Reconstruction of shear force in Atomic Force Microscopy from measured displacement of the cone-shaped cantilever tip

Abstract

In this paper, a dynamic model of reconstruction of the shear force $g(t)$ in the Atomic Force Microscopy (AFM) cantilever tip-sample interaction is proposed. 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 momentum generated by the transverse shear force $g(t)$. For the reconstruction of $g(t)$ the measured displacement $\nu (t):=u(\ell ,t)$ is used as an additional data. 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 through the solution of the adjoint problem is derived. This allows 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 allows to develop a numerical "gadget" for computational experiments of generic AFMs.

21 pages, 5 figures