Role of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers
RRole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers* [email protected]
Role of viscoelasticity on the dynamics and aggregation of chemicallyactive sphere-dimers
Soudamini Sahoo, a) Sunil Pratap Singh, b) and Snigdha Thakur* c) Department of Physics, Indian Institute of Science Education and Research Bhopal, Bhopal, India,462066
The impact of complex media on the dynamics of active swimmers has gained a thriving interest in the researchcommunity for their prominent applications in various fields. This paper investigates the effect of viscoelasticity onthe dynamics and aggregation of chemically powered sphere-dimers by using a coarse-grained hybrid mesoscopicsimulation technique. The sphere-dimers perform active motion by virtue of the concentration gradient around theswimmer’s surface, produced by the chemical reaction at one end of the dimer. We observe that the fluid elasticityenhances translational and rotational motion of a single dimer, however for a pair of dimers, the clustering in a particularalignment is more pronounced. In case of multiple dimers, the kinetics of cluster formation along with their propulsivenature are presented in detail. The key factors influencing the enhanced motility and the aggregation of dimers are theconcentration gradients, hydrodynamic coupling and the microstructures present in the system.
I. INTRODUCTION
Active swimmers have attracted tremendous interest in sci-entific community owing to their ability to exploit surround-ing energy for performing various biological and mechanicalfunctions . Such systems include inherently active unitswhich drive them out of equilibrium. In most of the casesthe communication between them sets up through a num-ber of complex signaling mechanisms like interactions amongneighbors , hydrodynamic interactions , chemotaxis ,quorum sensing , etc. A suspension of active natural en-tities like bacteria, spermatozoa and their synthetic analogueare known to display very intriguing phenomena like swarm-ing , pattern formation , vortex formation , surfaceaccumulation , phase transitions , etc. Such uniqueproperties make the individual as well as collective dynamicsof active particles very different from their passive counter-part . Fascinated by various peculiar phenomena in thesenatural microswimmers, a new realm of chemically pow-ered synthetic motors started over a decade ago , wherethe associated spacial-and temporal-asymmetry is the originof self-propulsion. A series of distinct schemes have beenevolved to design self-propelled synthetic motors, most com-mon among them is the phoretic mechanisms like diffusio-phoresis , thermophoresis , electrophoresis . These sys-tems also display nontrivial scenarios of an individual as wellas collective motion similar to the natural motors .Microswimmers encounter diverse fluidic environment dur-ing their course of motion . The comprehensive studies onthe effect of fluid reveal that the rheology of the surroundingmedium can strongly affect the motility of microsswimmers An increase in translational velocity but a decreasein the rotational diffusion in viscoelastic fluids compared toNewtonian fluids for the
E.coli has been established . It was a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] also demonstrated that the fluid elasticity of cervical mucuspromotes the collective swimming of sperms, which can con-tribute to an enhanced probability of successful fertilisation .In the context of artificial microswimmers, most of the stud-ies so far are limited to Newtonian fluid . A fewrecent investigations have demonstrated the role of viscoelas-tic , shear thinning , and shear thickening fluidson the self-propulsion. A report on diffusiophoretic activecolloid in a polymeric solution demonstrates restricted activemotion, i.e. , faster diffusion, lower motility, and shorter per-sistence length of directed motion . A mesoscale hydrody-namic simulation of a spherical squirmer immersed in a solu-tion of self-avoiding polymers shows a remarkable enhance-ment of the rotational diffusion of squirmer . This qualita-tively agrees to the observed experiments on the enhancementof rotational diffusivity of self-propelled Janus particle in vis-coelastic fluid .This article is an attempt to elucidate the collective behav-ior of an anisotropic active