Effect of substrate cooling and droplet shape andcomposition on the droplet evaporation and the depositionof particlesbyVahid BazarganM.A.Sc., Mechanical Engineering, The University of British Columbia, 2008B.Sc., Mechanical Engineering, Sharif University of Technology, 2006B.Sc., Chemical & Petroleum Engineering, Sharif University of Technology, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University Of British Columbia(Vancouver)March 2014c? Vahid Bazargan, 2014AbstractSessile droplets are liquid droplets resting on a flat substrate. During the evapora-tion of small sessile droplets, the contact line of the droplet undergoes two differentstages: pinned stage with fixed contact area and de-pinned stage with fixed con-tact angle. An evaporation with a pinned contact line produces a flow inside thedroplet toward the contact line. This flow carries particles and deposits them nearthe contact line. This causes the commonly observed ?coffee-ring? phenomenon.This thesis provides a study of the evaporation process and the evaporation-inducedflow of sessile droplet and brings insights into the deposition of particles from col-loidal suspensions.Here we first study the evaporation of small sessile droplets and discuss the impor-tance of the thermal conductivity of the substrate on the evaporation process. Weshow how current evaporation models produce a significant error for droplet sizesbelow 500 ?m. Our model includes thermal effects, in particular, it includes thethermal conductivity of the substrate that provides heat to the droplet to balancethe latent heat of evaporation. It considers the whole time of evaporation with thepinned and the de-pinned stages by defining a virtual movement of the contact linethat is related to the evolution of the contact angle and is based on experimentalresults. Our model is in agreement with experimental results for droplets smallerthan 500 ?m with an error below 2%.Furthermore, we study the evaporation of line droplets with finite sizes and discussthe complex behavior of the contact lines during evaporation. We apply an energyformulation and show that the contact line starts receding from the two ends ofline droplets with a contact angle above the receding contact angle of sphericaldroplets. And then we show the evaporation-induced flow inside the line droplets.iiFinally, we discuss the behavior of the contact line under the presence of surfactantand discuss the Marangoni flow effects on the deposition of the particles. We showthat the thermal Marangoni effect affects the amount of the particles depositednear the contact line, where a lower substrate temperature corresponds to a largeramount of particles depositing near the contact line.iiiPrefaceThis thesis is original, independent work by the author, Vahid Bazargan. This workwas supported by NSERC through the Strategic Projects Grant program and theDiscovery Grant program, The University of British Columbia through the FourYear Fellowship program, and the Howard Webster Foundation.The experimental setup in Chapter 3 was designed and built primarily by myself,except for the temperature control system with a Peltier element in Section 3.2.2that I used a code written by Simon Beyer and for the Graphical User Interfacecode in Section 3.2.1 that was written by Eton Leun as part of his EECE 496course project. A licensed copy of FLOW-3D software was used for numericalsimulation. The numerical model in Section 3.3 could not have been completedwithout the useful suggestions of Dan Milano.A version of Chapter 4 is submitted for publication. I performed all parts of theresearch and I wrote the manuscript.This work in Chapter 5 was carried out in conjunction with Charles Rabideau. Hesolved the energy equation for the line droplets in Section 5.2.1. A version of thischapter is submitted for publication. I am responsible for all major areas of conceptformation and analysis, as well as the majority of manuscript composition.A version of Chapter 6 is in preparation to be submitted for publication. I amresponsible for conducting all parts of the research and preparing the manuscript.The entire manuscript was prepared using the LATEXpackage.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Sessile droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . 52 Evaporation of sessile droplets . . . . . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Basic models of evaporation . . . . . . . . . . . . . . . . . . . . 82.3 Evaporation-induced flow . . . . . . . . . . . . . . . . . . . . . . 182.4 Drying of colloidal suspension droplets . . . . . . . . . . . . . . 262.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29v3 Experimental and numerical approach . . . . . . . . . . . . . . . . 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Printing stage . . . . . . . . . . . . . . . . . . . . . . . . 323.2.2 Droplet visualization and measurements . . . . . . . . . . 333.3 Numerical approach . . . . . . . . . . . . . . . . . . . . . . . . . 364 Effect of substrate conductivity on the evaporation of small sessiledroplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Numerical model . . . . . . . . . . . . . . . . . . . . . . . . . . 444.3 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 Evaporation of line droplets on a substrate . . . . . . . . . . . . . . 535.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2 Model development . . . . . . . . . . . . . . . . . . . . . . . . . 555.2.1 Contact line behavior . . . . . . . . . . . . . . . . . . . . 565.2.2 Velocity field . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706 Effect of surfactants and substrate temperature on contact line be-havior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 756.3.1 Effect of surfactant and particle concentration on the con-tact line behavior . . . . . . . . . . . . . . . . . . . . . . 766.3.2 Effect of substrate temperature on the contact line behaviorand particle deposition . . . . . . . . . . . . . . . . . . . 80vi6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . 877.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91A Geometrical calculations of the sessile droplets . . . . . . . . . . . . 98A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98A.2 Spherical cap calculations . . . . . . . . . . . . . . . . . . . . . . 98A.3 Sectional parameters across the droplet . . . . . . . . . . . . . . . 101viiList of TablesTable 2.1 Coefficient of Snow?s finite series in Equation 2.1 [1] . . . . . 9Table 3.1 Coefficients for pressure and temperature at the interface inEquation 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 39Table 3.2 The safety factor coefficient for each physics module to ensurenumerical stability. . . . . . . . . . . . . . . . . . . . . . . . 41viiiList of FiguresFigure 1.1 A water droplet with a radius of 1 mm resting on a glass sub-strate. The surface of the droplet takes on a spherical capshape. The contact angle ? is defined by the balance of theinterfacial forces. . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 2.1 Evaporation modes of sessile droplets on a substrate: (a) evap-oration at constant contact angle (de-pinned stage) and (b)evaporation at constant contact area (pinned stage) . . . . . . 8Figure 2.2 A sessil droplet with its image can be profiled as the equicon-vex lens formed by two intersecting spheres with radius of a. . 9Figure 2.3 The droplet life time for both evaporation modes derived fromEquation 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . 10Figure 2.4 A probability of escape for vapor molecules at two differentsites of the surface of the droplet for diffusion controlled evap-oration. The random walk path initiated from a vapor moleculeis more likely to result in a return to the surface if the startingpoint is further away from the edge of the droplet. . . . . . . . 11Figure 2.5 Schematic of the sessile droplet on a substrate. The evapora-tion rate at the surface of the droplet is enhanced toward theedge of the droplet. . . . . . . . . . . . . . . . . . . . . . . . 12ixFigure 2.6 The domain mesh (a) and the solution of the Laplace equationfor diffusion of the water vapor molecule with the concentra-tion of Cv = 1.9?10?8 g/mm3 at the surface of the droplet intothe ambient air with the relative humidity of 55%, i.e. ? = 0.55(b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.7 The evaporation flux, J for a water droplet with the radius ofR = 0.17 mm into the ambient air with the relative humidity of55% at room temperature for different contact angles of ? =90? and ? = 60?. . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.8 (a) The rate of change of the contact angle and the evapora-tion mass flux during the evaporation at constant contact area(fully pinned stage) with the initial contact angle of ?c = 40;(b) contact angle during evaporation versus t?. . . . . . . . . . 17Figure 2.9 An internal flow is generated inside the sessile droplet as aresult of evaporation that carries the fluid toward the corners:(a) top view of a ring deposition after drying a coffee droplet,(b) outward evaporation-induced flow inside a sessile droplet . 19Figure 2.10 The average evaporation-induced flow in a vertical fluid col-umn at a distance r from the center. The flux leaving the topsurface is equal the to local evaporation rate at the surface J(r, t). 20Figure 2.11 The air-liquid interfaces moves as a result of the evaporation.Using the conservation of mass for an infinitesimal element atthe surface, the velocity of the fluid normal to the boundary,~vnwas found by subtracting the evaporation flux, ~J/? from theinterface velocity ~Un. . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.12 The velocity field (left) and streamlines (right) of potentialflow inside a half-cylinder as a result of evaporation. . . . . . 23Figure 2.13 The radial velocity component from potential flow (left) andStokes flow (right). . . . . . . . . . . . . . . . . . . . . . . . 24Figure 2.14 Evaporation-induced flow carries particles toward the cornersand forms a deposit phase near the contact line of a sessiledroplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28xFigure 2.15 Marangoni flow produces an inward flow that counters the out-ward evaporation-induced flow for sessile droplets during thepinned stage. . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.1 The portable micro printing setup. A motorized linear stagefrom Zaber Technologies Inc. was used to control the placeand speed of the micro nozzle. . . . . . . . . . . . . . . . . . 32Figure 3.2 The experimental setup for investigating the contact line andcontact angle during the evaporation. The experiment was il-luminated using LED based background lighting. . . . . . . . 34Figure 3.3 The horizontal velocity near the substrate was observed usingan inverted microscope stage and the velocity was measuredusing PIV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 3.4 The temperature of the substrate is controlled using a Peltierdevice in a closed loop temperature control circuit. The signalfrom the temperature sensor attached to the substrate is used toprovide the feedback signal to the controller. . . . . . . . . . 36Figure 3.5 Temperature measurement of the surface of the droplet usingan IR camera. During evaporation at room temperature, theenthalpy of the liquid near the surface reduces to transformliquid molecules into the vapor phase. That causes a drop ofthe temperature at the surface of the droplet during evaporation. 37Figure 3.6 The schematic of the mesh structure using FAVOR techniquein FLOW-3D. . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.1 The radius and contact angle of a water droplet on an acryliccoated plastic slide during evaporation as a function of timeshowing two modes of evaporation, at 25?C and relative hu-midity of 55%. . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.2 The contact angle and initial radius of water droplets on Cu,glass and plastic slides. All the measurements were within amargin of error of less than 2%. . . . . . . . . . . . . . . . . 46Figure 4.3 Droplet volume over time for different substrates. . . . . . . . 46xiFigure 4.4 Total evaporation time, as measured by experiments (gray) andas predicted by the Popov model (black). . . . . . . . . . . . 47Figure 4.5 Comparing the measured evaporation rates with the valuesfrom the basic model from Equation 4.1. . . . . . . . . . . . . 47Figure 4.6 Temperature contours inside the substrate adjacent to the droplet 48Figure 4.7 The effect of substrate cooling on the evaporation rate, the ba-sic model shows the same value for all substrates. . . . . . . . 48Figure 4.8 The schematic of the heat transfer through an element acrossthe droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 5.1 Line droplet from (a) side-view and (b) top-view . . . . . . . 56Figure 5.2 Length L and contact angle ? of a line droplet with the initialaspect ratio of L0/R0 = 10. . . . . . . . . . . . . . . . . . . . 64Figure 5.3 The change of the contact angle ? and the line length L withdroplet volume for a shrinking droplet with an initial aspectratio of L0/R0 = 10. . . . . . . . . . . . . . . . . . . . . . . 65Figure 5.4 The evaporation flux on the centerline of the droplet and thespherical end for the droplet with the initial aspect ratio ofL0/R0 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Figure 5.5 The evaporation flux across the line droplet of the shrinkingdroplet for an initial aspect ratio of L0/R0 = 10 at L = 8.86 mm. 66Figure 5.6 The contour plots of the evaporation induced flow magnitudein m/s in a line droplet with the initial aspect ratio of L0/R0 = 10. 67Figure 5.7 The outward velocity vectors across the droplet at (a) the zxplane and (b) zy plane. Cross sections are taken at the middleof the droplet. . . . . . . . . . . . . . . . . . . . . . . . . . . 67Figure 5.8 The contact angle versus the aspect ratio for the fully and par-tially pinned stages of the evaporation. The relationship be-tween these two in the partially pinned stage can be seen to beindependent of the initial aspect ratio and radius. The initialand receding contact angles are 60? and 20?, respectively. . . . 68xiiFigure 5.9 The contact angle as a function of aspect ratio in the partiallypinned stage of line droplet evaporation for line droplets fordifferent receding contact angles. R0 and L0 are fixed to 1 mmand 10 mm, respectively. . . . . . . . . . . . . . . . . . . . . 69Figure 6.1 Interfacial forces on the contact line during the evaporation ofa sessile droplet. . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 6.2 Signal properties of the micro dispenser unit. The tip of thenozzle was 1 mm above the substrate and we used 50 cycles toachieve a droplet with the diameter of 250 ?m. . . . . . . . . 74Figure 6.3 Measured radius and contact angle of a sessile droplet of a col-loidal suspension containing S1 and surfactant at concentra-tions of 0.05 wt% and 0.0005 wt%, respectively, during evap-oration at T = 24 ?C. . . . . . . . . . . . . . . . . . . . . . . 76Figure 6.4 The effect of the concentration of S2 on the evaporation stages. 77Figure 6.5 The effect of low concentrations of surfactant on the behaviorof the contact line. . . . . . . . . . . . . . . . . . . . . . . . 78Figure 6.6 Normalized radius and contact angle measurement of the mix-tures with higher amount of surfactants. . . . . . . . . . . . . 79Figure 6.7 The effect of the concentration of S1 on evaporation stages. . 79Figure 6.8 The effect of the substrate temperature on the behavior of thecontact line for solutions (a) without and (b) with surfactantand particles. . . . . . . . . . . . . . . . . . . . . . . . . . . 81Figure 6.9 Particle deposition from a sessile droplet with 0.05% S1 and0.0005% surfactant at different substrate temperatures. Theinitial radius of the droplets was R0 = 250 ?m. . . . . . . . . 82Figure 6.10 The normalized width of particle rings for droplets at differentsubstrate temperature; the size of the error bars is the standarddeviation of the average ring width for roughly 20 dropletswith 0.05 wt% S1 and 0.0005 wt% surfactant for each tem-perature as shown partly in Figure 6.9. . . . . . . . . . . . . . 83xiiiFigure 6.11 Evaporation induced flow for cold (a) and hot (b) substrates;Marangoni flow at the surface of the droplet carries the par-ticles to the contact line with a cold substrate while it drivesthem away from the contact line with a hot substrate. . . . . . 85Figure A.1 Schematic of a sessile droplet on a substrate . . . . . . . . . . 99Figure A.2 Sectional parameters of a droplet in a sessile droplet. . . . . . 