Multiple power-law viscoelastic relaxation in time and frequency domains with atomic force microscopy

Complex viscoelastic materials exhibit power law (PL) relaxations, as opposed to simple materials described by exponential decays. Other interesting materials, like living cells, hold a universal double PL behavior whose exponents depend on the health and type of the cells. Usually, only dynamic assays are considered capable to study such viscoelastic relaxation mechanisms. In this work, we propose analytical responses with single or multiple power-law relaxation behavior by generalizing classical viscoelastic models in terms of fractional derivatives of arbitrary order alpha (0 <= alpha <= 1). In addition, we demonstrate that simple atomic force microscopy force curves are powerful methods to directly observe the viscoelastic relaxation of such complex materials. In order to validate our findings, we compare the viscoelastic relaxation exponents measured directly from simple force curves (SFCs) with those measured with dynamic techniques in both living cells and polyacrylamide gels. We believe the fractional models unveiled here describe a variety of complex materials and may be used (with SFCs) to explore sophisticated viscoelastic phenomena.