Abstract
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.
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