Shear-induced phase behavior and topological defects in two-dimensional crystals

We investigate through numerical simulations how a two-dimensional crystal yields and flows under an applied shear. We focus on a range that allows us to both address the response in the limit of an infinitesimal shear rate and describe the phase behavior of the system at a finite shear rate. In doing so, we carefully discuss the role of the topological defects and the finite-size effects. We map out the whole phase diagram of the flowing steady state in the plane formed by temperature and shear rate. Shear-induced melting of the two-dimensional crystal is found to proceed in two steps: first, the solid loses long-range bond-orientational order and flows, even for an infinitesimal shear rate (in the thermodynamic limit). The resulting flowing hexatic phase then melts to a flowing, rather isotropic, liquid at a finite shear rate that depends on temperature. Finally, at a high shear rate, a third regime corresponding to a strongly anisotropic stringlike flowing phase appears.

Figure 1

(a) Phase diagram of a sheared two-dimensional crystal in its flowing steady state in the plane of the shear rate $\dot{\gamma }$ and the temperature $T$. Red dots indicate the phase points corresponding to the snapshots displayed in (b)–(d). In regime I, we observe a plastic flow with nucleated free dislocations and hexatic quasi-long-range order (QLRO). A representative snapshot is shown in (b) for $T=0.003$ and $\dot{\gamma }=2\times {10}^{-4}$. Blue, white, and red particles have 5, 6, and 7 neighbors, respectively, and a pair of red and blue particles form a dislocation. In regime II, the dislocations are unbound and free disclinations, shown as isolated red and blue particles, are nucleated. Concomitantly, bond-orientational order has a short-ranged, exponential, spatial decay and the system is in a flowing liquid phase. A representative snapshot is given in (c) for $T=0.003$ and $\dot{\gamma }=1\times {10}^{-2}$. One can see an isolated disclination with 7 neighbors, as indicated by a circle. In regime III, we observe a stringlike flow in which particles mostly move along lanes following the direction of shear. The system is then strongly anisotropic. The corresponding snapshot is shown in (d) for $T=0.003$ and $\dot{\gamma }=4\times {10}^{-1}$ and an inset illustrates representative particle trajectories in the bulk of the system over a strain change $\mathrm{\Delta }\gamma =1.2$.

Figure 2

Flow curves for a crystal of $N=14400$ particles under uniform simple shear. (a) Log-log plot of the averaged shear stress $\overline{\sigma }$ vs the shear rate $\dot{\gamma }$ for several temperatures. (b) Zoom-in plot of (a).

Figure 3

(a) Log-log plot of the effective viscosity $\eta =\overline{\sigma }/\dot{\gamma }$ as a function of the shear rate $\dot{\gamma }$ for the same data as in Fig. 2. The dashed straight line shows the dependence $\eta \sim {\dot{\gamma }}^{-1}$. (b) Log-log plot of the effective viscosity as a function of the density of dislocations ${\rho }_{\mathrm{disl}}$. The dashed line corresponds to $\eta \sim {\rho }_{\mathrm{disl}}^{-1}$.

Figure 4

(a) Averaged square modulus of the bond-orientational order parameter, $\overline{|{\psi }_6{|}^2}$, as a function of shear rate for various temperatures and system sizes. Triangles (with dotted line), diamonds (dashed line), and circles (solid line) correspond to data for $N=900$, 3600, and $14400$, respectively. (b) Spatial decay of the bond-orientational correlation function $g_6\left(r\right)$ for $T=0.0030, N=14400$, and a wide range of $\dot{\gamma }$. The grey dashed line represents the bound imposed on a power-law decay by the KTHNY theory, $g_6\left(r\right)\sim r^{-1/4}$.

Figure 5

Probability to find at least one free disclination in the system, $p_{\text{disc}}$, as a function of $\dot{\gamma }$ and $T$ for $N=14400$ particles.

Figure 6

Real part of the bond-orientational order parameter $\Re \left\{{\psi }_6\right\}$ obtained in a single trajectory as a function of the shear strain $\gamma$ for a fixed shear rate $\dot{\gamma }=1\times {10}^{-3}$ and temperature $T=0.0030$ (corresponding to regime I). Different system sizes $N$ are shown.

Figure 7

Crystal-like rotation as seen from real-space snapshots (top) and the associated instantaneous static structure factor $S_{\gamma }\left(\mathbf{k}\right)$ (bottom) for strain values $\gamma =9.0,9.7,10.1$, and 10.5 (from left to right) which correspond to the maximum, the decreasing section, the minimum and the increasing section of the oscillation shown in Fig. 6. The snapshots are colored according to the value of the real part of the local bond-orientational order parameter, $\Re \left\{{\varphi }_{6,j}\right\}$. The system size is $N=3600$, the temperature $T=0.0030$, and the shear rate $\dot{\gamma }=1\times {10}^{-3}$ (regime I).

