Rotational and translational diffusion in an interacting active dumbbell system

1.8 MB·English

by  Leticia F. Cugliandolo

Rotational and translational diffusion in an interacting active dumbbell system
Leticia F. Cugliandolo,1 Giuseppe Gonnella,2,∗ and Antonio Suma3
1Sorbonne Universit´es, Universit´e Pierre et Marie Curie - Paris VI,
Laboratoire de Physique Th´eorique et Hautes E´nergies,
4 Place Jussieu, 75252 Paris Cedex 05, France
2Dipartimento di Fisica, Universita` di Bari and
5 INFN, Sezione di Bari, via Amendola 173, Bari, I-70126, Italy†
1 3SISSA - Scuola Internazionale Superiore di Studi Avanzati,
0 Via Bonomea 265, 34136 Trieste Italy‡
2
We study the dynamical properties of a two-dimensional ensemble of self–propelled dumbbells
y
with only repulsive interactions. This model undergoes a phase transition between a homogeneous
a
and a segregated phase and we focus on the former. We analyse the translational and rotational
M
mean square displacements in terms of the P´eclet number, describing the relative role of active
forces and thermal fluctuations, and of particle density. We find that the four distinct regimes of
8
the translational mean square displacement of the single active dumbbell survive at finite density
1
forparametersthatleadtoaseparationoftime-scales. WeestablishtheP´ecletnumberanddensity

dependence of the diffusion constant in the last diffusive regime. We prove that the ratio between
]
h thediffusionconstantanditsvalueforthesingledumbbelldependsontemperatureandactiveforce
c onlythroughtheP´ecletnumberatalldensitiesexplored. Wealsostudytherotationalmeansquare
e displacement proving the existence of a rich behavior with intermediate regimes only appearing
m at finite density. The ratio of the rotational late-time diffusion constant and its vanishing density
- limit depends on the P´eclet number and density only. At low P´eclet number it is a monotonically
t decreasing function of density. At high P´eclet number it first increases to reach a maximum and
a
next decreases as a function of density. We interpret the latter result advocating the presence of
t
s large-scalefluctuationsclosetothetransition,atlargeenoughdensity,thatfavourcoherentrotation
.
t inhibiting, however, rotational motion for even larger packingfractions.
a
m
PACSnumbers: 05.70.Ln,47.63.Gd,66.10.C-
-
d
n
I. INTRODUCTION The dynamical properties of a suspension are signifi-
o
cantlyaffectedbyself-propulsion. Forinstance,thecom-
c
[ Active matter includes different kinds of self-driven parisonofthediffusionconstantD ofadilute solutionof
systemswhichlive,orfunction, farfromthermodynamic passivesphericalcolloidswiththe oneofrun-and-tumble
2
bacteria shows that the Stokes-Einstein formula and the
equilibrium, by continuously converting internal energy
v
fluctuation-dissipationtheoremdonotholdfortheactive
4 sourcesintoworkormovement[1,2]. Natureoffersmany
system [13, 29]. Indeed, assuming the Stokes-Einstein
5 examples of this kind of condensed matter, at very dif-
0 ferent scales: the cytoskeleton, bacterial colonies and al- relation, D is given by D = kBT/(3πησ) where T is
4 gae suspensions, bird flocks and schools of fish are just the temperature, η the fluid viscosity, and σ the diam-
0 eter of the colloids. The diffusion coefficient in a three-
some among others [1–10]. Self-propelled units can also
1. be artificially realized in the laboratory in many differ- dimensionalsystemofrun-and-tumblebacteria[5]iseval-
0 ent ways, for example, by surface treatment of colloidal uated as DA ≈lr2un/(6τ) where τ is the durationof each
5 runandl itslength. Usingτ 1s,l 10 30µm,
particles [11]. run run
1 water viscosity η 10−3 Pa s, a≈nd σ 1 µ≈m, o−ne finds
Active matter is inherently out of equilibrium and ex-
v: hibits non-trivial properties that have no analogue in DA/D 102. ≈ ≈
≈
i
X passive, equilibrium materials. For example, large scale This simple order-of-magnitude argument suggests to
coherent motion and self-organised dynamic structures analyse more carefully how the diffusive behaviour is af-
r
a havebeenobservedincoloniesofbacteriainthe absence fected by self-propulsion. Actually, several experimental
ofanyattractiveinteraction[12–18]. Inaddition,aphase studies addressed this question. Wu and Libchaber con-
separation into an aggregate and a gas-like phase has sidered the mean-square displacement of passive tracers
been found in theoretical models [19–27] and, recently, coupled to a dilute suspension of Escherichia coli and
alsoinexperiments[28]onsuspensionsofself-motilepar- found that an initial super-diffusive behaviour crosses
ticles only subject to steric interactions. over to normal diffusion at late times [13]. The super-
diffusive behaviour was interpreted as being due to the
presence of coherent structures in the bacterial bath. A
similar study was carried on by Leptos et al. on a sus-
∗ [email protected] pension of algae. These authors found a linear time-
† [email protected] dependence ofthe passivetracer’smean-squaredisplace-
‡ [email protected] ment at all measured time-lags [30]. The tracer’s diffu-
2
siveconstantwasfoundtodependlinearlyonthedensity presented. Thephasediagramisanalysedintermsofthe
of swimmers [30] while, reducing the dimensionality to a P´eclet number. This is a preliminary step needed to fix
film-likegeometry,thedensity-dependencewasenhanced the region of the parameter space to be considered for
tothepower3/2[31]. Resultsonthediffusioncoefficient the measurements of the diffusive properties. Then the
of tracers in contact with bacterial suspensions with hy- translational and rotational mean-square displacements
drodynamicsplayingarelevantroleweregivenin[32–34]. in the homogeneous phase are studied in all their dy-
Other studies focused on the mean-square displace- namic regimes. Special emphasis is put on the analysis
ment of the active particles themselves. A linear de- of the parameter dependence of the diffusion coefficient
pendence of the diffusion constant of the active swim- in the asymptotic limit. A discussion will complete the
mers on the so-called active flux (active swimmers den- paper in Section V.
