Crystallization and melting of bacteria colonies and Brownian bugs

Motivated by the existence of remarkably ordered cluster arrays of bacteria colonies growing in Petri dishes and related problems, we study the spontaneous emergence of clustering and patterns in a simple nonequilibrium system: the individual-based interacting Brownian bug model. We map this discrete model into a continuous Langevin equation which is the starting point for our extensive numerical analyses. For the two-dimensional case we report on the spontaneous generation of localized clusters of activity as well as a melting-freezing transition from a disordered or isotropic phase to an ordered one characterized by hexagonal patterns. We study in detail the analogies and differences with the well-established Kosterlitz-Thouless-Halperin-Nelson-Young theory of equilibrium melting, as well as with another competing theory. For that, we study translational and orientational correlations and perform a careful defect analysis. We find a nonstandard one-stage, defect-mediated transition whose nature is only partially elucidated.

Figure 1

Upper panels: Snapshots of two-dimensional IBB model in its stationary state (time $t=2\times {10}^5$ Monte Carlo steps) in the (a) disordered active phase $\mu =1.0$ and (b) the ordered or patterned active one $\mu =2.0$; other parameter values are $N_s=50$, $R=0.1$, $D={10}^{-5}$. Lower panels: Snapshots of the two-dimensional Langevin Eq. (2) in its stationary state for $\mu =2.0$, $D=0.25$, $R=10$, $N_s=50$ in the (c) disordered active phase $\sigma =2.5$ and (d) the ordered or patterned active one $\sigma =1.5$. Observe that the two snapshots to the right [(b) and (d)] have a crystal-like hexagonal ordering, absent in the other two [(a) and (c)]. Also, the crystal-like packing is more evident in the continuous model for the chosen parameters, but no qualitative difference exist between the upper and the lower panels.

Figure 2

(Color online) Main: cluster mobility per unit time (squares and left axis) and mean cluster center of mass velocity (circles and right axis) as a function of $\sigma$ for $L=256$. Inset: Number of clusters per surface unit (multiplied by ${10}^3$) versus $\sigma$, for $L=256$.

Figure 3

(Color online) Main plot: Structure function $S\left(k\right)$ for three different values of the noise amplitude $\sigma$ and $L=512$. The peak around $k_0\approx 42$ becomes more pronounced as we go deeper into the ordered phase. Inset: $S\left(k_0\right)$ versus $\sigma$ for $L=256$ and $L=512$.

Figure 4

(Color online) Schematic presentation of the main predictions of the KTHNY theory for two-dimensional melting. See the main text for more details.

Figure 5

(Color online) Radial correlation function $g\left(r\right)$ as a function of $r$ for different values of $\sigma$ and linear size $L=512$.

Figure 6

(Color online) Correlation length $\xi$ and exponent $\eta$, obtained by fitting $g\left(r\right)$ in Fig. 5 to Eq. (9). Note the abrupt increase of the correlation length around $\sigma =2.39$ from a small value above to a large saturating value (roughly equal to $L/4=128$, i.e., half of the maximum radius). The exponent $\eta$ changes continuously, in remarkable agreement with the KTHNY predictions.

Figure 7

(Color online) Main plot: Stationary orientational order parameter ${\psi }_6$ as a function of $\sigma$ for two different system sizes $L=256$ and 512. Inset: Orientational susceptibility ${\chi }_6$ as a function of $\sigma$ for $L=256$ and $L=512$.

Figure 8

(Color online) Probability distribution functions of ${\left|{\psi }_6\left(i\right)\right|}^2$ for different values of $\sigma$ and linear size 512.

Figure 9

(Color online) Log-log plot of the (spherically averaged) orientational correlation function $g_6\left(r\right)$ for different values of $\sigma$ and linear size 512.

Figure 10

(Color online) Orientational susceptibility ${\chi }_6$ and correlation length ${\xi }_6$ in the isotropic phase. The solid lines represent the best fit to the stretched exponential function, Eq. (16).

Figure 11

(Color online) Defect maps at four different values of the noise amplitude $\sigma$ and size $L=512$. Lines define a Voronoi tessellation; fivefold, sixfold, and sevenfold clusters are marked with black circles, central dots, and red stars, respectively. The (few) empty polygons correspond to fourfold, eightfold, and ninefold defects. (i) Upper left panel (low noise intensity $\sigma =2.30$); only localized quadrupoles appear. (ii) Upper right panel ($\sigma =2.38$ near the critical point); a proliferation of isolated dipoles and dislocations can be observed and a small number of isolated monopoles and disclinations can also be observed. (iii) Lower left panel ($\sigma =2.41$ slightly above the critical point); although isolated dipoles and monopoles exist, there is a clear tendency to form defect condensates with stringlike geometry. (iv) Lower right panel ($\sigma =2.60$ well above the transition point); defects proliferate and the tendency to form stringlike structures is maintained.

Figure 12

(Color online) Main: Accumulated density of defects of the various possible types as a function of the noise amplitude $\sigma$. Inset: Relative isolated dislocation and disclination concentrations versus $\sigma$.

Figure 13

(Color online). Dislocation density (left axis and circles) and total defect density (right axis and squares) in logarithmic scale, as a function of the isolated disclination density $\left(L=512\right)$.