In nanomechanical mass spectrometry, sensing devices
are commonly
placed in the vacuum environment and a stream of analytes is directed
toward the sensor surface for measurement. Beam structures, such as
double-clamped nanobeams and nanocantilevers, are commonly used due
to their low inertial mass and the simplicity of the analytical models
for mass extraction. The drawback of such structures is their low
capture areas, compromising the capture efficiency and throughput
of this technique. Bi-axisymmetric resonators, such as ultrathin square
or circular membranes, arise as an optimal geometry to maximize capture
efficiency while minimizing the device inertial mass. However, these
structures present degenerate mechanical modes, whose frequency perturbations
upon analyte adsorption are not well described by commonly used models.
Furthermore, prior knowledge of the vibration mode shapes of the sensor
is crucial for the correct calculation of the analyte’s mass,
and the mode shape of degenerate modes may change significantly after
every adsorption event. In this work, we present an accurate analytical
theory to describe the effect of mass adsorption on the degenerate
modes of square membrane resonators and propose two different methods
based on the new theory to update the vibration mode shapes after
every adsorption event. Finally, we illustrate the problem experimentally
obtaining the mass and adsorption position of individualEscherichia coliK-12 bacterial cells on commercial
square silicon nitride membranes fabricated with very small tolerances.