Algorithm to enforce uniform density in liquid atomistic subdomains with specular boundaries

An important component in hybrid atomistic-continuum (HAC) modeling of liquids is the proper termination of the atomistic subdomain (${\Omega }_{\mathrm{A}}$). When the HAC model is based on the Schwarz domain decomposition method, the total number of particles in ${\Omega }_{\mathrm{A}}$ is conserved using specular boundaries in an overlap region, where information is exchanged with the continuum subdomain (${\Omega }_{\mathrm{C}}$). This non-periodic termination of ${\Omega }_{\mathrm{A}}$ fails to account for forces otherwise included in a periodic system, through the minimum image convention. The absence of such forces results in spurious density fluctuations adjacent to the specular boundaries. In this work, we present a new boundary force algorithm that establishes a uniform density profile at non-periodic boundaries of ${\Omega }_{\mathrm{A}}$. We also examine the effects of the non-periodic termination of ${\Omega }_{\mathrm{A}}$ on the liquid properties. The algorithm relies on force measurements carried out over a spatially discretized atomistic subdomain. It is relatively straightforward to implement and can be seamlessly extended to higher dimensions.

Figure 1

Top: Depiction of the HAC model overlap region (${\Omega }_{\mathrm{O}}$) where information is exchanged between ${\Omega }_{\mathrm{A}}$ and ${\Omega }_{\mathrm{C}}$. Also shown are subdomain boundaries ${\Gamma }_{\mathrm{A}}$, and ${\Gamma }_{\mathrm{C}}$, respectively. Bottom: Specular $y$-$z$ plane terminating ${\Omega }_{\mathrm{A}}$, and its corresponding region of lost interactions for a particle $i$ within a distance $r_c$ from the boundary ${\Gamma }_{\mathrm{A}}$.

Figure 2

Mean density profile comparison for periodic (PB) and non-periodic (NPB) ${\Gamma }_{\mathrm{A}}$, respectively. Only reflections are applied in the NPB case with no boundary force.

Figure 3

Average bin forces for: (a) $F_{x\mathrm{bin},k}^{*}$, (b) $F_{y\mathrm{bin},k}^{*}$, and (c) $F_{z\mathrm{bin},k}^{*}$, with periodic (PB) and non-periodic (NPB) ${\Gamma }_{\mathrm{A}}$, respectively.

Figure 4

Evolution of ${\mathbf{F}}_{\mathrm{b},k}$ along the $x$ axis (normal to the NPB ${\Gamma }_{\mathrm{A}}$) for selected bins (a) $k=$ {1, 16, 19, 27, 32, 80}, and (b) $k=$ {557, 605, 610, 618, 621, 636}. Forces shown are exponentially smoothed following Eq. (2), and are reversed (multiplied by $-1$), i.e., they are displayed in their applied form. There are 1000 samples in total taken once every time period of $2.3\tau$.

Figure 5

Mean boundary force $\langle F_{x\mathrm{b},k}^{*}\rangle$ for the NPB case shown for bins within $r_c$ from the specular boundary ${\Gamma }_{\mathrm{A}}$ at: (a) $-0.5L_x$, and (b) $0.5L_x$. Bins from Fig. 4 are indicated with markers. Average (c) density, and (d) temperature profiles for the NPB case under the application of $F_{x\mathrm{b},k}$.

Figure 6

One-dimensional mean-squared displacement (${\mathrm{MSD}}_x$): (a) for the PB case and selected bins $k:5$, 30, 40, and 60 from the NPB case with the boundary force $F_{x\mathrm{b},k}$ applied via Eq. (2); (b) Points of deviation from the periodic ${\mathrm{MSD}}_x$ for the selected bins.

Figure 7

Mean-squared displacement (${\mathrm{MSD}}_x$): (a) comparison between bin $k=1$ for the NPB case and that of the PB case; (b) Short period ${\mathrm{MSD}}_x$ with the vertical line dividing the lower and higher diffusion regions for NPB bin $k=1$.

Figure 8

The self-space-time correlation function $G_s$ for the NPB case. (a) $G_s$ vs $x^{*}$ for $t^{*}=0–0.0207$. (b) $G_s$ vs $t^{*}$ for $x^{*}=0–0.054$.

Figure 9

Comparison between $G_s$ for the NPB case and its short time approximation ${\tilde{G}}_s$ vs $x^{*}$.

Figure 10

Shear velocity profile comparison for the PB and the NPB cases. Regions of ${\mathrm{set}}_1$ and ${\mathrm{set}}_2$ where velocity swapping takes place are highlighted.

Figure 11

Two-dimensional density contours for ${\Omega }_{\mathrm{A}}$ with additional $x$-$z$ specular planes: (a) particle reflections only, (b) with boundary forces $F_{x\mathrm{b},k}^{*}$, and $F_{y\mathrm{b},k}^{*}$ applied independently.

Figure 12

Average boundary force contours with additional specular planes inserted at $\pm 0.5L_y$ for (a) $\langle F_{x\mathrm{b},k}^{*}\rangle$, and (b) $\langle F_{y\mathrm{b},k}^{*}\rangle$, respectively.