Figure 8

Instantaneous value of the bond-orientational correlation function $g_{6,\gamma }\left(r\right)$ for several values of the strain $\gamma$ (solid colored lines) and its value averaged over a period (black dashed line) for a system of $N=14400$ particles at $T=0.0030$ and $\dot{\gamma }={10}^{-3}$ (regime I).

Figure 9

Mean square displacement $\mathrm{\Delta }{y}^2\left(\gamma \right)$ along the direction perpendicular to the shear for one trajectory in the steady state as a function of the strain $\gamma$ for two different shear rates, $\dot{\gamma }={10}^{-3}$ (a) and $0.8$ (b), at a temperature $T=0.0030$. $\gamma$ is measured from a configuration in the steady state. The top panel corresponds to regime I and the bottom one to regime III.

Figure 10

Radial distribution function $g\left(r\right)$ for a system of $N=14400$ particles at a temperature $T=0.0030$ and for several shear rates $\dot{\gamma }$ covering the three regimes of flow. The data for different $\dot{\gamma }$ are shifted along the $y$ axis for clarity.

Figure 11

Averaged square modulus of the fourfold bond-orientational order parameter, $\overline{|{\psi }_4{|}^2}$, as a function of shear rate for a temperature $T=0.0030$ and several system sizes. Triangles, diamonds, and circles correspond to data for $N=900$, 3600, and $14400$, respectively.

Figure 12

(a) Flow curve of the two-dimensional crystal undergoing the SLLOD dynamics at $T=0.0001$ for a wide range of the strain rate for $N=3600$ and $10000$. (b) Corresponding effective viscosity. The black dashed line represents the divergence of the viscosity as a power law, $\eta \sim {\dot{\gamma }}^{-1}$.

Figure 13

System-size dependence of ${min}_{\dot{\gamma }}\left\{\overline{|{\psi }_6{|}^2}\right\}$, the minimum value over $\dot{\gamma }$ reached by $\overline{|{\psi }_6{|}^2}$ in Fig. 4, for various temperatures below the putative $T_{m,\mathrm{sol}}$. The dashed and dotted lines indicates a $L^{-1/4}$ and a $L^{-1}$ dependence, respectively.

Figure 14

Flow curves obtained from the Brownian dynamics for the averaged shear stress $\overline{\sigma }$ (a) and the effective viscosity $\eta$ (b) for several system sizes $N$. Triangles (with dotted line), diamonds (dashed line), and circles (solid line) correspond to data for $N=900$, 3600, and $14400$, respectively.

Figure 15

Density of free dislocations, ${\rho }_{\mathrm{disl}}$, as a function of the shear rate $\dot{\gamma }$ for $N=14400$. The dashed straight line corresponds to ${\rho }_{\mathrm{disl}}\sim \dot{\gamma }$.

Figure 16

Viscosity $\eta$ of the system as a function of the dislocation density ${\rho }_{\mathrm{disl}}$ for various system sizes. Triangles (with dotted line), diamonds (dashed line), and circles (solid line) correspond to data for $N=900$, 3600, and $14400$, respectively.

Figure 17

Radial distribution function $g\left(r\right)$ for systems with $N=3600$ (dashed curve) and $14400$ (solid curve) particles for various values of $T$ and $\dot{\gamma }$. $g\left(r\right)$'s are shifted vertically by hand for clarity.

Figure 18

Sixfold bond-orientational correlation function, $g_6\left(r\right)$, for a system of $N=3600$ (dashed curves) and $14400$ (solid curves) particles. The gray dashed straight lines in the background represent the upper bound imposed on the exponent ${\eta }_6$ of the power-law decay for a hexatic phase by the KTHNY theory, $g_6\left(r\right)\sim r^{-1/4}$.

Figure 19

Orientational correlation function $g_6\left(r\right)$ for a system size of $N=14400$ (solid-lines) and $N=57600$ (dash-dotted lines) in the vicinity of the transition between regimes I and II for several temperatures.

Figure 20

Structure factor for the direction transverse to the flow, $S_{\mathrm{T}}\left(k_y\right)$, for a system of $N=14400$ particles and various temperatures and shear rates. As $\dot{\gamma }$ increases at low enough temperature, sharp primary and secondary peaks appear near $k_y\approx 7.7$ and $k_y\approx 15.4$.