sity times their mean velocity) was found for different
kinds of swimmers in contact with a solid wall [35]. The
simulations of Hern´andez-Ort´ız et al. show ballistic be-
haviourcrossingovertonormaldiffusionfortheswimmer II. THE MODEL
andpassivetracerparticlesthoughwithdifferentdensity-
dependenceofthediffusionconstantinthelatterregime.
A dumbbell is a diatomic molecule formed by two
The swimmer’s diffusion constant decreases with den-
spherical colloids with diameter σ and mass m linked
d d
sity at low swimmer density while it increases at large
together. The atomic positions are noted r and r in a
1 2
values; the diffusion constant of the passive tracer has,
Cartesian system of coordinates fixed to the laboratory.
instead,amonotonicdependencewithdensityanditcon-
The colloids are subject to internal and external forces.
sistently increases with it [15]. The Lattice Boltzmann
Typically,one assumes that there is anelastic link be-
studyin[36]alsoshowsacross-overbetweenballisticand
tween the colloids modeled by the finite extensible non-
diffusive behaviour at relative high self-propelled parti-
linear elastic force
cledensityandasuper-diffusiveregimeassociatedtothe
formation of large scale clusters at low densities.
kr
A detailed study of diffusion properties in models for F = (1)
fene −1 (r2/r2)
self-propelled particles at different densities for the in- − 0
terestingcaseswhereself-propulsionhasbeenrecognised
to be aninteractioncapableofproducingaphasetransi- with k > 0. The denominator ensures that the spheres
tionislackingintheliterature. Someresultsforthecase cannot go beyond the distance r0 with r the distance
of self-propelled polar disks have been presented in [20]. between their centres of mass. An additional repulsive
ThepaperbyGr´egoireetal.[37],whoconsideredamodel force is added to ensure that the two colloids do not
with particle interaction that favors alignment `a la Vic- overlap. This is the Weeks-Chandler-Anderson (WCA)
sek,canalsobementionedinthisrespect. Theseauthors potential [41]
interpretedthe super-diffusive behaviourof[13] in terms
ofthecross-overfoundintheirmodelclosetothecritical
V (r) V (r ) r <r
point. Vwca(r)= 0LJ − LJ c r >rc (2)
c
Swimmers typically have elongated shape. We have (cid:26)
therefore decided to study the diffusive properties of ro-
with
tationalandtranslationaldegreesoffreedominasystem
ofactivedumbbells. Thismodelwasintroducedin[38]to
σ 12 σ 6
describe the experimental behaviour of a bacterial bath V (r)=4ǫ d d , (3)
LJ
coupled to colloidal tracers. A first study of its phase r − r
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
diagramappearedin[24,25]whereitwasshownthatfor
certain (relatively high) densities and active forces the where ǫ is an energy scale and r is the minimum of the
c
system phase separates into dense and loose spatial re- Lennard-Jones potential, r =21/6σ .
c d
gions. Alaterwork[39]focusedonthedumbbelleffective
The active forces are polar. They act along the main
temperaturedefined intermsofafluctuation-dissipation molecularaxisnˆ,areconstantinmoduluspointinginthe
relation[40]. However,adetailedanalysisofthe transla-
samedirectionforthetwospheresbelongingtothe same
tional and rotational mean-square displacements in the
molecule [42], and read
full range of time-delays and varying the three more rel-
evant parameters (temperature, activity and density) si-
F =F nˆ . (4)
multaneously,wasnotperformedyet. We presentsucha act act
complete analysis in the homogeneous phase here.
The paper is organised as follows. In Section II the We take the interaction between the spheres in differ-
dumbbell model is reviewed. The diffusion behavior of a ent dumbbells to be purely repulsive and of the same
singledumbbell is describedinSectionIII.InSectionIV WCA form as for the two colloids composing one dumb-
thenumericalresultsfortheinteractingactivesystemare bell.
3
The dynamic equations for one dumbbell are Thisparameterisalsoameasureoftheratiobetweenthe
work done by the active force in translating the center
m ¨r (t)= γr˙ (t)+F (r )+η
d i − i fene i,i+1 i of mass of the molecule by a distance of 2σd, and the
2N ∂Vij r thermal energy scale. Another important parameter is
− ∂rwcarij +Facti , (5) the active Reynolds number
ij ij
j=0
X
j6=i m F
d act
md¨ri+1(t)=−γr˙i+1(t)−Ffene(ri,i+1)+ηi+1 React = σdγ2 , (11)
2N ∂Vi+1,j r
wca i+1,j +F , (6) defined in analogy with the usual hydrodynamic
− ∂r r acti
j=0 i+1,j i+1,j Reynolds number Re = Lv/ν, where ν is the kine-
X
j6=i+1 matic viscosity of a given fluid, representing the ratio
withi=1,3,...2N 1,r =r r ,r = r andVij betweeninertialandviscousforces. HerewesetL σd,
− ij i− j ij | ij| wca ≡ v F /γ and ν γσ2/m . →
Vwca(rij) with Vwca defined in Eq. (2). Once the active → act → d d
force is attached to a molecule a sense of back and forth
atoms is attributed to them; F is directed from the
act
ithcolloid(tail)to the i+1thcolloid(head). The active III. A SINGLE DUMBBELL
forcesareappliedtoallmoleculesinthesampleduringall
theirdynamic evolution. Fact changesdirectiontogether Before studying the interacting problem with numer-
with the molecule’s rotation. ical simulations in Sec. IV, we derive analytically the
Thecouplingtothethermalbathismodelledasusual, translational and rotational mean-square displacements
with a friction and a noise term added to the equation of the single dumbbell.