102xivAcknowledgmentsThis thesis, as with everything I have achieved in my life, would not have beenpossible without the encouragement, help and support of many people. My firstdebt of gratitude is to my supervisor Prof. Boris Stoeber for his ongoing support,generous encouragement and sound instruction. I can think of no other way inwhich I could have spent my student life that would have brought such satisfactionas working with him. I count myself incredibly fortunate.I thank The University of British Columbia for support via the Four Year Fel-lowship program and the Howard Webster Foundation for support via R. HowardWebster Fellowships which has afforded me time to focus on this research. I alsothank National Science and Engineering Research Council for supporting this workthrough the Strategic Projects Grant and the Discovery Grant programs.I would like to thank Prof. Konrad Walus, Prof. Karen Cheung, Prof. HongshenMa and Prof. Mu Chiao for sharing their lab equipment, and for being generouswith their time and knowledge. I also thank Prof. Savvas Hatzikiriakos and Prof.Dana Grecov for their comments on sections of this thesis. And a special thanksto my good friend and roommate Charles Rabideau for his help and detailed com-ments on chapter five.I owe a great debt to my friends for their encouragement and help. I would like tothank especially my wonderful friends at Green College, one of the best places inthe world and a place where I have encountered beautiful ideas and friendship. Iwould also like to thank Green College principal, Prof. Mark Vessey who bright-ened my day on many occasions with his helpful encouragement and wonderfulspeeches.Finally, I would like to thank my Dad and my Mom for their support and their pa-xvtience during my prolonged absence, and my brothers, Hamid and Mohammad. Iwould like to acknowledge my grandmother here for her incredible help while mymother brought up triplets. My grandmother passed away while I was writing thisthesis. My heartfelt gratitude goes to her for her kindness and her patience, and forbeing relentless in seeking the best for us.xviDedicationTo Mohammad JahanaraxviiChapter 1Introduction1.1 Preliminary remarksTechnologies of the deposition of micro and nano particles from their suspensionshas gained great scientific interest in recent years. The small size of these particlesmakes them effectively a bridge between bulk materials and atomic or molecularstructures. Several methods have been proposed for structured deposition of microand nano particles including gas flow, magnetic alignment, electrospray, quasi-2Dnematic phase alignment, and electro spinning for different applications [2]. How-ever the scalability of these methods for the precised placing of particles over largeareas is still a challenge.One of the most conventional ways to achieve a desired pattern of coating or place-ment of micro and nano particles on a large surface has been the controlled de-position of these particles from suspensions [3?8]. That is mainly owing to thefact that is relatively easy to achieve stable suspensions of micro and nano parti-cles because Brownian motion and surface forces are strong enough to overcomedifferences in density. To achieve this goal, a technology for the systematic depo-sition of micro and nano particles from their suspending fluid is required. Sharmaet al. [3, 9] showed that the controlled evaporation of the droplets containing nanoparticles lead to the precise placement of these particles with a precision on theorder of a few nano meters. They generated lines of droplets in rectangular polarregions previously generated on a substrate. They showed that by pinning the line1droplet contact line to the polar pattern edges, a flow is generated during the evap-oration that carries the nano particles toward the edges and places them uniformlyonto the substrate. Dugas et al. [10] used the droplets of DNA solutions to formhomogeneous oligoprobe spot deposition. They used a mixture of surfactants inthe initial solution to control the behavior of the droplet shape during evaporationand to achieve the desired internal flow inside the droplet during evaporation. Thisflow generated during the evaporation of droplets was used by many researchers invarious applications [11?15]. Therefore, investigating the evaporation of dropletsand studying the internal flow during evaporation will be a major contribution tothe understanding of this problem and can lead to many applications for particledeposition that were never possible before.1.2 Sessile dropletsSessile droplets are liquid droplets resting on a substrate in still air. They are notconnected to or suspended from any fluid stream. The surface of the droplets isbounded by a gas phase that can usually be the atmospheric air. For droplets witha radius smaller than 1 mm, the effect of gravity is negligible and the shape of thedroplet is defined by the surface tension that tends to make a spherical cap shape,as shown in Figure 1.1. The contact angle of the droplet is defined by the balanceof the interfacial forces, i.e. fluid-gas (?FG), substrate-gas (?SG) and substrate-fluid(?SF ).Here we study the sessile droplets resting on a flat substrate that is indiffusibleto the liquid. While there is no diffusion through the substrate, depending on thecondition of the gas phase, a mass transfer between these the fluid and gas phasesmay occur that causes loss (evaporation) or accumulation (condensation) of thedroplet volume. At room temperature, if the ratio of liquid vapor pressure in air tothe saturated vapor pressure (known as ?relative humidity? if the liquid is water) isless than 1, the number of liquid molecules leaving the surface of the droplet anddiffusing into the ambient air is greater than the number of the vapor moleculesdiffusing into the droplet. This net transfer of liquid molecules to the air causesthe loss of liquid volume. Accordingly, the evaporation process follows these twoseparate steps:21 mmSpherical cap shapeDropletSubstrateFigure 1.1: A water droplet with a radius of 1 mm resting on a glass substrate.The surface of the droplet takes on a spherical cap shape. The contactangle ? is defined by the balance of the interfacial forces.1. Kinetics: where the liquid molecule at the fluid-gas interface receives theenthalpy of evaporation (a.k.a latent heat of evaporation), and transforms toa vapor molecule at the surface,2. Diffusion: where the vapor molecule at the surface leaves the surface andgoes to the ambient air.The evaporation happens after the completion of these two steps. If one process ismuch slower than the other, it controls the evaporation process. Hence the evapo-ration can be classified to two different types: Kinetically controlled evaporation:where the kinetic transformation happens at a slower rate than the diffusion, andDiffusion-controlled evaporation: where the diffusive relaxation of the saturatedvapor layer immediately above the drop defines the rate limiting step. In the for-mer case, the evaporation is uniform over the surface, while in the later case, aswill be shown in Chapter 2, the evaporation is strongly enhanced near the edge ofthe droplet.For a small water droplet with a radius less than 1 mm (as shown in Figure 1.1),the time for the diffusion process at the surface of the droplet scales with ? R2/D,where R is the radius of the droplet D is the vapor diffusivity in air ( 26.1 mm2/s).The time for the kinetic process, ?k, is the water molecule transformation time scaleand can be calculated from the Hertz-Knudsen equation [16]. For water molecules3at room temperature (? 24?C), ?k is in the order of 10?10 s which is much fasterthan the diffusion process (? 1 s). Thus, the assumption of the diffusion-controlledevaporation is justified.The enhanced evaporation rate of sessile droplets with diffusion-controlled evap-oration generated a non-uniform evaporation flux at the surface. This causes aninternal flow inside the droplet. The details of this flow will be studied in thenext chapter in Section 2.3. During evaporation of colloidal suspensions, this flowcarries micro and nano particles and eventually deposits them on the substrate.Hence, understanding the evaporation-induced flow during evaporation along withthe droplet shape during evaporation is a key part in understanding the depositionof colloidal suspensions.1.3 Research objectivesThe aim of this study is to investigate the diffusion controlled evaporation in smallsessile droplets in the order of few hundred microns in diameter. The main param-eters being particularly effective on the evaporation process of such small dropletswas the main focus on our study. This is a key point in understanding the de-position of the particles along with the understanding of the shape of the dropletduring evaporation. This objective will be achieved by solving the following threesub-problems:1. We first review the current studies on the evaporation of pure liquids on aflat substrate and study the models that have so far been used to calculatethe diffusion-controlled evaporation at the surface. We particularly show thestrong influence of the substrate cooling on the evaporation of small dropletsthat seems to be missing in the previous studies.2. The study of liquid line droplets generated by inkjet micro patterning will bethe next goal. This particularly will be useful for the growing area of inkjetprinting technology. We study the behavior of line droplets with finite aspectratio and will discuss the evaporation time as well as the internal flow duringthe evaporation. The internal flow produced in such elongated droplets isa complicated combination of the axial flow and the outward flow along4and across the droplet, respectively. The duration of such flows that areresponsible for deposition of particles will be studied.3. Furthermore, we study the shape of the droplets during evaporation of col-loidal solutions and study the effect of surfactants in the evaporation process.We will show how the small addition of surfactants can change the deposi-tion pattern. This will be of a great interest in many scientific applicationsthat require controlled deposition of particles on solid surfaces. We achievethis goal through creating appropriate substrate-liquid interactions, the pres-ence of surface roughness and chemical heterogeneity. We also study theeffect of substrate temperature in the deposition process and will discuss itseffect on the droplet shape and the outward flow during evaporation.These studies involve experimental and theoretical components. We will designa stage for printing the droplets and an optical setup for observing them duringevaporation. We will take a numerical approach to study the evaporation of ses-sile droplets by developing a model that includes the physics associated with theproblem.1.4 Organization of the thesisThis thesis is organized in the following order:Chapter 1: The sessile droplets are introduced and their significant applications inthe deposition processes are discussed.Chapter 2: The basic models of evaporation for sessile droplets are summarized,and their limitations are discussed. The evaporation-induced flow during diffusion-controlled evaporation using the basic models is studied through reviewing dif-ferent models including potential flow, Stokes flow and Navier-Stokes equations.Furthermore, the deposition of particles as a result of this flow and the effect ofMarangoni flow on the deposition is discussed.Chapter 3: The experimental setup for the studies in the next chapters is discussed.The visualization methods of small droplets for each part of the study is discussedand the measurement equipment is reviewed. Furthermore, the numerical approachto model the evaporation of the spherical sessile droplets is presented followed by5the assumptions in our study.Chapter 4: The effect of the substrate cooling on the evaporation of small dropletsis discussed. We particularly show how the basic models produce a significant errorin modeling the evaporation of small sessile droplets by using both experimentaland numerical approaches. A dimensionless number is presented that discusseswhere the substrate conductivity becomes important.Chapter 5: The evaporation of line droplets with different aspect ratio and theevaporation-induced flow inside these droplets are discussed.Chapter 6: The effect of surface tension gradients on evaporation of colloidal sus-pensions is discussed. We specifically show how the shape of the droplet duringevaporation can be controlled using surfactants. Furthermore, the effect of thermalMarangoni flow on the deposition is discussed.Chapter 7: The conclusions drawn from the investigation are presented and furtherdevelopments are discussed.6Chapter 2Evaporation of sessile droplets2.1 IntroductionThe evaporation of liquid drops on a substrate is a flow problem with a free sur-face and plays a key role in many scientific processes. This problem has beenextensively studied in the literature. Topics regarding the evaporation mechanism,the change of the droplet shape during evaporation, the evaporation-induced flow,drying and deposition patterns caused by the evaporation of colloidal suspensionshave been discussed for many different applications. During evaporation, the liquidmolecules transform to a vapor phase upon receiving the latent heat of evaporationand form a vapor layer at the surface of the droplet (kinetic stage). The vapormolecules at the surface diffuse into the ambient air (diffusion stage). Here wefocus on the evaporation mechanism and review the main evaporation models thathave been suggested for sessile droplets (Section 2.2). We then discuss the internalflow generated as a result of evaporation and will review the existing studies mod-eling this flow in Section 2.3. We then discuss the current models for the drying ofsessile droplets of colloidal suspensions and their deposition as a result of evapora-tion (Section 2.3). Eventually we address the limitation of the basic models and theimprovements that our model will give to the basic models (that will be discussedin the next chapter in Section 3.3).72.2 Basic models of evaporationThe first basic model for the evaporation of sessile droplets was pioneered by Pick-nett and Bexon in 1977 [1], where they predicted the evaporation of drops on non-absorbing surfaces by distinguishing two separate modes of evaporation: evapo-ration with a constant contact angle and evaporation with a constant contact area.These evaporation modes are shown in Figure 2.1.Substrate()(a) Constant contact angleSubstrate()(b) Constant contact areaFigure 2.1: Evaporation modes of sessile droplets on a substrate: (a) evapo-ration at constant contact angle (de-pinned stage) and (b) evaporation atconstant contact area (pinned stage)In the evaporation with constant contact angle (Figure 2.1a), the contact line movesand the contact area of the droplet shrinks (de-pinned contact line). In evaporationwith constant contact area (Figure 2.1b), the contact line stays fixed (pinned contactline) while the contact angle is decreasing during evaporation. They observed theseevaporation modes while studying droplets of methyl acetoacetate with a mass ofaround 1 mg on PTFE. They investigated these two evaporation modes both theo-retically and experimentally and developed a model that predicts the total evapora-tion rate for each mode of evaporation. They used the Maxwell analogy betweenthe electrostatic potential and the diffusive flux of a sphere in an infinite mediumand extended the model to find the evaporation flux of a sessile droplet. They cal-culated the capacitance of the equiconvex lens formed by the sessile droplet (shownin Figure 2.2) and used this to find the capacitance of the sessile droplet.The lens capacitance was twice of the capacitance of the sessile droplet. They cal-culated the capacitance for different apex angles of the lens (i.e, ? ) using Snow?s8SubstrateFigure 2.2: A sessil droplet with its image can be profiled as the equiconvexlens formed by two intersecting spheres with radius of a.summation of finite series solution [17]C (?) ={a(?1? +?2? 2??3? 3)if 0? ? ? 0.175 rada(?0 +?1? +?2? 2??3? 3 +?4? 