Traveling fronts in active-passive particle mixtures

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by  Adam Wysocki

Travelling fronts in active-passive particle mixtures
Adam Wysocki, Roland G. Winkler, Gerhard Gompper1
1Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation,
Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
(Dated: January 6, 2016)
The emergent dynamics in phase-separated mixtures of isometric active and passive Brownian
particles is studied numerically in two dimensions. A novel steady-state of well-defined traveling
fronts is observed, where the interface between the dense and the dilute phase propagates and
the bulk of both phases is (nearly) at rest. Two kind of interfaces, advancing and receding, are
formed by spontaneous symmtry breaking, induced by an instability of a planar interface due to
the formation of localized vortices. The propagation arises due to flux imbalance at the interface,
strongly resembling travelling fronts in reaction-diffusion systems. Above a threshold, the interface
6
velocity decreases linearly with increasing fraction of active particles.
1
0
2 PACSnumbers: 82.70.Dd,64.75.Xc

n
a Introduction – Generic models of active fluids divide between the two phases propagates.
J
into two main classes, systems of active Brownian par- The recent finding of a stable interface in pure, phase-
5
ticles (discs or spheres) [1, 2], which emphasize volume separated ABP fluids, together with a negative surface

] exclusion, and systems of anisotropic (or elongated) self- tension[14],questionsthemappingofanABPfluidonto
ft propelled particles, which emphasize alignment interac- an equilibrium fluid with attraction. The emergence of
o tions, like the Vicsek model [3, 4]. The most striking travellingfrontsinactive-passivemixturesclearlycontra-
s
. phenomenon of active Brownian particles (ABPs), ob- dicts the existence of an equivalent equilibrium system
at served in various experiments [5–8] and simulations [9– in this case. Instead, we propose an explanation of the
m 11], ismotility-inducedphaseseparation[12]. Thephase interface dynamics on the basis of well-studied models
- behavior and kinetics, like domain coarsening [10, 13] describing the growth of rough surfaces under far-from-
d orinterfacefluctuations[14], ofABPsresembleapassive equilibrium conditions [23, 24].
n
fluidwithattractiveinteraction. Moreover,ABPsexhibit
o Model – We simulate a mixture of NA active and NP
an intriguing collective dynamics with jets and swirls,
c passiveBrowniandisks(intotalN =N +N particles)
A P
[ which has been speculated to arise from interfacial sort- in a 2D simulation box of size L × L with periodic
x y
1 ing of ABPs with different orientations [11]. In contrast, boundary conditions. Their dynamics is overdamped,
thekineticsofmodelswithalignmentinteractionexhibits
v i.e., r˙ = V e +f /γ +ξ , where V is the propulsion
i 0 i i t i 0
0 variousmodesofcollectivemotion,forexamplelargepo- velocity along the polar axis e (V =0 for passive parti-
i 0
5 lar swarms, i.e., high-density polar bands travelling co- (cid:80)
cles), f = f is the force due to steric interactions,
8 herently through an isotropic background gas [3, 4]. i i(cid:54)=j ij
and ξ is a random velocity. The particles interact via
0 i
0 A natural extension of single-component ABP flu- the soft repulsive force fij = k(ai +aj −|rij|)rij/|rij|,
1. ids are mixtures of particles with, e.g., different activ- with rij =ri−ri if ai+aj <|rij| and fij =0 otherwise
0 ities [15–18], temperatures [19, 20], or diameters [21]. [9]. The discs are polydisperse in order to avoid crystal-
6 These models also exhibit activity-induced phase sepa- lizationandtheirradiiai areuniformlydistributedinthe
1 ration. We focus here on the dynamics of mixtures of interval [0.8a,1.2a] [9]. The zero-mean Gaussian white-
v: isometricactiveandpassiveBrownianparticles. Surpris- noise velocity ξi obeys (cid:104)ξi(t)ξi(t(cid:48))(cid:105) = 2Dtδij1δ(t−t(cid:48)),
Xi ingly, we find that these systems exhibit a novel and so where Dt = kBT/γt is the translational diffusion coeffi-
far unexplored type of collective motion in the phase- cient with thermal energy kBT and friction coefficient
ar separated state in the form of well-defined propagating γt. The propulsion direction ei undergoes a free ro-
fronts, whichcanbeeitherenrichedordepletedofactive tational diffusion with a diffusion constant Dr, where
particles, andareadvancingtowardorrecedingfromthe Dt/Dr = 4a2/3 holds for a no-slip sphere. The persis-
dense phase, respectively. The propagation arises due to tenceofswimmingischaracterisedbytheP´ecletnumber
fluximbalanceattheinterfacebetweenthedenseanddi- Pe=V0/(2aDr). The typical particle overlap due to ac-
lutephaseswithastrongresemblancetotravellingfronts tivity, γtV0/(2ak), is fixed to 0.01 [13]. Unless otherwise
in reaction-diffusion systems [22]. The selection of the noted, we consider systems with Pe=100 and an pack-
interface type (advancing or receding) happens by spon- ingfractionφ=(cid:80)Ni=1πa2i/(LxLy)=0.67,belowrandom
taneous symmetry breaking, induced by an instability of closed packing [9], and vary the fraction xA = NA/N of
a planar interface due to the formation of localized vor- ABPs. Moreover, lengths are expressed in units of 2a
tices. In contrast to the polar bands of the Vicsek model and time in units of 1/Dr.
