Investigating and Modeling the Rheology and Flow Instabilities of Thixotropic Yield Stress Fluids
by
Yufei Wei
A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Chemical Engineering) in the University of Michigan 2019
Doctoral Committee:
Professor Ronald G. Larson, Co-Chair Professor Michael J. Solomon, Co-Chair Professor Brian J. Love Professor Robert M. Ziff
Yufei Wei
ORCID iD: 0000-0002-5103-9015
© Yufei Wei 2019
Dedication
To my parents, Xiaowen, Tardis, and Tamaki.
ii Acknowledgements
I would like to thank my advisors, Prof. Ronald Larson and Prof. Michael Solomon. This thesis would not have been possible without your mentoring, guidance, and support in the past five years. Thank you for always being supportive, not only in helping me overcome setbacks in my research, but also encouraging me to pursue my career goals. You are not only my mentors but also my role models. I know for sure that I have become a better researcher and a better human being thanks to the experience of working with you.
I would like to thank Prof. Robert Ziff and Prof. Brian Love for serving on my dissertation committee. Thank you for being available for all my meetings and thank you for your feedbacks and advice on my research.
I would like to thank the wonderful members of the Larson group and the Solomon group. Thank you for always being so welcoming and helpful. I would also like to thank those who have worked with me in my projects – Qifan Huang, Sarika Mahimkar, Luofu Liu,
Srinivasa Tanmay Velidanda, Anukta Datta.
I would like to thank my friends in Ann Arbor – Tingwen Lo, Xingjian Ma, Tianhui Ma,
Chengwei Zhai, Tianyu Liu, Pengkai Kao, Shujie Chen. I will miss you all.
I would like to thank my cats, Tardis and Tamaki, for bringing me so much laughter and joy. You are two lovely creatures. I am glad that I have you in my life.
I would like to thank my parents for always being supportive and encouraging. You have done a lot for me. I love you.
iii At last, I would like to acknowledge Xiaowen Zhao, a very special friend in my life.
Thank you for bringing me so much laughter and so many happy memories. You are one of the most important persons in my life. I wish the best for you.
iv
Table of Contents
Dedication ii
Acknowledgements iii
List of Tables viii
List of Figures x
Abstract xxii
Chapter 1 State of Art 1
1.1 Thixotropy 1 1.1.1 Definition and History 1 1.1.2 Occurrence 2 1.1.3 Rheological Measurement 4 1.1.4 Representative Data sets 6 1.2 Modeling of Thixotropy 7 1.2.1 Overview 7 1.2.2 Structural Kinetics Models for Ideal Thixotropy 8 1.2.3 Incorporation of Viscoelasticity into Thixotropic Models 12 1.2.4 Incorporation of Advanced Plasticity into Thixotropic Models 14 1.3 Shear Banding in Thixotropic Fluids 18 1.4 Objectives and Outline 21 Chapter 2 Materials and Methods 22
2.1 Introduction 22 2.2 Materials 22 2.3 Rheological Experiments 23 2.3.1 Instruments 23
v 2.3.2 Steady-state Flow Curve 24 2.3.3 Step Tests 25 2.3.4 Intermittent Shear Tests 28 2.3.5 Flow Reversal Tests 29 2.3.6 Hysteresis Tests 30 2.3.7 Oscillatory Shear Tests 31 2.4 Particle Image Velocimetry (PIV) 33 2.4.1 Materials and Instruments 33 2.4.2 Algorithm 34 2.4.3 Method Validation 35 2.5 Conclusions 36 Chapter 3 Ideal thixotropy and Its Rheological Modeling 38
3.1 Introduction 38 3.2 Proposed Model 40 3.3 Decomposition of Stretched Exponential Function 42 3.3.1 Tikhonov regularization method 42 3.3.2 PDF Method 43 3.3.3 Comparison of the results of the two methods 45 3.4 Nonlinear Thixotropic Kinetics 47 3.5 Model Parameterization 50 3.6 Model Evaluation 53 3.6.1 Step Tests 53 3.6.2 Intermittent Shear Tests 55 3.6.3 Hysteresis Tests 58 3.7 Model Limitations 60 3.8 Predicting Non-monotonic Viscosity Bifurcation 61 3.9 Conclusions 66 Chapter 4 A Comprehensive Thixotropic Elasto-viscoplastic Model 69
4.1 Introduction 69 4.2 Proposed Model 71 4.2.1 Scalar Form 71 4.2.2 Tensorial Form 75 4.2.3 Summary of the Proposed Model 78
vi 4.2.4 Physical Interpretation of Structure Parameters 80 4.3 Model Parameterization 81 4.4 Model Evaluation 84 4.4.1 Qualitative comparison with the ML, IKH, and MDT models 84 4.4.2 Steady State Flow Curve 85 4.4.3 Step Tests 86 4.4.4 Intermittent Step Tests 89 4.4.5 Flow Reversal Tests 92 4.4.6 Large Amplitude Oscillatory Shear 94 4.4.7 Summary 98 4.5 Predictions of the Tensorial Model 100 4.5.1 Comparison with the Scalar Model 100 4.5.2 Comparison with the Tensorial IKH Model 102 4.6 Conclusions 104 Appendix 106 Chapter 5 Time-dependent Shear Bands in Transient Shear Flow 109
5.1 Introduction 109 5.2 Banding Dynamics in Shear Startup and Flow Reversal Tests 109 5.2.1 Shear Startup Tests 109 5.2.2 Flow Reversal Tests 113 5.3 Modeling 115 5.3.1 Proposed Model 115 5.3.2 Numerical Simulation of Banded Flow 118 5.4 Oscillation of Shear Bands Triggered by Geometry Misalignment 120 5.5 Conclusion 125 Chapter 6 Conclusions and Future Directions 126
6.1 Conclusions 126 6.2 Recommended Future Directions 128 Bibliography 130
vii List of Tables Table 1-1. Several representative definitions of thixotropy...... 1
Table 1-2. Recent examples of thixotropic materials...... 3
Table 1-3. A summary of several representative data sets ...... 6
Table 1-4. Expressions for the kinetic equation in some representative thixotropic models...... 10
Table 1-5. A summary of four models corresponding the four categories in Figure 1-4...... 13
Table 2-1. Components of the fumed silica suspension...... 23
Table 2-2. Details of the cone-and-plate geometries used in this thesis...... 24
Table 3-1. List of equations of the Tikhonov regularization method...... 43
Table 3-2. Forms of P , for different values ...... 44
Table 3-3. Parameter values for the RC Model and SC Model ...... 52
Table 3-4. Summary of the DM model...... 54
Table 4-1. Summary of the proposed model ...... 78
Table 4-2. The special values of model parameters for which the ML-IKH model reduces to
closely-related models...... 79
Table 4-3. Qualitative comparison of the ML, IKH (scalar form), MDT, and ML-IKH (scalar
form) models...... 83
Table 4-4. Parameter values of the ML-IKH, ML(SC), IKH-V, and MDT models for the
prediction of the rheological response of the 2.9 vol% fumed silica suspension. 85
viii Table 4-5. Parameters of the ML-IKH, ML, IKH-V, and MDT models for the prediction of the
LAOS results in reference [41]. Those parameters values are used in Figs. (7-8).
...... 98
Table 4-6. Evaluation of the MDT, ML, IKH-V, and ML-IKH models in predicting the simple
shear rheology of a model thixotropic fluid ...... 99
Table 4-7. Comparison of our model with the tensorial IKH model in the limit of no yield stress
( = 0) and small elastic deformation...... 103
.
ix List of Figures
Figure 1-1. Common procedures of rheological experiments. s denotes the control variable and
can be the shear rate, shear strain, or shear stress...... 4
Figure 1-2. Hysteresis loop of a 4 wt% Balangu solution. Shear rate first increases linearly with
time from 0 to 300 s − 1 within 6 min and then decreases at the same rate to 0 s −
1. (From Razavi etc. [33]) ...... 5
Figure 1-3. Illustration of early-time viscoelastic response (before 1 s) followed by late-time
thixotropic response in 2.9 vol% fumed silica suspension after a step-down from a
shear rate of 1 s–1 to the shear rates shown in the legend. Lines are model
prediction. (From Dullaert and Mewis [11], used with permission.)...... 12
Figure 1-4. Schematics of four representative forms of incorporating viscoelasticity into
thixotropic models: (a) Bingham-type model with yield stress replaced by an
elastic stress; (b) Maxwell-type model with λ -dependent modulus and/or
viscosity; (c) Oldroyd-type model with stress contribution from the medium and
internal structures; (d) a hybrid of the Bingham- and Maxwell-type models. In
each branch, the dotted line indicates the portion of the strain that does not affect
the stress...... 14
Figure 1-5. Mechanical analog of a simple TEVP model. is the spring constant, and are
respectively the elastic and plastic portion of the total strain . = + and
= + ...... 15
x Figure 1-6. Mechanical analog of a simple kinematic hardening TEVP model...... 17
Figure 1-7. (a) Hysteretic flow curves of a 2.5 wt% laponite suspension obtained by first
decreasing the shear rate from 103 to 10-3 s-1 in 90 logarithmically spaced steps
of duration 15.5 seconds and then increasing over the same range. Insert: velocity
profile in a Couette geometry. (b) Hysteresis loop area ( ) versus . (c) Overall
degree of banding ( ) versus . (From Radhakrishnan et al. [84], used with
permission. Data were initially reported in Divoux et al. [73].) ...... 20
Figure 2-1. Transmission electron microscope micrograph of dry Aerosil® R972 fumed silica
(from [86])...... 23
Figure 2-2. Schematics of (a) AR-G2 rheometer, (b) ARES rheometer, and (c) MCR702
rheometer...... 24
Figure 2-3. Steady-state flow curve of the 2.9 vol% fumed silica suspension, plotted as (a) stress
vs shear rate, (b) apparent viscosity versus shear rate ...... 25
Figure 2-4. Rheological response of the fumed silica suspension in three step-rate tests: (a) the
shear rate jumps from 0 to 0.1 s−1, (b) 0.1 to 1 s−1, (c) 1 to 9 s−1...... 26
Figure 2-5. (a) Response of a silicone oil (viscosity, 0.948 Pa·s) in a step-stress test (initial stress,
0 Pa; final stress 5 Pa). Circles: experimental data; line: calculation of Eq. (2-3)
and Eq.(2-4). The value of lag is determined from the best fit to be lag = 7 ms.
(b) Response of the fumed silica suspension in a step-stress test (initial stress, 3
Pa; final stress 11 Pa)...... 27
Figure 2-6. Experimental data of a set of intermittent shear tests. (a) Imposed shear rate histories:
the sample is first sheared at 100 s−1 until reaching steady state. The shear rate
then steps down to 0.5 s−1 and is held constant for ∆ . Finally, at = 0, shear rate
xi steps up to 5 s−1 until reaching steady state. (b) Rheological responses. , shear
stress; ∞, the steady-state value of stress. Stress shows non-monotonic
evolutions for intermediate ∆ ...... 29
Figure 2-7. Rheological responses of the fumed silica suspension in a group of flow reversal
tests. The subplot in (a) shows the shear rate histories. and ∞ denote
respectively the shear stress and steady-state value of stress. The strain is defined
as = 0 . The vertical green line indicates the position where = 3. The low-
shear-rate tests reach steady state at ≈ 3...... 30
Figure 2-8. A hysteresis test by ramping the shear rate linearly with time from 1 to 10 s−1 within
10 s and then back to 1 s−1 at the same ramping rate. Prior to the hysteresis test,
the sample is subjected a steady shear at 1 s−1...... 31
Figure 2-9. A large amplitude oscillatory shear test ( 0 = 10, = 0.1 rad/s) of a thixotropic
fumed silica suspension. (a) Reduced shear strain 0 versus time. (b) Reduced
shear rate 0 versus time. (c) Rheological response plotted as the reduced stress
0 versus time. Data are adopted from Armstrong et. al [41]...... 32
Figure 2-10. The elastic (a) and viscous (b) projections of the Lissajous–Bowditch curve for the
data shown in Figure 2-9. Data are adopted from Armstrong et. al [41] ...... 32
Figure 2-11. Instruments for rheo-PIV measurements...... 33
Figure 2-12. An image frame before (a) and after (b) CLAHE ...... 34
Figure 2-13. Cross-correlation of the rescaled BIT shown ...... 35
Figure 2-14. Velocity profiles, , , in a Newtonian fluid under constant shear at 0.2 s−1.
Calculations are based on a pair of frames. Black symbols are experimental
results with ndiv = 1, 5, 10, and 20 pixels. Red lines are the linear fit of the
xii experimental data. The four sets of results are shifted vertically with the shifting
factors indicated in the annotation. MAE: mean absolute error. wall: the velocity
of the fluid layer on the moving wall...... 36
Figure 2-15 Histogram of , the velocity along the gradient direction. Red line is the fit of
normal distribution...... 36
Figure 3-1 Illustration of the PDF method (β = 0.5). ‘Cutoff’: fraction of the exponential
spectrum used to generate the stretched exponential function; DL: lower limit of
D; DU: upper limit of D. The probability density function P(D,1⁄2) is integrated
over a set of evenly distributed intervals on a log scale for D. The integrals are
normalized, which gives the set of ...... 44
Figure 3-2. Algorithm flowchart of the 3.3.1 Tikhonov regularization method and the PDF
method...... 45
Figure 3-3. Comparison of the results of two regularization methods. (a-c): = 1/3; (d-f): =
1/2; (g-i): = 2/3. Graph (a), (d), (g), P(D,β). Graph (b), (e), (h) values
calculated from the PDF method and the Tikhonov regularization method. Graph
(c), (f), (i) Stretched exponential curve and approximation curves calculated from
the PDF method and the Tikhonov regularization method...... 46
Figure 3-4. ln(λ) vs time in a step-rate test (0 to 50 s–1) and in a step-stress test (0 to 20 Pa)...... 48
Figure 3-5. λ(1–n) (left axis, n = 2; right axis, n = 3) vs time in (a) a step-rate test (0 to 50 s–1) and
(b) a step-stress test (0 to 20 Pa)...... 48
Figure 3-6. Results of build-up tests (a) ln(σ∞ – σ) vs time in step-shear-rate tests, initial shear
–1 rate 100 s ; (b) ln(η∞ – η) vs time data in step-stress tests, initial stress 50 Pa; (c)
|slope| vs final shear rate with best power law fit (fitting equation, slope =
xiii 2 + 3), given by b = 0.51; (d) |slope| vs. final stress and power law fit
(fitting equation, slope = 2 − 0 + 3), best fit of b = 0.53...... 49
Figure 3-7. Experimental data (symbols) used to estimate model parameters and model
calculation. (a) Steady state flow curve of the fumed silica dispersion, including
results of both stress sweep tests and shear-rate sweep tests. Box in the upper
right, σ > 3σy0 and > 3 0, region of ideal thixotropy. (b) Step-stress tests
results, initial stress 11 Pa. The lines in (b) include the effects of inertia, which
produces the steep response for times less than t = 0.1 s...... 51
Figure 3-8. Model predictions for (a) step-shear-rate tests and (b) step-stress tests (σo, initial
stress; σt, final stress). (c), (d): predictions of the DM model with α = 0.5 and α =
0 in the time-dependent pre-factor, 1 α, for a particular final shear rate in (a) and
a particular final shear stress in (b)...... 54
Figure 3-9. Experimental results (symbols in b) and model predictions (lines in c-d) of a group of
intermittent step tests. (a) Test protocol. The sample is pre-sheared at 100 s-1. The
shear rate then steps down to 0.5 s-1 and is held constant for ∆t. At t = 0, shear rate
steps up to 5 s-1 until the stress reaches steady state. (b) Experimental results for
different ∆t values as indicated in the legend. (c) Predictions of the SC model with
multiple thixotropic timescales. (d) Predictions of the SC model with a single
thixotropic timescale ( is set to be unity)...... 57
Figure 3-10. Experimental results (symbols in b) and model predictions (lines in c-d) of another
group of intermittent step tests. (a) Test protocol. The sample is pre-sheared at 0.1
s-1. The shear rate then steps up to 3 s-1 and is held constant for ∆t. At t = 0, shear
rate steps down to 2 s-1 until the stress reaches steady state. (b) Experimental
xiv results for different ∆t values as indicated in the legend. (c) Predictions of the SC
model with multiple thixotropic timescales. (d) Predictions of the SC model with
a single thixotropic timescale ( is set to be unity)...... 58
Figure 3-11. Model predictions for (a-c) shear rate ramp tests and (d-f) stress ramp tests.
Procedure in shear rate ramp tests: initial shear rate, 1 s-1; maximum shear rate,
10 s-1; shear rate ramping mode, linear; ramp time, (a) 10 s, (b) 30 s, (c) 90 s.
Procedure in stress ramp tests: initial stress, 2 Pa; maximum stress, 10 Pa; stress
ramping mode, linear; ramp time, (d) 10 s, (e) 30 s, (f) 90 s...... 60
Figure 3-12. SC Model predictions of step tests with small initial shear rate or stress. (a) Step
shear-rate tests from an initial shear rate of 0.1 s-1; (b) step stress test from 1.5 Pa.
...... 61
Figure 3-13. (a, b) (Redrawn from Coussot et al. [101] with permission) Rheological response of
a Bentonite suspension: change of shear rate with time at different stress levels
applied after a rest period of duration following preshearing. (a) = 0; (b)
= 20 s. (c, d) Predictions of the new model for similar flow histories: change
in dimensionless shear rates ( ) over dimensionless time ( ) for different
dimensionless stresses levels ( ) after a rest period of duration following
preshearing. Stress levels increase from top to bottom. (c), = 0; (d) =
0.05...... 63
Figure 4-1. Mechanical analog of the proposed model...... 74
Figure 4-2. Experimental data used to estimate model parameters (except for h and in the
proposed model and the IKH-V model) and model fits (lines, using parameter
values in Table 4-4): (a) steady state flow curve, including data measured on a
xv stress-controlled rheometer, AR-G2, and a strain-controlled rheometer, ARES; (b)
Step-shear-rate tests from a shear rate of 0.1 s-1 to different final shear rates...... 86
Figure 4-3. Model predictions of (a) a step-shear-rate test (from 3 s-1 to 9 s-1) and (b) two step-
stress tests (from 3, 9 Pa to 7 Pa). Experimental data are adopted from the
reference [112]. These are independent results that have not been used to
parameterize the models. The rapid change at early times (time < 0.1 s) in (b) are
caused by the inertial response of the rheometer, which is explicitly included in
the calculated response as explained in Chapter 2.3.3...... 87
Figure 4-4. Model predictions for step tests over wide-ranging test conditions: (a) step-shear-rate
tests and the predictions of the ML-IKH model in RC (dashed lines) and SC forms
(solid lines); (b) the same experimental data as in (a) with predictions of the IKH
model (dashed lines) and MDT model (solid lines); (c) step-stress tests (σo, initial
stress; σt, final stress) with the predictions of the ML-IKH model; (d) the same
experimental data as in (c) with predictions of the IKH and MDT models. The
initial steep response in the step-stress tests is due to inertial effects, which are
incorporated into the model predictions...... 89
Figure 4-5. Experimental results (symbols) and model predictions (lines in b-f) of a group of
intermittent step tests: (a) shear rate histories. The sample is pre-sheared at 100 s–
1. The shear rate then steps down to 0.5 s-1 and is held constant for ∆t. At t = 0,
shear rate steps up to 5 s–1 until the stress reaches steady state. The legend is for
the experimental data in (b-f), which also show the predictions of respectively the
IKH-V, MDT, ML-IKH (SC, = 0.65), ML-IKH (SC, = 1), and ML (SC,
= 0.65) models. σ(t) denotes shear stress and σ∞ denotes the steady-state value.
xvi The subplots in (b-f) show σ(t)/σ∞ vs. time in the test with ∆t = 20 s. The dotted
oval encloses the region of non-monotonic thixotropic relaxation...... 90
Figure 4-6. Experimental results (a), model fits (black lines) and predictions (lines in color) (b)
of flow reversal tests with shear rates indicated in the legend. The subplot in (a)
shows the shear rate histories. σ(t) and σ∞ denote respectively the shear stress and
steady-state value of stress. The strain is defined as = 0 . The vertical green
line indicates the position where γ = 3. The low-shear-rate tests reach steady state
at γ ≈ 3. The predictions in (b) are only shown over the range of strain for which
experimental data are available. kh, q, and m (in the IKH-V model) are estimated
in the test with 0 = 0.01 s − 1. The IKH-V and ML-IKH (in SC form) models
give similar predictions in all tests (for simplicity, only two typical predictions of
the IKH-V model are shown here). The ML model (in SC form) predicts no
transient response in all tests...... 93
Figure 4-7. The elastic (a) and viscous (b) projections of the Lissajous–Bowditch curve for =
0.1 rad/s and 0 = 10. The subplot in (a) shows the strain histories with four
points marked out. Experimental data and model predictions are both linearly
rescaled from –1 to 1. The experimental data (black lines) are adopted from [41].
Panel (i) of (b) shows the complete experimental data. Panels (ii-iv) show only
half of the symmetric LAOS results for clarity. (c-d) The response of A, predicted
by the ML-IKH model, plotted against strain and shear rate, respectively. (e-f) the
response of λ, predicted the ML-IKH model, plotted against strain and shear rate,
respectively...... 96
xvii Figure 4-8. Pipkin diagram: (a) elastic projection in which reduced stress is plotted against strain;
(b) viscous projection, in which reduced stress is plotted against strain rate.
Experimental data (black lines) are adopted from [41]. Model predictions are
calculated using the parameter values in Table 4-5. Model predictions and
experimental data are normalized by the maximum of the latter. This figure shows
the quantitative and qualitative discrepancies between the model predictions and
experimental results. The dashed boxes contain the results presented in Figure
4-7(a,b)...... 98
Figure 4-9. Comparison of the 1-2 component of the tensorial form (dash black lines) and the
scalar model (solid red lines) in four shear-start-up tests: the evolution of (a) shear
stress, (b) , (c) back stress, taken as back, 12 in the tensorial model and h in
the scalar model. The shear rates in (i-iv) are respectively 0.1, 1, 10, 100 s-1.
Model calculation is generated using the parameter values for the SC form in
Table III except that h and take different values, respectively, 0.3 Pa and 3, in
the tensorial model, while for the scalar model, these values, taken from Table
4-4, are 0.195 Pa and 1, respectively...... 101
Figure 4-10. Predictions of the back strain in shear-start-up tests. Solid lines are 2A12 in the ML-
IKH model and dashed lines are A12 in the IKH model, in both cases in the limit
of no yield stress and rigid plasticity. q equals 0.01 for lines A and a; 0.1 for B
and b; 1 for C and c; and 10 for D and d. The shear stress predicted by the IKH
model is + ℎA12. The oscillation of shear stress is significant when q and
ℎ are small...... 104
xviii Figure 5-1. Evolutions of the stress and DOB in a shear startup test ( 0 = 0.01 s − 1). Dashed
lines indicate the strain values at which the instantaneous velocity profiles are
extracted and shown in Figure 5-3. The insert shows the test procedure. The
–1 sample is first sheared at 200 s for 2 min. The shear rate then decreases to zero
within 20 s. After a rest period of duration 10 min, the shear rate jumps instantly
to 0 and is held constant for a duration of so that the total strain tot =
0 = 180...... 111
Figure 5-2. Rheological response of the 2.9 vol% fumed silica suspension in an amplitude sweep
test with a frequency of 1 rad/s. G' and G'' denotes respectively the elastic and
loss moduli. The circle marks out the crossover point where G' = G''...... 111
Figure 5-3. Instantaneous velocity profiles (normalized by H and vwall) at the strain values as
indicated in Figure 5-1...... 112
Figure 5-4. Space-strain plot of the velocity profile (normalized by H and vwall) in the shear
startup test with 0 = 0.01 s − 1. Background color indicates the level of velocity
according to the color bar on the right...... 112
Figure 5-5. Rheological response (a) and banding dynamics (b) in a set of shear startup
experiments...... 113
Figure 5-6. (a) Evolutions of the stress, normalized by the absolute stress value before flow
reversal, and the DOB in a set of flow reversal tests. (b) Instantaneous velocity
profiles in the flow reversal test with 0 = 0.01 s − 1. Dashed black line is the
velocity profile before shear reversal. Lines in color are the velocity profiles after
shear reversal with corresponding strain values indicated in the legend...... 114
xix Figure 5-7. Space-strain plot of the velocity profile (normalized by H and vwall) in the flow
reversal test with 0 = 0.01 s − 1. Background color indicates the level of
velocity according to the color bar on the right...... 115
Figure 5-8. Steady-state flow curve determined from the bulk rheometry (black symbols) and
local velocimetry (red symbols). The insert plot shows model predictions with
solid line indicating the linearly stable portion and the dashed line indicating the
unstable portion...... 118
Figure 5-9. Numerical simulation of a shear startup test followed by a sudden flow reversal. (a)
and (b) shows the rheological response and the banding dynamics in respectively
the shear startup and flow reversal tests. (c) and (d) shows the space-strain plot of
the velocity profiles in respectively the shear startup and flow reversal tests. ... 119
Figure 5-10 (a-e) Oscillation of shear bands under a constant apparent shear rate of 0.075 s–1
while varying the value of cone. Corresponding rotation periods of the cone and
plate are given in annotations. Inserts show the coefficients of the discrete Fourier
transforms of the DOB transients. (f) Rheological response in the five
experiments of (a-e)...... 121
Figure 5-11. A sample image of the field of view for PIV calculation in the test of Figure 5-10
(e)...... 122
Figure 5-12 Evidence of traveling shear bands. Results are from the test of Figure 5-10 (e). (a)
The evolution of Δ . (b) left , 1 and right , 1 with 1 = 10 min. (c)
left , 2, left , 2 − Δ , and right , 2 with 2 = 410 min and Δ = 34 s. 122
xx Figure 5-13. Velocity profiles in a Newtonian fluid under constant shear-rate flow ( = 0.2 s −
1). (a) 30 velocity profiles equally spaced in time as the bottom plate rotates for
one cycle. (b) velocity fluctuation at five different y position across the gap. ... 123
Figure 5-14. Variation of the positions of the geometry walls as the cone and plate rotate at a
constant speed...... 124
Figure 5-15 Illustration of the geometry misalignment...... 124
.
xxi Abstract
Many complex fluids exhibit thixotropy – their viscosity, moduli, and yield stress gradually decrease under flow and slowly build up after the cessation of flow. Thixotropic fluids are very common in nature and industry. Examples include colloidal suspensions, gels, paints, pastes, crude oils, adhesives, personal care products, and many food materials. A thorough understanding of their rheological properties and flow behaviors is very important in the product development, process flow simulation, and product quality control of those materials. The goal of this thesis is to advance the understanding of the rheology and flow instability of thixotropic fluids.
Thixotropy often occurs simultaneously with other rheological phenomena such as yielding, viscoelasticity, and aging, which poses a great challenge in both experimental and theoretical studies. We first focus on the “ideal,” purely viscous, thixotropic response, which dominates for shear histories involving only relatively high shear rates. We show that to predict quantitatively an ideal thixotropic response, it is critical to introduce multiple timescales for thixotropic evolution and use nonlinear kinetic equations. We propose a novel “multi-lambda”
(ML) model that introduces a spectrum of timescales at the cost of only one additional model parameter. We examine this model against a wide range of shear-rate histories involving steady state, step shear rate, step stress, shear rate ramp, and stress ramp. The model shows quantitative predictions of the experimental data.
We next generalize the ML model and combine it with an isotropic kinematic hardening
(IKH) model to predict complex thixotropic elasto-viscoplastic (TEVP) responses. We evaluate
xxii this new model and compare its predictions with three representative TEVP models over a wide range of test conditions. We also provide a tensorial formulation of the ML-IKH model that is frame-invariant, obeys the second law of thermodynamics, and can reproduce the quantitative predictions of the scalar version.
