3.4 Analysis of clustering
3.4.3 Cluster analysis
A third method to quantify the clustering is based on a cluster analysis as it is done in the context of percolation theory. As the particles move off-lattice, we define that two
3.4 Analysis of clustering
α= 0.83 clustered α= 0.51 non-clustered
numberfluctuations∆n
mean number of particles hni
10−1 100 101 102 103 104
100 102 104
Figure 3.13: Two example curves for the analysis of number fluctuations: The mea-surements are given as symbols while the lines show the results of linear fits of
∆n ∼ hniα. The measurements in both examples show a power-law behavior. The first one (×) belongs to a non-clustered system where the exponent is close to the equilibrium value of 0.5. The second curve (◦) shows giant number fluctuations with an exponent close to the theoretically predicted value of 5/6 (figure to be published inBreier et al.,2017).
nematic, clustered isotropic, homogeneous
clustersizedistributions
cluster size s
100 101 102 103 104
10−2 100 102 104
Figure 3.14: Typical examples of the cluster size distribution: The results of two different simulations are given. The blue circles correspond to a simulation in the isotropic phase without clustering while the yellow crosses represent a system in the nematic phase with clusters.
particles belong to the same cluster if their centers are closer than a cutoff distancerc. In a given configuration of the system, the so-defined clusters are identified and their size s is measured as the number of particles in a given cluster. The distribution of sizes is called the cluster size distribution n(s). A significant result is obtained by averaging the cluster size distributions of several snapshots of the same system in the steady-state. Typical examples are given in Fig. 3.14. The cluster size distribution of a given system is (among others) characterized by the normalized second moment:
N2 ≡
P
sn(s)s2
P
sn(s)s . (3.79)
4 Structure formation by
self-propelled point particles
In this chapter we investigate our model of nematically aligning, self-propelled par-ticles. It is described in detail in Section 3.1.1. The simulations are performed in a cubic domain with PBCs, if not stated otherwise. The typical input parameters are given in Tab. 4.1.
parameter value
number of particles N = 663 = 287496
side lengths of simulations box L=Lx=Ly =Lz =q3N/ρ nematic interaction range = 1
self-propulsion speed v0 = 0.5 nematic relaxation constant γ = 0.1
rotational diffusion Dr=γ/P =η2∆t/2
time step ∆t= 0.1
Table 4.1: Input parameters for point-like self-propelled particles (if not stated oth-erwise).
globalnematicorder parameterS
Figure 4.1: Nonequilibrium phase diagram of self-propelled point particles with ne-matic interaction. The global nene-matic order parameter S is shown in color as a function of Péclet number and number density ρ. Different symbols refer to differ-ent states of the system as observed from the steady-state configurations. The bold symbols denote the systems from which the snapshots are given in Fig. 4.2 (graph modified from Breier et al., 2016).
4.1 Phase diagram and snapshots
The nonequilibrium phase diagram of the system is given in Fig. 4.1. The nematic order parameter in the steady-state is measured as a function of rotational Péclet number (Eq. 3.55) and number density ρ = N/L3. The system is evolved until a steady state is very well established (typically more than 105time steps) and then the order parameters in the steady state are measured (typically averaged over another 105 time steps). The nematic interaction range is the fundamental length scale in the system ( = 1) and hence the number density has the same value as the non-dimensional number densityρ3. For a constant number density and increasingP, we see a clear transition from the isotropic phase (denoted by4,S →0) to the nematic phase (denoted by ♦, S → 1). Typical snapshots of these two phases are given in Figs. 4.2a and 4.2b.
The nematic and isotropic phases are separated by a rather narrow transitional domain in the phase diagram where coexistence of nematic and isotropic domains in the system is observed (denoted by◦). Depending on the precise value ofP, this phase coexistence can typically have three different shapes (see Fig. 4.2c): For the smallest value of P typically a nematically ordered, rather dense cylinder occurs in a dilute, isotropic gas. This cylinder percolates through the PBCs and the nematic director is aligned with the cylinder axis. When the Péclet number is slightly increased, the cylinder grows such that it connects itself also through a second dimension thus leading to a nematically ordered, dense layer in an isotropic, dilute surrounding. The local director within the layer is perpendicular to the layer normal and typically aligned along one of the box axes. Finally, for the largest values ofP for which phase
4.1 Phase diagram and snapshots
(a) Isotropic state (4,P = 2.47,ρ= 1.125)
localnematicorder parameterSloc
0 0.5 1
(b) Nematic state (♦,P = 19.5,ρ= 1)
(c) Phase coexistence (◦,P = [8; 8.68; 7.4], ρ= [0.125; 0.125; 0.25])
Figure 4.2: Snapshots of the steady-state configurations of the self-propelled particles.
The corresponding points are given by the bold symbols in Fig.4.1. A small fraction (0.35%) of the simulated particles is shown with the orientation vectors as small arrows. The simulation box is subdivided into 203boxes to calculate the local nematic order parameterSloc (represented by the color).
coexistence is still visible, the emerging structure is the inverse of the first phase coexistence snapshot: Now a dilute, disordered cylinder is surrounded by a dense, nematically ordered surrounding. Here, the nematic director is again parallel to the cylinder axis. The cylinder axis (or the nematic director in the case of the nematic layer) is mostly aligned along one of the box axes in the examined snapshots of the phase coexistence state. This is an effect of the PBCs: When a nematically ordered cylinder forms, it can most easily connect to itself when it is aligned along one of the axis. This is a positive feedback because it stabilizes itself. In principle, however, the cylinder axis could be at any angle inside the box (but fulfilling thePBCs).
In the nematic phase (close to the isotropic-nematic phase transition) another struc-ture of the particles was found: Polarized, dense waves which travel through the box (denoted byF). We will discuss them in detail in Section 4.3.
Finally and most interestingly, a chiral structure occurs at different places in the nematic phase (see scattered squares in Fig. 4.1). This is investigated in detail in Breier et al. (2016) and in Section 4.4.