Dynamic response of a cracked atomic force microscope cantilever used for nanomachining
© Lee and Chang; licensee Springer. 2012
Received: 24 September 2011
Accepted: 15 February 2012
Published: 15 February 2012
The vibration behavior of an atomic force microscope [AFM] cantilever with a crack during the nanomachining process is studied. The cantilever is divided into two segments by the crack, and a rotational spring is used to simulate the crack. The two individual governing equations of transverse vibration for the cracked cantilever can be expressed. However, the corresponding boundary conditions are coupled because of the crack interaction. Analytical expressions for the vibration displacement and natural frequency of the cracked cantilever are obtained. In addition, the effects of crack flexibility, crack location, and tip length on the vibration displacement of the cantilever are analyzed. Results show that the crack occurs in the AFM cantilever that can significantly affect its vibration response.
PACS: 07.79.Lh; 62.20.mt; 62.25.Jk
Since its invention in 1986 , the atomic force microscope [AFM] has become a very powerful tool for studying the surface characteristics of diverse materials on a micro- and nanoscale level [2–4]. In addition, atomic force microscopy can also be applied to nanoscale lithography. AFM-based nanolithography technique has a very high potential for nanofabrication [5–8]. The nanolithography research can be roughly divided into three categories: (1) local electrochemical reactions of silicon and metal, (2) direct atomic and molecular manipulation, and (3) direct nanomachining of the material. The direct nanomachining of material structures has so far been useful for fabricating nanodevices.
The nanomachining technique has been the interest of many researchers. Horng  studied the flexural vibration responses of a rectangular AFM cantilever subjected to a cutting force using the modal superposition method. Voigt et al.  utilized a chip cantilever system for material processing tasks on the micro- and nanometer scales by the flexural-torsional resonance mode. Recently, Zhu et al.  investigated the AFM-based nanometric cutting process of copper using molecular dynamics simulation.
Cracks may occur in the AFM cantilever during the nanomachining experiments or in the fabrication of cantilever . The cantilever with cracks will affect its performance when used. The finite element method and finite difference method can be used to analyze the vibration response of a beam with cracks. For example, Sinha and Friswell  investigated the vibration behavior of a free-free beam with a breathing crack using a finite element model. Dorogoy and Banks-Sills  utilized a finite difference method to study the effect of contact and friction on the disk specimens containing a crack that are subjected to concentrated loads. Based on our literature survey, however, there has been no investigation on the mechanical properties of an AFM cantilever with cracks. The vibration response of a cantilever is related to the processing quality. In this paper, the vibration behavior of the cracked cantilever during the nanomachining process is studied. An analytical expression of vibration displacement of the cracked cantilever is obtained. Both the finite element and finite difference methods are a numerical method. However, the method adopted in this paper is an analytical model. In general, the solution obtained using the analytical method is more accurate than that of the numerical method. In addition, the effects of crack flexibility and crack location on the displacement are investigated.
An AFM probe is used to machine the specimen, and it is considered as a cantilever beam which has Young's modulus E, moment of inertia I, density ρ, the uniform cross-section A, and length L. When the machining is in progress, the cantilever tip contacts with the specimen and induces a vertical reaction force, F y (t), and a horizontal reaction force, F x (t), which are a function of time t. Assuming that the reaction forces are on the tip end, the product of the horizontal force and the tip length can form the equivalent moment exerted on the cantilever. The cutting system can be modeled as a flexural vibration motion of the cantilever. The motion is a partial differential equation, and its transverse displacement is dependent on time t and the spatial coordinate X.
where X is the distance along the center of the cantilever, t is time, and Y1(X,t) and Y2(X, t) are the transverse displacement of both segments, respectively.
where W is the rotational spring constant, and I stands for the mass moment of inertia.
where H and m are the tip height and mass, respectively; c is the distance between the lower edge of the cantilever and centroid of the cross section. The boundary condition of the cantilever at X = 0 is assumed a fixed end; then, the boundary conditions given by Equations 7 and 8 correspond to conditions of zero displacement and zero slope. The commercial cantilevers are fabricated with a holder providing the base to which the cantilever is suspended. The holder is assumed to be rigid, and the vertical and horizontal reaction forces and fixing moment at the built-in fixed end of the cantilever are to be neglected in the analysis. In addition, the boundary conditions that are given by Equations 9 and 10 correspond to the moment and the force balanced between the beam and a combination of the linear tip-sample stiffness at X = L, respectively. F y (t) and F x (t) which are both functions of time t are denoted the vertical and horizontal cutting force on the sample under the normal and lateral direction, respectively. The first term on right side in Equation 9 represents the moment due to the lateral tip-sample cutting force, and the second term denotes the moment due to the mass moment of inertia of the tip. Similarly, the first term on the right side in Equation 10 is the normal tip-sample interaction force, and the second term is the inertia force of tip mass . The relationship between F x and F y can be expressed as . The relationship is obtained from a geometrical relation for a cone-shape tip, and θ is a half-conic angle .
where Ω is the excitation frequency of the cutting force, and P is the arbitrary constant.
where are the dimensionless constants relevant to the cutting force in the vertical direction.
where B1...B8 are arbitrary constants, γ is the wave number, and γ4 = Λ2.
The above frequency equation can also be obtained from the result of Wu et al. .
Results and discussion
The vibration displacement of the cantilever significantly increased when the crack was near the fixed end.
The displacement of the cracked cantilever increased with increasing value of crack flexibility.
A higher vibration displacement of the cracked cantilever was obtained when the tip length was larger.
The authors wish to thank the National Science Council of the Republic of China in Taiwan for providing financial support for this study under Projects NSC 99-2221-E-168-019 and NSC 100-2221-E-168-018.
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