Shear banding often occurs in complex fluids, which greatly alters the local flow fields and hence the bulk rheological response. Understanding this phenomenon is crucial to interpret the rheological data correctly and is important in the design of processing flows that enhance or mitigate it as needed. Nevertheless, it has been unclear how shear bands occur and evolve in thixotropic fluids. We design a novel particle image velocimetry method to study the banding dynamics in a thixotropic fumed silica suspension under transient shear. We find that in shear startup experiments, the material exhibits a critical shear rate at around 0.2 s−1, below which shear bands can occur. The banding dynamics show a strong dependence on shear strain – the flow remains uniform for strain smaller than unity and bands quickly grow when the strain exceeds unity. Upon a sudden reversal of the shearing direction, shear bands show a non- monotonic evolution – shear bands grow slightly for strain less than 0.15, then gradually homogenize until the strain reaches around 1.5, after which bands slowly grow. We propose a simple thixotropic kinematic hardening model that can qualitatively capture the main features of the banding dynamics.
xxiii Chapter 1 State of Art
1.1 Thixotropy
1.1.1 Definition and History
Thixotropy is one of the oldest and, at the same time, most challenging phenomena in rheology and colloid science. Studies on thixotropy began with the birth of rheology. And yet it is still a hot topic at present. The concept of thixotropy was first introduced a century ago by
Peterfi [1] to describe the phenomenon that some gels, consisting of aqueous Fe2O3 dispersions, can be liquified into sols by shaking the container and these sols solidify again in the absence of mechanical agitation [2]. This sol-gel transition can be repeated for many times.
Table 1-1. Several representative definitions of thixotropy. A Dictionary of Food and Nutrition [4] The reversible property of a material that enables it to stiffen rapidly on standing; on agitation it becomes a viscous fluid. A Dictionary of Physics (6 ed.) [5] The viscosity depends not only on the velocity gradient but also on the time for which it has been applied. These liquids are said to exhibit thixotropy. The faster a thixotropic liquid moves the less viscous it becomes. IUPAC [6] The application of a finite shear to a system after a long rest may result in a decrease of the viscosity or the consistency. If the decrease persists when the shear is discontinued, this behaviour is called work softening (or shear breakdown), whereas if the original viscosity or consistency is recovered this behaviour is called thixotropy. Jan Mewis (1979) [7] The continuous decrease of apparent viscosity with time under shear and the subsequent recovery of viscosity when the flow is discontinued Jan Mewis and Norman J. Wagner (2009) [8] The continuous decrease of viscosity with time when flow is applied to a sample that has been previously at rest and the subsequent recovery of viscosity in time when the flow is discontinued.
1 The term “thixotropy” is derived from the Greek words θίξις (thixis: stirring, shaking)
and τρέπω (trepo: turning or changing) as a description of the gradual and reversible sol-gel
transition due to mechanical agitation. The formal definition of thixotropy has changed since this
first version. As noted by Barnes [3], the definitions of thixotropy often fall into two kinds - one
in its original sense, which focuses on the reproducible gel-sol (solid-liquid) transition, the other
one focuses on the flow-induced gradual decrease of viscosity, yield stress or moduli and the
buildup of those properties in the absence of flow. The academia mainly uses the second
category of definitions since 1970s while the industries still use first definition very often. Table
1-1 provides several representative definitions of thixotropy from different resources.
1.1.2 Occurrence
Thixotropy has been identified in many structured fluids, lists of which, along with
references to the relevant publications, can be found in refs. [7-9]. As Mewis and Wagner [8]
pointed out, the term “thixotropy” is often used in a general sense to describe time-dependent
shear-thinning behavior and the buildup of viscosity, yield stress, or modulus after cessation of
shear flow, and some recent works still follow this convention. Thus, some of materials
described as “thixotropic” might, in fact, be better thought of as nonlinearly elastic, or
viscoelastic aging materials. With this caveat in mind, below is a list of fluids identified by
various authors as thixotropic, drawn from the reviews described earlier:
Colloidal suspensions: paints, coatings, inks, clay slurries, pharmaceuticals, cosmetics,
agricultural chemicals, magnetic coatings, drilling muds, metal suspensions
Slurries: coal/lignite, mining slurries
Nanoparticle suspensions: asphaltene-containing crude oil
Emulsions: personal care products
2 Foams: mousses
Crystalline systems: waxy crude oils, waxes, butter/margarine, chocolate
Polymers: associating solutions/melts, starch/gums, sauces, unsaturated polyesters,
nanocomposites
Fibrous suspensions: tomato ketchup, fruit pulps
Biological systems: fermentation broths, sewage sludges, bloods
Table 1-2 updates the examples given in these earlier reviews by including some new
examples presented since 2010.
Table 1-2. Recent examples of thixotropic materials. Materials Reference Remarks Colloidal dispersions, suspensions, and gels The differentiation of stem cells can be regulated by polyethylene glycol-silica gel [10] controlling the yield stress of the thixotropic gel. A model thixotropic fluid developed by Dullaert and Mewis [12] whose rheological behavior has fumed silica suspended in [11-14] been studied intensively. Several quantitative paraffin oil models have been proposed and compared to experimental data for this fluid. alumina colloidal gel [15] A material for 3D printing. Shear-induced structural anisotropy gives a unique nanoparticle gel [16] signature in large amplitude oscillatory shear tests. One of the few studies on squeeze flow of laponite suspension [17] thixotropic suspensions. Microgels block copolymer microgel [18] An oil-based 3D-printing support medium. Mostly used as a model simple yield stress fluid with no significant thixotropy. A recent study [20] Carbopol gel [19, 20] however shows that strong mechanical agitation during preparation can change its rheological behavior, suggesting that it shows thixotropy. Emulsions The loaded emulsion is a thixotropic fluid while the normal emulsion without bentonite is a simple silicon oil-water emulsion [21] yield-stress fluid. The thixotropic and normal loaded with bentonite clay emulsions show different shear-localization behaviors. acrylic emulsion paint [22] For exterior wall coatings. The formation of emulsions significantly changes the rheology of crude oil, which is a challenge in water-in-crude-oil emulsions [23] the drilling, producing, transporting, and processing of crude oils. Polymeric materials Xanthan gum gel [24] Used as electrolyte in dye-sensitized
3 solar cells. Other examples foams [25, 26] Thixotropy is important in foam stability. waxy crude oil [27, 28] Combined experimental and modeling studies. A review on thixotropic modeling of drilling mud drilling mud [29] and crude oil. peptide-based organogel [30] Self-healing material. Microstructural origin of thixotropy in cement cementitious materials [31] pastes is explored. Examples include ketchup, mayonnaise, gums, food products [32-36] chocolate, purees, sauces Red blood cells aggregate and form weak structures human blood [37, 38] that break down under shear flow. Hydroxystearate forms large aggregates in mineral grease [39] oil. Fumed silica is mixed with propellant to form a fuels [40] thixotropic gelled fuel.
1.1.3 Rheological Measurement
Thixotropic fluids often show very complicated rheological responses. Depending on the
test procedure, they may display one or more characteristic rheological features, including
thixotropy, viscoelasticity, yielding, aging, and kinematic hardening. Here, we summarize some
common test procedures.
Figure 1-1. Common procedures of rheological experiments. s denotes the control variable and can be the shear rate, shear strain, or shear stress. A common test procedure is applying a stepwise change of the shear rate or stress, as shown
in Figure 1-1(a), and measuring the subsequent viscosity transients. A typical thixotropic
response is that the viscosity gradually decreases after a sudden increase of the shear rate or
4 stress and the viscosity slowly builds up if the shear rate or stress suddenly drops. The viscosity
evolutions reflect changes in internal structures – increase in viscosity indicates growth of
internal structures while decrease in viscosity implies structural breakdown. By changing the
initial and final shear rates or stress, one can study the impact of the test conditions on the
viscosity evolutions. Step-stress tests are especially useful for materials exhibiting an apparent
yield stress (which include almost all thixotropic fluids). Note that inertia effects govern the fast
response in step-stress tests and need to be distinguished from the constitutive response of testing
materials. Several consecutive step tests, as shown in Figure 1-1(b), can be used to gain deeper
insight about the thixotropic phenomenon.
Figure 1-2. Hysteresis loop of a 4 wt% Balangu solution. Shear rate first increases linearly with time from 0 to 300 s within 6 min and then decreases at the same rate to 0 s . (From Razavi etc. [33]) Hysteresis experiments is another technique often used to study thixotropy. In this test, the
shear rate or stress first increases with time and then decreases, as shown in Figure 1-1(c). A
thixotropic material usually shows a hysteresis loop when the transient data are plotted as stress
versus shear rate because the rheological response lags behind the change of the shear rate or
stress. Figure 1-2 shows an example of a Balangu solution in a hysteresis test. Note that the
shape of the loop depends on not only the rheological properties of the material but also the test
conditions – the range of the shear rate or stress ( ), the ramping rate (Δ ), and the initial
5 condition. Hysteresis tests are useful to evaluate constitutive models, but it does not provide as clear information as those in step tests.
Oscillatory shear tests, as shown in Figure 1-1(d), are another group of useful experiments.
A sinusoidal wave of shear strain or stress is imposed on the testing materials. One can vary the initial condition, the wave amplitude ( ), and the frequency ( ). Experiments at small amplitudes provide information about the evolution of moduli. Large amplitudes are usually adopted to probe the response of nonlinear viscoelasticity and thixotropy.
Table 1-3. A summary of several representative data sets Materials Rheological tests References Remarks steady-state flow curve fumed silica shear-rate jump Dullaert et multiple formulations suspension multiple rate jumps al. [11-14] shear reversal steady-state flow curve shear rate jump fumed silica frequency sweep Armstrong single formulation suspension LAOS et al. [41] data easily accessible shear rate ramp shear reversal steady-state flow curve shear-rate jump fumed silica multiple rate jumps single formulation This thesis. suspension stress jump data easily accessible shear-rate ramp shear reversal steady-state flow curve shear-rate jump carbon black Armstrong frequency sweep single formulation suspension et al. [42] shear reversal shear-rate ramp steady-state flow curve Dimitriou et waxy crude oil shear rate jump local velocimetry al. [27] LAOS steady-state flow curve Geri et al. multiple formulations waxy crude oil shear-rate jump [28] multiple temperatures steady-state flow curve Paredes et multiple formulations emulsion stress jump al. [21] local velocimetry multiple formulations steady-state flow curve Paredes et Carbopol gel local velocimetry shear rate jump al. [21] measurements steady-state flow curve Thurston et blood shear-rate jump multiple samples al. [43] frequency sweep
1.1.4 Representative Data sets
6 Comprehensive rheological data sets are very important in the studies of thixotropy and its modeling. Despite many studies on thixotropic materials, only a few works report comprehensive data sets that are suitable for model validation. In Table 1-3, we list several representative works that provide well-defined data sets. To be included in this table, the data sets need to contain at least a steady state flow curve, multiple transient tests such as shear-rate-jump tests, and at least one of the following features:
1) flow protocols covering a wide range of shear histories;
2) data in easily accessible forms;
3) test fluids formulated in multiple compositions;
4) different temperatures;
5) rheology with simultaneous local velocimetry.
1.2 Modeling of Thixotropy
1.2.1 Overview
The numerous models developed to describe thixotropy fall mainly into three groups
[45]: continuum mechanical models, structural kinetics models, and microstructural models.
Continuum mechanical models are usually written in tensor forms and the thixotropic phenomenon are introduced by a memory function (e.g. the model of Stickel and Powell [46]).
Continuum mechanical models are purely empirical and lack connections with the underlying physical mechanisms. Continuum mechanical models are rarely used to predict rheological data.
Structural kinetic models are most widely used for thixotropic materials. In structural kinetic models, rheological parameters are given as functions of some structure parameters whose microstructural bases are unclear, but whose behavior under flow is specified by a kinetic equation [8, 47, 48]. Structural kinetic models capture the changes of internal structures through
7 a semi-empirical approach with distinct terms phenomenologically accounting for the deformation, breakage, and formation of microstructures. Structural kinetic models usually involve a large number (around 10) of model parameters and provide quantitative predictions of experimental data.
Microstructural models seek to calculate the macroscopic properties based on a physical mechanism involving an explicitly identified microstructure [49]. Population balance model
(PBM) is a major class of microstructural models. In PBMs, rheological properties such as viscosity are linked to a population distribution function (PDF) which describes the instantaneous structural state – the size distribution of aggregates. The evolution of the PDF is described by a kinetic equation derived based on the aggregation and breakage kinetics of internal structures. Microstructural models require fewer model parameters than structural kinetics models do and can provide direct structural information such as the size distribution of aggregates. Despite the advantages, there is only a few microstructural models available in the literature and they can only qualitatively predict experimental data, especially in transient rheological experiments.
1.2.2 Structural Kinetics Models for Ideal Thixotropy
In the simplest case, one can use a purely viscous model to describe the rheology of thixotropic materials. A simple and general form such an ideal thixotropic model can be written as
, (1-1) where denotes the shear stress, ̇ the shear rate, ( ) the viscosity which is usually an increasing function of the structure parameter . indicates the instantaneous degree of internal structures. The rate of change of is described by
8 d m f n g 1 , (1-2) dt
Here, is a flow parameter, representing the influence of flow on the structural build-up and break-down, (which can be ̇, , combination of both, or more complicated definitions as often used in thixotropic elasto-viscoplastic models). and are the exponents on in the break- down and build-up terms. ( ) and ( ) are usually positive and increasing functions of .
Most thixotropic kinetic equations can be reduced to the following format:
d m m ka n k b 1 k 1 , (1-3) dt 1 2 3 where , and are, respectively, the rate coefficients of the break-down term, the shear- induced build-up term, and the Brownian build-up term; and are the orders of the dependence of the first two of these terms on the flow parameter .
Table 1-4 lists some representative kinetic equations in thixotropic models from the literature. Most of them are linear with and have two terms, the breakdown term and the
Brownian term. The models in references [11] and [41] have a shear-induced build-up term.
When the shear-rate histories involve changes of shearing directions, the absolute value of shear rate is often used as the flow parameter. Armstrong et al. [41], Apostolidis et al. [50] as well as
Dimitriou and McKinley [27] use the plastic contribution of the shear rate, ̇ , as the flow parameter, implying that the elastic deformation does not impact the thixotropic kinetics. ̇ reduces to ̇ in ideal thixotropic models.
The break-down term and the shear-induced build-up term usually have power-law dependencies on the flow parameter. The model proposed by de Souza Mendes and Thompson
[51] is an exception: in this model the function f(σ) is obtained from the steady-state relation between λ and stress, σ. The models developed by Teng, Zhang [52] and Dullaert, Mewis [11]
9 have a time-dependent pre-factor to introduce stretched exponential relaxation (which will be
discussed later in this thesis).
Table 1-4. Expressions for the kinetic equation in some representative thixotropic models. Authors Break-down Build-up (Shear) Build-up (Brownian)
Armstrong et al. [41] − ̇ ̇ (1 − ) (1 − )
. Apostolidis et al. [50] − ̇ - (1 − )
Dimitriou, McKinley [27] − ̇ - (1 − )
Blackwell, Ewoldt [53] − | ̇| - (1 − )
Teng, Zhang [52] − ( ̇ ) (1 + ) - (1 − )(1 + )
de Souza Mendes, − ( ) - ( − ) Thompson [51]
. Dullaert, Mewis [11] − ̇ ̇ (1 − ) (1 − )
Mujumdar et al. [54] − ̇ - (1 − )
Instead of following the format of Eq. (1-2), some works [55-58] that focus solely on
shear-rate-step tests adopt an alternative approach to express the structural kinetics. ⁄ is
written as a function of the difference between the instantaneous value of and its steady-state
value at a constant shear rate. The kinetic equation in break-down tests (where the initial shear
rate ̇ is smaller than the final shear rate ̇ ) and in build-up tests (where ̇ > ̇ ) take different
forms:
a b if , d kbreak ss t 0 t (1-4) d dt kc t if . grow ss 0 t
10 Here , , , , , are empirical constants and ( ̇) is an empirical function. Note that those equations are applicable only in sing-step-shear-rate tests. They better be called fitting equations instead of constitutive models.
Now consider a simple constitutive model proposed by Moore [59]:
0 , d (1-5) k k 1 . dt 1 3
Here, the constant 1⁄ can be considered a thixotropic time constant , whose value determines how long one must wait to reach steady state after start-up of flow, or to reach a unique rest state after cessation of flow. The product Th ≡ ̇⁄ = ̇ can be defined as the “thixotropy number” that controls the strength of the thixotropic response in flow, with low values of Th implying weak thixotropy [54]. In a step-shear-rate test where ̇ jumps from ̇ to
̇ , the solution to Eq. (1-5) is that
1 1 1 t exp k3 1 Tht , (1-6) 1 Th0 1 Tht 1 Th t where Th and Th are respective the thixotropy number before and after the shear-rate jump. is proportional to and therefore also follows a pure exponential relaxation. Note that the characteristic timescale is independent of previous shear-rate histories.
Eqs. (1-1) and (1-2) can be easily converted to tensor forms. Taking the model of Moore
[59] as an example, the tensorial equivalent of Eq. (1-5) can be written as
τ 20D , d (1-7) k k 1 , dt 1 3
11 where is the extra stress tensor, = ( + ) is the rate of deformation tensor, and Γ =
√2 : is an invariant of .
1.2.3 Incorporation of Viscoelasticity into Thixotropic Models
Figure 1-3. Illustration of early-time viscoelastic response (before 1 s) followed by late-time thixotropic response in 2.9 vol% fumed silica suspension after a step-down from a shear rate of 1 s–1 to the shear rates shown in the legend. Lines are model prediction. (From Dullaert and Mewis [11], used with permission.)
Many thixotropic fluids are simultaneously viscoelastic. Viscoelasticity manifests itself in
step-shear-rate tests where the stress first exhibits an instantaneous jump, followed by a
relatively fast (viscoelastic) relaxation that subsequently transitions to a slower (thixotropic) one
towards the opposite direction. Figure 1-3 shows an example of a step-shear-rate test. After the
stepwise decrease of shear rate, stress first decreases towards a minimum and then gradually
grows to a plateau. The contribution of elastic stress to the total stress can be directly measured
in flow cessation tests where flow suddenly stops after a period of pre-shear [14]. The residual
stress that does not instantly vanish when shearing ceases is the elastic stress.
12 Table 1-5. A summary of four models corresponding the four categories in Figure 1-4. An example of Figure 1-4 (a), the model of Mujumdar et al. [54]
n GK e 1 m d , if e c (pre-yield) e dt m 0, if e c (post-yield) 0, if 0 d e k1 k 3 1 ; dt , if e 0 Model parameters: G , K , n , m , , k , k c 1 3 An example of Figure 1-4 (b), the model of Dullaert and Mewis [11] m e p, G e , A p q A sign A p , k 1 1 k 3 p pmax 0,sign k h A k h A k y p
Model parameters: G , q , m , k1 , k 3 , k h , k y , p
An example of Figure 1-4 (c), the model of Blackwell and Ewoldt [53]. The original model was proposed in tensor forms. Presented below is its equivalent scalar form.
GGAA d k k 1 dt 1 3 Model parameters: G , , , k , k A 1 3 An example of Figure 1-4 (d), the model of Armstrong et al. [41]
m e p, e p e p max , max min CO ,1 , CO y 0G 0
a d G k G G, k tˆ 1 1 tˆ , fG f 0 Brown r 1 p r 2 p n n GKK 2 1 sign f e ST p p p Model parameters: m , G , , k , k , tˆ , tˆ , a, d, n , n , K , K 0y 0 G Brown r 1 r 2 1 2 ST
Thixotropic models need to incorporate viscoelasticity to capture such behaviors. Figure 1-4 is a summary of the representative approaches. Figure 1-4 (a) is derived from the Bingham model, with the viscosity written as a function of λ, and the yield stress replaced with an elastic stress (σe) that is a function of both λ and ̅ . ̅ is a strain-like structure parameter. An additional kinetic equation is needed to describe the evolution of ̅ . Examples of constitutive equations of the type of Figure 1-4 (a) are the models of Mujumdar et al. [54], and Dullaert and Mewis [11].
13
Figure 1-4. Schematics of four representative forms of incorporating viscoelasticity into thixotropic models: (a) Bingham-type model with yield stress replaced by an elastic stress; (b) Maxwell-type model with λ -dependent modulus and/or viscosity; (c) Oldroyd-type model with stress contribution from the medium and internal structures; (d) a hybrid of the Bingham- and Maxwell-type models. In each branch, the dotted line indicates the portion of the strain that does not affect the stress. Figure 1-4 (b) is derived from the Maxwell model by assuming a λ-dependent viscosity,
( ), and/or modulus, ( ), each of which contributes to the strain, , which is additively
divided into the viscous and elastic strains, and respectively. Examples of models of this
form were proposed by Labanda and Llorens [60], de Souza Mendes [61], and Dimitriou and
McKinley [27]. Figure 1-4 (c) is derived from the Oldroyd-B (or Jeffreys) model, in that it
contains a parallel viscous Newtonian stress that is structure independent, along with the
structure-dependent viscoelastic Maxwell component. Examples are the models of de Souza
Mendes and Thompson [51], Blackwell and Ewoldt [53], and the model in this work. Figure 1-4
(d) is a hybrid of the Bingham- and Maxwell-type models which assumes an additive
decomposition of both stress and strain. The elastic stress is a function of the elastic strain; and
the viscous stress is a function of the viscous shear rate, with both also functions of λ. Examples
are the models of Armstrong et al. [41] and Mwasame et al. [62].
1.2.4 Incorporation of Advanced Plasticity into Thixotropic Models
14 Almost all thixotropic materials exhibit yielding behavior. In the simplest cases, a
yielding response is expressed through a Bingham yield stress which depends on :
0, if y ; y (1-8) sign , if y ,
where ( ) is the yield stress and ( ) is the plastic viscosity. ( ) and ( ) are often
assumed to be linear functions of while more advanced forms are sometimes used for better
predictions. Such a Bingham-type model as described by (1-8) is often called a rigid plasticity
model [63] as there is no elastic deformation prior to yielding. Eq. (1-8) can be generalized
through the approaches described in Figure 1-4 to incorporate elasticity, resulting in a thixotropic
elasto-viscoplastic (TEVP) model. Figure 1-5 shows an example which is a combination of Eq.
(1-8) and the Maxwell model.
Figure 1-5. Mechanical analog of a simple TEVP model. is the spring constant, and are respectively the elastic and plastic portion of the total strain . = + and ̇ = ̇ + ̇ . The constitutive relationship between and ̇ is expressed as
. (1-9) G p
where ̇ is determined by the following “flow rule”
max 0, y sign . (1-10) p
15 The term sign( ) ensures that the plastic deformation occurs in the same direction as the stress, which is the so-called “codirectionality hypothesis” commonly adopted in the field of plasticity
[64]. Eq. (1-10) defines a yield surface in stress space, − ( ), ( ) . For stress in this yield surface, ̇ = 0 and Eq. (1-9) reduces to
G , (1-11) indicating purely elastic response prior yielding.
Even an elastic Bingham-type thixotropy model is often too simple to describe the complex rheologic responses of real fluids, especially those displaying Bauschinger effect [65].
The Bauschinger effect refers to an anisotropic yield stress resulting from plastic deformation.
Many metallic materials [65-69] show this phenomenon under cyclic loadings. Dimitriou et al. reported recently [27, 70] that some soft materials like Carbopol gel and thixotropic waxy crude oil exhibit similar behaviors under oscillatory shear. They further showed that kinematic hardening models, well-studied in the plasticity literature, can be incorporated into thixotropic models to predict the rheological response of those soft materials under large amplitude oscillatory shear.
“Kinematic hardening” are commonly adopted in the plasticity literature to model the
Bauschinger effect. Kinematic hardening models describe this phenomenon through a changing yield surface, the center of which moves as plastic deformation occurs, such that the material becomes harder to deform (exhibits a larger yield stress) in the direction of past deformation, but easier to deform in the reverse direction.
16
Figure 1-6. Mechanical analog of a simple kinematic hardening TEVP model. In kinematic hardening models (Figure 1-6), the flow rule takes a slightly different form than Eq. (1-10):
max 0, eff y sign , (1-12) p eff where is the effective stress driving the plastic flow and it is defined as follows:
eff back . (1-13)
is the additional direction-dependent “back stress” needed to overcome the yield stress during straining and it tracks the center of the yield surface. A simple elastic constitutive equation for the back-stress is the linear relationship:
back CA, (1-14) where is the “back strain” and is therefore the modulus associated with the back strain. A simple constitutive equation [65] for the back strain is
dA qA , (1-15) dt p p where is a model constant. The first term in the above produces a linear growth of with the plastic strain while the second leads to a negative exponential approach to saturation; i.e., =
[1 − exp − ]/ . More complicated forms of Eq. (1-15) are sometimes used to achieve
17 quantitative predictions [71]. For example, the second term in Eq. (1-15) can be replaced with a power-law function of , resulting in
dA m q A sign A . (1-16) dt
As reported by Dimitriou and McKinley [27], this modification is important in predicting the dynamic rheology of waxy crude oils ( = 0.25).
1.3 Shear Banding in Thixotropic Fluids
Thixotropic materials often show shear banding [21], in which an initially homogeneous fluid separates into layers of different shear rates under an imposed shear flow. Shear banding greatly alters the local shear histories and impacts the interpretation of the bulk rheological properties. Shear banding also potentially has a major influence on industrial applications involving processing flows of those complex fluids. For example, shear banding may cause unexpected dead zones during mixing processes and inconsistent qualities of the products.
At present, there are only a few studies (e.g. refs [21, 27, 72, 73]) on shear banding in thixotropic materials in contrast to the numerous works about other complex fluids such as wormlike micellar solutions [74, 75], lyotropic lamellar phases [76-78], polymer solutions [79-
81], and jammed systems [73, 82]. In this section, we briefly review the studies on shear banding in thixotropic fluids.
It has been reported in several works [21, 27, 72] that thixotropic materials often exhibit a critical shear rate below whichbelow which shear banding occurs. Paredes et al. [21] studied the banding behavior of a thixotropic emulsion and a normal emulsion, the latter of which was a simple yield stress material without thixotropy. The authors found that when wall-slip was suppressed by roughening the geometry surface, the normal emulsion did not exhibit shear
18 banding while the thixotropic emulsion formed a banded flow when sheared below a critical shear rate. Martin and Hu [72] found such a critical shear rate in Laponite suspensions and reported that aged Laponite suspensions show transient shear banding – bands that grow shortly after shear startup and then disappear as the fluid evolves towards a uniform shear rate.
Dimitriou and McKinley [27] confirmed the existence of a critical shear rate for banded flow in waxy crude oils, below which shear bands and wall-slip velocity fluctuate irregularly at a constant overall shear rate. The recent theoretical study of Jain et al. [83] showed that inertia plays an important role in both the short-term banding dynamics in shear start-up experiments and also the shear bands at steady states. In shear start-up experiments, inertia seeds heterogeneity which further grows due to inhomogeneous aging and rejuvenation. The critical shear rate determined from the constitutive flow curve alone was found to be an insufficient criterion for steady-state banding.
While most works focus on steady shear bands, interest in time-dependent shear bands has been growing in recent years. The study of Divoux et al. [73] provides insights into the relationship between time scales of rheology and of shear banding. The authors measured the flow curve and local velocity profile of several soft materials including both simple yield stress fluids and thixotropic fluids. In their flow protocol, the shear rate was first swept down from 10 to 10 s in 90 logarithmically spaced steps of duration and then swept up over the same range. Divoux et al. [73] found that both the hysteresis area in the flow curves ( ) and the overall degree of banding ( ) depend strongly on . For thixotropic materials such as laponite suspensions, and exhibit a maximum at a critical value of , as shown in Figure 1-7(b-c).
The authors also observed this phenomenon in mayonnaise and carbon black suspensions. This robust maximum indicates that both the bulk rheology and the banding dynamics depend
19 strongly on an intrinsic material restructuring time. (For simple yield stress fluids such as
Carbopol microgel, this timescale is very small therefore and decrease monotonically as
increases.)
Figure 1-7. (a) Hysteretic flow curves of a 2.5 wt% laponite suspension obtained by first decreasing the shear rate from 103 to 10-3 s-1 in 90 logarithmically spaced steps of duration 15.5 seconds and then increasing over the same range. Insert: velocity profile in a Couette geometry. (b) Hysteresis loop area ( ) versus . (c) Overall degree of banding ( ) versus . (From Radhakrishnan et al. [84], used with permission. Data were initially reported in Divoux et al. [73].) Recently, Radhakrishnan et al. [84] studied this phenomenon theoretically based on the
fluidity models and the soft glassy rheology model [85], which qualitatively capture the
dependence of and on . According to the models, this phenomenon is triggered by the
non-monotonic underlying stationary constitutive curve and the strongly time-dependent
thixotropic response.
In another recent contribution, Garcia-Sandoval et al. [74] modeled band formation in
worm-like micellar solutions using a viscoelastic (upper-convected Maxwell) constitutive
equation whose relaxation time and viscosity are controlled by a fluidity parameter analogous to
20 an inverse structure parameter 1⁄ . The evolution of the fluidity is governed by relaxation and breakdown terms, similar to those in standard thixotropy models, as well as a diffusion term that penalizes gradients in fluidity. The resulting theory predicts formation and migration of bands after start-up of shear.
