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# Toward an Improved Method for Determining the Hamaker Constant of Solid Materials Using Atomic Force Microscopy

Particle adhesion plays a significant role in a wide range of industries and applications including pharmaceuticals, semiconductors, explosive detection systems, and the processing of bulk solid materials. For neutrally charged materials, particle adhesion arises primarily due to the dominate attractive van der Waals (vdW) force; the study of vdW forces is of particular interest because they are always present in a system. The strength of the vdW force between a pair of interacting materials is quantified by the Hamaker constant, A. Several theoretical and experimental techniques have been developed to quantify the Hamaker constant of a material including Lifshitz theory, the surface force apparatus, centrifuge technique, contact angle goniometer, and atomic force microscope (AFM). The AFM is of particular interest because it can be used to accurately measure electrostatic attraction, liquid bridging, and vdW interactions across a broad range of solid materials at the nanometer length scale. The development of a new AFM-based method for determining the Hamaker constant of solid materials is the focus of this work.

Several AFM-based methods have been proposed to estimate the Hamaker constant of a solid material, each using different parts of the AFM deflection curve which is generated from an AFM force experiment. In particular, for the approach-to-contact region of the deflection curve, previous work established a connection between the Hamaker constant and the deflection of the cantilever at first contact with the surface, d_{c}. While d_{c} is well-defined experimentally, the estimation of A from the (average) value of d_{c},although consistent with known results, was nonetheless found to introduce a significant degree of uncertainty in the reported value. Inherent material surface roughness has since been shown to be the primary reason for this large uncertainty. Thus, the overall goal of this work is to develop an updated approach-to-contact method to directly account for material surface roughness, thereby providing accurate estimates of the Hamaker constant with a significant reduction in uncertainty for a broad range of solid materials.

First, a novel vdW force model is derived describing the interaction between an AFM cantilever, treated as an effective sphere, and a surface of arbitrary roughness. The underlying material surface geometry and surface roughness is modeled directly using a surface height function, such that the vdW force is computed using the new expression over the full domain of the surface. Then, d_{c} is determined for any point along the surface from the limit of stability of the cantilever, or critical point, as it approaches the surface. Because of surface roughness, different values of d_{c} will be obtained as the tip accesses separate surface positions. As such, a characteristic distribution of d_{c}-values, or a d_{c}-distribution, is observed for a given surface, providing a signature of the underlying surface roughness. A study is completed to understand the effects of surface geometry on the resulting d_{c}-distributions for several model surfaces, or surface height functions.

The aforementioned vdW force model is derived from the limit of quasi-static behavior of the cantilever, in which the cantilever is assumed to always be in mechanical equilibrium up until the critical point and at which the tip then immediately “jumps” into contact with the surface. In practice, however, the cantilever approaches the surface at a finite approach speed, such that mechanical equilibrium cannot truly be maintained (due to inertial effects). Therefore, a model describing the dynamic behavior of the cantilever tip is presented. An effective Hamaker constant is obtained for a particular surface and cantilever approach speed by minimizing the relative entropy, which is a quantitative metric used to determine the “closeness” of two probability distribution functions, between the d_{c}-distributions generated from the dynamic model and quasi-static limit. The effective A approaches the “true” A at sufficiently slow cantilever approach speeds, and this trend is validated computationally for various model surfaces. Therefore, the behavior of the cantilever is well-described by the quasi-static model and so d_{c}-values obtained experimentally can be properly compared with those predicted using the previously-developed quasi-static model.

Finally, a robust method to extract an accurate value of A from an experimentally obtained d_{c}-distribution for a particular substrate is developed. By inputting a range of A-values, the Hamaker constant of a given substrate, with a given surface roughness, is estimated by minimizing the relative entropy between the experimental (or true) and model-predicted (the surface with its given roughness) d_{c}-distributions. The self-Hamaker constant, A_{11}, of three experimental substrates – amorphous silica, stainless steel, and sapphire – is determined over a range of experimental surfaces with varying topographies. This provides a true test of the method since surface roughness is taken directly into account, and so the self-Hamaker constant, with a small degree of inevitable experimental error, should be constant across each surface for a particular substrate. The A_{11 } values were found to be in excellent agreement between surfaces comprised of the same substrate, and the average A_{11 }value across all plates for a particular substrate agreed very well with those found in the literature derived from the rigorous theoretical Lifshitz theory.