Analysis the effect of different geometries of AFM's cantilever on the dynamic behavior and the critical forces of three-dimensional manipulation.

An important challenge when using an atomic force microscope (AFM) is to be able to control the force exerted by the AFM for performing various tasks. Nevertheless, the exerted force is proportional to the deflection of the AFM cantilever, which itself is affected by a cantilever's stiffness coefficient. Many papers have been published so far on the methods of obtaining the stiffness coefficients of AFM cantilevers in 2D; however, a comprehensive model is yet to be presented on 3D cantilever motion. The discrepancies between the equations of the 2D and 3D analysis are due to the number and direction of forces and moments that are applied to a cantilever. Moreover, in the 3D analysis, contrary to the 2D analysis, due to the interaction between the forces and moments applied on a cantilever, its stiffness values cannot be separately expressed for each direction; and instead, a stiffness matrix should be used to correctly derive the relevant equations. In this paper, 3D stiffness coefficient matrices have been obtained for three common cantilever geometries including the rectangular, V-shape and dagger-shape cantilevers. The obtained equations are validated by two methods. In the first approach, the Finite Element Method is combined with the cantilever deflection values computed by using the obtained stiffness matrices. In the second approach, by reducing the problem's parameters, the forces applied on a cantilever along different directions are compared with each other in 2D and 3D cases. Then the 3D manipulation of a stiff nanoparticle is modeled and simulated by using the stiffness matrices obtained for the three cantilever geometries. The obtained results indicate that during the manipulation process, the dagger-shaped and rectangular cantilevers exert the maximum and minimum amounts of forces on the stiff nanoparticle, respectively. Also, by examining the effects of different probe tip geometries, it is realized that a probe tip of cylindrical geometry exerts the smallest force on a biological nanoparticle. Therefore, the rectangular cantilever is a more suitable geometry for preventing the exertion of excessive force and the possible damage of such nanoparticle.