Heat jet approach for finite temperature atomic simulations of triangular lattice

Abstract

In this paper, we propose a heat jet approach for atomic simulations at finite temperature of a triangular lattice. First we design a matching boundary condition by carefully examining a residual function based on the lattice dispersion relation. It leads to a two-way boundary condition, where prescribed incoming waves are included with a source term. Meanwhile, we adopt a phonon representation to determine Fourier mode amplitudes. The heat jet approach is then formulated by combining the two-way boundary condition and the phonon representation of heat source. Numerical tests of a tube-shaped computational domain illustrate the accuracy and effectiveness in simultaneously resolving thermal fluctuations and non-thermal motion at a given temperature.

Appendices

Appendix 1

In designing matching boundary conditions, the selection of involved atoms is crucial. The choice in this study is made after many tries, in addition to the experiences we gained through the study of other lattices. In the following, we choose two different ways with the same number of atoms as in (8).

First, we choose more atoms in the horizontal direction as shown in Fig. 18. The boundary condition reads

$$\begin{aligned} {\dot{u}}_{m,1}= & {} -a_2\left( {\dot{u}}_{m-1,2}+{\dot{u}}_{m+1,2}\right) -a_3\left( {\dot{u}}_{m-3,2}+{\dot{u}}_{m+3,2}\right) \nonumber \\&-\,a_4{\dot{u}}_{m,3}-a_5\left( {\dot{u}}_{m-2,3}+{\dot{u}}_{m+2,3}\right) \nonumber \\&-\,a_6\left( {\dot{u}}_{m-4,3}+{\dot{u}}_{m+4,3}\right) \nonumber \\&+\,b_1u_{m,1}+b_2\left( u_{m-1,2}+u_{m+1,2}\right) \nonumber \\&+\,b_3\left( u_{m-3,2}+u_{m+3,2}\right) \nonumber \\&+\,b_4u_{m,3}+b_5\left( u_{m-2,3}+u_{m+2,3}\right) \nonumber \\&+\,b_6\left( u_{m-4,3}+u_{m+4,3}\right) , \end{aligned}$$

(33)

with

$$\begin{aligned} a_2= & {} -0.042277, a_3=-0.414167, \nonumber \\ a_4= & {} -1.481894, a_5=-0.122950, a_6=0.252025, \nonumber \\ b_1= & {} -2.341130, b_2=4.560714, \nonumber \\ b_3= & {} -1.415865, b_4=-3.758910,\nonumber \\ b_5= & {} -0.186008, b_6=0.091179. \end{aligned}$$

(34)

In Fig. 19, we observe a reflection coefficient much larger than that for (8). This boundary condition performs reasonably well for oblique waves (e.g. in the direction $\pm \pi /6$), however, it can not suppress reflection as well for waves with large wave numbers, even at normal incidence.

Fig. 20

Schematic plot for the lattice around the bottom boundary, with more atoms in the vertical direction

Fig. 21

Reflection coefficient modulus for the bottom boundary condition, with more atoms in the vertical direction

If instead we choose more atoms in the vertical direction as shown in Fig. 20. The boundary condition reads

$$\begin{aligned} {\dot{u}}_{m,1}= & {} -a_2\left( {\dot{u}}_{m-1,2}+{\dot{u}}_{m+1,2}\right) -a_3{\dot{u}}_{m,3}\nonumber \\&-\,a_4\left( {\dot{u}}_{m-1,4}+{\dot{u}}_{m+1,4}\right) -a_5{\dot{u}}_{m,5}\nonumber \\&-\,a_6\left( {\dot{u}}_{m-1,6}+{\dot{u}}_{m+1,6}\right) +b_1u_{m,1}\nonumber \\&+\,b_2\left( u_{m-1,2}+u_{m+1,2}\right) +b_3u_{m,3} \nonumber \\&+\,b_4\left( u_{m-1,4}+u_{m+1,4}\right) \nonumber \\&+\,b_5u_{m,5}+b_6\left( u_{m-1,6}+u_{m+1,6}\right) . \end{aligned}$$

(35)

with

$$\begin{aligned} a_2= & {} -2.019414, a_3=4.543445, \nonumber \\ a_4= & {} -4.622717, a_5=5.040833, a_6=1.429083,\nonumber \\ b_1= & {} -1.691923, b_2=4.864710, \nonumber \\ b_3= & {} -15.841764,\nonumber \\ b_4= & {} 10.850438, b_5=-26.558221, b_6=6.330807. \end{aligned}$$

(36)

The reflection coefficient is displayed in Fig. 21. For an oblique incident wave with large angle, it appears to be unstable with the reflection coefficient modulus bigger than 1.

We remark that in case of inputs from all boundaries, both choices require special treatment for more corner atoms, whose involved atoms go out of the lattice under simulation.

From these comparisons, it is clear that the choice of boundary atoms in Fig. 3 is appropriate, with which the boundary condition (8) well treats both normal incidence with large wave numbers, and large angle incidence.

Fig. 22

Dislocation propagation under an external force $f=0.03$ with heat jet approcah

Fig. 23

Dislocation propagation under an external force $f=0.03$: a displacement u(0); b displacement u(550); c instantaneous temperature T(0); d instantaneous temperature T(550)

Appendix 2

The heat jet approach applies to weakly nonlinear lattices. This has been demonstrated in [23] for a one-dimensional chain with Morse potential. Here we present one more example, which models dislocation motion at finite temperature.

Similar to [30, 31], we use the Frenkel-Kontorova potential

$$\begin{aligned} U(x)=\displaystyle \sum _i \left[ \frac{k_1}{2}(x_i-x_{i-1}-a)^2 + \frac{k_2}{2}\cos (x_i-a i) \right] . \end{aligned}$$

(37)

Here, $x_i=u_i+a i$ is the position of i-th atom, and $u_i$ is its displacement away from equilibrium.

The dimensionless governing equation is

$$\begin{aligned} {\ddot{u}}_i=k_1 \left( u_{i-1}-2u_i+u_{i+1} \right) - k_2 \sin u_i + f \end{aligned}$$

(38)

Following [30], the characteristic mass, length and time are chosen as $m_c=26.98amu$, $L_c=3.253A$ and $t_c=2 \times 10^{-13}s$, respectively. The elastic constants are $k_1=1$ and $k_2=0.7$. We rescale temperature by $k_B t_c^2/m_c L_c^2$. The phonon heat source temperature 200K hence gives $T_L=T_R=0.0232$. A constant force $f=0.03$ is applied to each atom. See Fig. 22 and Fig. 23, the dislocation moves from right to left. The amplitudes near the dislocation becomes larger due to the reflection of incident wave reaching at the dislocation. The instantaneous temperature forms a gradient near dislocation. The results are comparable to those by MS-NEMD simulation (Fig. 3 and Fig. 4 in [30]). We remark that we do not perform multiscale decomposition for the motion.

This example shows that the heat jet approach can be applied to non-equilibrium MD to simulate a dislocation in a nonlinear chain.