of motion. γ is the friction coefficient and we do not The equation of motion for the position of the centre
distinguish friction along the main molecular axis and of mass, r =(r +r )/2, of a single dumbbell is
cm 1 2
transverse to it, as done in some publications [43]. The
noise η is a Gaussian random variable with 2m ¨r (t)= 2γr˙ (t)+2F (t)+ξ(t) (12)
d cm cm act
−
η (t) =0, (7)
h ia i with the new noise ξ(t) η (t)+η (t) with vanishing
η (t)η (t′) =2γk Tδ δ δ(t t′), (8) ≡ 1 2
ia jb B ij ab average, ξ (t) =0, and correlation
h i − a
h i
with k the Boltzmann constantandT the temperature
B ξ (t)ξ (t′) =4γk T δ δ(t t′). (13)
of the equilibrium environment in which the dumbbells a b B ab
h i −
move. a and b label the coordinates in d dimensional
ThisistheLangevinequationofapoint-likeparticlewith
space. Aneffectiverotationalmotionisgeneratedbythe
mass2m ,underaforce2F ,andincontactwithabath
random torque due to the white noise acting indepen- d act
with friction coefficient 2γ at temperature T.
dently on the two beads. We consider initial condi-
The equationof motionfor the relative positionof the
tions at time t=0 such that the initial angle θ =θ(0),
0
two monomers, r=r r , is
randomly distributed between [ π,π], has zero mean 1 2
−
−
[θ ] =0.
0 ic
m ¨r(t)= γr˙(t)+2F (t)+ζ(t) (14)
The surface fraction is d int
−
Sd with the new noise ζ(t) = η (t) η (t) having zero av-
φ=N (9) 1 − 2
S erage, ζ (t) =0, and correlation
a
h i
with S the area occupied by an individual dumbbell,
d ζ (t)ζ (t′) =4γk T δ δ(t t′). (15)
S the total area of the box where the dumbbells move h a b i B ab −
and N their total number. The spring is supposed to
Note that the noises ξ and ζ are independent,
be massless and void of surface. Therefore, in d = 2,
ξ (t)ζ (t′) =0,foralla,batalltimes. F includesthe
Sd = πσd2/2. We impose periodic boundary conditions ehlaasticbandirepulsive forces internal to theintsingle dumb-
on the two directions.
bell.
The P´eclet number, Pe, is a dimensionless ratio be-
Equation (14) controls the molecule’s elongation and
tween the advective transport rate and the diffusive
itsrotationalmotionwhileEq.(12)determinesthetrans-
transport rate. For particle flow one defines it as Pe =
lational properties of the dumbbell. The internal force
Lv/D, with L a typical length, v a typical velocity, and
F affectstheelongationofthemoleculewhilethether-
D a typical diffusion constant. We choose L σ , int
d
v F /γ and D Dpd = k T/(2γ) of the p→assive malnoiseaddsfluctuationstoitbut,moreimportantly,it
→ act → cm B applies an effective torque and induces rotations. Equa-
dumbbell to be derived below; then,
tions(12)and(14)arecoupledbythefactthatF acts
act
2σ F along the axis of the molecule, the orientation of which
d act
Pe= . (10)
k T changes in time in the presence of thermal fluctuations.
B
4
A. Elongation and rotation whereallfactorsareevaluatedatthesametimeandD =
2γk T. The explicit calculation yields
B
Let us call uˆ the instantaneous unit vector pointing
k
d r
from monomer 1 to monomer 2 along the axis of the γ h i =2 F +Dγ−1 r−1 , (25)
int
molecule, θ the angle between uˆ and an axis fixed to dt h i h i
k
the laboratory, and uˆ a unit vector that is perpendic- d θ
ular to uˆ at all time⊥s. Using r˙ = r˙uˆ +ruˆ˙ , with r γ dhti =0. (26)
k k k
the modulus of r, uˆ˙ = θ˙uˆ and uˆ˙ = θ˙uˆ (note
k ⊥ ⊥ − k The firstequationis independent ofθ but it involvesthe
that we use here the Stratonovich discretisation scheme
average of different functions of r. The second equation
of stochastic differential equations [44] and we are thus
implies θ = θ = θ(0) and, as we will take random
entitled to apply the usual rules of calculus) [45], one h i 0
initial conditions with average [θ ] = 0, then [ θ ] =
has 0 ic h iic
0. The statistics of θ can be further analysed from the
m (r¨ rθ˙2)= γr˙+2F +ζ , (16) equation for the angular variance
d int k
− −
m (2r˙θ˙+rθ¨)= γrθ˙+ζ , (17)
d ⊥ d
− γ θ2 =2Dγ−1 r−2 . (27)
wherewedecomposedthenoiseintotheparallelandper- dth i h i
pendicular directions, ζ = ζ uˆ +ζ uˆ . The relations
k k ⊥ ⊥ If we assume that r does not fluctuate aroundσ , other-
between the unit vectors in the fixed laboratory and the d
wisestated,thatthemoleculeisapproximatelyrigid[48],
co-moving frame are given by
uˆ =cosθ uˆ sinθ uˆ , r σ , (28)
x k− ⊥ ≈ d
uˆ =sinθ uˆ +cosθ uˆ . (18)
y k ⊥
this equation implies angular diffusion
With this, for any noise we write
θ2 =θ2+2D t (29)
ζ = (ζx cosθ+ζy sinθ) uˆk h i 0 R
+( ζ sinθ+ζ cosθ) uˆ , (19)
x y ⊥ with the angular diffusion constant
−
and
D 2k T
B
D = = . (30)
ζk =ζx cosθ+ζy sinθ , R γ2σd2 γσd2
ζ = ζ sinθ+ζ cosθ . (20)
⊥ x y
−
Thesametechniquecanbeusedtocomputeallmoments
The system of equations (16)-(17) for r and θ cannot ofthe angularvariableandthus showthatitisGaussian
be solved exactly. We will assume that the internal and distributed within the rigid molecule approximation.
