Transport in active systems crowded by obstacles

Mu-Jie Huang,∗ Jeremy Schofield,† and Raymond Kapral‡

Chemical Physics Theory Group, Department of Chemistry,University of Toronto, Toronto, Ontario M5S 3H6, Canada

(Dated: July 8, 2016)

The reactive and diffusive dynamics of a single chemically-powered Janus motor in a crowdedmedium of moving but passive obstacles is investigated using molecular simulation. It is found thatthe reaction rate of the catalytic motor reaction decreases in a crowded medium as the volume frac-tion of obstacles increases as a result of a reduction in the Smoluchowski diffusion-controlled reactionrate coefficient that contributes to the overall reaction rate. A continuum model is constructed andanalyzed to interpret the dependence of the steady-state reaction rate observed in simulations onthe volume fraction of obstacles in the system. The steady-state concentration fields of reactant andproduct are shown to be sensitive to the local structure of obstacles around the Janus motor. It isdemonstrated that the active motor exhibits enhanced diffusive motion at long times with a diffu-sion constant that decreases as the volume fraction of crowding species increases. In addition, thedynamical properties of a passive tracer particle in a system containing many active Janus motorsis studied to investigate how an active environment influences the transport of non-active species.The diffusivity of a passive tracer particle in an active medium is found to be enhanced in systemswith forward-moving Janus motors due to the cooperative dynamics of these motors.

I. INTRODUCTION

It is well known that molecular crowding can signifi-cantly alter the static and dynamical properties of sys-tems.1–6 Since the interior of a biological cell is highlycrowded by various macromolecular species, numerousstudies of the effects of crowding on the structure andtransport properties in the cell have been carried out.These investigations have revealed the existence of a widevariety of crowding effects; for example, they have shownthat the mean square displacement of particles in thecell often adopts a subdiffusive character on intermedi-ate time scales and that diffusion coefficients may besubstantially reduced by crowding. In vitro experimentsand theoretical studies have also examined the effectsof crowding on diffusion and other transport properties,and a substantial literature exists that documents thiswork.7–11

Usually the crowding agents are assumed to be pas-sive objects which are subject only to thermal fluctua-tions due to the environment in which they are immersed.When the crowding agents are themselves active and canmove autonomously, one expects and finds that the ef-fects of crowding manifest themselves in different ways.Experiments on passive tracers in suspensions of swim-ming microorganisms have been carried out and showthat the diffusion coefficients of tracers are modified bythe flow fields generated by the swimmers.12–14 Calcula-tions based on hydrodynamic equations have been em-ployed to analyse the effects of these flow fields on tracerdynamics.15,16 In an analogous manner, the motions ofswimming organisms and particles are altered in systemswith passive obstacles, and these altered motions havebeen studied both experimentally and theoretically.17–19

In this article we present the results of investigationsof reaction and diffusion in two types of system: a dif-fusiophoretic Janus motor in a suspension of mobile but

passive spherical obstacles, and a passive spherical parti-cle in a suspension of chemically-powered Janus motors.This investigation has several features that distinguish itfrom other research on related systems. Our simulationsare carried out at a particle-based level that accountsfor the chemical dynamics giving rise to many-body con-centration fields, hydrodynamics fluid flow and thermalfluctuations. Since the active objects are propelled by adiffusiophoretic mechanism that involves chemical reac-tions occurring asymmetrically on the motor surface, thereaction rates that are responsible for motion of either themotor in the passive suspension or the active suspensionitself are sensitive to the extent of crowding. The chemi-cal gradients that give rise to motor motion are also thedominant factor determining the long range interactionsamong the motors. Hydrodynamic interactions play asmaller role than for the swimming miroorganisms dis-cussed above, where they provide the major mechanismfor the observed collective behavior.

The outline of the paper is as follows: a brief descrip-tion of the microscopic model for a chemically-poweredJanus motor immersed in a fluid of reactive particles andobstacles is given in Sec. II. The active component in themodel is based on the asymmetric catalysis of fuel par-ticles into product particles at the surface of the Janusmotor and the dynamics conserves the overall mass, mo-mentum and energy of the system while explicitly in-corporating the effects of thermal fluctuations and hy-drodynamic flow. In Sec. III, a coarse grain continuummodel for the system is presented and the reaction ratefor the conversion of fuel into product is derived and com-pared to simulation results in systems where the motor iscrowded by a variable number of passive spherical parti-cles. The structural and dynamical properties of a singleJanus motor in the crowded environment are also dis-cussed. In Sec. IV the effects of an active medium on thedynamics of a passive tracer particle is discussed. The

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conclusions of the paper are given in Sec. V.