1.4 Objectives and Outline
This thesis aims to advance the understanding of the rheology and flow instability of thixotropic fluids. Chapter 2 introduces the materials and methods of this thesis. In the remaining chapters, we begin with our study on ideal thixotropy. As discussed earlier, most thixotropic materials show ideal thixotropy when the shear history involves only relatively high shear rates.
Being able to quantitatively predict ideal thixotropy is the first step toward developing more advanced models. Our first objective, therefore, is to develop an ideal thixotropy model that has quantitative predictions for varied shear histories, especially those involving multiple jumps of the shear rate or stress. Chapter 3 is devoted to this task. Our second objective is to generalize the ideal thixotropy model and develop a more advanced model for thixotropic elasto-viscoplastic behaviors. We present this work in Chapter 4. Our third objective is to investigate and model the time-dependent shear bands in thixotropic fluids. Chapter 5 focuses on this problem. We end this thesis in Chapter 6 where we summarize our findings and discuss future directions.
21 Chapter 2 Materials and Methods
2.1 Introduction
The purpose of this chapter is to introduce the materials, experimental methods, and numerical analysis techniques used in this thesis. We use a model thixotropic fluid, a fumed silica suspension, as our testing fluid. Details concerning this fluid are given in Chapter 2.2. In
Chapter 2.3, we introduce the rheological experiments that will be used later to develop and examine constitutive models. Chapter 2.4 explains our particle image velocimetry method for studying time-dependent shear bands in the fumed silica suspension.
2.2 Materials
The thixotropic fluid used in this thesis is similar to the one studied in refs. [12, 41]. It has three components: a hydrophobic fumed silica (Aerosil® R972, provided by Evonik Industries,
Germany), a highly refined paraffin oil (18512 Sigma-Aldrich) and a low molecular weight polybutene (Indopol H-25, Mn=635, Mw/Mn=2.1, provided by INEOS, Switzerland). Transmission electron microscopy shows (Figure 2-1, from [86]) that the primary particles (16 nm in diameter) are aggregated into micron-sized agglomerates. The particles are dispersed in a Newtonian medium, a mixer of 31 wt.% Indopol H-25 with 69 wt.% paraffin oil. This oil blend has a density of 855 kg/m–3 and a viscosity of 0.24 Pa·s at 25 °C. Fumed silica has a weight percent of 7.14 %
(corresponding to a volume fraction of 2.9 %). The sample was premixed for 1 min on a Fisher
Scientific Analog Vortex Mixer and then homogenized using a Cole-Palmer ultrasonic processor
(10 min, 500W, 20 kHz). The sample was then heated to 80 °C in a vacuum oven to eliminate air
22 bubbles. Finally, the sample was put on a Thermo Scientific Bottle Roller for slow aging of twenty days. (The sample showed a slow irreversible decay of stress in constant-shear-rate tests performed right after mixing. After aging of twenty days, the sample showed good reproducibility in following experiments.)
Figure 2-1. Transmission electron microscope micrograph of dry Aerosil® R972 fumed silica (from [86]).
Table 2-1. Components of the fumed silica suspension. Component Weight percent Properties Source
Density: 2200 kg ∙ m Evonik Industries, Fumed silica 7.1% Shape: Spherical Germany Dispersion: Monodisperse
Viscosity: 0.16 Pa ∙ S Paraffin oil 64.1% Sigma-Aldrich Density: 827 kg ∙ m
Indopol H-25, by by Polybutene 28.8% M =635, M / M =2.1 n w n INEOS, Switzerland
2.3 Rheological Experiments
2.3.1 Instruments
Three rheometers are used in this thesis. They are the AR-G2 rheometer (stress- controlled, made by TA Instruments), the ARES rheometer (rate-controlled, made by Rheometric
Scientific), and the MCR702 rheometer (Anton Paar) which can operate in both stress- and rate- controlled modes. Figure 2-2 shows the schematics of the three rheometers. The rate-controlled experiments in Chapters 3 and 4 are performed on the ARES rheometer. The stress-controlled
23 experiments in the two chapters are performed on the AR-G2 rheometer. The experiments in
Chapter 5 are performed on the MCR702 rheometer which is equipped with a bottom motor and a fluorescent microscope for simultaneous rheometry and velocimetry measurement. Cone-and- plate geometries are used throughout the thesis. Table 2-2 lists the major specifications of the geometries.
Figure 2-2. Schematics of (a) AR-G2 rheometer, (b) ARES rheometer, and (c) MCR702 rheometer.
Table 2-2. Details of the cone-and-plate geometries used in this thesis. Rheometer Diameter Cone angle Truncation gap Geometry material
AR-G2 60 mm 2° 48 μm Stainless steel
ARES 50 mm 2.29° (0.04 rad) 50 μm Titanium
MCR702 50 mm 2° 210 μm Stainless steel
2.3.2 Steady-state Flow Curve
Steady-state flow curve is a standard measurement to characterizes the rheological response at steady states. To measure the flow curve, the shear rate or stress is kept constant at a series of values (usually following an increasing or decreasing order) until the viscosity practically stops changing with time. An empirical criterion is specified to determines whether the testing material has reached the steady state. On the AR-G2 rheometer and the MCR702 rheometer, the criterion is that
24 t t t 0.5% (2-1) t t
for at least 3 consecutive time intervals of duration Δ = 30 s. On the ARES rheometer, the
criterion is ≥ 10 min with = 0 corresponding to the time point when the shear rate jumps to
the current value. To better measure the stead state flow curve, the shear rate or stress first ramps
up a series of values and then ramps down the same set of values in a reverse order. Figure 2-3
shows the flow curve of the fumed silica suspension measured on the ARG2 rheometer and the
ARES rheometer.
Figure 2-3. Steady-state flow curve of the 2.9 vol% fumed silica suspension, plotted as (a) stress vs shear rate, (b) apparent viscosity versus shear rate 2.3.3 Step Tests
A stepwise change of the shear rate or stress is the simplest test procedure to probe the
transient TEVP rheology. A sudden increase of the shear rate or stress leads to the deformation
and breakage of internal structures while a decrease of the shear rate or stress leads to the
relaxation and for mation of internal structures. The structural changes are difficult to observe
directly but lead to distinct rheological features. Figure 2-4 shows the rheological response of the
fumed silica suspension in three step-shear-rate tests. In Figure 2-4(a), the shear rate jumps from
0 to 0.1 s−1. After the sudden shear start-up, the sample first deforms elastically until yielding,
25 and then the internal structures gradually break down, leading to the large stress overshoot and
the subsequent decay. In Figure 2-4(b), the shear rate jumps from 0.1 to 1 s−1. The sample does
not exhibit yielding response due to the preshear at 0.1 s−1 which leads to partial breakage of
internal structures. After the stepwise increase of the shear rate, the stress overshoot, therefore, is
much smaller compared to that in Figure 2-4(a) and it occurs at an earlier time. In Figure 2-4(c),
the shear rate jumps from 1 to 9 s−1. The stress shows a monotonic decay, i.e., ideal thixotropic
response, which implies that the internal structures under 1 s−1 are rigid and weak.
Figure 2-4. Rheological response of the fumed silica suspension in three step-rate tests: (a) the shear rate jumps from 0 to 0.1 s−1, (b) 0.1 to 1 s−1, (c) 1 to 9 s−1. Besides of the example in Figure 2-4, we vary the test condition in a wide range to better
understand the TEVP rheology. It is worth noting that analysis of the fast transients in step-stress
tests requires a correction for the system’s inertia. Inertia in step stress tests manifests itself in a
continuous change of shear rate after the instant stepwise change of stress and in inertio-elastic
ringing (damped oscillation around the true steady value). Figure 2-5 shows the responses of a
Newtonian fluid and the thixotropic fumed silica suspension in step-stress tests.
26
Figure 2-5. (a) Response of a silicone oil (viscosity, 0.948 Pa·s) in a step-stress test (initial stress, 0 Pa; final stress 5 Pa). Circles: experimental data; line: calculation of Eq. (2-3) and Eq.(2-4). The value of is determined from the best fit to be = 7 ms. (b) Response of the fumed silica suspension in a step-stress test (initial stress, 3 Pa; final stress 11 Pa). The inertia effects can be calculated from the following equation:
d IMMM , (2-2) tdt in s f
where It is the total moment of inertia of the moving plate and sample, is the rotational
velocity, is the input torque exerted by motor as specified by the test procedure, is the
torque generated by sample deformation and is the drag torque caused by the friction in the
air bearing. The instrument variables ( and ) can be converted to sample variables ( and γ̇)
via the following relations: = , and ̇ = ̇ where is torque, and ̇ are constants
set by the geometry. Eq. (2-2) can be rewritten as follows:
d IF f F , (2-3) t mdt in s c m
where = / ̇ and is the friction coefficient for the air bearing.
The constitutive equation for and an initial condition of the shear rate are needed to
solve Eq. (2-3). In step-stress tests, the initial condition follows
t 0 for t t lag , (2-4)
27 where ̇ is the initial shear rate and is the time lag needed for the shear rate to change after a stepwise change of the stress. The cause for this time lag is not clear, but is typically about 7 ms for the AR-G2 rheometer.
The values of these constants for the 60 mm cone in the AR-G2 rheometer are: ̇ =
28.877, = 17684 m , = 612.38 m , = 42.345 μN ∙ m ∙ s , = 0.28281 μN ∙ m ∙
s ∙ rad , = 7 ms. ̇ = 1/ tan α, = 3⁄(2π ) and = / ̇ , where is the cone angle and is the radius of the cone. Fluid inertia is negligible. and are determined by doing a blank run with no sample using the instrument control software. is fitted from the results of a step-stress test applied to a Newtonian silicone oil (viscosity, 0.95 Pa·s), as shown in Figure
2-5(a).
2.3.4 Intermittent Shear Tests
We use intermittent shear tests to demonstrate the existence of multiple timescales in thixotropic evolutions. In intermittent shear tests, the shear rate jumps multiple times, and the jumps are executed before a steady state is reached. In a group of intermittent shear tests, we change the duration of the intermediate step while keeping the initial, intermediate, and final shear rates unchanged. Figure 2-6 shows the rheological responses of the fumed silica suspension in a group of intermittent shear tests. For intermediate values of ∆ , the stress exhibits non- monotonic evolutions while the shear rate remains constant at 5 s–1. Note that the non-monotonic stress evolutions should be ascribed to thixotropic response instead of viscoelasticity due to the long timescales. As shown in Chapter 2.3.3, in the absence of yielding, viscoelasticity occurs at around 0.1 s after a stepwise change of the shear rate. The non-monotonic stress evolution in
Figure 2-6, however, occurs over 0 < < 300 s. A more detailed analysis and modeling of intermittent shear tests are presented in Chapters 3 and 4.
28
Figure 2-6. Experimental data of a set of intermittent shear tests. (a) Imposed shear rate histories: the sample is first sheared at 100 s−1 until reaching steady state. The shear rate then steps down to 0.5 s−1 and is held constant for ∆ . Finally, at = 0, shear rate steps up to 5 s−1 until reaching steady state. (b) Rheological responses. ( ), shear stress; , the steady-state value of stress. Stress shows non-monotonic evolutions for intermediate ∆ . 2.3.5 Flow Reversal Tests
Flow reversal tests are performed to probe shear-induced anisotropic rheological response.
In a flow reversal test, after an attainment of a steady state under shearing, an instantaneous
reversal of the shear direction is imposed, with the magnitude of the shear rate held fixed. For a
Newtonian fluid in a flow reversal test, the sign of the shear stress instantaneously changes while
its magnitude remains constant. A Maxwell fluid, on the other hand, shows that the stress after
the reversal continuously evolves to its opposite value following an exponential function with a
characteristic time independent of the shear rate. For some colloidal suspensions that form
anisotropic structures under shear, the stress exhibits a gradual relaxation that scales with strain
[87, 88]. TEVP fluids, however, show different rheological behaviors.
29
Figure 2-7. Rheological responses of the fumed silica suspension in a group of flow reversal tests. The subplot in (a) shows the shear rate histories. ( ) and denote respectively the shear stress and steady-state value of stress. The strain is defined as = ̇ . The vertical green line indicates the position where = 3. The low-shear-rate tests reach steady state at ≈ 3. Figure 2-7 shows the rheological response of the fumed silica suspension in several flow
reversal tests (performed on the ARES rheometer). The shear rate ranges from 0.01 s−1 to 10 s−1.
The experimental data are plotted as the reduced stress versus strain. Similar results have been
reported in reference [89]. In flow reversal tests with shear rates of 0.1 s−1 or lower, the reduced
stress can be scaled with the strain to yield a master curve. The stress transients follow a two-step
relaxation that takes approximately three strain units to reach the plateau at steady state. As the
shear rate increases above 0.1 s−1, the initial steep response is no longer detectable and the stress
transients deviate from the master curve.
2.3.6 Hysteresis Tests
We also use hysteresis tests to examine constitutive models. Introduction about hysteresis
tests can be found earlier in Chapter 1.1.3. Figure 2-8 below shows an example in which the
shear rate first increases from 1 and 10 s−1 within 10 s and then decreases to 1 s−1. The sample
shows a hysteresis loop due to thixotropy. In Chapter 3, we examine our model against both
shear-rate-ramp tests and stress-ramp tests with varied test conditions.
30
Figure 2-8. A hysteresis test by ramping the shear rate linearly with time from 1 to 10 s−1 within 10 s and then back to 1 s−1 at the same ramping rate. Prior to the hysteresis test, the sample is subjected a steady shear at 1 s−1. 2.3.7 Oscillatory Shear Tests
In Chapter 4, we will examine our model along with several representative TEVP models
against a comprehensive data set of oscillatory shear tests. The data set is reported in the recent
work of Armstrong et. al [41]. This section briefly reviews their test procedures. In an oscillatory
shear test, the testing fluid is subjected to a sinusoidal strain ( ) with an amplitude of and a
frequency of ,
t 0 sin t . (2-5)
And therefore the shear rate follows
t 0 cos t , (2-6)
where ̇ = . When is small, the stress also follows a sinusoidal function,
t 0 sin t . (2-7)
The shift angle indicates the relative significance of the elastic and viscous responses. For a
Hookean elastic solid, = 0, and for a Newtonian fluid, = 90°. 0 < < 90° indicates
viscoelastic materials.
31
Figure 2-9. A large amplitude oscillatory shear test ( = 10, = 0.1 rad/s) of a thixotropic fumed silica suspension. (a) Reduced shear strain ⁄ versus time. (b) Reduced shear rate ̇⁄ ̇ versus time. (c) Rheological response plotted as the reduced stress ⁄ versus time. Data are adopted from Armstrong et. al [41].
Figure 2-10. The elastic (a) and viscous (b) projections of the Lissajous–Bowditch curve for the data shown in Figure 2-9. Data are adopted from Armstrong et. al [41]
When is large, the oscillatory shear cause significant changes in internal structures,
and the stress shows strong deviation from sinusoidal wave. Figure 2-9 shows an example
adopted from the work of Armstrong et. al [41]. Data from large oscillatory shear tests are often
presented in the form of a Lissajous-Bowditch plot with an elastic projection (stress versus
strain) and a viscous projection (stress versus shear rate), as shown in Figure 2-10. The
32 rheological response depends strongly on and . More results and their comparison with constitutive models are presented in Chapter 4.
2.4 Particle Image Velocimetry (PIV)
2.4.1 Materials and Instruments
To measure the local velocimetry during rheological measurement, we seed fluorescent particles (0.1 vol%, radius = 3 μm, Fluoresbrite-YG, Polysciences) into the fumed silica suspension introduced in Chapter 2.2. The suspension is then homogenized using a Cole-Palmer ultrasonic processor (5 min, 500W, 20 kHz).
Figure 2-11. Instruments for rheo-PIV measurements. We use the MCR 702 rheometer with cone and plate geometry (cone angle 2°, radius 50 mm, truncation gap 210 μm) to measure the rheology. A fluorescence microscope (Module 1, band pass 450-490 nm, manufactured by Anton Paar) that views the sample through its edge meniscus is used to record movements of tracer particles in the velocity – velocity gradient plane. Figure 2-11 shows the instrument setup and a schematic.
The MCR 702 rheometer allows the cone and plate to rotate simultaneously at opposite directions with rotational speeds of respectively ∆Ω and (1 − )∆Ω. Here, the rotational speed difference ∆Ω is proportional to the apparent shear rate ̇ which is defined by
33 the test procedure. is the speed balance coefficient ( = 1, only the cone is rotating;
= 0, only the plate is rotating).
2.4.2 Algorithm
Figure 2-12. An image frame before (a) and after (b) CLAHE This section provides details of our algorithm for the time-resolved PIV. The 1D velocity profile at time , ( , ), is calculated based on a pair of images captured at ( , + ∆ ). We first crop the images and keep only the region sandwiched between the upper cone ( = 0) and bottom plate ( = ). After cropping, the images have a size of 1210 × 1038 pixels (height × width), which corresponds to a physical dimension of 0.87 × 0.75 mm. We then use the contrast- limited adaptive histogram equalization (CLAHE) algorithm [90] to increase image contrast
(Figure 2-12).
We index all pixels according to their coordinates on the image (x = 0, 1, …, 1038, and y
= 0, 1, …, 1210; the origin is set as the pixel on the left upper corner). Next, we divide the pair of frames into adjacent horizontal stripes with a height of pixels. We calculate the brightness intensity fluctuation (BIF) along x in each stripe, ( ), according to
34 nk 1 Ik x I( x , y ), (2-8) y n k 1 where k = 1, 2, etc. is the index of the stripes and I(x,y) is the brightness intensity of the pixel at
(x, y). I(x,y) = 0 and 255 for respectively black and white pixels. ( ) is then linearly rescaled into the range of [−1, 1].
The cross-correlation of the rescaled BIF of the two stripes reaches a maximum at a shift value that equals to the average displacement (∆ ) of the tracer particles in the stripes. The average displacement can be further refined with sub-pixel precision by fitting the adjacent data points around the peak with the Gaussian function, as shown in Figure 2-13. The average velocity of particles in the stripe equals ∆ /∆ . Repeating this process for all stripes across the gap gives the velocity profile at = .
Figure 2-13. Cross-correlation of the rescaled BIT shown 2.4.3 Method Validation
This PIV algorithm has high spatial resolutions. The stripe thickness can be as small as a single pixel ( = 1). Figure 2-14 shows the instantaneous velocity profiles ( , ) in a
Newtonian fluid under constant shear at ̇ = 0.2 s . The experimental results are very close to
35 the theoretical values. The mean absolute error (MAE) is as small as 0.2% of the wall velocity.
The velocity along the gradient direction is negligible, as shown in Figure 2-15.
−1 Figure 2-14. Velocity profiles, ( , ), in a Newtonian fluid under constant shear at 0.2 s . Calculations are based on a pair of frames. Black symbols are experimental results with ndiv = 1, 5, 10, and 20 pixels. Red lines are the linear fit of the experimental data. The four sets of results are shifted vertically with the shifting factors indicated in the annotation. MAE: mean absolute error. : the velocity of the fluid layer on the moving wall.
Figure 2-15 Histogram of , the velocity along the gradient direction. Red line is the fit of normal distribution. 2.5 Conclusions
This chapter introduces the materials, instruments, rheological test procedures, and the
PIV technique used in this thesis. We study the rheological properties of a thixotropic fumed
silica suspension under a wide range of shear histories, including steady state, step shear rate,
step stress, intermittent shear, flow reversal, and large amplitude oscillatory shear. The
36 comprehensive data sets will be used to develop and evaluate constitutive models for thixotropic fluids, which will be presented in the following chapters. We develop a new PIV algorithm to study the temporal evolution of the local velocimetry during rheological experiments. This method can resolve velocity variations as small as 0.2% of the wall velocity and gives a velocity value every 0.7 μm across the gap 0 ≤ y ≤ H ≈ 873 μm.
37 Chapter 3 Ideal thixotropy and Its Rheological Modeling1 3.1 Introduction
Thixotropic fluids often exhibit simultaneously thixotropy, viscoelasticity, yielding, and other rheological behaviors. But their behaviors at high-shear-rates are usually close to ideal thixotropy – a reversible time-dependent shear-thinning viscous response [91]. This chapter focuses on this regime. Predicting ideal thixotropic response is the first step in developing more comprehensive thixotropic models that include viscoelasticity and advanced plasticity. In addition, a purely viscous model is useful for process flow simulations for which the inclusion of elastic terms can be expensive and/or challenging.
Structural kinetics models (see Chapter 1.2.2 for a review) predict thixotropy using a viscosity that depends on a structure parameter, and a thixotropic kinetic equation. Most models assume a following form
, (3-1)
d f g 1 , (3-2) dt where ( ) denotes the viscosity, the structure parameter, and the flow parameter (which can be shear rate, ̇; shear stress, σ; or some combination of both). f(ϕ) and g(ϕ) are usually positive and increasing functions of ϕ. (g(ϕ) is constant when the build-up rate is independent of
ϕ). For a thixotropic model with a shear-rate-controlled ( = ̇) linear kinetic equation (n, m =
This chapter is based on ref. [Wei, Solomon, Larson. J. Rheol. 60(6). 1301–1315 (2016)].
38 1), in a step-shear-rate test, the model predicts that λ follows a pure exponential relaxation and its relaxation time (or rate) is a function only of the final shear rate. Stress, if linear with λ, ashould also show a pure exponential relaxation. However, real fluids often show a stretched exponential relaxation and the relaxation time (or rate) depends on both initial and final shear rates in step-shear-rate tests [11]. Predicting these two features of thixotropic rheology in a single model is the objective of this chapter.
The deviation of the stress transients from a pure exponential relaxation was noticed in the 1970s [92]. This problem may be addressed by introducing multiple structure parameters and kinetic equations [92-94]; however, this method is rarely used due to difficulties in parameterizing the equations. Dullaert and Mewis [11], on the other hand, added a time- dependent pre-factor, 1⁄tα, to the kinetic equation, which generates a stretched exponential relaxation of λ (and consequently stress). Teng and Zhang [52] similarly used 1⁄(1+γα) as pre- factor (where γ is the total strain). Adding a strain- or time-dependent pre-factor to the kinetic equation is not general in the sense that such a formulation can be only used in experiments such as step tests for which an initial time point can be designated. For arbitrary flows, however, pre- factors cannot depend explicitly on time without violating time-invariance symmetry [95].
In addition, the literature is unclear about what causes the relaxation time (or rate) to depend on initial shear rate in step-shear-rate tests. Dullaert and Mewis [11] addressed this effect by coupling the transient response of elastic stress with that of viscous stress. According to this approach, the relaxation time should be independent of the initial shear rates in step-shear-rate tests if the elastic effects are negligible.
Focusing on the two aforementioned problems, this chapter seeks to answer the following questions: How may the stretched exponential relaxation behavior of thixotropic fluids be
39 accurately and parsimoniously modeled? What non-linear terms are needed to capture the complex dependence of thixotropic relaxation times on initial and final shear rates in step tests and how should these terms be parameterized? What is the relative performance of the kinetic evolution equations that incorporate different variables – such as shear stress versus shear rate – as the flow parameter?
3.2 Proposed Model
We propose a method for modeling thixotropic fluids with stretched exponential and nonlinear dynamic rheological behaviors. While viscoelastic effects are neglected to maintain simplicity, they can be incorporated into the models using existing methods [11, 41, 50, 51, 54].
We construct the constitutive equation for stress based on the Bingham model, assuming that the yield stress is proportional to λ and the viscosity is linear in λ:
y0 thi n (3-3) where σy0 is the apparent yield stress determined from the flow curve, ηthi is the thixotropic viscosity increment, and ηn is the limiting high-shear-rate viscosity. The model reduces to a
Newtonian fluid when λ = 0. Different from previous thixotropic models, the structure parameter
λ in this new model is a linear combination of a spectrum of ‘substructure-parameters’:
N Ci i , (3-4) i1 where Ci is the coefficient and N is the number of λi. All λi have the same steady state value at the same shear rate or stress; however, their time derivatives differ, and are set by dimensionless pre- factors Di:
d m m i D ka n k b 1 k 1 (3-5) dt i1 i 2 i 3 i
40 Here, k1, k2 and k3 are respectively the rate constants of the break-down term, the shear-induced build-up term and the Brownian build-up term; a and b are the orders of the dependence of the flow parameter ϕ; and n and m are the orders of the dependence of λ in the break-down term and build-up terms.
We assume that the distribution of relaxation times follows a stretched exponential with a constant stretching factor β. The values of { }, { }, and , therefore, are determined by
N Di t expt Ci e , (3-6) i1
Eq. (3-6) states that a stretched exponential function can be approximated by a linear combination of a series of simple exponentials. Here, { } are the dimensionless relaxation rates of the simple exponentials and { } are corresponding weights. β is a model parameter estimated from experimental results. The values of N and { } are specified based on the requirements of the accuracy, or resolution, of the discretization of the continuous relaxation spectrum. The values of { } are then calculated using the method described in Chapter 3.3. The model reduces to a conventional single-structure-parameter model when β = 1 and consequently both { } and
{ } are single values, each equals to unity. Through this method, which we refer to as the Multi-
Lambda Approach, we introduce multiple structure parameters while keeping the number of model parameters small (the equation for each λi has the same values of the parameters of k1, k2, k3, n, m, a, b).
Two models can be constructed after specifying whether the flow parameter, ϕ, corresponds to a shear-rate-controlled model (RC Model) or a stress-controlled one (SC Model):
RC Model: , (3-7) SC Model: y0 .
41 The SC model uses σ – σy0 as the flow parameter. Consequently, λ approaches unity in the zero- shear-rate limit where σ = σy0. However, using σ – σy0 as the flow parameter sets a limit on the application of the SC Model: σ cannot be less than σy0, otherwise ϕ is negative and the first two terms in Eq. (3-5) are not sensible. This is a pitfall to avoid when using this model and Eq. is an expedient solution to this problem.
y0 if y0 , (3-8) 0 if y0 .
3.3 Decomposition of Stretched Exponential Function
The stretched exponential function can be decomposed into a spectrum of simple exponential functions, written in discrete form, as in Eq. (3-6) [96], or in continuous form, as in
Eq. (3-9) [97]. Here, P( , ) is the probability density function (PDF) of this exponential spectrum.
Dt expt P D , e dD (3-9) 0
Two approaches are developed here to calculate { }. The first one is the Tikhonov regularization method [96, 98]. For this method, the set { } is calculated by solving a system of linear equations. The second method calculates { } from the probability density function
P( , ) and is abbreviated as the “PDF method” in the following.
3.3.1 Tikhonov regularization method
The first step in this method is to construct a linear system of equations. A time vector
= [ , , … , ] is first generated with each element being a specific time point. Next, the signal vector, , is calculated from a stretched exponential function. The stretching factor is determined from experimental results.
42 Each element of results from a simple exponential decay with a relaxation rate . can be determined simply through trial and error or using the algorithm given in Figure 3-2. We suggest that should have at least three elements distributed uniformly on a log scale over an interval not smaller than 10-1~101. The number of modes one needs to use to achieve a given accuracy is strongly influenced by the value of . For larger , fewer elements of and a narrower the interval are required for a given accuracy. When = 1, is simply a scalar, unity
1.
Matrix is calculated according to Eq. (3-14). The solution to Eq. (3-15) gives the coefficients vector { }. Eq. (3-15), however, is ill-posed, and is sensitive to the noise in [96].
Various methods of regularization can be used to solve this problem. The Tikhonov regularization [98] replaces Eq. (3-15) by a regularized system, Eq. (3-17) where is the regularization factor. An optimized value of can be found by solving the minimization problem given by Eq.(3-18). Then can be obtained by solving Eq. (3-17).