viscousforcesaresuchthattheinertialcontributions(all
It is interesting to compare our expression for the ro-
terms proportional to m ) can be neglected. We then
d tational diffusion constant, that for the sake of clarity
have we call Ddumb in this paragraph, with that for self-
R
propelled hard rods, as described in the Langevin ap-
γr˙ =2F +ζ cosθ+ζ sinθ , (21)
int x y
proach by Baskaran and Marchetti [43]. In the model
γrθ˙ = ζx sinθ+ζy cosθ . (22) of Ref. [43] the rotational diffusion constant is given by
−
Drod = (k Tm)/(Iγ) where I is the moment of inertia
Puttingtogetherr andθ intoavectory=(r,θ), thisset R B
calculated along the main axis of the rod and γ is a fric-
of equations reads
tion coefficient. If we identify the latter γ with our fric-
tioncoefficient,andwetakeforthedumbbellthemoment
y˙ =h [y]+g [y] ζ (23)
α α αβ β
of inertia of two point-like particles with mass m = m
d
where the index β is a Cartesian one, ζ1 = ζx and ζ2 = and diameter σd = 2R, I = 2mR2, then our expression
ζ , and the index α yields y = r and y = θ. The for the rotational diffusion coefficient coincides with the
y 1 2
components of the vector h and the matrix g can be one in [43]. Consider now a rod of length l with the
easily read from Eqs.(21) and (22). In the last term the same aspect ratio (l = 4R, σ = 2R) and the same total
noise appears multiplying a function of the stochastic mass 2md ofthe dumbbell. One finds I =2mdR2A with
variable y. A ≈ 2.94 [43], so that DRdumb = ADRrod > DRrod. On the
One can now average Eqs. (21) and (22) over the other hand, in the limit of a very long rod (l R) one
Cartesian white noise by using the rules of Stratonovich has I = ml2/6 and DRdumb = (DRrodl2)/(12R2≫), so that
stochastic calculus described in [44, 46, 47]: DRdumb ≫ DRrod as expected. Therefore the comparison
with the model of Ref [43] suggests that the rotational
∂ diffusion coefficient of a dumbbell is always larger than
g ζ =D g g (24)
h αβ βi h νβ∂y αβi the one of a rod in a suspension.
ν
5
B. The center of mass In the limit t t ,
I
≫
r2 (t)=4Dpd t
Wenowfocusonthestatisticalpropertiesofthecentre h△ cmi cm
ofmasspositionandvelocitythatdependupontheactive Fact 2 2 1 e−DRt
+ t − , (38)
force. One readily solves Eq. (12) γ D − D
(cid:18) (cid:19) R (cid:18) R (cid:19)
rcm(t)= r0+ v0md v0md e−mγdt where
γ − γ k T
(cid:18) (cid:19) Dpd = B (39)
+ 1 tdt′ [1 e−mγd(t−t′)][2Fact(t′)+ξ(t′)],(31) cm 2γ
2γ −
Z0 is the diffusion constant in the passive limit, F = 0,
vcm(t)=v0 e−mγdt see [39]. This equation presents several time scaalcets and
+2m1 tdt′ e−mγd(t−t′) [2Fact(t′)+ξ(t′)], (32) lpimreistesn.te(dSiimnil[a4r9].c)alFcourlattIionstfor atanoancetivfiendesllipsoid were
d Z0 ≪ ≪
with r0 =rcm(0) and v0 =vcm(0). r2 (t)=4Dpd t+ Fact 2t2 , (40)
FromEq.(32)andthanksto[ cosθ ]ic =[ sinθ ]ic =0 h△ cmi cm γ
h i h i (cid:18) (cid:19)
onefinds [ v ] =0and, aftersomelongbutstraight-
cm ic
forward inhtegraitions, that can still be split into the passive diffusive limit
r2 =4Dpd t for t t <t∗, and a ballistic regime
h△ cmi cm I ≪
F2 r2 = (F /γ)2 t2 for t∗ < t t , where the time
2md[hvc2mxi]ic =kBT + γ(t−1a+ctt−1) (33) hs△calecmt∗iis givaecnt by ≪ a
I a
4Dpdγ2 2k Tγ 4 2 σ2
with t∗ = cm = B = d
md Fa2ct Fa2ct (cid:18)Pe(cid:19) 4Dcpmd
tI = γ , 4 2
= t . (41)
a
γσ2 σ2 Pe
t =D−1 = d = d , (34) (cid:18) (cid:19)
a R 2kBT 4Dcpmd Note that these two intermediate regimes do not exist if
the parameters are such that t∗ < t or t∗ > t . They
inthe longtime limit, beyondt . The time scalest and I a
I I can also be easily confused with super-diffusion tα with
t are independent of the active force, and they are the
a 1 < α < 2 if they are not well separated (t t∗ t ).
usualinertialtimeandacharacteristictimeassociatedto I ≃ ≃ a
See [39] and Fig. 4 below for more details. In the large
rotationaldiffusioninthepassivedumbbell,respectively.
