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

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 ρ_A(x)u_tt +μ(x)u_t+(r(x)u_xx+κ(x)u_xxt)_xx=0, (x,t)∈(0,ℓ)x(0,T), subject to the following initial, u(x,0)=0, ut(x,0)=0 and boundary, u(0,t)=0, ux(0,t)=0, (r(x)u_xx(x,t)+κ(x)u_xxt)_x=ℓ=M(t), (-(r(x)u_xx+κ(x)u_xxt)_x)_x=ℓ=g(t) conditions, where M(t):=2hcosθg(t)/π is the momentum generated by the transverse shear force g(t). For the reconstruction of g(t) the measured displacement v(t):=u(ℓ,t) is used as an additional data. The least square functional ... 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 v(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. [ABSTRACT FROM AUTHOR]