Low-dimensional intrinsic material functions for nonlinear viscoelasticity

Abstract

Rheological material functions are used to form our conceptual understanding of a material response. For a nonlinear rheological response, the possible deformation protocols and material measures span a high-dimensional space. Here, we use asymptotic expansions to outline low-dimensional measures for describing leading-order nonlinear responses in large amplitude oscillatory shear (LAOS). This amplitude-intrinsic regime is sometimes called medium amplitude oscillatory shear (MAOS). These intrinsic nonlinear material functions are only a function of oscillatory frequency, and not amplitude. Such measures have been suggested in the past, but here, we clarify what measures exist and give physically meaningful interpretations. Both shear strain control (LAOStrain) and shear stress control (LAOStress) protocols are considered, and nomenclature is introduced to encode the physical interpretations. We report the first experimental measurement of all four intrinsic shear nonlinearities of LAOStrain. For the polymeric hydrogel (polyvinyl alcohol - Borax) we observe typical integer power function asymptotics. The magnitudes and signs of the intrinsic nonlinear fingerprints are used to conceptually model the mechanical response and to infer molecular and microscale features of the material.

Appendix: Constitutive models used

Corotational Maxwell model

The specific choice of this constitutive model is motivated by the simplicity in its definition and the existence of analytical solutions for the higher harmonics in the shear stress response under LAOS (Giacomin et al. 2011). Another unique feature of this model is that the shear stress is naturally nonlinear, due to the corotational derivative. The two-constant corotational Maxwell model is defined as follows:

$$ \underline{\underline \sigma} +\lambda\frac{\mathcal{D}\;\underline{\underline \sigma} } {\mathcal{D}\;t}=\eta_0\;\underline{\underline {\dot{\gamma}}} $$

(41)

where the high-frequency plateau modulus is ${G=\eta _0} /\lambda$ and the time derivative

$$ \frac{\mathcal{D}\;\underline{\underline \sigma}} {\mathcal{D}\;t}=\frac{D\,\underline{\underline \sigma}} {Dt}+\frac{1}{2}\left\{ {\underline{\underline \omega} \cdot \underline{\underline \sigma} -\underline{\underline \sigma} \cdot \underline{\underline \omega} } \right\} $$

(42)

is the Jaumann derivative defined through the material derivative $D\underline {\underline \sigma } /{Dt=\partial \underline {\underline \sigma }} / {\partial t}+\left ({\underline {\textrm v} \cdot \underline {\nabla } }\right )\underline {\underline \sigma }$, and

$$ {\dot{\underline{\underline \gamma}}} =\left( {\underline{\nabla} \,\underline{\textrm{v}} } \right)+\left( {\underline{\nabla} \,\underline{\textrm{v}} } \right)^T $$

(43)

is the rate of strain tensor and

$$ \underline{\underline \omega} =\left( {\underline{\nabla}\, \underline{\textrm{v}} } \right)-\left( {\underline{\nabla}\, \underline{\textrm{v}} } \right)^T $$

(44)

is the vorticity tensor. For an oscillatory shear flow of the form

$$ \dot{\gamma} ( t )=\left( {{\begin{array}{*{20}c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array}} } \right)\dot{\gamma}_0 \cos \omega t, $$

(45)

Giacomin et al. (2011) represent the shear stress response as a power series expansion in the higher powers of strain-rate amplitude

$$\begin{array}{*{20}l} \sigma_{12} \left( {t;\omega ,\dot{\gamma}_0} \right)=\sum\limits_{j:\;\text{odd}} \sum\limits_{n:\;\text{odd}}^j \dot{\gamma}_0^j &\left\{ {\eta_{jn}^{\prime} \left( \omega \right)\cos \left( {n\omega t} \right)}\right.\\ &\,+\left.{\eta_{jn}^{\prime \prime} \left( \omega \right)\sin ( {n\omega t} )} \right\} \end{array} $$

