Intermittent contact: tapping or hammering?

Abstract

The tip of an atomic force microscope in intermittent contact with a sample surface was numerically simulated. The model for the tip-sample system was that of the simple harmonic oscillator for the cantilever, Maugis continuum mechanics when the tip was in contact with the sample, and either the van der Waals or capillary force if the tip was out of contact with the sample. Of particular interest were (i) the instantaneous pressures beneath the tip, (ii) the contributions from the tip-sample interaction to the damping of the cantilever, and (iii) the role of capillary forces in determining the cantilever response. We found an estimate for the pressure underneath the tip, that there are multiple sources for cantilever damping, and that the behavior due to the capillary force is complex.The utility of intermittent-contact mode atomic force microscopy (IC AFM) [1] lies in its ability to lower, and possibly eliminate, lateral forces that are applied to the sample by the tip during contact mode. This enables imaging of particulate matter that is not well-bound to the sample surface, and prevents inadvertant modification of some polymers.The tip tapping the sample surface is a very nonlinear event: for most of its oscillation cycle, the cantilever is free from external forces, then as it nears the sample, long-ranged attractive or repulsive forces may modify its behavior, and finally a brief strong repulsive shock is exerted upon it. Under these conditions, it is not possible to find a reasonable analytical solution to the system's equation of motion, and therefore numerical simulations of the tip-sample dynamics are necessary.Our first results concerning numerical IC AFM simulations were recently published [2]. The major conclusions from that paper are that the model -one differential equation with five parameters relating to the tip-sample interaction (modulus, curvature, work of adhesion, interatomic distance, and damping) -replicates the experimental data in most circumstances. The unfortunate consequence is that four of the five parameters must be determined by other means in order to interpret phase-contrast images and then one unknown parameter might be extracted from the data. More pleasing is the result that the cantilever motion is relatively insensitive to the specific tip-sample interaction and that this allows one to approximate constant high set-point amplitude images as the sample topography.Previously, we used van der Waals interactions as the long-range forces; here we also discuss the capillary force. It is our intent in this publication to build upon our previous work and address the following questions. (i) What are the instantaneous pressures beneath the tip? (ii) What are the contributions to the damping of the cantilever from the tipsample interaction? (iii) How do liquid layers on the surfaces influence the cantilever response? The modelThe essential second-order differential equation used as a starting point for the model is a driven, damped harmonic oscillator, with additional terms to describ...

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“…Many numerical or analytical models [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] were developed to understand the behavior of the cantilever. These models solve the equation of the cantilever displacement when the tip interacts with highly non-linear gradient forces.…”

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