A New “Quasi-Dynamic” Method for Determining the Hamaker Constant of Solids Using an Atomic Force Microscope

In order to minimize the effects of surface roughness and deformation, a new method for estimating the Hamaker constant, A, of solids using the approach-to-contact regime of an atomic force microscope (AFM) is presented. First, a previous "jump-into-contact" quasi-static method for determining A from AFM measurements is analyzed and then extended to include various AFM tip-surface force models of interest. Then, to test the efficacy of the "jump-into-contact" method, a dynamic model of the AFM tip motion is developed. For finite AFM cantilever-surface approach speeds, a true "jump" point, or limit of stability, is found not to appear, and the quasi-static model fails to represent the dynamic tip behavior at close tip-surface separations. Hence, a new "quasi-dynamic" method for estimating A is proposed that uses the dynamically well-defined deflection at which the tip and surface first come into contact, d_c, instead of the dynamically ill-defined "jump" point. With the new method, an apparent Hamaker constant, A_app, is calculated from d_c and a corresponding quasi-static-based equation. Since A_app depends on the cantilever's approach speed, v_c, and the AFM's sampling resolution, δ, a double extrapolation procedure is used to determine A_app in the quasi-static (v_c → 0) and continuous sampling (δ → 0) limits, thereby recovering the "true" value of A. The accuracy of the new method is validated using simulated AFM data. To enable the experimental implementation of this method, a new dimensionless parameter τ is introduced to guide cantilever selection and the AFM operating conditions. The value of τ quantifies how close a given cantilever is to its quasi-static limit for a chosen cantilever-surface approach speed. For sufficiently small values of τ (i.e., a cantilever that effectively behaves "quasi-statically"), simulated data indicate that A_app will be within ∼3% or less of the inputted value of the Hamaker constant. This implies that Hamaker constants can be reliably estimated using a single measurement taken with an appropriately chosen cantilever and a slow, yet practical, approach speed (with no extrapolation required). This result is confirmed by the very good agreement found between the experimental AFM results obtained using this new method and previously reported predictions of A for amorphous silica, polystyrene, and α-Al₂O₃ substrates obtained using the Lifshitz method.