the same but a small hump in sec-ond peak appears at r/rp = 2

√3, which is due to the

presence of hexagonal close packed structure (HCP) in

the clusters. Also for S > 6, higher order peaks are morepronounced. The zoom in plot in Fig. 3(a) (inset) showsthe enlarged second peak for three S = 6, 8 and 10. Aswe increase, v = 300, the location of the first peak shiftsat distance smaller than r/rp ≈ 2, which suggests over-lapping particles due to soft repulsive interaction. Thesecond peak of gpp(r) appears at r/rp ≈ 2

√3, for size

ratio S > 6, hence HCP structure of clusters. Also thedistinct higher order peaks are found for larger size ratioS > 8. Again the inset Fig.3(b) shows the zoom in sec-ond peak of the gpp(r). On further increasing v = 600,the first and second peak of gpp(r), systematically shiftstowards the small scaled distance, hence more overlap-ping particle clusters, the distinct nature of higher or-der peaks decreases on increasing S. Hence periodic-ity of clusters weakens on increasing S for large activityv > 300. The inset Fig. 3(c) shows the zoom in sec-ond peak, where the structure in second peak has beendisappeared for large activity v = 600.

Hence using RDF, we find the periodic HCP nature ofparticle clusters first increases on increasing activity andsize ratio and again for very large v = 600 and S = 10,overlapping large clusters are found with weaker HCPstructure. Now we further characterise the characteris-tics of large clusters by calculating the cluster size dis-tribution (CSD) of different size clusters.

B. Cluster size distribution (CSD)

To further understand the characteristics of clusterswe calculate the probability distribution function ofdifferent sized clusters. A cluster is defined as set ofparticles connected by a distance smaller or equal to2rp (diameter of the particle). A cluster of size n hasn −particles connected cluster. Then we calculate thenumber of different sized cluster in the total system. Inthis manner, all particles are part of a single particlecluster. Hence number of clusters of size n = 1 istotal number of particles. We further calculate thefraction of cluster of size n, or cluster size distribution(CSD). In Fig. 4 we plot the normalised cluster sizedistribution (CSD) P (n)/P (1), where P (i) is obtainedfrom the counting all the clusters of size i = 1, 2, ..., n.In Fig. 4(a) we plot the CSD for activity v = 100 forthree different size ratios S = 6, 8 and 10. For smallactivity, v < 300 and size ratio S < 8, the CSD decayexponentially at n > 20. Hence for small activity andsize ratio we find small clusters and particles are wellseparated from each other (as shown in Fig. 4(b)).Hence weak clusters as found in the RDF, gpp(r) plotas shown in Fig. 3(a). As we increase size ratio S > 6,for v = 100, clustering increases and CSD decay as apower law. The power approaches −2.5 for large sizeratio S = 10. The real space snapshots of particlesshow the enhanced clustering on increasing S. Formoderate activity v = 300, CSD for small size ratioS = 6, is exponential with large exponential tail and for

4

0 10 20 30 40 50ρ

p

0.001

0.01

0.1P

(ρ p

)

0 10 20 30 40ρ

p

0.0001

0.001

0.01

0.1

P(ρ

p)

(h) (i)

FIG. 2. (color online) The real space local density of passive particles at difefrent times. (a-c) are for size ratio 10 andv= 100, 300 and 600 respectively. (d-f) are for size ratio 6 and v= 100, 300 and 600 respectively. Lowest panel is for time,t = 0.05, mid panel is for t = 0.5 and upper most panel is for t = 30.0. The lower color plot represents phase separation orderparameter (PSOP) in the plane of size ratio and activity (S, v). Probability distribution function (PDF) of local density ofpassive particles is plotted for two different size ratios 10 and 6 ((h) and (i)) respectively and for three v= 100, 300 and 600.

0 2 4 6 8r/r

p

0

5

10

gpp(r

)

0 2 4 6 8r/r

p

0

5

10

15

gpp(r

)

0 2 4 6 8r/r

p

0

5

10

15

gpp(r

)

4r/r

p

2

4

6

gp

p(r

)

4r/r

p

2

4

6

gp

p(r

)

4r/r

p

2

4

6g

pp(r

)

(a) (b) (c)

FIG. 3. (color online) Passive-passive radial distribution function (RDF), gpp(r) is plotted for different parameters. (a-c) isfor fixed activity and varying size ratio. v is varied for three different values, 100, 300 and 600. (a) is for v = 100 (b) is forv = 300 and (c) is for v = 600 and size ratio S is taken 6, 8 and 10 respectively.

large size ratio S > 6, the exponential tail approachesto power law tail −2. Also the real space snapshotsshow the clustering increases on increasing S. But forlarger size ratio S = 10, particles start to overlap andHCP structure weakens as shown in Fig. 3(b). As wefurther increase activity v = 600, the CSD is power lawfor all size ratios S = 6, 8 and 10 and power slowlyconverges to −2 for largest size ratio 10. The power −2suggest that for large activity passive particles form thepercolating clusters spanning the whole system [27].

