Experimental results for Dynamics of an interfacial bubble controls adhesion mechanics in a van der Waals heterostructure

<p>These are the COMSOL Multiphysics file used for our FEM calculations of frequency dispersion with gate voltage (<span class="math-tex">\(V_g^{dc}\)</span>) and stress at different values of <span class="math-tex">\(V_g^{dc}\)</span> for different conditions of graphene-hBN heterostructure interface.&nbsp;For the bubble growth rate equation, we considered linear gate-voltage dependence. Bubble growth will be restricted as the bubble diameter reaches the diameter of the drum. So the growth is conditional, it increases as the gate voltage increases with a constant slope till its radius reaches the drum radius (<span class="math-tex">\(R_f \)</span>); its governing equation is <span class="math-tex">\(R_{\textrm{bub}}(V_g^{\textrm{dc}})=R_i+mV_g^{\textrm{dc}} for R_{bub}\leq R_f\)</span>&nbsp;. Bubble growth stops at <span class="math-tex">\(R_{bub} ≈ R_f \)</span>, where <span class="math-tex">\(R_i \)</span>, <span class="math-tex">\(R_f \)</span>are the initial and final radii of the bubble and m is the increase in bubble radius per unit change in gate voltage&nbsp;(<span class="math-tex">\(µm/V\)</span>).</p>