Dynamical transitions and sliding friction of the phase-field-crystal model with pinning

We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.

Figure 1

Temperature dependence of the (a) scaled structure-factor peak $S\left(k_c\right)/N_p$; (b) specific heat $C$, and mobility $\mu$ for the model without an external driving force.

Figure 2

Velocity response as a function of applied force. Arrows correspond to critical values $f_a$, $f_b$, and $f_c$.

Figure 3

Scaled structure-factor peak $S\left(Q\right)/N_p$ as a function of applied force. Here, $\vec{Q}$ stands for the primary reciprocal lattice vector for either the $c\left(2\times 2\right)$ phase $\left({\vec{k}}_c\right)$ or the hexagonal phase $\left({\vec{k}}_h\right)$. Filled and open symbols correspond to increasing and decreasing forces, respectively.

Figure 4

Fraction of density peaks $p_k$ with coordination number $k$ (4,5,6, and 7 nearest neighbors) for increasing applied force.

Figure 5

Fraction of density peaks $p_k$ with coordination number $k$ for decreasing applied force.

Figure 6

(Color online) Snapshot of the density field in the sliding state at $f_x=0.0525$.

Figure 7

Configuration of the density peak locations with corresponding coordination numbers for the density plot in Fig. 6.

Figure 8

(Color online) Snapshots of the density field for increasing times after starting from the moving state of Fig. 6 and reaching the pinned commensurate state, at a force $f_x=0.045$ just below $f_b$. Times are in units of ${10}^3$ time steps.