Table 3-1. List of equations of the Tikhonov regularization method. T T [t1 , t 2 ... tn ] (3-10)
T b [b1 , b 2 ... bn ] (3-11)
biexp t i (3-12)
T D [DD1 , 2 ... Dm ] (3-13)
Dj t i Aij e (3-14) Ax = b (3-15)
T x [CC1 , 2 ... Cn ] (3-16) ATT A + I = A b (3-17) min{Ax - b + x } (3-18)
3.3.2 PDF Method
The PDF method is based on a continuous distribution of simple exponentials that is fully characterized by the PDF P( , ) which is a function of and . The analytical expression for
43 P( , ) have been discussed in detail by Johnston [97]. For = 1, P( , ) is a Dirac δ
function at = 1 and the stretched exponential reduces to a simple exponential. For 0< < 1,
P( , ) is expressed by Eq. (3-19). The simplified expressions for = 1/3, = 1/2 and =
2/3 are given in Table 3-2. Here, Ai( ) is the Airy function; Ai ( ) is its derivative; and K ( )
⁄ ⁄ is the modified Bessel function of the second kind; = (3 ) and = (3 )
Table 3-2. Forms of P( , ) for different values
1 PD , eu cos /2 cos Du u sin / 2 du (3-19) 0
1 4 1 2 PD , 3 z1 Ai z 1 K 1/3 (3-20) 3 3 D 3/2 27D
1 1 1 P,D e 4D (3-21) 2 4 D3
2 2 1 7/4 PD , 6 z2 exp 2 Ai z 2 Ai z 2 (3-22) 3 27D z2
Figure 3-1 Illustration of the PDF method (β = 0.5). ‘Cutoff’: fraction of the exponential spectrum used to generate the stretched exponential function; DL: lower limit of D; DU: upper limit of D. The probability density function P(D,1⁄2) is integrated over a set of evenly distributed intervals on a log scale for D. The integrals are normalized, which gives the set of { } The integration of P( , ) over a given integral of is used to calculate the coefficient
for the designated over this integral. Figure 3-1 shows an example for = 1/2. First, the
“cutoff ratio” R, representing the fraction of the continuous distribution to be captured between
the maximum and minimum values of D, and the number of simple exponentials N, are set. Then
44 is found which maximizes P( , 1⁄2). Then the largest and the smallest are found so
that ∫ P( , 1/2) ≥ . Since P( , 1⁄2) is nearly symmetric on a log scale for D, a set of continuous intervals is generated uniformly on a log scale between and . P( , 1/2) is integrated over each interval, giving the set of { }. { } is then normalized. The values of R and
N used in this article are, respectively, 99.5 % and 10. The algorithm flowchart is shown in
Figure 3-2. The vector generated in this algorithm can also be used in the Tikhonov regularization method.
Figure 3-2. Algorithm flowchart of the 3.3.1 Tikhonov regularization method and the PDF method. 3.3.3 Comparison of the results of the two methods
45 Both the Tikhonov regularization method and PDF method are here tested for three
values of : 1/3, 1/2 and 2/3. Ten simple exponentials are used to approximate the stretched
exponential. The results show that the coefficients calculated using the two methods are in good
agreement and the stretched exponential can be approximated accurately.
t YSEF e . (3-23)
10 Di st YPDFor Y Tikhnov C i e . (3-24) i1
Figure 3-3. Comparison of the results of two regularization methods. (a-c): = 1/3; (d-f): = 1/2; (g-i): = 2/3. Graph (a), (d), (g), P(D,β). Graph (b), (e), (h) { } values calculated from the PDF method and the Tikhonov regularization method. Graph (c), (f), (i) Stretched exponential curve and approximation curves calculated from the PDF method and the Tikhonov regularization method.
46 3.4 Nonlinear Thixotropic Kinetics
One assumption of our model is that the structural parameters follows Eq. (3-25) which is the same as Eq. (3-5) given earlier.
d m m i D ka n k b 1 k 1 (3-25) dt i1 i 2 i 3 i
Previous thixotropic models [11, 27, 41, 99] either assign some values for n, m, and b or determine their value through multivariate regression. Here, we present a more effective method based on analyzing the fast transients in step tests.
We assume that the fast transient response after a stepwise change of the shear rate or stress mainly comes from a single element of the structure parameters. To determine n, a relatively large shear rate (50 s–1 in a step-shear-rate test) or stress (20 Pa in a step-stress test) is suddenly imposed on a sample that was previously fully rested at t = 0. At t = 0, therefore, λ = 1, and the last two terms in Eq. (3-25) are zero. ln(λ) should be proportional to t if n = 1. For n > 1,
λ(1–n) should be linear in t. Since the models assume a linear relationship between viscosity and λ, we find that
t t = (3-26) o where ηo is the initial value of viscosity (at t = 0.1 s in a step-shear-rate test and at t = 0.3 s in a step-stress test) and η∞ is the steady-state value of viscosity. The nonlinearity between ln(λ) and time within the first 1.2 seconds after the change of shear rate or stress, as shown in Figure 3-4, demonstrates that n ≠ 1.
47
Figure 3-4. ln(λ) vs time in a step-rate test (0 to 50 s–1) and in a step-stress test (0 to 20 Pa). Figure 3-5 shows, on the other hand, that, in a step-shear-rate test, λ(1–n) is linear with
time when n = 3; while in the step-stress test, λ(1–n) is linear with time when n = 2. The reason for
this difference in exponents is that stress has a first-order dependence on λ as a result of the
assumed linearity between stress and λ and after converting a stress-controlled kinetic equation
of λ into a shear-rate-controlled one, the apparent order of λ in the break-down term should
therefore increase by one power.
Figure 3-5. λ(1–n) (left axis, n = 2; right axis, n = 3) vs time in (a) a step-rate test (0 to 50 s–1) and (b) a step-stress test (0 to 20 Pa). The exponent m can be determined in a series of build-up tests: in step-shear-rate tests,
with shear rate stepping from a high value of 100 s–1 to various final values ranging from 0.25 s–1
48 to 10 s–1; and in step-stress tests, with stress from a high value of 50 Pa to various final values
ranging from 2 Pa to 8 Pa. In the early period after the step, λ is close to zero and the build-up
term dominates. The evolution of λ in the early period should be approximately a single
exponential relaxation. Based on the linearity assumption between stress and λ, the evolution of
both viscosity and stress should also be exponential. Consequently, ln(η∞ – η) and ln(σ∞ – σ)
should be linear with time in early period, where η∞ is the equilibrium value of viscosity and σ∞
that of stress. Moreover, ln(λ∞ – λ) ~ t, ln(η∞ – η) ~ t and ln(σ∞ – σ) ~ t have the same slope, –
b (k2ϕ + k3).
–1 Figure 3-6. Results of build-up tests (a) ln(σ∞ – σ) vs time in step-shear-rate tests, initial shear rate 100 s ; (b) ln(η∞ – η) vs time data in step-stress tests, initial stress 50 Pa; (c) |slope| vs final shear rate with best power law fit (fitting equation, |slope| = ̇ + ), given by b = 0.51; (d) |slope| vs. final stress and power law fit (fitting equation, |slope| = − + ), best fit of b = 0.53.
49 Figure 3-6(a-b) shows the plots of ln(σ∞ – σ) and ln(η∞ – η) against time. The original stress transient data in the step shear rate tests are smoothed using a locally weighted scatterplot smoothing (LOWESS) method to better reveal the relation between ln(σ∞ – σ) and time. It is clear in Figure 3-6 (a-b) that both ln(σ∞ – σ) and ln(η∞ – η) are approximately linear with time, which demonstrates that m ≈ 1. The absolute value of the slope increases as the initial shear rate or stress increases, which is a clear manifestation of shear-induced structure building, i.e. k2 and b are both positive. The best fit of |slope| versus the final shear rate or stress from a power law function shows that b = 0.51 in step-shear-rate tests and b = 0.53 in step-stress tests. This is in good agreement with the theoretical study of van de Ven and Mason [100] that the shear-induced build-up rate is proportional to the square root of shear rate. Dullaert and Mewis also took this relationship in their model [11].
3.5 Model Parameterization
Each model (rate or stress controlled) has a total number of 11 parameters: k1, k2, k3, a, b, m, n, σy0, ηthi, ηn, β. In the previous section, we demonstrated that n = 3 in the shear-rate- controlled model (RC Model); n = 2 in the stress-controlled model (SC Model); m = 1 and b =
0.5 in both models.
The remaining eight model parameters are estimated from the experimental results in
Figure 3-7. Figure 3-7(a), in particular, shows the steady state flow curve of the fumed silica dispersion. The fumed silica dispersion shows an apparent yield stress of about 0.7 Pa and a limiting high shear rate viscosity 0.4 Pa·s. The ideal thixotropy region of the fumed silica suspension is in a regime in which the viscous contribution to the total stress dominates. The former limit is empirically specified as σ > 3σy0 and ̇ > ̇ 3 , which is based on Dullaert and Mewis’ study for a similar fluid [14]. Our experimental results also show that no elastic
50 response can be detected in step tests within this region. Figure 3-7(b) shows a set of step-stress tests data (initial stress 11 Pa). Inertia effects dominate in the first 0.1 second after which the response is close to ideal thixotropy.
Figure 3-7. Experimental data (symbols) used to estimate model parameters and model calculation. (a) Steady state flow curve of the fumed silica dispersion, including results of both stress sweep tests and shear-rate sweep tests. Box in the upper right, σ > 3σy0 and ̇ > ̇ 3 , region of ideal thixotropy. (b) Step-stress tests results, initial stress 11 Pa. The lines in (b) include the effects of inertia, which produces the steep response for times less than t = 0.1 s.
These model parameters can be estimated through the following procedure. All λi have the same value λss at the same shear rate or stress at steady state, as defined in Eq. (3-27), where
K1 = k1/k3 and K2 = k2/k3. Therefore, the stress vs shear rate relationship at steady state is fully determined by six parameters: K1, K2, a, σy0, ηthi, ηn. The inverse problem, i.e. estimating the six parameters from the steady state flow curve, however, is ill-posed. Among the six parameters,
σy0 can be easily determined by extrapolating the data in the low-shear-rate region and ηn can be estimated from the data in the high-shear-rate region. As for K1, K2, a, and ηthi, their values are estimated from the steady state flow curve together with the step-stress tests data shown in
Figure 3-8(b).
a n 0.5 KK1 ss 2 1 ss 1 ss 0 (3-27)
ss f(,,,,,,) K 1 K 2 a y0 thi n (3-28)
51 Table 3-3. Parameter values for the RC Model and SC Model Model Parameters RC Model SC Model
σy0 0.700 Pa 0.687 Pa
ηthi 1.30 Pa·s 1.00 Pa·s
ηn 0.443 Pa·s 0.457 Pa·s a 2.30 2.02 b 0.500 0.500 n 3 2 m 1 1 a -a K1 5.00 s 1.72 Pa b -b K2 2.68 s 1.79 Pa -3 -1 -3 -1 k3 3.00×10 s 7.00×10 s β 0.5 0.5 Other Parameters 7.94×10-2, 3.16×10-1, 1.26, 5.01, 1 1 2 { } 2.00×10 , 7.94×10 , 3.16×10 , 1.26×103, 5.01×103 7.29×10-2, 3.02×10-1, 2.90×10-1, 1.59×10-1, -2 -2 -2 -2 { } 8.95×10 , 2.63×10 , 1.98×10 , 1.98×10 , 1.98×10-2
Once K1, K2, a, and ηthi are determined, then k3 and β are estimated. k3 determines the mean Brownian build-up rate (and once its value is known, allows determination of the break- down and shear-induced build-up rate coefficients k1 and k2 through K1 = k1/k3 and K2 = k2/k3). β determines the distribution of relaxation times. { } and { } are pre-generated for a set of values of β (0.1, 0.2, … , 1.0) using the method described in Chapter 3.3. k3, β, { }, and { } are determined through a best fit of a set of step-test data shown in Figure 3-7(b). The values for all model parameters, { }, and { } are listed in Table 3-3. It is no surprise that the RC Model and
SC Model have similar parameter values. The choice of the flow parameter, shear rate or stress, only influences the shear-induced transient response. Consequently, the parameters σy0, ηn, m, k3 have similar values in the two models. ηthi, a, b, n should be sensitive to the choice of the flow parameter. But for this specific fluid, they have similar values. β, { }, and { } have the same values in both models because we assume β is constant for this suspension and independent of the flow parameter.
52 In summary, we used the following experimental data to parameterize the models: the fast response (t < 1.2 s) in a step-shear-rate test (0 to 50 s–1) and a step-stress test (0 to 20 Pa); the fast response (t < 5) in several step-shear-rate tests (100 s–1 to different final shear rates) and step-stress tests (50 Pa to different final stress); the steady state flow curve; and the results of four step stress tests (11 Pa to 3, 5, 7, 9 Pa). Experiments that can then be used to test the model are step-shear-rate and step-stress tests with intermediate shear rate and stress, continuous ramp tests with shear rate, or stress linearly changing with time.
3.6 Model Evaluation
In this section, we test the RC Model and SC Model in experiments that have not been used in model parameterization, including step-shear-rate tests with initial or final shear rate of 3, 5, 7 or 9 s-1; step-stress tests with initial or final stress of 3, 5, 7 or 9 Pa; shear rate ramp tests with shear rate changing continuously from 1 to 10 s-1 and then back to 1 s-1; shear stress ramp tests with stress changing continuously from 2 to 10 Pa and then back to 2 Pa; two groups of intermittent shear tests that demonstrate the necessity of introducing multiple thixotropic timescales.
3.6.1 Step Tests
Figure 3-8 shows the model prediction and experimental data in a set of step tests. The best prediction of Dullaert and Mewis’ model [11] (DM model) are also provided for comparison. The experimental data in Figure 3-7 are used to parameterize the DM model. Figure
3-8 (a) shows the results for step-shear-rate tests (initial shear rates 3, 5, 7, 9 s-1, final shear rates
5, 7, 9 s-1) and corresponding predictions of our models. The stress transients are predicted adequately although both models slightly overpredict the stresses. Figure 3-8(b) shows the results of step-stress tests (initial or final stresses, 3, 5, 7, 9 Pa). Both step-up and step-down tests
53 are predicted very well. The RC Model and the SC Model give nearly the same predictions in
step-down tests while the SC Model yields better predictions in step-up tests.
Figure 3-8. Model predictions for (a) step-shear-rate tests and (b) step-stress tests (σo, initial stress; σt, final stress). (c), (d): predictions of the DM model with α = 0.5 and α = 0 in the time-dependent pre-factor, 1⁄ , for a particular final shear rate in (a) and a particular final shear stress in (b). Figure 3-8(c-d) show part of the same results as that in Figure 3-8(a-b) and the
predictions of the original DM model as well as a version of it we tested in which the time-
dependent pre-factor had suppressed (α = 0 in the pre-factor 1⁄ ). The latter is representative of
models that have a linear kinetic equation for λ. Comparing the response of the three models
shows that capturing the stretched-exponential response is critical to predicting the transient
response in step tests. The DM model, with a pre-factor 1⁄ in the kinetic equation for λ (and
54 which therefore produces a stretched exponential, but not time invariant, response in these flows), has fair overall predictions but it still suffers from the drawback that it does not accurately capture the dependence of stress (or shear rate) transients on the initial shear rate (or stress) in step tests. The SC model, which includes both multiple thixotropic timescales and nonlinear kinetic equations, show better predictions than the DM model. Note that the DM model is applicable only in single-step tests due to the time-dependent pre-factor while the SC model can be used in arbitrary flow histories, as is demonstrated next.
Table 3-4. Summary of the DM model.
G0 e thi
d k e 4 ss e dt t
d 1 0.5 k1 k 2 1 k 3 1 i dt t
ss ss G 0 c ss thi
k0.5 k 2 3 ss 0.5 k1 k 2 k 3
Model parameters: G0 , thi , , c , k 1 , k 2 , k 3 , k4 ,
3.6.2 Intermittent Shear Tests
Adding a time-dependent pre-factor to the kinetic equation for λ causes the formulation to lose time-invariance symmetry. Consequently, while it is suitable to describe single-step-change experiments, it is not appropriate in general for modeling materials with stretched-exponential relaxation. The model of Dullaert and Mewis [11] and that of Teng and Zhang [52] are two such examples. In addition to the lack of time-invariance, the previous single-lambda models predict a monotonic thixotropic evolution because they include only a single structure parameter for thixotropy and under a constant shear rate, the thixotropic evolution is governed by either structural break-down or build-up. The experiments (Figure 3-9 and Figure 3-10), however,
55 show that under constant shear rate, multiple quick jumps of the shear rate can lead to subsequent non-monotonic thixotropic evolution even though the shear rate remains constant.
In the strain history of Figure 3-9(a), the sample is first sheared at 100 s-1 until reaching steady state. The shear rate then steps to 0.5 s-1 and is held constant for ∆t; finally, at t = 0, the shear rate steps up to 5 s-1 until reaching steady state. For some intermediate values of ∆t, the stress shows a non-monotonic response even though the shear rate is constant at 5 s-1. Figure
3-10 shows another group of intermittent shear experiments with slightly different shear histories. Similar non-monotonic transient behavior of shear rate in multiple-stress-jump tests has been observed by Coussot et al. [101] in a thixotropic bentonite suspension. Some microgel pastes have also been reported to show non-monotonic evolution of strain in similar step-stress experiments [102, 103]. Cloitre [102] discussed conceptually the physical origin of such non- monotonic behaviors in microgel pastes: various spatial configurations with a broad distribution of relaxation times were formed during the rest period and the collective response of these configurations led to the initial rejuvenation and subsequent aging in the last step. An analogy between microgel pastes to thixotropic fluids can be made to explain the aforementioned non- monotonic stress and shear rate transients in thixotropic systems. In the experiments of Figure
3-9(a), in the second step, structures with different relaxation times rebuild at different rates. At intermediate times, structures with short relaxation times can build up enough that they can break down again as the shear rate is stepped to 5 s-1. However, the slow-relaxing structures have not re-built much and so continue to rebuild after the step rate change is executed. Consequently, the stress first decreases and then increases, as shown in Figure 3-9(b), and vice versa for the test of
Figure 3-10(a).
56
Figure 3-9. Experimental results (symbols in b) and model predictions (lines in c-d) of a group of intermittent step tests. (a) Test protocol. The sample is pre-sheared at 100 s-1. The shear rate then steps down to 0.5 s-1 and is held constant for ∆t. At t = 0, shear rate steps up to 5 s-1 until the stress reaches steady state. (b) Experimental results for different ∆t values as indicated in the legend. (c) Predictions of the SC model with multiple thixotropic timescales. (d) Predictions of the SC model with a single thixotropic timescale ( is set to be unity). The non-monotonic responses in Figure 3-9 and Figure 3-10 are successfully captured
through the Multi-Lambda Approach introduced earlier in Chapter 3.2. Figure 3-9(c) and Figure
3-10(c) show the predictions of the SC model using the model parameters listed in Table 3-3.
Consider the test procedure of Figure 3-9(a), the mode shows that, during the intermediate step,
all structure parameters { } grow toward the steady state value at ̇ = 0.5 s . { } have
different growth rate set by { }. For intermediate values of ∆ , those with large grow
beyond the steady state value at ̇ = 5 s , and therefore once ̇ jumps to 5 s , those
decrease with time. Those with slower kinematics, however, do not grow too much during
57 ̇ = 0.5 s and they keep growing then ̇ jumps to 5 s . Because = ∑ and is linear in
, first decreases and then increases. Similar analysis applies to the results shown in Figure
3-10. In Figure 3-9(d) and Figure 3-10(d), we show the results of the SC model with the value of
set to unity. This simplified SC model represent previous single-lambda models that are unable
to capture the non-monotonic stress evolutions.
Figure 3-10. Experimental results (symbols in b) and model predictions (lines in c-d) of another group of intermittent step tests. (a) Test protocol. The sample is pre-sheared at 0.1 s-1. The shear rate then steps up to 3 s-1 and is held constant for ∆t. At t = 0, shear rate steps down to 2 s-1 until the stress reaches steady state. (b) Experimental results for different ∆t values as indicated in the legend. (c) Predictions of the SC model with multiple thixotropic timescales. (d) Predictions of the SC model with a single thixotropic timescale ( is set to be unity). 3.6.3 Hysteresis Tests
58 Both the RC Model and SC Model are next examined in hysteresis tests where the shear rate or stress is first linearly increased to a maximum value and then decreased back to the initial value. This test represents an example of an arbitrary flow history, such as might be encountered in a process flow simulation.
The stress vs shear rate plot of thixotropic fluids in a hysteresis test usually shows a loop with the structure parameter larger in the ascending period than that in the descending period due to the finite time for internal structures to break or build. The discrepancy in internal structures between the ascending and descending periods generates thixotropic hysteresis loops. The loop converges to the steady state flow curve for infinite ramp time.
We use the following test procedures in continuous ramp tests: for shear-rate ramps, the sample is first pre-sheared at 1 s-1 and then subjected to a linearly increasing shear rate from 1 s-1 to 10 s-1 after which shear rate is decreased to 1 s-1 within the same amount of time as that expended during the ascending period. The length of time of the ascending or descending period is referred to as the ramp time. Figure 3-11(a-c) gives the model predictions and experimental data for three shear-rate ramp tests with ramp times of 10 s, 30 s, and 90 s respectively. For the stress ramp tests, the initial stress is 2 Pa and the maximum stress is 10 Pa. Figure 3-11(d-f) are the model predictions and experimental data for three stress ramp tests with ramp times of 10 s,
30 s and 90 s respectively. The RC Model overpredicts the ascending curve in both shear rate ramp tests and stress ramp tests. Its prediction of the descending curve is very accurate, however.
The SC Model predicts all the data in Figure 3-11 very well.
59
Figure 3-11. Model predictions for (a-c) shear rate ramp tests and (d-f) stress ramp tests. Procedure in shear rate ramp tests: initial shear rate, 1 s-1; maximum shear rate, 10 s-1; shear rate ramping mode, linear; ramp time, (a) 10 s, (b) 30 s, (c) 90 s. Procedure in stress ramp tests: initial stress, 2 Pa; maximum stress, 10 Pa; stress ramping mode, linear; ramp time, (d) 10 s, (e) 30 s, (f) 90 s. 3.7 Model Limitations
Assuming ideal thixotropy and neglecting other complex responses such as
viscoelasticity will lead to poor predictions when the shear rate or shear stress is small, i.e.,
outside of the region of ideal thixotropy. Referring to Eq. (3-3), the stress has two components:
the yield stress, σy0λ, and the viscous stress, ( + ) ̇. The yield stress cannot change
instantly because the rate of change of λ is always finite for finite shear rate while the viscous
stress, however, is proportional to the shear rate and therefore a sudden change in shear rate
should produce an instantaneous change in the viscous stress. In a step-shear-rate test therefore,
the yield stress does not change instantly; the viscous stress, however, steps proportionally with
shear rate. In a step-stress test, the yield stress remains constant at the instant of the stepwise
change of stress; the shear rate, therefore, steps correspondingly so that the total stress sums up
60 to the final stress. However, this prediction does not agree with the experimental results. Figure
3-12 shows that the model predicts poorly in step tests with small initial shear rate (i.e. 0.1 s-1) or
small initial stress (i.e. 1.5 Pa). In the step-shear-rate tests of Figure 3-12(a), the model fails to
predict the initial viscoelastic response and underpredicts the instantaneous change of stress. In
the experiments of Figure 3-12(b) where stress steps from 1.5 Pa (twice the apparent yield
stress), the model overpredicts the instantaneous change of shear rate. Further modification of
the models is therefore needed to predict the response in low shear rate (or stress) region, which
is the focus of the next chapter.
Figure 3-12. SC Model predictions of step tests with small initial shear rate or stress. (a) Step shear-rate tests from an initial shear rate of 0.1 s-1; (b) step stress test from 1.5 Pa. 3.8 Predicting Non-monotonic Viscosity Bifurcation
In this section, we apply the multi-lamba approach (MLA) to the model of Coussot et al.
[101] to predict the non-monotonic time evolution of shear rate in multiple-step stress-jump tests.
Coussot et al. [101] suggested introducing multiple thixotropic timescales to predict this
phenomenon; however, no such introduction has been available or reported to date. Although
other data sets [13, 41] might be equally well suited to application of the MLA, we here select
the Coussot et al [101] data set because it highlights interesting non-monotonic behavior in
61 rejuvenation; such behavior is a hallmark of the phenominology that the MLA is well-suited to quantitatively model.
Coussot et al. [101] studied the rheological behaviors of a Bentonite suspension (solid volume fraction of 4.5%) in a group of stress-jump experiments. The test protocol includes 1) preshearing at a relatively large stress, 2) resting for a duration of , and 3) shearing again at different lower stress levels than the preshear stress. The steady shear rate during preshear at a
fixed stress is large (≈ 400 s ). Figure 3-13(a) shows the results of the experiments with =
0. After the sudden decrease of the shear stress, the viscsosity gradually builds up and therefore the shear rate decreases. The viscosity bifurcates at about 6 Pa. That is, for higher stresses than 6
Pa, the material flows continuously with the shear rate slowly decreasing. For stresses lower than this, the shear rate eventually falls rapidly below 10 s ; i.e., flow ceases in a practical sense.
The rheological responses are drastically different if there is a short delay time between the preshear and the final step, as shown in Figure 3-13(b) for = 20 s. In this case, the viscosity bifurcates at about 12 Pa, so that for stresses lower than this value, flow eventually stops.
Interestingly, for stresses within the range (6 ~ 12 Pa), the shear rate first decreases and then rapidly collapses, indicating that at a constant stress shear rejuvenation and aging can coexist – shear rejuvenation dominates the short-term relaxation and aging dominates the long-term evolution.
62
Figure 3-13. (a, b) (Redrawn from Coussot et al. [101] with permission) Rheological response of a Bentonite suspension: change of shear rate with time at different stress levels applied after a rest period of duration following preshearing. (a) = 0; (b) = 20 s. (c, d) Predictions of the new model for similar flow histories: change in dimensionless shear rates ( ) over dimensionless time ( ) for different dimensionless stresses levels ( ) after a rest period of duration following preshearing. Stress levels increase from top to bottom. (c), = 0; (d) = 0.05. Coussot et al. [101] proposed a simple constitutive model for viscosity-bifurcating
TYSFs that qualitatively captures most of their experimental findings except for the non-
monotonic time evolution of shear rates shown in Figure 3-13(b). Their model involves the stress
expression
0 f , (3-29)
where denotes the shear stress, the asympotic value of the viscosity, ̇ the shear rate, the
structure parameter, and ( ) an unknown function of . ( ) satisfies two constraints: 1) it is
63 an increasing function of , and 2) (0) = 1. Coussot et al. [101] found that ( ) = exp( ) qualitatively agrees with their experiments . Their expression for the evolution of is
d 1 , (3-30) dt T0 where is the characteristic timescale of aging and is a model parameter. In this model, the value of is unbounded. When ̇ = 0, continuously builds up to infinity. Eqs. (3-29) and
(3-30) can be written in dimensionless forms by defining the dimensionless stress = ⁄ , the dimensionless shear rate = ̇, and the dimensionless time = ⁄ :
(a) exp , d (3-31) (b) 1 . d
At steady state, the flow curve follows
exp 1 . (3-32)
The flow curve has a negative slope for < 1. In a constant stress test at = , the time evolution of follows
d 1 . (3-33) dexp 0
always evolves monotonically – either towards the steady state value at or towards infinity.
Regardless of the initial value of , a non-monotonic evolution of requires ⁄ to have non- unique values at certain values, which is impossible for Eq. (3-33). At = , ( ) =
⁄exp[ ( )]. ( ) therefore also evolves monotonically, and the model cannot predict the non-monotonic behavior shown in Figure 3-13(b).
64 To predict the non-monotonic phenomenon, we generalize this simple model using the
MLA: we decompose into a group of sub-structure parameters according to = ∑ and rewrite the kinetic equation as
d i D 1 . (3-34) d i i
As discussed in Chapter 3.3, , , and are not model parameters, but are obtained from a stretched exponential distribution according to
N exps Ci exp D i s , (3-35) i1 where is a dummy time variable and is the stretching exponent, the only new model parameter we introduce. Note that in this new model, in single-step-rate test, ( ) follows a stretched exponential function, but ( ) does not because ( ) ∝ exp[ ( )] ̇. Nevertheless, when the change of ( ) over time is small, e.g. (0)⁄ (∞) ∈ (0.1, 10), ( ) can be closely approximated by a stretched exponential.
Eqs. (3a, 6-8) specify the dimensionless form of this new model. When = 1, = =
= 1, and the new model reduces to the original form of Eq. (3-31). When < 1, one can use arbitrarily many modes for the approximation described by Eq. (3-35); however, the smaller is, the more modes are needed for an accurate representation of the stretched exponential. For =
0.5, 10 modes can give a very close approximation. The method for calculating and for a given value of is discussed in Chapter 3.3. We reiterate that even with the selection of 10 modes, the MLA still only adds a single parameter, namely , to the set that most be specified.
