Validity of point-mass model in off-resonance dynamic atomic force microscopy

Abstract

The quantitative measurement of viscoelasticity of nano-scale entities is an important goal of nanotechnology research and there is considerable progress with advent of dynamic atomic force microscopy. The hydrodynamics of cantilever, the force sensor in AFM measurements, plays a pivotal role in quantitative estimates of nano-scale viscoelasticity. The point-mass (PM) model, wherein the AFM cantilever is approximated as a point-mass with mass-less spring is widely used in dynamic AFM analysis and its validity, particularly in liquid environments, is debated. It is suggested that the cantilever must be treated as a continuous rectangular beam to obtain accurate estimates of nano-scale viscoelasticity of materials it is probing. Here, we derived equations, which relate stiffness and damping coefficient of the material under investigation to measured parameters, by approximating cantilever as a point-mass and also considering the full geometric details. These equations are derived for both tip-excited as well as base-excited cantilevers. We have performed off-resonance dynamic atomic force spectroscopy on a single protein molecule to investigate the validity of widely used PM model. We performed measurements with AFMs equipped with different cantilever excitation methods as well as detection schemes to measure cantilever response. The data was analyzed using both, continuous beam model and the PM model. We found that both models yield same results when the experiments are performed in truly off-resonance regime with small amplitudes and the cantilever stiffness is much higher than the interaction stiffness. Our findings suggest that a simple PM approximation based model is adequate to describe the dynamics, provided care is taken while performing experiments so that the approximations used in these models are valid.

Quantitative Elasticity of Flexible Polymer Chains Using Interferometer-Based AFM

“…Cantilever amplitude and phase difference between cantilever drive and actual oscillations at the tip are measured using an interferometer and recorded usina g lock-in amplifier. As shown in Figure 4 , phase difference is close to zero and featureless due to negligible contribution made by dissipation in off-resonance conditions [ 23 , 31 , 39 ]. However, amplitude signals show characteristic non-linear features every time a polymer is picked up.…”

“…This condition, therefore, forms a check on the correctness of our measurement. We have recently shown that for off-resonance operation, phase difference in the displacement signal z is 0 [ 23 , 31 ]. The amplitude signal gives the stiffness of the polymer molecule and both amplitude and phase are measured with a lock-in amplifier.…”

Viscoelasticity of single folded proteins using dynamic atomic force microscopy

“…The point-mass model seems to be a gross oversimplification for a continuous beam oscillating in liquids; 65,77,84 however, it has been shown that it is adequate in certain experimental parameter space. 66 For a fixed separation of the base and the substrate, the above equation can also be written for z(t), the deviation of the tip from its equilibrium position.…”

“…Typically, tip amplitude of 5 nm and cantilever stiffness of 0.1 N m À1 will exert a force modulation larger than 500 pN. 66,87 Due to these concerns, dynamic AFM measurements of proteins have been restricted to off-resonance operation. 88,89 It is also worthwhile to note that solution to eqn (10) is valid only when stiffness and damping coefficient are constant over the oscillation amplitude of the tip.…”

“…[62][63][64][65] These developments include use of fast cantilevers 6,43 and magnetic excitation of the tip end of the cantilever. 66,67 The less common, new interferometer based detection scheme, unlike conventional AFM, measures cantilever displacement, with enhanced sensitivity, at the tip end and not its bending. 47,68 It allows use of stiff cantilevers, which relax faster in liquids and have high resonance frequencies.…”