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We present a new comprehensive mathematical model of the cone-shaped cantilever tip-sample interaction in Atomic Force Microscopy (AFM). The importance of such AFMs with cone-shaped cantilevers can be appreciated when its ability to provide high-resolution information at the nanoscale is recalled. It is an indispensable tool in a wide range of scientific and industrial fields. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation $ \rho_A(x)u_{tt} $ $ +\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx} = 0 $, $ (x, t)\in (0, \ell)\times (0, T) $, subject to the following initial, $ u(x, 0) = 0 $, $ u_t(x, 0) = 0 $ and boundary, $ u(0, t) = 0 $, $ u_{x}(0, t) = 0 $, $ \left (r(x)u_{xx}(x, t)+\kappa(x)u_{xxt} \right)_{x = \ell} = M(t) $, $ \left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x\right)_{x = \ell} = g(t) $ conditions, where $ M(t): = 2h\cos \theta\, g(t)/\pi $ is the moment generated by the transverse shear force $ g(t) $. Based on this model, we propose an inversion algorithm for the reconstruction of an unknown shear force in the AFM cantilever. The measured displacement $ \nu(t): = u(\ell, t) $ is used as additional data for the reconstruction of the shear force $ g(t) $. The least square functional $ J(F) = \frac{1}{2}\Vert u(\ell, \cdot)-\nu \Vert_{L^2(0, T)}^2 $ is introduced and an explicit gradient formula for the Fréchet derivative of the cost functional is derived via the weak solution of the adjoint problem. Additionally, the geometric parameters of the cone-shaped tip are explicitly contained in this formula. This enables us to construct a gradient based numerical algorithm for the reconstructions of the shear force from noise free as well as from random noisy measured output $ \nu (t) $. Computational experiments show that the proposed algorithm is very fast and robust. This creates the basis for developing a numerical "gadget" for computational experiments with generic AFMs. |
