Scaling of cluster growth for coagulating active particles

Scaling of cluster growth for coagulating active particles

Peet Cremer Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany    Hartmut Löwen Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany
July 12, 2019

Cluster growth in a coagulating system of active particles (such as microswimmers in a solvent) is studied by theory and simulation. In contrast to passive systems, the net velocity of a cluster can have various scalings dependent on the propulsion mechanism and alignment of individual particles. Additionally, the persistence length of the cluster trajectory typically increases with size. As a consequence, a growing cluster collects neighbouring particles in a very efficient way and thus amplifies its growth further. This results in unusual large growth exponents for the scaling of the cluster size with time and, for certain conditions, even leads to “explosive” cluster growth where the cluster becomes macroscopic in a finite amount of time.

64.75.Xc, 82.70.Dd

I Introduction

Phase separation of a homogeneous state into two distinct bulk phases is not only relevant for many technological processes, but also constitutes a classical problem of nonequilibrium statistical mechanics Binder and Stauffer (1976); Onuki (2002); Tanaka (2000); Löwen (1997). For ordinary fluids and solids, the separation process is usually triggered by an initial fluctuation from which a critical nucleus arises. This initial cluster grows and ripens according to different scaling laws Krug (1997). Typically, the extension of the cluster increases with a power law of time, where the exponent depends on the growth process and the dimensionality of the system. For ordinary (passive) systems, varies in the range between and Bray (2002). More recently, both for mesoscopic colloidal suspensions Aarts et al. (2005) and for complex plasmas Wysocki et al. (2010), the phase separation process has been studied by observing the individual particle trajectories, giving insight into the microscopic (i.e. particle–resolved) mechanisms of the separation process Ivlev et al. (2012).

While the physics of the phase separation processes is by now well–studied and understood for inert, passive particles, there is recent work demonstrating that similar separation and clustering processes occur for an ensemble of microswimmers. The latter can be regarded as active particles in a solvent (experiencing a Stokes drag) with an internal propulsion mechanism. In fact, there are widely different realizations of such active particles, ranging from swimming bacteria to artificial self–propelled colloidal particles Vicsek and Zafeiris (2012); Romanczuk et al. (2012); Cates (2012).

Basically, two different separation processes in active systems occur. First, clustering can be purely motility induced Cates and Tailleur (2013), such that it vanishes if the self–propulsion is removed as recently demonstrated Palacci et al. (2013); Peruani et al. (2006); Ishikawa and Pedley (2008); Wensink and Löwen (2008); Redner et al. (2013a); Buttinoni et al. (2013); Bialké et al. (2013); Fily and Marchetti (2012); Yang et al. (2008). The simplest variant is a swarm of self–propelled particles resulting in an overall moving cluster. Second, there is already phase separation in the unpropelled, passive system which is then altered due to the drive. This was considered e.g. for a self–propelled Lennard–Jones system with attractive particle interactions Theurkauff et al. (2012); Grégoire et al. (2003). Attraction can hardly be avoided in metal–capped colloidal swimmers due to the mutual van–der–Waals forces Theurkauff et al. (2012). However, for active particles, the dynamical evolution of cluster growth, as characterised by a nontrivial growth exponent , has only rarely been considered apart from very recent studies Redner et al. (2013a, b); Wysocki et al. (2013); Méhes et al. (2012); Peruani and Bär (2013).

In this paper, phase separation is investigated in a situation, where active particles irreversibly coagulate with each other on contact, resulting in a compact aggregate. Irreversible coagulation is well understood for passive particles Meakin (1984); Meakin et al. (1985a, b); Kolb et al. (1983); Meakin et al. (1986) and the scaling of cluster size with time has been studied as well Trizac and Hansen (1996); Carnevale et al. (1990); Kolb (1984). We show here that activity of particles enables qualitatively different and novel cluster growth behaviour. Due to the self–propulsion, clusters perform a persistent random walk Soto and Golestanian (2014) in contrast to the typical diffusive motion of passive particles. This allows a cluster of active particles to effectively “sweep up” smaller clusters, which self–accelerates and amplifies cluster growth considerably further. Our theoretical analysis and computer simulation show that the cluster growth scaling exponent can not only be considerably larger for active particles than the known values for passive particles, but that there is even a scenario of “explosive” cluster growth. We refer to the term “explosion” if the cluster reaches macroscopic size in a finite amount of time. Such explosive behaviour was found earlier in the context of gelation kinetics (see, e.g. Ben-Naim and Krapivsky (2003); Herrero et al. (2000); Singh and Rodgers (1996); Dongen and Ernst (1988)) and in phase separation in external fields, like gravity Falkovich et al. (2002).

