Active crystals and their stability

A recently introduced active phase field crystal model describes the formation of ordered resting and traveling crystals in systems of self-propelled particles. Increasing the active drive, a resting crystal can be forced to perform collectively ordered migration as a single traveling object. We demonstrate here that these ordered migrating structures are linearly stable. In other words, during migration, the single-crystalline texture together with the globally ordered collective motion is preserved even on large length scales. Furthermore, we consider self-propelled particles on a substrate that are surrounded by a thin fluid film. We find that in this case the resulting hydrodynamic interactions can destabilize the order.

Figure 1

Snapshots of the order parameter fields illustrating the different phases that appear at $(\varepsilon ,C_1,C_4,D_r)=(-0.98,0.2,0,0.5)$ for increasing active drive $v_0$. The mean density is (a)–(d) $\overline{\psi }=-0.4$ and (e) and (f) $\overline{\psi }=0.4$. In the first four panels, the active drive $v_0$ of the individual particles increases from (a) $v_0=0.1$ via (b) $v_0=0.5$ and (c) $v_0=1$ to (d) $v_0=1.9$. In all snapshots, brighter color corresponds to higher densities ${\psi }_1$, whereas the thin bright needles illustrate the polarization field $\mathbf{P}$ pointing from the thick to the thin ends. Starting from (a) a resting hexagonal structure, increasing the active drive first leads to (b) a traveling hexagonal texture, then (c) a traveling quadratic structure, and finally (d) traveling lamellae. (e) and (f) Inverse traveling structures corresponding to (b) and (c), respectively, i.e., a traveling honeycomb and a traveling inverted quadratic texture. The thick bright arrows indicate the predominant direction of migration of the structures in (b)–(f). Only a fraction of the numerical calculation box is shown.

Figure 2

Collective migration speed $v_m$ of the single crystals as a function of the active drive $v_0$ of the individual particles at $(\varepsilon ,\overline{\psi },D_r)=(-0.98,-0.4,0.5)$. For $C_1<0$ and $C_4>0$ (squares) the directions of active drive of the individual particles order spontaneously and the structures travel collectively with $v_m=v_0$. A nonzero threshold appears for $C_1=0$ and $C_4>0$ (rhombi). At $C_1>0$ and $C_4\ge 0$ (circles) the directions of active drive of the individual particles do not order spontaneously. Then the structures are at rest ($v_m=0$) below a finite threshold value $v_{0,c}$; they start to travel collectively with nonzero $v_m<v_0$ above the threshold.

Figure 3

The left column shows snapshots of crystalline structures at $(\varepsilon ,\overline{\psi },C_1,C_4,D_r)=(-0.98,-0.4,0.2,0,0.5)$ and (a) $v_0=0.5$ for a traveling hexagonal crystal, (b) $v_0=0.7$ for a traveling rhombic crystal, and (c) $v_0=0.9$ for a traveling quadratic crystal. Only a fraction of the numerical calculation box is shown. The right column shows corresponding power spectra in two-dimensional Fourier space. A distorted hexagonal lattice structure is obtained in Fourier space in (d) from the traveling hexagonal case. (e) Power spectrum for the traveling rhombic crystal. (f) Finally, the square pattern in real space transforms itself into a square pattern of the two-dimensional power spectrum. The intensity of the peaks in the power spectrum is represented by their brightness and was partially rescaled for better visualization. Circles mark the directions of the reciprocal lattice vectors of smallest magnitude ${\mathbf{G}}_1$. The examples of linear stability analysis against linear dynamical instabilities that are shown in Fig. 5 were performed in these directions.

Figure 4

Linear stability of the structures presented in Fig. 3 against instabilities of small wave numbers in the comoving frame. The analyzed regions in $\mathbf{q}$ space contain the first Brillouin zones of the textures in Fig. 3, respectively. Surface plots show the logarithm of the determinant $\mathcal{D}\left(\mathbf{q}\right)$ of the stability matrix in the analyzed $\mathbf{q}$ regions. It has been rescaled for technical reasons in each panel, so the absolute values should not be interpreted. Below the surfaces, the shadow plots represent colored projections of the surfaces into the plane. Here $G_1=\parallel {\mathbf{G}}_1\parallel$, $q_{\parallel }$ is the component of $\mathbf{q}$ parallel to ${\mathbf{G}}_1$, and $q_{\perp }$ is its component in the perpendicular direction, where ${\mathbf{G}}_1$ is the reciprocal lattice vector of smallest magnitude. A linear instability would be indicated by a vanishing $\mathcal{D}\left(\mathbf{q}\right)$, however, we observe nonzero values in all probed instances (and no sign changes). We thus conclude that the structures are linearly stable against static instabilities in the comoving frame.