II. DYNAMICAL MODEL

A coarse-grain microscopic description is used to in-vestigate the dynamics of the systems considered in thisstudy. We consider two types of systems: a single activeJanus motor in a solution crowded by mobile sphericalpassive particles, and a passive spherical particle in a so-lution crowded by active Janus motors. The evolutionof the entire system, active Janus motors, passive par-ticles and solvent particles, is carried out by moleculardynamics for solute particles in a binary solvent mediumof A and B particles governed by multiparticle collisiondynamics20–23.

u

A

BRC

N R↵A

FIG. 1. The Janus motor is composed of catalytic (C) andnoncatalytic (N) hemispherical surfaces. The chemical reac-tion, A → B, takes place on the C surface. The orientationof the Janus motor is indicated as u, where the radius of themotor is R and the collision radius with solvent particles oftype α is Rα.

The Janus motor is composed of catalytic (C) and non-catalytic (N) hemispheres (see Fig. 1). Whenever an Aparticle collides with the catalytic surface of the Janusmotor, a chemical reaction converting fuel particles A toproduct particles B takes place. No reactions occur onthe noncatalytic surface of the Janus motor and on thesurface of a passive particle. We adopt a hard collisionmodel, discussed in detail elsewhere,24 for the interac-tions of the solvent particles with Janus motor. Briefly,solvent particles of type α = A,B interact with the Janusmotor through hard potentials, WJα(r) =∞ for r < Rα,and WJα(r) = 0 for r ≥ Rα, where Rα denotes the colli-sion radius. We denote the larger of the radii by R, theeffective radius of the motor. While fuel and product par-ticles may interact with the Janus motors through differ-ent potentials, all solvent particles interact with a passiveobstacle with the same potential WP , where WP (r) =∞if r < R and WP (r) = 0 if r ≥ R. The solvent par-ticles undergo modified bounce-back collisions at radiiRα for the active Janus motors and at R for the pas-sive spheres, and after a collision the relative velocity be-tween the sphere and the solvent particles is completelyreversed. Activity of the Janus motors characterized by

directed motion along or away from the director u ariseswhenever RA 6= RB , with forward motion along the di-rector occurring when RB < RA

24. In addition to theconversion of A particles into B particles at the catalyticsurface of the Janus motor, bulk reactions in which Bparticles are converted back into A particles with rateconstant k2 are incorporated via reactive multiparticlecollision dynamic25. The bulk reactions allow the estab-lishment of a non-equilibrium steady-state with non-zeroconcentrations of fuel particles in the system.

While the interactions between solvent particles andsolute spherical particles are modeled using hard poten-tials, any two spheres interact with each other througha soft-repulsive Lennard-Jones potential, VLJ(r) =4ε[(σ/r)12−(σ/r)6+1/4], when their distance r < 21/6σ,where ε is the interaction strength. Further details of thesimulation method and the system parameters are pro-vided in the Appendix. Results are reported in dimen-sionless units that are also specified in this Appendix.

III. SINGLE MOTOR IN A PASSIVE,CROWDED MEDIUM

We first consider a single Janus motor operating in amedium with a variable number of Np passive obstacles(illustrated in Fig. 2(a)) to investigate how the catalyticrate of reactions occurring at the surface of the Janusparticle and the activity of the Janus particle depend onthe volume fraction of obstacles in the system.

(b)(a)

FIG. 2. An instantaneous configuration drawn from the simu-lation of (a) a single Janus motor moving in a passive medium(gray spheres) and of (b) a single passive sphere in an activeJanus-motor medium, where the light blue dots are productparticles. In panel (b) only 1/10 of the product particles areshown to aid visualization.

A. Reaction dynamics of a Janus motor

In the usual continuum description for Janus particlemotion the chemical concentration fields are describedby reaction-diffusion equations and the flow fields by theStokes equations. Under steady state conditions the ve-locity of the Janus motor along motor axis u is given

3

by24,26–28

Vu =kBT

ηΛ〈u ·∇scB(R, s)〉S , (1)

where s denotes coordinates on the surface with radiusR, the angle brackets signify a surface average, η is thefluid viscosity and Λ = 1

2 (R2A−R2

B) for our hard collisionmodel. A knowledge of the product (or fuel) concentra-tion field is needed to compute the Janus motor velocityand the structure of this field is determined by reactionson the Janus motor surface and in the bulk.