4)if 0.175? ? ? pi rad, (2.1)where a is the radius of the sphere and ?i and ?i are coefficient of the polynomialapproximation of the series and their values are shown in Table 2.1.Table 2.1: Coefficient of Snow?s finite series in Equation 2.1 [1]Parameter Value?1 0.6366?2 0.09591?3 -0.06144?0 0.00008957?1 0.6333?2 0.1160?3 -0.08878?4 0.01033Using the Maxwell analogy, Picknett and Bexon derived the evaporation rate at anycontact angle of the droplet, ? , for both evaporation modes asm? = 2piD(c0? ci)C (?), (2.2)9where m? is the evaporation mass flux, D is the molecular diffusion constant of thevapor in air, and c0 and ci are the vapor concentrations close to the lens surface andfar from the sphere, respectively. They found this formulation for quasi-stationaryMaxwellian conditions of diffusion controlled evaporation in an infinite medium.From Equation 2.2, they predicted the lifetime of a droplet with the initial mass ofM0 and initial contact angle of ?0, and concluded that the evaporation with constantcontact area (fully pinned stage) leads to a relatively shorter lifetime compared tothe evaporation with constant contact angle (fully de-pinned stage). The schematicof the evaporation profiles for both evaporation modes achieved by Picknett andBexon through theoretical formulation of Equation 2.2 is shown in Figure 2.3.TimeDrop massFigure 2.3: The droplet life time for both evaporation modes derived fromEquation 2.2.This was the main model used extensively in the literature to approximate the evap-oration rate for the diffusion controlled evaporation. In 2001, Deegan et al. addedan improvement to the model while studying the deposition of colloids from col-loidal suspensions sessile droplets along the contact line [18]. They developed amodel that could predict the local evaporation rate at the surface of a fully pinnedsessile droplet. For the first time, they specifically reported a higher evaporationrate near the contact line of the droplet. The higher probability of escape for vapormolecules near the edge of the droplet was explained using random walk paths asshown in Figure 2.4. A molecule of vapor near the edge of the droplet, has more10SubstrateReturning vapor moleculeNon-returning vapor moleculeFigure 2.4: A probability of escape for vapor molecules at two different sitesof the surface of the droplet for diffusion controlled evaporation. Therandom walk path initiated from a vapor molecule is more likely to re-sult in a return to the surface if the starting point is further away fromthe edge of the droplet.space to depart from the surface, while the random walk path suggest a higherprobability of return for a vapor molecule at the center of the droplet. To modelthis enhanced evaporation rate toward the edge of the drop, Deegan et al. used thesame analogy between a sessile droplet evaporation rate and a charged conductorelectric field with the same shape (Figure 2.2). The concluded that the evaporationdiverges near the contact line of the drop in the same way the electric field doesnear a sharp corners. Hence, the evaporation profile on the surface of the dropletclose to the contact line can be presented asJ(r,?)? (R? r)?? (2.3)where ? = (pi?2?)/(2pi?2?), R is the droplet radius and r is the horizontaldistance from the center of the droplet as shown in Figure 2.5. Using this analogy,the evaporation rate profile over the entire surface was found as:J(r,?) =?D(ci? co)(sin(?2)+?2(x+ cos?)3)? ?0P?1/2+n?(x)?cosh(??) tanh((pi??)?)cosh(pi?) d?)(2.4)11Substrate?EvaporationFluidAir Figure 2.5: Schematic of the sessile droplet on a substrate. The evaporationrate at the surface of the droplet is enhanced toward the edge of thedroplet.where P?1/2+n?(x) is the Legendre function of the first kind and x is a parameterranging from 0 to 1 and can be calculated fromr =R?1? x2x+ cos? . (2.5)Deegan et al. suggested a separation of variables for the evaporation rate profile atthe surface of the droplet that approximates Equation 2.4 asJ(r,?)? J(0,?)(1?( rR)2)??. (2.6)Equations 2.3 and 2.6 show that for diffusion controlled evaporation, the evapora-tion increases toward the corners when r? R.Hu and Larson [19] continued this study and investigated the enhanced evaporationrate near the corner by solving the boundary value problem numerically and com-pared their results with experiments. The mathematical model for the diffusioncontrolled evaporation that they developed was based on solving for the quasi-steady state diffusion of the saturated vapor at the surface of the droplet into the12ambient air. Modeling the diffusion of the vapor molecules then yields?2C =?C? t , (2.7)where C is the vapor concentration around the droplet and t is time. In the caseof water droplets, the evaporation process is slow and the time that it takes for thevapor to adjust the concentration at the surface is fast compared to the diffusiontime scale. Hence the evaporation can be considered to be quasi-steady and thetime derivative vanishes. The diffusion model then changes to the Laplace equation?2C = 0. (2.8)A non-homogeneous Dirichlet boundary condition (i.e. ci 6= c0) drives the vaporfrom the free surface of the droplet to the ambient air and evaporation occurs. Fora spherical water droplet with the parameters shown in Figure 2.5, the boundaryconditions in 2D can be written as:1 At the surface of the droplet, the concentration of the vapor is equal to thesaturated vapor concentration:C|inter f ace = ci = Cv|Tinter f ace . (2.9)2 Far from the surface of the droplet, we haveC|ambient = co = ? Cv|Tambient , (2.10)where ? is the relative humidity of water in air.3 No diffusion through the substrate:?C = 0. (2.11)Similar boundary conditions can be written for line droplets in 3D (Equations 5.22to 5.24) in Chapter 5.13SubstrateDroplet Ambient air(a)Substrate(b)Figure 2.6: The domain mesh (a) and the solution of the Laplace equation fordiffusion of the water vapor molecule with the concentration of Cv =1.9?10?8 g/mm3 at the surface of the droplet into the ambient air withthe relative humidity of 55%, i.e. ? = 0.55 (b).The evaporation flux~J(x,y,z) = D~?C (2.12)can be calculated from the vapor concentration solutions of the Laplace equationwith the boundary conditions above (Equations 2.9 to 2.11) using Finite ElementMethod (FEM) with the mesh structure shown in Figure 2.6a are shown in Figure2.6b. The evaporation flux using Equation 2.12 is shown in Figure 2.7 for twodifferent contact angles of ? = 90? and ? = 60?. As shown in Figure 2.7, theevaporation profile is uniform over the surface when the contact angle is equalto 90? (hemisphere). When the contact angle is reduced below 90? as shown fora droplet with the contact angle of 60?, the evaporation is enhanced toward thecorners until it becomes singular at the contact line. This is consistent with theprevious results by Deegan et al. [18].Hu and Larson solved Equation 2.8, with the boundary conditions (Equations 2.9to 2.11) numerically and suggested a formula for the total evaporation mass flux:dMdt=?piRD(1??)Cv(0.27? 2 +1.30). (2.13)Popov [20] solved the Laplace equation explicitly and found the evaporation massflux asdMdt=?piRD(1??)Cv f (?), (2.14)140.00 0.04 0.08 0.12 0.16 0.200.02.0x10-54.0x10-56.0x10-58.0x10-51.0x10-4 60 degree 90 degreeJ (g/mm2)r (mm)Figure 2.7: The evaporation flux, J for a water droplet with the radius of R =0.17 mm into the ambient air with the relative humidity of 55% at roomtemperature for different contact angles of ? = 90? and ? = 60?.wheref (?) = sin?1+ cos? +4??01+ cosh(2??)sinh(2pi?) tanh [(pi??)?]d?. (2.15)By writing the droplet volume in terms of ? (from Equation A.13) and inserting itinto Equation 2.14, the rate of change of the contact angle for a fully pinned droplet(i.e. R remains constant during evaporation) was found asd?dt=?D(1??)Cv?R2 (1+ cos?)2 f (?), (2.16)where ? is the density of the fluid. Also, by rewriting Equation 2.16 asd?(1+ cos?)2 f (?)=?D(1??)Cv?R2 dt, (2.17)15the total evaporation time can be calculated asTf =?R2D(1??)Cv?c?0d?(1+ cos?)2 f (?), (2.18)where ?c is initial equilibrium contact angle of the droplet. By definingt? =(1??)Cv?tR2/D(2.19)andM? =M?R3 , (2.20)Equations 2.14 and 2.16 are simplified todM?dt?=?pi f (?) (2.21)andd??dt?=?(1+ cos?)2 f (?), (2.22)and become independent of the environmental conditions of the evaporation. Thesimplified rate of change in the contact angle and the evaporation mass flux (de-fined in Equations 2.19 and 2.20) is shown in Figure 2.8a. Also, the change of thecontact angle versus simplified time (defined in Equation 2.22) was found by solv-ing the ordinary differential equation numerically (Equation 2.17) and is shown inFigure 2.8b. Popov estimated ?dM?/dt? = pi f (?) ?= 4 for small contact angles (ascan be seen from Figure 2.8a) and was able to calculate the evaporation mass fluxfrom Equation 2.14 asdMdt=?4RD(1??)Cv, (2.23)and the rate of the contact angle change from Equation 2.16 asd?dt=?16D(1??)Cv?piR2 . (2.24)16(a)(b)Figure 2.8: (a) The rate of change of the contact angle and the evaporationmass flux during the evaporation at constant contact area (fully pinnedstage) with the initial contact angle of ?c = 40; (b) contact angle duringevaporation versus t?.17This yields the total evaporation timeTf =pi?R2?16D(1??)Cv(2.25)and the droplet massM =pi?R3?4(1?tTf)(2.26)as a function of time. Equation 2.26 shows the linear time dependence of mass dur-ing most of the evaporation process. The basic evaporation models, showed goodagreement with the experiments at various evaporation conditions [1, 19?22]. Huand Larson specifically showed that their evaporation model predicts the experi-mental results with an error around 10 % for droplets with 0.5 mm < R < 1 mm inradius. They also observed that the experimental results deviate significantly fromthe basic evaporation models by up to 25% for small droplets with R < 0.5 mm.2.3 Evaporation-induced flowDuring the evaporation of sessile droplets with a pinned contact line, the evapora-tion rate increases toward the corner of the droplets. The amount of fluid removedby the evaporation needs to be replenished to keep the substrate wet and to keepthe contact line pinned. Therefore, a flow is generated from the center toward thecontact line. This viscous flow for sessile droplets with a pinned contact line, thatis also referred to in the literature as the ?evaporation-induced flow?, has alwaysan outward direction. For colloidal suspensions, this internal flow carries the sus-pension particles with the fluid toward the contact line where they deposit in a ringshape [18]. This is often referred to as the ?coffee-stain? effect (Figure 2.9a).Deegan et al. [18] quantified the evaporation-induced flow for the first time usingthe evaporation model in Equation 2.6. The average velocity of a column shown inFigure 2.10 was found from the evaporation rate through mass conservation? ?hr? t =??1r? (rhrvr)? r ? J(r, t)?1+(?hr? r)2, (2.27)18(a)SubstrateEvaporation(b)Figure 2.9: An internal flow is generated inside the sessile droplet as a resultof evaporation that carries the fluid toward the corners: (a) top view ofa ring deposition after drying a coffee droplet, (b) outward evaporation-induced flow inside a sessile dropletwhere hm and hm are the column heights at the center and at the distance r from thecenter, respectively. With J(0,?) from Equation 2.4, and with hr versus R, r and ?from Equation A.17, the average velocity of the column vr in 2.27 was calculated.Tarasevich [23] suggested the potential flow assumption to solve the evaporation-induced velocity field for a hemispherical droplet and showed, to the best of ourknowledge for the first time, the qualitative picture of the flow. The potential flowwas solved for the hemisphere (with constant evaporation flux J0) asJ(r,?) = J(0,pi/2) = J0 =?piD(1??)cvR. (2.28)The potential flow field equation?2? = 0, (2.29)19SubstrateEvaporationAir ( , )( , )Figure 2.10: The average evaporation-induced flow in a vertical fluid columnat a distance r from the center. The flux leaving the top surface is equalthe to local evaporation rate at the surface J(r, t).where ? is the velocity potential then was solved analytically with the followingboundary conditions:1 Geometrical symmetry at r = 0 gives??? r = 0. (2.30)2 No-flow through the substrate condition at z = 0 gives:??? z = 0. (2.31)3 The kinematic boundary condition at the surface of the droplet is achievedby writing the conservation of mass~vn = ~Un?~J? , (2.32)where ~vn is the velocity of the fluid and ~Un is the normal velocity of thesurface as shown for a liquid element at the surface in Figure 2.11.20EvaporationSubstrateFigure 2.11: The air-liquid interfaces moves as a result of the evaporation.Using the conservation of mass for an infinitesimal element at the sur-face, the velocity of the fluid normal to the boundary,~vn was found bysubtracting the evaporation flux, ~J/? from the interface velocity ~Un.Tarasevich approximated the velocity of the interface during evaporation for ahemispherical droplet as ???~Un???=U0 cos?, (2.33)whereU0 =?hm? t . (2.34)is the velocity of the interface at the center and ? is an angle shown in Figure2.11. Using the simplified assumption of a constant flux over the surface of thehemisphere, we can write?Jds = J0(2piR2) = ?dVdt, (2.35)where dV/dt can be approximated as piR2U0. Hence,U0 =2J0? . (2.36)21By inserting U0 into Equation 2.33 and then Equation 2.32, we will find the normalcomponent of the fluid velocity at the interface of a hemisphere as:~vn =?J0? (1?2cos?) (2.37)or~vn =?J0?(1?2r?r2 +h2r). (2.38)Petsi and Burganos [24] extended this work and calculated the evaporation-inducedflow by solving the velocity potential formulation for a half-cylinder on a substrate.They assumed Equation 2.29 with the boundary conditions in Equations 2.30 to2.32 for both fully pinned and fully de-pinned droplets. By assuming a constantevaporation flux at the surface (J(0,pi/2) = J0), they solved the potential field an-alytically for the fully pinned droplet and suggested the potential field as?(r,?) = J0R???m=1(r/R)2m cos(2m?)m(4m2?1). (2.39)The velocity field and streamlines produced by the potential flow of Equation 2.39are shown in Figure 2.12. They approximated the potential field of Equation 2.39with a closed-form expression?(r,?)? J0?(r2 cos(2?)3R2+r4 cos(4?)30R4)(2.40)with 99% accuracy except for the area close to the air-fluid interface.They solved the potential flow for the fully de-pinned half-cylinder droplet of in-finite length (assuming that the contact angle, ? = pi/2 remains constant duringevaporation) with the constant evaporation flux at the surface of the droplet J0.The boundary conditions in Equations 2.30 and 2.31 will be the same for fully de-pinned droplets. However, the mass balance at the air-fluid interface of dropletswith a moving contact line at constant evaporation flux yields a constant interfacevelocity???~Un???=dRdt=?J0? . (2.41)22SubstrateFigure 2.12: The velocity field (left) and streamlines (right) of potential flowinside a half-cylinder as a result of evaporation.Later on, they extended this work by solving the potential flow for any contactangle for both fully pinned and fully de-pinned evaporation modes [25]. To dothis, ~Un in Equation 2.33 was calculated for any contact angle as???~Un???= JTOTsin?(cos? ???1? (r/R)2 sin2 ?)2R(sin? ?? cos?) , (2.42)where JTOT is the total evaporation rate per unit length of the cylinder and ? isdefined as? ={1 if 0? ? ? pi/2?1 if pi/2? ? ? pi. (2.43)Masoud and Felske [26] solved the velocity field using Stokes flow when inertialforces are negligible for both spherical and cylindrical droplets with the initial ra-dius less than 1 mm where the Reynolds number is around 10?3. They solved theStokes flow field for both fully pinned and fully de-pinned sessile droplets at anycontact angle considering the non-uniform evaporation flux over the surface. Thevelocity stream function then yields the biharmonic equation?4? = 0. (2.44)23Their results showed good qualitative agreement with the velocity field describedby potential flow. They specifically showed that while the vertical velocity com-ponents (vz) are similar for both solutions, the radial velocity components (vr) ofthe solutions give rather different velocity profiles (as shown in Figure 2.13). Themain reason behind this difference is the no slip boundary condition that is valid forStokes flow field (for a viscous fluid) that results in a zero velocity at the substratesurface.Substrate Potential flow Stokes flowFigure 2.13: The radial velocity component from potential flow (left) andStokes flow (right).Hu and Larson [27] solved the full Navier-Stokes equation for small droplets withdiffusion controlled evaporation. Assuming an incompressible quasi-steady flowwith negligible convective acceleration for a spherical droplet shown in Figure 2.5,yields the continuity equation1r?vr? r +? z? z = 0 (2.45)and the momentum equations in r and z direction1??P? r =?? r(1r? (rvr)? r)+? 2vr? z2 ,1??P? z =?? r(r?vzr)+? 2vz? z2 , (2.46)where P is the pressure and ? is the viscosity of the fluid. They solved the Navier-Stokes equation with the following boundary conditions and assumptions:241 For thin droplets the lubrication theory assumption can be used to estimatethe vertical velocity profile asvz = ?1z2 +?2z+?3, (2.47)where ?i are the coefficient that need to be determined using the boundaryconditions.2 The zero shear-stress boundary condition at the air-fluid interface yields?vr? z????z=hr=??vz? r????z=hr. (2.48)3 The kinetic boundary condition at the surface isvz|z=hr =?hr? t +J(r, t)? . (2.49)Hu and Larson used an approximation for the rate of height change over timeas?hr? t =?hm? t(1? (r/R)2)= 2dMdt1? (r/R)2?piR2 . (2.50)4 The no-slip boundary condition at the substrate surface for the fully pinnedevaporation mode givesv(r,z)|z=0 = 0, (2.51)and ?3 = 0 in Equation 2.47.5 For a flat droplet, they assumed that?vz? r ??? r (vz|z=hr) . (2.52)256 By assuming a linear rate of decrease of the droplet height during evapora-tion, then the rate of height change at the center was written as?hm? t ??HTf, (2.53)where H is the initial height of the droplet, i.e. H = hm|t=0.With these boundary conditions and assumptions, Hu and Larson derived an ex-pression for the dimensionless vertical and radial velocitiesv?r =3811? t?1r?((1? r?2)? (1? r?2)??)((z?/h?)2?2(z?/h?))+r?H2h?R2(J?? (1? r?2)???1 +1)((z?/h?)?3/2(z?/h?)2)(2.54)andv?z =3411? t?(1+? (1? r?2)???1)(1/3(z?3/h?2)? (z?2/h?))+3211? t?((1? r?2)? (1? r?2)??)