In detail, the growth exponent depends on the scaling of the total propulsion force of a cluster with its size, the persistence of the cluster trajectory and the dimension . We present several cases for the scaling of the total propulsion force of a cluster, which is determined by the type of swimmer, the fraction of particles contributing to the propulsion and the alignment of particles. If the cluster is driven by aligned surface particles only, in dimensions we find up to exponential growth. Uncorrelated contribution of all particles leads to algebraic growth with up to . Finally, if all particles in the cluster propel the cluster in the same direction “explosive” growth becomes possible. These results apply for the case that clusters possess a compact structure. Additionally, we also consider the case where the growing cluster is fractal and discuss briefly the scaling implications on the growth laws. All our predictions are verifiable in experiments for self–propelled particles with very strong van–der–Waals attraction, e.g. as prepared in Ref. Theurkauff et al. (2012), phoretic attraction Palacci et al. (2013) or dipolar interaction Baraban et al. (2013).

Ii Scaling theory

We perform our scaling theory in a general –dimensional space () and assume that self–propelled particles irreversibly coagulate and form clusters with member particles and radius such that . In the following, we refer to as the cluster size. The cluster formation process is described in a simplified way insofar as we consider compact clusters only and distinguish between different extreme cases. Once the particles contribute to the cluster, they stay fixed and their direction of self–propulsion (or orientation) is frozen. One may therefore distinguish two basic cases, one, where all orientations of cluster particles are completely uncorrelated and another where all directions are perfectly aligned. The first case occurs if the orientational reordering is frozen–in during coagulation (as realised for rough spheres) while the latter case arises if there is a considerable alignment interaction during the coagulation process (as realised for example for rod–like artificial swimmers or bacteria). The next basic distinction concerns the particles which really contribute to the overall self–propulsion of the cluster. Here we also discuss two extreme cases: either all cluster particles contribute in the same way or only particles at the cluster boundary contribute. The first case is realised for two–dimensional catalytic swimmers on a substrate which are embedded in a bulk liquid such that there is enough fuel all over the cluster. It also occurs for coagulation of passive colloidal particles in gravity Falkovich et al. (2002). The second case of cluster surface activity is realised for three–dimensional catalytic swimmers where a fuel–depletion zone is created inside the cluster which reduces the push of inside–particles Paxton et al. (2004); Ebbens et al. (2012); Golestanian et al. (2007). Moreover, catalytic swimmers move along the gradient of the chemical which also results in surface activity of the growing cluster Paxton et al. (2004); Ebbens et al. (2012); Golestanian et al. (2007). Surface cluster activity also occurs due to hydrodynamics for pushers and pullers. When swimming in a tight formation, the propulsion of particles can be cancelled by the flow created by the swimmers behind them. Consequently, only the particles in the rear of the cluster contribute to the total propulsion force Cisneros et al. (2007), which again scales with the surface of the cluster.

A single swimmer is propagated formally by an internal force 111It has been argued that real swimmers are force free and therefore do not directly feel a Stokes drag. However, the equations of motion can be expressed by a formal force which is proportional to the Stokes drag, see Golestanian (2008). Therefore, all scaling relations are unaffected by assuming an effective Stokes drag propulsion force. which is compensated by the Stokes drag at low Reynolds number resulting in a constant propagation velocity . All these individual forces () add up to give the total force acting on the cluster of size and putting it into motion with a velocity . This force is balanced by the Stokes drag acting on the cluster which scales in both and Kim and Karrila (1991) as . This after all yields different scalings for with a nontrivial exponent such that


