Complex Inkjets: Particles, Polymers and Non-Linear Driving Claire McIlroy Submitted in accordance with the requirements for the degree of Doctor of Philosophy The University of Leeds Department of Applied Mathematics October 2014 The candidate confirms that the work submitted is her own, except where work which has formed part of jointly authored publications has been included. The contribution of the candidate and the other authors to this work has been explicitly indicated below. The candidate confirms that appropriate credit has been given within the thesis where reference has been made to the work of others. The work in Chapter 3, Section 2 of the thesis has appeared in publication as follows: C. McIlroy and O. G. Harlen Modelling Capillary Break-up of Particulate Suspensions Physics of Fluids 26 (2014) 033101. I am the principal author of this paper. I discussed the idea with my supervisor, Oliver Harlen, who provided guidance and comments on the drafts. The work in Chapter 4 of the thesis has appeared in publication as follows: C. McIlroy, O. G. Harlen and N. F. Morrison Modelling the jetting of dilute polymer solution in drop-on-demand inkjet printing Journal of Non-Newtonian Fluid Mechanics 201 (2013) 17-28. I am the principal author of this paper. The model is based on previous publication: Hoath et al. Jetting Behaviour of polymer solutions in drop-on-demand inkjet printing Journal of Rheology 56 (2012) 1109. Neil Morrison provided the DoD axisymmetric jetting model. I discussed the idea with my supervisor, Oliver Harlen, who provided guidance and comments on the drafts. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. c 2014 The University of Leeds and Claire McIlroy. The right of Claire McIlroy to be identified as Author of this work has been asserted by her in accordance with the Copyright, Designs and Patents Act 1988. iii iv For Mum and Dad. `Mathematicians are born not made.' - Henri Poincar´e v vi Acknowledgements `If I have been able to see further, it was only because I stood on the shoulders of giants' - Isaac Newton Foremost, I would like to express my sincere gratitude to my supervisor Oliver Harlen for his continuous support throughout my studies. I could not imagine a better advisor and mentor for my PhD study. His motivation and enthusiasm have made my PhD experience truly enjoyable and I have been inspired to continue academic research. I also thank my co-supervisor Mark Kelmanson. His insight gave a different perspective to my research and I have appreciated his advice and encouragement. In particular, I would like to thank Mark for his generous offer to implement his `linked-list' numerical programme into my work. This work was supported by the UK Engineering and Physical Sciences Research (EP- SRC) through grant number EP/H018913/1 Innovation in Industrial Inkjet Technology (I4T). I found the collaboration with both academic and industrial partners within the I4T project highly motivational and their feedback invaluable. In particular, I would like to thank Neil Morrison, Steve Hoath and Rafa Casterj`on-Pitafor many helpful discussions and contributions to my work. I would also like to thank Christian Clasen and his PhD student Wouter Mathues from the University of Leuven for many inspiring conversations, insight and collaboration of their experimental data. The School of Mathematics at the University of Leeds has provided years of first-class tutelage, as well as a social and friendly environment to work within. It has been a pleasure to study in the department and I have thoroughly enjoyed my time here. In particular, I would like to thank Laurence Hawke from the polymer group for helpful discussions. Finally, I would like to thank my family for their love and unremitting support and my friends, who have made life in Leeds a truly amazing and unforgettable experience. Thank you. vii viii Publications C. McIlroy and O. G. Harlen Modelling Capillary Break-up of Particulate Suspensions Physics of Fluids 26 (2014) 033101. C. McIlroy, O. G. Harlen and N. F. Morrison Modelling the jetting of dilute polymer solution in drop-on-demand inkjet printing Journal of Non-Newtonian Fluid Mechanics 201 (2013) 17-28. Publications that do not appear explicitly in the thesis are as follows: S.D. Hoath, D.C.Vadillo, O.G. Harlen, C. McIlroy, N.F. Morrison W-K. Hsiao, T.R. Tuladhar S. Jung, G.D. Martin, I.M. Hutchings Inkjet printing of weakly elastic polymer solutions Journal of Non-Newtonian Fluid Mechanics 205 (2014) 1-10. S.D. Hoath, J.R. Castrejon-Pita, W.K. Hsiao, S. Jung, G.D. Martin, I.M. Hutchings, T.R. Tuladhar, D.C. Vadillo, S.A. Butler, M.R. Mackley, C. McIlroy, N.F. Morrison, O.G. Harlen, H.N. Yow Jetting of Complex Fluids Journal of Imaging Science and Technology 57 (2013) 040403. Research