The University of Chicago a Study of the Evaporative
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THE UNIVERSITY OF CHICAGO A STUDY OF THE EVAPORATIVE DEPOSITION PROCESS: PIPES AND TRUNCATED TRANSPORT DYNAMICS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY RUI ZHENG CHICAGO, ILLINOIS SEPTEMBER 2007 Copyright °c 2007 by Rui Zheng All rights reserved To people from whom I have learned and bene¯ted; To the University of Chicago where I rediscovered myself. ABSTRACT We consider contact line deposition and pattern formation of a pinned evaporating thin drop. We identify and focus on the transport dynamics truncated by the maximal concentration, proposed by Dupont [1], as the single deposition mechanism. The trun- cated process, formalized as \pipe models", admits a characteristic moving shock front solution that has a robust functional form and depends only on local conditions. By applying the models, we solve the deposition process and describe the deposit density pro¯le in di®erent asymptotic regimes. In particular, near the contact line the density pro¯le follows a scaling law that is proportional to the square root of the concentration ratio, and the maximal deposit density/thickness occurs at about 2=3 of the total dry- ing time for uniform evaporation and 1=2 for di®usion-controlled evaporation. Away from the contact line, we for the ¯rst time identify the power-law decay of the deposit pro¯le with respect to the radial distance. In comparison, our work is consistent with and extends previous results [7]. We also predict features of the depinning process and multiple-ring patterns within Dupont model, and our predictions are consistent with empirical evidence. PACS. 47.55.Dz Drops and Bubbles - 68.03.Fg Evaporation and condensation - 81.15.- z Methods of deposition of ¯lms and coatings; ¯lm growth and epitaxy iv ACKNOWLEDGMENTS This work is the presentation of one of several research projects pursued under the supervision of Professor Thomas A. Witten at the Department of Physics, University of Chicago. It also serves to summarize my education experience and intellectual development at the University. Tom has been a great mentor to me in both sciences and wisdoms of everyday life. Many pursuits would not be accomplished, nor would any success be achieved, without his constant guidance, encouragement, and understanding. I am deeply a®ected by his unrelenting passion for the truth of nature, the joy of life, and the purity of mind. At the same time, I owe a lot to the close cooperation and communications among Tom's research group, which includes these former and current members: Vladimir Belyi, Jungren Huang, Nathan Krapf, Tao Liang, Yuri Popov, and Quan Zhang. I have been fortunate enough to enjoy the most intense intellectualism that the Uni- versity of Chicago has to o®er. I was able to listen to and learn from many great minds from various disciplines across the University community, and they helped to hone my wits and enrich my life. I thus pay my sincerest gratitude to Gary Becker, Peter Constantin, Todd Dupont, Peter Freund, Ilya Gruzberg, Lars Hansen, Je®rey Harvey, James Heckman, Heinrich Jaeger, Leo Kadano®, David Kutasov, Gene Mazenko, Kevin Murphy, Roger Myerson, Raghavan Narasimhan, Raghuram Rajan, Lenya Ryzhik, Thomas Sargent, Mary Silber, Robert Townsend, Shankar Venkataramani, Carlos Wag- ner, and Paul Wiegmann. I also like to thank Van Bistrow, Mark Chantell, Kate Cleary, Charlotte Coles, Joseph O'Gallagher, Stuart Gazes, Qiti Guo, and especially Nobuko McNeill for their support and help at the department. There are a lot of former and current members of the department who helped v vi to make my life more enjoyable. They are Ruoyu Bao, Xiang Cheng, Ishai Ben- Dov, Seung-Yeop Lee, Zumin Luo, Arjun Menon, Xiaohui Qu, Maria Ratajczak, Ilia Rushkin, Jing Shu, Xinliang Xu, Nataliya Yufa, Xibo Zhang, Yi Zheng, and Ling-Nan Zou. Finally I shall thank my closest friends Jingshi Hu and Ming Zhang. Jingshi, Ming, and I have spent the past ten years together in pursuit of physics. In this great instance of convergence of lives, we di®er and we compete, but most importantly we have learned to share. TABLE OF CONTENTS ABSTRACT . iv ACKNOWLEDGMENTS . v LIST OF FIGURES . ix CHAPTER 1 INTRODUCTION . 1 2 PIPE MODELS . 8 2.1 A Uniform Pipe . 8 2.2 A More Realistic Pipe: Non-Singular Case . 10 2.3 A More Realistic Pipe: Singular Case . 15 2.4 Truncated Transport Dynamics and the Shock Front . 16 3 EVAPORATIVE DEPOSITION WITH UNIFORM EVAPORATION . 20 3.1 Review of Evaporating Thin Drop Hydrodynamics . 21 3.2 Solute Volume Concentration Pro¯le . 24 3.2.1 Concentration Pro¯le in the Transport Region . 24 3.2.2 Concentration Pro¯le in the Deposition Region . 