Extreme Spontaneous Deformations of Active Crystals

We demonstrate that two-dimensional crystals made of active particles can experience extremely large spontaneous deformations without melting. Using particles mostly interacting via pairwise repulsive forces, we show that such active crystals maintain long-range bond order and algebraically decaying positional order, but with an exponent $\eta$ not limited by the $\frac{1}{3}$ bound given by the (equilibrium) KTHNY theory. We rationalize our findings using linear elastic theory and show the existence of two well-defined effective temperatures quantifying respectively large-scale deformations and bond-order fluctuations. The root of these phenomena lies in the sole time-persistence of the intrinsic axes of particles, and they should thus be observed in many different situations.

Figure 1

Model defined by Eqs. (1) and (2) with $d_0=D_r=1$, ${\mu }_r=1.5$. (a) Phase diagram in the $(\kappa ,s_0)$ plane (details about its elaboration in [48]). (b) Positional order correlation function $\widehat{g}(r)$ in the crystal phase ($\kappa =0.35$, $s_0=0.05$, various system sizes). (c) Same as (b) but for the bond order correlation function ${\widehat{g}}_6(r)$. (d) Decay exponent $\eta$, extracted from plots similar to (b), as a function of $\kappa$ at various $s_0$ values; the dashed vertical lines indicate our estimates of the melting transition. (e) Linear variation of $\eta$ with $s_0^2$ at various $\kappa$ values; inset: same data rescaled by $a^2$, where $a$ was extracted from noise spectra such as in Fig. 2 (see main text).

Figure 2

(a) Spatial spectra of $\mathbf{s}$ for various $\kappa$ values. Black dashed curves are fits by the function $⟨|{\tilde{\mathbf{s}}}^2(k){|}^2⟩=s_0^2{D}_r\rho /(a+b{k}^2)$ obtained by solving Eq. (7) [48], from which we extract estimates of $a$. Inset: variation of $a$ with $\kappa$. (b) Spatial spectra $⟨|\mathbf{u}(k){|}^2⟩$ for $\kappa =0.3$, 0.35, 0.4, 0.44, 0.48, 0.52, 0.56 from bottom up (lower curves) and their collapse $⟨|\mathbf{u}(k){|}^2⟩/c^{*}(\kappa )$ vs $k/k^{*}(\kappa )$ obtained by choosing appropriate scaling parameters $c^{*}(\kappa )$ and $k^{*}(\kappa )$. (c) variation with $\kappa$ of $|\widehat{\mathbf{G}}{|}^2{c}^{*}{k}^{*2}/4\pi$ ($|\widehat{\mathbf{G}}{|}^2=4{\pi }^2$) and $\eta$ measured as in Fig. 1 from the decay of $\widehat{g}(r)$; inset: $c^{*}(\kappa )$ and $k^{*}(\kappa )$ values used to collapse spectra in (b). (d) $-\log {\widehat{g}}_6{|}_{r\to \infty }$ vs $s_0^2$ for $D_r=1$ and $\kappa =0$ (gray stars), $\kappa =0.17$, 0.26, 0.35, 0.44 (down triangles from right to left), and for various $D_r=\frac{1}{8},\frac{1}{4},\frac{1}{2},2,5,10,20,40,80,160,320,640$ at $\kappa =0$ (up triangles) (solid lines are near perfect fits to linear functions of $s_0^2$) (e) variation with $D_r$ (top) and $\kappa$ (bottom) of $s_0^2$-rescaled temperatures $t_6$ and $t_S$ extracted from fits shown in (d) and from corresponding estimates of $\eta$ (see text). [$L=384$, $s_0=0.05$ in (a), $s_0=0.01$ in (b)–(e)].

Figure 3

Highly distorted configuration in the crystal phase for which the effective decay exponent $\eta \simeq 14$ ($\kappa =0.56$, $s_0=0.01$, $L=1536$). (a) Displacement field $\mathbf{u}$ (color is orientation, brightness is magnitude) with superimposed contour lines drawn for integer values of $|\mathbf{u}|$ in units of lattice spacing. (b) Hexatic bond order field ${\psi }_6$ [color scheme as in (a)]. (c) Instantaneous static structure factor $S(\mathbf{q})={\rho }^{*}(\mathbf{q})\rho (\mathbf{q})$ where $\rho (\mathbf{q})=⟨{\mathrm{e}}^{i\mathbf{q}·{\mathbf{r}}_j}{⟩}_j$ (color in linear scale); the inset is a zoom on the region of the first quasi-Bragg peak. (d) Time-averaged static structure factor $⟨S(q_y){⟩}_t$ measured at $q_x=0$ superimposed on that obtained for $\kappa =0.35$ for which $\eta \simeq 0.027$. Inset: zoom on the first peak located at $q=q^{*}$ showing an algebraic divergence $|q_y-q^{*}{|}^{\eta -2}$ only for $\kappa =0.35$.

Figure 4

Crystal phase of active Brownian particles interacting via a repulsive WCA potential (initial conditions as in other figures, $\rho {\sigma }^2=1.25$, $\epsilon ={\mu }_r=1$). (a) $\widehat{g}(r)$ in the crystal phase ($s_0=8$, various system sizes), showing a decay exponent $\eta \simeq 0.58$; inset: $\eta$ vs $s_0^2/D_r$ for various $D_r$ and $s_0$ values ($L=768$). (b) Same as (a) but for the bond order correlation function ${\widehat{g}}_6(r)$; inset: ${\widehat{g}}_6^{\infty }$ vs $s_0^2/D_r$.