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Published : 03.04.2016 | Author : admin | Categories : Astrology Cancer
Soft Condensed Matter, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands. In recent years a significant amount of research has been focused on active matter systems, whose individual units are able to convert internal energy or energy from the local environment into their own motion (see ref.
However, even though experimentalists gain increasingly better control over the realization of active systems, the physics community lacks a fundamental understanding of the laws that govern their collective behavior. A simple step towards complexity is the addition of attractive interactions between the particles, which was shown to also lead to clustering of particles.
However, the transition from homogeneity to clustering in active attractive systems has not yet been clearly linked with the known phase behaviour of the corresponding equilibrium systems.
In the present work we study a three-dimensional model of self-propelled Brownian particles that interact via a Lennard-Jones potential.
Moreover, we find that the interplay between attraction and self-propulsive motion in three dimensions gives rise to a highly dynamic, percolating network. In Section II we describe the model, the dynamics implemented for our simulations and the analysis methods used in the subsequent parts of the article. In this paper we examine the behaviour of self-propelled, attractive particles immersed in a solvent. The axis of self-propulsion is subject to rotational diffusion and in our simulations its motion always obeys the overdamped rotational Langevin equation (6)where Dr denotes the rotational diffusion coefficient and the random vector satisfies relations analogous to eqn (3) and (4). The behavior of the system was probed as the following dimensionless parameters were systematically varied: the strength of the Lennard-Jones potential = ??, the magnitude of the propulsion force = ?Fp?, the rotational diffusion coefficient Dr? and the density of the system . For sets of parameters where the system is close to crystallization, the system was additionally initialized from a gaseous state that contains a large (face-centered-cubic) crystalline cluster.
We used a Voronoi construction to construct local density histograms of the system.40 By calculating the volume of the Voronoi cells we were able to estimate the local density of particles.
In order to distinguish the percolating network state from the bulk gas–liquid coexistence region we used the following criterion.
Second, we calculated the probability of having a cluster percolating simultaneously in all three dimensions in the system. When the passive particles are replaced by active particles by introducing self-propulsion, deviations from the equilibrium behaviour are expected.
For all values of Dr? we consider, we find that the system forms a homogeneous fluid for a sufficiently low value of (see Fig.
In order to better quantify the aggregation of particles, we obtained density histograms for the systems we examined.
Setting the rotational diffusion coefficient equal to the value dictated by the Stokes–Einstein relation, namely Dr? = 3, results in the state diagram shown in Fig. Interestingly, the stability domain of the percolating network increases monotonously with decreasing rotational diffusion coefficient.
One of the most striking differences between a gas–liquid coexistence state and the novel percolating network state is the compactness of the dense clusters. In the fast rotational diffusion regime, the liquid cluster is indeed compact, resulting in a small surface-to-volume ratio, which decreases further with increasing system size, as expected.
To gain further insight into the properties of the percolating network state, we study the pairwise correlations between particles. Additionally, we calculate the normalized orientation correlation function C1(r), defined as (8)where the prime on the summation sign indicates the terms for which i = j are not included.
In light of the above conclusions, we propose here a possible mechanism that accounts for the formation of the percolating network state. Now, if we increase the system density and instead of a fixed direction of self-propulsion we allow the particles to slowly rotate (corresponding to a low rotational diffusion coefficient), then this argument should continue to hold, e.g. As Dr? increases, the persistence length of the self-propelled motion of the particles decreases. In the present work, we employed computer simulations to study the self-assembly of a system of self-propelled Brownian particles that interact via the truncated and shifted Lennard-Jones potential. We determined state diagrams of the overdamped system for various rates of rotational diffusion of the self-propelled motion of the particles. We subsequently discussed the unique properties of the percolating state, and presented evidence of a transition from gas–liquid to a percolating network with decreasing rotational diffusion.