To gain insight into the reaction dynamics of the cat-alytic conversion of A particles to B particles on the sur-face of the motor, we consider the time evolution of theconcentration fields starting from an initial state con-taining all fuel. Under conditions where the velocity ofthe motor is small fluid advection is negligible and thetime-dependent fuel concentration cA(r, t) and productconcentrations cB(r, t) in a coordinate frame centered onthe Janus motor satisfy reaction-diffusion equations,

∂cA(r, t)

∂t= D0∇2cA(r, t) + k2cB(r, t),

∂cB(r, t)

∂t= D0∇2cB(r, t)− k2cB(r, t), (2)

where D0 is the common diffusion coefficient for the Aand B solvent species. The reaction-diffusion equationmust be solved subject to the radiation boundary condi-tion on the Janus motor at a radial distance r = R,

D0r ·∇cA(r, θ, t)∣∣r=R

=k0

4πR2cA(R, θ, t)Θ(θ), (3)

where k0 is an intrinsic rate constant for reaction on thecatalytic surface, and Θ(θ) is the characteristic functionthat is unity on the catalytic hemisphere (0 < θ < π/2)and zero on the noncatalytic hemisphere (π/2 < θ < π).In Eq. 3, the polar angle θ is defined relative to the orien-tational director axis u (see Fig. 1). Far from the Janusmotor the boundary condition is limr→∞ cA(r, t) = c0(t),where we have allowed for the possibility of time depen-dence in c0(t). It follows from the nature of the reactionat the Janus motor surface and the bulk reaction that thesum of the concentrations is fixed, and we assume thatthis relation holds locally, cA(r, t) + cB(r, t) = c0(t). Us-ing these conditions, we may focus on the solution of thereaction-diffusion equation for cB(r, t). It is convenientto Laplace transform the reaction-diffusion equation forthe concentration of product, leading to

(∇2 − ν2)cB(r, z) = 0, (4)

where cB(r, z) =∫∞0dt e−ztcB(r, t), cB(r, t = 0) = 0,

and ν2(z) = α2 + κ2 with α2 = z/D0 and κ2 = k2/D0.The solution for the Laplace transform of the concentra-tion of product particles is axisymmetric around u andis given by24

cB(r, θ, z) = c0(z)∑`

a`(z)f`(r, z)P`(µ), (5)

where P`(µ) is a Legendre polynomial with µ = cos θ,and the radial function f`(r, z) is given by

f`(r, z) =K`+ 1

2(νr)

√νr

√νR

K`+ 12(νR)

, (6)

where K`+ 12(νr) is a modified Bessel function of the

second kind. The coefficients a` can be determinedby solving a set of linear equations and are given bya`(z) =

∑m(M−1)`mEm, where

M`m =2Q`

2`+ 1δ`m +

k0kD

∫ 1

0

duPm(µ)P`(µ), (7)

Em =k0kD

∫ 1

0

duPm(µ), (8)

where Q` = νRK`+ 32(νR)/K`+ 1

2(νR) − ` and kD =

4πD0R is the Smoluchowski diffusion-controlled reactionrate coefficient29,30. In the limit z → 0 one recovers thesteady state concentration fields discussed earlier24, andusing these solutions the steady state Janus motor veloc-ity is given by Vu = kBT

ηc03RΛa1.

We are interested in the net rate of change of the num-ber of product particles NB(t) in the system at time tand this quantity can be obtained by integrating the timederivative of the product concentration cB(r, t) over thevolume outside the Janus motor. From Eq. (2), we findthat

dNB(t)

dt= D0

∫dr δ(r −R)r · ∇cA(r, t)

−k2NB(t), (9)

where integration by parts and the fixed sum of con-centrations has been used to evaluate the first term onthe right. The contribution to the reaction rate dueto chemical reactions on the Janus motor surface isRJ(t) = D0

∫dr δ(r − R)r · ∇cA(r, t), while k2NB(t) is

the contribution to the change in the amount of productdue to reactions in the bulk of the solution. Consideringthe Laplace transform of RJ(t) and using the solution ofthe reaction-diffusion equation (5), we obtain