(1/2(z?2/h?2)?1/3z?3/h?3)h?m?H2R2(J?? (1? r?2)???1 +1)(z?2? (z?3/h?))+r?H2R2J?? (? +1)(1? r?2)?? ?2(z?2? (z?3/h?))?r?H2R2(J??(1? r?2)???1+1)(z?3/h?2)h?m, (2.55)where v?r = vrTf /R, v?z = vzTf /H, r? = r/R, z? = z/H, h? = hr/H, h?m = hm/H and ?and Tf were previously defined in Equations 2.6 and 2.25, respectively.2.4 Drying of colloidal suspension dropletsDuring the evaporation of sessile droplets from colloidal suspensions, the internalflow carries particles and they precipitate until the droplet fully dries. The depo-sition of the particles from the colloidal suspensions as a result of evaporation hasbeen extensively studied in various geometries for different suspensions and sol-vents. However the first mathematical model for deposition of colloids as a result26of evaporation-induced flow was suggested by Deegan et al. [18]. They showedthat how evaporation generates an outwards flow inside the droplet which can causeparticles to form a ring deposition near the contact line as the droplet dries. Theycalculated this flow for a vertical liquid column within the droplet shown in Fig-ure 2.10. They assumed a uniform concentration in the vertical column during theevaporation of a sessile droplet in still air with fully pinned contact line. Therefore,the mass conservation equation for particles gives?? t (crhr)+1r?? r (rcrhrvr) = 0, (2.56)where cr is the concentration of the particles in the vertical column. The mass ofthe deposition ring then was found by subtracting the mass of the particles in thedroplet at a later time from the mass of particles at t = 0. For thin droplets, theysuggested the mass of the deposition ring to change in time asmR = m0(1? (1? t/Tf )(1+? )/2)2/(1+? )(2.57)where mR and m0 are the deposition ring and the total mass of the particles at t = 0.Knowing the mass of the ring at any time, they suggested a formula for the widthof the deposition band based on experiments as a function of time as? (t) = mRt2/(1+? )2? tan?(2?piR2?? tan?(dV/dt)(1+? ))2/(1+? ), (2.58)where ? is the number of particles per until volume at t = 0.Popov [20] continued this and using the basic model of evaporation, and suggesteda model to calculate the width of the deposition ring. In this model, a growingdeposit phase of particles was defined that is formed by a viscous particle flowas a result of evaporation (Figure 2.14). By writing the global and local massconservation of particles and liquid, they were able to estimate the width of thedeposition band as? (t) =??c0R(3t?)2/327/3. (2.59)27Substrate Deposit phaseFigure 2.14: Evaporation-induced flow carries particles toward the cornersand forms a deposit phase near the contact line of a sessile droplet.Bhardwaj et al. [28] and Petsi and Burganos [29] investigated the pattern formationof microsphere in colloidal suspensions using Lagrangian and Brownian dynamics.Park and Moon [30] demonstrated a controlled ring deposition of a uniform particlelayer within picoliter sessile droplets. They showed that surfactant mixed-solventswill result in vanishing ring deposition. They also found a critical concentrationof particles above which the ring deposition is reduced [30]. This is in contrast toPopov?s model that predicts a wider ring deposition at higher particle concentra-tions. The main reason of the ring deposition suppression at higher concentrationswas the Marangoni flow that produced a backward flow that countered the outwardevaporation-induced flow [30]. This Marangoni flow was produced because of thesurface tension gradients that can be generated conceivably as a result of particleconcentration or a thermal gradient inside the droplet.Hu and Larson [27] studied the thermal Marangoni flow during quasi-steady stateevaporation of a sessile droplets by assuming the convective heat transfer to be neg-ligible and solved the conductive heat transfer across the droplet using the Laplaceequation for temperature. They showed that the latent heat of evaporation inducesan inward thermal Marangoni flow near the fluid-air interface that carries the par-ticles toward the center. The velocity profile under the presence of the Marangonieffect is shown in Figure 2.15. At the surface of the droplet, the Marangoni flowis zero at the center (due to axisymmetry) and is enhanced toward the contact line.This inward flow acts against the outward evaporation-induced flow at the surfaceuntil canceling it out at the stagnation point beyond which the direction of the sur-28face flow reverses (Figure 2.15). They confirmed the effect experimentally later byshowing that the inward Marangoni flow reduces the ring deposition [31].Substrate Stagnation point Figure 2.15: Marangoni flow produces an inward flow that counters the out-ward evaporation-induced flow for sessile droplets during the pinnedstage.2.5 ConclusionThe basic models of evaporation for a sessile droplet on a substrate assume isother-mal diffusion of vapor into the ambient air. By assuming isothermal diffusion,i.e. cv|interface ? cv|ambient air, the temperature only affects the problem by chang-ing cv and D. Using this assumption, the Laplace equation was solved for thediffusion-controlled evaporation at a quasi-steady state. The total evaporation timewas then calculated by finding the amount of fluid removed by evaporation in asmall time step and then by integrating these until the entire evaporation of thedroplet. The basic models showed an enhanced evaporation rate near the contactlines for droplets with ? < 90?. However, while these models predict fairly wellthe evaporation of droplets with 0.5 mm < R < 1 mm, they produce significanterrors for smaller droplets. While to date, the main reason behind this significanterror for small droplets is not known, Hu and Larson [19] rejected the effect ofthe substrate and the droplet thermal conductivities as the maximum temperaturechange across the droplet during evaporation is less than 0.02 ?C for droplets withR < 1 mm, which has negligible effect on cv and D. In Chapter 4, we study theeffect of temperature and the important effect of substrate conductivity on evapo-ration of small droplets and we will discuss that while this temperature change has29negligible effect on cv and D, it defines the rate limiting step of heat transfer to thesurface to provide the latent heat of evaporation.The enhanced evaporation near the contact line induces a flow inside the droplet.The basic theories that model this flow were discussed. The evaporation-inducedflow solutions for a spherical droplet and an infinite cylinder (liquid line), for bothevaporation modes (i.e. fully pinned and fully de-pinned) were shown. The com-parison of the Navier-Stokes, Stokes flow, Lubrication theory and potential flowshows that a qualitative agreement between the results exists. The no-slip bound-ary condition for viscous flow causes the main difference between the results of theinviscid and viscous flows.Finally, the deposition of particles of colloidal suspensions during evaporation as aresult of internal flows shows that a ring deposition forms during the evaporation offully-pinned droplets. The main studies that model the deposition ring as a resultof the evaporation-induced flow was shown, and the suppression of the ring bandas a result of Marangoni flow was discussed.30Chapter 3Experimental and numericalapproach3.1 IntroductionThe study of small sessile droplets in the order of 50 ?m < R < 1 mm needs carefuldeposition and measurement techniques. The small size of the droplets increasestheir sensitivity to environmental condition that could produce a significant errorin the measurement and interpretation of the data. In this chapter, the methods thatwere used in our experiments to print the droplets at the micro scale, to control theambient conditions, to visualize the droplet during evaporation and to measure thedeposition pattern after evaporation will be explained (Section 3.2).As mentioned in the previous chapter, the current evaporation methods produce asignificant error for small droplets with R < 500 ?m. In this chapter, the numer-ical approach that we used to overcome these limitations will be presented anddiscussed (Section 3.3).313.2 Experimental setup3.2.1 Printing stageTo print spherical and cylindrical droplets, a portable micro patterning stage wasdesigned. The system consist of: 1) a motorized linear stage for printing the linedroplets, 2) a micro nozzle for printing droplets by dispensing liquid from a con-tainer onto the substrate, 3) a micro positioner to adjust the distance of the nozzlefrom the substrate, 4) a pressure manifold that controls the pressure of the liquidinside the micro nozzle. The printing stage setup is shown in Figure 3.1. The(1) Motorized linear stage(3) Micro PositionerMicroscope objective lens connected to a CCD Camera(2) NozzleSubstrate(4) Pressure manifoldFigure 3.1: The portable micro printing setup. A motorized linear stage fromZaber Technologies Inc. was used to control the place and speed of themicro nozzle.motorized linear stage is the model T-LSM050A-KT03 from Zaber TechnologiesInc. (Vancouver, BC V6P6P2, Canada) with a 50 mm range of travel. MicroFab(Plano, TX 75074 USA) MJ-ABP-01 dispensing nozzles with a diameter of 80 mmwere used for printing droplets. The model 2210-CE amplifier from Trek Inc. (NY14103, United States) is used to increase the power of the signal to the micro noz-zle. A computer code was written in C] to synchronize the printing sequence withthe motion of the stage so that the desired length and width of the line droplets oran array of point droplets can be printed. The micro positioner is the model MT45-13-X-MS measuring stage from the OWIS GmbH (Im Gaisgraben 7, 79219Staufen, Germany), with central micrometer with a measuring range of 12 mm.32The PID pressure controller is the system MFCS-8C 1000 from Fluigent(94800Villejuif, France).3.2.2 Droplet visualization and measurementsTo understand the behavior of small droplets, precise visualization methods andcareful measurements of parameters are required. Here we show the tools that areused to capture the droplet shape (the contact line and the contact angle) duringevaporation while measuring and controlling the environmental conditions such ashumidity and temperature.Contact angle measurementAs described earlier, one of the key points in understanding the evaporation of ses-sile droplets and determining the deposition sites of colloidal suspensions is thebehavior of the contact line, i.e. whether it is pinned or de-pinned. In our exper-iments we are using colloidal suspensions of nano particles at different particlesconcentrations. In evaporation of colloidal suspensions, depending on the initialconcentration of the solutes, we observed a combination of both pinned and de-pinned for the contact line. A series of experimental measurements is conducted tofind the receding contact angle ?r for different substrates and for different types ofparticles at different sizes. The experimental data collected in this section will beused as the material properties and coefficients for the numerical approach in Sec-tion 3.3. To investigate these parameters, we use a Theta Lite optical tensiometerfrom Attension/Biolin Scientific (SE-107 24 Stockholm, Sweden) to measure thecontact angle and radius of the droplet during the evaporation. Images are taken at1 Hz with the resolution of 640? 480 pixels using a Firewire digital video cam-era with the speed of 60 fps (frames per second) . Images are processed usingthe OneAttension software and the Young-Laplace fitting method is used to findthe droplet profile. The portable printing stage (Figure 3.1) is used to dispensecontrolled volumes of the liquid onto the substrate to generate droplets at differentinitial radii.33CCD Camera LED LampSubstrateFigure 3.2: The experimental setup for investigating the contact line and con-tact angle during the evaporation. The experiment was illuminated usingLED based background lighting.Velocity measurementTo measure the evaporation-induced velocity field of sessile droplets, several tech-niques have been used by different researchers. For spherical droplets, a crosssectional view across the center gives a sufficient understanding of the velocityprofile due to the axisymmetric shape of the droplet [11]. Kang et al. suggesteda quantitative visualization technique to correct the effect of light refraction nearthe surface of the droplet [32]. However, to capture the flow inside droplets with asmall diameter (R < 500 ?m) a rather large lens with high magnification is requiredto observe the flow. We use an inverted microscope image (with the objective lensmagnification of 40?) to capture the horizontal flow near the substrate while ob-serving the contact line movement in real time. FS04F fluorescent microspheresbeads from Bangs Laboratories, Inc. (IN 46038, United States) are used for flowvisualization. The images of microspheres are captured at two different frame ratesof 50 fps (using a Imager sCMOS Camera - with the resolution of 5.5 Mpixels) and2000 fps (using a Phantom M4 high-speed camera). The horizontal velocity fieldwas then calculated using micro particle image velocimetry (PIV) through Davis8.0 software from Lavision (MI 48197, United States) as shown in Figure 3.3.341mm/st = 5 secFigure 3.3: The horizontal velocity near the substrate was observed using aninverted microscope stage and the velocity was measured using PIV.Temperature and humidity measurementAs mentioned in Chapter 2, ambient temperature and humidity play an importantrole in evaporation rate. Also, in Chapter 3 and Chapter 6.1, we will discuss the im-portant influence of the substrate temperature on the evaporation of sessile dropletsand the deposition at the contact line in colloidal suspensions. The humidity andtemperature was measured using a SHTC1 humidity sensor and a STS21 tempera-ture sensor from Sensirion (8712 Staefa ZH, Switzerland).To control the temperature of the substrate a closed loop temperature control sys-tem is designed as shown in Figure 3.4. A CZ1-1.0-127-1.27 Peltier device fromTellurex (MI 49686, United States) is attached from the cold side to the substrateto control the temperature while the excessive heat is dissipated from the hot side.We use a temperature sensor mounted on the Peltier element, a power supply, anda data acquisition unit (DAQ) and CPU to communicate between each. Usingthis self-regulating setup, a temperature range of 15 ?C to 40 ?C with an accuracyof ?0.1?C for the substrate was achieved. The temperature of the surface of the35CCD CameraPeltier elementHeat sinkTemperature sensorMicro nozzlePositionerFigure 3.4: The temperature of the substrate is controlled using a Peltier de-vice in a closed loop temperature control circuit. The signal from thetemperature sensor attached to the substrate is used to provide the feed-back signal to the controller.droplet during evaporation is measured using IR-TCM-384 infrared thermographiccamera from Jenoptik (07743 Jena, Germany) as shown in Figure 3.5.Deposition measurementTo measure the ring deposition after drying of the sessile droplet, two differentpieces of equipment are used. The Wyko NT1100 Optical Profiling System is usedto measure the shape of the whole ring deposition from suspensions of non-coatedparticles. Suspensions of coated polystyrene particles deposit in thin film layersand a finer resolution is needed to measure the deposition ring. For this purpose,the OLS4000 laser confocal microscope from Olympus is used.3.3 Numerical approachAs we discussed in Chapter 2, the basic evaporation models solve the Laplace equa-tion assuming a quasi-steady state, isothermal diffusion controlled evaporation fora fully pinned droplet. We will show in Chapter 4 and Chapter 6 that this pro-duces a significant error for sessile droplets with a small initial radius R < 500 ?m.36Measurement lineTemperature profile along the lineDropletSubstrateFigure 3.5: Temperature measurement of the surface of the droplet using anIR camera. During evaporation at room temperature, the enthalpy of theliquid near the surface reduces to transform liquid molecules into thevapor phase. That causes a drop of the temperature at the surface of thedroplet during evaporation.Hence, we use a numerical approach that solves the full energy equation for alocally averaged temperature. The model is developed based on custom FLOW-3D sub-routines (version 10.0.1, Flow Science, Inc., Santa Fe, NM 87505, UnitedStates). FLOW-3D is a computational fluid dynamics software that defines multi-physics flow problems by creating separate sub-routines representing each physicsmodel. The sub-routines default equations can be customized and new parame-ters or equations can be added to each model through custom a code. To modelthe evaporation of a sessile droplet, we activate the phase change, surface tension,heat transfer, viscous flow and residue deposition physics models.In the phase change physics module in FLOW-3D, the inertia of the gas adjacent tothe liquid is assumed negligible. This approach has the advantage of reducing thecomputational effort as in most cases the details of the gas motion are unimportantfor the motion of the much heavier liquid. Thus, the free surface becomes oneof the external boundaries of the fluid region. A proper definition of the bound-ary conditions at the free surface then becomes important to achieve an accuratecapture of the free-surface dynamics. To model the phase change at the fluid-air in-terface, instead of solving the diffusion equation, a local vapor density and velocity37of the molecules at the surface of the droplet define the local evaporation fluxm?local = ??M2piRT?