On averaging force curves over heterogeneous surfaces in atomic force microscopy

Atomic force microscopy (AFM) can be used to study mechanics at the nanoscale. Biological surfaces and nanocomposites have typically heterogeneous surfaces, both mechanically and chemically. When studying such surfaces with AFM, one needs to collect a large amount of data to make statistically sound conclusions. It is time- and resource-consuming to process each force curve separately. The analysis of an averaged raw force data is a simple and time saving option, which also averages out the noise and measurement artifacts of the force curves being analyzed. Moreover, some biomedical applications require just an average number per biological cell. Here we investigate such averaging, study the possible artifacts due to the averaging, and demonstrate how to minimize or even to avoid them. We analyze two ways of doing the averaging: over the force data for each particular distance (method 1, the most commonly used way), and over the distances for each particular force (method 2). We derive the errors of the methods in finding to the true average rigidity modulus. We show that both methods are accurate (the error is <2%) when the heterogeneity of the surface rigidity is small (<50%). When the heterogeneity is large (>100×), method 2 underestimates the average rigidity modulus by a factor of 2, whereas the error of method 1 is only 15%. However, when analyzing the different surface chemistry, which reveals itself in the changing long-range forces, the accuracy of the methods behave oppositely: method 1 can produce a noticeable averaging artifact in the deriving of the long-range forces; whereas method 2 can be successfully used to derive the averaged long-range force parameters without artifacts. We exemplify our conclusions by the study of human cervical cancer and normal epithelial cells, which demonstrate different degrees of heterogeneity.