^{1}, Gareth H. McKinley

^{1,a)}and Randy H. Ewoldt

^{2}

### Abstract

In a previous paper [T. S. K. Ng and G. H. McKinley, J. Rheol.52(2), 417–449 (2008)], we demonstrated that gluten gels can best be understood as a polymericnetwork with a power-law frequency response that reflects the fractal structure of the gluten network. Large deformation tests in both transient shear and extension show that in the absence of rigid starch fillers these networks are also time-strain factorizable up to very large strain amplitudes . In the present work, we further explore the nonlinear rheological behavior of these critical gels by considering the material response obtained in large amplitude oscillatory shear over a wide range of strains and frequencies. We use a Lissajous representation to compare the measured material response with the predictions of a network theory that is consistent with the proposed molecular structure of gluten gels. In the linear viscoelastic regime, the Lissajous figures are elliptical as expected and can be quantitatively described by the same power-law relaxation parameters determined independently from earlier experiments. In the nonlinear regime, the Lissajous curves show two prominent additional features. First is a gradual softening of the network indicated by the rotation of the major axis of the stress ellipse. This feature is accounted for in the model by the inclusion of a simple nonlinear network destruction term that reflects the reduction in network connectivity as the polymer chains are increasingly stretched. Second, a distinct upturn in the viscoelastic stress is discernable at large strains. We show that this phenomenon can be modeled by considering the effects of finitely extensible segments in the elasticnetwork. We use this model to quantitatively predict the material response in other large amplitude transient flows such as the start-up of steady shear and transient uniaxial extension up until the onset of strongly nonlinear unsteady phenomena such as edge fracture in shear and sample rupture during extension.

The authors wish to acknowledge Kraft Foods for financial support of this work and Professor Eric Windhab from ETH-Zurich for encouraging us to explore the properties of gluten gels. The authors would also like to thank M. Padmanabhan for a number of stimulating conversations on the rheological properties of doughs.

I. INTRODUCTION

II. GLUTEN DOUGH PREPARATION

III. RHEOMETRY

IV. LINEAR VISCOELASTICITY AND NETWORK STRUCTURE OF GLUTEN GELS

V. NONLINEAR DEFORMATION OF GLUTEN GEL

A. LAOS

B. Network model

C. Frequency dependence of Lissajous figures

D. Transient oscillatory response

E. Comparison with other nonlinear deformations

VI. CONCLUSIONS

### Key Topics

- Gels
- 82.0
- Linear viscoelasticity
- 31.0
- Elasticity
- 24.0
- Elastic moduli
- 22.0
- Stress strain relations
- 15.0

## Figures

(a) Measured mixograph output (torque in arbitrary units) vs mixing time for a typical gluten dough. (b) Details of temporal oscillations in the measured torque signal from 700 to 720 s due to the periodic motion of the pins. (c) Power spectrum of the measured mixograph torque (solid line) from 700 to 800 s. Peaks correspond to harmonics calculated from the epitrochoidal motion of the four moving pins in relation to the three stationary pins in the mixograph bowl.

(a) Measured mixograph output (torque in arbitrary units) vs mixing time for a typical gluten dough. (b) Details of temporal oscillations in the measured torque signal from 700 to 720 s due to the periodic motion of the pins. (c) Power spectrum of the measured mixograph torque (solid line) from 700 to 800 s. Peaks correspond to harmonics calculated from the epitrochoidal motion of the four moving pins in relation to the three stationary pins in the mixograph bowl.

Storage and loss moduli of a gluten gel measured in small amplitude oscillation (circles) at . Solid lines represent predictions from Eqs. (7)–(11), the superposition of power-law gel and Rouse models: , , , . The response of a gluten gel dissolved in 8 M urea solution (triangles) and the corresponding reconstituted dough (squares) are also shown. The dissolved gluten shows a viscoelastic fluid-like behavior with significantly lower moduli. The gluten gel that is reformed by diluting the solution with water to wash out the urea has very similar viscoelastic properties to the original sample.