[3, 4], which travel as a whole, here only the interface Phase behavior–Amixtureofactiveandpassivediscs
2
-1 0 1Φ ticlesbehaveaseffectively’hot’(’cold’)particles. Sucha
400 mixtureexhibitsphaseseparationintoasolid-likecluster
(a)
δt=0 ofpassiveandagaseousphaseoftheactiveparticles[20].
δt=4
300 δt=8 Bulktravellingfronts–Thefocusofourpaperisonthe
kineticsofaphase-separatedactive-passivemixture. The
y 200 domain dynamics of a one-component ABP fluid in the
steadystateislimitedtofluctuatinginterfacesresembling
100 thermalcapillarywaves[14,25],exceptforthecoarsening
(b) kinetics[10,13]. Bycontrast,active-passivemixturesex-
0 hibit amazingly mobile or travelling interfaces in a large
0 200 400 0 200 400
x x regionofthePe−φ−xA parameterspace. Theinterface
0.35 2 propagationbecomesapparentfromFig.1(b), wherethe
(c) (d)
0.3 ωρ timeevolutionoftheinterfacepositionisshown, seealso
Γ
0.25 ωρρ=Vρq 1.5 movies in Ref. [25].
0.2 Γρ=Dρq2 In order to quantify our observations, we anal-
q 1 yse the density correlations at x = 0.5 and φ =
0.15 A
0.78 by the dynamic structure factor S(q,ω) =
0.1 0.5 (cid:82)∞ F(q,t)exp(iωt)dt, where q is the wavevector, ω is
−∞
0.05 the angular frequency, and F(q,t) = (cid:104)ρ (t)ρ (0)(cid:105)/N
q −q
0 0 is the correlation function of the Fourier components
-1 0 1 0 0.1 0.2 0.3
ω q ρq of the density [26]. The circularly averaged S(q,ω),
whichisaccessiblebyscatteringexperiments,isshownin
FIG. 1. (color online) (a) Snapshot of the segregation order Fig. 1(c). The structure factor exhibits peaks at the fre-
parameter field Φ(r) of a system of size Lx =Ly =400 with quencies±ωρ,withawidth,whichincreaseswithincreas-
N =160000 particles (φ=0.78) in the steady state at x =
A ing q. This suggests a damped propagative mode [26]
0.5. Φ = 1 (Φ = −1) corresponds to a pure active (passive)
related to the traveling interfaces indicated in Fig. 1(b).
phaseandΦ=0correspondstoauniformmixture. Theblack
We obtain the full dispersion relations ω (q) and Γ (q)
lineindicatestheinterfaceposition. (b)Timeevolutionofthe ρ ρ
interface position, see movies in [25]. (c) Dynamic structure by fitting F(q,0)exp(−Γρt)cos(ωρt) to the correspond-
factorS(ω,q)asafunctionoffrequencyωandwavenumberq. ing simulation data. As can be seen in Fig. 1(d), the
The shape of S(ω,q) indicates a damped propagative mode. peak positions follow the Brillouin-like dispersion rela-
(d) Frequency ωρ (also indicated in (c) by black dots) and tion ωρ =Vρq up to q ≈0.2 with a velocity Vρ ≈0.04V0.
