Analytical model of atomic-force-microscopy force curves in viscoelastic materials exhibiting power law relaxation

We propose an analytical model for the force-indentation relationship in viscoelastic materials exhibiting a power law relaxation described by an exponent n, where n = 1 represents the standard viscoelastic solid (SLS) model and n < 1 represents a fractional SLS model. To validate the model, we perform nanoindentation measurements of polyacrylamide gels with atomic force microscopy (AFM) force curves. We found exponents n < 1 that depend on the bisacrylamide concentration. We also demonstrate that the fitting of AFM force curves for varying load speeds can reproduce the dynamic viscoelastic properties of those gels measured with dynamic force modulation methods.