dimers in a complex medium.The self-propelled sphere-dimer comprises two connectedspheres, where one sphere consumes fuel in the environmentto generate a chemical gradient, and the other exploits thisgradients to perform directed motion. Our intention here is toinvestigate the role of viscoelasticity on the dynamics and ag-gregation of the active dimers. We use a mesoscopic simula-tion technique based on explicit solvent that provides strengthto probe the microstructures present in the system in great de-tail. This helps us to correlate the dynamics of the dimer withfluid micro-structural rearrangement in the course of its mo-tion. The diffusiophoretically active dimers are immersed in aviscoelastic solvent, which is modeled using finite-extensiblenonlinear elastic (FENE) dumbbells.We observe an enhancement in the translational and rota-tional motion of a single dimer in the viscoelastic fluid. Fur-ther, we demonstrate that the origin of such enhancement isrelated to various dynamical quantities such as microstruc-tural changes and hydrodynamic correlations present in theviscoelastic fluid. A substantial boost in the aggregation ofdimers is also noticed in the viscoelastic fluid as compared to a r X i v : . [ c ond - m a t . s o f t ] F e b ole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 2 ˆ RAI
A I
P R
FIG. 1. A schematic picture includes a dimer, few dumbbells, andpoints like solvent particles. The two big connected spheres repre-sent a dimer, where blue and orange spheres represent the active ( A ) and the inert ( I ) monomers, respectively. The green dumbbells andsolvent point-like spheres represent the reactant species ( R ) whereasthe red represent the product species ( P ) . its Newtonian counterpart. Systematic studies of aggregationare carried out by varying the dimer densities. The synergybetween the dimers emerges due to a combination of interac-tions like chemical gradient, hydrodynamic correlations, andfluid microstructures.The work is organized as follows. The model of a chem-ically active dimer along with the viscoelastic fluid and theirgoverning equations are described in Section II. Section IIIpresents the effect of medium viscoelasticity on the trans-lational and rotational dynamics of a single diffusiophoreticdimer, a pair-dimer, and the clustering dynamics in case ofmultiple dimers. Finally, the conclusions of the study aredrawn in Section IV. II. SIMULATION MODEL
The chemically active sphere-dimer consists of two identi-cal spheres of radii σ , connected by a harmonic potential , U d = κ d ( | R − R | − l d ) . Here, κ d , the spring constant ischosen in a way that the large fluctuations in the average bondlength of the dimer ( l d ) are suppressed. The dimer is embed-ded in a coarse-grained viscoelastic fluid mixture consisting ofpoint-like and dumbbell-like reactants ( R ) as shown in Fig. 1.The dimer is chemically active by virtue of one of the spheres,labeled as ‘ A ’ (Active), which takes part in a chemical reac-tion on its surface. The sphere ‘ A ’ catalyzes the irreversiblereaction R + A → P + A , whenever a reactant ‘ R ’ encounters‘ A ’ and leaves its boundary layer at r c = / σ . Notethat, when the point particle of a dumbbell-like reactant inter-act with A , the entire dumbbell get converted to a product typesolvent ( P ). The other sphere ‘ I ’ (inert) is chemically neutral.A schematic picture in Fig. 1 indicates the chemical reactionon ‘ A ’ that generates the product ‘ P ’ around its neighborhood. This results in an inhomogeneous distribution of ‘ R ’ and ‘ P ’around the inert sphere ‘ I ’.The interaction between the dimer and solvent moleculesare implemented through repulsive Lennard-Jones (LJ) poten-tial, U α S ( r ) = ε α S (cid:104)(cid:16) σ r (cid:17) − (cid:16) σ r (cid:17) + (cid:105) , r < r c (1) = , otherwise . Here, S = A or S = I and α = R or α = P to indicate differ-ent interactions between dimer and solvent. In particular, wetake ε RA = ε PA = ε RI = ε (cid:54) = ε PI , to characterize the energy pa-rameters between different pairs. This choice of the energyparameter and the asymmetry around the inert sphere is thekey ingredient for the self-propulsion of the sphere-dimer. Wechoose ε > ε PI to enable the dimer to move with the activesphere ( A ) at the front as explained in section III. For continu-ous propulsion of dimer, the continuous supply of reactant ‘ R ’shouldn’t be disrupted. Therefore, we perform a back conver-sion of ‘ P ’ to ‘ R ’, 4 σ away from every dimer’s center-of-massposition, to maintain the required concentration of ‘ R ’.In case of multi-dimers, the interaction between twomonomers separated by a distance r is implemented throughrepulsive Lennard-Jones (LJ) potential with interaction energy ε d and diameter 2 σ . The motion of a sphere-dimer and fluidparticles are updated by hybrid molecular-dynamics multi-particles collision (MD-MPC) technique .The fluid is modelled in the frame work of MPC dynam-ics , where usually the fluid particles are considered aspoint particles. To bring viscoelasticity in the medium, weconnect a few pairs of point-like solvent molecules through afinite-extensible nonlinear elastic (FENE) potential to makethem dumbbell-like, U ( r i , r i + ) = − κ R ln (cid:34) − (cid:18) r i − r i + R (cid:19) (cid:35) . (2)Here, κ is the spring constant and R is the maximum exten-sion of the spring. We consider total N s solvent point particlesin solution, out of which N b forms the dumbbells. The extentof viscoelasticity in the medium is altered by varying N b = N s . N b = N b = N s have only dumbbell-like particles resulting in a vis-coelastic fluid (read as 100% in figures). We also consider thefluid to be a mixture of point-like and dumbbell-like particles.We define f d = ( N b / N s ) ×
100 to represent the percentage ofpoint-like particles which form the dumbbells.The positions and velocities of the solvent particles are up-dated by the MPC rules, which involve streaming and colli-sion steps. The streaming step performed at every MD time( ∆ t ) comprises updating the positions and velocities of solventvia Newton’s equation of motion. The collision step is carriedout at a time interval of τ . Here all the solvent particles aresorted into a grid of cubic cells of size a where they interactwith other molecules. The relative velocities of the particles ( δ v i = v i − v cm ) with respect to the centre-of-mass velocity ( v cm ) of the cell are rotated about a random axis by an angleole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 3 γ . The velocity of the i th particle after collision is given as v i ( t + τ ) = v cm ( t ) + ℜ ( γ ) δ v i , where ℜ ( γ ) is the rotation ma-trix. Random grid shifting is performed before the collisionstep to ensure the Galilean invariance when mean free path ofthe solvent particles is smaller than the MPC cell size . Simulation parameters:
The physical quantities are re-ported in units of thermal energy k B T , mass of the solvent m , and the collision cell length a . Time is scaled in units of τ m = (cid:112) ma / K B T , velocity in units of a / τ m , the spring con-stants κ , and κ d are in units of k B T / a . The MD integrationis performed at a fixed time interval, ∆ t = × − . Othersimulation parameters are, radii of the dimer spheres σ = . m d ≈
320 (ensures the density matching with sol-vent), bond length of the dimer l d = .
78, and spring constant k d = ε RA = ε PA = ε RI = k B T , ε PI = . k B T and ε d = k B T . The MPC simulation parameters are, rota-tion angle, γ = ◦ , the average number density of the fluid ρ = / a , and the collision time τ is varied in the range of10 − to 10 − to ensure viscosity matching of different typesof fluids. Unless mentioned, the zero-shear viscosity of thefluid is η s ≈
30. All the simulations are performed in thecubical box of dimension L s = . R =
20 and κ = . . a . Results have been averaged overmore than 30 independent ensembles. The Reynolds number ( Re ) for the fluid lies in the range of 10 − to 10 − , similarly,Schmidt number ( Sc ) is in the range of 10 to 10 ensuring thefluidic behavior . We have considered three different con-centration of the dimers with N d = , ( − , − ) . In few results, thetime is scaled as t / τ s where τ s (= L d / (cid:104) V p (cid:105) ) is the time takenby the dimer to travel it’s own length ( L d ≈ .
0) in case of aNewtonian fluid.
III. RESULTS
The solvent properties are gradually varied from Newto-nian to viscoelastic by changing the fraction of dumbbell-likesolvent particle while maintaining a nearly constant viscosity.The physical quantities like translational, rotational motion ofthe dimers and solvent density profile contribute to a great ex-tent to our understanding of their dynamics and assembly.
