Through-thickness coupling of diffusion and viscoelastic relaxation in the transient bending of bilayer microcantilevers

RESUMO

Static-mode microcantilever sensors detect chemical analytes through the bending of a bilayer beam whose polymer coating swells upon absorption. The transient bending is usually modeled as if the coating absorbed analyte uniformly, which is adequate for thin coatings but breaks down as the coating thickens and a moving diffusion front couples to depth-dependent viscoelastic relaxation. We resolve this coupling with a purpose-built finite-element solver requiring only one linear solve per time step, and show that the through-thickness structure controls the transient response in ways uniform-uptake models cannot capture. Verified against a published analytical solution, the solver reproduces the full transition from a pronounced curvature overshoot (up to nearly twice the steady-state value) when diffusion is fast relative to relaxation to a monotonic rise when it is slow. In the intermediate-thickness regime the analytical solution cannot reach, the thickness maximizing the steady-state signal and the thickness preserving the transient overshoot are governed by different physics, one geometric and one dynamic, so one coating cannot be optimized for both. The overshoot further survives realistic surface mass-transfer resistance, justifying the idealized surface boundary condition used throughout.