Figure 5

Examples for the linear stability of the structures presented in Fig. 3 against linear dynamical instabilities. The panels show stability probes performed in $\mathbf{q}$ space along the direction of the reciprocal lattice vector of smallest magnitude ${\mathbf{G}}_1$. In addition, the frequency $\omega$ was varied in an interval centered around $-\mathbf{q}·{\mathbf{v}}_{\mathbf{m}}$ at each wave vector $\mathbf{q}$, with ${\mathbf{v}}_{\mathbf{m}}$ the collective migration velocity. We define $q=\parallel \mathbf{q}\parallel$, $G_1=\parallel {\mathbf{G}}_1\parallel$, and $\tilde{\omega }=\omega /|{\mathbf{G}}_1·{\mathbf{v}}_{\mathbf{m}}|$. Surface plots show the logarithm of the determinant $\mathcal{D}(\mathbf{q},\tilde{\omega })$ of the stability matrix in the analyzed $\mathbf{q}$-$\tilde{\omega }$ regions. It has been rescaled for technical reasons in each panel, so the absolute values should not be interpreted. Below the surfaces, the shadow plots represent colored projections of the surfaces into the plane. Our examples correspond to the directions indicated by the circles in the power spectra of Fig. 3 for (a) $v_0=0.5$, (b) $v_0=0.7$, and (c) $v_0=0.9$. A linear instability would be indicated by a vanishing $\mathcal{D}(\mathbf{q},\tilde{\omega })$ (or a sign change). Apparently, the structures are linearly stable against linear dynamical instabilities.

Figure 6

Hydrodynamic interactions for the two-dimensional self-propelled motion on or close to a surface in a surrounding fluid film at $(\overline{\psi },\varepsilon ,C_1,C_4,v_0)=(-0.4,-0.98,0.2,0,0.7)$. The top panels (a)–(c) show the density field ${\psi }_1$ and the polarization field $\mathbf{P}$ in the same way as in Fig. 1. We illustrate the corresponding fluid flow fields $\mathbf{u}$ in the bottom panels (d), (f), and (g) by needles pointing from the thick to the thin white ends. The friction parameter $\alpha$ of the fluid film decreases from the left to the right column: (a), (d), and (e) $\alpha =300$; (b) and (f) $\alpha =10$; and (c) and (g) $\alpha =1.5$. (a) and (d) Hydrodynamic interactions are weak at high fluid friction, where the flow remains mostly localized around the density peaks. (c) and (g) On the contrary, hydrodynamic interactions are strong at low fluid friction, where the whole fluid can be set into motion nearly homogeneously and the fluid flow extends between the density peaks. Panel (e) highlights the backflow regions occurring in (d) around the density peaks by rescaling the magnitude of the depicted flow vectors. A zoom into the region around one single-density peak is included for illustration. Only a fraction of the numerical calculation box is shown in each case.

Figure 7

Collective migration speed $v_m$ of the single crystals as a function of the active drive $v_0$ of the individual particles in the presence of hydrodynamic interactions at $(\overline{\psi },\varepsilon ,C_1,C_4)=(-0.4,-0.98,0.2,0)$. For low hydrodynamic interactions, corresponding to a large fluid friction parameter $\alpha$, the collective migration speed is lower. For strong hydrodynamic interactions, corresponding to a low fluid friction parameter $\alpha$, the collective migration speed is enhanced. In the latter case the particles can push each other by setting the fluid between them into motion. The critical value of the self-propulsion velocity $v_{0,c}$ remains unchanged by the presence of the hydrodynamic interactions.

Figure 8

Hydrodynamic interactions can destabilize a collectively migrating active single crystal at $(\overline{\psi },\varepsilon ,C_1,C_4,v_0)=(-0.4,-0.98,0.2,0,1)$. (a) Negligible hydrodynamic interactions at strong fluid friction ($\alpha =300$) allow the emergence of a perfect traveling active single crystal. (b) Strong hydrodynamic interactions at low fluid friction ($\alpha =1.5$) destabilize this texture.