Quantitative elasticity evaluation by contact resonance in an atomic force microscope

Abstract

We present a local elasticity measurement from the resonance frequency of a cantilever vibrated at its base. A sharp tip on top of the cantilever is in contact with the sample surface at a servo-controlled constant static force. The advantage of a continuum model for the cantilever vibration over the localized-spring model in the elasticity measurement is proved quantitatively. As an application, Young's modulus of soda lime glass was measured and a smooth radial gradient of Young's modulus of about 29% was observed for the first time on a cross section of a carbon fiber of diameter 7 µm. Further improvement of the independent shear stiffness measurement was proposed by a theoretical analysis of torsional vibration with a boundary condition for the elastically supported tip.Recently, local elasticity measurement methods using a vibrating cantilever of an atomic force microscope (AFM) [1] based on the contact mechanics [2] have been developed. They depend either on the deflection measurement in linear [3, 4] and nonlinear [5, 6] regimes or on the resonance frequency measurement [8][9][10][11][12]. It has been shown in the latter approach that a soft cantilever can induce elastic deformation on relatively stiff materials when it is vibrated at higher order modes [9-12] because of the acceleration effect. However, quantitative measurement is still difficult for the following reasons. Although a very soft cantilever can be used to evaluate stiff materials, in some approaches the tip on such a cantilever is easily trapped by the surface moisture film and then the reproducibility of the measurement is usually poor. On the other hand, a very stiff cantilever (> 100 N/m) can be used to measure the local elasticity of stiff materials in the force-modulation mode [3] and the force-curve method [13], but the large deformation of the sample causes problems [13]. Finally, it is always necessary to assume Poisson's ratio in * order to obtain Young's modulus. In this paper we present an approach to solve these problems. The principle of Young's modulus measurements by using resonance of the deflection vibrationThe deflection vibration of the cantilever shown in Fig. 1 is employed for the measurement of Young's modulus. First, we briefly summarize the theory of cantilever vibration [10,12]. The equation of motion for deflection vibration iswhere E is Young's modulus, ρ is the density and t the thickness of the cantilever. The boundary condition at x = 0 is z(0) = 0 and dz/dx = 0 whereas that at x = L (L is the length of cantilever) iswhere w is the width of cantilever and s is the contact stiffness, or force gradient in the contact mode. The left-hand side of (2) is the shear force of the cantilever and the righthand side is the restoring force from the sample. Solving (1) with (2) we have obtained the frequency equation [10]where α = (48π 2 ρL 4 /Et 2 ) 1/4 f 1/2 (frequency constant) and k = Ewt 3 /4L 3 (cantilever stiffness). The frequency constant α of the lowest three modes are plotted in Fig. 2a as a function of the ratio o...

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“…FMM is a SFM-based mode where the tip in direct contact with the surface is forced to vibrate at a high frequency (contact resonance frequency typically of several 100's of KHz) in order to evaluate the mechanical response of the material [27][28][29]. Any variation in the contact resonance frequency is related to the local sample stiffness, i.e., a positive shift in frequency corresponds to stiffer mechanical properties.…”

Wavelet Transforms in Dynamic Atomic Force Spectroscopy

“…The frequency shift can be followed starting from the very beginning of the cantilever interaction with the surface and with good temporal resolution. The single measure is taken in approximately 100 ms. With an appropriate analysis, it would be possible to study in detail both the adhesion forces dynamics of the cantilever (Espinosa-Beltrán et al, 2009;Yamanaka & Nakano, 1998) and the elasticity parameters (e.g the Young's modulus) from the contact region (Hertz contact dynamics) (Dupas et al, 2001;Espinosa-Beltrán et al, 2009;Rabe et al, 1996;Vairac et al, 2003).…”

“…With a suitable model this technique could allow a thorough measurement of the adhesion force properties (Drobek et al, 2001;Espinosa-Beltrán et al, 2009;Yamanaka & Nakano, 1998). Analytical and numerical models describing the free cantilever-vibration as well as the contact-resonances are well known and provide quantitative evaluation when complete contact-resonance spectra are measured.…”

Ultrasonic Studies of the Fundamental Mechanisms of Recrystallization and Sintering of Metals

“…For clarity, we will limit the discussion to the approach used in AFAM. The specific details of the UAFM approach are somewhat different, although equally valid [YAM98,YAM99].…”