RJ(z) = kD(1 + νR)a0(z)c0(z) (10)

≡ k(z)c0(z),

where the last line of this equation defines the z-

dependent rate kernel k(z) for the reaction on the Janusmotor. From Eq. (10) and the form of a0(z) one may

show that limz→∞ k(z) = k0/2 ≡ k0J , so that it is con-

venient to write k(z) = k0J + ∆k(z).The time-dependent rate follows from the inverse

Laplace transform of Eq. (10) and is given by

RJ(t) =

∫ t

0

dτ k(τ)c0(t− τ), (11)

and the corresponding time-dependent rate kernel canbe written as the sun of two terms,31 one of which is a

4

singular part proportional to k0J , the effective intrinsicrate constant for the Janus motor, and a non-singularcontribution, ∆k(t):

k(τ) = 2k0Jδ(τ) + ∆k(t). (12)

From this expression we see that the value of k0J can bedetermined from the initial value of the time-dependentreaction rate. Taking c0(t) = c0 to be constant, we findthat RJ(t) = kJ(t)c0, where the time-dependent ratecoefficient is defined as

kJ(t) =

∫ t

0

dτ k(τ). (13)

In the long time limit the system approaches a steadystate, and the corresponding rate coefficient approaches

limt→∞ kJ(t) = k(z = 0) ≡ kJ , the rate coefficient ob-served in steady-state conditions.

To gain additional insight into the structure of kJ , wemay approximate a0(z = 0) by a0(z = 0) ' E0/M00 =k0J [k0J + kD(1 + κR)]−1, to obtain,

kJ 'k0JkD(1 + κR)

k0J + kD(1 + κR), (14)

which reduces to the result of Collins and Kimball32 inthe absence of the bulk reaction converting B particlesinto fuel particles since κ = 0.

B. Simulation of reaction rate constants

Turning our attention to crowded systems, we considerthe rate at which product B particles are produced bythe catalytic reaction on the forward-moving Janus mo-tor where RB < RA. The steady-state reaction rateRJ = kJc0 can be computed by counting the total num-ber of fuel particles that enter the surface of the cat-alytic hemisphere per unit time since all of these parti-cles are converted to product particles in the irreversiblereaction on the motor surface. Comparisons of reactionrates can be made between active Janus motors and re-active but passive (non-propelled) Janus spheres whereRA = RB = R and no propulsion is possible. The resultsare listed in Table I for active (RAJ ) and passive (RPJ )Janus motors in a crowded passive medium for variousvalues of the volume fraction φ of obstacles. Althoughthese two reaction rates are within the statistical uncer-tainty, their means satisfy RAJ > RPJ . For the Janusmotor simulations, when φ < 0.182 the Peclet numberPe = Vu(φ)R/D0(φ) > 0.7 is non-negligible. In thisregime the concentration fields of the solvent species areperturbed by the directed motion of the motor and thisinfluences the rate of chemical reactions on the motorsurface.

In order to compare simulation results with the theo-retical expressions for the reaction rate and rate coeffi-cient, we assume that these quantities can be calculated

using the development given above in the absence of ob-stacles but that k0J(φ) and kD(φ) depend on the obstaclevolume fraction. The rate constant k0J can be determinedin simulations from the initial value of the reaction ratefor a pure system of fuel particles. For a system con-taining a single Janus motor, the initial rate k0J can beestimated from the average value of −(dNA(t)/dt)/cA(t)computed from the first ten MD steps over multiple real-izations of the dynamics starting from a pure solvent offuel particles in the system. In Table I, the rate constantk0J(φ) obtained in this manner for various values of φ islisted. The initial rate constant increases with volumefraction, which can be explained by the fact that in acrowded environment fuel particles are confined in thespace between obstacles so that initially fuel particlesare non-uniformly distributed with an enhancement offuel particles near the surface of the Janus particle. Thisenhancement in fuel particle density leads to a highercollision frequency with the catalytic hemisphere of theJanus motor relative to a system without obstacles inwhich the fuel particles are uniformly distributed.33 Itis well known that crowding can alter solvent diffusioncoefficients, so one might expect that the Smoluchowskicontribution kD to the rate coefficient kJ will acquireφ dependence through its dependence on the diffusioncoefficient, kD(φ) = 4πRD0(φ), where D0(φ) is the φ-dependent diffusion constant of the solvent in the pres-ence of obstacles.