Pe limit one has t∗ t . In the last time-lag regime
WealsoobservethatinthepassivelimitEq.(33)reduces ≪ a
t t , we recover normal diffusion,
to the equipartition theorem for the kinetic energy of a ≫ a
point-like particle havingthe totalmass ofthe dumbbell r2 (t)=4D t, (42)
2m . As, typically, t t , one has h△ cmi A
d a I
≫
with the diffusion coefficient
m F2
2m [ v2 ] k T + d act . (35)
d h cmxiic ≃ B γ2 k T 1 F σ 2
B act d
D (F ,T,φ=0)= 1+ . (43)
A act
2γ 2 k T
Withasimilarcalculation,startingnowfromEq.(31), " (cid:18) B (cid:19) #
we calculate the mean-square displacement (MSD)
Intermsofthediffusionconstantofthecenterofmassof
the passive dumbbell and the P´eclet number the above
h∆r2cmi(t)=[h(rcm(t+t0)−rcm(t0))2i]ic (36) equation reads DA(Pe,φ=0)=Dcpmd (1+Pe2/8).
Inthe figureswith numericalresultsforthe finite den-
with [...] the average over initial conditions at time
ic sity problem shown in the next Section we include data
t=0. t isasufficientlylongtimeafterpreparationsuch
0 for φ=0 that correspond to the single dumbbell limit.
that the stationary dynamics have been established and
the mean-square displacement is therefore independent
of t . Henceforth, t denotes time-delay.
0 IV. FINITE DENSITY SYSTEMS
In the limit t t ,
I
≪
r2 (t)=2 v2 t2 , (37) InthisSectionwepresentournumericalresults. Wefo-
h△ cmi h cmxi cus on three issues: the phase diagram,the translational
where v2 = [ v2 ] is the velocity given in diffusion properties and the rotational diffusion proper-
h cmxi h cmxiic
Eq. (33). The factor two is due to the sum over the ties. Details on the numerical method used for solving
two Cartesian directions. the dynamical equations (5) are given in [39]. We set
6
m = σ = k = ǫ = 1 in proper physical units, and packing fractions φ = 0.3, 0.4, 0.6. We found the same
d d B
r = 1.5, k = 30, γ = 10, assuring over-damped motion values for the densities of the separated phases at φ =
0
and negligible dumbbell vibrations. Depending on the 0.4, 0.6 and Pe = 100, 200, and at φ = 0.3 and Pe
plotsweusedbetween15000and20000dumbbellsinthe = 200. At φ = 0.3 and Pe 100 the effects of the
≤
simulations. presence of the spinodal line require a more elaborate
analysis of the phase diagram, as discussed in [25]. In
Table I we report these density values for the cases with
A. The phase diagram Fact =1, 0.5, 0.1, 0.05andPe=100, 200. Asobserved,
the coexistence values get closer for smaller active forces
even though the P´eclet number remains the same.
Aspectsofthephasediagramandthedynamicsofthis
systemwerealreadyestablishedin[24,25]. Itwasshown
in these papers that at sufficiently low temperature and
largeactiveforcethesystemphaseseparatesintogas-like
Pe Fact =0.05 Fact =0.1 Fact =0.5 Fact =1
spatial regions and clusters of agglomerateddumbbells.
200 0.37 0.34 0.21 0.049
0.70 0.71 0.80 0.890
The model has three important energy scales, ǫ, k T 100 0.44 0.41 0.37 0.096
B
andF σ . Thereisanotherenergyscalerelatedtothe 0.66 0.68 0.77 0.870
act d
elastic constant k but, since we consider stiff dumbbells
in this paper, we do not vary k. Keeping also ǫ and the TABLE I. Density values of the two coexisting phases mea-
suredfromthehistogramsinFig.1 atφ=0.5. Thefirstand
other parameters listed at the beginning of this section
the third lines refer to the dilute phase while the two other
fixed,wewilltrytodeterminewhetherthephasediagram
lines correspond to the aggregated phase. Similar values are
dependsuponk T andF σ separatelyoronlythough
B act d obtained at φ=0.4,0.6 (Pe = 100, 200) and at φ=0.3 (Pe
theirratio,i.e. throughtheP´ecletnumberPe,asusually
=200).
assumed [22]. The other free parameter to be varied is
the global density of the sample, φ.
In Fig. 2 we show four snapshots of the systemconfig-
In the four panels in Fig. 1 we show the probability uration. The active force is F = 0.1 in all panels and
act
distribution function, ρ, of the local density, φloc, for temperature is increased from left to right and from top
four values of the active force, Fact = 0.05, 0.1, 0.5, 1. tobottom. Theconfigurationintheupper-leftpanel(Pe
Each panel contains results for the same set of P´eclet = 200) shows phase separation with large scale clusters
numbers Pe(T,F ) obtained by different combinations
act while the configuration in the lower-right panel (Pe =
of temperature and active force. The system has global 20) is clearly homogeneous. The case Pe = 100 is in the
packing fraction φ = 0.5. We used the same operative
segregated phase while the one for Pe = 66 is close to
definition of the local density as in [24, 25]. We divided critical.
the fullsysteminsquareplaquetteswithlinearsize10σ
d
thatismuchsmallerthanthelinearsizeofthefullsample
and big enough to sample φ correctly. We improved
loc
thestatisticsbysamplingovermanydifferentrunsofthe
same kind of system.
