Abstract
The interaction between a rapidly oscillating atomic force microscope tip and a soft material surface is described using both elastic and viscous forces with a moving surface model. We derive the simplest form of this model, motivating it as a way to capture the impact dynamics of the tip and sample with an interaction consisting of two components: interfacial or surface force, and bulk or volumetric force. Analytic solutions to the piece-wise linear model identify characteristic time constants, providing a physical explanation of the hysteresis observed in the measured dynamic force quadrature curves. Numerical simulation is used to fit the model to experimental data and excellent agreement is found with a variety of different samples. The model parameters form a dimensionless impact-rheology factor, giving a quantitative physical number to characterize a viscoelastic surface that does not depend on the tip shape or cantilever frequency.
Highlights
An increasingly important application of atomic force microscopy (AFM) is the characterization of viscoelastic materials and interfaces, such as cell membranes and tendons [1,2,3,4], polymer blends and composites [5,6,7,8,9,10,11], liquid-gas and liquid-solid interfaces [12,13], and suspended membranes [14]
Viscous force may even dominate over elastic force with soft materials, and a proper characterization of the material must rely on a dynamic measurement that distinguishes viscous force from elastic force
To display the results of both experiment and theory, we show dynamic-force-quadrature curves obtained with a a technique called “intermodulation atomic fore microscopy” (ImAFM) [20]
Summary
An increasingly important application of atomic force microscopy (AFM) is the characterization of viscoelastic materials and interfaces, such as cell membranes and tendons [1,2,3,4], polymer blends and composites [5,6,7,8,9,10,11], liquid-gas and liquid-solid interfaces [12,13], and suspended membranes [14]. One can use finite-element methods with linear [26] or nonlinear force-displacement relations that account for an attractive tip-sample force [27] Independent of these bulk viscoelastic models, tipsample force in AFM may result from interfacial energy or surface tension γ. For a soft material with E ∼ 3 MPa forming a contact with relatively low interfacial energy γTS ∼ 30 mN/m and no additional surface stress (in which case Υ = γTS), we find L ∼ 10 nm, the typical radius of an AFM tip. If we disregard the inertia of the sample contained in the very small interaction volume, our interaction force depends on this one additional “hidden” dynamic variable: FTS(t) = f (z(t), z(t), zs(t)) This type of model was introduced by Cantrell and Cantrell [32] to account for externally forced oscillations of the sample in the context of ultrasonic AFM. Our measurements show that the impact-rheology factor is independent of the oscillation frequency and the tip geometry
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