presented at NIP Conferences in Digital Printing Technology and Digital Fabri- cation include: S.D. Hoath, T.R. Tuladhar, D.C. Vadillo, S.A. Butler, M.R. Mackley, C. McIlroy, O.G. Harlen, W. Hsiao, I.M. Hutchings Jetting Complex Fluids containing Pigments and Resins Proc. IS&T'2 NIP30: 30th Intl. Conf. in Digital Printing Technology and Digital Fabri- cation (IS&T, Philadelphia, PA, 2014) 30-33 S.D. Hoath, D.C. Vadillo, O.G. Harlen, C. McIlroy, N.F. Morrison, W. Hsiao, T.R. Tuladhar, S. Jung, G.D. Martin, I.M. Hutchings DoD Inkjet Printing of Weakly Elastic Polymer Solutions Proc. IS&T'2 NIP30: 30th Intl. Conf. in Digital Printing Technology and Digital Fabrication (IS&T, Philadelphia, PA, 2014) 152-156 S.D. Hoath, I.M. Hutchings, O.G. Harlen, C. McIlroy, N.F. Morrison Regimes of Poly- mer Behaviour in Drop-on-Demand Ink-Jetting Proc IS&T's NIP28: 28th Intl. Conf. in Digital Printing Technology and Digital Fabrication (IS&T, Spingfield, VA, 2012) 408-411 ix x Abstract Can inkjet technology revolutionise manufacturing processes as we know them? By extending the existing benefits of inkjet methods to attain the speed, coverage and material diversity of conventional printing, we can transform inkjet from its present status as a niche technology into a mainstream process, with the UK as a major player. However, we require a better understanding of the science underlying the formation of small droplets and the effect of complex additives. First, we highlight key inkjetting methods and discuss well-known effects that particles and polymers have on jet evolution. We describe how jetting and filament-thinning exper- iments can be used to measure key characterisation parameters and how these techniques can be modelled via an established simulation method. Second, we review the literature exploring jet stability and break-up, including the Rayleigh stability analysis and universal self-similar thinning laws. In Chapter 3, we develop a simple one-dimensional model. First, we model particulate effects on the decay of a liquid bridge and identify three thinning regimes. In particular, we describe a mechanism for acceleration, which agrees quantitatively with experiments. In contrast, the addition of viscoelasticity retards thinning processes and delays break- up. Our viscoelastic jetting model demonstrates the theoretical exponential thinning law, `beads-on-string' structures and is in quantitative agreement with axisymmetric simula- tions. In Chapter 4, we develop a simplified drop-on-demand jetting model to predict the printability of polymer solutions. We demonstrate three known jetting regimes and the predicted `jettable' concentration threshold is in quantitative agreement with experimental data. Using axisymmetric simulations, we identify a `pre-stretch' mechanism that is able to fully extend polymers within the nozzle. Consequently, we show that molecules can undergo central scission due to high strain rates at the nozzle exit. In Chapter 5, we simulate a one-dimensional continuous inkjet using an adaptive mesh technique. We explore non-linear behaviour caused by finite-amplitude modulations in the driving velocity profile, where jet stability deviates from Rayleigh behaviour. We identify a modulation range where pinching becomes `inverted', occurring upstream of the filament connecting the main drops, rather than downstream. This behaviour can be controlled by the addition of a second harmonic to the initial driving signal. Our results are compared to full axisymmetric simulations in order to incorporate the effects of nozzle geometry. xi xii Contents Acknowledgements . vii Publications . ix Abstract . xi 1 Inkjet Printing Complex Fluids 1 1.1 Inkjet Technology . 1 1.1.1 History and Development of Inkjet Printing . 1 1.1.2 Mechanisms for Drop Generation . 3 1.1.3 Fluid Properties of Standard Inks . 5 1.1.4 Jetting Complex Fluids . 7 1.2 Theory of Polymeric Fluids . 9 1.2.1 Modelling Polymer Molecules . 9 1.2.2 A Dumbbell Approximation . 11 1.2.3 General Beads-Springs Models . 14 1.2.4 The Rouse Model for Relaxation Time . 15 1.2.5 The Zimm Model for Dilute Solutions . 17 1.3 Measuring Rheological Properties . 20 1.3.1 The Capillary-Thinning Rheometer . 20 1.3.2 Rheometry using Inkjet Technology . 22 1.3.3 Polymer Behaviour in Extensional Flow . 22 1.4 Modelling and Simulating Jets . 25 1.4.1 Long-Wavelength Approximation for Newtonian Jets . 25 1.4.2 Full Numerical Simulations . 28 1.4.3 A Drop-on-Demand Inkjet Simulation . 29 1.4.4 A Continuous Inkjet Simulation . 31 1.5 Thesis Outline . 33 2 Stability and Break-up of Liquid Jets 35 2.1 History of Studying Liquid Jets . 35 2.2 The Initial Capillary Instability . 37 2.2.1 The Classic Rayleigh Stability Analysis . 37 2.2.2 A Dispersion Relation for Viscous Jets .