Determining the effects of surface elasticity and surface stress by measuring the shifts of resonant frequencies

Abstract

Both surface elasticity and surface stress can result in changes of resonant frequencies of a micro/nanostructure. There are infinite combinations of surface elasticity and surface stress that can cause the same variation for one resonant frequency. However, as shown in this study, there is only one combination resulting in the same variations for two resonant frequencies, which thus provides an efficient and practical method of determining the effects of both surface elasticity and surface stress other than an atomistic simulation. The errors caused by the different models of surface stress and mode shape change due to axial loading are also discussed.

Surface effect on resonant properties of nanowires predicted by an elastic theory for nanomaterials

“…36,37 Chiu and Chen 38 further discussed the effect of the surface bending modulus on the resonant frequency of nanowires. Recently, Zhang et al 39 analyzed the motion equation of vibrating nanowires, based on which contributions of surface elasticity and surface stress to the resonant frequency can be abstracted, respectively.…”

Detecting the stiffness and mass of biochemical adsorbates by a resonator sensor

“…(13) applies to all resonant frequencies, which is somewhat surprising. As discussed later, this fact has the implication that the method of multiple resonant frequencies [23,24] versus t a /t c is plotted in Fig. 2.…”

“…We can continue the same procedure for other higher resonant frequencies, which will generate the same intersection line, too. This fact in essence is to say that the method of using the shifts of different resonant frequencies [23,24] to solve the inverse problem does not work here. The method [23,24] works because the responses of different resonant frequencies to surface stress (modeled as an axial load) are different.…”

“…Due to the presence of the adsorbate layer, the mode shapes of the system vary as a function of the layer dimensions and locations. In the numerical methods such as Rayleigh-Ritz method [11,12] and Galerkin method [23,24], the accuracy of the computed eigenfrequencies depends on the accuracy of the mode shape [23]. A tiny deviation from the exact mode shape causes a tiny relative error in the eigenfrequency computation.…”

“…The adsorbate stiffness always increases the resonant frequency. Because surface stress can be either tensile or compressive [21,22], it can either increase or decrease the stiffness [19,23,24]. It is worth mentioning here that the heated debates on whether surface stress can be modelled as an axial load on a cantilever beam, which can thus change the stiffness, are still being exchanged [23,25,26].…”