Storage and loss moduli of a gluten gel measured in small amplitude oscillation (circles) at . Solid lines represent predictions from Eqs. (7)–(11), the superposition of power-law gel and Rouse models: , , , . The response of a gluten gel dissolved in 8 M urea solution (triangles) and the corresponding reconstituted dough (squares) are also shown. The dissolved gluten shows a viscoelastic fluid-like behavior with significantly lower moduli. The gluten gel that is reformed by diluting the solution with water to wash out the urea has very similar viscoelastic properties to the original sample.

Lissajous–Bowditch curves after 12 oscillatory cycles for a gluten gel at fixed frequency with (a) ; (b) ; (c) ; (d) and 6.00. As the imposed strain amplitude increases from (a) to (d), the magnitude of the maximum stress grows and the axes are rescaled. Experimental data are plotted as open symbols. The decomposed elastic stresses are shown in the lower panel as a dotted line for . Predictions from the nonlinear generalized gel model described in the text are plotted as a solid line for each strain amplitude.

Lissajous–Bowditch curves after 12 oscillatory cycles for a gluten gel at fixed frequency with (a) ; (b) ; (c) ; (d) and 6.00. As the imposed strain amplitude increases from (a) to (d), the magnitude of the maximum stress grows and the axes are rescaled. Experimental data are plotted as open symbols. The decomposed elastic stresses are shown in the lower panel as a dotted line for . Predictions from the nonlinear generalized gel model described in the text are plotted as a solid line for each strain amplitude.

Dynamic moduli of gluten gel undergoing a strain-sweep (ARES rheometer, controlled-strain mode) at . The elastic moduli and are the first and third in-phase Fourier coefficients of the output signal; is typically quoted as the “storage modulus”; and are the small and large strain moduli, respectively [see Eqs. (15) and (16)], and is the stiffening ratio defined in Eq. (19).

Dynamic moduli of gluten gel undergoing a strain-sweep (ARES rheometer, controlled-strain mode) at . The elastic moduli and are the first and third in-phase Fourier coefficients of the output signal; is typically quoted as the “storage modulus”; and are the small and large strain moduli, respectively [see Eqs. (15) and (16)], and is the stiffening ratio defined in Eq. (19).

Normalized elastic stress extracted from the Lissajous curves for at and .

Normalized elastic stress extracted from the Lissajous curves for at and .

Rheological fingerprint of a gluten gel in LAOS flow. The shapes and maximum stress amplitude of each Lissajous curve are presented in a Pipkin space spanning , . In each case, the data represent the sixth cycle of the LAOS test, and there are at least 80 experimental data points in each cycle. The relative error [defined in Eq. (30)] is given on the top left of each sub-figure, while the maximum stress associated with each limit cycle is displayed at the bottom right. The solid lines represent the predictions from the FENE network model discussed in Sec. V B.

Rheological fingerprint of a gluten gel in LAOS flow. The shapes and maximum stress amplitude of each Lissajous curve are presented in a Pipkin space spanning , . In each case, the data represent the sixth cycle of the LAOS test, and there are at least 80 experimental data points in each cycle. The relative error [defined in Eq. (30)] is given on the top left of each sub-figure, while the maximum stress associated with each limit cycle is displayed at the bottom right. The solid lines represent the predictions from the FENE network model discussed in Sec. V B.

Summary of constitutive parameters in the FENE network model: (a) The critical gel-like behavior observed in linear viscoelastic tests is approximated by a series of Maxwell relaxation modes; . (b) Non-linear functions in the network model. A FENE type spring law is used as the non-linear modulus and is characterized by the finite-extensibility limit . The rate of junction destruction, , is characterized by two additional parameters; describes an increase in the network destruction rate at intermediate microstructural deformations; modifies the FENE function by allowing the junction points to be destroyed close to, but before, the finite-extensibility limit is reached. (c) Lissajous figures for LAOS tests on a gluten dough at . The quasi-linear formulation with and grossly overpredicts the stress. Inclusion of the term gives strain-softening (clockwise rotation of ellipse). A small degree of strain-stiffening is apparent at large strain when the non-linear FENE spring law is introduced.

