Theoretical modelling and implementation of elastic modulus measurement at the nanoscale using atomic force microscope

Quantitative studies of mechanical behaviour and primarily elastic modulus are essential for material science at the nanometer scale. AFM nanoindentation is the most promising approach to address the problem. In our study we perform AFM-based nanoindentation (deflection-versus-distance curves) on a set of polymer materials with microscopic moduli ranging from 1 MPa to 10 GPa. The measurements were done with probes of different tip shapes and force levels from 100 nN to 3 mu N. The tip geometry was evaluated from TEM and SEM micrographs and piecewise linearly interpolated for the use of analysis software; probe spring constant was determined from thermal tune data. The comparative analysis of nanoindentation data was carried out using models of Sneddon and Oliver-Pharr. We derived Sneddon's integrals in closed form for any practical tip shape using a piecewise linear interpolation. Oliver-Pharr's method to account for plasticity for the unloading curve was adapted for Sneddon's integrals. An interactive software implementation with both models was developed and applied.