Two-dimensional localized states in an active phase-field-crystal model

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Figure 1

Sketch of the numerical setup: Two-dimensional structures that are reflection symmetric with respect to the $x$ axis are computed on a reduced domain $[0,L_x]\times [0,L_y/2]$ as indicated by the colored area. The density field $\psi$ and the $x$-component $P_x$ of the polarization satisfy Neumann boundary conditions at $y=0$ while Dirichlet boundary conditions apply to the $y$-component $P_y$ of the polarization: $P_y=0$ at $y=0$. All fields are periodic in the $x$ direction, i.e., only motion in the $x$ direction is allowed. For visualization, the entire domain ${\mathrm{\Omega }}_{\exp }=[0,L_x]\times [-L_y/2,L_y/2]$, indicated by the colored and grayscale regions, is used.

Figure 2

(Left) Bifurcation diagram showing branches of homogeneous, periodic, and localized steady states of the passive PFC model ($v_0=0$) on a rectangular domain. Shown is the $L^2$-norm ${\left|\right|\psi \left|\right|}_2$ as a function of the mean density $\overline{\psi }$. Stable and unstable states are shown as solid and dotted lines, respectively. The liquid phase (gray horizontal line) is destabilized at $\overline{\psi }\approx -0.55$ and an unstable branch of periodic hexagonal patterns (black line, cf. location IV) emerges subcritically. In a secondary bifurcation, a branch of LS (blue line) is created. After various folds responsible for repeated gain and loss of stability, the LS branch terminates on the same branch of periodic hexagons from which it bifurcated. (Right) Selected solution profiles $\psi \left(\mathbf{r}\right)$ at locations labeled I to IV in the left panel. The domain size is $2L_{\mathrm{a}}\times 4L_{\mathrm{c}}$ with where $L_{\mathrm{a}}=2L_{\mathrm{c}}/\sqrt{3}$ is the side length of a triangle and $L_{\mathrm{c}}=2\pi$ is its height and the critical wavelength. The remaining parameter is $\epsilon =-0.98$.

Figure 3

(Left) Bifurcation diagram showing branches of homogeneous (gray line), periodic (black line), targetlike (red line), and localized (blue line) steady states of the passive PFC model ($v_0=0$) on a hexagonal domain. Shown is the norm ${\left|\right|\psi \left|\right|}_2$ as a function of the mean density $\overline{\psi }$. (Right) Selected solution profiles $\psi \left(\mathbf{r}\right)$ at locations labeled I to IV in the left panel. The hexagonal domain has side length $3L_{\mathrm{a}}$. Remaining line styles and parameters are as in Fig. 2.

Figure 4

(Left) Bifurcation diagram showing suppressed snaking at lower values of $\epsilon$, here $\epsilon =-1.5$, in the passive PFC model ($v_0=0$). In contrast to $\epsilon =-0.98$ (Figs. 2 and 3) the crystalline patches do not grow by adding layers of density peaks. Instead a single peak grows into an elongated structure and subsequently forms a dumbbell-shaped two-peak state. Asymmetric states are omitted. The domain size is $2L_{\mathrm{a}}\times 2L_{\mathrm{c}}$. Remaining line styles and parameters are as in Fig. 2.

Figure 5

(Left) Bifurcation diagram of resting and traveling one-peak LS at mean density $\overline{\psi }=-0.9$ showing the $L^2$ norm of $\psi$ as a function of the activity $v_0$. Resting and traveling states are indicated by blue and orange lines, respectively. Branches of stable and unstable states are shown as solid and dotted lines. At $v_{\mathrm{c}}\approx 0.15$, a stable resting LS undergoes a drift pitchfork bifurcation and a branch of traveling localized states (TLS) emerges. The region of existence of the TLS is limited by a fold at $v_0\approx 0.24$. The panels on the right show (top) selected solution profile $\psi \left(\mathbf{r}\right)$ at $v_0=0.1$ (only part of the domain is shown); (center) the drift velocity $c$ vs $v_0$. Above $v_{\mathrm{c}}\approx 0.15$, the velocity increases as $\sqrt{v_0-v_{\mathrm{c}}}$. Deviations from a sharp onset of motion are due to lattice effects. (Bottom) The difference ${\left|\right|\psi \left|\right|}_2^2-{\left|\right|\mathbf{P}\left|\right|}_2^2$ crosses zero at the drift pitchfork bifurcation. Note that, in the left panel, ${\left|\right|\psi \left|\right|}_2$ times the area $V$ is plotted for clarity as for 2D domains the norm of LS tends to be very small. The domain size is $V=60\times 30$. Remaining parameters are $\epsilon =-1.5$, $C_1=0.1$, $C_2=0$, and $D_{\mathrm{r}}=0.5$.