We now focus more on the individual forces which constitute . As discussed before, a fraction of the particles in a cluster can be rendered inactive, implying for all inactive particles. Apart from this we assume an additional overall reduction of the nonvanishing with the cluster size. We describe this reduction by assuming a further scaling law with a general exponent . The exponent vanishes for pushers and pullers Cisneros et al. (2007); Behkam and Sitti (2008) and for surface tension driven self–propelled droplets Thutupalli et al. (2011) 222 Individual surface tension driven swimmers have a propagation velocity which scales with the square of the radius Thutupalli et al. (2011). If these droplet particles merge to a larger droplet when clustering, case LABEL:sub@fig.TotalForceScaling_all_align is realised and this scaling carries over to a cluster such that . This leads to . However, if the clustering cannot be associated with a merging of the droplet particles, they behave like pushers or pullers. The latter are known to realise case LABEL:sub@fig.TotalForceScaling_surf_align with the scaling Cisneros et al. (2007) and thus again possess by definition. . However, there are other situations where the effective individual forces of contributing particles depend on the cluster size such that an overall reduction is relevant. Nontrivial values for can be estimated by relating the scaling of the velocity of an individual particle with its radius to the scaling of with via Eq. (1). Phoretic particles in are propelled by a gradient generated on surface sites and their velocity is usually independent of the particle radius in three dimensions Golestanian et al. (2007). Ideally, the contributions of surface sites are aligned parallel and add up. Hence, should not depend on in this situation which yields . Likewise, the velocity of phoretic particles in a fuel–scarce environment is known to depend inversely on the particle radius Ebbens et al. (2012), implying for aligned surface contributions and thus .

Figure 1: Four cases for the scaling exponent of the total propulsion force with cluster size in spatial dimensions. The arrows denote the directions of single particle contribution forces.

Let us discuss the previously introduced four cases (see Fig. 1) in more detail. For each of the four cases, one can simply compute the exponent for any prescribed as follows: We define a further exponent which measures the particles contributing to the cluster propulsion such that . Insertion into Eq. (1) yields . Contribution of all particles in case LABEL:sub@fig.TotalForceScaling_all_align means that , while the case LABEL:sub@fig.TotalForceScaling_surf_align where only surface particles contribute corresponds to . Random alignment of the particles imposes a factor leading to in case LABEL:sub@fig.TotalForceScaling_all_random and in case LABEL:sub@fig.TotalForceScaling_surf_random. These exponents are included in Fig. 1.

We now introduce a simple sweeping argument for active particles leading to scaling laws for the cluster size as a function of time . Consider a typical cluster of size travelling through the system which has a uniform number density of particles on average, no matter whether they are members of small clusters or noncoagulated, individual particles. Therefore, any inhomogeneities and local fluctuations in the particle and cluster distribution are neglected 333These approximations can be abandoned in a more sophisticated Smoluchowski coagulation equation approach Dongen and Ernst (1988), which leads to the same scaling laws. Still, scaling itself is an assumption in the Smoluchowski approach which demands further numerical tests.. If denotes the volume in -dimensional space which is covered by a cluster of size moving with velocity during a time , we assume that all individual particles in this volume are irreversibly swept by the cluster. Differentially in time this implies


Two limiting cases can be discriminated. In the so–called ballistic regime, the persistence is so high that the cluster trajectory appears straight on the length scale the cluster possesses itself such that the rate of the swept volume is . This will occur in any case if the cluster becomes so large that rotational diffusion is suppressed Vicsek et al. (1995); Zheng et al. (2013). In the complementary case the cluster moves diffusively. Then the effective diffusion constant of a random walk with step velocity scales as , such that the volume swept out is given by Veshchunov (2011) . Insertion into Eq. (2) yields ordinary differential equations for leading to our main result:


for the ballistic regime, and


for the diffusive regime, where is the initial cluster size and is a positive amplitude prefactor. The last case of Eq. (3) corresponds to an explosive growth scenario, where the cluster size diverges after a finite time . Please note that is never realised in the diffusive regime, as the cluster size would explode, which necessarily puts the system into the ballistic regime. When measuring size in terms of the cluster radius the algebraic growth exponents of are given by in the ballistic regime and in the diffusive regime, which can be very large when is close to but below .

Iii Simulation

Using computer simulation, we investigate the cluster growth for various values of and different persistence lengths of cluster trajectories. The scaling of the total cluster force with an exponent from Eq. (1) is an input in the simulation. Nevertheless, the final scaling of the cluster size with time as predicted by Eqs. (3) and (4) is not an input but an output. Therefore, this final scaling behaviour is tested by our simulations. Moreover, the crossover to a possible ultimate scaling for finite clusters can be addressed and computed in a simulation.