Summary of constitutive parameters in the FENE network model: (a) The critical gel-like behavior observed in linear viscoelastic tests is approximated by a series of Maxwell relaxation modes; . (b) Non-linear functions in the network model. A FENE type spring law is used as the non-linear modulus and is characterized by the finite-extensibility limit . The rate of junction destruction, , is characterized by two additional parameters; describes an increase in the network destruction rate at intermediate microstructural deformations; modifies the FENE function by allowing the junction points to be destroyed close to, but before, the finite-extensibility limit is reached. (c) Lissajous figures for LAOS tests on a gluten dough at . The quasi-linear formulation with and grossly overpredicts the stress. Inclusion of the term gives strain-softening (clockwise rotation of ellipse). A small degree of strain-stiffening is apparent at large strain when the non-linear FENE spring law is introduced.

Lissajous curves of a gluten gel performed at large amplitude for a range of frequencies and at and the prediction of the FENE network model.

Lissajous curves of a gluten gel performed at large amplitude for a range of frequencies and at and the prediction of the FENE network model.

Transient behavior of gluten gel. (a) Lissajous figures depicting material response when the strain amplitude of oscillation is increased from to , and then back to again at a fixed frequency of ; (b) enlarged view of the approach back to the small strain limit cycle for ; (c) temporal response of the scaled oscillatory stress or apparent modulus illustrating the transient nature of the network. The FENE network correctly predicts the evolution in the oscillatory stress when the strain amplitude is increased and complete recovery of the gluten gel when the strain amplitude is reduced.

Transient behavior of gluten gel. (a) Lissajous figures depicting material response when the strain amplitude of oscillation is increased from to , and then back to again at a fixed frequency of ; (b) enlarged view of the approach back to the small strain limit cycle for ; (c) temporal response of the scaled oscillatory stress or apparent modulus illustrating the transient nature of the network. The FENE network correctly predicts the evolution in the oscillatory stress when the strain amplitude is increased and complete recovery of the gluten gel when the strain amplitude is reduced.

Start-up of steady shear flow at . The power-law growth in the shear stress and also in the normal stress difference is well predicted by the FENE network model up to , after which the non-linearity of the model overpredicts the measured overshoot in the normal stress difference and the shear stress. Improvements to the predictions can be made by using the modified network model; however, accurate constitutive modeling at large strains is difficult because the sample deformation deviates substantially from viscometric flow at torsional strains .

Start-up of steady shear flow at . The power-law growth in the shear stress and also in the normal stress difference is well predicted by the FENE network model up to , after which the non-linearity of the model overpredicts the measured overshoot in the normal stress difference and the shear stress. Improvements to the predictions can be made by using the modified network model; however, accurate constitutive modeling at large strains is difficult because the sample deformation deviates substantially from viscometric flow at torsional strains .

Transient extensional stress difference upon inception of uni-axial elongation for deformation rates ; experimental data are limited by sample rupture. Two experimental runs were performed at each strain rate to ensure reproducibility. The form of the quasi-linear simulation is shown for reference at . The FENE network overpredicts the magnitude of stress; this can be remedied by a simple modification; see Eq. (33). The strain and time to rupture correspond closely to those predicted by the finite-extensibility limit determined from LAOS with and .

Transient extensional stress difference upon inception of uni-axial elongation for deformation rates ; experimental data are limited by sample rupture. Two experimental runs were performed at each strain rate to ensure reproducibility. The form of the quasi-linear simulation is shown for reference at . The FENE network overpredicts the magnitude of stress; this can be remedied by a simple modification; see Eq. (33). The strain and time to rupture correspond closely to those predicted by the finite-extensibility limit determined from LAOS with and .

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