(46)

and obtain analytical expressions for the shear stress (their Eq. 58) and the viscosity coefficients $\eta _{jn}^{\prime } ( \omega )$ and $\eta _{jn}^{\prime \prime } ( \omega )$ up to the fifth harmonic (their Eqs. 160–171). Their shear stress expansion includes Weissenberg number $\left ( {\text {Wi}=\dot {\gamma }_0 \lambda =\gamma _0\text {De}} \right )$ as front factors at various order (Wi$^{2}$ and Wi$^{4})$. This can be equivalently interpreted as an expansion with respect to strain-rate amplitude $\dot {\gamma }_0$ or strain amplitude $\gamma _{0}$. Rheological material functions are defined with respect to these experimental variables, rather than dimensionless variables that require a priori knowledge of the material (here, the relaxation time $\lambda$). Those authors choose the strain-rate expansion, e.g., as given in their Eqs. 160–171, which results in expressions that depend on both De and $\lambda$ (the timescale $\lambda$ remains in the front factors).

We choose the strain amplitude expansion, leaving De $=$ $\lambda \omega$ in the front factors, which gives material functions that are only a function of De. This is a simpler representation, which we prefer. This is found most directly from the dynamic viscosity coefficients (their Eqs. 160–171). We convert these viscosity coefficients to our notation and define the normalized intrinsic nonlinearities as a function of De,

$$ \frac{\left[e_1 \right]}{G}=\omega^3\eta_{13}^{\prime \prime} =-\frac{3}{2}\text{De}^4\left( {\frac{1}{\left( {1+\text{De}^2} \right)\left( {1+4\text{De}^2} \right)}} \right), $$

(47)

$$ \frac{\left[{v_1}\right]}{\eta_0} =\omega^2\eta_{13}^{\prime} =-\frac{3}{4}\text{De}^2\left( {\frac{1}{\left( {1+\text{De}^2} \right)\left( {1+4\text{De}^2} \right)}} \right), $$

(48)

$$ \begin{array}{rll} \frac{\left[ {e_3} \right]}{G}&=-\omega^3\eta_{33}^{\prime \prime}\\ &=+\frac{3}{2}\text{De}^4\\ &\quad\times\left( {\frac{\left( {1-\text{De}^2} \right)}{\left( {1+\text{De}^2} \right)\left( {1+4\text{De}^2} \right)\left( {1+9\text{De}^2} \right)}} \right), \end{array} $$

(49)

$$ \begin{array}{rll} \frac{\left[{v_3}\right]}{\eta_0} &=\omega^2\eta_{33}^{\prime}\\ &=-\frac{1}{4}De^2\\ &\quad\times\left( {\frac{\left( {1-11\text{De}^2} \right)}{\left( {1+\text{De}^2} \right)\left( {1+4\text{De}^2} \right)\left( {1+9\text{De}^2} \right)}} \right). \end{array} $$

(50)

The above measures are normalized by the plateau modulus G and steady shear viscosity $\eta _{0 } = G\lambda$ which follow from the Maxwell-type linear viscoelastic response. It can be seen from Eqs. 47–50 that these intrinsic nonlinearities do not depend on any nonlinear parameter and are only a function of the Deborah number De. Equations 47–50 are used in Fig. 2.

Giesekus model

The single-mode Giesekus model is defined by a tensor equation in polymeric stress $\underline {\underline \sigma }$ as (Bird et al. 1987)

$$ \frac{1}{\lambda} \underline{\underline \sigma} +\underline{\underline \sigma} _{( 1 )} +\frac{\alpha} {\lambda G}\underline{\underline \sigma} \cdot \underline{\underline \sigma} =G\underline{\underline {\dot{\gamma}} } $$

(51)

where

$$ \underline{\underline \sigma} _{( 1 )} =\frac{\partial} {\partial t}\underline{\underline \sigma} +\left( {\underline{\textrm{v}} \cdot \underline{\nabla} } \right)\underline{\underline \sigma} -\left( {\underline{\nabla} \underline{\textrm{v}} } \right)^T\cdot \underline{\underline \sigma} -\underline{\underline \sigma} \cdot \underline{\nabla} \underline{\textrm{v}} $$