Finally, we noted that the ratio of the strength of attraction over the magnitude of self-propulsion does not solely characterize our system. Last but not least, as demonstrated in Section IIIC, the exact dynamics of a theoretical model are of importance not only for the quantitative but also for the qualitative results it generates. Administrators can allow users to update their own information in Active Directory without the interference of help desk or admin. Administrator can customize and assign a unique Self Update layout consisting of selected fields to a particular group or OU or to the entire domain.Most essential fields of personal and official information can be set mandatory so that end users can not neglect those fields. We examine the state diagrams of the system for different rotational diffusion coefficients of the self-propelled motion of the particles. Moreover, structural properties of the clustered state have not been examined and compared to the well-studied gas–liquid phase separation.
The choice of the potential qualitatively accounts for the steric repulsion and the short range attraction that are present in many colloidal systems. As we will show in this paper, this percolating network has many similarities to living clusters, observed in ref. In Section IIIA we present the state diagrams of the system, and in IIIB we focus on properties of the percolating network state.
We consider a three-dimensional system, consisting of spherical particles (colloids) in a periodic cubic box of length L. We use two distinct expressions to describe the translational motion of the individual colloidal particles inside the solvent, namely the underdamped and overdamped Langevin equations. A time step of dt = 3 ? 10?5? was used for the numeric integration of the equations of motion and the simulations ran for at least 106?, so that we get sufficient statistics for the system.
Nevertheless, by following the evolution of the total potential energy of the system with time, we observed that after a short transient period there was no energy drift. First, we considered two particles as clustered when their center of mass distance was less than 1.2?. To determine whether a cluster percolates in a given direction we duplicated the system in that direction, doubling the number of particles.
To explore these deviations, we study the behaviour of the system as a function of the rotational diffusion coefficient Dr? while keeping the self-propulsion force and temperature fixed.
We found that in most cases the density histograms transitioned from a unimodal to a bimodal curve as the strength of attraction is increased, a transition that indicates passing from a homogeneous state into a coexistence state.
2 Local density histograms for a system with total density ??3 = 0.381 and magnitude of self-propulsion = 50.
3 (a)–(d) State diagrams of the active Lennard-Jones system with rotational diffusion coefficients Dr? = 30, 9, 3, and 0.3 respectively. 3(a), for a rotational diffusion coefficient Dr? = 30 (ten times larger than the value dictated by the Stokes–Einstein relation), the behaviour of the system is very similar to the phase diagram seen for passive systems. In a gas–liquid coexistence state, the system evolves in order to minimize the surface area of the cluster, resulting in compact spherical or cylindrical geometries. 4 Average ratio of the number of surface particles NS over the number of particles NV in the biggest cluster of the system, as a function of the rotational diffusion coefficient Dr?.
5 Radial distribution function g(r) (a), and normalized orientation correlation function C1(r) (b) for different values of the rotational diffusion coefficient. Consider a dilute system of self-propelled particles with an attraction strength at least strong enough to cause gas–liquid phase separation in the absence of self-propulsion. In this case, the attractive force is able to aggregate particles with larger differences in the orientation of the axes of self-propulsion.


6(a) we plot the degree of clustering ? as a function of the rotational diffusion coefficient Dr? at constant Pagg near the percolating network to gas–liquid transition.
We found that for fast rotational diffusion, the properties of the state diagram bore strong similarities to the phase diagram of the equilibrium Lennard-Jones system. By examining the correlations between the orientations of the axes of self-propulsion of the particles, we found significant local alignment in the percolating state. First, we note once more that this state is caused by the synergy between attraction and self-propulsion, so we expect it to be present in three dimensional systems for a wide variety of attractive potentials and propulsion mechanisms. Detailed comparisons between different theoretical models, as well as actual experimental systems, such as ref.
For fast rotational diffusion, the state diagram exhibits a strong similarity to that of the equilibrium Lennard-Jones fluid.
By tuning the rotational diffusion rate of the particles, we are able to continuously move the system from the regime of fast rotational diffusion, where strong similarities with the equilibrium behavior are expected,35 to small values of the diffusion rate where non-equilibrium features arise.