Transport properties of fluid
Before exploring the dynamics of dimers in viscoelastic flu-ids, here we discuss some of the essential properties of thefluid consisting of FENE dumbbells. We consider fluids hav-ing different percentage of dumbbells and to maintain theirzero-shear viscosity, we tune the MPC collision time ( τ ). Thezero-shear viscosity η s is computed from the non-equilibriumsimulations by imposing linear shear flow where the stresstensor varies linearly with viscosity in linear-response regime( η s = σ xy / ˙ γ ) . This is calculated by using Lees-Edwards boundary condition for Newtonian and different viscoelas-tic systems where the applied shear rate is ˙ γ = . τ for Newtonian and vari-ous viscoelastic fluids which results in η s = TABLE I. A constant shear viscosity η s =
30 is maintained, whilevarying concentrations of dumbbells ( f d ), at the following choice offluid parameters. f d τ To quantify the elastic strength of the fluids for different f d ,we have computed the variation of storage modulus ( G (cid:48) ) asa function of f d . This is performed by imposing an oscilla-tory strain as, γ = γ sin ( ω t ) where γ is the strain amplitudeand ω is frequency of oscillation. The storage modulus ( G (cid:48) ) and loss modulus ( G (cid:48)(cid:48) ) can be measured in the simulations byfollowing the stress-tensor expression σ xy ( t ) = γ [ G (cid:48) ( ω ) sin ( ω t ) + G (cid:48)(cid:48) ( ω ) cos ( ω t )] . (3) G ′ f d -2 -1 -3 -2 G ′′ ω FIG. 2. Variation of storage modulus ( G (cid:48) ) for viscoelastic fluids hav-ing different f d with γ = 0.2 and ω = 0.1. Inset: Loss modulus ( G (cid:48)(cid:48) ) as a function of ω for a viscoelastic fluid ( f d = γ = 0.2.The black dashed line is a fit of G (cid:48)(cid:48) = η s ω , η s = The storage modulus is identified at sin ( ω t ) = ± cos ( ω t ) = ±
1. The mea-sured values are shown in Fig. 2, where the storage modulusinitially increases with f d following a saturation for f d ≥ G (cid:48)(cid:48) ) with ω . Asexpected in regime of ω < . G (cid:48)(cid:48) follow a linear behaviorand the viscosity calculated in this region ( η s = G (cid:48)(cid:48) ( ω ) ω ≈ G (cid:48) and G (cid:48)(cid:48) , 40 or more cycles are consideredand for each ensemble for better statistics.ole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 4 Case I: Single active dimer
The self-propulsion of chemically active sphere-dimerhas been extensively studied in the context of Newtonianfluid . The chemically active sphere ‘ A ’ converts reac-tants ‘ R ’ species to the product ‘ P ’, thereby generating a sym-metric distribution of ‘ R ’ and ‘ P ’ species around ‘ A ’. How-ever, the diffusion of solvent species produces a concentrationasymmetry of ‘ R ’ and ‘ P ’ in the vicinity of inert sphere ‘ I ’(see Fig. 1). The combination of non-equilibrium concentra-tion gradient and the interaction potential difference of ‘ R ’ and‘ P ’ with ‘ I ’ causes the directed motion of the dimer, whichis also known as self-diffusiophoresis . The direction ofthe dimer’s motion depend on the choice of ε and ε PI . For ε > ε PI , the inert sphere experience a net force from the sol-vent in a direction of ˆ R AI , therefore it moves with A sphere atthe front end. In this part, our goal is to highlight the effect ofviscoelasticity on the propulsion dynamics of a single dimer. 〈 V p 〉 f d 〈 V p 〉 f d FIG. 3. The average directed velocity (cid:104) V P (cid:105) of the dimer is plotted forvarious f d when the entire reactant dumbbell take part in the chem-ical interaction with ‘ A ’ . The inset shows the same but when onlythe reactant molecule linked to the dumbbell interact with ‘ A ’ andconverts to ‘ P ’. To quantify the translational motion of a single dimer, wecalculate directed speed of the dimer along its axis ( ˆ R AI ) , de-fined as V p = V cmv · ˆ R AI where V cmv is center-of-mass velocity.Figure 3 shows the variation of (cid:104) V P (cid:105) with increasing dumbbellpercentage ( f d ). The averaging ( (cid:104) (cid:105) ) is performed over timeand ensembles. It is quite evident that the viscoelasticity of themedium enhances the average directed velocity of the dimerto approximately twice as compared to the Newtonian fluid.Moreover the directed velocity exhibits a non-linear increasewith the elasticity of the medium.As mentioned in the section II, when a reactant point parti-cle linked with a dumbbell interact with ‘ A ’, the entire dumb-bell get converted to ‘ P ’ type solvent. Hence, it is expectedto have higher reaction rate with increase of f d . To observethe solo effect of the viscoelasticity over simple fluid, wecompute directed speed for the same reaction rate. Here, the ρ p0 θ R P
