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D. Platz, A. Gesing, U. Schmid:
"Modelling Vibrational Modes Plates in Fluids for Applications in High-Speed Atomic Force Microscopy";
Talk: 8th Multifrequency AFM conference, Madrid; 10-27-2020 - 10-30-2020; in: "8th Multifrequency AFM conference, Book of Abstracts", (2020), 33.

Novel probe designs played a vital role in the development of high-speed atomic force
microscopy (AFM). Often new probe designs evolved from slender cantilever beam designs
which are modeled with one-dimensional beam theory. Recently, it has been demonstrated that
non-conventional vibrational modes in two-dimensional plate structures as depicted in figure 1
exhibit extraordinary high quality factors in liquids while their resonance frequency can easily
be tuned by changing the plate width [1]. Due to these properties high-speed AFM methods for
imaging biological samples in liquids would greatly benefit from plate-based probes. However,
a theory for predicting quality factors of plate modes in liquids has been missing and theory for
determining the fluid damping of slender beams in fluids is not applicable to plates. Here, we
present a modelling approach for determining the dynamic response of plates immersed in
fluids. The model comprises an elastic plate and a viscous fluid around the plate. For modelling
the elastic plate, we use the Kirchhoff-Love plate equation which we solve with the method of
finite elements. The finite element solution is obtained with a continuous/discontinuous
Galerkin method which allows for the use of Lagrange-type elements while weakly imposing
the physically required continuity conditions to the solution. We determine the fluid flow from
an integral formulation of a Stokes flow and couple the resulting forces at the fluid-plate
interface to the Kirchhoff Love equation. Using this method, we determine the spectral response
of driven plate modes in gases and fluids as exemplified in figure 2 and compare the results
with the spectral response of slender beam structures. Moreover, the resonance frequencies and
quality factors of plate modes in different fluids are predicted. The presented method establishes
a theoretical framework for the design of novel AFM which utilize vibrational modes in two-
dimensional plate resonators and paves the way for practical methods for calibrating plate
modes in AFM.