We show the predictions of the new model for similar flow protocols as those in Figure
3-13(a, b) in Figure 3-13(c, d). Here, we focus on a parsimonious formulation and hence make only qualitative comparisons between this new model and the experiments. We arbitrarily set
65 = 0.5 and use 10 modes for the thixotropic kinetics. Figure 3-13(c) shows the predictions of the dimensionless shear rate ( ) after instantly jumps from a high value to several lower levels. The impact of multiple structure parameters is not obvious in this case because = 0 and therefore all values evolve upwards. The influence of multiple structure parameters can be clearly seen if a short rest period is introduced between the pre-shear and the subsequent lower stress. During rest, the ’s with large values rapidly build up and after re-start of the stress they break down, causing the initial rise of . Those ’s with small values, however, do not build up much during resting. They have longer memories of the flow history. At stress levels lower than the preshear stress, they continue to slowly build up and cause the long-term decrease of . The predictions of this simple model agree nicely with experiments, as shown in Figure
3-13.
To summarize, in this section, we introduced multiple modes of thixotropic kinetics to the model of Coussot et al. [101] for a viscosity-bifurcating yield-stress fluid to predict the non- monotonic time evolution of shear rates in multiple stress-jump tests that their original model cannot capture. Our modification requires only one additional model parameter and allows the original model to now qualitatively capture the phenomenon. This example, together with the studies discussed earlier in this chapter, strongly demonstrates the necessity of including multiple structural kinetics in thixotropic models. This inclusion is especially important for complex shear histories that extend the applicability of single-lambda models beyond tests such as single step jumps in shear rate or stress.
3.9 Conclusions
We have presented two versions of an ideal (i.e. purely viscous, non-elastic) thixotropic model with a spectrum of structure parameters and nonlinear kinetic equations. We used shear
66 rate as the flow parameter in the kinetic equations of the shear-rate controlled version (RC
Model); and used the difference between the total stress and the apparent yield stress as the flow parameter in the stress-controlled version (SC Model). We developed a multi-lambda approach to introduce multiple structure parameters and kinetic equations without increasing the number of model parameters (except for β) compared to the conventional single-structure-parameter models. In our approach, the conventional structure parameter λ is taken to be a linear combination of ‘substructure parameters’ (λ = ∑ ) each of which evolves according to its own equation that differs from the others only in the dimensionless pre-factor Di. The set of { } is generated as an approximate discretization of the continuous relaxation spectrum with the number of terms in the discretization set by the desired accuracy. The relaxation spectrum was assumed to be governed by a stretched exponential relaxation with a constant stretch factoring, β, which determines the set { }. The set of weight coefficients { } was calculated for a given β and a set of { }.
We used the model to describe the rheological behavior of a weakly aggregated fumed silica dispersion, which in the relatively high shear-rate region has behavior close to that of ideal thixotropy, i.e., purely viscous, though time-dependent. We presented an extensive data set including results of steady-state rate sweep tests, step-shear-rate tests, step-stress tests, shear- rate-ramp tests and stress-ramp tests. A method for correcting for inertia effects in stress- controlled tests is also provided. The suspension showed stretched exponential relaxation of stress (or shear rate) in step tests and the relaxation rate is dependent on the initial value of shear rate or stress. The suspension also exhibited non-monotonic transients of stress in constant-shear- rate tests with some special flow histories. We demonstrated that stretched exponential relaxation and the non-monotonic transients can be modeled simply by multiple structure parameters and
67 that the relaxation time’s dependence on the initial shear rate or stress in step tests is captured by a nonlinear break-down term in the kinetic equation for λ. We further showed that, if λ is linearly related to viscosity (or stress), then the power-law exponent on λ in the break-down term is three for the RC Model and two for the SC Model. We also showed that the order (or power-law exponent) of λ in the build-up term is unity and that the build-up rate is approximately proportional to the square root of the final shear rate or stress.
After determining the model parameters, we tested the two versions of the model against the experimental results. Both models give good predictions of the experimental data in the relatively high shear rate (or stress) region, where viscoelastic effects are small. The prediction of the SC Model is the better of the two, however. The models give poor predictions at low shear rates (or stresses) due to the omission of viscoelastic elements in the model. Existing methods of introducing viscoelastic effects can be incorporated into our models, however, to extend their range of validity. We also showed that the multi-lambda approach can be applied to the model of
Coussot et al. [101] to predict the non-monotonic time evolution of shear rate in multiple-step stress-jump tests in a Bentonite suspension.
68 Chapter 4 A Comprehensive Thixotropic Elasto-viscoplastic Model2
4.1 Introduction
Thixotropic materials show complex rheology due to the coexistence of thixotropy, viscoelasticity, and plasticity. The term “thixotropic elasto-viscoplastic” (TEVP) is often used to describe such complex rheological behavior. TEVP rheology often involve simultaneously different timescales of the thixotropic, viscoelastic, and plastic evolutions, exhibiting distinct rheological behaviors depending on the test conditions (e.g. amplitudes of strain, strain rate, and stress). Ideal thixotropy models, such as our ML model presented in Chapter 3, lack the applicability in capturing the complex TEVP rheology. For example, the ML model successfully captures the nonlinear and stretched-exponential thixotropic evolution but it lacks the component of viscoelasticity and therefore gives poor predictions in transient tests when the shear rate is small. The recent isotropic kinematic hardening (IKH) model [27] shows good agreement with the rheology of waxy crude oils in several hysteresis tests, shear start-up tests, and LAOS tests.
However, a comprehensive examination of the IKH model in other rheological experiments is lacking. The recent work of Armstrong et. al. [41] presents the modified Delaware thixotropic
(MDT) models which was examined in a range of step-shear-rate tests, flow reversal tests, and
LAOS tests. It successfully captures several important features of TEVP rheology including elastic stress contribution in steady state, thixotropic buildup and breakdown, and viscoelasticity
This chapter is based on ref. [Wei, Solomon, Larson. J. Rheol. 62(1). 321–342 (2018)].
69 in LAOS tests. But it lacks the feature of multiple thixotropic time scales, which is important to accurately predict the thixotropic response; it also shows poor predictions in flow reversal tests and some LAOS tests.
Still lacking, therefore, is a comprehensive model that can give quantitative predictions over wide-ranging test conditions. Developing such a robust model would be important in applications that may involve all the above-mentioned or more complex flow conditions.
Another limitation of the available thixotropic models is that they are written mainly in scalar forms and are therefore directly applicable only to simple shear flows. The relatively few [27,
104-106] that have been written in tensorial form have for the most part not been tested against experimental results. Thus, a well-tested, quantitative tensorial constitutive model for thixotropic fluids is currently lacking.
The purpose of this chapter is to fill this gap. We generalize the thixotropic-only ML model presented in Chapter 3 and combine it with the IKH model. We call this new model the
ML-IKH model. We present the model in both a scalar and a tensorial form, the latter of which is in Eulerian form, frame-invariant, and compliant with the second law of thermodynamics. We outline the procedure for model parameterization. And we test the proposed model, in comparison with the ML, IKH, and MDT models, over a wide range of simple shearing flows, including steady state shear, unidirectional step tests, intermittent step tests, flow reversal tests, and LAOS tests. We discuss the model components that are critical to successful prediction of those rheological tests. We also provide a tensorial formulation of the ML-IKH model that is frame-invariant, obeys the second law of thermodynamics, and can reproduce the predictions of the scalar version.
70 4.2 Proposed Model
TEVP rheology involves four important aspects, namely: 1) multiple thixotropic structure parameters, 2) nonlinear thixotropic evolution equations, 3) viscoelastic model components, and
4) a yield stress with kinematic hardening. In this section, we introduce the ML-IKH model which includes these four aspects. (In comparison, the ML model includes only aspects 1 and 2; the IKH model contains only aspects 3 and 4; and the MDT model only has aspect 3. The ML-
IKH model generalizes the ML model by amalgamating it with the IKH model [27], and making some additional modifications, as discussed shortly. We propose the model in both scalar and tensorial forms.
4.2.1 Scalar Form
The total stress is composed of contributions from the medium (σm) and from internal structures (σs):
m s . (4-1)
In this work, we consider only thixotropic suspensions with a Newtonian continuous phase.
Assuming that the system has a constant viscosity of ηm when all internal structures are broken, we write σm as follows:
m m . (4-2)
As a common assumption in elasto-viscoplastic models [27, 41, 107], the total shear strain is additively decomposed into the elastic strain (γe) and plastic strain (γp):
e p e p . (4-3)
Here, γe represents the elastic deformation of the internal structures and γp represents the plastic distortion caused by, for example, the relative movement of internal structures with respect to the
71 medium. ̇ and ̇ are, respectively, the elastic and plastic strain rate. We assume linear elasticity and a constant modulus, taking σs is proportional to γe:
s G e . (4-4)
Next, we introduce the equations determining ̇ . ̇ can be written as the product of its sign
(np) and its absolute value ( ̇ ):
p n p p . (4-5)
np indicates the direction of the plastic deformation while ̇ is the magnitude of its rate. Both np and ̇ are determined by σeff which is defined as
eff s back . (4-6)
As is common in plasticity theory [64], we assume that the plastic deformation is codirectional with the effective stress, i.e.
npsign eff . (4-7)
̇ is determined as follows:
0 if eff k y , p eff k y (4-8) if eff k y , thi where ηthi is the plastic viscosity.
The model can be written in a compact form as follows:
k max 0,eff y . s eff thi (4-9) eff where = ⁄ is the viscoelastic relaxation time in the fully structured state. Notice that when = 0, = 0, and is removed from this model (by setting = 1), this model reduces
72 to the Jeffrey model; when = 0, = 0, and is absent, this model reduces to the Bingham model.
We next introduce the evolution equations for and to complete this model. We assume that follows the Armstrong-Frederick kinematic hardening rule [65]:
Ap qA p . (4-10)
Following the ML approach, is expressed as a linear combination of :
N Ci i , (4-11) i1 where Ci is the coefficient λi and N is the number of thixotropic relaxation modes. The kinetic equation for each λi follows
d m m i D ka n k b 1 k 1 . (4-12) dt i1 i 2 i 3 i
It has been discussed in Chapter 3 that the thixotropic build-up rate is linear in (m = 1) if a linear relationship between and viscosity or yield stress is adopted. In addition, the flow- induced aggregation rate has a square-root dependence (b = 0.5) on the shear rate or stress. This relationship is first proposed by van de Ven and Mason [100] in their theoretical study on the aggregation kinetics of colloidal dispersions at low Peclet numbers. The build-up terms with m =
1 and b = 0.5 were first adopted by Dullaert and Mewis [11] in their structural kinetics model.
The parameters a and n in the break-down term of Eq. (4-12) are two adjusting model parameters. The nonlinearity of λi in the breakdown term captures the positive correlation between the thixotropic relaxation times and the initial shear rate or stress in step tests, which improves model predictions in these tests. The best-fit value of n depends on the choice of ϕ (i.e., stress or shear rate) and can be experimentally determined from the fast response in shear start- up tests at large shear rates or stress, as discussed in Chapter 3. The previous ML model assumes
73 Eq. (4-12) either takes a stress-controlled (SC) form (ϕ = ) or a rate-controlled (RC) one (ϕ =
̇). In the ML-IKH model, with the assumption that thixotropy occurs only upon plastic deformation, ϕ therefore is written as follows:
SC form: max 0, eff k y , (4-13) RC form: . p
Prior to yielding, ̇ = 0 and | | ≤ . ϕ therefore equals zero in both the SC and RC forms, and thus the break-down terms in Eq. (4-12) disappear.
Note that the RC and SC forms are two similar but different models. They make different constitutive assumptions about how flow conditions influence the thixotropic kinetics. We refer the readers to ref. [108] for a conceptual discussion about the choice of flow parameters, and ref.
[109] for a numerical comparison between different choices. A third option b ased on energy consideration has also been adopted [52, 110].
Figure 4-1. Mechanical analog of the proposed model. A mechanical analog can be drawn for the proposed model, as shown in Figure 4-1. Prior to yielding, ̇ = 0 and this model behaves as a Kelvin-Voigt viscoelastic solid. Upon yielding, the model predicts a Jeffreys-type viscoelastic behavior with a yield stress in unidirectional shearing given by + and in the opposite direction of − + . As λ decreases, the
74 yield surface shrinks and the viscoelastic relaxation time ( ⁄ ) decreases. In limit of ̇ =
+∞ and = 0, this model reduces to a Newtonian fluid with a viscosity of .
4.2.2 Tensorial Form
In this section, we present the tensorial formulation of the proposed model that can be used in arbitrary 3D flow histories. We write the model in the Eulerian form based on an additive decomposition of the rate of deformation tensor. Compared with the Lagrangian form that is based on the multiplicative decomposition of the deformation gradient tensor, and therefore requires the definition of a fixed reference state with respect to which total strain is defined, the
Eulerian form is more suitable for most fluid flow simulations of thixotropic materials. In addition, we assume that the elastic deformation is small while the plastic deformation and rotation might be large. This assumption greatly simplifies the tensorial constitutive equations and is valid for weakly-aggregated thixotropic suspensions in which the internal structures can withstand only small elastic strains before breaking.
As in Eq. (4-1), the total extra stress tensor (τ) is divided into contributions from the medium (τm) and from the internal structure (τs), with τm proportional to the rate of deformation tensor (D):
τ = τm τ s , (4-14)
τm 2 mD . (4-15)
Here, D is the sum of its elastic and plastic components (De and Dp):
DDDe p . (4-16)
Below, Eq. (4-17) expresses τs in a differential and frame-invariant form.
▽ (4-17) τs 2GDe
75 The symbol “▽” denotes the upper-convected time derivative. Its operation on an arbitrary tensor, S, is defined by Eq. (4-18).
▽ T (4-18) S= S v SS v , where the over-dot denotes the material time derivative, the superscript “T” means transpose, and ∇v is the velocity gradient tensor.
The effective stress tensor (τeff), the counterpart of in the 1D model, is defined as
τ effτ s κ back , (4-19) where is an intermediate variable that depends on the back stress tensor ( = ):
2 2 2 κ back= τ backτ back kh AA . (4-20) kh
Here, A is the 3D generalization of the scalar kinematic hardening parameter, A. is introduced to avoid the aphysical oscillation of A in simple shear flows (see details in Chapter
4.5). This method was first proposed by Tsakmakis [111] in his finite plasticity model.
Assuming the von Mises yield criterion, the 3D generalization of the plastic flow rule in the scalar form are written as follows:
0 if ky , D k (4-21) p y τef f if ky , 2thi where is the equivalent shear stress, defined as
τDD: τ eff eff . (4-22) 2
76 The superscript “D” means the deviatoric part of the tensor; e.g., = − trace( ) ⁄3, where is the unit tensor. The “:” denotes the scalar product (or double dot product) of two tensors.
The tensorial equivalent of Eq. (4-9) is written as
ky (4-23) τs+ max 0, τ eff 2thiD , where again = ⁄ is the viscoelastic relaxation time in the fully structured state. The evolution equation for A is given by Eq. (4-24):
ο (4-24) ADAADDAp + p + p qd p ,
A is symmetric and equals to the zero tensor in the initial undeformed state. This kinetic equation satisfies the second law of thermodynamics and does not bring aphysical oscillation in simple shear flows (see Chapter 4.5 for further discussion). The symbol “o” over A denotes the Jaumann derivative. Its operation on an arbitrary tensor, , is defined by Eq. (4-25):
ο SWSSW=S , (4-25)
Here, = ( − ( ) )⁄2 is the vorticity tensor. dp is the magnitude of Dp, defined by Eq.
(4-26):
DD: d p p . (4-26) p 2
The expression for and the kinetic equation for are the same as those in the scalar model except that the flow parameter ϕ is now written as a function of or dp, as shown in Eq. (4-27). ϕ is therefore a scalar isotropic function of τeff or Dp and therefore independent of an arbitrary rotation imposed on the frame.
77 SC form: max 0, ky , (4-27) RC form: 2dp .
4.2.3 Summary of the Proposed Model
Table 4-1. Summary of the proposed model Scalar constitutive equations Tensorial constitutive equations
m s τ =2m D τ s
eff k y ky smax 0, eff thi τs+max 0, τ eff 2thi D eff
eff s back τeffτ s κ back
2 back k h A κ back= τ back2τ backkh , τ back kh A
0 if eff k y 0 if ky D k p eff k y p y τeff signeff if eff k y if ky thi 2thi
ο Ap qA p A DAADDAp + p + p qd p
SC form: max 0, k eff y SC form: max 0, ky RC form: RC form: 2d p p
N Ci i i1
d i D ka n k 0.5 1 k 1 dt i1 i 2 i 3 i
N Di t expt Ci e i1 Model parameters (12 in total):
kh k 1 k 2 governing steady state: a , n ky , , , , m , thi q k3 k 3
controlling transient response: k3 , , q , G
Table 4-1 is a summary of the proposed model, which has 12 parameters. (In comparison, the thixotropic only ML model has 9 parameters, the IKH model has 8, and the MDT model has
78 13, cf. Table 4-3. Qualitative comparison of the ML, IKH (scalar form), MDT, and ML-IKH
(scalar form) models..) The parameters governing the relationship between stress and shear rate
at steady state are , , , , ⁄ , ⁄ , , and .
Table 4-2. The special values of model parameters for which the ML-IKH model reduces to closely-related models.
Model parameters = 1,2,3 − 1 (and is set to be unity) Oldroyd-B model = 0 = 0 = 0 - Saramito’s model [107] = 0 = 0 - Upper-Convected Maxwell = 0 = 0 = 0 - = 0 model ML model in [112] = 0 = 0 IKH model* = 0 = 0 Mono-lambda ideal = 0 = 0 = 0 = 0 thixotropic models Bingham model = 0 = 0 = 0 - Newtonian fluid = 0 = 0 = 0 = 0 - no no back no yield no thixotropic model single Implication of zero values elasticity stress stress component * With additional requirements: = 0, = 1, = 1, = ̇ , replacing with , and using a different evolution equation for A or A (see detailed comparison in the Chapter 4.4).
The scalar model predicts that the values of , , and at steady state are as follows:
sign A (4-28) ss q
k1 k 2 ss f ,,,, a n (4-29) k3 k 3
kh ss sign ssk y ss thi m (4-30) q
The function f in Eq. (4-29) represents the solution of Eq. (4-12) at steady state where
⁄ = 0 for all .
79 The parameters governing the transient response are , , , and . directly influences the time scale of thixotropic relaxation. determines the extent of deviation from single exponential relaxation. For = 1, the stretched-exponential reduces to a single exponential and only a single is needed, which reduces the model to a single-lambda model. determines the plastic strain at which reaches steady state. / governs the time scale of the viscoelastic relaxation. The model reduces to several different models in certain limits, as described in Table 4-2.
4.2.4 Physical Interpretation of Structure Parameters
In the ML-IKH model, { }, A, and γe account phenomenologically for, respectively, the instantaneous degree of structures (in the isotropic sense), characteristic orientation, and elastic deformation of the internal structures. In single-lambda thixotropic models, the structure parameter is often interpreted as the instantaneous number of links (or other structural units) normalized by the maximum number in a fully-structured state [48, 113]. As a first approximation, it is assumed that all links have the same strength and break or form at the same rate. Most thixotropic models assume a first-order mechanism of the breakage, which results in a linear kinetic equation for λ. The use of a set of { } is a generalization of such description. We envision { } as the relative numbers of internal links of different types specified by their relative rates of breakage or formation (the Di values). The yield stress and viscosity a related to the linear combination of λi with the weights following a stretched exponential distribution.
As discussed in references [27, 114], A quantifies the level of structural anisotropy, or characteristic orientation of internal structures, which gives rise to the back stress. A = 0 implies randomly orientated microstructures and A = 1/q gives the steady-state anisotropic orientation in simple shear flows. In 3D deformations, the scalar parameter A is replaced by the tensor A.
80 (Since our model accounts for multiple types of structures through the multiple λi values, it might make sense to include multiple A values, one corresponding to each type of structure, and possibly each having a different value of q. However, for simplicity, we forego this refinement here.) Shear-induced anisotropic microstructures of thixotropic fluids have been experimentally observed in Laponite suspensions [115], fumed silica suspensions [116], waxy crude oils [27], etc. A review is given by Vermant and Solomon [117]
The internal structures withstand a certain amount of elastic deformation before breaking, and the relaxation of this elastic deformation results in the viscoelastic response of thixotropic materials. γe can be interpreted as the characteristic elastic strain of internal structures. One 3D generalization of γe is the tensor strain ( − ) where is the elastic Finger tensor, defined as
= ∙ . Here is the elastic component of the deformation gradient (F) according to the
Kroner Decomposition [118], = ∙ , where is the plastic deformation gradient. A detailed discussion about this decomposition is provided in the Appendix.
4.3 Model Parameterization
The parameterization of the proposed model requires the following experimental data: the steady state flow curve, a flow reversal test with small shear rate, and several step-shear-rate tests with relatively small initial shear rates. For example, in this study, we use the experimental
data in Fig. 3 and the flow reversal test with γ̇ = 0.01 s in Fig. 6(a) to parameterize the model.
As discussed before, the model predicts a viscosity of ηm in the fully unstructured state and a yield stress of (kh/q+ ky) in the totally structured state. Therefore, ηm can be directly estimated from the high-shear-rate limit of the steady state flow curve and (kh/q+ ky) from the zero-shear- rate limit.
81 In a flow reversal test where the shearing direction instantaneously changes while the shear rate remains the same, the model predicts a two-step relaxation of shear stress. This two-step relaxation is caused by the evolution of γe, the viscoelastic relaxation, and that of A, the kinematic hardening relaxation. In the case where the shear rate is small and viscous stress is insignificant, the amount of stress that undergoes the kinematic hardening relaxation, as predicted by this model, equals to kh/q, which can be estimated experimentally. Next, the value of q, governing the plastic strain at which A reaches steady state in a flow reversal test, can be also determined in the same flow reversal test through best fit.
The remaining parameters are divided into two groups – one governing the steady state response, including a, n, ⁄ , ⁄ and ηthi; the other one governing the transient response, including k3, β, and G. Parameters in the first group are first determined from the best fit of the steady state flow curve and those in the second group from the best fit of several step-shear-rate tests (in this study, we use three step tests from 0.1 s-1, as shown in Fig. 3b). Iteration may be needed to adjust the boundaries of the parameters in the first group, particularly ⁄ , ⁄ and ηthi. Alternatively, the power-law exponent n in the shear-induced structure break-up term can be determined from the fast response in shear startup tests at large shear rates or stresses. In such tests, the viscous stress dominates, and the model predicts an apparent viscosity of
( + ). Therefore, the transients of λ can be estimated from the measured viscosity. In the very short period after the stepwise change of shear rate or stress, if = 1, the model predicts a linear relationship between ln and time; while if ≠ 1, a linear relationship is predicted between and time. The value of may be different for the SC and RC forms. More details about the estimation of are given in our previous work [112].
82 The { } and { } values as well as N, the number of “substructure parameters”, are
determined from the value of β. They are calculated to best approximate the stretched
exponential using a limited number of mono-exponentials. More details are provided in Chapter
3.3.
Table 4-3. Qualitative comparison of the ML, IKH (scalar form), MDT, and ML-IKH (scalar form) models. ML IKH MDT ML-IKH Number of parameters 9 8 13 12 Linear Linear elastic Pre-yield response Rigid solid Linear elastic solid viscoelastic solid solid Response in fully- Newtonian Maxwell Newtonian Newtonian fluid unstructured state fluid fluid fluid Modified Armstrong- Kinematic hardening None Armstrong- None Frederick Frederick Prediction of delayed no no no no yielding Predictions in step-rate tests: (1) instantaneous stress yes no yes yes jump (2) fast elastic relaxation no yes no yes with an initial extremum (3) stretched-exponential yes no no yes thixotropic relaxation Summary of the IKH model m e p, G e , A p q A sign A p , k 1 1 k 3 p pmax 0,sign k h A k h A k y p
Model parameters: G , q , m , k1 , k 3 , k h , k y , p Summary of the MDT model m e p, e p e p max , max min CO ,1 , CO y 0G 0
a d G k G G, k tˆ 1 1 tˆ , fG f 0 Brown r 1 p r 2 p
n n GKK 2 1 sign f e ST p p p ˆ ˆ Model parameters: m , G0 , y 0 , k G , k Brown , t r 1 , t r 2 , a ,d,,,, n1 n 2 K ST K
83 4.4 Model Evaluation
4.4.1 Qualitative comparison with the ML, IKH, and MDT models
We first qualitatively compare the ML-IKH model with several representative models – the ML, IKH, and MDT models. As reviewed in Chapter 1, TEVP materials behave like viscoelastic solids before yielding and viscoelastic fluids after yielding. Moreover, their response approaches ideal thixotropy at high shear rates (where the required shear rate level may differ in different systems. For the fumed silica suspensions reported in [14, 41, 112], ̇ > 5 s ). The thixotropic relaxation has multiple time scales and follows a stretched exponential in step tests.
In general step-shear-rate tests, the stress exhibits an instant jump, followed by a fast elastic relaxation with an initial extremum and a subsequent slow thixotropic relaxation. The ML, IKH, and MDT models predict only partially these rheological behaviors. All four models fail to predict the delayed yielding behavior that some TEVP materials exhibit. We refer the readers to
[51, 119-121] for models that address the delayed yielding phenomenon.
Next, we test the proposed model, in both SC and RC forms, against experimental results with shear histories including steady state, step shear rate, step stress, intermittent step shear rate, flow reversal, and LAOS. We also investigate the predictions of the ML, IKH, and MDT models for comparison. There are other models in the literature that fall into the category of TEVP models. One important example is the “unified-approach” (UA) model of de Souza Mendes and
Thompson [51]. It has been shown that the UA model qualitatively captures the TEVP rheology of the fumed silica suspension but lacks quantitative agreement [41]. For brevity, we do not include it in this work.
We discuss the critical model components and corresponding improvements in predicting each type of tests. The LAOS data is adopted from the work of Armstrong et al. [41] and all
84 other data are from this thesis. Cone and plate geometries are used in all rheological experiments.
Both data sets are from fumed silica suspensions (a hydrophobic fumed silica, Aerosil® R972, is
dispersed in a blend of paraffin oil and low-molecular-weight polymer). The two fumed silica
suspensions were synthesized according to the same formula reported in reference [12] except
that they contain different polymer ingredients.
Note that in this section the original IKH model (summarized in Table 4-3) is modified
by replacing the constant plastic viscosity, , with + . This modification is necessary
because the original IKH model predicts that thixotropic response vanishes when the viscous
stress dominates, which does not agree with the rheology of the fumed silica suspension [13, 14,
41, 112] (e.g., the original IKH model would predict a nearly constant viscosity of in the tests
of Fig. 4, contradicting the experimental results). This modification introduces one additional
parameter. In the following discussion, we refer to this slightly modified model as IKH-V model.
Table 4-4. Parameter values of the ML-IKH, ML(SC), IKH-V, and MDT models for the prediction of the rheological response of the 2.9 vol% fumed silica suspension. Parameters SC form RC form ML(SC) IKH-V model MDT model
0.434 Pa 0.458 Pa 0.68 Pa 0.51 Pa = 0.8 Pa 0.195 Pa 0.205 Pa 0 0.22 Pa = 0.41 Pa ∙ s 1 1 0 1 = 1 60 Pa 70 Pa +∞ 100 Pa = 100 Pa 2.325 Pa ∙ s 1.840 Pa ∙ s 1.74 Pa ∙ s 1.29 Pa ∙ s = 0.52 = 1 0.430 Pa ∙ s 0.425 Pa ∙ s 0.45 Pa ∙ s 0.56 Pa ∙ s = 1.76 Pa ∙ s 0.65 0.65 0.65 = 0.7 2.537 1.758 1.9 = 0.3 2 3 2 = 8.7 × 10 s ̂ = 3.42 s 0.042 Pa s 0.105 s 0.043 Pa s 6.5 × 10 ̂ = 0.18 s . . . 3 × 10 Pa s 3 × 10 s 0.0104 Pa s 7.4 × 10 s = 8.2 × 10 s 6 × 10 s 1.6 × 10 s 0.01 s = 1
4.4.2 Steady State Flow Curve
85 Figure 4-2 (a) shows the steady state flow curve of the fumed silica suspension reported
in ref. [112] and the fitting results of the ML-IKH model in both SC and RC forms, the IKH-V,
MDT, and ML (in its SC form) models. The steady state flow curve can be fitted accurately by
all models.
Table 4-4 lists the parameter values that generate the model calculations. The MDT
model is parameterized from the best fit of the experimental data in Figure 4-2. Besides these
data, the parameterization of the IKH-V and ML-IKH models also requires the experimental
results of the flow reversal test with a shear rate of 0.01 s-1 in Figure 4-6.