The position of the center of mass of the ith particle at time t is given by the vector i(t). The underdamped Langevin equation is given by (2)where m is the particle's mass, ? is the damping coefficient and Fp denotes the magnitude of the self-propelling force. Nevertheless, the rotational diffusion coefficient is considered as an independent parameter in our study, similar to previous theoretical work.35–37 The reason for this extra degree of freedom is that individual particles in experimental active systems, such as bacterial colonies,15 are often subject to athermal rotational diffusion. We observed high crystallization and melting barriers in many cases, which caused difficulties in identifying the true state of the system, as the simulations would have to run for a very long time. At high temperatures, Lennard-Jones systems exhibit a single first-order phase transition from a fluid to a face-centered-cubic crystal as the density of the system is increased.
Note that in the limit of fast rotational diffusion, Dr? > ?, the persistence length of the particles goes to zero and the active force acts effectively as translational diffusion.35 As a result, we expect the behaviour in this limit to coincide with the behaviour in the equilibrium (passive) system, but with a modified interaction strength.
In (a) the system is in a homogeneous fluid state, in (b) there is liquid–gas coexistence, in (c) a percolating network state is found, and in (d) a crystal coexists with a gas.
The system transitions with increasing attractive strength from a homogeneous fluid state to a gas–liquid coexistence state and eventually to a gas–crystal coexistence state. This new state consists of a dynamic network of clustered particles coexisting with a gas phase and in static images resembles an equilibrium system which has undergone spinodal decomposition (see the ESI† Movie S1). Here, we find that the region where the percolating network state occurs is increased at the expense of the gas–liquid coexistence region. The percolating network has now completely replaced the gas–liquid region – according to our simulations, the system transitions directly from the percolating network state to a gas–crystal coexistence state.
3 was robust with respect to the system size, we simulated a few state points for larger and smaller systems, consisting of N = 21952 and 2197 particles respectively. In the percolating network, the active system appears to almost attempt to maximize the surface area, resulting in a highly branched network. 5(a), we plot the radial distribution function g(r) for four different values of Dr?, at the same density and interaction strength as was used for Fig.
This function is equal to unity if all the axes of self-propulsion of particles are aligned, and equal to zero if all particle orientations are uncorrelated. Now assume that the magnitude of self-propulsion is stronger than the attractions > , and that the axis of self-propulsion associated with each particle is pointing in a fixed, random direction (Dr? = 0).
This results in highly dynamic aggregates with groups of particles frequently attaching and detaching, and neighbouring particles displaying high degrees of orientational correlation as seen in Fig. This process ultimately leads to a transition to the (bulk) gas–liquid phase coexistence region (Fig. To examine whether this collapse also occurs in our system, we calculated the degree of clustering ?, introduced in ref. However, as the rotational diffusion was decreased, new features arose due to the interplay between self-propulsion and attraction. A possible mechanism was proposed in order to explain the formation of the percolating network. Moreover, our simulations suggest that this novel state is present for low density systems and experimentally relevant rotational diffusion, so a search for this state in real colloidal systems is feasible. 31, are thus extremely valuable and needed, in order to deepen our understanding of active matter systems.
If an employee changes his phone number or moves to some other location, his new information has to be updated in Active Directory. As we decrease the rotational diffusion coefficient, the state diagram is slowly transformed. By varying the strength of the attraction and the swimming velocity of the particles, they showed that the self-propulsion can have two opposing effects for a given strength of attraction – for slow swimmers it can break aggregations caused by the attractive force, while it can induce aggregation for fast enough swimmers. Thus, we are able to construct a series of state diagrams that evolve from a diagram similar to the well-established Lennard-Jones phase diagram to diagrams with novel properties. To each particle i, we associate a three-dimensional unit vector i(t) that identifies the direction in which the self-propelling force propels the particle at a given time.
Note that is a unit-variance random vector, with mean value and variation (3) (4)where 3 is the unit matrix in three dimensions. 33, we quantified the ratio between the strength of attraction and the magnitude of self-propulsion by the aggregation propensity (7) The number of particles for all simulations in the underdamped regime was 1728, in order to compare directly our results with ref. When the probability of percolation was found higher than a certain threshold that was density-dependent, we identified the system as being in the percolating cluster state. Upon lowering the temperature, a critical temperature is reached where a second phase transition appears separating the fluid phase into gas and liquid phases. The subfigures (a)–(d) correspond to different rotational diffusion coefficients as indicated. Data points correspond to local maxima of density histograms, which we identify as the local densities ?l of the coexisting phases. Moreover, the binodal envelope is similar to that of the equilibrium system, in the sense that the value of attraction alone dictates the densities of the two coexisting phases.
