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

These are the COMSOL Multiphysics for our FEM calculations of frequency dispersion with gate voltage (\(V_g^{dc}\)) and stress at different values of \(V_g^{dc}\) for different conditions of graphene-hBN heterostructure interface. 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 (\(R_f \)); its governing equation is \(R_{\textrm{bub}}(V_g^{\textrm{dc}})=R_i+mV_g^{\textrm{dc}} for R_{bub}\leq R_f\) . Bubble growth stops at \(R_{bub} ≈ R_f \), where \(R_i \), \(R_f \)are the initial and final radii of the bubble and m is the increase in bubble radius per unit change in gate voltage (\(µm/V\)).