A I θ ◦ ◦ FIG. 4. The normalised density of P around the inert sphere fordifferent kinds of fluid (semi-log in y-axis). The inset shows aschematic picture defining θ . chemical reaction converts to ‘ P ’ only that part of dumbbellwhich physically interacts with ‘ A ’. We found directed speed (cid:104) V P (cid:105) = . f d =
100 which isnearly 50% increase in (cid:104) V P (cid:105) compared to the Newtonian fluidfor the same viscosity (see inset of Fig 3). This clearly indi-cates the effect of elasticity on the enhancement of directedspeed of the dimer. Henceforth, for the simplicity we willstick to the case where the entire dumbbell is involved in thechemical reaction (see Fig. 1).It is established that for diffusophoertic motion, the concen-tration gradient of the fluid around the dimer plays a key rolein dictating its propulsion behavior . Therefore, to com-prehend the translational motion of the dimer, we computethe concentration gradient of product species ( P ) around inertsphere ( I ). Figure 4 shows the normalised local density, ρ p = ρ p / ρ ( ρ being the bulk density) of P around the inert sphere.For θ ≥ ◦ , the variation in ρ p with θ exhibits a faster decayfor f d = ρ p in inset of Fig. 5. The vari-ation of ρ p in a XY plane of thickness 2 a passing through thecenter of the dimer is shown in the color-map. A higher degreeof asymmetry around the dimer, in case of viscoelastic fluid,is obvious from the colormap. Important to note here that vis-coelastic medium has a steeper gradient around inert sphere( I ), which might be responsible for the enhanced directed ve-locity. Similar microstructural rearrangements of fluid ele-ments has also been addressed in our previous work .The local environment may impact the rotational motion ofthe dimer along with translational motion. To quantify therotational motion, we calculate the orientational correlationfunction, C φ ( t ) = (cid:104) ˆ R AI ( t ) · ˆ R AI ( ) (cid:105) . Figure 5 compares C φ ( t ) of the dimer for various f d . It is evident from the correla-tion that the rotational dynamics of dimer is faster with theincreasing degree of elasticity in the medium. The orienta-ole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 5 -1 C φ ( t ) t/ τ s -6 -3 0 3 6X-6-3036 Y AI (b) -6 -3 0 3 6X-6-3036 Y AI (a) FIG. 5. The relaxation of orientation vector as its auto-correlationfunction ( C φ (t)) of the dimer in different kinds of fluid with viscosity η s =
60 (semi-log in y-axis). Inset: The density of P solvents arounda dimer in a plane of thickness 2 a passing through the dimer. tional relaxation time τ R , obtained from the exponential decayof C φ ( t ) ∼ e − t τ R , is shown in Table II. Such enhancement inrotational dynamics of active particles in viscoelastic fluid hasalso been observed in recent studies . TABLE II. The scaled orientational relaxation time τ R / τ s with f d . f d τ R / τ s Case II: A pair of active dimers
We intend to present a detailed study of the collective dy-namics of chemically active dimers in viscoelastic medium.Before attempting the multi-dimer case, we first report thecharacteristic dynamics of a pair of dimers, which will en-lighten our understanding of collective dynamics. The inter-dimer center-of-mass position separation ( r d ) with time ismeasured in Fig. 6 for a viscoelastic fluid with f d = r d ≈ r d < r d )and alignment angle ( φ ). The probability distribution P ( r d ) for the dimers to be at a distance r d for both Newtonian (0%)and viscoelastic (100%) fluid is shown in Fig. 7. A peak in P ( r d ) at r d ≈ = σ is observed in both cases. This peak cor-responds to the structure, where the center of the two dimersare at their diameter distance apart like in the inset (b) of (a) (b) A AI I
A AI I r d t/ τ s FIG. 6. Variation of the distance between two dimer’s center-of-massposition ( r d ) with time for three different simulation realisations. Theinset shows the observed self-assembled configurations in case of apair of dimers in the viscoelastic fluid. Fig. 6. Notice that the height of the peak is much stronger incase of the viscoelastic medium. Further, the Newtonian fluiddoesn’t have any cross configurations with r d < .