However, in contrast to such a state, the percolating network we observe is clearly not kinetically trapped (see the ESI† Movie S2). Additionally, in the percolating network region, attraction does not solely dictate the densities of the coexisting states anymore. One way to characterize this difference is by looking at the ratio of surface to volume of these large aggregates.
For Dr? = 30, the crystal and the gas coexist, for Dr? = 9 and 3, the liquid and the gas coexist and for Dr? = 0.3, the system is in the percolating network state. When two particles collide there are two possible scenarios – if their axes of self-propulsion are pointing in a similar direction, then the attraction will cause them to aggregate and travel together. That is, a new state was observed between the fluid phase and gas–liquid coexistence, which we identified as a highly dynamic, percolating network state. If it is not done, co-workers may retrieve the incorrect information during a needy situation and results in unnecessary burdens.
Specifically, the liquid–gas coexistence region is gradually replaced by a highly dynamic percolating network state.
For intermediate swimming velocities, the steady state of the system was identified as a homogeneous fluid phase. This new state is accompanied by an unexpected result – the emergence of local alignment of the axes of self-propulsion of the particles despite the absence of an aligning mechanism. The forces that appear on the right-hand side of eqn (2) are, from left to right, the force due to particle interactions, the drag force, the self-propelling force and a stochastic force. Nevertheless, the results obtained for the parameter space points presented in this article have been thoroughly verified.
This threshold was used for the necessary distinction between strong fluctuations of the liquid phase that can temporarily percolate in three dimensions and the percolating network structure.


At even lower temperatures a triple point appears below which the liquid phase disappears and only the gas and crystal remain. For a high rotational diffusion coefficient, the aggregated phase appears as the liquid in a classical liquid–gas phase separation, namely, the liquid phase is organized such that the surface of the cluster is minimal (Fig. We subsequently used the local maxima of the density histograms, which we identified as the local densities of the coexisting phases ?l, to construct the state diagram of the system for different rotational diffusion coefficients, see Fig. The low-density curves do not fall exactly on top of each other for high values of attraction due to surface effects that are discussed at the end of this section. In contrast, the peaks in the local density histograms also depend on the total density of the system. 4 we find that the transition between the network state and the gas–liquid separation occurs at a rotational diffusion of around Dr? ? 2. For the highest value of Dr?, the system exhibits a gas–crystal phase separation, and the radial distribution function shows sharp peaks characteristic of the crystalline order. In contrast, if the axes of self-propulsion are pointing in sufficiently different directions, the particles will overcome the attraction and move away from each other. For sufficiently high density (such as those studied in this paper), these aggregates become completely system spanning, and the majority of particles are connected to a single network, as seen in our simulations. A similar transition from the percolating network state to a gas–liquid coexistence state can take place by fixing the persistence length of the particles and increasing the attraction (Fig.
6(b) we do not see a collapse when plotting ? as a function of Pagg at fixed rotational diffusion Dr? for a wider range of state points.
That state consisted of interconnected but mobile clusters that created a system-spanning network. We thank Wiebke Albrecht, Guido Avvisati, Tommaso Comparin and Frank Smallenburg for careful reading of the manuscript. Even the few updates for every week is considered to be wastage of time for the administrator when calculated for the whole year. We find significant local alignment of the particles in the percolating network state despite the absence of aligning interactions, and propose a simple mechanism to justify the formation of this novel state. A first study of a similar model in three dimensions has been done by Mognetti et al.33 The focus of this work was mainly on the clustering properties of the system – as the strength of attraction is increased, the system passes from a homogeneous state to a clustered state caused by the attractive interactions. The drag and stochastic forces account for the constant collisions between the colloidal particles and the molecules of the solvent.
