Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory

Abstract

A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuated by a piezoelectric layer laminated in one side.

Microbeams with surface and piezoelectric effects in AFM

“…It was introduced a matrix formulation for those microbeams models that include rotatory inertia and shear deformations of the Timoshenko model. This formulation has also been used with microbeams partially covered by piezoelectric materials leading to the study of multispan beams (Claeyssen et al, 2013).…”

“…where 0 denotes the 2 × 2 null matrix and I the 2 × 2 identity matrix. We have that h(t − τ, x, ξ) acts as an integrating factor in Lagrange's adjoint method (Claeyssen et al, 2013). It turns out the dynamic response…”

“…where J is a term containing effects of the initialvalue Green function with values of v at the boundary (Claeyssen et al, 2013).…”

Efeitos de superfície no espectro do Modelo de Timoshenko

“…O momento de inércia daárea da seção tranversalé I = 2bh 3 3 , sendo A = 2bh aárea da seção transversal. Diferentemente do modelo clássico, no modelo proposto em [6], o qualé baseado no modelo contínuo de Gurtin-Murdoch [5], considerase que a viga tem uma superfície elástica de espessura nula perfeitamente colada ao seu volume principal (bulk ).…”

“…Esses podem ser sensores, atuadores, máquinas, e eletrônicos caracterizados em nanoescala, cujas configurações podem fazer uso de micro e nanovigas. O interesse das ciências em geral por essas escalas menores, têm também impulsionado o desenvolvimento de tecnologias tais como microscópios de alta precisão, por exemplo microscópios de força atômica, e transdutores micro-eletro-mecânicos como plataforma para sensores químicos e biológicos [9], [6], [3], [7].…”