(Pv?Pair), (3.1)where M is the molecular weight of a vapor molecule, R is the vapor gas constant,Pv and Pair are respectively saturation and air pressure, T? is the local average of theliquid temperature along the surface and ? is an accommodation coefficient thatdescribes the probability of a vapor molecule at the liquid surface being detachedfrom the surface. This accommodation coefficient is considered to be a functionof the phase properties and is derived from experiments. The value of the phasechange accommodation coefficient ? is typically in the range of 0.001 to 0.1. Weassumed a vaporization at the free surface of a fluid (droplet) at a constant satu-ration state defined by a void region (ambient air). The diffusion at the interfacedefines the rate limiting step of the evaporation while heat and mass transfer mayoccur between the liquid and the surrounding liquid (void region) and the solid sur-faces. The vapor pressure in Equation 3.1 is then computed as a function of the lo-cal fluid temperature using a user-defined saturation curve. A Clausius-Clapeyronrelation was used to find the vapor pressure Pv versus the temperature Tinter f ace :Pv = c1e(?LRTinter f ace)(3.2)where L is the latent heat of vaporization and c1 is a constant of the vapor. Whilethe pressure (Pa) and temperature (Tambient) of the void is considered constant,Tinter f ace can be different than Tambient allowing heat transfer across the interface.The evaporation parameters in Equation 3.2 for evaporation of water molecules atroom temperature are shown in Table 3.1.For the heat transfer between fluid and air with the substrate, a first order inter-nal energy advection is assumed and the energy equation is solved for each phase.Viscous-heating as a result of evaporation-induced flow is considered negligible.We also assume this internal flow to be laminar.To model the surface tension effect, the initial contact angle of the droplet is de-fined from Equation 6.1. A linear variation of interfacial tension with temperature38Table 3.1: Coefficients for pressure and temperature at the interface in Equa-tion 3.2.Parameter Value (CGS)c1 31690L 2.4423?1010R 8.3145?107was considered (i.e. d?FG/dT was constant) so that the thermal Marangoni flowcan be modeled. For the behavior of the contact line, we use a model that includesboth pinned and de-pinned stages during the evaporation of colloidal suspensions.In the model, we define a receding contact angle, ?r, that is determined from ex-periments. The behavior of the droplet then is considered to be in two differentphases:1. When the contact angle is above ?r the velocity of the fluid at the contactline is zero and the contact area of the droplet is constant (pinned stage).2. When the contact line reached ?r, the contact line moved while keeping thecontact angle constant (de-pinned stage).To pin the contact line when the contact angle is above ?r, we customize the sur-face tension model by creating a phantom obstacle that is attached to the contactline to keep the contact line pinned. This is achieved by creating a sub-componentaround the droplet that limits the spreading of the fluid and then assigning a wet-ting contact angle inside this region. The phantom sub-component basically setsthe velocities inside the sub-component to zero so no fluid can enter. Also, thephantom sub-component does not have surface area so it does not interact with thesurface tension model equations.To model the deposition of the particles, we assumed non-cohesive particles thatneither do interact with the flow nor with each other. The mass flow rate of ascalar entering the domain was then equal to the specified particle concentrationtimes volume flow rate of the source fluid at the source. During the evaporation,the particle concentration will automatically be increased by removing fluid at thefree surface of the fluid. If a surface element is less than half filled with liquid,then the concentration of the scalar will also occur in the principal neighbor of the39Void (ambient air)DropletSubstrateFigure 3.6: The schematic of the mesh structure using FAVOR technique inFLOW-3D.surface element to the extent that the concentration change is spread over a regionequal to half the thickness of the surface element. To find the deposition profile, theresidue model in FLOW-3D was activated by setting a parameter defining a criticalconcentration for the particles, Ccritical . Once the suspension is concentrated to itscritical concentration near a solid surface, then any further concentration results ina residue on the substrate that is fixed (not moving).The geometry was constructed using the Fractional Area/Volume Obstacle Repre-sentation (FAVOR) technique. In this method, the open area fraction inside a cellis computed along with the open volume fraction for a fixed mesh and then thefluid geometry is constructed based on the volume fraction information from cells(shown in Figure 3.6). This approach offers a simple and accurate way to representfree surfaces in the domain without requiring a body-fitted grid. Evaporation maytake place only in cells containing a free surface. All temperatures are initially inequilibrium and equal to room temperature. Simulations are run for the maximumtime step sizes of 2?10?5, 1?10?5 and 1?10?6 seconds and maximum temper-ature differences between the fluid and air during the evaporation are consistentwith 2% error. We use structured rectangular grids for the FAVOR to improve theaccuracy and the stability of the numerical solutions. The rectangular grids withthe sizes of 4 ?m, 2 ?m and 1 ?m are used and the mesh dependency of less than1% is observed for the maximum phase change rate and 0.5% for the maximumtemperature differences. The time step size in our simulation is controlled by theconvergence and the stability. The convergence of the solution was controlled by40Table 3.2: The safety factor coefficient for each physics module to ensurenumerical stability.Parameter Value? f 0.45?s 0.5?p 1?h 0.45setting the implicit iteration method for each solver and with the maximum numberof iteration counts of 500. The numerical stability is checked by two criteria:1. There should be no flow across more than one computational cell in one timestep. This is checked by setting the coefficient ? f so that:time step < ? f ?min[?rvr,?zvz](3.3)where ?r and ?z are the rectangular cell sizes in r and z direction. When othermodules exist in the model, additional coefficients of ?s, ?p and ?h are usedto account for the instability caused by surface tensions, phase change andheat transfer, respectively. The value of these coefficients in our simulationis mentioned in Table 3.2.2. A cell with a large free surface area and a small volume could restrict thetime step to small values if there is significant flow in this cell. This isautomatically checked by an algorithm in FLOW-3D through monitoring themesh locations and controlling the time step.41Chapter 4Effect of substrate conductivityon the evaporation of small sessiledroplets4.1 IntroductionSessile droplets are liquid drops that are located on top of a substrate surface. Theevaporation of sessile droplets on substrates has been widely discussed in the lit-erature due to the wide occurrence of this phenomenon in a variety of applicationsincluding inkjet printing [33], wetting [7], deposition processes [34] and bio assays[35]. In recent work the hydrodynamic flow generated during the evaporation ofsuch droplets has been used to place and align nano particles, e.g. carbon nanotubes (CNTs) and nano wires [33]. Hence the understanding of the evaporationprocess of sessile droplets and investigating the important parameters affecting theevaporation process and flow dynamics is important for such applications.During the evaporation of small sessile droplets on a surface, the contact line ofthe droplet undergoes two different stages: the pinned stage, with a fixed contactline, and the de-pinned stage with a moving contact line [1]. Figure 4.1 shows thecontact angle and the radius of a water droplet measured over time, indicating thesetwo stages. When the contact line is pinned, evaporation reduces the fluid volume42Figure 4.1: The radius and contact angle of a water droplet on an acryliccoated plastic slide during evaporation as a function of time showingtwo modes of evaporation, at 25?C and relative humidity of 55%.at a constant footprint, leading to a reducing contact angle of the droplet (pinnedstage). When the contact line reaches a certain contact angle, usually called a re-ceding contact angle, ?r, the contact line starts to move while the contact angleremains almost constant (de-pinned stage). The behavior of the contact line duringeach stage affects the hydrodynamic flow inside the droplets [20]. The evapora-tion flux of a sessile droplet in air is controlled by the rate at which the thin vaporlayer above the surface diffuses into the ambient air [19]. It can be shown that thediffusion of the vapor into the ambient air is slow and it can be considered as aquasi-steady state.Hu and Larson [19] suggested a basic model for diffusion, solved the Laplace equa-tion of mass transfer with this assumption, and derived an equation for the evap-oration flux at the surface of the droplet. In their model, they treated the localtemperature gradient across the fluid due to latent heat of evaporation as negligibleand solved the mass transfer equations at constant temperature, i.e. ambient tem-perature. This, combined with neglecting the effect of this local temperature gra-dient on vapor saturation concentration, leads to neglecting the effect of substrateconductivity; one would expect that heat from a substrate with a high thermal con-43ductivity can reduce the effect of fluid cooling caused by evaporation.Popov [20] also used the assumption of a constant temperature and developed a ba-sic model and found that the term for the evaporation rate divided by the droplet?sinitial radius?(dVdt)R0?4DCv(1??)? (4.1)is independent of the droplet shape and is only a function of the phase change prop-erties with the initial radius R0, volume V , time t, vapor saturation concentrationCv, the diffusion coefficient of water vapor in air D, the ambient humidity of thevapor ? and the density of the fluid ? .While this basic model showed good agreement with droplets with a diameter of 1mm and above, there was a considerable error for small droplets [19]. Dunn et al.[36] showed the strong influence of the substrate conductivity and droplet coolingon the evaporation and tried to improve the basic model by considering Newton?slaw of cooling and inserting the effect of buoyancy of the vapor in the atmosphere.Although their model was in good qualitative agreement, it under-predicts the ex-perimental results. Here we present a model that includes the thermal effects ofthe phase change by adopting the cavity-liquid model presented by Plesset andProsperetti [37]. Also in the basic model, it is assumed that the contact line isalways remaining pinned. That could produce a significant error when a consid-erable time of the evaporation process occurs in the de-pinning stage (Figure 4.1).In our model, the movement of the contact line is included so that it matches theexperimental results.4.2 Numerical modelThe numerical simulation is performed using the software Flow-3D (version10.0.1, Flow Science, Inc., Santa Fe, NM 87505, United States). It solves theNavier-Stokes equation for the fluid, the energy equation for the fluid, the ambi-ent air and the substrate, and solves the phase change equation for the air-liquidinterface based on the cavity-liquid model presented in [37]. The schematic of theproblem is shown in Figure 2.5. The no-slip boundary condition was used on thesubstrate surface and the movement of the contact line is following the concept that44was explained earlier in Figure 4.1). All temperatures are initially at equilibriumand equal to room temperature.4.3 MaterialsWe used a Theta Lite optical tensiometer to measure the contact angle and radiusof the droplet during the evaporation. Images were taken at 1 Hz. To print thedroplets on the surface we used a MicroFab MJ-ABP-01 dispensing nozzle withthe diameter of 80 ?m. To show the effect of the substrate cooling on evaporationof small droplets, we chose four different substrates with different thermal con-ductivities: Copper (Cu) with K = 400 W/mK, glass with K = 1.05 W/mK, treatedpolystyrene coated plastic slides (Plastic Y) with K = 0.03 W/mK and treated UVTacrylic coated plastic slides (Plastic B) with K = 0.02 W/mK and printed the samevolume of distilled water on them.4.4 ResultsFigure 4.2 shows the initial contact angle and radius of the 45 nL sessile waterdroplets printed on each substrate. For all of the substrates, a combination of thepinning and de-pinning was observed during the evaporation, similar to Figure 4.1.The volume of the droplets during the evaporation is shown in Figure 4.3. The totalevaporation of the droplet, t f , was defined as the time-intercept of the linear line,fitted to the data of droplet volume over time and the average evaporation rate wasdefined as the slope of this line.The measured evaporation time of water droplets deviate consistently from thehigher values predicted by the basic model [20], and the gap is particularly highfor more thermally conductive substrates, i.e. Cu and glass, as shown in Figure 4.4.To investigate whether this discrepancy in the evaporation rate is related to thermaleffects, the evaporation rate from the Popov model (Equation 4.1) which is inde-pendent of the shape of the droplet was compared with experiments (Figure 4.5).The basic model predicts the same value for all substrates and depends only onvapor diffusivity into the ambient air.To show the effect of substrate cooling, the numerical simulation based on themodel for diffusion described earlier (Equation 3.1) was developed. Substrates at45Figure 4.2: The contact angle and initial radius of water droplets on Cu, glassand plastic slides. All the measurements were within a margin of errorof less than 2%.0 20 40 60 800.000.010.020.030.040.05 Cu Glass Plastic B Plastic YV (L)Time (s)dVdtftFigure 4.3: Droplet volume over time for different substrates.46Cu Glass Plastic Y Plastic B020406080100tf (s)Material Experiment TheoryFigure 4.4: Total evaporation time, as measured by experiments (gray) and aspredicted by the Popov model (black).Cu Glass Plastic Y Plastic B01000200030004000(dV/dt)/R (m2/s)Material Experiment TheoryFigure 4.5: Comparing the measured evaporation rates with the values fromthe basic model from Equation 4.1.different thermal conductivities were chosen and a water droplet with a fixed initialradius and contact angle was modeled. During the evaporation, the more conduc-tive substrate was able to supply heat to the droplet, increasing the evaporation rate.Figure 4.6 shows the plot of the temperature inside the substrate for a substrate withthe thermal conductivity of K = 100 W/(mK). Since the simulated droplets havethe same initial geometry, the only parameter that could produce the difference inevaporation rate should be the thermal effects. The effect of the substrate coolingon evaporation rate is shown in Figure 4.7 by comparing the numerical simulationresults with experiments and theory.47Figure 4.6: Temperature contours inside the substrate adjacent to the droplet1E-3 0.01 0.1 1 10 10010001500200025003000 Theory Simulation Experiment(dV/dt)/R ((m)2/s)Conductivity (W/(mK))Figure 4.7: The effect of substrate cooling on the evaporation rate, the basicmodel shows the same value for all substrates.4.5 DiscussionThe results show the effect of substrate cooling on the evaporation of small dropletsat room temperature. Previously Hu and Larson [19] showed that using the ba-sic model for diffusion leads to a considerable error for small droplets less than1 mm in diameter. To investigate the influence of substrate conductivity for smalldroplets, we consider a droplet with initial radius R0 on a substrate as shown inFigure 2.5. The total amount of heat necessary for evaporation can be calculatedby multiplying the latent heat of evaporation, L by the evaporation fluxQ? =L m?, (4.2)48Figure 4.8: The schematic of the heat transfer through an element across thedroplet.where the evaporation flux, m? can be approximated as [19]m??