MODELLING AND SIMULATION OF THE MECHANICAL BEHAVIOR OF HYDROGELS ZHANG XIAOXIN (B. Eng., Tongji University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 - 1 - Acknowledgements I first wish to express my sincere appreciation to my supervisor, A/P Zhang Yongwei, from Department of Materials Science and Engineering of NUS, for his guidance, advice, support and encouragement throughout the four years. Insightful discussion with him provides most of inspiration in this thesis. The range of knowledge and interests he provided was invaluable for the successful completion of this work. Working with him is one of the luckiest things in my life. I extend thanks to my co-supervisors, A/P Zeng Kaiyang from Mechanical Engineering and A/P Li Jun from Bio-Engineering, for sharing their experience and knowledge in experiments. Dr. Liu Zishun from Institute of High Performance Computing also deserves thanks for his inspiring discussion and guidance for numerical simulations. I would also like to acknowledge Dr. Guo Tianfu from Institute of High Performance Computing for reviewing part of the analysis and paper writing. Thanks also go to Department of Materials Science and Engineering of NUS for providing me the scholarship and devices, the Institute of High Performance Computing and the Institute of Materials Research and Engineering for providing resources to accomplish this work. Many friends have added much in my life in NUS. I would like to thank my senior Zhang Chunyu who taught me the first lesson in modeling and simulation. I would also like to thank Wu Zhaoxuan and Sun Lu for their supports and I collaboration within the four years. Koh Tiongsoon and Huang Zhijun also deserve thanks for their aid and suggestion in my initial way of research, and Wang Yu, Yuan Du, Gong Ying and Hu Guangxia, for their friendship. Finally, I would like to thank my family. Without their love and support, none of this would have been possible. To my grandma, my dad, my mom and especially my grandpa, I own you forever. Thank all of you. Zhang Xiaoxin National university of Singapore August 2010 II Contents Acknowledgements........................................................................................................ I Contents .......................................................................................................................III List of symbols.............................................................................................................VI List of figures................................................................................................................X Summary...................................................................................................................XVI Chapter 1: Introduction..................................................................................................1 1.1 Swelling behavior of hydrogels....................................................................1 1.1.1 Neutral hydrogel swelling equilibrium ..............................................3 1.1.2 Ionic hydrogel swelling equilibrium..................................................5 1.1.3 Physical hydrogel swelling ................................................................7 1.1.4 Patterns formation in hydrogel structures..........................................8 1.2 Motivations.................................................................................................11 1.3 Objectives...................................................................................................13 Chapter 2: Computational Methods.............................................................................16 2.1 Introduction ................................................................................................16 2.2 Existing models ..........................................................................................16 2.2.1 One phase model..............................................................................16 2.2.2 Multi-phase model ...........................................................................20 2.2.3 Transport model ...............................................................................23 2.2.4 Thermodynamics model...................................................................25 III 2.3 Model implementation ...............................................................................33 2.4 Instability investigations.............................................................................37 2.4.1 Introduction......................................................................................37 2.4.2 FEM approach..................................................................................37 Chapter 3: Hydrogel microlenses under concentrated loadings ..................................43 3.1 Introduction ................................................................................................43 3.2 Hydrogel model and numerical simulation ................................................45 3.3 Simulation results.......................................................................................46 3.3.1 Vertical inward indentation..............................................................47 3.3.2 Vertical outward indentation............................................................56 3.3.3 Lateral indentation ...........................................................................59 3.4 Discussion ..................................................................................................63 3.5 Conclusion..................................................................................................64 Chapter 4: Formation of gears through buckling multilayered film-hydrogel structures ......................................................................................................................................66 4.1 Introduction ................................................................................................66 4.2 Numerical simulation of gear formation ....................................................67 4.3 Three types of gears ...................................................................................69 4.4 Teeth number..............................................................................................73 4.5 Teeth amplitude ..........................................................................................77 4.6 Discussion and conclusion .........................................................................78 Chapter 5: Buckling behavior of a hydrogel plate with a space-varying cross-linking IV density under swelling .................................................................................................80 5.1 Introduction ................................................................................................80 5.2 Numerical simulations-three types of buckling patterns............................82 5.3 Simulation results.......................................................................................84 5.4 Scaling analysis ..........................................................................................90 5.5 Patterns for cylindrical hydrogel tube ........................................................94 5.6 Discussion ..................................................................................................96 5.7 Conclusion ..................................................................................................97 Chapter 6: Conclusion and future work.......................................................................99 6.1 Conclusion..................................................................................................99 6.2 Future work ..............................................................................................101 Bibliography ..............................................................................................................103 V List of symbols N Effective polymer chains per unit volume E Young’s modulus of hydrogel s Lame constant of hydrogel s Shear modulus of hydrogel Osmotic pressure k Boltzmann constant T Temperature N A Avogadro’s constant K Donnan coefficient F Faraday constant Flory Huggins interaction parameter v Volume of water molecule c Molar concentration of polymer units z Valence of the fixed charges Csalt Salt concentration in the external solution Stretch in principle directions s Stretch due to swelling m Stretch due to mechanical loading chain Principal stretch J Vo l u me st r a in VI W Energy of the hydrogel G Energy of the system m Number of links in polymer chains P Additional pressure by boundary condition S Cauchy stress Density of each phase Chemical potential f Frictional coefficient per unit volume vi Velocity of i phases Volume fraction of each phase e Strain tensor I Identity tensor Osmotic coefficient M Molecular weight Activity of each phase Electric potential D Water diffusivity 0 Dielectric constant iK Nominal stress with respect to dry states 'iK Nominal stress with respect to swollen states FiK Deformation gradient with respect to dry states FiK ' Deformation gradient with respect to swollen states VII I1 First invariance with respect to dry states I1 ' First invariance with respect to swollen states X K Coordinates of material point in reference configuration xi Coordinates of material point in deformed configuration