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Vibrating-Wire Rheometry Western University Scholarship@Western Electronic Thesis and Dissertation Repository 8-20-2018 2:30 PM Vibrating-wire Rheometry Cameron C. Hopkins The University of Western Ontario Supervisor de Bruyn, John R. The University of Western Ontario Graduate Program in Physics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Cameron C. Hopkins 2018 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Statistical, Nonlinear, and Soft Matter Physics Commons Recommended Citation Hopkins, Cameron C., "Vibrating-wire Rheometry" (2018). Electronic Thesis and Dissertation Repository. 5584. https://ir.lib.uwo.ca/etd/5584 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract This thesis consists of two projects on the behaviour of a novel vibrating-wire rheometer and a third project studying the gelation dynamics of aqueous solutions of Pluronic F127. In the first study, we use COMSOL to perform two-dimensional simulations of the oscillations of a wire in Newtonian and shear-thinning fluids. Our results show that the resonant behaviour of the wire agrees well with the theory of a wire vibrating in Newtonian fluids. In shear-thinning fluids, we find resonant behaviour similar to that in Newtonian fluids. In addition, we find that the shear-rate and viscosity in the fluid vary significantly in both space and time. We find that the resonant behaviour of the wire can be well described by the theory of a wire vibrating in a Newtonian fluid if the viscosity in the theory is set equal to the viscosity averaged over the cir- cumference of the wire and over one period of the wire’s oscillation at the resonant frequency. In the second study, we present the design and operation of our vibrating-wire rheometer and use it to measure the properties of Newtonian and viscoelastic fluids. We find that our de- vice can accurately measure small viscosities. In homogeneous polymer solutions, we find that the viscoelastic moduli measured using our device are consistent with measurements at lower frequencies using a shear rheometer. In addition, we find that our device can measure the microrheological properties of an aging Laponite clay suspension that is heterogeneous on the micron scale. In the third project, we study the gelation dynamics of aqueous solutions of Pluronic F127. Using shear rheometry we find that when slowly heating the solutions, they un- dergo a transition from sol to gel around room temperature, followed by a gel to sol transition at a higher temperature. Our results show that these transitions take place over a temperature range of a few degrees. When cooling the solutions, the reverse transitions occur over a larger range of temperature and the width in temperature of the gel phase is larger. At temperatures near the phase transitions, we find that the rheological relaxation time becomes very long. i Keywords: Rheology, rheometry, viscometry, vibrating wire, complex fluids, Pluronic, COMSOL ii Co-Authorship Statement Chapter 2 has been published as: Cameron C. Hopkins and John R. de Bruyn, Vibrating Wire Rheometry, Journal of Non-Newtonian Fluid Mechanics, 238, 2016. I did all of the experiments, data analysis and writing of early drafts of the paper. John R. de Bruyn supervised the project and provided editorial comments on the paper. A version of chapter 3 is being prepared for submission. I did all of the simulations, data analysis, and writing of the early drafts of the paper. John R. de Bruyn supervised the project and provided editorial comments. A version of chapter 4 will be submitted for publication in the Journal of Rheology. I did all of the experiments, data analysis, and writing of the early drafts of the paper. John R. de Bruyn supervised the project and provided editorial comments. iii Acknowledgments First, I must thank my advisor, Dr. John R. de Bruyn. I have grown considerably, both personally and professionally, over my time here in no small part thanks to his mentorship and patience. His guidance over the years has shaped me into an effective scientist. I feel like I am well prepared for a future in science thanks to him. I gratefully acknowledge the funding received towards my PhD from the National Sciences and Engineering Research Council, Ontario Graduate Scholarship, University of Western On- tario, and the Canada Foundation for Innovation. The work in chapter 3 was made possible by the facilities of the Shared Hierarchical