Toward an Improved Method for Determining the Hamaker Constant of Solid Materials Using Atomic Force Microscopy. II. Dynamic Analysis and Preliminary Validation

Abstract

The further development of an existing approach-to-contact atomic force microscopy (AFM) method for determining the Hamaker constant, A, of a solid nondeformable material is presented. Upon explicitly accounting for the surface roughness of the given substrate, an estimate of A is directly obtained from the resulting distribution of deflections of the AFM cantilever tip at first contact with the surface, d c. We explore the effects of surface roughness on the resulting d c distributions for several model surfaces. We also analyze the dynamic behavior of the cantilever tip for a range of cantilever approach speeds. At sufficiently slow approach speeds, the dynamic d c distributions are nearly identical to those obtained in the quasi-static limit. Consequently, a method is proposed whereby the relative entropy between the dynamic and quasi-static distributions is used to estimate the value of A. Finally, an experimental test of the method is performed in which the self-Hamaker constant of an amorphous silica surface is determined to be in excellent agreement with the literature established value. Because the surface topography is directly included in the method, the uncertainty in the estimation of the self-Hamaker constant is also significantly reduced, compared to other similar approaches.

Toward an Improved Method for Determining the Hamaker Constant of Solid Materials Using Atomic Force Microscopy. III: Experimental Validation of a New Approach-to-Contact Method

“…First, an AFM-based experimental d c -distribution is generated for a given substrate, obtained at slow enough cantilever approach speeds such that inertial effects are unimportant. (As confirmed in ref , inertial effects are negligible for approach speeds less than or equal to about 200 nm/s.) This experimentally obtained distribution is then considered to be the “true” distribution for which the self-Hamaker constant of the chosen substrate is unknown.…”

“…The relative entropy equals zero when the two distributions are identical, while a nonzero value provides a measure of how much “information is lost” between these two distributions . As shown in ref , the relative entropy exhibits in most cases a reasonably sharp minimum at a particular value of A input , which is then identified as the effective Hamaker constant, A eff , between the tip and the substrate (again strictly speaking, A eff,12 ). In other words, A eff is the estimated value of the Hamaker constant that presumably gave rise to the experimentally obtained “true” distribution.…”

“…(We also note that due to optical interference and electronic background effects of the AFM instrument’s data collection, a linear regression is used in order to identify the tip’s undeflected state . Along with the sampling resolution of the AFM, the error of this instrument noise was found to be around 0.1–0.2 nm, much smaller than most of the obtained deflections, and falling within the histogram “binning effects” that arise when generating the d c -distributions. , )…”

“…Thus, upon reaching the critical point, the tip immediately jumps into contact with the surface from which d c can be determined. The complete details for determining the resulting value of d c at a particular location along the substrate for a given R , spring constant k c of the cantilever, and the chosen A input are provided in refs and . These quasistatic distributions are now considered to be the “approximating” distributions.…”

“…The current authors recently presented , a framework for an improved experimental AFM-based method for estimating A with relatively high accuracy from the approach-to-contact portion of the deflection curve. The new method, in contrast to previous approaches, explicitly accounted for the inherent surface roughness of the given substrate under consideration.…”