Figure 6

Density $\psi \left(\mathbf{r}\right)$ and polarization $\mathbf{P}\left(\mathbf{r}\right)$ profile shown as a color map overlaid with white arrows for (a) a resting ($v_0=0.13$) and (b) a traveling $(v_0=0.22>v_{\mathrm{c}})$ one-peak LS. Note that in (a) the +1 defect of the polarization field coincides with the density maximum (and the net polarization is zero), while in (b) they are shifted with respect to one another as the front-back symmetry is broken. The shift corresponds to a net polarization, i.e., net propulsion to the left, with $c\approx -0.19$. Only a part of the computational domain is shown. The remaining parameters are as in Fig. 5.

Figure 7

(Left) Bifurcation diagram of a resting two-peak LS (cf. Fig. 4) at mean density $\overline{\psi }=-0.9$ showing the $L^2$ norm of $\psi$ as a function of the activity $v_0$. (Right) Selected density profiles $\psi \left(\mathbf{r}\right)$ at $v_0=0.03$ (top) on the upper part of the branch of left-right symmetric states (blue line), $v_0=0.09$ (middle) on the branch of left-right asymmetric states (black line), and $v_0=0.04$ (bottom) on the lower part of the branch of the left-right symmetric states. In contrast to the one-peak LS, only resting two-peak LS exist at this mean density as the saddle-node bifurcation of the symmetric states is located at $v_0<v_{\mathrm{c}}\approx 0.15$. The remaining line styles, parameters, and the domain size are as in Fig. 5.

Figure 8

Summary bifurcation diagram showing ${\left|\right|\psi \left|\right|}_2$ vs $v_0$ for resting and traveling dumbbell-shaped two-peak LS at mean density $\overline{\psi }=-0.8$. Blue and black lines indicate branches of resting symmetric and asymmetric LS, respectively. At $v_0\approx 0.15$, states traveling in different directions (orange branches) emerge in various drift bifurcations. See Figs. 9 and 11 for details and selected solution profiles. Thin gray lines correspond to branches of resting and traveling one-peak LS. Remaining line styles, parameters, and the domain size are as in Fig. 5.

Figure 9

(Top left) Shown is a subset of the bifurcation curves from Fig. 8, namely, the traveling dumbbell-shaped two-peak LS that move parallel to their long axis, the one-peak states, and the resting two-peak states. The right panels show selected density profiles $\psi \left(\mathbf{r}\right)$ at points labeled I to IV in the main panel. The resting two-peak LS are destabilized with respect to parallel motion in drift-pitchfork bifurcations at $v_{\mathrm{c}}\approx 0.16$, marked by black circles. On the orange branches of traveling LS, state III (state II) travels with the larger (smaller) density peak at the front. The asymmetric steady solution is destabilized in a drift-transcritical bifurcation marked by the circle on the black branch. Here two branches of TLS with opposite drift velocities emerge. The lower left panels show the drift velocity $c$ as a function of $v_0$ and the measure ${\left|\right|\psi \left|\right|}_2^2-{\left|\right|\mathbf{P}\left|\right|}_2^2$ that crosses zero at the respective onsets of motion. The remaining line styles, parameters, and the domain size are as in Fig. 8.

Figure 10

Shown is a magnification of Fig. 9 completing the branch of steady asymmetric LS (black dashed line). Panels I to VI present selected density profiles $\psi \left(\mathbf{r}\right)$ at points labeled I to VI in the main panel. The remaining line styles, parameters, and the domain size are as in Fig. 9.

Figure 11

(Top left) Shown is a subset of bifurcation curves from Fig. 8, namely, the traveling dumbbell-shaped two-peak LS that move perpendicular to their long axis, the one-peak states, and the resting two-peak states. The remaining panels, line styles, symbols, parameters, and the domain size are as in Fig. 9.

Figure 12

A sequence of bifurcation diagrams ${\left|\right|\psi \left|\right|}_2$ vs $v_0$ showing how traveling one-peak LS (orange line) come into existence with increasing mean density $\overline{\psi }$ (from top left to bottom right) at $\epsilon =-0.98$ (corresponding to the value used in [38]). Two drift-pitchfork bifurcations appear simultaneously at the saddle-node bifurcation of the branch of resting LS (blue line). Increasing $\overline{\psi }$ further expands the range of existence of traveling LS toward larger $v_0$ and the drift-pitchfork bifurcations separate. The onset of motion is always at $v_{\mathrm{c}}\approx 0.15$. Remaining line styles, parameters, and the domain size are as in Fig. 5.