In detail, the particles and compact clusters in these simulations are modelled as spherical droplets with radius , so that the total volume of all member particles is conserved. Initially, single particles start at random positions in a periodic simulation box with velocity and random direction. Particle collision events are predicted and on contact, particles merge at their centre of mass, forming larger clusters which again merge when colliding. The velocity of clusters is assigned to . To model changes in the travelling direction of clusters in a general way, we use the following approach. After a reorientation time step , a deviation from the current cluster direction is sampled for each member particle and the new cluster direction is taken as the average of all the member particle deviations. Then the collision events for the new time step are predicted. When two clusters merge, we weight each cluster with its number of member particles in the direction of the merged cluster. Since the averaging process is a biased random walk, the persistence length of cluster trajectories increases with cluster size . We sample the direction deviations of each particle from a von Mises–Fisher distribution Fisher et al. (1987) with concentration parameter , which is used as an input parameter. This distribution plays the role of the normal distribution on the –dimensional unit sphere and is similar to an inverse variance and determines the persistence of the trajectories such that we refer to as the persistence parameter in the following. Both, the ballistic and diffusive regime, can be gained as extreme limits or .

The single particle radius defines the length scale in the system, while the time a single particle requires to travel its own radius is used as time scale. We chose , which is sufficiently small when using a collision event prediction scheme. The packing fraction is taken as with initial particles. We vary the scaling exponent of the total propulsion force as well as the persistence parameter in dimensions. Since at late times the system is depleted of particles and the anticipated scaling laws clearly cannot be observed any more, we terminate the simulation as soon as a cluster reaches a size of , where is the box length. Typical snapshots from a simulation are shown in Fig. 2.




Figure 2: Typical simulation snapshots at various times, here in a two–dimensional simulation of total particles at packing fraction with and persistence parameter . Snapshot LABEL:sub@fig.snapshot_2148 shows the situation just before the simulation is terminated.

Figures (a)a and (b)b show the evolution of the mean cluster size in two– and three–dimensional simulations in the diffusive regime () for various values of . Fits of with Eq. (4) possess the predicted algebraic scaling. Data in the ballistic regime are shown in Figs. (c)c and (d)d. The predicted scaling exponent for the algebraic growth is verified for all values of .

For the case of , the predicted exponential growth in both regimes is reproduced by the simulations and the prefactor in the ballistic regime is significantly larger than in the diffusive regime, see the slope of the semilogarithmic plots in Fig. (e)e. The slope of the plot for steadily increases until it reaches the level for as clusters in this system need to grow first to enter the ballistic regime. Finally, for in three dimensions explosive cluster growth is documented in Fig. (f)f which confirms the predicted scaling .

Figure 3: Cluster size evolution obtained from simulations in dimensions with various values of the persistence parameter and the total propulsion force scaling exponent . Algebraic growth in the diffusive regime LABEL:sub@fig.DiffusiveAlgebraic_2d, LABEL:sub@fig.DiffusiveAlgebraic_3d as well as in the ballistic regime LABEL:sub@fig.BallisticAlgebraic_2d, LABEL:sub@fig.BallisticAlgebraic_3d occurs with the predicted exponents as indicated in the plots. Exponential growth LABEL:sub@fig.Exponential_3d in the ballistic regime occurs faster than in the diffusive regime as indicated by the much higher slope. For high persistence and high force scaling, explosive cluster growth occurs LABEL:sub@fig.Explosive. The dashed lines are fits using Eqs. (3) and (4) respectively.

Iv Fractal aggregates

Our results can be further extended to clusters which lack the reorganisation mechanism leading to a compact shape. When particles and clusters simply stick to each other on the first point of contact, the resulting shapes possess a ramified and fractal structure Meakin (1984); Kolb et al. (1983).

The size of such aggregates can be conveniently described by the radius of gyration which replaces and is approximately proportional to the hydrodynamic radius van Saarloos (1987); Lattuada et al. (2003). A structure with fractal dimension () then implies the scaling . Therefore, the analogue to Eq. (1) for fractal clusters is


We have performed additional simulations implementing irreversible sticking of particles at the point of contact. Equation (5) is taken into account by assigning the cluster velocity to . Therefore, the radius of gyration of each cluster has to be tracked throughout the simulation. Apart from this, the simulation follows the same procedure as in III. Figure 4 shows typical snapshots confirming the ramified structure of the aggregates. We have determined the fractal dimension from the simulation data for the cluster structure. Results for are presented in the legends of Fig. 5.




Figure 4: Typical snapshots of a two–dimensional simulation of fractal clustering using the same parameters as in Fig. 2, except for the persistence parameter which is now chosen to be .