Aca- demic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada, and the provision of the COMSOL Multiphysics license by CMC Microsystems. I am grateful for the guidance my advisory committee, Dr. Silvia Mittler and Dr. Michael G. Cottom, have given me over the years. Frank Van Sas and Brian Dalrymple manufactured the parts for my vibrating-wire rheome- ter, and Doug Hie helped develop the electronics. Without their expertise this project would not have been possible. I want to thank fellow graduate student Nirosh Getangama, and former colleagues Yang Liu and Masha Goiko, for their helpful discussions over the course of my PhD. Thank you to my parents and brother for their unconditional support and encouragement over the duration of my studies. Finally, I must thank my wife, Ellie, for her unwavering support and patience over the last year. I would not have been able to do this without her. iv Contents Abstract i Co-Authorship Statement iii Acknowledgements iv List of Figures viii List of Tables xviii List of Symbols xix List of Abbreviations xxi 1 Introduction 1 1.1 Fluids . 1 1.1.1 Shear Thinning and Shear Thickening . 3 1.1.2 Yield Stress . 5 1.1.3 Viscoelasticity . 7 1.1.4 Small-Amplitude Oscillatory Shear . 8 1.1.5 Shear Rheometry . 11 1.2 Summary of Present Work . 12 1.2.1 Overview . 12 v 1.2.2 Numerical Simulations of the Forced Oscillations of a Wire in a Fluid . 13 1.2.3 Vibrating-wire Rheometry . 13 1.2.4 Gelation Dynamics of Aqueous Solutions of Pluronic F127 . 14 1.2.5 Appendices . 14 Bibliography . 15 2 Numerical Simulations of the Forced Oscillations of a Wire in a Fluid 18 2.1 Introduction . 18 2.2 Theory . 20 2.3 Geometry . 21 2.4 Model . 23 2.5 Results . 29 2.5.1 Newtonian Fluids . 29 2.5.2 Shear-thinning fluids . 34 2.6 Discussion . 46 2.7 Conclusion . 48 Bibliography . 49 3 Vibrating-wire Rheometry 52 3.1 Introduction . 52 3.2 Theory . 54 3.2.1 Newtonian fluids . 54 3.2.2 Viscoelastic fluids . 56 3.3 Experiment . 56 3.3.1 Vibrating-wire rheometer . 56 3.3.2 Rheometric Measurements . 59 3.3.3 Sample Preparation . 60 3.4 Results . 61 vi 3.4.1 Newtonian fluids . 61 3.4.2 Viscoelastic polymer solutions . 62 3.4.3 Aging and gelation of a clay suspension . 65 3.5 Discussion and Conclusion . 69 Bibliography . 70 4 Gelation Dynamics of Aqueous Solutions of Pluronic F127 73 4.1 Introduction . 73 4.2 Materials and Methods . 76 4.3 Results . 78 4.3.1 Dynamic Rheology . 78 4.3.2 Phase Transitions . 81 4.3.3 Rheological Relaxation . 83 4.4 Discussion . 92 4.5 Conclusions . 97 Bibliography . 98 5 Discussion, Conclusions, and Future Work 101 Bibliography . 105 Appendix A Additional Measurements and Analysis 106 A.1 Temperature Dependence . 106 A.2 Dependence of the resonance curve on G0 and G00 . 111 Bibliography . 112 Appendix B Vibrating-wire Rheometry of Aqueous Solutions of Pluronic F127 113 Curriculum Vitae 120 vii List of Figures 1.1 A cross-section segment of a material subject to a shear stress τ, with the bot- tom edge held stationary. The material will experience a strain γ = dl=l. 2 1.2 Comparison of the viscosity vs. shear rate predicted by the Carreau model and the power-law model. For both models, n = 0:5, while for the Carreau model, −3 −4 η1 = 10 Pa·s, η0 = 1 Pa·s, λ = 10 s. ..................... 6 1.3 Graphical schematic of the Maxwell model of a viscoelastic fluid, consisting of a viscous dashpot and an elastic spring connected in series. λM = η/G is the characteristic relaxation time of the model. 7 0 00 5 1.4 G and G vs. ! for the Maxwell model with λM = 0:1 s and η = 10 Pa·s. The −1 crossover frequency ! = λM is indicated with an arrow. 10 1.5 Common geometries used for shear rheometry. (a) Cone-and-Plate, (b) Parallel Plate, and (c) the Couette, or concentric cylinder, geometry. [2] . 11 2.1 (a) The simulation geometry. Fluid space is coloured blue. The tungsten wire is shaded gray. (b) The simulation mesh. (c) A magnified view of the mesh near the wire. The region shown in (c) is approximately square, with side length 16Rw. 21 2.2 Plots of viscosity vs. shear rate for the Carreau model with η0 = 1 Pa·s η1 = 10−3 Pa·s, and several different values of λ and n. 26 viii 2.3 The x-components of the (a) position, (b) velocity, and (c) acceleration of the center of the wire as a function of time for a simulation of a wire vibrating in a Newtonian fluid with η = 3 × 10−3 Pa·s and f = 900 Hz. The lines connect the data points to aid the eye. Note the difference in units between the plots.
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