Parametric tip model and force–distance relation for Hamaker constant determination from atomic force microscopy

Hamaker constants and dispersion forces interactions of materials are of increasing interest and the advent of atomic force microscopy (AFM) force measurements represents a new opportunity for quantitative studies of these interactions. A critical problem is the determination of a force–distance relation for realistic AFM probes. Due to the inadequacies of existing power-law sphere–plane models to describe the probe–sample system, we present a new parametric tip force–distance relation (PT/FDR). A surface integration method is developed to compute the interactions between arbitrarily shaped bodies. The method is based on the Hamaker pairwise integration in a continuous fashion, reducing the six-dimensional integration to a four-dimensional scheme. With this method, the PT/FDR is obtained and a nonlinear fitting routine is used to extract the model parameters and the Hamaker constant from AFM force–distance data. From the sensitivity analysis of the fitting of synthesized AFM force–distance data, one finds that, for large tip radius (compared to separation), the force is proportional to the product of the Hamaker constant and tip radius. Unique determination of the Hamaker constant can be achieved if a small radius tip is used in the AFM scan. By fitting to literature data, the effectiveness of the PT/FDR is shown.