Nonlinear dynamics and force spectroscopy in dynamic atomic force microscopy
Abstract
As one branch of atomic force microscopy (AFM), dynamic atomic force microscopy (Dynamic AFM) uses a resonating probe with frequency (FM-AFM) or amplitude modulation (AM-AFM) to measure sample topography and material properties of nanostructures with nanometer resolution. Under the influence of tip-sample interaction forces, the dynamics of the probe become highly nonlinear and can affect the imaging stability and interaction forces. On the other hand, the tip-sample interaction forces can also be extracted from the nonlinear response, opening up new possibilities for nanoscale material property measurements. This thesis combines analytical, experimental, and computational works to (a) investigate the nonlinear response of AFM cantilevers, (b) understand and develop approximate scaling laws for peak interaction forces in dynamic AFM, and to (c) reconstruct the tip-sample interactions from conveniently acquired experimental data. First the nonlinear frequency response of an AFM microcantilever interacting in AM-AFM with the sample is investigated systematically through theoretical, numerical, analytical and experimental analyses. A discretized dynamic model with Derjaguin-Müller-Toporov (DMT) contact mechanics interaction potential is developed to explore various nonlinear phenomena in AM-AFM. Accurate tip vibration responses are computed numerically using the DDASKR routine with root finding algorithm in Fortran for highly nonlinear and non-smooth differential equations. The periodic averaging method is implemented to analytically predict the tip vibration nonlinear response. Numerical computations, analytical predictions using the averaging method, and experimental measurements all match excellently for large setpoint amplitude ratios. The results indicate that attractive van der Waals interactions lead to a initial softening nonlinearity of the periodic solution response while the repulsive DMT interactions lead to a subsequent hardening nonlinear response. Next the theoretical approach above is used to tackle a long-standing theoretical question in dynamic AFM. Determining the peak interaction force between an oscillating nanoscale tip and a sample surface has been a fundamental yet elusive goal in dynamic atomic force microscopy. Closed form analytical expressions are derived using nonlinear asymptotic theory for the peak attractive and repulsive forces that approximate with a high degree of accuracy the numerically simulated peak forces under ambient or vacuum conditions. Scaling laws involving van der Waals, chemical forces, nanoscale elasticity and oscillator parameters are identified to demonstrate approximate similitude for the peak interaction forces under practical operating conditions. Finally, a detailed investigation is performed on the feasibility of reconstructing tip-sample interaction forces from data typically acquired in AM-AFM. Dynamic force spectroscopy, that is reconstruction of the interaction forces from the experimentally measured frequency vs distance curve, has become a standard routine for FM-AFM. However, to date a general method for dynamic force spectroscopy for AM-AFM has not been demonstrated. Based on the harmonic balance method and the Chebyshev polynomial expansion method, it is shown that the conservative tip sample interaction forces can be reconstructed and used to experimentally determine the peak interaction forces in AM-AFM using standard measurements. The results are especially important for nanoscale dynamic material analysis (Nano DMA) and peak interaction force estimation and optimization for AM-AFM.
Degree
Ph.D.
Advisors
Raman, Purdue University.
Subject Area
Mechanical engineering
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