At low Pe the system is homogeneous and ρ is peaked
around φ =φ=0.5. The critical Pe at which the sys-
loc
tem starts aggregating is approximately the same in all B. Translational diffusion properties
panels, Pe 65. Around this value the density distribu-
≃
tionρnotonlybecomesasymmetricbutstartsdeveloping
a second peak at φloc > 0.5 that characterises the dense In Ref. [13] the diffusion properties of a tracer im-
phase in the system. Snapshots of typical configurations mersed in a bacterial bath were monitored. A cross-over
at Fact = 0.1 and four values of Pe are shown in Fig. 2. betweenasuper-diffusiveregimeatshorttime-delaysand
ThelocationofthecentralpeakatPelessthanthecriti- a diffusive regime at long time-delays was reported. The
calvalue isindependent ofallparameters(apartfromφ) cross-over time was found to increase linearly with the
while the location of the peak at φ > 0.5 is situated density of the active medium, showing that the cross-
loc
at different values of φloc for different Fact and the same overis not due to the tracer’s inertia but to the dynami-
Pe (compare the different panels in Fig. 1). The reason cal properties of the bacterialbath. We explore here the
for this is that the strength of the interactions between sameissuesbyfocusingontheMSDofthecenterofmass
the dumbbells under different F is different as F /ǫ of the dumbbells, defined in Eq. (36). We will consider,
act act
varieswithFact. Alargeractiveforcepermitsthedumb- for the rest of the paper, sufficiently low P´eclet numbers
bellsto bemorecompact,whilealoweronefavorslooser such that the system will always be in the homogenous
clusters. phase even though fluctuation effects can be relevant, as
Werepeatedtheanalysisaboveforthecaseswithtotal we will see.
7
Pe(0.0005,0.05)=200 Pe(0.001,0.1)=200
30 Pe(0.001,0.05)=100 30 Pe(0.002,0.1)=100
Pe(0.0015,0.05)=66.6 Pe(0.003,0.1)=66.6
Pe(0.002,0.05)=50 F =0.05 Pe(0.004,0.1)=50 F =0.1
Pe(0.0025,0.05)=40 act Pe(0.005,0.1)=40 act
Pe(0.005,0.05)=20 Pe(0.01,0.1)=20
)c 20 )c 20
o o
φl φl
ρ( ρ(
10 10
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
φ φ
loc loc
Pe(0.005,0.5)=200 Pe(0.01,1)=200
30 Pe(0.01,0.5)=100 30 Pe(0.02,1)=100
Pe(0.015,0.5)=66.6 Pe(0.03,1)=66.6
Pe(0.02,0.5)=50 F =0.5 Pe(0.04,1)=50 F =1
Pe(0.025,0.5)=40 act Pe(0.05,1)=40 act
Pe(0.05,0.5)=20 Pe(0.1,1)=20
)c 20 )c 20
o o
φl φl
ρ( ρ(
10 10
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
φ φ
loc loc
FIG. 1. Probability distributions for the local density φloc at P´eclet numbers given in the keys in the form Pe(T,Fact) for
different temperatures, and Fact =0.05, 0.1, 0.5, 1 in thedifferent panels. The global density of the system is φ=0.5.
1. Dumbbell trajectories 2. Four dynamic regimes
In Fig. 4 we show the center of mass MSD normalised
by time-delay in such a way that normal diffusion ap-
pears as a plateau. The four panels display data at four
SeveralsingledumbbelltrajectoriesareshowninFig.3 temperatures, T = 0.005, 0.01, 0.05, 0.1, all under the
for different values of the temperature and global den- same active force F = 0.1. Each panel has five curves
act
sity, under the same active force F =0.1. The trajec- init,correspondingtofivedifferentdensitiesgiveninthe
act
tories correspond to a total time interval that includes key. In all cases m = 1 and γ = 10 implying t = 0.1.
d I
the late diffusive regime (see below). At low tempera- The characteristic times t , t∗, t are shown with small
I a
ture and global density (T = 0.005 and φ = 0.1, up- vertical arrows in each panel. These plots show several
per left panel) we see periods of long directional motion. interesting features:
These are reduced at higher global density (φ = 0.4 up- – In all cases there is a first ballistic regime (the dashed
per right panel). Increasing temperature at φ = 0.1 segment close to the data is a guide-to-the-eye) with a
(T = 0.05 and T = 0.5 lower left and right panels, re- pre-factor that is independent of φ and increases with
spectively) the trajectories become more similar to the temperatureasgivenbyEq.(37)(Thecaset t ofthe
I
≪
typical ones of passive diffusion. While the trajectories single dumbbell.)
are very stretched at T = 0.005, they become the most – Next, the dynamics slow down and, depending on T
compact in the intermediate case at T =0.05 and again and φ, the normalised mean-square displacement may
quite stretched in the last case at T = 0.5. This be- attain a plateau associated to normal diffusion (low T)
haviour corresponds to the non monotonic behavior of or even decrease, suggesting sub-diffusion. (The case
the translational diffusion constant of Eq. (43) in terms t t t∗ t of the single dumbbell.)
