(363a) A New Method for Determining the Hamaker Constant of a Solid With Atomic Force Microscopy

The determination of the Hamaker constant, A, of a solid material with a flat surface using atomic force microscopy (AFM) is often done based on the following steps. First, data of the dynamic AFM tip deflection under the influence of the attractive force between the tip and the solid and for a given cantilever approach speed are obtained. Second, a quasi-static analysis is used to fit the dynamic data, which enables one to infer the jump-into-contact distance Δd and hence the value of A. Here, we evaluate the validity of this quasi-static method and develop an improved procedure for determining A which acknowledges explicitly that the tip motion is dynamic, and not quasi-static.

We begin by correcting an earlier quasi-static method based on the “jump-into-contact” distance for the simplest sphere-plane model. This analysis is further extended to three additional tip-surface models. The sensitivity of these models in the quasi-static analysis is investigated. Moreover, a mechanically equivalent energy-based analysis is reported. The energy approach offers additional insights on the mechanical stability of the tip as it approaches the solid surface quasi-statically. Next, a rigorous dynamic analysis of the tip motion is developed to determine the time-dependent tip deflection. For close contact distances with the surface, the tip motion is found to deviate from the predictions of the quasi-static analysis except at the limit of very low or zero cantilever speeds. This dynamic analysis also indicates that a “jump-into-contact” distance, which is defined in the quasi-static analysis, cannot be uniquely determined from the dynamic data. Hence, a new method is needed if the quasi-static model is still to be used when interpreting the dynamic data.

We introduce a novel approach whereby an energy analysis is used to approximate the dynamic data with a quasi-static model. An iterative scheme is introduced, in which an apparent Hamaker constant, A_app, is determined from the dynamic data and the quasi-static model. This A_app depends on the cantilever speed and the spatial resolution δ of the data points. In the limit of zero cantilever speed, A_app should converge to the value of A obtained from the quasi-static model. For proper use of the new method, one therefore needs to obtain dynamic data at resolutions of δ < Δd and for several decreasing cantilever speeds. We test the method using simulated data, and investigate the effects of the number of data points sampled from the dynamic curve of the tip motion. High sampling rates, small values of δ and low cantilever speeds are shown to yield the most accurate estimates of A. Finally, the new method is applied to dynamic AFM data for a model system of glass (amorphous SiO₂).