The effects of the finite size of the system on our results are discussed, when considered relevant, in the next section. Furthermore, for each point in the parameter space, a minimum of two simulations was performed, starting from two different initial configurations: one where the particles were on a cubic lattice that spanned the entire system, and one where all the particles were part of a dense liquid slab.
3(b) we see that the system now transitions with increasing strength of attraction from a homogeneous fluid state to a percolating network and then to a gas–liquid coexistence state.
Additionally, the density of the dense phase (percolating network of liquid clusters, liquid or crystal) was only slightly affected by the system size. Note that this transition becomes sharper with increasing system size, characteristic of a phase transition. For Dr? = 9 and 3, the system forms a gas–liquid separation, resulting in much weaker peaks in g(r).
In all cases, we see that for r < ?, there is a negative correlation between the orientations of the particles.
After a large number of collisions, this process will ultimately create a collection of clusters, with each cluster containing particles with similar orientations. This picture agrees well with movies from our simulations (see ESI† Movies S1 and S2) and accounts for both the local alignment of particles (Fig.
Finally, we checked to see if the collapse would occur if we use underdamped dynamics in place of overdamped dynamics for the translational degrees of freedom. Finally, for slow rotational diffusion the (bulk) gas–liquid coexistence disappeared and the system transitioned from the percolating network state directly to gas–crystal coexistence. So we can save the administrator's valuable time by delegating the employees to update their own contact information whenever it is changed. These two initial conditions were chosen in order to study any possible effect the starting configuration could have on the steady state of the system. However, for a total density higher than ??3 = 0.5 this criterion failed as the liquid cluster always percolated. However, for slower rotational diffusion, the aggregated phase is much less compact as shown in Fig. Labels are as follows: F indicates the fluid, G–L gas–liquid coexistence, G–X gas–crystal coexistence and PN the percolating network state. Finally, for Dr? = 0.3, we observe the percolating network state, for which the radial distribution function looks essentially the same as that of the gas–liquid separation.
This is expected, as the only way for particles to be significantly closer than r = ? is for them to be pushing towards each other. 5) and the lack of system size dependence of the structural properties of the percolating network (Fig. We found that after the transient period, the average potential energy in both simulations converged to the same value, which indicates that the system indeed relaxed at the same state and we can safely identify the regime of quasi-constant potential energy with the steady state. 1(c), and frequently spans the entire system or forms “living” clusters as described in ref.
2 some of the peaks in the gas phase are very low compared to the peaks of the liquid phase, indicating that only a small fraction of our system consisted of gas particles. These deviations are expected, since the first peak of the local density histogram is complicated by the presence of surface particles in addition to the gas particles.
Additionally, as expected, the crystalline state shows significant statistical noise, due to small values of the denominator in eqn (8). JiJi Self Update reduces the work load of administrator and makes Active Directory accurate. For a limited number of simulations we also looked at the time evolution of the degree of clustering and the local density histograms, both of which are described in detail later in this paper, and we found that in the regime of quasi-constant energy these structural functions were also only subject to fluctuations and there were no major changes. We identified as fluid the state where only a single peak is visible in the local density histogram. We conclude that the data collapse found by Mognetti et al.33 does not occur in our system.
States which exhibited two peaks but showed no signs of global phase separation were identified as percolating network states, and states which exhibited a clear phase separation were marked are either gas–liquid or gas–crystal, depending on whether the high density phase is crystalline. Thus, the size of the largest cluster does not fluctuate significantly, and the dynamic changes in the shape of the percolating network do not seem to affect the presented results. 5 is the strong local alignment of particles in the percolating state in comparison to the other states. We attribute this discrepancy to the difference in the applied dynamics, and more specifically to the different mechanisms that rotate the particles in the two different systems.
All the above ensure that potential energy is a reasonable indicator of when the system reaches the steady state. Note that we have not probed the exact positions of the critical or triple points in any of the diagrams presented in Fig. As there is no explicit aligning torque in the model, this is surprising, and indicates that particles with similar orientations tend to stay in closer proximity in the percolating network. Additionally, the boundaries presented in black dashed lines are simply approximate state boundaries.



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