0, whereasits viscoelastic counterpart does possess such configurations.As seen from Fig. 6, the observed cross configurations in vis-coelastic fluid is a short-lived which finally leads to a stableanti-parallel configuration at r d ≈ P ( r d ) r d P ( φ ) φ FIG. 7. The probability distribution function of dimer’s separation r d for both Newtonian and viscoelastic fluids. The inset contains theplot for the probability distribution of angle between the dimer’s axis( φ ) for r d ≤ a . The local ordering of dimers, quantified with an averageangle φ between them when r d ≤
5, provides more insightinto these configurations. The inset of Fig. 7 illustrates theprobability distribution ( P ( φ ) ) of angle φ . Two peaks in theviscoelastic fluid at φ ≈ ◦ and 90 ◦ , indicate the preferenceole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 6of dimer pair to be in an anti-parallel and cross configura-tions, respectively, substantiating the main plot of Fig. 7. Thestrong peaks at φ ≈ ◦ indicate that the dimer pair favorsthe anti-parallel configuration for both types of fluid. Suchanti-parallel alignment can be interpreted in terms of the ef-fective attractive interaction between the active sphere ( A ) ofone dimer with the inert sphere ( I ) of other, mediated by theproduct particles . This effective attraction arises due to thechoice of ε > ε PI . In a system of multiple dimers an inertsphere will always get influenced by the presence of a chem-ical gradient irrespective of which active sphere produces it.Therefore, the I spheres gets influenced by both A s simultane-ously. Which results in a net attraction to the two dimer andhence an anti-parallel configuration.The distribution of φ confirms the absence of intermediatecross configuration in the Newtonian fluid. The above resultsdemonstrates that the presence of elasticity in the mediummodifies the effective interactions leading to a stronger self-assembly.Next, we focus on the dynamical behavior of the activepair-dimer. An interesting dynamical feature due to the cor-relation between the dimer pair is captured in the mean-squared-displacement (MSD), (cid:104) ∆ R ( t ) (cid:105) = (cid:104) ( R ( t ) − R ( )) (cid:105) ,where R ( t ) is the centre-of-mass position of the dimer at time t . Typically, (cid:104) ∆ R ( t ) (cid:105) ∝ t β where β = β > β > . Figure 8 presents MSD for a crossconfiguration ( r d < r d ≈ r d > β > r d > r d < r d ≈
4) the transport of dimer pair exhibit a diffusive behaviorwith β = -2 -1 -2 -1 〈 ∆ R 〉 t/ τ s r d < 3r d ≈ d > 5 FIG. 8. The average mean-squared displacement of a dimer in differ-ent structures of cluster in viscoelastic fluid.
To unravel the MSD behavior of pair assembly, we take acloser look at the density distribution of the product P aroundthe pair. It is quite evident from the color-map (inset of Fig. 9)that the asymmetry of the product ( P ), essential for the propul- sion of dimers is lost as soon as the dimer-pair assemble in ananti-parallel configuration, leading to their diffusive behav-ior. However, the cross configurations are able to maintainthe product ( P ) asymmetry (due to the arrangement of chemi-cally active sphere ( A ) on one side) and, hence exhibit directedmotion. This color-map further validates by the mean normal-ized product density ( ρ p ) in the vicinity of inert sphere ( I )(see Fig. 9). It is apparent that for anti-parallel configuration( r d ≈ P diminishes due to thepresence of other active sphere ( A ). ρ p0 θ r d < 3r d ≈ d > 5 -6 -3 0 3 6 9X-6-30369 Y A I I A -6 -3 0 3 6X-6-3036 Y A I I A (b) (a) FIG. 9. The normalised density ρ p of P around the inert spherefor different spatial distances in the viscoelastic fluid. Inset: Thecolor map for the density distribution of product particles around thedimers for viscoelastic fluid. (a) Top view of a cross aggregation, (b)anti-parallel structure. To summarize, the concentration gradient of product par-ticles facilitates a long-range attraction among the inert andactive spheres of different dimers. When the dimers cometogether by long-range attraction, they arrange themselves intheir favorable anti-parallel configurations. However, in somecases before achieving the anti-parallel configuration a short-lived cross configuration is also observed in the viscoelasticfluid.
