On the Measurements of Rigidity Modulus of Soft Materials in Nanoindentation Experiments at Small Depth

It is of interest to measure the modulus of rigidity at small indentation depths for many systems, such as thin films, nanocomposites, biomaterials, etc. Depth-dependence of the rigidity modulus of homogeneous soft materials is broadly observed in nanoindentation experiments. Typically, the modulus reaches its bulk value only when the indentation depth becomes relatively large. Nature of this effect (we suggest to call this "skin-effect" for short) is not well understood. It is not even clear if this is a real effect or an artifact. Here we present the results of precise indentation measurements based on the use of atomic force microscopy (AFM), which suggest that the skin-effect may be an artifact. It can be eliminated, and the bulk modulus can be measured at nanometer indentations if one (a) takes into account adhesion between the indenter and surface of interest, and (b) operates mostly within the linear stress strain regime. To demonstrate it, we used three AFM probes of well-defined geometry (radii of the apex were 22, 810, and 1030 nm) to study the indentation of three different polymers of the bulk rigidity of 0.6-0.7 GPa (polyurethanes) and 2.8 GPa (polystyrene). The obtained force indentation curves were processed through the Oliver-Pharr, Hertz, Johnson-Kendall-Roberts (JKR) and Derjaguin-Muller-Toporov (DMT) models. We found that the skin-effect disappeared when using dull (810 and 1030 nm) probes and processing the force-deformation data with either of the adhesion models (JKR or DMT). Moreover, the measured moduli were independent of the indentation depth. The values of the rigidity modulus were very close to the bulk values starting from the indentations of 2-3 nm. Such a small indentation seems to be the smallest one for soft materials at which the bulk modulus has been reached. When using the sharp (22 nm) probe, we were not able to reach the bulk moduli up to the maximum possible indentation allowed by the instrument 90 nm. The other sources of possible error in the modulus measurements are discussed. We conclude that the skin-effect originates mainly at both nonlinearity of stress strain relation (occurs when using excessively sharp probes) and if the probe surface adhesion is not taken into account (like in Oliver-Pharr and Hertz models).