Figure 4-2. Experimental data used to estimate model parameters (except for and in the proposed model and the IKH-V model) and model fits (lines, using parameter values in Table 4-4): (a) steady state flow curve, including data measured on a stress-controlled rheometer, AR-G2, and a strain-controlled rheometer, ARES; (b) Step-shear- rate tests from a shear rate of 0.1 s-1 to different final shear rates. 4.4.3 Step Tests
Step tests refer to the experiments where the shear rate or stress undergoes a stepwise
change after a period of steady shear. Figure 4-2(b) shows the experimental results and model
fits in three step-shear-rate tests from 0.1 s-1 to different final shear rates. The stress exhibits a
fast elastic relaxation with an initial maximum followed by a gradual thixotropic relaxation. The
thixotropic response follows a stretched-exponential relaxation. The ML(SC) and ML-IKH
models successfully capture this behavior through the ML approach. The IKH-V model and
86 MDT model, however, always show a single exponential thixotropic relaxation, which results in
the sharper transition of stress transients than the experimental results. The IKH-V and ML-IKH
models fit the initial increase and maximum of stress while the ML(SC) and MDT models do
not. The SC form of the ML-IKH model yields slightly better fitting results than does the RC
form.
Figure 4-3. Model predictions of (a) a step-shear-rate test (from 3 s-1 to 9 s-1) and (b) two step-stress tests (from 3, 9 Pa to 7 Pa). Experimental data are adopted from the reference [112]. These are independent results that have not been used to parameterize the models. The rapid change at early times (time < 0.1 s) in (b) are caused by the inertial response of the rheometer, which is explicitly included in the calculated response as explained in Chapter 2.3.3. In step tests with relatively large initial shear rates or stress, the fast elastic response is
insignificant, and the test material behaves like an ideal thixotropic fluid. While the stress in
step-shear-rate tests shows an instantaneous jump, the measured viscosity changes continuously.
In this region of ideal thixotropy, the thixotropic-only ML model and the ML-IKH model have a
similar level of agreement with experimental results, as shown in Figure 4-3. (The SC and RC
forms of the ML-IKH model have very similar predictions, as shown in Fig. A1. For brevity, in
Figure 4-3, we show the predictions of the SC form only.)
Figure 4-3(a) shows the model predictions in a step-shear-rate test from 3 to 9 s-1.
Compared with their results in Figure 4-2(b), the IKH-V model and MDT model show better
predictions in this test. The MDT model agrees with experimental results well although it still
87 shows a mono-exponential stress relaxation. The IKH-V model can only qualitatively capture the thixotropic response. The ML and ML-IKH models slightly overpredict the stress but it agrees with the experimental results better than do the other two models, due to the stretched- exponential relaxation in it.
The results of two step-stress tests (from 3 Pa and 9 Pa to 7 Pa) are given in Figure
4-3(b). Instrument inertia causes the steep response for times less than 0.1 seconds.
Compensation for the inertia effects is incorporated in the model calculations follwing the approach described in Chapter 2.3.3. Both the IKH-V and MDT models show significant deviations from the experimental data while the ML and ML-IKH models give more accurate predictions.
Next, we test the models in step tests over a range of test conditions. Figure 4-4 (a-b) plot the model predictions in 9 step-shear-rate tests. The RC and SC forms of the ML-IKH model give similar predictions that are both in quantitative agreement with the experimental data, as shown in Figure 4-4(a). The MDT model predicts those tests well but tends to predict narrower stress transients than the experimental results. The IKH model provides only qualitatively correct predictions. Figure 4-4(c-d) plot the model predictions of 12 step-stress tests. The ML-IKH model accurately predicts those step-stress tests while the other two models show only qualitative agreement with experimental data.
Figure 4-4, along with Figure 4-2(b) and Figure 4-3, shows that the ML-IKH model can quantitatively predict a wide range of step tests while the MDT model and IKH-V model fail to predict as accurately over this range. In the ideal thixotropy region, multiple thixotropic structure parameters (corresponding model parameter: β) and their nonlinear kinetic equations (a, n) are two key factors needed to predict accurately these step tests. For step tests with small initial
88 shear rates, viscoelasticity (G) is important in capturing the fast elastic response. The kinematic
hardening component (kh, q, m) does not contribute to the transient response because the shearing
direction remains constant.
Figure 4-4. Model predictions for step tests over wide-ranging test conditions: (a) step-shear-rate tests and the predictions of the ML-IKH model in RC (dashed lines) and SC forms (solid lines); (b) the same experimental data as in (a) with predictions of the IKH model (dashed lines) and MDT model (solid lines); (c) step-stress tests (σo, initial stress; σt, final stress) with the predictions of the ML-IKH model; (d) the same experimental data as in (c) with predictions of the IKH and MDT models. The initial steep response in the step-stress tests is due to inertial effects, which are incorporated into the model predictions. 4.4.4 Intermittent Step Tests
89
Figure 4-5. Experimental results (symbols) and model predictions (lines in b-f) of a group of intermittent step tests: (a) shear rate histories. The sample is pre-sheared at 100 s–1. The shear rate then steps down to 0.5 s–1 and is held constant for ∆t. At t = 0, shear rate steps up to 5 s–1 until the stress reaches steady state. The legend is for the experimental data in (b-f), which also show the predictions of respectively the IKH-V, MDT, ML-IKH (SC, = 0.65), ML-IKH (SC, = 1), and ML (SC, = 0.65) models. σ(t) denotes shear stress and σ∞ denotes the steady- state value. The subplots in (b-f) show σ(t)/σ∞ vs. time in the test with ∆t = 20 s. The dotted oval encloses the region of non-monotonic thixotropic relaxation. Intermittent step tests provide more insights about the thixotropic phenomenon. A single
jump of the shear rate or stress can be used to examine the viscosity evolution from one steady
state to another one. Intermittent step tests consist of multiple jumps of the shear rate. Figure 4-5
(a) shows such example. The shear rate first jumps from a steady shear at 100 s–1 to 0.5 s–1, after
which the stress evolves towards the steady-state value at 0.5 s–1. Before the stress reaches the
new steady-state value, a second jump of the shear rate is executed. With this test procedure, we
can examine stress evolution starting from a non-equilibrium state.
Figure 4-5 (b-f) present the experimental data and model predictions in a set of
intermittent shear tests following the procedure as shown in Figure 4-5 (a). The stress transients
90 for time > 0 are sensitive to the value of ∆t up to about 10 minutes. The stress evolves monotonically when ∆t is small or large while for some intermediate values of ∆t, non- monotonic thixotropic evolution (stress transients for time > 0.1 s) occurs – stress first decreases, then builds up, and eventually reaches steady state.
Such non-monotonic thixotropic relaxations should be distinguished from the elastic response, which has a much shorter time scale (< 0.1 s). The ML Approach successfully captures this behavior. The rate of change of each λi is proportional to the pre-factor Di. In the test of Fig.
-1 5(a), initially all λi have the same steady state values at 100 s . After the shear rate steps down to
-1 0.5 s , the { } build up at different rates. For intermediate ∆t values, the λi values with large Di
–1 build up enough that they then decrease as the shear rate steps to 5 s . The λi with small Di, however, have not re-built much and so continue to increase after the stepwise change of shear rate. Consequently, in the period of t > 0, the stress first decreases and then increases.
The IKH-V and MDT models incorporate a single structure parameter for thixotropic evolution and therefore show poor agreement with experimental results, as shown in Figure 4-5
(b, c). Figure 4-5(d) shows that the ML-IKH model captures both the non-monotonic thixotropic evolution and the dependence of the stress transients on ∆t. (For brevity, we show only the predictions of SC form. The RC form has similar level of agreement). In Figure 4-5(e) we show the predictions of the ML-IKH model with the same parameters values except that β is set to unity, so that N = Di = Ci = 1, and the ML-IKH model reduces to a single-lambda model. Its predictions in this limit, like those of the IKH-V and MDT models, show a monotonic thixotropic relaxation when the shear rate is constant. And the overall predictions are worse than those in Figure 4-5(c). Note that the initial elastic response is only slightly influenced by the value of β. The elastic response might also lead to non-monotonic stress transients but the elastic
91 response has a much smaller time scale than the thixotropic relaxation. Figure 4-5(f) shows the predictions of the thixotropic-only ML model. The ML and ML-IKH models have a similar level of agreement with experimental results, which indicates that the ML approach is the dominating model component in predicting the test of Figure 4-5(a). For shear histories that involve further smaller shear rates than 0.5 s–1 the differences between ML and ML-IKH model predictions increase. The latter gives better predictions because viscoelastic evolution is significant when low shear rates are involved and the ML model lacks the component accounting for viscoelasticity.
In summary, the ML Approach (corresponding model parameter, β), and the viscoelastic model component (G) if low shear rates are involved, are critical to qualitatively predict intermittent step tests. The ML Approach successfully predicts the non-monotonic thixotropic evolution. The effects of nonlinear thixotropic evolution equation (a, n) cannot be clearly seen here. The kinematic hardening component (kh, q, m) does not contribute to the transient response because the shearing direction remains the same.
4.4.5 Flow Reversal Tests
Flow reversal tests are performed to examine the shear-induced anisotropic rheological response. In a flow reversal test, after an attainment of a steady state under shearing, an instantaneous reversal of the shear direction is imposed, with the magnitude of the shear rate held fixed.
For a Newtonian fluid in a flow reversal test, the sign of the shear stress instantaneously changes while its magnitude remains constant. A Maxwell fluid, on the other hand, predicts that the stress after the reversal continuously evolves to its opposite value following an exponential function with a constant characteristic time. For some colloidal suspensions that form anisotropic
92 structures under shear, the stress following flow reversal exhibits a gradual relaxation that scales
with strain [87, 88]. TEVP fluids, e.g., the thixotropic fumed silica suspension in this study,
however, show different rheological behaviors, as shown in Figure 4-6.
Figure 4-6. Experimental results (a), model fits (black lines) and predictions (lines in color) (b) of flow reversal tests with shear rates indicated in the legend. The subplot in (a) shows the shear rate histories. σ(t) and σ∞ denote respectively the shear stress and steady-state value of stress. The strain is defined as = ̇ . The vertical green line indicates the position where γ = 3. The low-shear-rate tests reach steady state at γ ≈ 3. The predictions in (b) are only shown over the range of strain for which experimental data are available. kh, q, and m (in the IKH-V model) are estimated in the test with ̇ = 0.01 s . The IKH-V and ML-IKH (in SC form) models give similar predictions in all tests (for simplicity, only two typical predictions of the IKH-V model are shown here). The ML model (in SC form) predicts no transient response in all tests. In Figure 4-6 we compare the experimental results with model fits (for ̇ = 0.01 s ) or
independent predictions (for all ̇ > 0.01 s ) in several flow reversal tests, each identified by a
specific color of symbols and lines. Figure 4-6 (a) shows experimental data plotted as reduced
stress versus strain in flow reversal tests with shear rates ranging from 0.01 s-1 to 10 s-1. Similar
results have been reported in reference [89]. In flow reversal tests with shear rates of 0.1 s-1 or
lower, the reduced stress can be scaled with the strain to yield a master curve. The stress
transients follow a two-step relaxation that takes approximately three strain units to reach the
plateau at steady state. As the shear rate increases above 0.1 s-1, the initial steep response is no
longer detectable and the reduced stress transients deviate from the master curve.
93 Figure 4-6 (b) shows the model fits and predictions corresponding to the experimental
results of Figure 4-6 (a). The flow reversal test with ̇ = 0.01 s is used to fix the values of kh, q, and m. The optimal values of q and m are both unity in the IKH-V and ML-IKH models. (The
RC and SC forms give similar predictions. For brevity, we show only the predictions of the SC form.) The two models yield similar results in all the flow reversal tests of Figure 4-6(a), both predicting the two-step relaxation in good agreement with experimental data. In Figure 4-6(b), for clarity, we show only two typical predictions of the IKH-V model.
Since the absolute value of the shear rate remains constant in a flow reversal test, the predicted thixotropic effect is insignificant. The transient response results mainly from the viscoelastic relaxation and the plastic evolution (for example, kinematic hardening effects). In the IKH-V and ML-IKH models, the evolution of γe, the viscoelastic relaxation, captures the initial steep response; and the transients in A, which describe the kinematic hardening effects, allows both models to predict well the subsequent gradual stress response that scales with strain.
The two models predict the rise of the initial plateau as shear rate increases. But they over- predict the initial plateau and under-predict the amount of strain it takes for the stress to reach steady state. The discrepancies are significant for large shear rates, possibly because the viscosity is also anisotropic and evolves in reversing flows with a smaller time (or strain) scale than that of the back-stress evolution. The MDT model captures only the elastic effects and therefore has poor predictions in flow reversal tests. The ML model lacks components of both viscoelasticity and kinematic hardening. It, therefore, predicts no transient response in flow reversal tests.
4.4.6 Large Amplitude Oscillatory Shear
In this section, we test the ML, ML-IKH, IKH-V, and MDT models in LAOS tests where both the shearing direction and the shear rate change periodically. The strain history of a large
94 amplitude oscillatory shear (LAOS) test is a sinusoidal wave with a frequency of ω and amplitude of :
t 0 sin t . (4-31)
The experimental data are originally reported in ref. [41]. The test fluid is a fumed silica suspension and the rheological tests were performed on an ARES-G2 rheometer. To parameterize the ML, ML-IKH, and IKH-V models, we used the following experimental data
[41]: a steady state flow curve, step-shear-rate tests with initial shear rate of 5 s-1 and 0.1 s-1, and a flow reversal test (–1 to 1 s–1). The predictions of the MDT model are calculated using the original parameter values given in [41]. Table V lists all parameter values. For this set of experimental data, the RC forms of the ML and ML-IKH models have better agreement with experimental data than the SC forms. For brevity, in Figure 4-7 and Figure 4-8, we show predictions of only the RC forms of the ML and ML-IKH models.
The test fluid shows complex rheological behaviors in LAOS tests due to the coexistence of thixotropic, viscoelastic, and plastic response. Their prominence vary depending on the values of and . Figure 4-7 shows the results of a LAOS test with = 0.1 rad/s and = 10. This
test has a Deborah number of = ⁄ ≈ 3 × 10 , at which the elastic effects are insignificant. We chose this test condition for detailed discussion because it is suitable to examine the model components accounting for thixotropy and kinematic hardening. We compare the models in more test conditions in Fig. 8.
95
Figure 4-7. The elastic (a) and viscous (b) projections of the Lissajous–Bowditch curve for = 0.1 rad/s and = 10. The subplot in (a) shows the strain histories with four points marked out. Experimental data and model predictions are both linearly rescaled from –1 to 1. The experimental data (black lines) are adopted from [41]. Panel (i) of (b) shows the complete experimental data. Panels (ii-iv) show only half of the symmetric LAOS results for clarity. (c-d) The response of A, predicted by the ML-IKH model, plotted against strain and shear rate, respectively. (e-f) the response of λ, predicted the ML-IKH model, plotted against strain and shear rate, respectively. Figure 4-7(a,b) plot respectively the elastic (reduced stress vs. strain) and viscous
(reduced stress vs. shear rate) projection of the Lissajous–Bowditch curve. Experimental data
and model predictions are both normalized into the range –1 to 1 to better show the level of
qualitative agreement. The quantitative discrepancies are shown in Fig. 8. Four points in the
strain histories are marked out for the following discussion.
Because De is small, the transient response results mainly from kinematic hardening and
thixotropy. Kinematic hardening leads to the stress transients within around three strain units
after the shearing direction changes, as seen in Figure 4-7(a). The thixotropic effects can be seen
in Figure 4-7(b): from the point a to b, the slope increases as the shear rate decreases to zero,
96 which indicates a build-up of the viscosity; and the slope decreases again for the strain histories from the point b to c, as the shear rate reaches its negative maximum. A hysteresis loop is formed because the evolution of internal structures lags behind the change of the shear rate. The ML and
MDT models lack the component of kinematic hardening. They, therefore, yield poor agreement with the experimental data in Figure 4-7(b). The original IKH model includes kinematic hardening but it fails to quantitatively capture the thixotropic response due to the assumption of a constant plastic viscosity. In comparison, the IKH-V and ML-IKH models capture both kinematic hardening (through the evolution of A) and thixotropy (through the response of λ) and therefore give better predictions than the other three models.
The model predictions of the LAOS tests for additional test conditions are presented in
Figure 4-8. The IKH-V and ML-IKH models have very similar predictions in the LAOS tests although they show distinct results in step tests. MDT, IKH-V, and ML-IKH models all have accurate predictions when is large (in the ideal thixotropy region) but fail to predict experimental results, even qualitatively, when = 0.1. The models fail because they over- simplify the complex viscoelastic response. The test fluid shows complex nonlinear viscoelastic response with multiple time scales while the models assume linear viscoelasticity with a single viscoelastic relaxation time. As increases, the model predictions improve. In the tests for =
1 and 10, the kinematic hardening effect is significant. The IKH-V and ML-IKH models capture this response and show better agreement with experimental results than the MDT model does.
97 (a) (b) IKH-V MDT 1000 ML-IKH Exp. data
100
10
1
0. 1
0.01 0. 1 1 10 0.01 0. 1 1 10 rad/s Figure 4-8. Pipkin diagram: (a) elastic projection in which reduced stress is plotted against strain; (b) viscous projection, in which reduced stress is plotted against strain rate. Experimental data (black lines) are adopted from [41]. Model predictions are calculated using the parameter values in Table 4-5. Model predictions and experimental data are normalized by the maximum of the latter. This figure shows the quantitative and qualitative discrepancies between the model predictions and experimental results. The dashed boxes contain the results presented in Figure 4-7(a,b). Table 4-5. Parameters of the ML-IKH, ML, IKH-V, and MDT models for the prediction of the LAOS results in reference [41]. Those parameters values are used in Figs. (7-8). Parameters ML-IKH (RC form) ML (RC form) IKH-V MDT
7.66 Pa 10.96 Pa = 11 Pa 3.30 Pa 0 = 1.17 Pa ∙ s 1 0 = 1 700 Pa +∞ = 7.88 Pa = 450 Pa 20.0 Pa ∙ s 16.4 Pa ∙ s = 3.36 = −1.5 = 1 = 0.81 1.22 Pa ∙ s 1.33 Pa ∙ s 0. 5 0.5 = 1 = 11.21 Pa ∙ s 1.29 1.45 = 700 Pa = 1.53 = 16.32 Pa ∙ s = 0.63 1 1 = 1.32 Pa ∙ s = 0.28 s 0.496 s 0.24 s . . = 1.18 ̂ = 0.77 s 0.178 s 0.2 s = 0.36 s ̂ = 2.0 s = 0.09 s 0.2 s 0.1 s Values are adopted from [41].
4.4.7 Summary
98 Table 4-6. Evaluation of the MDT, ML, IKH-V, and ML-IKH models in predicting the simple shear rheology of a model thixotropic fluid MDT ML IKH-V ML-IKH Remarks The original IKH model is modified by adding Number of parameters 13 9 9 12 one model parameter, as explained in the beginning of Sec. V. Steady state flow curve E E E E
Step tests with small MDT fails to predict the initial elastic M M M G shear rate or stress response. Step tests with large IKH-V and MDT fail to predict the stretched- M G M G shear rate or stress exponential thixotropic relaxation.
IKH-V and MDT give poor predictions for Intermittent step tests P M P G intermediate ∆
The IKH-V and ML-IKH models show good Flow reversal tests P P G G predictions for small shear rates.
LAOS tests (large The IKH-V and ML-IKH models have very M M G G ) similar predictions. LAOS tests (small For the same ω value, model predictions P P P P ) improve as increases. * E: excellent; G: good; M: moderate; P: poor.
Table 4-6 assesses the predictive ability of the MDT, ML, IKH-V, and ML-IKH models
for transient shear rheology of thixotropic fluids. The ML-IKH model has four key aspects that
result in the quantitative predictions of the complex TEVP rheology. They are 1) a spectrum of
thixotropic structure parameters (corresponding model parameter: β), 2) nonlinear kinetic
equations for thixotropic evolution (a, n), 3) the Armstrong-Frederick kinematic hardening rule
( , q, m), and 4) the inclusion of viscoelasticity (G). Feature 1 generates stretched-exponential
thixotropic evolution in step tests and is critical in capturing the non-monotonic stress transients
in intermittent step tests. Feature 2 enables the model to quantitatively predict step tests with
various test conditions. Feature 3 is critical in predicting the transient response induced by the
change of shearing direction. Feature 4 is important in capturing the fast transient response
resulting from viscoelasticity. In applications where only some of the features are of interest or
importance, the model can be conveniently simplified by setting corresponding model
parameters to specific values, as summarized in Table 4-2.
99 4.5 Predictions of the Tensorial Model
4.5.1 Comparison with the Scalar Model
Here, we demonstrate that the proposed tensorial model can reproduce the quantitative predictions of the scalar model. According to Eq. (4-23), the viscoelastic relaxation time is
⁄ . In simple shear flows, an effective Weissenberg number (Wieff) can be defined as
Wi thi . (4-32) eff G
For infinitesimal Wieff, ≈ , and the flow parameter, ϕ, and consequently λ, are equal to those in the scalar model. Our model allows Wieff to remain small even for large shear rates because as shear rate increases, λ decreases. With some parameter values, e.g. n = 1 and a > 1.5 in the RC form, Wieff approaches zero for infinitely large ̇.
With the same parameter values and finite Wieff, the shear stress and back strain predicted by the tensorial model are not identical to those of the scalar model. Consider the case where θ =
0, for which Eqs. (4-23) and (4-24) reduce to
y0 max 0, τ eff 2 thi D , (4-33)
(4-34) ADA qdp .
In simple shear flows, the components of and are
0 0 AA11 12 0 τ eff 0 0 , A A 12 0 0 . (4-35) 0 0 0 0 0 0
is related to ̇ through
sig n ( ) y0 thi . (4-36)
Therefore,
100 12 back,12 sig n ( ) y0 thi m . (4-37)
where , = (1 + 2 ). The evolution of and follows Eq. (4-38):
A112 A 12 q A 11 , 2 (4-38) A12 q A 12. 2 2
In comparison, for the scalar model, ̇ = ̇ − ̇ and
kh A sig n ( ) y0 th i m . (4-39)
Figure 4-9. Comparison of the 1-2 component of the tensorial form (dash black lines) and the scalar model (solid red lines) in four shear-start-up tests: the evolution of (a) shear stress, (b) , (c) back stress, taken as , in the -1 tensorial model and in the scalar model. The shear rates in (i-iv) are respectively 0.1, 1, 10, 100 s . Model calculation is generated using the parameter values for the SC form in Table III except that and take different values, respectively, 0.3 Pa and 3, in the tensorial model, while for the scalar model, these values, taken from Table 4-4, are 0.195 Pa and 1, respectively.
With different values of kh and q, the tensorial evolution equation for A can reproduce the
predictions of its scalar equivalent. To show this, Figure 4-9 plots the predictions of the tensorial
and scalar models in shear-start-up tests, with model parameters for both taken from Table IV for
the SC form, except that for the tensorial model we take kh = 0.3 Pa and q = 3. The kh and q are
determined so that , and have close results in flow histories with small Wi.
Figure 4-9 shows that as the shear rate increases, , deviates from and exhibits
a maximum that increases with increasing shear rate. This discrepancy occurs because as shear
rate and therefore Wieff increase, Eq. (4-35) is no longer valid and the first term in Eq. (4-23) is
101 significant. Nevertheless, the stress difference resulting from this difference between , and
remains insignificant relative to the viscous stress (for example, in test iv, the difference between , and is less than 0.3 Pa while the total shear stress is about 50 Pa).
Consequently, the tensorial and scalar models give very similar predictions of shear stress and λ, as shown in Figure 4-9(a-b). The tensorial model, therefore, can reproduce the scalar model’s quantitative predictions shown in Chapter 4.4.
4.5.2 Comparison with the Tensorial IKH Model
Here, we briefly compare our tensorial ML-IKH model with the tensorial IKH model, given in the supporting information of ref. [27]. The tensorial IKH model is a 3D generalization of its scalar form based on the multiplicative decomposition of the deformation gradient tensor.
It involves stress and strain measures defined in a “reference configuration” and an “intermediate configuration”. Such formulation belongs to the Lagrangian form and is more suitable for predicting the solid mechanics of TEVP materials. The tensorial ML-IKH model is based on the additive decomposition of the rate of deformation tensor. It does not depend on a reference configuration. The Cauchy stress tensor, its rate of change, the rate of deformation tensor, and the kinematic hardening variable can be written in a simple functional form as discussed in
Chapter 4.2.2. This formulation belongs to the Eulerian form and is more suitable for most fluid flow simulations of TEVP fluids. The tensorial IKH model is in principle applicable for finite and large elastic deformations while the tensorial ML-IKH model assumes infinitesimal elastic deformation. This assumption is valid for weakly-aggregated suspensions and it greatly simplifies the constitutive equations. With this assumption, the tensorial IKH model can also be written in Eulerian form. But it still has a drawback in that it predicts an unphysical oscillatory shear stress in simple shear flows for certain values of model parameters and shear rates. The
102 oscillation results from its kinematic hardening rule, adopted from the theoretical work of
Henann and Anand [64]. This disadvantage is however absent in the ML-IKH model.
Table 4-7. Comparison of our model with the tensorial IKH model in the limit of no yield stress ( = 0) and small elastic deformation. Additional limits ML-IKH IKH model
τ τs 2 mD ο τ +τeff 2D τs+τ eff 2 thi D τeffτ τ back=2 D p None τeffτ κ back=2 thi D p τ kA = k ln A 2 back h h κback k h A +2 A ο ο A= A Dp D p A qd p A ln A A= A Dp D p A + D p qd p A
τ =2 D+κ τ =2 D + τback Rigid plasticity thi m back ( = 0) A= Dqd A A =qd A ln A is symmetric and unimodular. = A is symmetric. A = 0 in the un- ln is symmetric and deviatoric. = Remarks: deformed reference configuration 1 and = 0 in the un-deformed reference configuration. For simplicity, consider the limit of zero yield stress and small elastic deformation. The tensorial IKH model in this limit can be written in a simple Eulerian form, as summarized in
Table VII. In the limit of = 0, the two models further reduce to rigid, i.e., non-elastic, plastic models.
Figure 4-10 plots the predictions of 2A by our model and its equivalent, A , by the
IKH model in the limit of rigid plasticity in shear-start-up tests. For small q values, the IKH model predicts a damped oscillation of A and therefore of the shear stress ( ̇ + ℎA12). As q increases, this oscillation vanishes, and the two models show the same predictions. The oscillation of shear stress is significant when both q and ̇⁄ are small. Note that q = 1 is used in this study to model the rheology of the fumed silica suspension, and it is yet unclear what the range of values that q takes in experimental TEVP systems.
103
Figure 4-10. Predictions of the back strain in shear-start-up tests. Solid lines are 2A in the ML-IKH model and dashed lines are A in the IKH model, in both cases in the limit of no yield stress and rigid plasticity. q equals 0.01 for lines A and a; 0.1 for B and b; 1 for C and c; and 10 for D and d. The shear stress predicted by the IKH model is ( ̇ + A ). The oscillation of shear stress is significant when q and ̇⁄ are small. 4.6 Conclusions
The primary goal of this chapter is to develop a comprehensive constitutive model for the
transient rheology of TEVP fluids. Such TEVP fluids show the following important rheological
features: 1) viscoelasticity and stretched-exponential thixotropic relaxation in step rate or step
stress tests, 2) a non-monotonic thixotropic response in intermittent flows, 3) kinematic
hardening, and 4) linear viscoelasticity before yielding. We proposed here a phenomenological
model that captures those rheological features by combining four key components that have not
previously been unified in a single model: 1) multiple thixotropic structure parameters, 2)
nonlinear thixotropic kinetic equations, 3) the Armstrong-Frederick kinematic hardening rule,
and 4) linear viscoelasticity. The proposed model is a generalization of our previous ML ideal
thixotropic model and merger of this with the IKH model. This new model has 12 parameters, 4
more than the IKH model, where the additional parameters introduce multiple thixotropic
structural parameters which allow the structure to relax in a stretched exponential form, as well
nonlinearities into the evolution of structure, and leading to more accurate predictions in step rate
104 or step stress tests. Such a large number of parameters is necessary to accurately capture the
TEVP rheology and is common in thixotropic models [11, 27, 41, 52, 54, 62].
