Abstract
The mass sensing superiority of a micro-/nano-mechanical resonator sensor over conventional mass spectrometry has been, or at least is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors, such as position and axial force, can also cause the shifts of resonant frequencies. The in situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated, and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of a mechanical resonator sensor for two reasons: reducing extra experimental equipment and achieving better mass sensing by considering more factors.
Highlights
Mass spectrometry is a widely-used analytical tool in biology and chemistry, which is expected to play an important role in proteomics [1,2]
The inverse problem is approximately solved by the Rayleigh–Ritz method by assuming that the beam/string strain energy does not change after mass loading/adsorption and is equal to the kinetic energy of the unloaded beam/string [44,45]
The general method presented in this study provides a straightforward and relatively fast way of solving the inverse problem, which should be of some help to mass sensing in real time
Summary
Mass spectrometry is a widely-used analytical tool in biology and chemistry, which is expected to play an important role in proteomics [1,2]. A rather straightforward method was presented to solve the inverse problem of determining the mass and position of an adsorbate on a beam [44] and a string [45] by the shifts of resonant frequencies. The inverse problem is approximately solved by the Rayleigh–Ritz method by assuming that the beam/string strain energy does not change after mass loading/adsorption and is (approximately) equal to the kinetic energy of the unloaded beam/string [44,45] It can be a good approximation in certain circumstance, the assumption in general is not valid, which could be the very reason why the method does not work when an adsorbate is (very) close to the cantilever clamped end or its mass is (very) small [44]. The general method presented in this study provides a straightforward and relatively fast way of solving the inverse problem, which should be of some help to mass sensing in real time
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