I a
≪ ≪ ≪
of temperature. It decreases going from T = 0.005 to – The dynamics accelerate next, with a second super-
T = 0.05 while it increases going from T = 0.05 to diffusiveregimeinwhichthecurvesforallφineachpanel
T = 0.5. The single dumbbell diffusion coefficient, as look approximately paralleland very close to ballistic at
calculated from Eq. (43),is D = 0.050, 0.0075, 0.025 T = 0.005, 0.01, 0.05. (The case t t∗ t t of
A I a
≪ ≪ ≪
for the cases at T =0.005, 0.05, 0.5, respectively. the single dumbbell.)
–Finally,thelatenormaldiffusiveregimeisreachedwith
allcurvessaturatingataplateauthatyieldsthedifferent
8
FIG. 2. Snapshots of the system for Fact = 0.1 (cfr. the upper right panel in Fig. 1), and T = 0.001, 0.002, 0.003, 0.01
correspondingtoPe=200, 100,bothphaseseparated,Pe=66,closetocritical, andPe=20,homogeneous(fromlefttoright
and from top to bottom). The global density is φ=0.5.
D coefficients. (The case t t∗, t t of the single also shows a good agreement between the values of Dφ
A I ≪ a ≪ R
dumbbell.) found in these fits and the values of the rotation diffu-
It is hard to assert whether the intermediate regime sioncoefficientD (F ,T,φ)comingfromthelatetime-
R act
is super-diffusive or simply ballistic as the time-scales t∗ delaydiffusiveregimeintherotationalMSDdiscussedin
and t are not sufficiently well separated (and not even Sec. IVC.
a
ordered as t∗ < t in the last panel). Moreover, in the
a The cross-overtime-delay between the last ballistic or
lasttwo panels(high T or lowPe)the diffusion-ballistic-
super-diffusive, and the diffusive regimes seems quite φ-
diffusionregimesaremixed,duetothefactthatthecon-
independent in the first two panels T = 0.005, 0.01 and
dition t∗ t is no longer satisfied. The effective slope
≪ a it increases, though rather weakly, with φ, in the last
intheintermediatesuper-diffusiveregimedecreaseswhen
twopanels, T =0.05, 0.1,see the inclined dashedline in
the density increases.
the last panel that is also a guide-to-the-eye. This cross-
A rathergoodfitof the finite density datain the limit
overtime-delayistheonethatwecouldassociateto the
Pe 1 and for time-delays such that t t∗ is achieved
≫ ≥ cross-over time between a superdiffusive regime and the
by using the single dumbbell expression in Eq. (38)
lastdiffusiveregimefoundintheexperiment in[13]. The
r2 (t)=4Dφ t 1−e−DRφt , (44) sotrrosunbg-edstiffeuffseivcteorefgdiemnes.ity is though on the first diffusive
h△ cmi A − Dφ
(cid:18) R (cid:19) In summary, no large qualitative change in the cen-
withoutthe firstterm(negligible ifPe 1)andupgrad- ter of mass MSD behavior is observed in the range
≫
ingtheremainingparameters,Dφ andDφ,tobedensity- φ [0,0.6]. There is just a natural slowing down of the
A R ∈
dependent fitting parameters, as done in [13, 20]. This dynamics with larger packing fractions that translates
is shown in Fig. 5 (left panel). For not that large values into a change from diffusive to sub-diffusive behavior in
ofPeonecouldrecoverthe remainingparameteranduse the second regime and a general decrease of the diffu-
instead r2 (t) = 4Dpd,φt+4Dφ(t 1−e−DRφt) with sion constant in the last regime for all Pe. We study the
h△ cmi cm A − Dφ dependence of the diffusion constant with Pe in detail
R
an additional fitting parameter. Figure 5 (right panel) below.
9
T=0.005, φ=0.1 T=0.005, φ=0.4
T=0.05, φ=0.1 T=0.5, φ=0.1
FIG.3. Severaltrajectories ofonebeadindifferentdumbbells. TheactiveforceisFact =0.1andtheevolutiontimeintervalis
t=10000 in allpanels. TheotherparametersareT =0.005 andφ=0.1, 0.4(upperpanels, left andright),andT =0.05, 0.5
andφ=0.1 (lowerpanels, left and right). Thevaluesofthesingle dumbbelldiffusion coefficient, ascalculated from Eq.(43),
are DA =0.050, 0.0075, 0.025 for thecases at T =0.005, 0.05, 0.5, respectively.
3. The late-epochs translation diffusion coefficient of the two colloids, that is r 0.96 σ . In the rest of
d
≈
this section we simply call σ the molecular length and
d
we take σ =0.96.
Let us now discuss the normal diffusive regime at d
longest time-lags. In [39] we studied the translational
diffusion coefficient D as a function of F and φ at
A act
fixed temperature. In particular, we compared the φ The error-bars are smaller than the symbol size and
dependence to the Tokuyama-Oppenheim law for col- we do not display them. The curves show a minimum
loids [50]. Here, we first examine, instead, the T and located at kBTmin = Factσd/√2 for φ = 0, that weakly
φ dependence of D for fixed active force, F . Then increases with φ. The two regimes, Pe 1 and Pe 1,
A act ≪ ≫
we consider how the dependence of DA from T and Fact stillexistandDAisdominatedbythermalfluctuationsin
can be re-expressed in terms of the P´eclet number. The theformerandbytheworkdonebytheactiveforceinthe
mainresultsforD obtainedin[39]willalsoberevisited latterasinthesingledumbbelllimit. Weseeasaturation
A
in this subsection. of DA at small values of T for φ > 0.2 and therefore
thebreakdownofthesingledumbbell1/(k T)behaviour
ThefirstquestionwewanttoansweriswhetherD de- B
A
at low temperatures. Instead, at high temperatures D
pendsonk T asforthesingledumbbellcase(φ=0),the A
B
seemstoretainthelineargrowthwithtemperatureofthe
functional form recalled in Eq. (43). For F σ /k T
act d B
≪ singledumbbellatleastforthe temperaturesusedinthe
1 such that the quadratic term can be neglected this
simulations.
