Abstract
Spring constant calibration of the atomic force microscope (AFM) cantilever is of fundamental importance for quantifying the force between the AFM cantilever tip and the sample. The calibration within the framework of thin plate theory undoubtedly has a higher accuracy and broader scope than that within the well-established beam theory. However, thin plate theory-based accurate analytic determination of the constant has been perceived as an extremely difficult issue. In this paper, we implement the thin plate theory-based analytic modeling for the static behavior of rectangular AFM cantilevers, which reveals that the three-dimensional effect and Poisson effect play important roles in accurate determination of the spring constants. A quantitative scaling law is found that the normalized spring constant depends only on the Poisson’s ratio, normalized dimension and normalized load coordinate. Both the literature and our refined finite element model validate the present results. The developed model is expected to serve as the benchmark for accurate calibration of rectangular AFM cantilevers.
Highlights
According to the dimensional method, the beam theory-based equation of spring constant for a rectangular cantilever is[19] kz
We examine the mechanical behavior of the rectangular atomic force microscope (AFM) cantilever by constructing the Hamiltonian variational principle from the original Hellinger-Reissner variational principle for the thin plate bending problem
We develop an up-to-date superposition method[24] to offer a rational way to accurately derive the analytic solution of the rectangular AFM cantilever with the length b and width a under a point load P
Summary
Our finding is to be validated by the well-accepted finite element method (FEM). The normalized load-point deflections Et 3W (a/2, 0)/(Pa2) are tabulated in Table 1 for a rectangular cantilever with the aspect ratio a/b = 1/5, 2/9, 1/4, 2/7, 1/3, 2/5, 1/2, 2/3, 1, and 2, respectively, and the Poisson’s ratio ν= 0, 0.25 and 0.4, respectively. The number of terms for the present series solution is taken such that the results converge up to the last significant figure of four To explain this observation as well as the mechanism of accuracy improvement of our model, we would like to interpret more on the difference between the two theories.
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