Incorporating Rheological Nonlinearity into Fractional Calculus Descriptions of Fractal Matter and Multi-Scale Complex Fluids

Abstract

In this paper, we use ideas from fractional calculus to study the rheological response of soft materials under steady-shearing flow conditions. The linear viscoelastic properties of many multi-scale complex fluids exhibit a power-law behavior that spans over many orders of magnitude in time or frequency, and we can accurately describe this linear viscoelastic rheology using fractional constitutive models. By measuring the non-linear response during large step strain deformations, we also demonstrate that this class of soft materials often follows a time-strain separability principle, which enables us to characterize their nonlinear response through an experimentally determined damping function. To model the nonlinear response of these materials, we incorporate the damping function with the integral formulation of a fractional viscoelastic constitutive model and develop an analytical framework that enables the calculation of material properties such as the rate-dependent shear viscosity measured in steady-state shearing flows. We focus on a general subclass of fractional constitutive equations, known as the Fractional Maxwell Model, and consider several different analytical forms for the damping function. Through analytical and computational evaluations of the shear viscosity, we show that for sufficiently strong damping functions, for example, an exponential decay of fluid memory with strain, the observed shear-thinning behavior follows a power-law response with exponents that are set by the power-law indices of the linear fractional model. For weak damping functions, however, the power-law index of the high shear rate viscosity is set by the terminal behavior of the damping function itself at large strains. In the limit of a very weak damping function, the theoretical formulation predicts an unbounded growth of the shear stress with time and a continuously growing transient viscosity function that does not converge to a meaningful steady-state value. By determining the leading terms in our analytical solution for the viscosity at both low and high shear rates, we construct an approximate analytic expression for the rate-dependent viscosity. An error analysis shows that, for each of the damping functions considered, this closed-form expression is accurate over a wide range of shear rates.

Kramers-Kronig Relations for Nonlinear Rheology: 2. Validation of Medium Amplitude Oscillatory Shear (MAOS) Measurements

“…Multiscale complex fluids such as polydisperse and/or branched polymer melts and solutions [69], structured food materials [70,71], the critical gel state in polymeric or colloidal gels [25,[72][73][74], etc., show power-law dependence of relaxation modulus, G(t) = St −n , over a certain range of timescales t. Here, n ∈ (0, 1) is the power-law exponent, and the quasi-property S has units of Pa•s n and characterizes material stiffness. The corresponding storage and loss moduli are [25],…”

“…The weak dependence of the G 33 and G 33 at low frequencies (ω n ) results in a non-integrable singularity in equation 7 at u = 0. In practice, this issue is moot because power-law behavior is confined to a finite domain of frequencies [69,72].…”

Viscoelastic scaling regimes for marginally-rigid fractal spring networks

“…This second factor is absent in [33]. Inserting τ = 6τ into (3) gives 3 2 6 1−µ = 1, or µ = 2 ln 3/ ln 6. Using the relation…”

“…It has been hypothesised that the broad distribution of relaxation times derives from a similarly broad distribution of structural length scales [3,13,15]. Such structure emerges naturally from cluster aggregation processes, which can produce a scale-invariant, or fractal, geometry up to a characteristic maximum length [16][17][18].…”

Viscoelastic Scaling Regimes for Marginally Rigid Fractal Spring Networks

“…Introduction.-Many soft matter and complex systems exhibit power law rheology over a broad frequency range, manifested as parallel scaling of the linear storage and loss moduli G ′ (ω) ∝ G ′′ (ω) ∝ ω ∆ [1][2][3], or equivalently a power-law relaxation spectrum [4][5][6]. Relating this scaling to the underlying causal mechanisms would guide the selection of synthesis pathways producing desirable material properties in a number of application domains [7,8], but is not yet generally possible.…”