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 [email protected] 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.