Figure 13

Morphological phase diagram for the aPFC model in 2D in the plane spanned by the activity $v_0$ and the mean density $\overline{\psi }$ as obtained through systematic time simulations. The region of stable liquid state is white, while crystalline structures of various size exist in the colored areas. The color bar indicates the number of density peaks formed in the domain of size $8L_{\mathrm{c}}\times 7L_{\mathrm{a}}$ with $L_{\mathrm{a}}=2L_{\mathrm{c}}/\sqrt{3}$ and $L_{\mathrm{c}}=2\pi$. Regions where resting and traveling LS exist are marked by shades of blue while domain-filling periodic patterns are shown as green (56 peaks). The various lines in the diagram, the initial conditions for the simulations, and the peak counting procedure are described in the text. The remaining parameters are $\epsilon =-1.5$, $C_1=0.1$, $C_2=0$, and $D_{\mathrm{r}}=0.5$ as used throughout Sec. 3b. The parameter increments between simulations are $\mathrm{\Delta }{v}_0=0.035$ and $\mathrm{\Delta }\overline{\psi }=0.0125$. See Fig. 14 for a magnification of the region close to the onset of motion and selected density profiles.

Figure 14

The large panel shows a magnification of the region close to the onset of motion in the morphological phase diagram in Fig. 13. The parameter increments between simulations are $\mathrm{\Delta }{v}_0=0.02$ and $\mathrm{\Delta }\overline{\psi }=0.025$. The small panels show selected density profiles $\psi \left(\mathbf{r}\right)$ at points labeled I to V in the phase diagram after the time simulations have converged. Arrows indicate direction of motion. Shown are (I) resting hexagonal pattern close to the transition to stripes, (II) traveling hexagonal pattern, (III) traveling cluster of hexagonal order, (IV) resting LS, and (V) traveling LS.

Figure 15

Bifurcation diagram showing slanted snaking of resting and traveling LS at $\epsilon =-0.98$. Shown is the norm ${\left|\right|\psi \left|\right|}_2$ as a function of the mean density $\overline{\psi }$ with activity fixed at the still relatively low value $v_0=0.151>v_{\mathrm{c}}$ that allows for the coexistence of resting and traveling states. Labels I to IV mark the location of the stable traveling LS shown on the right. The liquid phase (gray line) with norm zero is destabilized at $\overline{\psi }\approx -0.53$ and a branch of traveling periodic patterns (dark red) emerges. Close to the first primary bifurcation, a branch of resting crystals (black) emerges in another primary bifurcation. Resting and traveling LS (blue and orange), respectively, bifurcate in secondary bifurcations from these branches. Since $v_0>v_{\mathrm{c}}$, all resting solutions are unstable. The inset illustrates the small separation of the branches in terms of their norm. The lower panel shows the drift velocity $c$ as a function of $\overline{\psi }$. Since $c<0$, all TLS move to the left. The domain size is $2L_{\mathrm{a}}\times 4L_{\mathrm{c}}$ while the remaining line styles and parameters are as in Fig. 5.

Figure 16

Bifurcation diagram ${\left|\right|\psi \left|\right|}_2$ vs $\overline{\psi }$ for traveling hexagonal patterns and traveling LS at the relatively high activity $v_0=0.18$ where no resting states exist. Labels I to IV denote the locations of the stable traveling LS shown on the right. The bottom right panel shows the drift velocity $c$ as a function of $\overline{\psi }$. All states travel to the left. Domain size, line styles, and the remaining parameters are as in Fig. 15.

Figure 17

Selected snapshots of periodic density patterns $\psi \left(\mathbf{r}\right)$ at the fixed mean density $\overline{\psi }=-0.4$ as obtained for increasing activity by time simulation. (a) A resting hexagonal pattern at $v_0=0.25$, (b) a traveling hexagonal pattern at $v_0=0.3$, (c) a traveling rhombic pattern at $v_0=0.8$, and (d) a traveling stripe pattern at $v_0=1.5$. The respective directions of motion are indicated by white arrows. The domain size is $6L_{\mathrm{c}}\times 5L_{\mathrm{a}}$ while the remaining parameters are $\epsilon =-0.98$, $C_1=0.2$, $C_2=0$ as in Ref. [38].