Case III: Collection of dimers
The obvious extension of the work is the investigation ofcollective dynamics in both media. For this, we consider N d =
28 corresponding to a dilute solution of packing frac-tion 0 . A ) and inert ( I ) spheres ofdifferent dimers results formation of clusters. Similar interac-tion drives the clustering in the multiple dimers (see Fig. 10a-c). There are two noteworthy points here. Firstly, the inertspheres ( I ) are attracted towards active spheres ( A ) of otherdimers by virtue of the concentration field of ‘ P ’. And sec-ondly, the individual dimer likes to maintain it’s propulsionby keeping its ‘ A ’ sphere at the front while moving. The com-petition between these two factors results in aggregates whereole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 7 FIG. 10. A few representative clusters of (a) 3 dimers, (b) 4 dimersand (c) more than 6 dimers. Blue sphere- Active sphere, Orangesphere- Inert sphere. the active spheres ( A ) are often outward. Such structures arevery dynamic and are quantified in terms of the distribution ofcluster size and number of clusters.To quantify the aggregation, its kinetics is monitored bycomputing total number of clusters ‘ n ’ over time ( t ). A dimeris part of a cluster if it falls within a cut-off distance r c = . n as a function of time for the vis-coelastic fluid is revealed in Fig. 11. Initial increase of thenumber of clusters ( n ) is by virtue of the long-range attractionamong the dimers which is further followed by the coales-cence of clusters resulting in the decrease of ‘ n ’ in the long-time limit. In addition, we calculate the average number ofdimers in a cluster (cid:104) N p (cid:105) , defined as (cid:104) N p (cid:105) = (cid:104) N m / n (cid:105) where N m is the total number of dimers that are part of any cluster. n t/ τ s 〈 N p 〉 t/ τ s FIG. 11. The kinetics of total number of clusters ( n ) as function ofscaled time t / τ s . The inset shows variation of average cluster size( (cid:104) N p (cid:105) ) with time at f d =
0% and 100%.
For the viscoelastic fluid, the inset of Fig. 11 shows an in-crease in (cid:104) N p (cid:105) for t τ s >
1. However, before that, the (cid:104) N p (cid:105) re-mains roughly at two, indicating that even though the numberof clusters in the system ‘ n ’ is increasing, most of them areclusters with dimer pair. Beyond this time, the increase in (cid:104) N p (cid:105) , along with the dynamics of ‘ n ’ implies the evolution oflarger clusters. In contrast to this, the plateau in both ‘ n ’ and (cid:104) N p (cid:105) for the Newtonian fluid implies the existence of dimerpairs only for the mentioned simulation range.To probe the pronounced clustering of dimers, we calculatethe probability of finding a cluster of size N p . It is clear fromFig. 12 that for the viscoelastic fluid larger clusters exists inthe system with substantially higher probability, which is notthe case for Newtonian fluid even-though the fluids are at vis-cosity.Further, we quantify the spatial correlation of dimers bycalculating the radial distribution function defined as, g ( r ) = V ( r ) (cid:68) N d ∑ i (cid:54) = j = δ ( r − r i j ) (cid:69) . (4)Here r i j is the distance between center-of-mass of dimers and V ( r ) is the volume of the thin concentric spherical shell of in-ner radius r and outer radius r + δ r where δ r is 0 . a . The ra-dial distribution function displays various peaks, specificallyfor the viscoelastic fluid attributing to the presence of largeclusters as illustrated in inset of Fig.12. Therefore, the obser-vations related to ‘ n ’, P ( N p ) , and g ( r ) indicate that the elasticnature of the fluid promotes aggregation of active dimers.It is important to emphasize here that once the aggregatesare formed the propulsion efficiency of clusters starts to decay.To quantify this, we compute the mean-squared displacementof a single dimer with N d = V ( N p ) as a function of cluster size ( N p ). Figure 13 dis-plays the MSD of a dimer in both kinds of fluid. Apparently,the dynamics of the dimer on average gets slower in presenceof many dimers, which is consistent for both medium. It isnoteworthy that dynamics of single as well as pair dimer isfaster in case of viscoelastic fluid than the Newtonian, whichis analogous to our previous observations . However, in thecase of many dimers, the MSD is relatively slower for vis- -3 -2 -1
2 4 6 8 10 12 14 16 18 P ( N p ) N p g ( r ) r FIG. 12. The probability distribution of different cluster sizes in vis-coealstic and Newtonian fluid. The inset shows the radial distributionfunction ( g ( r ) ) of system for both kinds of fluid. ole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 8 -2 -1 -2 -1 〈 ∆ R 〉 t/ τ s (1, 100%)(1, 0%)(28, 0%)(28, 100%) FIG. 13. The MSD of a dimer with N d = N d =
28 in both kindsof fluid. The index ( N d , f d ) represent the number of dimers ( N d ) andthe percentage of the solvent used for viscoelastic fluid elements ( f d )in the system. The solid-lines illustrates the superdiffusive behaviorwith exponent β = .
