## 2 Materials and Methods
### Vacancy Flux Model  
Atomic migration is a term encompassing various processes of material transport within solid materials. In metals, atomic migration typically occurs through the movement \begin{table}
\begin{tabular}{l l} \hline \hline
**Testing Specifications** & \\ \hline Voltage & 0.6 V \\ Current type & DC \\ Hotplate temperature & 1577 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Testing conditions. Figure 3: Tungsten microhotplate. (**a**) Cross-section of schematic with marked area of focus; (**b**) image of heater arm at start of test at 1580 \({}^{\circ}\)C; (**c**) image of heater arm after 224 h, just before failure; (**d**) image of heater arm failing due to visibly increased current density; (**e**) image of heater arm after device failure: fracture on heater. of mobile defects, such as vacancies. This migration process is governed by diffusion laws, where atoms move by occupying neighboring vacancies. [PERSON]'s first law of diffusion relates diffusive flux \(J\) to the gradient of concentration \(c\). It suggests that the movement of particles from high to low concentration is directly proportional to the concentration gradient [25]:  
\[J=-D\frac{\partial c}{\partial x} \tag{2}\]  
The number of atoms diffusing per unit time across a unit area per unit concentration gradient is known as diffusivity or diffusion coefficient \(D\). In metals, diffusion typically follows the vacancy diffusion mechanism. Atoms move from one lattice site to another, provided they have sufficient energy to overcome a local activation energy barrier and that vacancies are available. The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy needed to move an atom into the vacancy. In addition to bulk diffusion, grain boundary diffusion plays a crucial role in metals. Here, net mass transport occurs along grain boundaries, which may act as sources or sinks for atomic fluxes. Grain boundary diffusion, which occurs through a vacancy mechanism, is significant across both small-angle and large-angle boundaries. Diffusivity for vacancy diffusion along the grain boundaries can be described by the Arrhenius law [20]:  
\[D=D_{0}exp\bigg{(}\frac{-E_{aDB}+\sigma_{hyd}\Omega}{kT}\bigg{)} \tag{3}\]  
where \(D_{0}\) is the pre-exponential factor and \(E_{aDB}\) is the activation energy for the diffusion along the grain boundaries. Hydrostatic stress \(\sigma_{hyd}\) and atomic volume \(\Omega\) take into account the effect of mechanical stress on diffusion [20]. In metals, atomic migration involves vacancy diffusion, where material transport is expressed in terms of vacancy flux \(\overset{\rightarrow}{J}_{v}\). When the vacancy flux diverges, vacancies can accumulate or diminish based on the divergence's sign, as governed by the continuity equation:  
\[\frac{\partial c_{v}}{\partial t}=-\
abla\overset{\rightarrow}{J}_{v}+G \tag{4}\]  
where \(c_{v}\) is the vacancy concentration and \(t\) corresponds to time. The source term \(G\) accounts for vacancy generation or annihilation and depends on vacancy relaxation time \(\tau\) relative to equilibrium vacancy concentration \(c_{v0}\):  
\[G=\frac{c_{v}-c_{veq}}{\tau}=\frac{c_{v}-c_{v0}\text{exp}\bigg{(}\frac{(1-f) \sigma_{hyd}\Omega}{kT}\bigg{)}}{\tau} \tag{5}\]  
where relaxation time \(\tau\) determines the rate at which vacancy concentration reaches a steady state, with smaller \(\tau\) values corresponding to faster stabilization, and \(f\) is a relaxation factor. The initial vacancy concentration, \(c_{v0}\), is defined as  
\[c_{v0}=N\ exp\bigg{(}\frac{-E_{av}}{kT}\bigg{)} \tag{6}\]  
where \(N\) is a total number of atomic sites and \(E_{av}\) is the vacancy formation energy. In metals, atomic migration, primarily through vacancy diffusion, contributes to total material transport and can be described in terms of vacancy flux, incorporating vacancy fluxes driven by external forces.