For limiting values of model parameters, the ML-IKH model reduces to several rheological models, such as the Oldroyd-B model (when thixotropy and plasticity are absent), the
Bingham model (when viscoelasticity, thixotropy, and back stress are absent), Saramito’s elastoviscoplastic model [107] (when thixotropy and back stress are absent), Moore’s ideal thixotropic model [122] (when viscoelasticity and back stress are absent), More examples are summarized in Table II. This flexibility allows the users to conveniently simplify it as needed for desired applications.
We comparatively examined this model over a wide range of test conditions for steady shear, step shear rate, step stress, intermittent shear, shear reversal, and LAOS experiments. The experimental data are from a well-studied thixotropic fumed silica suspension. The results of the comparative testing are summarized in Table 4-7. This comparative study demonstrates the success of the four key model components in capturing several typical rheological features of
TEVP materials. Multiple thixotropic structure parameters and their nonlinear kinetic equations are critical for predicting the rheology in step tests and intermittent shear tests. The Armstrong-
Frederick kinematic hardening rule is important in predicting the flow reversal tests. The model components for linear viscoelasticity capture the fast elastic response in step tests.
We also presented a tensorial form of this model based on the additive decomposition of the rate of deformation tensor. It is written in the Eulerian form and suitable for the fluid flow simulation of thixotropic fluids. We demonstrated that this tensorial model can reproduce the quantitative predictions of the scalar model and does not show unphysical oscillation of shear
105 stress in simple shearing flows. It is frame-invariant and satisfies the second law of thermodynamics (see Appendix for details).
This model has following limitations. First, the ML approach accounting for thixotropy assumes a constant stretching factor (β) while the experiments in ref. [38] have shown that β varies as test condition changes. Breakdown experiments with large final shear rates exhibit small β values (e.g., 0.3); conversely, build-up tests with small final shear rates give large β values (e.g., 1.5). This limitation may lead to poor predictions of the thixotropic evolution when the shear rate is small. Second, this model fails to predict the delayed yielding behavior that some TEVP materials exhibit. Third, this model assumes linear viscoelasticity with a single time scale for viscoelastic relaxation. The real TEVP fluids, however, often exhibit complex nonlinear viscoelasticity. Fourth, the model assumes that the yield stress depends on structural anisotropy
(through the Armstrong-Frederick kinematic hardening rule of back stress) but the viscosity does not, which needs to be further examined. Furthermore, the tensorial form of this model in principle can be used in arbitrary 3D flow conditions. But it has hitherto been tested only in simple shear flows because of the lack of non-shearing rheological experiments on TEVP materials. We leave it for further work to overcome these limitations.
Appendix
In this section, we demonstrate that the tensorial ML-IKH model satisfies the free-energy imbalance, i.e., obeys the second law of thermodynamics. The total free energy (Ψ) has the following form:
G 1 trBAA 3 k : , (4-40) 2e 2 h where the elastic energy is described the incompressible neo-Hookean model. Here the first and second terms on the right side are respectively the elastic (Ψ ) and plastic (Ψ ) free energies.
106 denotes the elastic finger tensor, defined as = ∙ , where is the elastic component of the deformation gradient according to the Kroner decomposition [118], = ∙ . Here, ,
, and are defined as
x x x FFF, p , e , X x X
x x x (4-41) FFFi,,. i i ij pij e ij Xj x j X j where X, , and x are respectively position vectors in the reference, intermediate, and current configurations. The velocity gradient tensor ( = ( ) ) as well as its elastic and plastic
components are respectively = ̇ ∙ , = ̇ ∙ , and = ∙ ̇ ∙ ∙ .
For incompressible flows, the free-energy imbalance requires that : − Ψ̇ > 0 is always valid, where = + . The rate of change of free energy is Ψ̇ = Ψ̇ + Ψ̇ where Ψ̇ =
: ̇ , and Ψ̇ = : ̇ . We expand ̇ as follows
TT BFFFFe e e e e , 1 T T T T FFFFFFFFe e e e e e e e , (4-42) T LBBLe e e e.
Therefore,
G T e δ :,LBBL e e e e 2
G T BLLe:, e e (4-43) 2
GBDe:, e
τs:.D e which gives Eq. (4-44).
τ::.D τsD p p (4-44)
We next derive the expression for : , which will be used in Eq. (4-46).
107 κback: D p
khA +2 A A : D p ,
T kA::, D + k A A W D W D h p h p p
T (4-45) τ :D +τ :WDAAWD , back p back p p =τback :D p +τ back :ADA p qd p , τback:.AA qd p
τs: D p p τs::,D p τ back A τs::::,D p κ backD p κ backD p τ back A (4-46) τeff:::,D p κ backD p τ back A
= τeff::,D p qd pτ back A
According to Eq. (29), : is nonnegative. And : = : ≥ 0. Therefore, the free energy imbalance ( : − Ψ̇ > 0) is always satisfied.
108 Chapter 5 Time-dependent Shear Bands in Transient Shear Flow 5.1 Introduction
Thixotropic fluids exhibit very complex rheological behaviors because of the rich internal structures that can deform, break, and restructure under flow. The strong couplings between internal structures and flow in thixotropic fluids often lead to shear banding that occurs in various flow regimes and greatly influences the rheology [73, 123-127]. Both the banding dynamics and rheology of thixotropic fluids involve a strong time-dependence. The interplay between time scales of rheology and shear banding in thixotropic fluids is drawing growing attention [73, 123-128], but remains far from being adequately understood.
The purpose of this chapter is to study and model the occurrence and evolution of shear bands in transient shear flow. Please refer to Chapter 1.3 for a brief review of shear banding in thixotropic fluids and Chapter 2.4 for the materials, instruments, and velocimetry algorithm used in this chapter.
5.2 Banding Dynamics in Shear Startup and Flow Reversal Tests
5.2.1 Shear Startup Tests
We first examine the evolution of the local velocimetry in shear startup tests. The test procedure consists of three steps – an initial high-shear-rate step ( ̇ = 200 s , duration = 2
–1 min), a rest period (0 s , 10 min), and a final step ( ̇ , ). The purpose of the first step is to create a reproducible starting state of the sample by shearing the materials at a large shear rate to break most of the internal structures. The internal structures slowly grow during the following 10
109 mins of zero shear rate. And after a sudden shear startup, the internal structures gradually deform and break down. The focus of this section is to investigate the rheological response and the evolution of shear bands after the sudden startup of the shear flow.
We simultaneously measure the rheological response, ( , ̇ ), and the local velocimetry,
( , , ̇ ) using the method introduced in Chapter 2.4. Here, y denotes the position along the velocity gradient direction in the field of view of the microscope. y = 0 corresponds to the fluid layer in contact with the upper cone, and y = H = 873 μm is the position of the bottom plate. To examine the banding dynamics, we use the following metric for the degree of banding (DOB):
H v(,,) y t y DOB 2 0 dy , (5-1) 0 vwall H where v = γ̇ H. By this definition, the DOB equals two times the total absolute area between the normalized velocity profile and that of a homogeneous flow (i.e., a straight line). The DOB equals zero for a uniform shear flow and unity for a plug-flow with vplug = 0 or vwall.
Figure 5-1 shows the evolutions of the stress and the DOB in a shear startup test with
̇ = 0.01 s . The stress first gradually increases and reaches a maximum at around 0.04 strain units. The stress then decreases but soon it starts to grow again until the second peak at around one strain unit, after which the stress exhibits a slow decay. The first peak and the subsequent decrease in the stress transients indicate a yielding response of the material, which agrees with the results of a dynamic strain sweep test (Figure 5-2). The crossover of the elastic and loss moduli curves occurs at a strain amplitude of about 0.04 strain units, very close to the strain value of the first stress overshoot in Figure 5-1. Following the theory presented in Chapter 4, the growth of the stress after the first stress overshoot can be presumably ascribed to kinematic hardening, and the subsequent slow decay can be explained as thixotropic response.
110
Figure 5-1. Evolutions of the stress and DOB in a shear startup test ( ̇ = 0.01 s ). Dashed lines indicate the strain values at which the instantaneous velocity profiles are plotted in Figure 5-3. The insert shows the test procedure. –1 The sample is first sheared at 200 s for 2 min. The shear rate then decreases to zero within 20 s. After a rest period of duration 10 min, the shear rate instantly jumps to ̇ and is held constant for a duration of so that the total strain = ̇ = 180.
Figure 5-2. Rheological response of the 2.9 vol% fumed silica suspension in an amplitude sweep test with a frequency of 1 rad/s. G' and G'' denotes respectively the elastic and loss moduli. The circle marks out the crossover point where G' = G''. The characteristic strains determined from the rheological response can also be seen in
the banding dynamics. For convenience of discussion, we marked out seven strain values ( ,
… ) as indicated by the dashed line in Figure 5-1. The instantaneous velocity profiles at those
strain values are shown in Figure 5-3. The results show that the flow remains uniform for <
, and the inhomogeneity slightly grows for < < . The flow inhomogeneity, however,
stops growing and decreases during and . After shear bands start to grow continuously
and eventually form two distinct bands, as shown in Figure 5-3(b). A space-time plot of the
111 velocity profiles is presented in Figure 5-4, which clearly shows how the flow instability grows in this experiment.
Figure 5-3. Instantaneous velocity profiles (normalized by H and vwall) at the strain values as indicated in Figure 5-1.
Figure 5-4. Space-strain plot of the velocity profile (normalized by H and vwall) in the shear startup test with ̇ = 0.01 s . Background color indicates the level of velocity according to the color bar on the right. Next, we discuss the rheological response and banding dynamics in a set of shear startup tests with varied shear rates. Figure 5-5 (a) shows that as the shear rate increases, the first stress overshoot rises and moves to larger strain values while the second stress overshoot becomes less obvious and reduces to a kink in the stress transients. Figure 5-5 (b) shows the evolutions of the
DOB in corresponding shear startup tests. As the shear rate increases, the overall level of the
DOB decreases. A critical shear rate exists between 0.1 and 0.25 s–1, and the flow remains uniform for shear rates above this critical value. In experiments with small apparent shear rates, most of the growth of the DOB take place after the strain exceeds unity. The results shown in
112 Figure 5-5 lead to a hypothesis that kinematic hardening tends to stabilize the flow while thixotropic response leads to the growth of bands. In the following sections, we test this hypothesis through flow reversal experiments and numerical simulations of a simple model.
Figure 5-5. Rheological response (a) and banding dynamics (b) in a set of shear startup experiments. 5.2.2 Flow Reversal Tests
We study the properties of the shear bands through a sudden reversal of the shearing direction at the end of shear startup tests. In purely viscous fluids, including ideal thixotropic fluids, the stress ( ) and the velocity profile ( , ) instantly flips to − ( ) and − ( , ) when the shearing direction changes (neglecting fluid inertia). The thixotropic fumed silica suspension, however, show different behaviors due to viscoelasticity and kinematic hardening. Figure 5-6 (a) shows the rheological response and the evolution of the DOB in a set of flow reversal experiments ( ̇ suddenly jumps from − ̇ to ̇ at = 0). Denoting the stress value before the flow reversal as , the reduced stress, − ⁄ , can be scaled with the strain to yield a master curve. In Chapter 4, we have shown that a kinematic hardening TEVP model can qualitatively predict the master curve. According to the MK-IKH model, the initial response ( < 0.04) is governed by elasticity and the subsequent slow rise of the reduced stress is governed by
113 kinematic hardening. Thixotropic response is negligible in flow reversal tests because the
absolute value of the shear rate remains unchanged.
Figure 5-6. (a) Evolutions of the stress, normalized by the absolute stress value before flow reversal, and the DOB in a set of flow reversal tests. (b) Instantaneous velocity profiles in the flow reversal test with ̇ = 0.01 s . Dashed black line is the velocity profile before shear reversal. Lines in color are the velocity profiles after shear reversal with corresponding strain values indicated in the legend. The banding dynamics also exhibit strong dependence on the strain. In the four flow
reversal experiments, the DOB first slightly increases for < 0.15, then decreases with the
strain until = 1.5, after which the DOB slowly rises and approaches the value before flow
reversal. Figure 5-6(b) shows the velocity profiles in the flow reversal test with ̇ = 0.01 s .
After flow reversal, the velocity profile shows a sudden but slight homogenization. The velocity
profile then evolves towards the shape before flow reversal. At = 0.15, a second
homogenization of the velocity profile starts. This homogenization is driven by the strain and
ends at around 1.5 strain units. During the period of > 0.15, shear bands grow, and the
114 velocity profile eventually reaches the same shape prior to flow reversal. Figure 5-7 below shows
a space-strain plot of the velocity profile in the flow reversal test.
Figure 5-7. Space-strain plot of the velocity profile (normalized by H and vwall) in the flow reversal test with ̇ = 0.01 s . Background color indicates the level of velocity according to the color bar on the right. The flow reversal experiments provide a support of the hypothesis that kinematic
hardening tends to homogenize the flow. Elasticity and kinematic hardening are two governing
rheological responses in flow reversal experiments. As shown in Figure 5-6 and Figure 5-7, the
banding dynamics depends mainly on kinematic hardening – the DOB decrease when the
kinematic hardening is significant and the DOB grows as the kinematic hardening slows down.
5.3 Modeling
5.3.1 Proposed Model
We present here a minimal model consisting of kinematic hardening and thixotropy and
show that it can qualitatively predict the main features of the banding dynamics discussed in the
previous section. We assume that ̇ depends on and two structural variables, and ,
according to the following constitutive equation:
sign max ,0 eff eff y , (5-2) thi n
115 where = − is an intermediate variable. Here, the kinematic hardening parameter and the thixotropic parameter conceptually account for two aspects of internal structures – respectively, the level of anisotropy and the overall degree of aggregation. Eq. (5-2) defines a anisotropic yield surface in stress space, − , + . The yield stress equals +
in the positive direction and − in the negative direction. The model predicts a rigid solid for stress within this yield surface.
The evolutions of follows
K k (1 ). (5-3)
The terms on the right side of Eq. (5-3) account for, respectively, the rate of structural breakage and formation. The evolutions of follows
A q kA A. (5-4)
Eq. (5-4) states that A changes because of a contribution of shear and its relaxation. Eq. (5-4) is slightly different from the original Armstrong-Frederick kinematic hardening equation in that it has an additional parameter which allows the anisotropic yield surface to relax and restore isotropy ( = 0) in the absent of shear flow. Eq. (5-4) reduces to the well-known Armstrong-
Frederick kinematic hardening rule [129] when equals zero. The original Armstrong-
Frederick kinematic hardening equation is adopted from models for metals or metallic glasses
[129] and assumes that the shift of yield surface persists indefinitely after cessation of flow. This description is however not appropriate for thixotropic fluids because after flow stops, the microstructure of these materials can rearrange and consequently restore, at least partially, the isotropic yield surface. The new parameter captures this relaxation process.
At steady state, depends on ̇ according to
116 ss CA ss sign ss y ss thi n , (5-5) where ( ̇) and ( ̇) are defined as follows:
Ass , (5-6) q kA
1 ss . (5-7) K k 1
The linear stability of a homogeneous flow under constant-stress conditions can be determined from the stability matrix M which is defined as
A AAAA ,,,, M A,,. (5-8) A AA,,,,
̇( , , ) and ̇( , , ) can be obtained by inserting Eq. (5-2) into respectively Eq. (5-4) and
Eq. (5-3). M can be evaluated at steady states. For any point ( ̇, ) on the steady state flow curve, it is linearly unstabe under constant stress conditions if M contains eigenvalues with positive real parts. This is a necessary but not sufficient condition for the occurrence of shear bands under constant stresses or constant apparent shear rates.
This model qualitatively captures important features seen in the experiments. Figure 5-8 shows the apparent steady state flow curve measured through bulk rheometry and the true steady states (stress versus local shear rate) determined through the rheo-PIV technique. Experiments show that there exists a region of shear rates that is inaccessible at steady states. The model captures this phenomenon with appropriate parameter values: σy = 0.72 Pa, C = 1 Pa, ηthi = 4
−1 −1 Pa·s, ηn = 0.9 Pa·s, kA = 0.0001 s , kλ = 0.001 s , K = 0.0025.
117
Figure 5-8. Steady-state flow curve determined from the bulk rheometry (black symbols) and local velocimetry (red symbols). The insert plot shows model predictions with solid line indicating the linearly stable portion and the dashed line indicating the unstable portion. 5.3.2 Numerical Simulation of Banded Flow
We solve the model over the velocity gradient direction to simulate the formation and
evolution of shear bands under transient shear flow. In this one-dimension model, ̇, , and
depend on both time and the position along the velocity gradient direction. Eq. (5-4) and Eq.
(5-3) need to be rewritten as follows
2 tA k A A D y A, (5-9)
2 t K k (1 ) D y . (5-10)
The spatial gradients and allow for smooth velocity profiles at scales larger than ∝
√ , which is physically due to interactions across adjacent streamlines. The apparent shear rate,
Γ, is set by the test procedure and equals the spatial average of the local shear rate ̇( , ):
1 H y,, t dy (5-11) H 0
where is the height of the gap between two boundaries. We solve this model using boundary
conditions = 0 and 〈 ̇〉 = ̇ and initial conditions ( , = 0) = ( ), ( , = 0) = 1 +
118 ( ), where ( ) and ( ) are the initial spatial heterogeneity of and . We set D = 10
2 μm /s, = sin( ), and = |sin( )|, where = = 0.1 and = =
25 ⁄ . We set = 1 mm. The other parameters take the same values as used in Figure 5-8.
We have noted that the quantitative details of the calculation depend strongly on ( ) and
( ) but the qualitative features in the evolution of the DOB do not.
Figure 5-9. Numerical simulation of a shear startup test followed by a sudden flow reversal. (a) and (b) shows the rheological response and the banding dynamics in respectively the shear startup and flow reversal tests. (c) and (d) shows the space-strain plot of the velocity profiles in respectively the shear startup and flow reversal tests. In Figure 5-9 we present the model predictions of a shear start-up test (Γ jumps from 0 to
0.01 s ) and a subsequent flow reversal test (Γ jumps from 0.01 to –0.01 s ). Figure 5-9 (a) shows the rheological response and the DOB evolution in the shear start-up test. Note that this model does not include the elastic component and therefore cannot capture the initial gradual rise and overshoot of the stress ≈ 0.04 as shown in Figure 5-1. This model qualitatively captures the kinematic hardening response (the increase of the stress for 0.2 < < 3) and the later thixotropic response (the stress decay for > 3). The model shows that the DOB does not grow until the kinematic hardening response slows down and the thixotropic response becomes
119 dominant. A space-strain plot of the velocity profile is shown in Figure 5-9 (c). The model predictions qualitatively agree with the experimental results in Figure 5-4.
The impact of the kinematic hardening response on the banding dynamics can be seen more clearly in the flow reversal test. Figure 5-9 (b) and (d) show that a sudden flow reversal lead to temporary homogenization of the banded flow. Shear bands grow again as the kinematic hardening response slows down.
5.4 Oscillation of Shear Bands Triggered by Geometry Misalignment
In the previous two sections, we showed that kinematic hardening and thixotropy are two main factors governing the banding dynamics. Kinematic hardening response tends to stabilize the flow while thixotropic response leads to the growth of shear bands. Next, we show that the thixotropic fumed silica suspension exhibits oscillatory banding dynamics under long periods of shear at constant apparent shear rates. As introduced in Chapter 2.4, the MCR 702 rheometer allows the upper and bottom geometries to counter-rotate. We denote the rotation speed of the cone as Ω and that of the plate as Ω . Due to the counter rotation of the cone and the plate, the apparent shear rate ̇ follows
0K cone plate , (5-12) where ̇ is a constant conversion factor set by the geometry, and its value is provided by the manufacturer of the instrument. The coefficient of the cone-plate rotation speed balance can be defined as
cone acone . (5-13) cone plate
= 0 indicates that the rotation of the cone is locked while the plate is rotating. =
100% indicates that only the cone rotates while the plate does not.
120
–1 Figure 5-10 (a-e) Oscillation of shear bands under a constant apparent shear rate of 0.075 s while varying . Corresponding rotation periods of the cone and plate are given in annotations. Inserts show the coefficients of the discrete Fourier transforms of the DOB transients. (f) Rheological response in the five experiments of (a-e). Figure 5-10 presents the banding dynamics and the rheological response of the 2.9 vol.%
fumed silica suspension in a set of constant-shear-rate experiments ( ̇ = 0.075 s ) with
different values (0, 25%, 50%, 75%, and 100% in respectively a - e). The rotation periods
of the cone (tcone) and the plate (tplate) are also provided in the annotations. The evolution of the
DOB exhibits distinct periodical oscillations. The periodicity of the DOB transients can be
determined through discrete Fourier transforms (inserts in Figure 5-10). The results show that the
DOB transients have a period of either tcone or tplate, whichever smaller. The DOB transients may
exhibit a shorter period of oscillation than the one set directly by the rotation of the geometry.
The five tests show significant difference in the DOB transients. In contrast, the rheological
responses are very similar, as shown in Figure 5-10 (f).
121
Figure 5-11. A sample image of the field of view for PIV calculation in the test of Figure 5-10 (e).
Figure 5-12 Evidence of traveling shear bands. Results are from the test of Figure 5-10 (e). (a) The evolution of Δ . (b) ( , ) and ( , ) with = 10 min. (c) ( , ), ( , − Δ ), and ( , ) with = 410 min and Δ = 34 s. The results in Figure 5-10 (a-e) suggest the possibility of flow heterogeneity along the
velocity direction. A traveling wave may form as the shear bands carried by the rotating
geometry cross through the velocimetry observation window. To test this hypothesis, we
calculate separately the velocity profiles in the left-most region and the right-most region (as
illustrated in Figure 5-11) and compare the difference between the two velocity profiles. This
calculation is based on the test of Figure 5-10 (e). Here, we use a slightly different setup from the
default one introduced in Chapter 2.4. This setup provides a field of view with a dimension of
1392 × 470 pixels, corresponding to 2.6 × 0.873 mm in real space. In this experiment, = 0
122 and therefore = 0, = 0.0655 mm⁄s. It takes 34 seconds for a tracer particle that
sticks to the rotating plate to travel from the left-most region to the right-most region.
Figure 5-12 (a) shows the evolution of Δ which is defined as
v vleft y H2, t v right y H 2, t , (5-14)
where ( , ) and ( , ) are the velocity profiles in respectively the leftmost region and
the rightmost region as illustrated in Figure 5-11. Δ is therefore a measure of the velocity
heterogeneity in the fluid layer (0 ≤ ≤ 2.6 mm, = ⁄2). Δ is small during the first 30
minutes after the shear start-up. An oscillation of Δ then occurs, and the amplitude of
oscillation gradually increases. Figure 5-12 (b) shows that ( , ) is very close to
( , ). In contrast, ( , ) and ( , ) exhibit noticeable difference. If our
hypothesis about the traveling bands is true, ( , − Δ ) and ( , ) will have very
close values, which is confirmed in Figure 5-12 (c).
Figure 5-13. Velocity profiles in a Newtonian fluid under constant shear-rate flow ( ̇ = 0.2 s ). (a) 30 velocity profiles equally spaced in time as the bottom plate rotates for one cycle. (b) velocity fluctuation at five different y position across the gap. We next measure the velocity profiles in a Newtonian fluid (mineral oil with a viscosity
of 0.9 Pa∙s at 25 ℃) under a constant shear rate of 0.2 s–1. Figure 5-13 (a) plots 30 velocity
123 profiles that are equally spaced in time as the bottom plate rotates for one cycle. The 30 velocity profiles closely fall onto a single straight line. A further examination of the velocities at several y positions reveals slight oscillations that are in phase with the rotation of the geometry. The velocimetry oscillation in the Newtonian fluid is, however, much smaller than that in the thixotropic fumed silica suspension.
Figure 5-14. Variation of the positions of the geometry walls as the cone and plate rotate at a constant speed.
Figure 5-15 Illustration of the geometry misalignment. The velocimetry oscillation in the Newtonian fluid indicates that the cone-plate geometry has a small amount of misalignment. We use a bright field microscope to characterize the variation of the wall positions of the cone and plate as they rotate at a constant speed. The wall position of the cone has a periodical variation of 5 µm around the mean. The bottom plate shows slightly larger variations, as shown in Figure 5-14. The variation of the geometry wall positions suggests that both the cone and the plate have slight misalignment as illustrated in Figure 5-15.
According to the data in Figure 5-14, the cone has a tilt angle ( ) of 0.012° and the plate has a tilt angle ( ) of 0.017°. Both values are within the range of manufacturing errors.
124 5.5 Conclusion
We studied how shear bands occur and grow in the thixotropic fumed silica suspension in shear start-up tests and flow reversal tests. The material exhibits a critical shear rate below which shear bands occur. In shear start-up tests below the critical shear rate, shear bands start to grow as the strain reaches around unity. We proposed a hypothesis that kinematic hardening and thixotropy are two main factors governing the banding dynamics, and the former response tend to stabilize the flow while the latter leads to the growth of shear bands. Flow reversal experiments provide support for this hypothesis.
We have constructed a simple thixotropic yield stress model to explain this phenomenon with two dynamical variables – one for thixotropy and the other one for kinematic hardening
(KH). As opposed to previous thixotropic KH models, our model allows the KH effect to decay after flow stops. This model successfully captures important features of experiments.
We found that under long periods of shearing at constant apparent shear rates, shear bands periodically oscillate. We confirmed that the velocimetry oscillation is triggered by the small geometry misalignment. Shear bands show two-dimensional heterogeneity. The velocity profile in the PIV observation window oscillates periodically as the shear bands, carried by the geometry, cross in and out of the field of view. We characterized the geometry misalignment and showed that it is within the range of manufacturing errors. The geometry misalignment causes very slight velocimetry oscillation in a Newtonian fluid. The fumed silica suspension, however, significantly amplifies the geometry misalignment and displays large oscillations of shear bands.
125 Chapter 6 Conclusions and Future Directions 6.1 Conclusions
This thesis provides in-depth studies on the rheology and shear banding of thixotropic fluids. Thixotropic fluids exhibit complex rheological behaviors, including thixotropy, viscoelasticity, yielding, and kinematic hardening. This thesis first focuses on developing constitutive models that can quantitatively predict the rheological behaviors of thixotropic fluids, and then seeks to understand how shear bands occur and evolve under transient shear histories.
Thixotropy is the dominating rheological response when shear histories involve only relatively high shear rates. Being able to quantitatively predict such ideal thixotropic response is the first step toward developing more advanced models. We show that under transient shear flow, thixotropic evolution exhibits multiple timescales. It is critical to introduce a set of thixotropic kinetic equations in order to quantitatively predict ideal thixotropic response. We proposed a novel approach to introduce a spectrum of thixotropic kinetic equations at the cost of only one additional model parameter. Based on the approach, we constructed a simple model, which we call “multi-lambda” (ML) model, and examine this model against experiments including steady state, step shear rate, step stress, shear rate ramp, and stress ramp tests. This model can quantitatively predict the experimental data.
Viscoelasticity, yielding, and kinematic hardening are important rheological behaviors when shear histories involve small shear rates or changes of shearing directions. Under such shear histories, the ML model, an ideal thixotropic model, gives poor predictions. To predict the complex thixotropic elasto-viscoplastic response, we generalize the ML model and combine it
126 with the isotropic kinematic hardening (IKH) model proposed by Dimitriou et al [2]. This new constitutive equation, which we call ML-IKH model, has the following features: 1) multiple thixotropic structure parameters that collectively exhibit a stretch-exponential thixotropic relaxation in step tests; 2) nonlinear thixotropic kinetic equations in both the shear rate or stress and the structure parameters; 3) incorporation of the Armstrong-Frederick kinematic hardening rule [3] for the evolution of yield stress; and 4) viscoelasticity.
We evaluate this 12-parameter model, discuss its four key features, and compare its predictions with those of the ML, IKH, and modified Delaware thixotropic (MDT) models [4] for two sets of experimental data [1, 4] of a thixotropic fumed silica suspension. The shear-rate histories include steady state, step shear rate, step stress, intermittent shear, flow reversal, and large amplitude oscillatory shear (LAOS). We show that in step tests the thixotropic and viscoelastic evolutions are dominant, while intermittent shear tests the multiple thixotropic timescales, and flow reversal tests the viscoelastic and plastic evolutions. The rheological responses in LAOS tests are more complex and involve all aspects of TEVP rheology. The four features quantitively capture different aspects of TEVP rheology. We also provide a tensorial formulation of the ML-IKH model that is frame-invariant, obeys the second law of thermodynamics, and can reproduce the predictions of the scalar version.