= 2piD(1??)CvR0. (4.3)This means that the total heat needed at the droplet surface for evaporation is lin-early proportional to the size of the droplet. The molecules at the surface of thedroplet get this amount of heat from the droplet itself. The droplet could either pro-vide the entire amount while its enthalpy decreases in an infinitesimal time step, orcould get part of this energy from the substrate, via substrate cooling. Figure 4.8shows the heat transfer through an infinitesimal element across the droplet.The balance of energy for the element yieldsQ?? Q?c = Ah?cp? T?? t (4.4)where T? is the average temperature along hm and Q?c is the heat conducted throughthe droplets from the fluid-substrate interface toward the surface of the droplet andis found as:Q?c = K` A?T? z????z=0, (4.5)where K` is thermal conductivity of the fluid. The temperature distribution insidethe droplet can be found by solving the partial differential equationK` A?T? z????z=0+Ah?cp? T?? t????z=hm= 2D(1??)cv R0L . (4.6)49To simplify the problem, we assume the temperature change inside the volume ele-ment happens as the superposition of two separate temperature changes due to ther-mal capacity and conductivity. We assume first that the temperature of the dropletis decreasing steadily and that the droplet loses heat similarly to a lump body bythe amount of ?T1. At the same time, the droplet uses the heat through conductionto increase its temperature by the maximum amount of ?T2 at the substrate-fluidinterface. This amount of heat could be entirely obtained through the substrateconductivity (full substrate-fluid coupling) or the droplet itself (insulated substrate-fluid boundary). Therefore the temperature inside the element across the dropletyields to a profile shown in Figure 4.8.The heat conducted through the droplet is helping the droplet to provide the amountof heat required for faster evaporation. If there is a full thermal coupling betweenthe substrate and the fluid, i.e. the same temperature of the fluid and substrate atthe fluid-substrate interface,Q?s = KsA?T?h????substrate= KsA?T1??T2hs, (4.7)where Ks is thermal conductivity of the substrate. This shows that, in case of fullthermal coupling of substrate-fluid, more heat can be provided through substrateconductivity when (?T1??T2) is higher. From equation 4.5 and 4.7 and assumingthat all of the conductivity heat is obtained from the substrate, we will have:K`?T2hm= Ks?T1??T2hs, (4.8)or?T2 = ?T1(1+K` hsKs hm)?1. (4.9)We can write the scale of the height of the droplet for small droplet as:hm = R0 tan(?2), (4.10)50and combine this with Equation 4.9 to write it versus droplet radius:?T2 = ?T1(1+K` hsKs R0 tan(?2))?1. (4.11)If we assume a constant thermal capacity at a certain height of the element shownin Figure 4.8, there will be an upper bound for the value of ?T1. Then we can seefrom Equation 4.8 that the value of (?T1??T2) is higher at lower droplet radii?T1??T2 = ?T2(K` hsKs R0 tan(?2)). (4.12)Hence, we conclude that when the size of the droplet shrinks, the influence of thesubstrate conductivity has a more important role for heat transfer into the system.Equation 4.12 suggests that in an infinitesimal time step with a full thermal cou-pling of substrate-fluid, heat transfer through the substrate to the droplet is easierwhen the conductivity of the substrate is higher. Moreover, from Equation 4.11 wecould see that the effect of the droplet radius size is stronger when the value of(K` hsKs R0 tan(?2)) 1, (4.13)orR0 tan(?2)(K` hsKs). (4.14)The initial contact angle is defined by the interfacial forces balance on the contactline and by assuming that is independent of the radius size, we can find the criticalradius of the droplet that below which the substrate cooling effect influences theevaporation timeR0(K` hsKs tan(?2)). (4.15)where ? is the initial contact angle of the droplet before evaporation starts as shownin Figure 4.2.514.6 ConclusionsOur experimental data shows that the substrate conductivity plays an important rolein the evaporation of small sessile droplets. Although the temperature differenceacross a small droplet is not large, the substrate conductivity becomes important.We developed a numerical model (details in Section 3.3) to include the effect of thethermal conductivity of the substrate. Our numerical model is in good agreementwith experimental data and supports the measurement results. While the theorypredicts the same evaporation rates for a sessile droplet on substrates with differ-ent thermal conductivities, our experimental measurements and numerical resultsshow a higher evaporation rate for a substrate with a higher thermal conductivity.Moreover, the numerical and experimental results converge to the results predictedby the theory when the conductivity of the substrate is low. Finally, we discussedthe substrate cooling effect of the droplet size and showed that for sessile droplets,below a critical radius the substrate conductivity becomes substantial.52Chapter 5Evaporation of line droplets on asubstrate5.1 IntroductionThe study of the evaporation of liquid droplets on a substrate is a classical fluid me-chanic topic. When droplets are freely sitting on a substrate, the liquid moleculesevaporate by transition into a vapor layer at the surface of the droplet. The evap-oration rate is controlled by the transfer rate across the liquid-vapor boundary(kinetically-controlled evaporation) or the diffusive relaxation of the saturated va-por layer (diffusion-controlled evaporation) [18, 19]. The droplet shape will changeduring the evaporation, depending on whether the evaporation happens with a fixedcontact line (pinned stage) or with a moving contact line (de-pinned stage) or acombination of both. Inside a so-called ?point droplet? with a spherical cap, theevaporation with pinned contact line induces an internal flow toward the contactline of the droplet [25, 27]. In the evaporation of colloidal suspensions, this flowcarries the solute particles to the corners leading to a deposition along the ring-shaped contact line and this phenomenon is often referred to as the ?coffee-ring?effect [18, 27, 38].For the evaporation of small droplets the effect of the gravitational forces is negli-gible and the shape of the droplet is defined by the balance of the interfacial forcesbetween the solid, liquid, and gas phase. Therefore, the evaporation-induced flow53in small droplets is mainly governed by the interplay of these forces during theevaporation. This makes the evaporation process and hence the deposition of thesolute particles more controllable [3]. There has recently been growing interest indeposition and transport processes at micro scales using the evaporation-inducedflow in sessile droplets [3, 9, 11, 13?15]. Sharma et al. [3, 9] used this flow for theplacement and alignment of carbon nanotubes with line droplets with a high aspectratio of 1000:1 (length to width).The micro flow induced by evaporation in small point droplets on a substrate hasbeen extensively studied before. Deegan et al. [18] developed a method to describethe diffusion controlled evaporation and used this model to find the average flowvelocity inside the droplet toward the contact line for small point droplets with afully pinned contact line. Tarasevich [23] assumed potential flow inside the dropletand a uniform evaporation rate over its surface and analytically solved the flow fieldfor fully pinned point droplets and found the velocity profile inside the droplet. Huand Larson [27] solved the micro flow for a droplet with a fully pinned contactline and a small height using the Navier-Stokes equation. Fischer [39] used lubri-cation theory to solve for the particle convection in an evaporation-induced flowwith either fully pinned or fully de-pinned contact lines. Popov [20] solved themomentum and mass equations locally and globally for fluid and solute particlesand developed a closed-form expression for particles deposition at the contact lineas a result of evaporation-induced flow. While an extensive study of point dropletsexists in the literature, the evaporation of droplets with other shapes including linedroplets is a rather untouched area.Line droplets of different aspect ratios are usually produced at a small scale byusing micro contact [3, 9] or inkjet printing methods [40?44]. The shape of linedroplets of small size, when the magnitude of gravity is negligible, is mainly de-fined by the interplay of the interfacial tension forces, similar to point droplets.However the behavior of the contact lines at the long sides and at the ends of a linedroplet is not necessarily the same. This introduces a higher level of complexityto the evaporation process. Particularly, unlike point droplets, it is possible for linedroplets to have simultaneously pinned contact lines at the sides and de-pinnedcontact lines at the ends during the evaporation. Petsi and Burganos solved thepotential flow for infinite line droplets of cylindrical shape by assuming that all the54contact lines at the sides and at the ends are pinned during the evaporation [24].They continued their study and solved the Stokes flow for any contact angle [25]when all contact lines are anchored. Masoud and Felske [26] solved the Stokesequation for infinite line droplets with both fully pinned or fully de-pinned sidecontact lines. However, to the best of our knowledge, there is no work in the liter-ature discussing droplets with finite ratio of length to width and there is no modelthat explains the different behaviour of the side and end contact lines during theevaporation of line droplets. All of the models that assume infinite line dropletsignore the line ends. However, relatively high evaporation rates occur at the endsand it is likely for the droplet to shrink primarily from the ends. In fact, the mod-els for infinite line droplets correspond to models for point droplets with a slightlydifferent evaporation rate distribution at the surface of the droplet.Here we develop a model that considers the effect of both side and end contactlines using energy methods, and we solve for the evaporation rate distribution andthe velocity field in 3D.5.2 Model developmentWhen a line droplet is printed on a substrate, the elongated droplet readily assumesthe same contact angle ? everywhere. The shape of the line droplet consist of acylindrical segment with the length L at the center connected to two spherical capsat both ends with the initial radius R of their footprint (Figure 5.1).Similar to the spherical droplet, at any given contact angle ? , of the line droplet,the geometry of the droplet results in a non-uniform evaporation flux at the surface.For non-rotationally symmetrical geometries such as line droplets, this leads to acomplex contact line behavior at the different locations around the droplet whichleads to complex flows inside the droplet.We first develop a model that describes the geometry of the droplet during evap-oration, including a discussion of the contact line behavior, using an energy min-imization approach (Section 5.2.1). Then, given the geometry of the droplet for agiven remaining fluid volume, we find the evaporation rate distribution assuming aquasi-steady state regime and diffusion controlled evaporation. Finally, we use the55AirFluidSubstratexz(a) Side-viewFluidSubstratexy(b) Top-viewFigure 5.1: Line droplet from (a) side-view and (b) top-viewpotential flow assumption to approximate the evaporation induced flow inside theline droplet (Section 5.2.2).5.2.1 Contact line behaviorUnder the assumption that the interior flow occurs on a much shorter time scalethan the evaporation [27], the shape of the droplet will be determined by minimiz-ing the energy associated with interfacial tension at a given volume [45, 46]. Ourbasic model will consider a droplet of a given volume deposited on a substrate withinterfacial tensions for the liquid-gas, liquid-substrate and substrate-gas interfaces.This global approach is equivalent to a local equilibrium where forces on the triplepoint between the liquid, gas and substrate are considered, but it will be more con-venient for less symmetric shapes.However this basic model does not allow for the often experimentally observed56hysteresis where the contact angle at the triple point differs for an advancing and areceding contact line. This behavior can be described by introducing an additionalproperty, different surface tensions for wetted and un-wetted substrate . Whenthe contact line is advancing, the droplet will be covering un-wetted substrate, butwhen the contact line recedes it uncovers wetted substrate. This difference in sur-face energy will change the balance of forces at the triple point and will result in adifference between the advancing and receding contact angle.This approach can be demonstrated with the well understood case of a droplet witha circular footprint. The minimum area surface with this footprint is a spherical capthat is described by two parameters, the radius of the footprint and the contact an-gle. Fix the zero point of the energy by setting the surface tension of the un-wettedsubstrate-gas interface to vanish. Then denote the surface tensions for the liquid-gas, liquid-substrate, wetted substrate-gas and un-wetted substrate-gas interfacesby ?FG, ?SF , ?SG,d , and ?SG,w respectively. Then the energy of a droplet with acircular footprint of radius R and a contact angle ? deposited onto an un-wettedsubstrate can be easily calculated asE = ?FG Acap(R,?)+?SF A f oot print(R) (5.1)whereAcap(R,?) = 2piR2 tan(?2)(5.2)andA f oot print(R) = piR2. (5.3)This energy must be minimized under the constraint that the volumeV (R,?) = piR33sin3 ?(2?3cos? + cos3 ?) (5.4)be fixed to a given V0. For this simple geometry, this minimization can be per-formed exactly and the contact angle is found to be independent of the size of the57droplet. This is the advancing contact angle,?A = arccos(??SF?FG). (5.5)For an existing droplet which is receding, the interfacial tension of the wetted sub-strate becomes relevant. If R0 is the initial radius of the footprint, the energy for asmaller droplet with footprint radius R isE = ?FG Acap(R,?)+?SF A f oot print(R)+?SG,w Awetted(R,R0) (5.6)whereAwetted = piR20?piR2. (5.7)Again, for this simple geometry this minimization can be performed exactly and thecontact angle is independent of the size of the droplet.This is the receding contactangle,?r = arccos(??SF ??SG,w?FG). (5.8)Notice that as long as the droplet is receding, R0 has no effect on its shape, as wecan split its energy,E(R,? ,R0) = Edroplet(R,?)+Ewetsubstrate(R0), (5.9)into a term that depends only on the current shape of the droplet,Edroplet(R,?) = ?FG Acap(R,?)+(?SF ??SG)A f oot print(R), (5.10)and a term with depends only on R0,Ewetsubstrate(R0) = ?SG,w A f oot print(R0). (5.11)The only effect of the wetted substrate is to change the effective liquid-substratesurface tension for a receding droplet from ?SF to ?SF ??SG. This means that theregion of the wetted substrate can recede with the contact line of the droplet with-58out affecting the droplet shape.This model includes the well-known phenomenon of pinning. When a droplet isdeposited onto un-wetted substrate, it will settle at the advancing contact angle. Ifa small amount of the liquid is removed, for example by evaporation, it will notbe energetically favorable for the droplet to expose the wetted substrate and so thecontact angle will decrease with a fixed footprint until it reaches the receding con-tact angle. This behavior can be verified by numerically minimizing the energyfrom Equation 5.6 in this intermediary regime.The advantage of this energy formulation is that it can be easily extended to morecomplicated footprints, here to the elongated droplet shown in Figure 5.1b as arectangle with circular ends.Then the minimal surface covering a volume of liquid occupying this footprint willbe a cylinder segment with spherical caps attached to the ends. This minimal sur-face has 3 parameters, R the radius of the circular part of the footprint or half thewidth of the rectangular part, L the length of the rectangular part and ? the contactangle. Because the surfaces attaching to the circular ends and the rectangular mid-dle must match smoothly at their interface, the contact angle must be the same allalong the liquid-gas-substrate triple point (the contact line) throughout the evapo-ration.This 3 parameter minimization problem is tractable to numeric minimization:E(R,L,? ,R0,L0) = ?FGAcap(R,L,?)+?SFA f oot print(R,L)+?SG,wAwetted(R,R0,L,L0), (5.12)where L0 is the initial length of the line droplet,Acap(R,L,?) = 2piR2 tan(?2)+2RL?sin? , (5.13)A f oot print(R,L) = piR2 +2LR, (5.14)andAwetted(R,R0,L,L0) = A f oot print(R0,L0)?A f oot print(R,L). (5.15)59This energy must be minimized under the constraint that the volume of the linedropletV (R,L,?) = piR33sin3 ?(2?3cos? + cos3 ?)+LR22sin2 ?(2? ? sin(2?)) (5.16)is fixed to a given V0.When numerically solving this problem, multiple stages of the evaporation can beseen. First there is a fully pinned stage, where the footprint is fixed and the contactangle decreases. Next there is a partially pinned stage where L and ? decrease butR is fixed, until L vanishes. Finally there is the fully de-pinned stage when ? isfixed and R decreases.Informed by these numerical results, the partially pinned stage can be studied byfixing R. This reduces the problem to a two parameter minimization which can besolved analytically. We find that:LR=f (? ,(?SF ??SG,w)/?FG)g(? ,(?SF ??SG,w)/?FG)(5.17)where f and g are functions of the contact angle and interfacial tensions and areequal tof (? ,(?SF ??SG,w)/?FG) = pi (cos? +(?SF ??SG,w)/?FG)(1? cos?)2 (5.18)andg(? ,(?SF ??SG,w)/?FG) = ? 