45 and 1 . N d =
28 and N d =
1, respectively. coelastic fluid. The slow down of dynamics can also be seenin Fig. 14 in terms of the cluster speed. Here, V ( N p ) = | V clcmv | and V clcmv = ∑ Npi = V i , cmv N p is the center-of-mass velocity of thecluster of size N p . Figure 14 shows that the speed of clusterdecreases as the cluster size grows, and it reaches an asymp-totic value beyond N p >
12. In the larger cluster, all the inertspheres aggregate together, thereby suppressing the concen-tration asymmetry around the cluster resulting in the sloweraverage speed. The concentration of the product particles ( P )around a dimer which is part of a cluster is shown in the insetof Fig. 14. It is evident from the color-map that the concen-tration symmetry around the dimer has been re-established bythe virtue of clustering, and hence a significant reduction incluster speed as displayed in Fig. 14.To conclude this part, we observe a drastic enhancement inthe aggregation of active dimers induced by the fluid elasticity.Such clustering re-enforces the concentration symmetry in theaggregate that eventually leads to slow motility of the dimers. IV. CONCLUSIONS
We have presented a systematic study of the dynamics ofchemically active motors with the help of a coarse-grainedhybrid MD-MPC simulation in the viscoelastic and Newto-nian fluids. The motors are modelled as sphere-dimers withone end as a catalytic sphere which consumes fuel and gen-erate product particles, whereas the other end responds to thegenerated product gradient thereby propelling the dimer. Wehave shown an enhancement in both translational and rota-tional motion of a single dimer in the presence of viscoelastic-ity. This is attributed to the higher concentration asymmetryand the microstructural rearrangement in the vicinity of thedimer. -6 -3 0 3 6 9 X -6-3036 Y A I V ( N p ) N p FIG. 14. The variation of average speed of a cluster with cluster size N p . Inset: The distribution of product solvent ( P ) around a dimer,which is a part of a cluster with N p = Further, we have presented the assembly and dynamics ofa pair-dimer. The pair-dimer assembles in anti-parallel andcrossed configurations with much higher probabilities in vis-coelastic fluid. The spontaneous assembly of pair-dimer re-sults in the diffusive motion for anti-parallel configurationwhile the cross configuration still exhibits the self-propulsivedynamics with nearly same speed as the single-dimer. Theobserved self-assembly and its dynamics can be understood interms of the inter-dimer long-range attraction induced by theself-generated concentration gradient. In the case of multipledimers, we have demonstrated kinetics of cluster formationexhibiting higher probabilities in viscoelastic fluid. Such ag-gregation however, results in slower translational dynamics ofthe cluster, which can be linked to the loss of fluid-asymmetryaround the dimers, required for propulsion.In summary, our work emphasized the relationship betweenthe micro-structural relaxation of the fluid to the motility, self-assembly and the other dynamical responses of the active col-loids. The present work will be helpful in providing insightsof the dynamics for various types of microswimmers drivenby concentration, temperature or electric field gradients in acomplex medium. In general self-propelled bodies encounterseveral types of complex environments during their course ofmotion . Understanding the impact of the complex sur-rounding media on their dynamics is an exciting domain of therecent research society. A plausible extension of the presentwork could be the investigation of effect of viscoelasticity onmicroswimmers in the presence of other external factors likefluid flow, effect of confinement, and mixture of active andpassive colloids.
ACKNOWLEDGMENTS
The computational work was performed at the HPC fa-cility in IISER Bhopal, India. ST and SPS (grant no:ole of viscoelasticity on the dynamics and aggregation of chemically active sphere-dimers 9YSS/2015/000230) acknowledge SERB, DST for funding.
Data Availability:
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