damping rate Γ as function of wavenumber. Lines indicate
ρ Inasimpleequilibriumfluid(Newtoniandynamics)such
ω = V q, with V ≈ 0.04V , and Γ = D q2, with D ≈
ρ ρ ρ 0 ρ ρ ρ avelocityisthespeedofsound[26],buthere,V isrelated
0.004D , where D is the diffusion constant of a free ABP. ρ
0 0
to the velocity of interface propagation. The decay rate
Γ (q)obeysΓ =D q2 uptoq ≈0.2,withthetransport
ρ ρ ρ
coefficient D ≈ 0.004D , where D = D +V2/(2D )
ρ 0 0 t 0 r
separates into a dense and a dilute phase at sufficiently is the diffusion constant of a free ABP. All modes are
large Pe, φ, and xA [16], very similar to a pure ABP stronglydampedforq >0.25,i.e,dense(ordilute)phase
fluid [10]; in addition, active and passive particles tend droplets with size smaller than 50a dissolve quickly.
to segregate inside the dense phase. This is illustrated
Stabilized travelling fronts – In order to study a prop-
in Fig. 1(a), where the segregation order parameter field
agating interface in more detail, we employ a quasi-
Φ(r) of a large phase-separated system with curved in-
one-dimensional setup of an elongated box of lengths
terfacesbetweenthedenseandthedilutephaseisshown.
L = 2L [14, 27] such that the interface favors to span
x y
ΦisdefinedasΦ(r)=(φ −φ )/(φ +φ )withcoarse-
A P A P the shorter box length, see Fig. 2(a). Given that the
grained packing density fields φ (r) and φ (r) of active
A P systemphase-separates,suchaconfigurationformsspon-
and passive particles, respectively [25]. The dilute phase
taneously and remains stable over long time. In Fig. 3
consist mainly of passive particles (Φ ≈ −1) and the
(a,b), we show the time evolution of the active-particle
bulk of the dense phase is a homogenous active-passive
packing profile φ (x,t) averaged over the y-coordinate.
A
mixture (Φ≈0) with small patches of enriched active or
The interface position in a pure active fluid performs
passiveparticles. Withintheinterfaceregion,weobserve
a diffusive motion [25]. By contrast, in our mixture
eitheranaccumulation(Φ≈1)oradepletion(Φ<0)of
thetranslationalsymmetryisbrokenandbothinterfaces
active particles.
propagate steadily in parallel (for the set up of Fig. 2)
Note that a completely different behavior appears for either to the right or to the left with equal probabil-
dilute solutions (not considered here), namely, for mean ity [25]. The travelling front is extremely stable within
free paths much larger than the persistence length of the typical simulation length of T = 5×103, however,
swimming(πσ/4φ(cid:29)V /D ),whereactive(passive)par- the steady propagation is occasionally interrupted by in-
0 r
3
-1 0 1 ω s
Q
100
(a) (b) (c)
80
60
y
40
20
0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
x x x
FIG. 2. (color online) (a) Snapshot of a mixture of active (red) and passive (blue) Brownian discs at x = 0.5 in a box of
A
size L = L /2 = 100. The interfaces between the two phases (dashed lines) travel by chance to the right . (b) Vorticity
x y
ω (r) = ∇×Q(r), where Q is the coarse-grained particle flux [25]. (c) Visualisation of the bulk flow. Particle positions are
Q
shownafterthetimelagδt=2,whereparticlesarecoloredaccordingtotheirinitialxposition,asindicatedbythecolorscale.
The solid line marks the isoline of s(r,δt) used for the stability analysis. See corresponding movies in Ref. [25].