The next focus of this thesis is to investigate and model the time-dependent shear bands under transient shear. We developed a new algorithm for particle image velocimetry to study shear banding at high spatial resolutions. The thixotropic fumed silica suspension exhibits a critical shear rate at around 0.2 s–1. Shear bands occur only when the apparent shear rate is smaller than this critical value. In shear start-up tests with small shear rates, the rheological response (stress versus strain) is successively governed by 1) viscoelasticity and yielding (0 <
127 < 0.2), 2) kinematic hardening (0.2 < < 3), and 3) thixotropy ( > 3). The growth of shear bands mainly occurs in the last period. Flow reversal tests were performed to study the transient response of the shear bands after a sudden reversal of the shearing direction. The overall degree of banding (DOB) shows a slight overshoot at ≈ 0.15, decreases during 0.15 < < 1.5, and then grows towards the level before the reversal of the shearing direction. We observed a distinct correlation between the banding dynamics and the rheological response. The slight overshoot of the DOB occurs as the rheology undergoes a transition from viscoelasticity to kinematic hardening. The DOB decreases when kinematic hardening becomes the dominating rheological response and increases again as kinematic hardening slows down.
We constructed a simple thixotropic yield stress model to explain the banding dynamics discussed above. This model contains two dynamical variables – one for thixotropy and the other one for kinematic hardening. This model shows that kinematic hardening tends to stabilize the flow while thixotropy leads to the growth of shear bands.
6.2 Recommended Future Directions
Recent years have seen rapid progress in the studies on thixotropy. Nevertheless, yawning gaps in the understanding of this phenomenon remain. The purpose of this section is to recommend important future works that are needed, including obtaining more data sets with comprehensive shear histories, cross-comparison of existing rheological models with various datasets, investigation of non-viscometric flows, detailed measurements of microstructures and local flow fields, and further development of microscopic models and simulations.
At present, there are only a few comprehensive rheological data sets. Detailed rheological characterization of various materials is recommended. A good data set should contain a wide range of shear histories, and we recommend the following to be included: steady shear, shear-
128 rate jumps, stress-jumps, ramps of the shear rate or stress, flow reversal, and oscillatory shear. It is also important to study how the rheology depends on factors including preparation procedure, formulation of the sample, and the temperature at which rheological measurements are performed.
Simultaneous local velocimetry during rheological measurements is recommended because shear bands can occur and alter the local shear histories. Shear banding, if present, means that rheological measurements of thixotropic phenomena are incomplete or even misleading, since they represent an average rheologocal response and do not necessarily apply to local fluid elements.
Most rheological models for thixotropic fluids are compared against only a single data set. It is important to examine those models in other data sets, especially those from different materials. Another recommended future direction is investigation of non-viscometric (i.e., non- shear) flows. The literature at present lacks data in non-shear flows. Without extensional or mixed flows, we remain ignorant of thixotropic behavior in anything other than shearing flows.
Another important future direction is the development of microstructure models and simulations. The beginnings of such works are underway, with the development of population balance models and the simulations using mesoscale models such as Dissipative Particle
Dynamics. Further works along this path are of high priority.
129 Bibliography
[1] Peterfi, T., Arch. Entwicklungsmech. Organ. 112, 680 (1927).
[2] Schalek, F. E., and Szegvary, A., ‘‘Ueber eisenoxydgallerten,’’ Colloid & Polymer Science 32, 318-319 (1923).
[3] Barnes, H. A., ‘‘Thixotropy—a review,’’ Journal of Non-Newtonian Fluid Mechanics 70, 1-33 (1997).
[4] Bender, A., and Bender, D., A dictionary of food and nutrition (Oxford University Press, 1995).
[5] “Newtonian fluid,” in Editor (ed)^(eds). Book Newtonian fluid (Oxford University Press, 2009).
[6] IUPAC, “Compendium of chemical terminology,” in Editor (ed)^(eds). Book Compendium of chemical terminology (Blackwell Scientific Publications, 1997).
[7] Mewis, J., ‘‘Thixotropy - a general review,’’ Journal of Non-Newtonian Fluid Mechanics 6, 1-20 (1979).
[8] Mewis, J., and Wagner, N. J., ‘‘Thixotropy,’’ Advances in Colloid and Interface Science 147-148, 214-227 (2009).
[9] Cheng, D. C. H., ‘‘Thixotropy,’’ International Journal of Cosmetic Science 9, 151-191 (1987).
[10] Pek, Y. S., Wan, A. C. A., and Ying, J. Y., ‘‘The effect of matrix stiffness on mesenchymal stem cell differentiation in a 3d thixotropic gel,’’ Biomaterials 31, 385-391 (2010).
[11] Dullaert, K., and Mewis, J., ‘‘A structural kinetics model for thixotropy,’’ Journal of Non-Newtonian Fluid Mechanics 139, 21-30 (2006).
[12] Dullaert, K., and Mewis, J., ‘‘A model system for thixotropy studies,’’ Rheologica Acta 45, 23-32 (2005).
[13] Dullaert, K., and Mewis, J., ‘‘Thixotropy: Build-up and breakdown curves during flow,’’ Journal of Rheology 49, 1213-1230 (2005).
130
[14] Dullaert, K., and Mewis, J., ‘‘Stress jumps on weakly flocculated dispersions: Steady state and transient results,’’ Journal of Colloid and Interface Science 287, 542-551 (2005).
[15] Zhu, C., and Smay, J. E., ‘‘Thixotropic rheology of concentrated alumina colloidal gels for solid freeform fabrication,’’ Journal of Rheology 55, 655-672 (2011).
[16] Min Kim, J., Eberle, A. P. R., Kate Gurnon, A., Porcar, L., and Wagner, N. J., ‘‘The microstructure and rheology of a model, thixotropic nanoparticle gel under steady shear and large amplitude oscillatory shear (laos),’’ Journal of Rheology 58, 1301-1328 (2014).
[17] Shaukat, A., Sharma, A., and Joshi, Y. M., ‘‘Squeeze flow behavior of (soft glassy) thixotropic material,’’ Journal of Non-Newtonian Fluid Mechanics 167, 9-17 (2012).
[18] O'Bryan, C. S., Bhattacharjee, T., Hart, S., Kabb, C. P., Schulze, K. D., Chilakala, I., Sumerlin, B. S., Sawyer, W. G., and Angelini, T. E., ‘‘Self-assembled micro-organogels for 3d printing silicone structures,’’ Science Advances 3 (2017).
[19] Blackwell, B. C., and Ewoldt, R. H., ‘‘Non-integer asymptotic scaling of a thixotropic- viscoelastic model in large-amplitude oscillatory shear,’’ Journal of Non-Newtonian Fluid Mechanics 227, 80-89 (2016).
[20] Dinkgreve, M., Fazilati, M., Denn, M. M., and Bonn, D., ‘‘Carbopol: From a simple to a thixotropic yield stress,’’ Journal of Rheology 62, 773-780 (2018).
[21] Paredes, J., Shahidzadeh-Bonn, N., and Bonn, D., ‘‘Shear banding in thixotropic and normal emulsions,’’ Journal of Physics-Condensed Matter 23, 8 (2011).
[22] Baldewa, B., and Joshi, Y. M., ‘‘Thixotropy and physical aging in acrylic emulsion paint,’’ Polymer Engineering and Science 51, 2085-2092 (2011).
[23] Evdokimov, I. N., and Losev, A. P., ‘‘Thixotropy in native petroleum emulsions,’’ Journal of Dispersion Science and Technology 32, 1206-1212 (2011).
[24] Park, S. J., Yoo, K., Kim, J. Y., Kim, J. Y., Lee, D. K., Kim, B., Kim, H., Kim, J. H., Cho, J., and Ko, M. J., ‘‘Water-based thixotropic polymer gel electrolyte for dye-sensitized solar cells,’’ Acs Nano 7, 4050-4056 (2013).
[25] Du, X., Zhao, L., Xiao, H. L., Liang, F., Chen, H., Wang, X., Wang, J. J., Qu, W. M., and Lei, Z. X., ‘‘Stability and shear thixotropy of multilayered film foam,’’ Colloid and Polymer Science 292, 2745-2751 (2014).
[26] Akpinar, S., Kusoglu, I. M., Ertugrul, O., and Onel, K., ‘‘Silicon carbide particle reinforced mullite composite foams,’’ Ceramics International 38, 6163-6169 (2012).
131
[27] Dimitriou, C. J., and McKinley, G. H., ‘‘A comprehensive constitutive law for waxy crude oil: A thixotropic yield stress fluid,’’ Soft Matter 10, 6619-6644 (2014).
[28] Geri, M., Venkatesan, R., Sambath, K., and McKinley, G. H., ‘‘Thermokinematic memory and the thixotropic elasto-viscoplasticity of waxy crude oils,’’ Journal of Rheology 61, 427-454 (2017).
[29] Livescu, S., ‘‘Mathematical modeling of thixotropic drilling mud and crude oil flow in wells and pipelines-a review,’’ Journal of Petroleum Science and Engineering 98-99, 174-184 (2012).
[30] Yu, X. D., Cao, X. H., Chen, L. M., Lan, H. C., Liu, B., and Yi, T., ‘‘Thixotropic and self-healing triggered reversible rheology switching in a peptide-based organogel with a cross- linked nano-ring pattern,’’ Soft Matter 8, 3329-3334 (2012).
[31] Roussel, N., Ovarlez, G., Garrault, S., and Brumaud, C., ‘‘The origins of thixotropy of fresh cement pastes,’’ Cement and Concrete Research 42, 148-157 (2012).
[32] Augusto, P. E. D., Falguera, V., Cristianini, M., and Ibarz, A., ‘‘Rheological behavior of tomato juice: Steady-state shear and time-dependent modeling,’’ Food and Bioprocess Technology 5, 1715-1723 (2012).
[33] Razavi, S. M. A., and Karazhiyan, H., ‘‘Flow properties and thixotropy of selected hydrocolloids: Experimental and modeling studies,’’ Food Hydrocolloids 23, 908-912 (2009).
[34] Toker, O. S., Karasu, S., Yilmaz, M. T., and Karaman, S., ‘‘Three interval thixotropy test (3itt) in food applications: A novel technique to determine structural regeneration of mayonnaise under different shear conditions,’’ Food Research International 70, 125-133 (2015).
[35] Wei, Y. X., Lin, Y. B., Xie, R., Xu, Y. F., Yao, J., and Zhang, J., ‘‘The flow behavior, thixotropy and dynamical viscoelasticity of fenugreek gum,’’ Journal of Food Engineering 166, 21-28 (2015).
[36] Yousefi, A. R., and Razavi, S. M. A., ‘‘Steady shear flow behavior and thixotropy of wheat starch gel: Impact of chemical modification, concentration and saliva addition,’’ Journal of Food Process Engineering 39, 31-43 (2016).
[37] Apostolidis, A. J., Armstrong, M. J., and Beris, A. N., ‘‘Modeling of human blood rheology in transient shear flows,’’ Journal of Rheology 59, 275-298 (2015).
[38] Horner, J. S., Armstrong, M. J., Wagner, N. J., and Beris, A. N., ‘‘Investigation of blood rheology under steady and unidirectional large amplitude oscillatory shear,’’ Journal of Rheology 62, 577-591 (2018).
132
[39] Paszkowski, M., and Olsztynska-Janus, S., ‘‘Grease thixotropy: Evaluation of grease microstructure change due to shear and relaxation,’’ Industrial Lubrication and Tribology 66, 223-237 (2014).
[40] Santos, P., Carignano, M., and Campanella, O., ‘‘Qualitative study of thixotropy in gelled hydrocarbon fuels,’’ Engineering Letters 19, 13-19 (2011).
[41] Armstrong, M. J., Beris, A. N., Rogers, S. A., and Wagner, N. J., ‘‘Dynamic shear rheology of a thixotropic suspension: Comparison of an improved structure-based model with large amplitude oscillatory shear experiments,’’ Journal of Rheology 60, 433-450 (2016).
[42] Armstrong, M. J., Beris, A. N., Rogers, S. A., and Wagner, N. J., ‘‘Dynamic shear rheology and structure kinetics modeling of a thixotropic carbon black suspension,’’ Rheologica Acta 56, 811-824 (2017).
[43] Thurston, G. B., ‘‘Rheological parameters for the viscosity viscoelasticity and thixotropy of blood,’’ Biorheology 16, 149-162 (1979).
[44] Bureau, M., Healy, J., Bourgoin, D., and Joly, M., ‘‘Rheological hysteresis of blood at low shear rate,’’ Biorheology 17, 191-203 (1980).
[45] Mewis, J., ‘‘Thixotropy - general-review,’’ Journal of Non-Newtonian Fluid Mechanics 6, 1-20 (1979).
[46] Stickel, J. J., Phillips, R. J., and Powell, R. L., ‘‘A constitutive model for microstructure and total stress in particulate suspensions,’’ Journal of Rheology 50, 379-413 (2006).
[47] Barnes, H. A., ‘‘Thixotropy - a review,’’ Journal of Non-Newtonian Fluid Mechanics 70, 1-33 (1997).
[48] Azikri de Deus, H. P., and Dupim, G. S. P., ‘‘On behavior of the thixotropic fluids,’’ Physics Letters A 377, 478-485 (2013).
[49] Potanin, A., De Rooij, R., Van den Ende, D., and Mellema, J., ‘‘Microrheological modeling of weakly aggregated dispersions,’’ The Journal of chemical physics 102, 5845-5853 (1995).
[50] Apostolidis, A. J., Armstrong, M. J., and Beris, A. N., ‘‘Modeling of human blood rheology in transient shear flows,’’ Journal of Rheology (1978-present) 59, 275-298 (2015).
[51] de Souza Mendes, P., and Thompson, R., ‘‘A unified approach to model elasto- viscoplastic thixotropic yield-stress materials and apparent yield-stress fluids,’’ Rheologica Acta 52, 673-694 (2013).
133
[52] Teng, H., and Zhang, J., ‘‘A new thixotropic model for waxy crude,’’ Rheologica Acta 52, 903-911 (2013).
[53] Blackwell, B. C., and Ewoldt, R. H., ‘‘A simple thixotropic-viscoelastic constitutive model produces unique signatures in large-amplitude oscillatory shear (laos),’’ Journal of Non- Newtonian Fluid Mechanics 208, 27-41 (2014).
[54] Mujumdar, A., Beris, A. N., and Metzner, A. B., ‘‘Transient phenomena in thixotropic systems,’’ Journal of Non-Newtonian Fluid Mechanics 102, 157-178 (2002).
[55] Montes, S., and White, J. L., ‘‘Rheological models of rubber-carbon black compounds: Low interaction viscoelastic models and high interaction thixotropic - plastic - viscoelastic models,’’ Journal of Non-Newtonian Fluid Mechanics 49, 277-298 (1993).
[56] Tiu, C., and Boger, D., ‘‘Complete rheological characterization of time‐dependent food products,’’ Journal of texture studies 5, 329-338 (1974).
[57] Nguyen, Q. D., and Boger, D. V., ‘‘Thixotropic behaviour of concentrated bauxite residue suspensions,’’ Rheologica Acta 24, 427-437 (1985).
[58] Nguyen, Q. D., Jensen, C. T., and Kristensen, P. G., ‘‘Experimental and modelling studies of the flow properties of maize and waxy maize starch pastes,’’ Chemical Engineering Journal 70, 165-171 (1998).
[59] Moore, F., ‘‘The rheology of ceramic slip and bodies,’’ Trans. Brit. Ceram. Soc. 58, 470- 492 (1959).
[60] Labanda, J., and Llorens, J., ‘‘A structural model for thixotropy of colloidal dispersions,’’ Rheologica Acta 45, 305-314 (2005).
[61] Mendes, P. R. d. S., ‘‘Modeling the thixotropic behavior of structured fluids,’’ Journal of Non-Newtonian Fluid Mechanics 164, 66-75 (2009).
[62] Mwasame, P. M., Beris, A. N., Diemer, R. B., and Wagner, N. J., ‘‘A constitutive equation for thixotropic suspensions with yield stress by coarse‐graining a population balance model,’’ AIChE Journal 63, 517-531 (2017).
[63] Casey, J., and Naghdi, P., ‘‘On the relationship between the eulerian and lagrangian descriptions of finite rigid plasticity,’’ Archive for rational mechanics and analysis 102, 351-375 (1988).
[64] Gurtin, M. E., Fried, E., and Anand, L., The mechanics and thermodynamics of continua (Cambridge University Press, 2010).
134
[65] Frederick, C. O., and Armstrong, P., ‘‘A mathematical representation of the multiaxial bauschinger effect,’’ Materials at High Temperatures 24, 1-26 (2007).
[66] Buckley, S., and Entwistle, K., ‘‘The bauschinger effect in super-pure aluminum single crystals and polycrystals,’’ Acta Metallurgica 4, 352-361 (1956).
[67] Chun, B., Jinn, J., and Lee, J., ‘‘Modeling the bauschinger effect for sheet metals, part i: Theory,’’ International Journal of Plasticity 18, 571-595 (2002).
[68] Milligan, R., Koo, W., and Davidson, T., ‘‘The bauschinger effect in a high-strength steel,’’ Journal of Basic Engineering 88, 480-488 (1966).
[69] Xiang, Y., and Vlassak, J., ‘‘Bauschinger and size effects in thin-film plasticity,’’ Acta Materialia 54, 5449-5460 (2006).
[70] Dimitriou, C. J., Ewoldt, R. H., and McKinley, G. H., ‘‘Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (laostress),’’ Journal of Rheology 57, 27-70 (2012).
[71] Jiang, Y., and Kurath, P., ‘‘Characteristics of the armstrong-frederick type plasticity models,’’ International Journal of Plasticity 12, 387-415 (1996).
[72] Martin, J. D., and Hu, Y. T., ‘‘Transient and steady-state shear banding in aging soft glassy materials,’’ Soft Matter 8, 6940-6949 (2012).
[73] Divoux, T., Grenard, V., and Manneville, S., ‘‘Rheological hysteresis in soft glassy materials,’’ Physical Review Letters 110, 018304 (2013).
[74] Garcia-Sandoval, J. P., Bautista, F., Puig, J. E., and Manero, O., ‘‘The interface migration in shear-banded micellar solutions,’’ Rheologica Acta 56, 765-778 (2017).
[75] Dimitriou, C. J., Casanellas, L., Ober, T. J., and McKinley, G. H., ‘‘Rheo-piv of a shear- banding wormlike micellar solution under large amplitude oscillatory shear,’’ Rheologica acta 51, 395-411 (2012).
[76] Eiser, E., Molino, F., Porte, G., and Diat, O., ‘‘Nonhomogeneous textures and banded flow in a soft cubic phase under shear,’’ Physical Review E 61, 6759 (2000).
[77] Møller, P., Rodts, S., Michels, M., and Bonn, D., ‘‘Shear banding and yield stress in soft glassy materials,’’ Physical Review E 77, 041507 (2008).
[78] Cloitre, M., Borrega, R., Monti, F., and Leibler, L., ‘‘Glassy dynamics and flow properties of soft colloidal pastes,’’ Physical review letters 90, 068303 (2003).
135
[79] Manneville, S., Colin, A., Waton, G., and Schosseler, F., ‘‘Wall slip, shear banding, and instability in the flow of a triblock copolymer micellar solution,’’ Physical Review E 75, 061502 (2007).
[80] Ravindranath, S., Wang, S.-Q., Olechnowicz, M., and Quirk, R. P., ‘‘Banding in simple steady shear of entangled polymer solutions,’’ Macromolecules 41, 2663-2670 (2008).
[81] Hayes, K. A., Buckley, M. R., Qi, H., Cohen, I., and Archer, L. A., ‘‘Constitutive curve and velocity profile in entangled polymers during start-up of steady shear flow,’’ Macromolecules 43, 4412-4417 (2010).
[82] Chaudhuri, P., Berthier, L., and Bocquet, L., ‘‘Inhomogeneous shear flows in soft jammed materials with tunable attractive forces,’’ Physical Review E 85, 021503 (2012).
[83] Jain, A., Singh, R., Kushwaha, L., Shankar, V., and Joshi, Y. M., ‘‘Role of inertia and thixotropy in start-up flows of aging soft materials: Transient dynamics and shear banding in a rate-controlled flow field,’’ Journal of Rheology 62, 1001-1016 (2018).
[84] Radhakrishnan, R., Divoux, T., Manneville, S., and Fielding, S. M., ‘‘Understanding rheological hysteresis in soft glassy materials,’’ Soft Matter 13, 1834-1852 (2017).
[85] Fielding, S. M., ‘‘Shear banding in soft glassy materials,’’ Reports on Progress in Physics 77, 15 (2014).
[86] Evonik, “Aerosil® - fumed silica technical overview,” in Editor (ed)^(eds). Book Aerosil® - fumed silica technical overview.
[87] Narumi, T., See, H., Honma, Y., Hasegawa, T., Takahashi, T., and Phan-Thien, N., ‘‘Transient response of concentrated suspensions after shear reversal,’’ Journal of Rheology 46, 295-305 (2002).
[88] Ugaz, V. M., Burghardt, W. R., Zhou, W., and Kornfield, J. A., ‘‘Transient molecular orientation and rheology in flow aligning thermotropic liquid crystalline polymers,’’ Journal of Rheology 45, 1029-1063 (2001).
[89] Dullaert, K., ‘‘Constitutive equations for thixotropic dispersions (constitutieve vergelijkingen voor thixotrope dispersies),’’ (2005).
[90] Zuiderveld, K., “Contrast limited adaptive histogram equalization,” 1994.
[91] Larson, R. G., ‘‘Constitutive equations for thixotropic fluids,’’ Journal of Rheology 59, 595-611 (2015).
[92] Godfrey, J., ‘‘Steady shear measurement of thixotropic fluid properties,’’ Rheologica Acta 12, 540-545 (1973). 136
[93] Joye, D. D., and Poehlein, G. W., ‘‘Characteristics of thixotropic behavior,’’ Transactions of The Society of Rheology 15, 51-61 (1971).
[94] Harris, J., ‘‘A continuum theory of time-dependent inelastic flow,’’ Rheologica Acta 6, 6-12 (1967).
[95] Holzapfel, G. A., Nonlinear solid mechanics Vol. 24 (Wiley Chichester, 2000).
[96] Kontogiorgos, V., Calculation of relaxation spectra from stress relaxation measurements (INTECH Open Access Publisher, 2010).
[97] Johnston, D. C., ‘‘Stretched exponential relaxation arising from a continuous sum of exponential decays,’’ Physical Review B 74 (2006).
[98] Tikhonov, A. N., and Arsenin, V. I. A. k., Solutions of ill-posed problems (Vh Winston, 1977).
[99] Denny, D. A., and Brodkey, R. S., ‘‘Kinetic interpretation of non‐newtonian flow,’’ Journal of Applied Physics 33, 2269-2274 (1962).
[100] Ven, T. G. M., and Mason, S. G., ‘‘The microrheology of colloidal dispersions,’’ Colloid and Polymer Science 255, 794-804.
[101] Coussot, P., Nguyen, Q. D., Huynh, H. T., and Bonn, D., ‘‘Viscosity bifurcation in thixotropic, yielding fluids,’’ Journal of Rheology 46, 573-589 (2002).
[102] Cloitre, M., Borrega, R., and Leibler, L., ‘‘Rheological aging and rejuvenation in microgel pastes,’’ Physical Review Letters 85, 4819 (2000).
[103] Purnomo, E. H., van den Ende, D., Mellema, J., and Mugele, F., ‘‘Rheological properties of aging thermosensitive suspensions,’’ Physical Review E 76, 021404 (2007).
[104] de Souza Mendes, P. R., ‘‘Thixotropic elasto-viscoplastic model for structured fluids,’’ Soft Matter 7, 2471-2483 (2011).
[105] Potanin, A., ‘‘3d simulations of the flow of thixotropic fluids, in large-gap couette and vane-cup geometries,’’ Journal of Non-Newtonian Fluid Mechanics 165, 299-312 (2010).
[106] Derksen, J. J., and Prashant, ‘‘Simulations of complex flow of thixotropic liquids,’’ Journal of Non-Newtonian Fluid Mechanics 160, 65-75 (2009).
[107] Saramito, P., ‘‘A new constitutive equation for elastoviscoplastic fluid flows,’’ Journal of Non-Newtonian Fluid Mechanics 145, 1-14 (2007).
137
[108] de Souza Mendes, P. R., and Thompson, R. L., ‘‘A critical overview of elasto- viscoplastic thixotropic modeling,’’ Journal of Non-Newtonian Fluid Mechanics 187-188, 8-15 (2012).
[109] Yziquel, F., Carreau, P. J., Moan, M., and Tanguy, P. A., ‘‘Rheological modeling of concentrated colloidal suspensions,’’ Journal of Non-Newtonian Fluid Mechanics 86, 133-155 (1999).
[110] Bautista, F., de Santos, J. M., Puig, J. E., and Manero, O., ‘‘Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model,’’ Journal of Non-Newtonian Fluid Mechanics 80, 93-113 (1999).
[111] Tsakmakis, C., ‘‘Original article kinematic hardening rules in finite plasticity part i: A constitutive approach,’’ Continuum Mechanics and Thermodynamics 8, 215-231 (1996).
[112] Wei, Y., Solomon, M. J., and Larson, R. G., ‘‘Quantitative nonlinear thixotropic model with stretched exponential response in transient shear flows,’’ Journal of Rheology 60 (2016).
[113] Goodeve, C. F., and Whitfield, G. W., ‘‘The measurement of thixotropy in absolute units,’’ Transactions of the Faraday Society 34, 511-520 (1938).
[114] Dimitriou, C. J., Ewoldt, R. H., and McKinley, G. H., ‘‘Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (laostress),’’ Journal of Rheology 57, 27-70 (2013).
[115] Pignon, F., Magnin, A., and Piau, J.-M., ‘‘Thixotropic behavior of clay dispersions: Combinations of scattering and rheometric techniques,’’ Journal of Rheology 42, 1349-1373 (1998).
[116] Armstrong, M., “Investigating and modeling the thixotropic behavior, microstructure, and rheology of complex material,” University of Delaware, (2015).
[117] Vermant, J., and Solomon, M., ‘‘Flow-induced structure in colloidal suspensions,’’ Journal of Physics: Condensed Matter 17, R187 (2005).
[118] Kröner, E., ‘‘Allgemeine kontinuumstheorie der versetzungen und eigenspannungen,’’ Archive for Rational Mechanics and Analysis 4, 273-334 (1959).
[119] Quemada, D., ‘‘Rheological modelling of complex fluids: Iv: Thixotropic and "thixoelastic" behaviour. Start-up and stress relaxation, creep tests and hysteresis cycles,’’ European Physical Journal-Applied Physics 5, 191-207 (1999).
[120] Oishi, C. M., Martins, F. P., and Thompson, R. L., ‘‘The “avalanche effect” of an elasto- viscoplastic thixotropic material on an inclined plane,’’ Journal of Non-Newtonian Fluid Mechanics 247, 165-177 (2017). 138
[121] Liu, W.-W., Peng, J., Zhu, K.-Q., and Zhang, J.-J., ‘‘A microstructural criterion for yielding: Linking thixotropy and the yield stress,’’ Rheologica Acta 55, 957-968 (2016).
[122] Moore, F., ‘‘The rheology of ceramic slips and bodies,’’ Trans. Br. Ceram. Soc 58, 470- 494 (1959).
[123] Fielding, S. M., ‘‘Triggers and signatures of shear banding in steady and time-dependent flows,’’ Journal of Rheology 60, 821-834 (2016).
[124] Moorcroft, R. L., Cates, M. E., and Fielding, S. M., ‘‘Age-dependent transient shear banding in soft glasses,’’ Physical Review Letters 106, 055502 (2011).
[125] Moorcroft, R. L., and Fielding, S. M., ‘‘Criteria for shear banding in time-dependent flows of complex fluids,’’ Physical Review Letters 110, 086001 (2013).
[126] Radhakrishnan, R., and Fielding, S. M., ‘‘Shear banding in large amplitude oscillatory shear (laostrain and laostress) of soft glassy materials,’’ Journal of Rheology 62, 559-576 (2018).
[127] Divoux, T., Tamarii, D., Barentin, C., and Manneville, S., ‘‘Transient shear banding in a simple yield stress fluid,’’ Physical Review Letters 104, 208301 (2010).
[128] Rogers, S. A., Vlassopoulos, D., and Callaghan, P. T., ‘‘Aging, yielding, and shear banding in soft colloidal glasses,’’ Physical Review Letters 100, 128304 (2008).
[129] Frederick, C. O., and Armstrong, P. J., ‘‘A mathematical representation of the multiaxial bauschinger effect,’’ Materials at High Temperatures 24, 11-26 (2007).
139