# Design of a Piezoelectrically Actuated Ultrananocrystalline Diamond (UNCD) Microantilever Biosensor
## 3 Results and Discussion
#### 3.1.2 Simulation of the Q Factor  
Although a decrease in oscillation amplitude is generally not expected in undamped simulations, in this study, amplitude decay was observed because thermoelastic damping was explicitly included in the model. This was achieved by coupling the Heat Transfer in Solids module with the Solid Mechanics module in COMSOL Multiphysics. Thermoelastic damping arises because of the irreversible heat flow between the compressed and extended regions of the oscillating microcantilever, which leads to energy dissipation over time.  
The decay in amplitude, therefore, reflects a physically meaningful damping mechanism, not just numerical damping from the solver. This inclusion allows for a more realistic estimation of the quality factor (\(Q\)) and validates the time domain analysis used in this study. The decay observed in the oscillation peaks (as shown in Figure 6a,b can thus be attributed primarily to the thermoelastic losses captured by the multiphysics coupling in the simulation.  
Using the amplitude decay between two consecutive microcantilever oscillation peaks shown in Figure 6a,b and applying Equations (8)-(10), the quality factor (\(Q\)) was computed, as summarized in Table 3. Figure 6a corresponds to the simulation performed with a Backward Differentiation Formula (BDF) solver of maximum order 2, while Figure 6b shows the results using a BDF solver with maximum order 5.  
The validation results suggest that using a lower maximum order for the BDF solver yields more accurate outcomes. This may be attributed to the fact that applying an electrical pulse excites not only the fundamental mode but also the higher-order vibrational modes of the microcantilever. A lower-order BDF solver introduces greater numerical damping, which helps suppress the influence of these higher modes, thereby enhancing the stability of the simulation and, in this case, improving the accuracy of the numerical model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference Quality** & **BDF Solver** & **Simulated Quality** & **Percent Difference** **(\%) from** **Experiment** \\
**Factor \(Q\) (kHz)** & **Maximum Order** & **Factor \(Q\) (kHz)** & **Ref. [27]** \\ \hline
1650 & 2 & 1647.4 & \(-1.57\) \\
1650 & 5 & 1466 & \(-11.1\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Results of the quality factor \(Q\) for the first resonant frequency. The minus sign in the percent difference indicates that the value is underestimated with the experimentally measured \(Q\) for the first resonant frequency of the microcantilever reported in [27].  
Figure 6: Microcantilever displacement vs. time for the metal/AlN/metal/NCD-integrated microcantilever, (**a**) using BDF with maximum order 2 and (**b**) using the BDF solver with maximum order 5. The red arrows show the points used for the calculation of the Q value.  
This observation is supported by the fact that the displacement at Point 1 in Figure 6a,b is identical, while the displacement at Point 2 differs between the two simulations, as shown in Table 4. Without numerical damping, both displacements at Point 2 should be equal. For our purpose, the order BDF solver introduces less numerical damping, which improves the \(Q\) factor estimation.