## 2 Materials and Methods
### Simulation Model Development  
The simulation model developed here is based on a tungsten microhotplate suspended above the substrate by six heaters, described in Section 2.1, with dimensions shown in Figure 1. The simulation model was created in COMSOL (version 6.2) using the following modules:  
* Solid Mechanics;
* Electric Currents;
* Heat Transfer in Solids;
* Multiphysics;
* Transport in Solids. The first four modules were used to accurately simulate the electro thermomechanical behavior of the system, including the resistive heating of the device, thermal expansion due to heating, conduction through the material, the radiative cooling of the plate, and mechanical fixed constraints at the clamps. The Transport in Solids module was used to introduce mass transport that is specified by [PERSON]'s law of diffusion. Vacancy generation and annihilation are controlled by the source term, and the vacancy fluxes are introduced by external flux nodes. As for the boundary conditions, the conditions of the simulated environment are presented in Table 3. The ambient conditions were set at room temperature. The voltage was applied to half of the electric pads and was set to 0.59 V (with the remainingpads held at ground) to reproduce the testing conditions and to reach a temperature of approximately 1580 \({}^{\circ}\)C of the device, shown in Figure 3. The mechanical fixed constraint was placed at all six clamps. Thermal pathways were limited to conduction through the material and radiative cooling. A list of the input parameters, their values and references can be found in Table 4.  
A list of the variables related to atomic migration and derived values from the simulation are shown in Table 5. For all formulas presented in this table, a complete list of the symbols and their definitions are provided in the Nomenclature section. The following material was selected and applied to whole geometry from the COMSOL Material Library:  
Material Library > Elements > Tungsten > Phase: solid > Variation: Binkele. With this choice of material, all material properties required from the simulations are specified in the material library as being temperature-dependent. To create a mesh that is suitable for simulating the system, an extra-fine mesh element size was chosen as the variables listed in Table 5 were dependent on hydrostatic stress, which requires a fine mesh to simulate accurately. The stress simulations are sensitive to the mesh size, and applying a sufficiently fine mesh is critical to ensure an accurate simulation of the terms related to stress. To validate the accuracy of the simulation, a mesh convergence study was performed, confirming that the chosen mesh resolution captured the stress gradients without numerical artifacts. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Units** & **Reference** \\ \hline Pre-exponential factor \(D_{0}\) & \(5.2\times 10^{-8}\) & \(\frac{\text{m}^{2}}{\text{s}}\) & [26] \\ Diffusion activation energy \(E_{aDB}\) & 5.2 & eV & [25; 27] \\ Vacancy formation activation & 3.3 & eV & [25; 28; 29] \\ energy \(E_{av}\) & 3.3 & eV & [25; 28; 29] \\ Molar mass of tungsten \(A_{W}\) & 183.84 & \(\frac{\text{g}}{\text{mol}}\) & [30] \\ Vacancy relaxation ratio \(f\) & 0.73 & - & [31] \\ Vacancy relaxation time \(\tau\) & \(10^{-4}\) & s & [32] \\ Atomic volume \(\Omega\) & \(2.48^{-29}\) & \(\frac{\text{m}^{3}}{\text{atom}}\) & [27] \\ Effective valence \(z^{*}\) & \(-20\) & - & [27] \\ Heat of transport \(Q^{*}\) & \(9.4\times 10^{-3}\) & eV & [27] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: A list of parameters specified for the simulation of atomic migration in a tungsten high-temperature MEMS. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Environmental Conditions** & **Value** & **Units** \\ \hline Ambient temperature \(T_{0}\) & 20 & \({}^{\circ}\)C \\ Reference temperature \(T_{ref}\) & 20 & \({}^{\circ}\)C \\ Applied voltage \(V_{0}\) & 0.59 & V \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Conditions of the simulated environment.