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 ρ A ( x ) u t t

+ μ ( x ) u t + ( r ( x ) u x x + κ ( x ) u x x t ) x x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 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 , ( r ( x ) u x x ( x , t ) + κ ( x ) u x x t ) x = ℓ = M ( t ) , ( − ( r ( x ) u x x + κ ( x ) u x x t ) x ) x = ℓ = g ( t ) conditions, where M ( t ) : = 2 h cos ⁡ θ

g ( t ) / π is the momentum generated by the transverse shear force g(t). For the reconstruction of g(t) the measured displacement ν ( t ) : = u ( ℓ , t ) is used as an additional data. The least square functional J ( F ) = 1 2 ∥ u ( ℓ , ⋅ ) − ν ∥ L 2 ( 0 , T ) 2 is introduced and an explicit gradient formula for the Fr\'echet 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 ν ( 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.