27 3.3 The Shock Front . 29 4 DEPOSIT DENSITY PROFILE WITH UNIFORM EVAPORATION . 32 4.1 Areal Density . 32 4.2 Asymptotic Time-Independent Deposition . 35 4.2.1 Deposition at Initial Drying Stage . 35 4.2.2 Deposition in Time-Independent Geometry . 37 4.3 Density Pro¯le in Spatial Asymptotic Regimes . 39 4.3.1 Near the Contact Line . 39 4.3.2 Toward the Center of the Drop . 43 5 DEPOSIT PROPERTIES WITH DIFFUSION-CONTROLLED EVAPORA- TION . 45 5.1 Hydrodynamics and Concentration Pro¯le . 45 5.2 Deposit Density Pro¯le . 47 6 COMPARISON WITH POPOV'S MODEL . 52 6.1 Thickness Pro¯le . 52 6.2 Asymptotic Deposit Properties . 55 vii viii 7 COMMENTS ON DEPINNING AND MULTIPLE-RING PATTERNS . 58 7.1 Depinning Con¯guration . 58 7.2 Formation of Multiple-Ring Patterns . 61 8 DISCUSSION . 66 8.1 Experimental Aspects of the Model . 66 8.2 Generalizations of the Model . 68 9 CONCLUSION . 71 APPENDIX A DIRECT DERIVATION OF THE SHOCK FRONT VELOCITY . 73 LIST OF FIGURES 1.1 Photographs of evaporative deposit patterns . 2 1.2 Time evolution of a pinned evaporating drop . 4 1.3 Numerical simulation of a ring-like deposit pattern . 5 1.4 Numerical results for the areal density pro¯le . 6 2.1 Sketch of a uniform pipe . 9 2.2 Sketch of a real pipe . 11 2.3 Sketch of the truncated dynamics solution . 18 3.1 Con¯guration of an evaporating thin drop . 21 3.2 Sketch of the de¯nition of m(r; t) ..................... 24 3.3 Volume concentration pro¯le in the transport region . 26 3.4 Shock front versus virtual front . 31 3.5 Time evolutions of the two fronts . 31 4.1 Areal density pro¯le . 33 4.2 Aeal density growth dynamics . 34 4.3 Sketch of a time-independent geometry . 38 4.4 Concentration pro¯le in a time-independent geometry . 39 4.5 Scaling property of the areal density pro¯le for small ±0 . 42 4.6 Scaling property of the areal density growth dynamics for small ±0 . 43 5.1 Deposition pro¯le with di®usion-controlled evaporation . 48 5.2 Deposition growth dynamics with di®usion-controlled evaporation . 48 6.1 Areal density pro¯le versus thickness pro¯le . 54 7.1 Photographs of multiple-ring patterns in Deegan's experiments . 64 A.1 Sketch of the de¯nition of M0 and Mt ................... 74 ix CHAPTER 1 INTRODUCTION Evaporative contact line deposition, also known as the \co®ee-drop e®ect", has been the subject of several recent papers [2, 3, 4, 5, 6, 7]. The physical problem originates from a simple phenomenon of everyday life: when a drop containing a solute such as co®ee dries on a surface, the solute is driven to the contact line and forms a characteristic ring pattern (Fig. 1.1). The basic mechanism behind the phenomenon is now well understood [2]: the drying drop is pinned at the contact line so that there must be an outward capillary flow to replace the liquid evaporating at the edge, and this flow carries solute particles to the contact line where the deposit is formed. This simple phenomenon is important in many scienti¯c and industrial applications [8, 9, 10, 11, 12, 13, 14]. The evaporation mechanism can create very ¯ne lines of deposit in a robust way that requires no explicit forming. It can be used to concentrate material strongly in a controllable way. It also creates capillary flow patterns which are useful for processing polyatomic solutes like DNA [15, 16, 17]. The deposition process and the resulting patterns are a®ected by various physical and chemical conditions. The wetting property of the substrate and surface roughness a®ect the geometry of an evaporating drop [18, 19]. Also, within the drop there can be a convective flow induced by gradient in temperature, surfactant concentration, or surface tension [20, 21, 22, 23], and it can lead to very complicated, even fractal- like, deposit patterns. Furthermore, some mechanisms, such as contact line depinning [5, 24], are still not clear, and there are no satisfactory explanations for multiple-ring deposit patterns [25, 26]. Several models have been established under the common assumptions of slow evapo- ration and creeping flow so that quasi-equilibrium dynamics is warranted: the dynamics 1 2 Figure 1.1: Evaporative deposit patterns. Left: photograph of a co®ee stain. Right: a video micrograph of one-micron polystyrene spheres migrating toward the contact line in evaporating water. (Courtesy Robert Deegan) is determined by the time evolution of the equilibrium properties of the evaporating system. These models use di®erent conditions and assumptions on such properties as geometry, evaporation pro¯le, and deposition criteria. In addition to circular thin drop, there have been studies on contact line deposition in an angular region [27, 28, 29], where the nontrivial geometry at the tip of the angle induces a crossover in deposit properties.