Model for mechanical properties nanoprobes
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S.P. Baker Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853
H.M. Pollock School of Physics and Chemistry, Lancaster University, Lancaster LA1 4YB, United Kingdom (Received 8 September 1999; accepted 26 June 2000)
Researchers may use several different instruments to determine chemical and mechanical properties of materials with nanometer-scale vertical, and occasionally, lateral, resolution. Three such instruments are the depth-sensing indenter, the atomic force microscope, and the surface forces apparatus. Until now, these methods were individually modeled, and an analysis of their mechanical response was never done in a general way. In this article, we show that these instruments can be treated as a class—a class that we call mechanical properties nanoprobes (MPNs)—that can be described by a single universal linear model. Using this model, we solved both the quasistatic and dynamic response as a function of excitation frequency and complex compliance using an electrical analog for the mechanical system. Earlier work did not find correct solutions for the amplitude and phase, did not examine the influence of finite stiffness in the head of the MPN, and overlooked the difference between a partial and full derivative and its influence on quasistatically acquired force curves. The equations here will allow scientists to correctly interpret their results concerning elastic and anelastic materials response, especially for low-modulus, high-damping samples such as polymers.
I. INTRODUCTION
The instruments of interest: depth-sensing indenters (DSIs),1,2 atomic force microscopes (AFMs),3,4 surface forces apparatuses (SFAs)5,6 and their relations, all share the capability for performing force spectroscopy, i.e., the measurement of force as a function of separation or indentation depth. When data are interpreted from these mechanical properties nanoprobes (MPNs), it is essential to know the instrument’s response as a function of (i) excitation frequency, (ii) the stiffness of the machine, (iii) the stiffness and damping of the force transducer, and (iv) whether force or displacement is controlled, in order to arrive at the desired quantities—the stiffness and damping of the tip–sample interaction. In the past, treatments of individual MPNs have appeared in the literature,7–11 but a general model for all MPNs has not yet been published. Although damping elements were present in the rheological diagrams, analyses of DSIs7,8 did not arrive at correct expressions for the amplitude and phase because the velocity difference across the sample’s damping element was never taken into account. Earlier analyses of AFMs and SFAs did not include machine stiffnesses.9–11 Previous treatments of 2006
http://journals.cambridge.org
J. Mater. Res., Vol. 15, No. 9, Sep 2000 Downloaded: 13 Mar 2015
MPNs share the same goal as we—to extract materials properties such as elastic modulus, hardness, and work of adhesion from the data. To date, methods for doing so have been developed separately for DSI
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