0 0.5 1φA(x,t) all packing fraction φ and activity Pe. However, Vfront
t234500000000 (a) (b) (mc) phase separated 000...000246 V/Vfront0 mamseguoprnmeaeormatjotueinsontontuawssbeloiyttvshedineVtch,ρreseoeaabesctetFasiivingew.e-dip3tahf(rrctoi)inmc.clerSefar(asqic,ntωgio)]nx,,Awwi[thihneraeampghaoaxoside-
100 nifor We calculate profiles of various quantities in a co-
0 u 0 moving frame. In a pure active fluid, the two inter-
0 100 2000 100 2000 0.5 1
x x xA=NA/N faces are equivalent and the dense phase is symmetri-
1 (e) cally surrounded by a corona of particles with their po-
0.8 0.2 lar vector pointing preferentially toward that phase, see
φα0.6 0 eix dashed lines in Fig. 3(d,e). In a mixture, this symme-
0.4 h try is broken and active particles preferentially accumu-
0.2 (d) xxAA==01.5 -0.2 late at one interface and deplete at the other. Similarly,
0 the x-polaristion (cid:104)e (cid:105) at both interfaces is different; it is
0 50 100 150 2000 50 100 150 200 x
x−Vfrontt x−Vfrontt larger at the side of preferred accumulation. Moreover,
0.02 0.8 (cid:104)e (cid:105) takes negative values in the dilute phase, accom-
(g) x
0 0.6 panied by a negative x-velocity of active particles, VA,
V/Vα0--00..0042 0.4 EQ cVau,sidnugeintotucornllisaionnesgabteitvweevenelopcaitsysivoef apnadssiavcetipvaertpiaclretsi-,
--00..0086 (f) αα==AP 0.2 clPes, see Fig. 3(f). The mass transport from the dense
0 into the dilute phase is characterized by the intensity of
0 50 100 150 2000 50 100 150 200
x−Vfrontt x−Vfrontt the particle flux vorticity EQ(x)=(cid:82)0LyωQ2(r)dy/Ly (see
detaileddiscussionbelow),whereω (r)isthecurlofthe
Q
FIG. 3. (color online) Space-time dependence of the active- particle flux Q(r). E is most pronounced within the
particle packing density φ (x,t) for (a) the mixture with Q
A interface region and, in case of a mixture, E is larger
x = 0.5 and (b) a pure active fluid with x = 1. The Q
A A
at the side of larger polarisation and active particle ac-
black lines indicate the center-of-mass position of φ (x,t).
A
(c)VelocitiesV (bullets)oftheinterfaceandV (circles) cumulation, see Fig. 3(g) and Fig. 2(b).
front ρ
extracted from S(q,ω), see Fig. 1, as function of xA. Phase DiscussionofinterfacesinpureABPfluids–Interfaces
separation appears for xA = 0.2. (d-g) Time averaged pro- in pure ABP fluids are not propagating, but show a dif-
filesmeasuredinacomovingframe,i.e.,relativetotheprop-
fusivedynamicsasevidentfromFig.3(b)andthemovies
agatingfront,(d)ofactive-andpassive-particlepackingfrac-
inRef.[25]. Withinthepictureofmotility-inducedphase
tionsφ (x)andφ (x),(e)oftheactive-particlex-polaristion
A P
separation, there is a balance between an active flux of
(cid:104)e (cid:105)(x),(f)ofactive-andpassive-particlevelocitiesalongthe
x
front propagation direction V (x) and V (x), and (g) of the particles from the low-density gas phase colliding with
A P
intensity of the particle flux vorticity E (x). We choose a the dense phase and a diffusive flux of particles leaving
Q
representation,wherethefrontpropagatestotheright. Solid the high-density phase due to rotational diffusion [10].
lines correspond to xA =0.5 and dashed lines to xA =1. This alone would generate a very rough and uncorre-
lated interface structure, like in random particle depo-
sition [23].
termittent large-scale rearrangements of the front. The However, activity also leads to smoothing of the inter-
front velocity V is nearly independent of the over- face. For an undulated interface, particles which bump
front
4
0.016
(a) (b) (c)
103
0.5 0.012
(V/V)(L/c)s0y00..000048 xA=1 2|h|Lqy110012 xx|hAAq==|102∝.5 q-2 Γh1100-10 xxΓAAh∝==10q.25.1
x =0.5 100 |h |2∝ q-3.3 Γ ∝q1.6
A q h
0
10-1 10-1 10-1
q q q
FIG. 4. (color online) (a) Growth velocities V (q) of the Fourier modes of an isoline of the scalar displacement field s(r,δt)
s
insidethedensephase,seeFig.2(c). (b)Spectrumoftheinterfacialheightfluctuations(cid:104)|h |2(cid:105)asafunctionofthewavenumber
q
q; results for different L at fixed L /L are shown. (c) Decay rate Γ (q) of the interfacial height autocorrelation function
y y x h
(cid:104)h (t)h (0)(cid:105). Different scaling regimes are indicated by lines.
q −q
into the interface slide into regions of high convexity decayforlargerq, seeFig.4(a). Thisconfirmsthevisual
[11, 28], thereby level out the interface on small scales impression in Fig. 2(b,c) of a characteristic length scale
and produce a local polarisation. In turn, this local po- of internal mass currents.
