Discrete-to-continuum modeling of weakly interacting incommensurate chains

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.

Figure 1

Reference (top) and deformed (bottom) configurations of the system of two chains in dimensional coordinates.

Figure 2

Reference (top) and deformed (bottom) configurations of the system of two chains in nondimensional coordinates.

Figure 3

Finding $d_{ij}$.

Figure 4

Definition of the lattice offset.

Figure 5

Discrete simulation result with $\sigma =1.0, \omega =1.0, h_1=1.0, h_2=0.99, k_b=100.0$, and $k_s=10.0$. The atoms on the rigid chain (not shown) occupy positions $(j,0)$, where $j$ is an integer. Inset (a) shows that in the commensurate regions, each atom falls into a potential well created by the rigid chain. Inset (b) shows how the wrinkle allows six atoms to span over five potential wells. The two lines of overlapping circles below the plot form the one-dimensional moiré pattern.

Figure 6

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=1.0$, and $k_b=1.0$. The parameter $h_2=1.01$ and $h_2=0.99$ on the left and right, respectively. The discrete simulation involves 100 atoms on the rigid chain, while there are 99 (left) and 101 (right) atoms on the deformable chain per period.

Figure 7

Details of atomistic structure near the wrinkle when $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=1.0$, and $k_b=1.0$. The parameter $h_2=1.01$ and $h_2=0.99$ on the left and right, respectively. The discrete simulation involves 100 atoms on the rigid chain, while there are 99 (left) and 101 (right) atoms on the deformable chain per period.

Figure 8

Discrete and continuum displacements for $\sigma =1.0, \omega =1.0, h_1=1.0, h_2=0.99, k_s=1.0$, and $k_b=1.0$. The horizontal and vertical displacements are shown on the left and right, respectively. The discrete simulation involves 100 atoms on the rigid chain, while there are 101 atoms on the deformable chain per period.

Figure 9

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=100.0$, and $k_b=1.0$. The parameter $h_2=1.01$ and $h_2=0.99$ on the left and right, respectively. The discrete simulation involves 100 atoms on the rigid chain, while there are 99 (left) and 101 (right) atoms on the deformable chain per period.

Figure 10

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=1.0$, and $k_b=100.0$. The parameter $h_2=1.01$ and $h_2=0.99$ on the left and right, respectively. The discrete simulation involves 100 atoms on the rigid chain, while there are 99 (left) and 101 (right) atoms on the deformable chain per period.

Figure 11

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=10.0$, and $k_b=1.0$. The parameter $h_2=1.003$ and $h_2=0.9967$ on the left and right, respectively. The discrete simulation involves 300 atoms on the rigid chain, while there are 299 (left) and 301 (right) atoms on the deformable chain per period.

Figure 12

Details of atomistic structure near the wrinkle when $\sigma =1.0, \omega =1.0, h_1=1.0, k_s=10.0$, and $k_b=1.0$. The parameter $h_2=1.003$ and $h_2=0.9967$ on the left and right, respectively. The discrete simulation involves 300 atoms on the rigid chain, while there are 299 (left) and 301 (right) atoms on the deformable chain per period.

Figure 13

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, h_2=0.9967, k_s=10.0$, and $k_b=100.0$. The plot on the left shows the simulation results over a full period, while the plot on the right shows the details of the atomistic structure near the wrinkle. The discrete simulation involves 300 atoms on the rigid chain and 301 atoms on the deformable chain per period.

Figure 14

Discrete and continuum simulation results with $\sigma =1.0, \omega =1.0, h_1=1.0, h_2=0.99, k_s=100.0$, and $k_b=1.0$. The discrete simulation involves 300 atoms on the rigid chain and 303 atoms on the deformable chain per period. There are three wrinkles to accommodate the three extra atoms on the chain. The two lines of overlapping circles below the plot form the one-dimensional moiré pattern.

Figure 15

van der Waals force on an atom of the deformable chain when this atom is the distance $y$ above the midpoint between two adjacent atoms of the rigid chain (dash-dotted line) or the distance $y$ directly above an atom of the rigid chain (solid line). The atom is in equilibrium when $f=0$.

Figure 16

Discrete and continuum simulation results with $\sigma =3.0, \omega =1.0, h_1=1.01, h_2=0.99, k_s=10.0$, and $k_b=100.0$. The discrete simulation involves 99 atoms on the rigid chain and 101 atoms on the deformable chain per period. Note that the vertical scale is approximately 100 times smaller than the vertical scale in Figs. 6–14.