The volume fraction dependence of the steady-statereaction rate, RTJ (φ) = kJ(φ)c0, can be computed usingthese results in the rate coefficient expression, kJ(φ) =kD(φ)(1 + κ(φ)R)a0, with a0 obtained by solving thefull set of linear equations combined with kD(φ) and

κ(φ) =√k2/D0(φ). The values of D0(φ), kD(φ) and

RTJ (φ) computed from simulations are listed in Table I.Since the expression for RTJ (φ) was derived under theconditions of vanishing Peclet number, it does not dependon the motor velocity and should be compared with sim-ulation results for the passive Janus sphere. The theoryunderestimates the simulation results. From the form ofRJ above Eq. (10) we see that it depends on the deriva-tive of the concentration field in the normal directionevaluated on the surface of the Janus sphere. Thus, itis sensitive to the structure of the fields in the bound-ary layer. Earlier investigations24 showed that there isa discrepancy between the concentration fields obtainedfrom simulation and continuum theory in the immediatevicinity of the Janus sphere, and this can account for theobserved difference in the theoretical predictions and thesimulation results.

C. Structural and dynamical properties of thecrowded active motor system

Structural and dynamical properties of the active mo-tor are also influenced by crowding obstacles in the sys-tem. To examine the influence of crowding on the struc-

5

TABLE I. Volume fraction dependent intrinsic rate constant k0J(φ), solvent diffusion constant D0(φ), and diffusion-controlledrate constant kD(φ). The steady-state reaction rate RJ as a function of volume fraction φ obtained from simulations for theactive (A) and passive (P) Janus motors and from theoretical estimates (T).

φ 0.03 0.061 0.091 0.121 0.151 0.182

k0J(φ) 16.38± 0.33 16.74± 0.64 17.17± 0.43 18.02± 0.28 18.41± 0.37 19.18± 0.66D0(φ) 0.069 0.068 0.067 0.066 0.065 0.064kD(φ) 2.168 2.137 2.111 2.081 2.051 2.025

RAJ (φ) 26.9± 1.5 26.7± 1.4 26.4± 1.5 25.8± 3.0 25.6± 1.5 25.2± 1.5RPJ (φ) 24.0± 1.3 24.0± 1.3 23.8± 1.3 23.5± 1.3 23.6± 1.2 23.5± 1.3RTJ (φ) 19.72 19.65 19.59 19.63 19.54 19.53

5 10 15 20r

4

6

8

10

12

n JΑ(r

)

5 10 15 20r0

0.5

1

1.5

2

2.5

3

g(r)

(a) (b)

FIG. 3. (a) The radial distribution function, g(r), betweenthe active Janus motor and the surrounding passive obstaclesand (b) the average density of the fuel particles as a functionof distance from the center of the active Janus motor in adilute (φ = 0.03, black solid curve) and a dense (φ = 0.18,red dashed curve) passive medium. The blue dashed curvesin (a) and (b) are for a passive Janus sphere in the densemedium.

tural properties of the motor system, we consider thedistribution of passive obstacles and fuel particles aroundthe motor. The radial distribution function of obstaclesaround the motor is,

g(r) =

⟨L3

4πr2NP

NP∑i=1

δ(rJi − r)⟩, (15)

where rJi = |rJ − ri| is the distance between the centerof the Janus motor and i-th the obstacle, and 〈· · · 〉 is thesteady-state ensemble average over trajectories generatedfrom the molecular dynamics. In Figure 3 (a), the radialdistribution g(r) computed from the simulations in dilute(φ = 0.03, black curve) and dense (φ = 0.18, red dashedcurve) media is shown. For a dilute system of obstacles,a sharp peak in the radial distribution function at r ≈ 6is evident followed by an uniform distribution of passiveobstacles beyond the distance r ≈ 10, whereas at thehigher volume fraction of obstacles the presence of anadditional peak at r ≈ 12 indicates long-range structuralordering of obstacles primarily due to packing effects.As a result of the packed configuration of obstacles, the

density of solvent species varies in the radial directionfrom the Janus motor. The steady-state density of fuelparticles around the Janus motor is given by,

nJA(r) =

⟨1

4πr2

NA(t)∑i=1

δ(rJi − r)⟩

(16)

where NA(t) is the instantaneous number of fuel particlesin the system, and rJi = |rJ −ri| is the distance betweenthe motor and the i-th fuel particle. In Figure 3 (b), wesee that the density of fuel particles for a dilute system(black curve) is lowest near the surface of the Janus motorat r ' R = 2.5 where the catalytic reactions occurs, andmonotonically increases to the asymptotic value c0(1−φ).However, as the volume fraction increases, obstacles forma packed configuration, and fuel particles are depleted atthe position occupied by the obstacles located aroundr = 6.

