The mathematics of oscillatory recovery rheology with applications to experiments, the Cox-Merz rules, and the nonlinear modeling of common amplitude sweep behaviors

Abstract

Oscillatory shear tests are frequently used to determine viscoelastic properties of complex fluids. Both the amplitude and frequency of the input signal can be independently varied, allowing rheologists to probe a wide range of material responses. Historically, most oscillatory tests have focused on the measurement and application of the total strain. However, the total strain is a composite parameter consisting of recoverable and unrecoverable components. Use of only the total strain therefore provides an incomplete description of the rheology. In this work, we provide a mathematical derivation for the determination of the recoverable and unrecoverable components in steady-state linear viscoelastic oscillatory flows via a simple experimental procedure. The relationship between the total strain and its components is fully explored and challenged in the context of how rheologists define moduli and common rheological models. These relations are demonstrated via experimental measurements on model viscoelastic and viscoplastic materials: wormlike micelles and Carbopol 980. Additionally, we show how the derived mathematics fully details the conditions where the Cox-Merz rules are valid in terms of recovery rheology. Finally, we demonstrate how a thorough understanding of the strain components can be used to create a simple nonlinear model that reproduces all common amplitude sweep behaviors.

Appendix

Several different terms describing various stress, strain, phase, and moduli contributions were introduced in this work that are similar in notation and may be confusing to the reader. This appendix provides definitions for each symbol used in this work for reference and clarity.

A₀—The amplitude of the total strain signal which consists of an arbitrary number of summed component waves.

A_n—The amplitude of the nth strain signal that comprises the total strain signal.

G′—The storage modulus. It is the sum of G′_solid and G′_fluid.

G′_solid—The solid-like contribution to the storage modulus. It is equivalent to G′ when δ_un = π/2.

G′_fluid—The fluid-like contribution to the storage modulus. It is zero when δ_un = π/2.

G″—The loss modulus. It is the sum of G″_solid and G″_fluid.

G″_solid—The solid-like contribution to the loss modulus.

G″_fluid—The fluid-like contribution to the loss modulus.

G*—The complex modulus.

|G*|—The magnitude of the complex modulus.

G_tot—The magnitude of the complex modulus.

G_rec—The modulus describing the resistance to recoverable deformation.

G′_rec—The modulus describing the resistance to elastic recoverable deformation.

G″_rec—The modulus describing the resistance to viscous recoverable deformation.

G_un—The modulus describing the resistance to unrecoverable deformation.

G_el—The modulus describing the resistance to elastic deformation. It is equivalent to G′_rec when δ_un = π/2.

G_vis—The modulus describing the resistance to viscous deformation. It has contributions from G″_rec and G_un.

t—Time.

W_s(ω)_avg—The average energy stored in a cycle of oscillation.

W_d(ω)_avg—The average energy dissipated in a cycle of oscillation.

γ—The total strain, this is an oscillatory time-dependent signal. It is comprised of all the strain contributions and is what one measures directly on a rheometer.

γ₀—The amplitude of the total strain signal.

γ_ss—The strain shift, the non-zero value around which the steady state total strain signal oscillates. It is equivalent to the unrecoverable strain amplitude at steady state if the applied stress is σ(t) = σ₀sin(ωt).

γ₀—The amplitude of the total strain signal.

γ_rec—The recoverable strain, this is an oscillatory time-dependent signal. It is the amount of strain recovered upon instantaneously applying zero stress at time t.

γ_rec,0—The amplitude of the recoverable strain signal.

γ_un—The unrecoverable strain, this is an oscillatory time-dependent signal. It is the amount of strain that persists upon instantaneously applying zero stress at time t after the strain shift is subtracted.

γ_un,0—The amplitude of the unrecoverable strain signal. It is equivalent to the strain shift at steady state.

$\dot{\upgamma }$—The strain rate.

${\dot{\upgamma }}_{un}$—The unrecoverable strain rate. In a steady shear flow, this is equivalent to the strain rate.

${\dot{\upgamma }}_{0}$—The amplitude of the total strain rate signal, equivalent to ωγ₀.

δ—The phase difference between the stress and total strain signals.

δ₀—The phase difference between the stress and the total strain signal which consists of an arbitrary number of summed component waves.

δ_n—The phase difference between the stress and the nth strain signal that comprises the total strain signal.

δ_rec—The phase difference between the stress and recoverable strain signals.

δ_un—The phase difference between the stress and unrecoverable strain signals, assumed to be π/2.

$\eta \left(\dot{\gamma }\right)$—The viscosity measured from a steady shear flow.

$\eta ^{\prime}$—The dynamic viscosity.

${\eta }_{{\text{c}}}$—The consistency as defined by Cox and Merz.

${\eta }_{{\text{un}}}$—The viscosity describing the resistance to unrecoverable rate of deformation.

$|{\eta }^{*}\left(\omega \right)|$—The magnitude of the complex viscosity.

σ—The total stress, this is an oscillatory time-dependent signal. It is comprised of all the stress contributions and is what one measures directly on a rheometer.

σ₀—The amplitude of the total stress signal.

σ_e—The elastic stress, this is an oscillatory time-dependent signal.

σ_v—The viscous stress, this is an oscillatory time-dependent signal.

ω—The angular frequency of oscillation.