(52)

is the upper convected time derivative, $\lambda$ the characteristic relaxation time, and G the plateau modulus. $\alpha$ is a dimensionless nonlinear parameter, also called the mobility factor, varying between 0 and 1. For a sinusoidal simple shear input $\gamma ( t )=\gamma _0 \sin \omega t$, Gurnon and Wagner (2012) represent the alternance state polymeric shear stress response as an asymptotic expansion in frequency (index $n)$ and strain (index $j)$

$$\begin{array}{*{20}l} \sigma \left( {t;\omega ,\gamma_0} \right)=\sum\limits_j^{\infty} \sum\limits_n^j &\left[ {A_n^{( j )} \left( {\omega ,\gamma_0} \right)\sin ( {n\omega t} )}\right.\\ &\,+\left.{B_n^{( j )} \left( {\omega ,\gamma_0} \right)\cos ( {n\omega t} )} \right] \end{array} $$

(53)

where $A_n^{( j )} \left ( {\omega ,\gamma _0} \right )$ and $B_n^{( j )} \left ( {\omega ,\gamma _0} \right )$ are coefficients for the in-phase and out-of-phase components of stress to the deformation input. Expanding Eq. 53 up to the third harmonic and third power of strain, they obtain analytical expressions for the coefficients as a function of the Weissenberg number and Deborah number. We take their asymptotic expressions and convert them to our notations, normalizing them with the linear viscoelastic plateau modulus G and steady shear viscosity $\eta$to obtain intrinsic material functions that only depend on De and $\alpha$,

$$ \begin{array}{rll} \frac{\left[ {e_1} \right]}{G}&=\frac{A_{12,1}^{( 3 )}} {\gamma _0^3} \\ &=\frac{\alpha \text{De}^4\left( {-21-41\text{De}^2-8\text{De}^4+4\alpha \left( {4+7\text{De}^2} \right)} \right)}{4\left( {1+\text{De}^2} \right)^3\left( {1+4\text{De}^2} \right)}, \end{array} $$

(54)

$$ \begin{array}{rll} \frac{\left[ {e_3} \right]}{G}&=-\frac{A_{12,3}^{( 3 )}} {\gamma _0^3} \\ &=-\frac{\alpha \text{De}^4\left( {-21-30\text{De}^2+51\text{De}^4+4\alpha \left( {4-17\text{De}^2+3\text{De}^4} \right)} \right)}{4\left( {1+\text{De}^2} \right)^3\left( {1+4\text{De}^2} \right)\left( {1+9\text{De}^2} \right)}, \end{array} $$

(55)

$$ \begin{array}{rll} \frac{\left[ {v_1} \right]}{\eta} &=\frac{B_{12,1}^{(3)}} {\omega \gamma_0^3} \\ &=-\frac{\alpha \text{De}^2\left({9+11\text{De}^2-10\text{De}^4+2\alpha \left( {-3-\text{De}^2+8\text{De}^4} \right)} \right)}{4\left( {1+\text{De}^2} \right)^3\left( {1+4\text{De}^2} \right)}, \end{array} $$

(56)

$$ \begin{array}{rll} &\frac{\left[{v_3} \right]}{\eta} \\ &=\frac{B_{12,3}^{(3)}}{\omega \gamma_0^3} \\ &=\frac{\alpha \text{De}^2\left({-3{\kern-0.5pt}+{\kern-0.5pt}48\text{De}^2{\kern-0.5pt}+{\kern-0.5pt}33\text{De}^4{\kern-0.5pt}-{\kern-0.5pt}18\text{De}^6{\kern-0.5pt}+{\kern-0.5pt}\alpha \left( {2{\kern-0.5pt}-{\kern-0.5pt}48\text{De}^{2}{\kern-0.5pt}+{\kern-0.5pt}46\text{De}^4} \right)} \right)}{4\left({1+\text{De}^2} \right)^3\left( {1+4\text{De}^2} \right)\left( {1+9\text{De}^2} \right)}. \end{array} $$

(57)

Equations 54–57 were used for Fig. 4.