In addition, motor dynamics affects the structuralproperties of the surrounding obstacles and fuel parti-cles. This is evident in the plots of g(r) and nJA(r)for a passive Janus sphere, indicated by the blue dashedcurves in Figure 3. In panel (a), comparing the activeand passive Janus particles, the position of the peak forthe passive sphere lies at larger separations, indicating alooser structural ordering of the surrounding obstacles.Correspondingly, in panel (b) one sees that the steady-state spherically-averaged fuel density in the vicinity ofa passive Janus sphere is higher than that for an activemotor, indicating a lower steady-state reaction rate asobserved in Table I.

Dynamical properties of the Janus motor are also af-fected by the presence of obstacles. First, we investigatethe crowding effects on motor dynamics on different timescales by computing the mean square displacement of theJanus motor, ∆L2(t) = 〈|rJ(t)− rJ(0)|2〉. In Fig. 4, themean square displacement of the motor in a dilute (blackcircles) and a dense (red squares) system of obstacles iscompared. In going from a dilute to a dense crowdingmedium, the long-time diffusive motions are found to besuppressed, as indicated by the decreasing enhanced dif-fusion constants with volume fraction listed in Table II.The theoretical expression for the mean square displace-

6

TABLE II. . Volume fraction dependent quantities obtained from simulations, where Vu(φ) is the propulsion speed along themotor axis u, De(φ) is the enhanced diffusion constant of the motor, and D′(φ) is the diffusion constant of the motor withoutpropulsion.

φ 0.03 0.061 0.091 0.121 0.151 0.182

Vu(φ) 0.0276 0.0263 0.0247 0.0227 0.0206 0.0184De(φ) 0.18± 0.06 0.17± 0.06 0.13± 0.03 0.12± 0.04 0.10± 0.03 0.07± 0.02D′(φ) 0.0030± 0.0001 0.0029± 0.0001 0.0026± 0.0001 0.0024± 0.0001 0.0021± 0.00005 0.0018± 0.00003

ment in the absence of obstacles is of the form34

∆L2(t) = 6Det− 2V 2u τ

2r (1− e−t/τr )

− 6kBT

Mτ2v (1− e−t/τv ), (17)

where Vu is the propulsion speed of the motor along itsorientational axis, and τv and τr are the velocity andmotor orientation relaxation times, respectively. In thetime regime τv � t � τr, the motor moves ballisticallyand ∆L2(t) ≈ V 2

u t2, whereas when t � τr, the motion

of the motor is diffusive and ∆L2(t) = 6Det. Here theenhanced diffusion constant is given by

De = D′ +1

3V 2u τr, (18)

where D′ is the diffusion constant of a motor withoutpropulsion. To compare the theoretically-predicted formof ∆L2(t) with simulation results for crowed systems weagain assume a similar functional form applies but withparameters that depend on the obstacle volume fraction.Thus, for each φ, the simulation values of D′, Vu, τvand τr are required. The diffusion constant D′ can beextracted from the ∆L2(t) of a passive particle of ra-dius R = RA = RB in a crowded system with volumefraction φ, and Vu = 〈V(t) · u(t)〉 can be determinedby the average of the instantaneous motor velocity V(t)projected along the motor axis. The velocity relaxationtime is τv(φ) = D′(φ)M(kBT )−1 < 2.0 for all volumefractions considered in this study, and by computing theorientational correlation function of the motor axis fromsimulation data, 〈u(t) ·u(0)〉 = e−t/τr and fitting the cor-relation function with a single exponential, the long-timeorientation relaxation time is estimated to be τr ≈ 600,which is much larger than τv(φ). The orientational re-laxation time is independent of φ due to the fact that theinteractions between motors are described by central po-tentials so that no angular momentum exchange occursin motor-motor collisions. Using the simulation values ofthese quantities (listed in Table II) in Eq. (18), the pre-dicted mean square displacement of a Janus motor in adilute (φ = 0.03, red solid curve) and a dense (φ = 0.18,red dashed curve) passive medium are in good agreementwith simulation results, as shown in Fig. 4.

