Seamlessly joining length scales: From atomistic thermal graphs to anisotropic continuum conductivity

Abstract

Thermal transport in complex solids is governed by local structure, defects, and anisotropy, yet most continuum models still rely on oversimplified, homogenized conductivities. Here, we bridge atomistic and continuum descriptions by building finite element (FE) models directly from the site-projected thermal conductivity (SPTC), an atomic-level decomposition of the Green-Kubo thermal conductivity. We introduce a new toolkit, the Simulator Collection for Atomic-to-Continuum Scales (SCACS), which uses a graph neural network to predict SPTC on large atomic structures, coarse-grain these fields into anisotropic conductivity tensors, and embeds them into the heat-flow FE equation with a customized, anisotropy-aware adaptive mesh refinement scheme. Applied to silicon nanostructures, the resulting FE models act as representative volume elements, reproduce bulk conductivities, and capture interfacial and defect-driven anisotropy while maintaining thermodynamic consistency. Additionally, SCACS predicts experimental conductance trends and fields. This innovation demonstrates a novel and general route for transferring atomistic transport information into device-scale thermal simulations with physics-based approximations.

Figures (5)

• Figure 1: Architecture and Training of sptc-ai. Empirical probability of high-SPTC outcome from binned distribution of the continuous structural descriptors, including (a) the coordination number (CN), (b) the root-mean-square deviation from the ideal tetrahedral angle ($\angle\mathrm{rms}$), (c) the mean (B$_\mu$) and (d) standard deviation (B$_\sigma$) of bond-lengths, and the (e) fractions of short (B$_\mathrm{S}$) and (f) long (B$_\mathrm{L}$) bonds. (g) Farthest point sampling score for all structures. (h) Mean absolute error (MAE) for the averaged directional SPTC ($\hat{\zeta}^\mathrm{c}$) and the directly predicted isotropic SPTC ($\hat{\zeta}^\mathrm{d}$).
• Figure 2: Performance of sptc-ai. The "NSPTC" colorbar indicates the normalized SPTC, ranging from 0 (blue) to 1 (red), while the "Structure class" colorbar encodes the OVITO-based local environment: amorphous (0; gray), cubic diamond (1; blue) and its first (2; cyan) and second (3; green) neighbors, and hexagonal diamond (4; orange) with its first (5; yellow) and second (6; lime) neighbors. (a) Model performance on held-out test structures: computed (top) and predicted (bottom) SPTC distributions for (left$\rightarrow$right) amorphous, defective, and purely crystalline Si. (b) Generalization of sptc-ai to an unseen amorphous–crystalline Si interface containing 2022 atoms, showing (i) structure class, (ii) predicted SPTC, and (iii) computed SPTC. (c--e) Structure class and predicted SPTC for (c) a 223,930-atom Si twin-grain-boundary nanowire, (d) a 134,923-atom polycrystalline Si nanowire, and (e) a 26,191-atom Si nanopillar. Panels (c) and (d) show mid-slices; the black rectangle in (e) highlights the zoomed mid-slice region of the nanopillar.
• Figure 3: Analysis of sptc2fem results for the amorphous–crystalline Si structure. (a) Anisotropic and isotropic SPTC (top) and coarse-grained conductivity (bottom); colormap shows normalized values. (b) Adaptive mesh refinement using the equal mass controller with Dörfler marking, showing coarse (left) and refined (right) meshes. (c, d) Dirichlet solve for thermal loading along $z$: magnitude of the temperature gradient and flux. (e) steady-state temperature for $\mathbf{K}(\mathbf{x})= \mathbf{1}$ and (F) for sptc-ai; arrows mark regions with differing temperature transitions. (f--l) Slices at $z = 30$ Å of directional fluxes, flux magnitude, temperature-gradient magnitude, and flux residuals $\eta$. (m) Histogram of continuity residual and (n) the AMR convergence.
• Figure 4: Analysis of the sptc2fem results for the other structures. (a) Twin grain boundary: isotropic conductivity (arrows mark EMC-refined regions), temperature gradient, and flux (left to right). (b) Polycrystalline Si: anisotropic SPTC (top) and coarse-grained conductivity (bottom); colormap shows normalized values, and (c) the magnitude of temperature gradient (top) and flux (bottom) for thermal loading along $x$, $y$, and $z$ (left to right). (d) Silicon nanopillar: magnitude of temperature gradient (left) and flux (right) for thermal loading along $x$, $y$, and $z$ (top to bottom).
• Figure 5: Experimental validation of the method. (a--d) Atomic-scale (left) and FEM-scale (right) 3C/2H nanowire models at annealing temperatures of 600, 700, 900, and 1000 $^\circ$C. The colormap spans low (black) to high (white) SPTC. (e) Experimental areal conductance BenAmor2019 compared with SPTC-derived values (W m$^{-2}$ K$^{-1}$); the inset shows the percent relative difference, $\Delta K = K_{\mathrm{exp}} - K_{\mathrm{eff}}$. (f--i) Mesh refinement showing the starting coarse (left) and final refined (right) meshes. The bottom panel shows convergence of the residual indicator threshold $\tau_\eta$ for each refinement series.