2 cos??? sin? (sin? ?2(?SF ??SG,w)/?FG cos?)? sin2 ? (cos? +2(?SF ??SG,w)/?FG) . (5.19)When L vanishes,cos? =?(?SF ??SG,w)/?FG = cos?r, (5.20)60the receding contact angle for the point droplet. Thus the partially de-pinned stagetransitions directly into the fully de-pinned stage of a point droplet when L reacheszero. The right hand side of Equation 5.17 has a divergence in theta for pi/4 ??(?SF ??SG,w)/?FG ? 1, which moves from ? = pi/2 at the start of this rangeto ? = 0 at the end. This means that within this range of parameters, this relationis invertible for an arbitrary aspect ratio and we can find a critical contact angle atwhich the partial depinning begins for any given aspect ratio.To illustrate this, consider a given initial aspect ratio and contact angle. Whenthe initial contact angle is higher than that given by the relation in Equation 5.17,then evaporation starts with a fully pinned stage at a fixed aspect ratio. Once thecontact angle meets that given by the relation in Equation 5.17, the partially pinnedstage begins where the contact angle decreases at the same time as the aspect ratio,while preserving this relation. Once L vanishes, the droplet will follow the fullyde-pinned stage of the point droplet.5.2.2 Velocity fieldTo calculate the flow field induced by evaporation, we first need to know the evapo-ration flux on the surface of the droplet. Assuming diffusion controlled evaporationand quasi-steady state diffusion, the evaporation flux is governed by the Laplaceequation?2C = 0, (5.21)where C is the vapor concentration around the droplet [19]. The concentrationoutside the droplet is found by solving this equation with the following boundaryconditions:at |x| (R+L/2) or z H or |y| R:C = ?Cv (5.22)where ? is the relative humidity and Cv is the saturated vapor concentration in air,at z = 0 there is no flux into the substrate?C = 0, (5.23)61and at the surface of the droplet the concentration is equal to the vapor saturationconcentrationC =Cv. (5.24)By knowing the concentration field, we can calculate the evaporation mass flux atany point on the surface of the droplet by~J(x,y,z) = D~?C, (5.25)where D is the diffusion coefficient of water vapor in air. Hence, we know theamount of fluid removed at the surface of the droplet and the geometry at eachquasi-steady state time step. While in principle the flow could be rotational, themain flow behavior for relatively slow shape changes in this quasi-static case canbe approximated by the potential flow formulation. For spherical droplets the ve-locity field and streamlines modeled by a potential flow matched well in magni-tude and shape with Stokes flow and the Navier-Stokes solutions [23, 24, 26, 27].Hence, since the evaporation-induced flow is in principle the same for droplets ofany size and footprint shape, it makes sense to similarly assume that the potentialflow describes the internal flow field for the quasi-static flow generated inside linedroplets at good approximation. The potential flow field described by ? yields?2? = 0, (5.26)with the following boundary conditions:Symmetry at x = 0:???x = 0 (5.27)Symmetry at y = 0???y = 0 (5.28)at z = 0 there is no flow through the substrate:??? z = 0 (5.29)62the kinematic boundary condition at the surface of the droplet, the normal velocityis equal to:(~n ????n)=~n ? ~J? +[~nx, ~ny, ~nz] ?[0, 0,?h? t], (5.30)where ? is the density of water, h is the local height of the droplet at any point,~n is a unit vector along the normal direction to the surface, and nx, ny, and nz arethe x, y and z components of the unit vector n, respectively. If we assume thatduring the evaporation, the droplet shape remains a spherical cap at both ends witha cylindrical segment at the center, we can relate (?h/? t) at any point of the dropletsurface (x,y,z) to the liquid line height, H, using geometrical constraints?h? t =??????H4?R44H3?(H2 +R22H)2? r2+R2 +H22H2??????H?+r(x? L2 )?(H2 +R22H)2? r2H(x?L/2) L?+??????R2H2H2 +R2?(H2 +R22H)2? r2?RH??????R?, (5.31)where r is the xy?planar projection of the distance of this point from the center lineof the cylindrical segment; this means r = y for |x| ? L/2 and r2 = (x?L/2)2 +y2for x > L/2.5.3 ResultsTo investigate the flow field inside line droplets generated by evaporation at roomtemperature of T = 24?C and relative humidity of ? = 0.5, we first choose a waterline droplet on a glass substrate with an initial aspect ratio of L0/R0 = 10 andinitial radius of R0 = 1. Then we follow step by step the model that was presented630 1 2 3 4 5 60246810 L V (mm3)L (mm)2530354045505560 (degree)Figure 5.2: Length L and contact angle ? of a line droplet with the initialaspect ratio of L0/R0 = 10.above for finding the velocity field. First, the contact angle ? and the length of theline droplet L are calculated from Equation 5.17 as the fluid volume is reduced byevaporation (Figure 5.2). The rate at which L and ? change with the fluid volume Vis calculated by finding their derivatives dL/dV and d?/dV from Equation 5.17 asshown in Figure 5.3. The rate of change with respect to time, i.e. L? and ?? , is foundby multiplying these values by the total evaporation rate from Equation 5.25. Then,the diffusion controlled evaporation is modeled by solving the Laplace equation.The local evaporation flux is calculated using Equation 5.25. The evaporation fluxacross the cylindrical and spherical parts is shown in Figure 5.4. The evaporationflux across the droplet along its length is obtained by Equation 5.25 and is shown inFigure 5.5. The total evaporation rate is found by integrating the local flux ~J(x,y,z)over the surface for each time step. Then the potential flow is solved with theboundary condition shown in Equations 5.27 to 5.30 (Figure 5.6).Using the potential flow function, we can find the velocity field. As shown in Figure6, the potential flow solution yields an outward velocity both along and across the640 1 2 3 4 5 61.82.02.2 dL/dV d /dVV (mm3)dL/dV (1/mm2)05101520 d/dV (degree/mm3)Figure 5.3: The change of the contact angle ? and the line length L withdroplet volume for a shrinking droplet with an initial aspect ratio ofL0/R0 = 10.droplet with a higher magnitude along the length of the line droplet.The quasi-steady state assumption dictated the outward velocity field pattern forline droplets with acute initial contact angles near the contact line for all aspectratios L/R during the evaporation similar to the velocity field in Figure 5.7 for aline droplet with the initial aspect ratio of L0/R0 = 10. Equations 5.17 to 5.19show that during the partially pinned stage of the evaporation, the contact angle ?depends only on the aspect ratio L/R of the droplet for given initial and recedingcontact angle parameters. The contact angle as the line droplet shrinks as a resultof evaporation versus the aspect ratio of the droplet is shown in Figure 5.8, whereis can be seen that for the partially pinned stage the aspect ratio and the contactangle follow a fixed relationship independent of the width of the droplet, 2R or theinitial aspect ratio.Finally, the effect of changing the interfacial tensions equivalent to changing thereceding contact angle in the point droplet was studied for a droplet with R0 = 1mm and L0/R0 = 10 mm. Figure 5.9 shows the effect this has on the relationship65Figure 5.4: The evaporation flux on the centerline of the droplet and thespherical end for the droplet with the initial aspect ratio of L0/R0 = 10.0 2 4 65x10-76x10-77x10-78x10-79x10-71x10-61x10-6J (g/mm)x (mm)Figure 5.5: The evaporation flux across the line droplet of the shrinkingdroplet for an initial aspect ratio of L0/R0 = 10 at L = 8.86 mm.66 3.056 10 1.321 10 Figure 5.6: The contour plots of the evaporation induced flow magnitude inm/s in a line droplet with the initial aspect ratio of L0/R0 = 10.xz(a) The velocity field in zx planeyz(b) The velocity field in zy planeFigure 5.7: The outward velocity vectors across the droplet at (a) the zx planeand (b) zy plane. Cross sections are taken at the middle of the droplet.670 5 10 15 20 25202530354045505560R0 = R*0, L0 = R*0 R0 = 10 R*0, L0 = 10 R*0 R0 = R*0, L0 = 2 R*0 R0 = 5 R*0, L0 = 10 R*0 R0 = R*0, L0 = 5 R*0 R0 = 2 R*0, L0 = 10 R*0 R0 = R*0L0 = 20 R*0 R0 = R*0L0 = 10 R*0 (degree)L/RFigure 5.8: The contact angle versus the aspect ratio for the fully and partiallypinned stages of the evaporation. The relationship between these two inthe partially pinned stage can be seen to be independent of the initialaspect ratio and radius. The initial and receding contact angles are 60?and 20?, respectively.between the contact angle and the aspect ratio during the partially pinned stage ofthe evaporation, for different receding contact angles.5.4 DiscussionOur results show that the pinning of the contact line of line droplets leads to the out-ward evaporation-induced flow toward the contact line as previously described forspherical and infinite cylindrical droplets [24?27]. However, for the line dropletswith aspect ratios of L/R > 1, the evaporation at the ends of the line is considerablyhigher compared to the center of the droplet (as shown in Figure 5.4). The higherevaporation rate requires more fluid to move toward these sites. Hence the outwardflow toward the ends along the droplet is higher as shown in Figure 5.7. Howeverthis flow does not necessarily balance the high evaporation rate. In fact, in Figure5.8 and Figure 5.9, we show that at the ends of the line droplet, the contact linestarts to recede at a much higher contact angle compared to the receding contact680 5 1020304050607080 r= 20 r= 25 r= 30 r= 40 (degree)L/R0Figure 5.9: The contact angle as a function of aspect ratio in the partiallypinned stage of line droplet evaporation for line droplets for differentreceding contact angles. R0 and L0 are fixed to 1 mm and 10 mm, re-spectively.angle of spherical droplets. This higher receding contact angle occurs while thewidth of the droplet remains constant, but the length of the droplet shrinks. Forhigher aspect ratio droplets, the contact angle at which the ends start to recede ishigher (as shown in Figure 5.8 and Figure 5.9) and the initial stage of a fully pinnedcontact line is consequently shorter. Therefore, the stronger flow toward the endsis opposed by the ends receding sooner for long droplets.Also, we can see from Equations 5.17 to 5.19 and Figures 5.8 and 5.9 that the con-tact angle at which the two end caps of the line droplets join each other (i.e. whenL = 0 and the footprint becomes circular) is the receding contact angle of the pointdroplet independent of the initial shape and initial equilibrium contact angle of thedroplet. This is of course subject to the assumptions of our model that shows goodagreement with the experimental results when applied to spherical droplets [47].695.5 ConclusionsHere we explored a new method that models the evaporation of line droplets withdifferent shapes and length to width ratios. Using a global method where the energyof interfacial tensions is minimized allows us to study more complicated dropletshapes in a straightforward manner. This method allows the de-coupling of theevaporation and momentum equations of the fluid and the contact line and hencecould be used for any droplet shape. Our results show that during the evaporationof line droplets, while the equations for fluid momentum and evaporation are sim-ilar to spherical sessile droplets in principle, the complex behavior of the contactline in different regions of the droplet affects the evaporation-induced flow fieldinside the droplet. Our model uses a new approach to study the evaporation of linedroplets and the evaporation rate and evaporation-induced flow results from ourmodel need to be verified by experiments in future work.In our method we used the potential flow assumption for simplicity and to show themain idea of the model. Including the viscous effects by solving the Navier-Stokesequations for the fluid would be an interesting direction for further investigation.However, this would result in a complex instability problem and the de-couplingof the contact line momentum equation from the fluid would be a challenge. De-pending on the scale of the droplet geometry and interfacial forces, a discontinuityin the line droplet is expected. Surface roughness effects could also be added to theenergy model in future work.70Chapter 6Effect of surfactants andsubstrate temperature on contactline behavior6.1 IntroductionAs mentioned previously, the initial contact angle of a sessile droplet with the sub-strate surface is defined by the balance of the interfacial tension forces at the contactline of the droplet. During the evaporation of sessile droplets of colloidal suspen-sions, particles deposit along the contact line as a result of ?coffee ring? effect [18].This is caused by the evaporation-driven flow evaporation inside the droplet whichcarries the solute near the contact line [19, 27]. The deposition of solute using theevaporation induced flow inside droplets has applications in a variety of areas suchas biology [10, 15, 48?50], separation processes [12], deposition processes [3, 6, 9]inkjet printing [4, 33, 41, 42, 51], and wetting processes [52?55].The contact line motion during evaporation is the key to understanding the flowgenerated during the evaporation of sessile droplets [24?26]. For a hydrophilicsubstrate and a droplet with a fixed contact line an outward flow is generated in-side the droplet, while a receding contact line results in an inward flow inside thedroplet [26]. During the evaporation of sessile droplets two separate stages for the71Figure 6.1: Interfacial forces on the contact line during the evaporation of asessile droplet.contact line behavior exist: evaporation with a fixed contact line (pinned stage) andevaporation with a moving contact line (de-pinned stage). During evaporation ofa droplet, in general both stages exist, starting with the pinned contact line withreducing contact angle followed by the de-pinned stage with moving contact lineand constant contact angle. The evaporation-induced flow for each stage has beendetermined separately and the flow dynamics have been discussed extensively [23?27]. Petsi and Burganos showed that for evaporation with constant evaporation flux(kinetically controlled evaporation) of the droplets with an initial contact angle ofless than pi/2, the pinned stage produces an outward flow while the de-pinned stagecreates an inward liquid flow [25]. Masoud and Felske analytically solved the inter-nal flow for droplets with diffusion controlled evaporation and showed that whilethe flow is always outward at any contact angle for droplets with a fully pinnedcontact line, the moving contact line creates a complex flow pattern including bothflow toward and away from the contact line [26]. However, in all of these stud-ies, only a single stage of the evaporation has been studied, not a combination ofthe stages. This is mainly due to the complexity that the transition from one stageto another imposes on the flow dynamics inside the droplet. This transition be-tween the stages results from the balance of the interfacial forces at the contactline. The three effective forces on the contact line during the evaporation actingon the substrate-gas (?SG), substrate-fluid (?SF ) and fluid-gas interfaces (?FG) areshown in Figure 6.1.Assuming a homogeneous, rigid, isotropic solid substrate surface that is imperme-able to fluid, and by neglecting the contribution of the droplet size on the equi-librium contact angle [56], we can write the Young?s equation for the ideal three72phase systems as?SG??SF ??FG cos? = 0, (6.1)where ? is the equilibrium contact angle. The adsorption of colloids or surfactantin colloidal suspensions in water at the fluid-gas interface or at the substrate-fluidinterface, could alter the surface tension and consequently the contact angle [57?59]. Okubo et al. showed that while the diameter of the silica microspheres atT = 24?C has no major effect on the surface tension at the fluid-gas interface fora particle concentration within the range of 0.0001-0.02 wt%, the surface tensionchanges due to the presence of polystyrene particles when the microsphere diame-ter is smaller than 0.2 ?m [59]. They related this phenomenon to the special order-ing of the particles near the fluid-gas interface as a result of the strong hydrophobicsurface of the polystyrene microspheres. However they reported that for particleslarger than 0.2 ?m in diameter the surface tension remains constant and equal tothe value of the pure water at different particle concentrations up to 0.1 wt%. Hy-drocarbon surfactants such as TWEEN R?20 or sodium dodecyl sulfate (SDS) havebeen shown to exhibit a higher value of ?SF and a lower surface tension than thepure liquid [60]. Also, the values of ?SF and ?FG change with temperature. Theinterfacial tension of water-air changes linearly with temperature with a sensitiv-ity of ?0.16 mN m?1 K?1 within the temperature range of 15 to 50?C [61]. Thesurface tension of the TWEEN R?20 and SDS surfactants in water has also beenshown to decrease linearly with temperature with a sensitivity independent of thesurfactant concentration and in the same range as for distilled water [62]. Also,the interfacial tension of water-air changes linearly due to these surfactants with asensitivity of ?0.01 mg L?1 mN m?1 within temperature range of 5 to 50?C.During the evaporation of colloidal suspension with surfactants, the internal flowchanges the initial homogeneity of the suspension by carrying the solute and sur-factants toward the contact line. By changing the concentration of the surfactantsnear the contact line, the surface tension at the contact line changes. This changesthe balance of the interfacial forces at the contact line during the evaporation.The effect of the temperature and surfactant concentration gradients on producinga Marangoni flow has been discussed previously. Hu and Larson showed that thiseffect suppresses the outward flow for droplets with a fully pinned contact line73Time (s)Voltage (V)21 ?s10 ms1 nsFigure 6.2: Signal properties of the micro dispenser unit. The tip of the noz-zle was 1 mm above the substrate and we used 50 cycles to achieve adroplet with the diameter of 250 ?m.during the whole time of evaporation [27, 31]. While this assumption is useful forstudying the Marangoni flow for a pinned droplet, in general it does not apply tothe entire evaporation process. Here we study the effect of colloid and surfactantconcentration on the contact line behavior during evaporation. We also show theeffect of the substrate temperature on the evaporation stages and eventually discussthe effect of substrate temperature on the width of the deposit near the contact line.6.2 MaterialsDistilled water was used to prepare the suspensions with two different solid mi-crosphere types: Fluoro-Max polystyrene beades (Bangs Laboratories, Inc., IN46038, United States) with a diameter of 1 ?m (S1) and surfactant-free whitecarboxyl/sulfate-fictionalized polystyrene beads (Life Technologies Inc., Burling-ton, ON, Canada) with a size of 0.6 ?m (S2). We used TWEEN R?20 as sur-factant at different concentrations well below its critical micelle concentration(Ccritical ? 