Non-Contact AFM Measurement of the Hamaker Constants of Solids: Calibrating Cantilever Geometries

The van der Waals (vdW) attractive force plays a significant role in several microscale and nanoscale phenomena, such as nanoparticle self-assembly, colloidal stability, and interfacial adhesion. The Hamaker constant, A, captures the effects of material composition on the vdW force, which is often needed as input for predicting the attractive interactions among particles and between particles and surfaces. Thus, developing accurate experimental methods to determine A reliably is important.

Experimental attempts to determine the strength of the vdW interaction using an atomic force microscope (AFM) are typically hindered by issues inherent to the cantilever tip-surface contact regime, such as surface roughness and deformation, and contact separation distance. Thus, we previously developed a new method for determining A of solids from the approach-to-contact regime of an AFM experiment.^1,2 In contrast to other non-contact methods that rely solely upon a quasi-static model of the tip deflection, the new method accounts for the inertial effects of the cantilever tip’s motion. Specifically, an apparent Hamaker constant is determined, the value of which depends on the cantilever approach speed and AFM sampling resolution. As the approach speed goes to zero and the sampling resolution approaches the ultra-fine limit, the apparent Hamaker constant becomes nearly identical to the system’s “true” Hamaker constant. The method was tested experimentally using silica and polystyrene test substrates, and was demonstrated to yield estimates of A for these materials that were in very good agreement with previously published Lifshitz calculations.

The accuracy of the non-contact method for measuring interaction forces relies heavily on the accuracy of the models for the geometry of the interacting surfaces that are used in the data interpretation. For the initial validation experiments of the new method, an AFM cantilever tip with a pyramidal body and a hemispherical cap was used. Although this geometry can be confirmed and the dimensions estimated via scanning electron microscopy (SEM), even high-resolution SEM analysis of the tip cannot provide sufficient detail to allow precise enough determination of the tip’s geometric parameters. In addition, the numerical methods required to evaluate the Hamaker constants with high accuracy for geometries like the pyramid with spherical cap can quickly become prohibitively complex.

Therefore, we propose an adaptation of our current method in which the geometric complexity of the cantilever tip is eliminated from the determination of the Hamaker constant by describing the tip as an ‘effective’ perfect sphere, thereby capturing all the geometric effects in the single dimension of this sphere. This new approach requires measurements of the interaction between the cantilever and one or more ideal surfaces. These ideal surfaces have vdW interactions with the tip that are very well established, and they are extremely smooth, so that there are minimal effects of the surface topography on the execution of the method. After the forces against these well-known surfaces are measured, the interactions are modeled by assuming a simple ‘effective’ shape for the cantilever tip. When the dimensions of this effective shape are captured properly, the Hamaker constants for the tip-surface interactions will be correctly calculated in all cases where this cantilever is applied. In this way, the dimensions of the effective tip shape are “calibrated” based on the attractive forces of the well-characterized ideal surfaces. If a sphere is chosen as the “effective” shape for the tip, the effective “radius of curvature”, R_eff, of the tip is therefore calibrated. Thus, this procedure utilizes a calibration surface of known A to generate a R_eff of a given tip which can then be used to determine the unknown A of another material of interest. We demonstrate the practicality and accuracy of this updated method by comparing the results with both the original pyramid model and Lifshitz approximations (when available) for flat substrates composed of silica, polystyrene, highly ordered pyrolytic graphite (HOPG), sapphire (α-Al₃O₂), and acrylonitrile butadiene styrene (ABS).

References

(1) Fronczak, S. G.; Dong, J.; Franses, E. I.; Beaudoin, S. P.; Corti, D. S. A New Method for Determining the Hamaker Constant of Solids Using an Atomic Force Microscope. 1. Theoretical Development. Langmuir 2016, Submitted.

(2) Fronczak, S. G.; Browne, C.; Beaudoin, S. P.; Corti, D. S. A New Method for Determining the Hamaker Constant of Solids Using an Atomic Force Microscope. 2. Experimental Considerations. Langmuir 2016, Submitted.