In addition to the drag, the radius of gyration also determines the collision cross–section of the cluster. Applying the same simple sweeping argument used in II, we obtain for the rate of the swept volume in the ballistic regime and in the diffusive regime. Obviously, the sweeping volumes of compact clusters are recovered when setting . Insertion into Eq. (2) then yields the scaling relations


for the ballistic regime where the abbreviation is used. The critical time for explosive growth is here. Note, that for these results are indistinguishable from Eq. (3) as does not depend on for .

For the diffusive regime we obtain the scaling law


with . Contrary to the behaviour in the ballistic regime, the threshold value for corresponding to exponential growth is raised as compared to compact clusters. Similarly, the algebraic growth exponents are lower for the same value of .

Figure 5: Fractal cluster size evolution obtained from simulations in dimensions with various values of the persistence parameter and the total propulsion force scaling exponent . For in the ballistic regime LABEL:sub@fig.BallisticAlgebraic_2d_fractal, the same scaling laws hold as in the compact case. Accordingly, the threshold for exponential growth in the ballistic regime LABEL:sub@fig.Exponential_2d_fractal remains at . In the diffusive regime for LABEL:sub@fig.DiffusiveAlgebraic_2d_fractal the fractal dimension enters the scaling laws and leads to slower growth as compared to the compact case. Thresholds in for explosive growth are lower for so that such growth behaviour occurs already for when LABEL:sub@fig.Explosive_fractal. The dashed lines are fits using Eqs. (6) and (7) respectively.

Computer simulation results verifying the growth behaviour for in the ballistic regime are shown in Fig. (a)a for the case of algebraic growth as well as in Fig. (b)b for exponential growth. In fact, for , the algebraic growth exponents and the threshold for exponential growth are the same as in the compact case. Conversely, for , Fig. (d)d shows explosive growth at high persistence not only for but also for , confirming the reduced threshold. Finally, the prediction that the algebraic growth exponents in the diffusive regime are lower than in the compact case due to the influence of the fractal dimension is confirmed in Fig. (c)c, where results for algebraic scaling in the diffusive regime for are shown.

V Conclusions

In conclusion, we have investigated the scaling of cluster size with time for active particles in a solvent that irreversibly coagulate on collision by using theory and simulation. The scaling laws heavily depend on the scaling exponent of the total propulsion force of clusters. We identify four main scenarios for the total propulsion force scaling. If all particles in a cluster are aligned and able to contribute, the fastest growth is possible. Completely uncorrelated directions of particles lead to a significantly weaker scaling. Furthermore, hydrodynamics, fuel scarcity or lack of a field gradient required for propulsion can lead to the situation that only particles on the surface of a cluster can contribute. These contributing particles can again be completely aligned or their directions can be completely uncorrelated. The scaling of the total propulsion force is then further modified by the details of the propulsion mechanism and the thereby implied scaling exponent of the single particle contribution force with the size of the cluster.

Another crucial ingredient is the persistence length of cluster trajectories. In the diffusive regime, the persistence length is much smaller than the extension of clusters. Clusters explore the system volume and encounter each other on a diffusive time scale. More efficient growth occurs in the ballistic regime applying for a persistence length much larger than the cluster extension, where clusters sweep through the system volume on their semi–ballistic trajectories. The ballistic regime should be more relevant for active particle clusters since the persistence length of trajectories of active particles is usually rather large and tends to increase with aggregate size.

We have verified these predictions in a simulation of compact clusters modelled as droplets that merge on contact in dimensions. The simulation data shows good agreement with the model scaling laws and gives the correct algebraic growth exponents or exponential growth corresponding to the various values of in both regimes.

Additionally, we extended our model to fractal clusters which show a different growth behaviour due to increased drag (hampering growth) and increased collision cross–section (enhancing growth). In the ballistic regime, the increased collision cross–section dominates the increased drag, leading to faster growth. However, in the diffusive regime, the increased drag dominates, resulting in a comparatively slower growth.

Given a sufficiently strong attraction between particles leading to irreversible coagulation, our findings are verifiable in experiments Theurkauff et al. (2012); Palacci et al. (2013); Baraban et al. (2013). Usually it is attempted to avoid attractions like van–der–Waals attraction appearing in metal–capped active particles. However, by intentionally enhancing the attraction to a level where particles cluster irreversibly the prerequisites of our theory can be met.

We thank Kurt Binder, Thomas Speck, Akira Onuki and Hajime Tanaka for helpful discussions. Financial support from the ERC Advanced Grant INTERCOCOS (Grant No. 267499) and the newly founded DFG Science Priority Program SPP 1726 is gratefully acknowledged.


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