101 102 103 104

t10-1

100

101

102

103

104

∆L2 (t)

FIG. 4. Mean square displacement, ∆L2(t), of the activeJanus motor in a crowded medium with volume fractions φ =0.03 (black circles) and φ = 0.18 (black squares). The solidred (φ = 0.03) and dashed red (φ = 0.18) curves are thevalues of ∆L2 obtained from Eq. (17) using the values of thetransport properties listed in Table II.

IV. A SINGLE PASSIVE SPHERE IN ANACTIVE MEDIUM

In the previous section the effects of a crowded en-vironment of passive obstacles on a single active motorwere investigated. We now consider systems in whichthe crowding agents themselves are active by embeddinga tracer particle in an active medium of Janus motors (seeFig. 2(b)). It has been shown that a single Janus motorcan move in the forward (+u) or backward (−u) direc-tion depending on the interaction potentials of the Janusparticle with the fuel and product particles. The co-operative behavior of collections of forward-moving andbackward-moving Janus particles is quite different; forexample transient clusters arising from interactions me-diated by concentration fields are observed in a collectionof forward-moving Janus motors, whereas in a system ofbackward-moving motors no significant directional andorientational orderings were found.24 Consequently thecollective behavior of these two active systems and thedynamical effects of the active environment on a pas-sive tracer particle immersed in the active systems areexpected to be qualitatively different.

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TABLE III. Volume fraction dependent tracer diffusion constant in an active medium with forward-moving (DF ) and backward-moving (DB) Janus motors.

φ 0.03 0.061 0.091 0.121 0.151 0.182

DF (φ) 0.0045± 0.0015 0.0054± 0.0017 0.0064± 0.0019 0.0056± 0.0025 0.0071± 0.0029 0.0068± 0.0026DB(φ) 0.0040± 0.0009 0.0052± 0.0020 0.0052± 0.0012 0.0050± 0.0017 0.0056± 0.0019 0.0052± 0.0014

5 7.5 10 12.5r

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

CVV

(r)

5 7.5 10 12.5r

(a) (b)

FIG. 5. Velocity correlation, CV V (r), between the single pas-sive sphere and the surrounding forward-moving (black cir-cles) or backward-moving (red squares) Janus motors withvolume fraction (a) φ = 0.03 and (b) φ = 0.18.

sTo investigate the influence of active motors on the

dynamics of the tracer particle, we compute the meansquare displacement of a tracer particle at long times todetermine the diffusion coefficients DF and DB of thetracer particle in an active medium with forward andbackward moving Janus motors, respectively. In the sys-tem with forward-moving Janus motors, DF (φ) is foundto increase with φ. Using the data listed in Table IIand III, one finds, in going from low (φ = 0.03) to high(φ = 0.18) volume fraction, the ratio DF (φ)/D′(φ) in-creases from around 1.5 to 4, indicating stronger tracerdiffusivity enhancement at high volume fractions. Similartrends are found when the active motors are backward-moving. However, we notice that at low volume fractionsφ the tracer particle exhibits similar diffusivity regard-less the medium properties, whereas at larger φ one findsDF (φ) > DB(φ) indicating stronger interactions betweenthe tracer and the forward-moving motors.

To understand the activity-dependent enhancement oftracer diffusivity, the correlation function CV V (r) be-tween the velocity of the tracer and that of the activemotors is computed according to,

CV V (r) =

⟨ NJ∑i=1

(vT · vi)δ(rTi − r)⟩/n(r) (19)

where vT = vT /|vT | and vi = vi/|vi| is the unit vectorof the velocity of the tracer particle and of the i-th ac-