Figure 18

The main panel shows the bifurcation diagram ${\left|\right|\psi \left|\right|}_2$ vs $v_0$ for hexagonal patterns oriented such that the onset of motion is parallel to an edge. Resting hexagons (blue line) are stable (solid line) until a drift-pitchfork bifurcation occurs at $v_{\mathrm{c}}\approx 0.3$ where a branch of stable traveling hexagonal patterns (orange line) emerges. The corresponding drift velocity $c$ is shown in the lower left panel while the lower right panel shows the measure ${\left|\right|\psi \left|\right|}_2^2-{\left|\right|\mathbf{P}\left|\right|}_2^2$ that crosses zero at the drift bifurcation. On the right selected solution profiles $\psi \left(\mathbf{r}\right)$ at the locations labeled I to IV in the bifurcation diagram are shown. Profiles II—IV travel in the $x$ direction to the right. The domain size is $V=3L_{\mathrm{a}}\times 4L_{\mathrm{c}}$ and the remaining parameters are as in Fig. 17.

Figure 19

Density $\psi \left(\mathbf{r}\right)$ and polarization $\mathbf{P}\left(\mathbf{r}\right)$ profiles in terms of a color map with overlaid white arrows, respectively, for (a) a resting ($v_0=0.25$) and (b) a traveling $(v_0=0.4>v_{\mathrm{c}})$ hexagonal pattern. In (a) the +1 defects of the polarization field coincide with the density maxima (and the net polarization is zero), while in (b) they are shifted with respect to one another, breaking the left-right symmetry. This shift generates a net polarization and results in net propulsion to the right with $c\approx 0.3$. Parameters and domain size are as in Fig. 18.

Figure 20

The main panel shows the bifurcation diagram ${\left|\right|\psi \left|\right|}_2$ vs $v_0$ for hexagonal patterns for the onset of motion perpendicular to an edge. On the right selected solution profiles $\psi \left(\mathbf{r}\right)$ at locations labeled I to IV in the bifurcation diagram are shown. Profiles II–IV travel in $x$ direction to the right. Resting hexagons (blue line) are stable (solid line) until a drift-pitchfork bifurcation at $v_{\mathrm{c}}\approx 0.3$ where a branch of stable traveling hexagonal patterns (orange line) emerges. With increasing $v_0$, the traveling hexagonal pattern (e.g., profile II) deforms into modulated stripes (e.g., profile III). The branch terminates on the horizontal branch of traveling stripes (dark green line, e.g., profile IV) that itself emerges in a drift-pitchfork bifurcation from an unstable branch of resting stripes (red line). The domain size is $V=4L_{\mathrm{c}}\times 3L_{\mathrm{a}}$ and the remaining parameters are as in Fig. 17.

Figure 21

The main panel shows the bifurcation diagram ${\left|\right|\psi \left|\right|}_2$ vs $v_0$ for square patterns oriented such that the motion is parallel to a diagonal of the square. On the right selected solution profiles $\psi \left(\mathbf{r}\right)$ at locations labeled I to III in the bifurcation diagram are shown. Profiles II–III travel in the $x$ direction to the right. Resting squares (black line) are unstable. At drift-pitchfork bifurcations at $v_{\mathrm{c}}\approx 0.3$ branches of unstable traveling square patterns (orange lines) emerge. At $v_0\approx 0.7$ traveling squares gain stability in a Hopf bifurcation. The domain size is $2\sqrt{2}{L}_{\mathrm{c}}\times 2\sqrt{2}{L}_{\mathrm{c}}$ and the remaining parameters are as in Fig. 17.

Figure 22

Combined bifurcation diagrams for the resting and traveling periodic states in Figs. 18, 20, and 21. Note that the domain sizes differ between the different branches. All resting patterns undergo drift-pitchfork instabilities at nearly identical values of $v_0$, $v_{\mathrm{c}}\approx 0.3$. Different traveling states coexist at large activity $v_0$.

Figure 23

Morphological phase diagram for the aPFC model accompanied by selected density profiles at locations labeled I–V obtained from systematic simulations. Large panel: Different states are characterized by the total number of density peaks that form in a rectangular domain of size $7L_{\mathrm{a}}\times 8L_{\mathrm{c}}$ as indicated by color coding. The various lines in the diagram, the initial conditions of the simulations, and the peak counting procedure are described in the text. The liquid state refers to a uniform density phase with zero peaks (white area). LS exist in the regions marked in blue. Periodic hexagonal patterns (green) fill the domain with 56 density peaks (I). Around $v_0>1$ the number of peaks slightly increases as resting hexagonal patterns (I) begin transforming towards traveling rhombic patterns (II) with a smaller wavelength allowing for more density peaks. Arrows indicate the direction of motion. At even higher $v_0$, traveling stripe patterns (III) arise. These remain spatially modulated so that individual peaks can still be located on each ridge and the number of density peaks does not drop. (IV) and (V) give examples of resting LS coexisting with the liquid phase. The remaining parameters are $\epsilon =-0.98$, $C_1=0.2$, $C_2=0$, and $D_{\mathrm{r}}=0.5$ as in [38].