equation implies the linear growth of D with k T as
A B
inthepassivelimit. Instead,whenthesecondtermdom- Figure 6 also shows that for the Pe that we used D
A
inates, i.e. forverysmallthermalenergywith respectto is a decreasing function of φ at all fixed temperatures.
theworkperformedbytheactiveforce,DA shoulddecay This fact can be better appreciated in the left panel in
as 1/(kBT) with a slope that is quadratic in Factσd. Fig.7, whereDA is plottedas afunction ofφforvarious
In Fig. 6 we display D as a function of T for various temperaturesgiveninthe key. (Recallthat the φ depen-
A
values of φ given in the key and F = 0.1. The theo- denceofD atfixedT andfordifferentactiveforceswas
act A
retical values for φ = 0 are included in the figure (with discussedin[39]whereitwasshownhowthe Tokuyama-
open triangles joined by a dotted curve). Here, we used Oppenheim [50] law of the passive system was simplified
the measured value for the distance between the centres under activation to a decay that is close to a simple ex-
10
10−1 10−1
10−2 Pe=40 10−2 Pe=20
4t 4t
〉/m10−3 〉/m10−3
2c 2c
r r
∆ ∆
〈 φ=0 〈 φ=0
10−4 00..12 10−4 00..12
0.3 0.3
0.4 0.4
0.5 0.5
10−150−3 10−2 10−1 100 101 102 103 104 10−150−3 10−2 10−1 100 101 102 103 104
t t
t t* t t t* t
I a I a
10−2 10−2
Pe=4 Pe=2
4t 4t
〉/m10−3 〉/m10−3
2c 2c
r r
∆ ∆
〈 φ=0 〈 φ=0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
10−140−3 10−2 10−1 100 101 102 103 104 10−140−3 10−2 10−1 100 101 102 103 104
t t
t t t t t*
I a I a
FIG.4. ThecenterofmassMSDnormalisedbytime-delay,foranactivesystemunderFact =0.1atT =0.005, 0.01, 0.05, 0.1
(Pe = 40, 20, 4, 2), at the different densities given in the key. The P´eclet number decreases from one panel to the other. It
inducesastrongqualitativechangeinh∆r2cmi,seethetextforadetaileddiscussion. Thetwodashesinthefirstpanelrepresent
the ballistic dependence ≃ t2. The dashed line in the last panel is a guide-to-the-eye for the density dependence of the last
cross-over time-delay that increases weakly with φ. The horizontal dotted lines at long times correspond to thevalues of the
singledumbbelldiffusionconstantDA fromEq.(43). Theverticalblackarrowsindicatethesingledumbbelltime-scalestI and
ta,whilethered arrows indicatethesingle dumbbellcharacteristic timet∗,for eachcase. Inthecasewith Pe=4thetimest∗
and ta coincide. The curves for φ= 0 are obtained from Eq. (38), valid in the limit t≫ tI =0.1; for this reason they start
from the middle of thegraph.
ponential. We will come back to this issue below.) andsinglepassivedumbbellD (F ,T,φ)/Dpd depends
A act cm
Thenon-monotonicityofD asafunctionofT already onlyontheP´ecletnumber,asitdoesforthesingledumb-
A
discussed in Fig. 6 is confirmed by the data presentation bell problem. With this aim, we fix Fact and we vary T,
in Fig. 7, with the minimum situated around T 0.07. and the values Pe = 4, 20, 40, 66 in each panel are ob-
≃
In the right panel we observe the opposite behaviour in tained from three different combinations of Fact and T.
the ratio D (F = 0.1,T,φ)/D (F = 0.1,T,0), first In all panels the collapse of data is very good. Note the
A act A act
growing for increasing T to reverse its trend at around change in concavity of the collapsed data that occurs at
T 0.05 0.07. Consistently with the behaviour found Pe = 20. This value is relatively far from the transition
≃ −
in [39], there are temperatures such that the data for between homogeneous and segregated phases estimated
the above ratio cross each other when the density is in- in [25, 39], and the system configurations are still ho-
creased, see for example T = 0.01, 0.1 (or Pe = 2, 20). mogeneous, see the last panel in Fig. 2, though with a
The right panel in Fig. 7 also shows that a very small distribution oflocaldensities, φloc, with a certainwidth,
density can have relevant effects on the behaviour of the see Fig. 1.
diffusion coefficient. These results suggest
We have repeated this analysis for a stronger active
force and we found that the results are consistent, with DA(Fact,T,φ)=kBT fA(Pe,φ) (45)
a cross-overtemperature that growswith F σ , as pre-
act d
dictedbythesingledumbbellequation,thoughwecannot with f (Pe,0) = (2γ)−1(1 + Pe2/8) =
A
assert that the dependence be linear. D (F ,T,0)/(k T) and f a decreasing non-linear
A act B A
function of φ at fixed Pe. This relation is equivalent to
Next, we analyse in Fig. 8 whether the ratio of dif- DA(Fact,T,φ) fA(Pe,φ)
= . (46)
fusion coefficients of the active system at finite density D (F ,T,0) f (Pe,0)
A act A