V. GROWTH KINETICS

After understanding the steady state properties of themixture, we study the kinetics of phase separation ofathermal passive particles. We study the kinetics ofgrowing cluster by calculating the mass of the largestcluster at different times, m(t). In Figure 5, we plot themean mass of the largest cluster < m(t) > for two dif-ferent size ratios S = 10 and 6 and for different v. where< .. > is mean over 50 independent realisations. In Fig.5(a) we plot < m(t) > for size ratio 10 for three differ-

5

100n

0.001

0.01

P(n

)/P

(1)

(a) (b) (c) (d)

slope=-2.5

100n

0.001

0.01

P(n

)/P

(1) slope=-2.0

(e) (f) (g) (h)

100n

0.001

0.01

P(n

)/P

(1)

slope=-2.0

(i) (j) (k) (l)

FIG. 4. (color online) Cluster size distribution (CSD) and real space snapshots of part of the system are plotted for differentparameters. Different vertical panels (a-d) are for fixed v = 100, (e-h) are for v = 300 and (i-l) are for v = 600 and black,red and green colors are for size ratios 6, 8 and 10 respectively. CSD is shown in (a),(e) and (i) for v = 100, 300 and 600respectively and for size ratio 6, 8 and 10. Symbols have the same meaning as in Fig. 3

ent activities v = 100, 300 and 600. For all activities andS = 10, at very early time t < 10−2, m(t) grows withtime and then for an intermediate times t (10−2, 10),growth becomes slow and < m(t) > develops a plateauand again starts to grow for late times t ≥ 10. Theplateaus region decreases with increasing activity. Thelate time growth of < m(t) >∼ t1/3 for all activities. InFig. 5(b) we plot the < m(t) > vs. t for size ratio S = 6.The growth of < m(t) > shows the same behaviour as forS = 10, only the plateaus region is increased for S = 6.For largest activity v and size ratio S = 10, < m(t) >increases monotonically with time t with m(t) ∼ t1/3.

VI. CONCLUSION

We have studied the phase separation and ordering ki-netics of a binary mixture of passive and active Brownianparticles on a two-dimensional substrate with periodicboundary condition. The passive particles are smallerin size comparison to the ABPs. The ABP moves alongthe direction of their heading and both types of particlesinteract through a short range soft core repulsive inter-action. Hence the dynamics of passive particles are onlydue to interaction force among the particles. The sys-tem is studied for various size ratios S and activities v of

0.01 1 100t

10

100

<m

(t)>

0.01 1 100t

10

100<

m(t

)>

slope=

1/3

slop

e=1/

3

(a) (b)

FIG. 5. (color online) Mean mass of the largest cluster <m(t) > is plotted for activities v = 100, 300 and 600. (a)is for size ratio 10 and (b) is for size ratio 6. Black, redand green show the activities 100, 300 and 600 respectively.Dashed line is the line with slope 1/3

active particles. We focus our study on the steady stateand kinetics of small passive particles in the presence ofbig passive particles. Starting from the random homo-geneous state, the clustering of small passive particle ismeasured by calculating phase separation order param-eter (PSOP). The PSOP is small for size ratio S < 6and small activity v < 300, whereas for large size ratioS > 8 and higher activity v > 300, PSOP approaches∼ 1. The clusters of passive particles are random smallclusters for small size ratio and activity and HCP struc-tures are formed for intermediate size ratio S = 8 and

6activity v = 300 and then overlapping clusters are foundfor large size ratio and activity S > 8 and v = 300. TheCluster size distribution decays exponentially for smallsize ratio and activity and approaches to a power lawdecay with exponent −2 for large size ratio and activ-ity. The power law decay with power −2 indicates theformation of connected clusters as in [27] for large sizeratios and activities.We have also calculated the kinetics of growing cluster ofpassive particles. The mean mass of the largest clustergrows with time as a power law. The growth law ismuch smaller than the corresponding equilibirumsystem with conserved growth kinetics [29–33].Hence our study gives the steady state of collection ofpassive particles moving under the effect of dynamics ofactive Brownian particle.It focuses on the steady stateand kinetics for binary mixture where passive particlesare much smaller than the ABPs. The system resem-

bles the effect of big microorganism moving in passivemedium.

VII. CONFLICTS OF INTEREST

There are no conflicts of interest to declare.

VIII. ACKNOWLEDGEMENT

The support and the resources provided by PARAMShivay Facility under the National Supercomputing Mis-sion, Government of India at the Indian Institute ofTechnology, Varanasi are gratefully acknowledged. Com-puting facility at Indian Institute of Technology(BHU),Varanasi is gratefully acknowledged.

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