AFM Indentation on Highly Heterogeneous Materials Using Different Indenter Geometries

Abstract

Hertzian mechanics is the most frequently used theory for data processing in Atomic Force Microscopy (AFM) indentation experiments on soft biological samples, due to its simplicity and significant scientific results previously published. For instance, using the Hertz model, it has been proven that there are significant differences in the mechanical properties of normal and cancerous tissues and that cancer cells’ invasive properties are correlated with their nanomechanical properties. However, many scientists are skeptical regarding the applicability of the Hertz theory to biological materials, as they are highly heterogeneous. The main critical question to be addressed is “what do we calculate” when fitting the force-indentation data to Hertz equations. Previous studies have shown that when using cylindrical, parabolic, or conical indenters, the fitting parameter is the average Young’s modulus. In this paper, it is demonstrated that it is also valid to fit equations derived from Hertzian mechanics to force-indentation data when testing soft, heterogeneous samples for any indenter geometry. The fitting factor calculated through this approach always represents the average Young’s modulus for a specific indentation depth. Therefore, Hertzian mechanics can be extended to soft heterogeneous materials, regardless of the indenter’s shape.

Keywords

• hertz model
• heterogeneous samples
• fitting
• depth-dependent behavior
• biological materials
• deep spherical indentations