larisation induces an internal mass flow inside the dense
Interface correlations and relaxation – We analyse the
phase such that alternating vortices of opposite vorticity
structure and dynamics of the interface by a Fourier
emergewithintheinterfaceregionduetomassconserva-
transform of its fluctuations. From the Fourier am-
tion, see Fig. 3(g) and movies in Ref. [25]. As a result,
plitudes h [25], we obtain the interface structure fac-
q
randomly oriented particles emerge from the bulk and tor S(q) = (cid:104)|h |2(cid:105) and the autocorrelation function
q
cause an evaporation of the interface protrusions.
(cid:104)h (t)h (0)(cid:105), which we fit by S(q)exp(−Γ t) to ob-
q −q h
tain the damping rate Γ (q) as function of q. We ob-
h
Discussion of interfaces in active-passive mixtures –
serve a length-scale-dependent scaling S ∝q−(1+2α) and
Thispicturechangesconsiderablyinmixtures,wherethe
Γ ∝ qz, where α and z are the roughness and the dy-
h
interfacesarepropagatingduetothedynamicalcoupling
namic exponent, respectively [29]. We find α ≈ 1/2 and
of active and passive particles. Imagine a perturbation z ≈ 1.6 on large scales, q (cid:46) 0.1, and α ≈ 1 and z ≈ 2
such that the polarisation in the right interface region of on intermediate scales, 0.1 (cid:46) q (cid:46) 0.4, for static as well
Fig. 2(a) is larger then that in the left one. Hence, more
astravelinginterfaces,seeFig.4(b,c). Incomparison,an
active particles leave the right interface, due to a larger
overdamped fluid interface with thermally excited capil-
vorticity, and the collision-induced flux of passive parti-
lary waves in equilibrium has α = 1/2 [27] and z = 1
cles from right to left exceeds the opposite flux. Now, a
[30] for q <0.6. However, physically more related is the
corona of passive particles covers the dense phase in the
Edward-Wilkinson model [23, 24], for non-equilibrium
left interface region and inhibits the further discharge of
interface growth – where random particle arrival leads
active particles. This is supported by the shape of the
to interface roughening, while lateral motion (e.g., due
profiles φ (x), φ (x), (cid:104)e (cid:105)(x), and E (x) in Fig. 3. The
A P x Q to gravity) yields interface smoothing – with exponents
overall picture that the traveling front is a consequence
α =1/2 and z =2. If additionally local growth perpen-
of a flux imbalance and that the particle transport out
diculartotheinterfaceispresent,asintheKardar-Parisi-
of the interface is dominated by vortex formation and
Zhangmodel,thedynamicexponentz =3/2isexpected
not by rotational diffusion is in line with the fact that
[23, 24], very close to the exponents characterizing the
V is independent of Pe, if V is fixed and D is var-
front 0 r interface behavior of our active particle fluids.
ied via k T. In order to quantify the vorticity-induced
B
Conclusions – Active-particle systems display many
mass transport, we use the horizontal positions x (t) of
i
unexpectedfeatures–bothstaticanddynamic. Wehave
all particles i (at time t) to construct a scalar displace-
shownthatthelarge-scaleinterfacestructureinmixtures
mentfields(r,δt)[25]. Thetemporalevolutionofs(r,δt)
is similar to that at equilibrium, however, the dynamics,
then indicates the particle convection, see Fig. 2(c) and
like interface relaxation or front propagation, exhibits
movies in Ref. [25]. We choose a isoline of s(r,δt) inside
strong nonequilibrium characteristics. Our results call
thedensephase;thisisolineisinitiallyflat,butroughens
for an experimental investigation over a wide range of
as a function of time. This process is monitored by the
concentrations and activities. Active-passive mixtures
Fouriermodesofthisisoline, similartoastabilityanaly-
could be realized experimentally by active colloids [5–8],
sis of the Rayleigh-Taylor instability. The Fourier-mode
vibrated polar disks [31] or even robots [32, 33].
amplitudes grow with constant velocity V (q) at short
s
times. The growth velocities first increase with increas- We thank A. Varghese and J. Horbach for helpful
ing q, reach a maximum at q ≈ 0.1, and exhibit a fast discussions. The support by the DFG priority pro-
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