tive motor, respectively, and n(r) = 〈∑NJ

i=1 δ(rTi − r)〉 isthe average number of tracer-motor pairs with separation

rTi = |rT − ri| at r. In Figures 5(a) and (b) the veloc-ity correlation functions CV V (r) in dilute (φ = 0.03) anddense (φ = 0.18) media of forward-moving (black circles)and backward-moving (red squares) motors are shown.From these plots, it is evident that there is no signifi-cant correlation at any separation distance between thevelocities of the tracer and motor particles at low volumefractions. On the other hand at higher volume fractions,positive correlations in the tracer and motor velocitiesare observed at both short (r ' 5) and intermediate sep-arations (5 < r < 7.5) for the forward-moving motors, in-dicating that the tracer is moving along with the motors,whereas a decrease of CV V (r) at small separations is ob-served for the system with backward-moving motors. Inour previous study24 while no collective motion was seenfor a system of backward-moving motors, transient clus-ters of increasing size with volume fraction were observedin a system of forward-moving Janus motors. The re-sults of CV V (r) and the volume fraction dependent clus-ter size for forward-moving Janus motors suggest thatthe passive particle is encapsulated by and moves col-lectively with forward-moving Janus motors in a denseactive medium. This transport enhances the tracer dif-fusivity, and a larger effective diffusion coefficient is ob-served in the forward-moving system than that seen in amedium of backward-moving motors.

V. CONCLUSION

The dynamics and catalytic behavior of a Janus mo-tor in a suspension of passive obstacles was investigatedthrough simulations of a microscopic model. A contin-uum model for the reaction rate of the conversion of fuelto product particles at the catalytic surface of the Janusmotor was constructed and compared in the steady-stateregime with the reaction rate observed in simulations.The reaction rate is found to be influenced by the activityof the motor and by the packing structure of surroundingobstacles. The dynamics of the Janus motor was exam-ined from the simulation trajectories and the crossoverbehavior of the mean square displacement from ballisticto diffusive motion was observed in accordance with the-oretical models. The average propulsion speed of the mo-tor and the enhanced diffusion of the motor at long timesarising from its activity were found to be suppressed dueto crowding. In contrast, the diffusion of a passive tracerwas found to be enhanced in a crowded environment ofactive obstacles, with the strongest enhancements occur-

8

ring at higher volume fractions. The enhancement ofthe tracer diffusion was found to be larger in an activemedium of forward-moving motors than in backward-moving motors due to the formation of moving transientclusters in the former that carry the tracer particle in adirected fashion.

These observations suggest concentration-mediated in-teractions among diffusiphoretic Janus motors play animportant role not only in their collective behavior butalso in their ability to transport material. In particu-lar, our study demonstrates how motor dynamics in acrowded environment can facilitate the transport of pas-sive species such as macromolecules.

Appendix A: Simulation method and parameters

Simulations are carried out in a periodic cubic box withlinear size L. The system contains NS = NA +NB pointparticles of solvent with NA fuel and NB product parti-cles, and the mass of each particle ism. Janus motors andpassive particles are modeled as spheres of radius R, vol-ume VJ = 4

3πR3 and mass M = VJc0m with c0 = 10 the

number of solvent particles per unit cubic cell of linearsize a, and moment of inertia I = 2

5MR2. There are in

total N = NJ+Np solute particles, made up of NJ Janusmotors and Np passive spherical particles, leading to a so-lute volume fraction of φ = (NVJ)/L3. The total numberof solvent particles in the system is NS = c0(1−φ)L3. Tomaintain the system in a non-equilibrium steady state,the reactive multiparticle collision dynamics algorithm is

employed25, where the reaction, Bk2→ A, takes place in

the bulk solution with bulk reaction rate k2 = 0.01. Grid-shifting is employed to ensure Galilean invariance35,36.The time evolution of the entire system is carried outusing a hybrid MD-MPCD scheme.24

The results are presented in dimensionless units wheremass is in units of m, length in units of a, energies inunits of kBT and time in units of t0 =

√ma2/kBT . In

these units, we have R = 2.5, L = 60, M ≈ 655, andI ≈ 1636. The multiparticle collision time was set toτ = 0.1, with a molecular dynamics step size of δt = 0.01.The collision radii in the modified bounce back collisionwith a forward-moving Janus motors are RA = 2.5 andRB = 2.4 and with a backward-moving motor are RA =2.4 and RB = 2.5, whereas solvent particles interact withpassive particles through the same radius R = 2.5. Therepulsive interaction among spherical has the strengthε = 1 and distance σ = 6.

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(1949).33 In the absence of crowding, the intrinsic rate constant may

be estimated using simple kinetic theory as the productof the collision cross section and the mean speed of thefuel particles: k0J = R2

A

√2πkBT/m. In crowded systems

the collision rate depends on φ, so that the rate constantcan be approximated as k0J(φ) = p(φ)k0J where the factorp(φ) accounts for the modification of the collision rate due

9

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