An analytic model for accurate spring constant calibration of rectangular atomic force microscope cantilevers

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.

Scientific RepoRts | 5:15828 | DOi: 10.1038/srep15828 According to the dimensional method, the beam theory-based equation of spring constant for a rectangular cantilever is 19 where E is the Young's modulus of the material, a, b and t are the width, length and thickness of the cantilever, respectively. It is well known that equation (1) ignores the bowing of the cantilever across the width thus is only applicable to the cantilevers with a << b. For relatively wide cantilevers, the plate theory should be used, instead of the beam theory, to obtain more accurate results. However, the accurate analytic solution to the governing equation of a rectangular cantilever plate was extremely difficult to obtain due to the complexity of the mathematical model. Accordingly, the approximate/numerical solutions had to be developed to calibrate the AFM spring constants on a case-by-case basis. Obviously, the analytic solutions are necessary to capture the essence of the problem by quantitatively realizing the relation among the key parameters/quantities, which cannot be realized by an approximate/numerical solution.
According to the classical Kirchhoff thin plate theory 22  where ( , ) x y denote the coordinates in the plane where the plate lies, ( , ) W x y is the transverse deflection of the plate mid-plane, ( , ) q x y is the distributed transverse load, and ν = / ( − ) D Et [12 1 ] 3 2 is the plate flexural stiffness in which ν is the Poisson's ratio. Using the classical methods, analytic solution of equation (2) can only be obtained for a rectangular plate with at least a pair of opposite edges simply supported. For the rectangular AFM cantilever which is modeled as the plate with one clamped edge and three free edges, i.e. the cantilever plate, the accurate analytic modeling was unavailable. Urgent need to address this issue motivates the present work.
We examine the mechanical behavior of the rectangular AFM cantilever by constructing the Hamiltonian variational principle from the original Hellinger-Reissner variational principle for the thin plate bending problem. The corresponding Hamiltonian system-based governing matrix equation is derived (see supplementary information for details) as 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  We thus find a scaling law in determining the spring constants k z : Equation (5) clearly shows that the spring constant k z , normalized by / Et a 3 2 , depends only on the Poisson's ratio ν, normalized dimension φ and normalized load coordinate y 0 . To illustrate this dependence, for an end-tip-loaded rectangular cantilever (i.e. = y 0 0 ) is plotted versus ν and φ for different φ and ν, respectively, as shown in Fig. 2a,b.
Our finding is to be validated by the well-accepted finite element method (FEM). The normalized load-point deflections 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. Comparison is shown with FEM by ABAQUS software package where the 4-node general-purpose shell element S4R and uniform mesh with grid size . a 0 0025 are employed. The number of terms for the present series solution is taken such that the results converge up to the last significant figure of four. It is evident in Table 1 that our analytic solutions agree perfectly with those by FEM which are regarded as the benchmarks in view of the absence of comparable analytic solutions, and our plate theory-based results give obvious accuracy improvement on the beam theory 25 . It is noted that the approximate methods had to be developed in the past when the theoretical analysis was needed 20 , which, however, cannot yield the results as accurate as presented here. It should be pointed out that the case with / = a b 2 in Table 1 is rarely encountered in practice, but we still present the results in order to demonstrate the better accuracy as well as the broader applicability of the plate theory which better describes the behavior of the cantilevers at any scales, i.e. at the scales which are not merely restricted to the AFM cantilevers. The error variations of beam theory with ν and φ for different φ and ν, respectively, . are plotted in Fig. 3a,b so that one can quickly assess what situations might yield a significant correction. It is interesting to observe in both Table 1 and Fig. 3 that there are very small errors for the cantilevers with ν = 0 under relatively lower / a b. 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. From the physical point of view, for the cantilever as depicted in Fig. 1, the beam theory actually describes such a cantilever plate with zero Poisson's ratio: the uniformly distributed line load with the intensity of / P a is applied along = y y 0 and no constraint is imposed in the x direction while no rotation around the y axis is allowed for the edges = which reduces to the beam model solution. Therefore, compared with the three-dimensional model by the plate theory which incorporates the Poisson effect, the beam theory yields a much more simplified plane model and ignores the Poisson effect, which is numerically revealed in Table 1 by the independence of the beam theory-based results on ν under the same / a b. The ability of the plate theory to depict the field distribution of mechanical quantities over a cantilever is clearly reflected in Fig. 4a by comparison with the beam theory in Fig. 4b. Furthermore, it is seen from both Table 1 and Fig. 3b that the plate theory would predict either an increase or a decrease in the spring constant over the beam theory, but  actually there is the important insight to be drawn that the plate theory always predicts a decrease in the spring constant over the beam theory when the Poisson ratio ν = 0. This can be explained by a simple inference. As illustrated in Fig. 5, for a cantilever with ν = 0, the plate model under a point load P at the central line (Fig. 5a) gives a lower spring constant than the same plate under a uniformly distributed line load with the intensity of / P a (Fig. 5b), because the former deflection at the load point is definitely large than the latter at the same point. On the other hand, this latter model (Fig. 5b) yields a lower spring constant than the same model plus the constraint that no rotation around the two side edges is allowed (Fig. 5c), because adding the constraint would enhance the stiffness. We have shown that the plate model in Fig. 5c is equal to the beam model under the same point load P as in Fig. 5a (see Fig. 5d). Therefore, if the spring constants of Fig. 5a-d are denoted by k a , k b , k c , and k d , respectively, we conclude that We use the developed solutions to calibrate the spring constants for real cantilevers. Three sets of end-tip-loaded rectangular cantilevers, fabricated from the Perspex 21 (PMMA, Young's modulus = E 3 GPa, Poisson's ratio ν = 0.35), with the length = b 20 cm, thickness = t 3 mm and width = a 9.31, 6.53 and 3.29 cm, respectively, are considered. As shown in Table 2, the normalized spring constants /( ) k b Et z 2 3 by the present analytic solution agree very well with the numerical results from ref. 21. We also examine several commercial rectangular AFM cantilevers in Table 3, with = E 169 GPa and ν = 0.408 26 , of which the dimensions a, b and t are listed in the table. We emphasize that the significant accuracy improvement (e.g. 5.7%) can be achieved by the present model even though the cantilevers tend to those with a<<b.  Table 2. Normalized spring constants of several end-tip-loaded rectangular cantilevers fabricated from Perspex.
In conclusion, we have explored an analytic approach to accurate spring constant calibration of rectangular AFM cantilevers based on the thin plate theory, by which the importance of the three-dimensional effect as well as Poisson effect is confirmed. The obtained solutions eliminate the errors caused by the classical beam theory, and hold for rectangular cantilevers with any aspect ratio / a b. Deformation of an AFM cantilever involves seven load, material and geometry quantities: the point load P, Poisson's ratio ν, Young's modulus E, length b, width a, thickness t and load position y 0 . The scaling law in equation (5), verified by FEM, shows that the normalized spring constant depends only on three normalized quantities, i.e., ν, φ and y 0 . This scaling law could serve as the theoretical basis for analytically calibrating the spring constants of rectangular AFM cantilevers. It should be noted that in some applications the AFM cantilevers are coated with three-dimensional layers, and the influence of these layers on the spring constant calibration is not negligible. That is especially important for the applications that need to collect currents simultaneously, such as the conductive AFM [27][28][29] . One possible simple treatment is using the equivalent stiffness which appropriately incorporates the effect of the layers 30 . Our ongoing work is to obtain accurate enough equivalent stiffness so that the present model is applicable to the cantilevers with layers as well.  Table 3. Spring constants of several commercial rectangular cantilevers at end tips.