Development of an accurate simplified approach for data processing in AFM indentation experiments

Abstract

Atomic Force Microscopy (AFM) nanoindentation is the most effective method for determining the mechanical properties of soft biological materials and biomaterials at the nanoscale, with significant applications in many areas, including cancer diagnosis. However, a major drawback of this method is the complexity of the experimental procedure and data processing, which requires several calibration steps.To avoid this complexity, the AFM tip is usually approximated as a perfect cone. In this case, $F=c{h}^2$, where F is the applied force, ℎ is the indentation depth, and $c$ is a constant that depends on both the cone’s half-angle and the material's properties. However, since AFM tips are pyramidal with a rounded tip apex (or similar to a truncated cone in some cases), the conical approximation may lead to non-negligible errors. Although equations exist that relate the applied force, indentation depth, and the sample’s Young’s modulus for real indenters, they are rarely used because they do not directly relate the applied force to the indentation depth (i.e., the fitting process is much more complicated compared to the conical approximation). In this paper, a new, accurate, simplified approach for data processing is proposed, based on fitting the force–indentation data to a quadratic equation of the form: $F=c_2{h}^2+c_{1}h$. It is proven that the parameter $c_2$ is independent of the tip apex properties. On the other hand, the parameter $c_1$ depends on the material properties, the cone’s half angle, and the shape and dimensions of the tip apex. Simulated force-indentation data from sphero-conical and blunted pyramidal indenters, along with real experimental data from lung tissues, are processed using the proposed approach. The key result is that Young’s modulus can be accurately determined using only the $c_2$ parameter; therefore, tip characterization can be avoided.