0.01 wt%). The concentration of solid microspheres of S1 and S2 andthe concentration of surfactants in the colloidal suspension are Cs1, Cs2 and Csur,respectively. The printing setup was described in Section 3.2. The dispensing noz-zle was actuated with the signal described in Figure 6.2. A Peltier device wasused under the substrate (polystyrene coated plastic slide from Ted Pella Inc., CA96049-2477, United States) to set the temperature and closed-loop control was usedto maintain a constant temperature (details in Section 3.2.2).746.3 Results and discussionWe investigate the contact line behavior of the droplets with particles and surfac-tant during evaporation in two different studies. We first study the effect of thesurfactant and particles concentration on the pinned stage and the de-pinned stageof the droplet during evaporation at room temperature in Section 6.3.1. Then weshow the effect of the substrate temperature on the contact line behavior of thesedroplets during the evaporation in Section 6.3.2. We study this effect when thesubstrate temperature is cooled below the ambient temperature. We also discussthe effect of the substrate temperature on the deposition of the particle ring afterdrying of the droplets with particles and surfactant.By assuming an ideal three phase system, i.e. the solid surface is homogeneousand smooth, we can neglect the effect of the initial droplet size, R0, on the reced-ing contact line and the contact angle [56, 63?65]. Under these conditions andby neglecting the thermal and surfactant induced Marangoni effects, the behaviorof the contact line represented by the normalized radius versus normalized timecan be well scaled for all droplet sizes if we assume a negligible variation of thesurface tension with the size of the droplet [56]. Therefore, to show the effectof colloids and surfactant concentration, we consider the normalized radius of thedroplet r(t)/R0 for an initial radius of R0 = 250 ?m for all droplets as a function ofthe evaporation time, normalized with the time over the total evaporation durationt/t f .Each data series represents the average of roughly 20 experiments that were con-ducted under the same conditions with a relative standard deviation of less than 1%.The minimum contact angle measurable by our imaging device is 8? and the datapoints with lower contact angles are not shown. At low concentration of the colloidparticles (below 1%) and surfactants (below 0.05%) no major change in the initialcontact angle of the droplets is observed (with an error less than 0.1%). Therefore,the total evaporation time, t f for the droplets at low concentrations colloids andsurfactants is the same and equal to 60 s (with 1% error).750 10 20 30 40 50190210230250 Radius Contact AngleTime (s)Radius (m)102030405060 Contact Angle (?)Figure 6.3: Measured radius and contact angle of a sessile droplet of a col-loidal suspension containing S1 and surfactant at concentrations of 0.05wt% and 0.0005 wt%, respectively, during evaporation at T = 24 ?C.6.3.1 Effect of surfactant and particle concentration on the contactline behaviorThe effect of the solute particles and surfactant on the contact line during evap-oration is assessed by measuring the radius and the contact angle of the dropletover time. As an example, Figure 6.3 shows the radius and the contact angle of adroplet containing S1 with the concentration Cs1 = 0.05 wt% and surfactant withthe concentration Csur = 0.0005 wt%. The contact line of this droplet is pinnedduring the entire evaporation process, indicated by the constant droplet radius andthe constantly decreasing contact angle.The effect of the concentration of S2 on the behavior of the contact line is shown inFigure 6.4 with the normalized radius of the droplet with respect to the normalizedtime at different concentration of S2 from 0.014 to 0.8 wt%. The fluid was surfac-tant free and the liquid droplets evaporated at room temperature. No major changein the behavior of the contact line is observed at this range of concentrations thatagrees with the results obtained by Okubo et al. [59]. To show the effect of sur-760.0 0.2 0.4 0.6 0.8 1.00.600.650.700.750.800.850.900.951.00t/tfcs1 = csur = 0% Water cs2 = 0.014% cs2 = 0.044% cs2 = 0.4% cs2 = 0.8%r/R0Figure 6.4: The effect of the concentration of S2 on the evaporation stages.factant on the contact line and the contact angle behavior, solutions with differentconcentrations of surfactant up to 0.0005% were prepared. Figure 6.5 shows thenormalized radius versus normalized time of these droplets. The evaporation startswith a pinned stage when the normalized radius of the droplet is equal to 1. Afterroughly half of the evaporation time passes (i.e. t/t f ?= 0.5), the contact line startsto move and the normalized radius is reduced for low surfactant concentrations. Atsurfactant concentrations above 0.0002% the contact line stays fully pinned.However, while increasing the surfactant concentration lengthens the duration ofthe pinned-stage of the droplet and pins its contact line fully, at concentrations nearcsur = 0.002 wt% and above, the surfactant significantly affects the initial dropletgeometry. The initial contact angle of the droplet decreases by about 10? whenthe concentration of the surfactant is increased from 0.0005 wt% to 0.001 wt% asshown in Figure 6.6.The effect of a combination of particles and surfactant on the contact line of thesessile droplets evaporating at room temperature is shown in Figure 6.7. The trend770.0 0.2 0.4 0.6 0.8 1.00.600.650.700.750.800.850.900.951.00t/tf Water csur = 0.0002% csur = 0.0004% csur = 0.0005%r/R0cs1 = cs2 = 0%Figure 6.5: The effect of low concentrations of surfactant on the behavior ofthe contact line.observed is similar to the effect of surfactant alone shown in Figure 6.5, and theparticles do not seem to affect the contact line behavior as in Figure 6.6. The ex-periments suggest that the presence of solute solid microspheres of different sizes(S1 and S2) at concentrations up to 0.05% for S1 and 0.8% for S2 has no majoreffect on the behavior of the contact line (Figures 6.7 and 6.4, respectively), whilethe presence of surfactants increases the duration of the pinned stage during evapo-ration. This can be easily understood by impact of the presence of these solutes onthe interfacial tensions. While low concentrations of microspheres do not changethe surface tension, adding surfactant reduces it. This means that the surfactantreduces the surface tension as the main driving force for de-pinning, and thus thecontact line stays pinned for a longer period during evaporation.780.0 0.2 0.4 0.6 0.8 1.00.600.650.700.750.800.850.900.951.00 r/R for csur = 0.0005% r/R for csur = 0.001% CA for csur = 0.0005% CA for csur = 0.001%r/R0t/tf0102030405060 Contact angle (o)cs1 = cs2 = 0%Figure 6.6: Normalized radius and contact angle measurement of the mix-tures with higher amount of surfactants.0.0 0.2 0.4 0.6 0.8 1.00.600.650.700.750.800.850.900.951.00t/tf Water cs1 = 0.01%, csur = 0.0001% cs1 = 0.03%, csur =0.0003% cs1 = 0.05%, csur =0.0005%r/R0cs2 = 0%Figure 6.7: The effect of the concentration of S1 on evaporation stages.796.3.2 Effect of substrate temperature on the contact line behaviorand particle depositionThe effect of the substrate temperature on the contact line behavior is shown forsolutions without and with surfactant and particles (S1) in Figure 6.8, at an ambienttemperature of 24 ?C and relative humidity of 49.3%. The total evaporation time islonger at lower substrate temperatures. The normalized radius of the droplet versusthe normalized time during evaporation of water droplets at different temperaturesis shown in Figure 6.8a. The same parameters are shown for the droplets withsurfactant and particles (S1) in Figure 6.8b; according to the previous section weassume that the contact line behavior is affected by the surfactant alone and notby the particles. Particle deposition can therefore be studied without the particlesaffecting the internal flow inside droplets. For pure water, the temperature has anegligible effect on the evaporation modes as shown in Figure 6.8a. This is mainlydue to the low sensitivity of the surface tension to temperature for water, and thethermal gradient across the droplet does not significantly affect the behavior of thecontact line. When surfactant is added to the droplets, we observe no major changein the duration of the fully pinned stage at lower substrate temperature. However,at lower substrate temperature the de-pinned stage leads to a smaller final dropletradius as shown in Figure 6.8b. This is mainly because of the surface tension, thatis the main force of de-pinning, is higher near the contact line when the substratetemperature is lower [62].Figure 6.9 shows the deposition of particles (S1) near the contact line as a result ofevaporation of droplets in Figure 6.8b. The particles deposit in one single layer onthe substrate. We measured the width of the particle ring for droplet suspensionswith 0.05 wt% S1 and 0.0005 wt% surfactant, ? at different substrate temperatures.The normalized width of particle deposition ?/R after evaporation as a function ofsubstrate temperature is shown in Figure 6.10.The effect of a lower substrate temperature on particle deposition at the contactline requires explanation. According to Figure 6.8b, the contact line stays pinnedfor the same normalized duration at all temperature. It will therefore stayed pinnedfor longer in absolute time at lower temperature. Therefore, the outward flow thatcarries particles to the contact line during evaporation persist for a longer amountof time. At the same time, the outward velocity magnitude is smaller for substrates800.0 0.2 0.4 0.6 0.8 1.00.30.40.50.60.70.80.91.0t/tf T=18oC T=22oC T=24oCr/R0Water, cs1 = cs2 = csur = 0(a)0.0 0.2 0.4 0.6 0.8 1.00.30.40.50.60.70.80.91.0t/tfcs1 = 0.05%, cs2 = 0%, csur = 0.0005%T=18oCT=22oCT=24oCr/R0(b)Figure 6.8: The effect of the substrate temperature on the behavior of the con-tact line for solutions (a) without and (b) with surfactant and particles.8120 ?m(a) T = 18?C20 ?m(b) T = 22?C20 ?m(c) T = 24?CFigure 6.9: Particle deposition from a sessile droplet with 0.05% S1 and0.0005% surfactant at different substrate temperatures. The initial ra-dius of the droplets was R0 = 250 ?m.8218 19 20 21 22 23 240.0400.0450.0500.0550.0600.0650.0700.0750.080/RSubstrate temperature (degree)Figure 6.10: The normalized width of particle rings for droplets at differentsubstrate temperature; the size of the error bars is the standard devia-tion of the average ring width for roughly 20 droplets with 0.05 wt%S1 and 0.0005 wt% surfactant for each temperature as shown partly inFigure 6.9.at lower temperature due to the reduced evaporation rate. These two effects shouldcancel each other. However the particle ring has a larger width for sessile dropletsuspensions at lower substrate temperature (Figure 6.9 and Figure 6.10).The first apparent reason for a wider ring at lower temperature may seem to be thelonger evaporation time and thus a longer pinned stage for the substrate at lowertemperature. However, if we assume that all of the ring deposition happens at thepinned stage, we could see the total depositionm deposition ?(dVdt)t, (6.2)at time t where (dV/dt) is the evaporation rate. Approximating the evaporationrate with V/t f , we see that the ring deposition is only a function of the normalized83time. This means that although the evaporation time for ring deposition is faster atlower temperatures, the average outward flow velocity is slower such that it leadsto the same deposition mass. To explain the wide deposition band for lower tem-perature substrate, we need to account for the two types of flow for small droplets(r < 1 mm when gravity is negligible): viscous outward flow and Marangoni flowat the droplet surface.For droplets with a pinned contact line, the evaporation-induced viscous flow to-ward the contact line keeps the substrate wet until the contact line by replacing thefluid that evaporates near the contact line. Hence, the viscous flow generated in adroplet with pinned contact line in the diffusion controlled evaporation is alwaysoutward [18, 19, 27]. However, the direction of the Marangoni flow at the surfaceof the droplet is somehow more complex. As mentioned earlier, for droplets withsurfactant, the surface tension increases when the temperature decreases. When thesubstrate is at a lower temperature than the environment, the droplet surface is at itslowest temperature near the contact line. The surface tension is higher at a lowertemperature and therefore a Marangoni flow is produced from the center toward thecontact line at the surface of the droplet (Figure 6.11a). For evaporation at roomtemperature, the substrate temperature is not above the ambient temperature andevaporation would lower the temperature at the contact line. However, the temper-ature difference (?T ) will be much smaller compared to the cooled substrate. It isworth pointing out that for a hot substrate at a temperature well above the ambi-ent temperature, the Marangoni flow at the surface is in the opposite direction asshown in Figure 6.11b. This is due to the higher temperature near the contact linethat leads to a lower surface tension, and generates a Marangoni flow at the surfacetoward the center of the droplet.The magnitude of the Marangoni flow compared to the outward viscous flow canbe compared using the dimensionless thermal Marangoni numberMa =??Tt f?R , (6.3)where ? is the temperature sensitivity of the surface tension, ?T is the temperaturedifference across the droplet and ? is the viscosity of the fluid. Deegan et al.[18] estimated that for a droplet evaporating at room temperature the amount of84Cold Substrate(a)Hot Substrate(b)Figure 6.11: Evaporation induced flow for cold (a) and hot (b) substrates;Marangoni flow at the surface of the droplet carries the particles to thecontact line with a cold substrate while it drives them away from thecontact line with a hot substrate.deposit carried by the Marangoni flow is around 10% of the total deposit. Whenthe substrate temperature is reduced, the thermal Marangoni number is higher andthe flow is stronger. Therefore more deposit is carried toward the contact line andleads to a ring deposition with a larger width (Figure 6.10).6.4 ConclusionsThe effect of colloids and surfactant on the evaporation of sessile water dropletswas investigated. Microspheres of two different sizes (0.6 ?m and 1 ?m) were usedto prepare suspensions at different concentrations, and the evaporation of dropletswas studied at concentrations below 1%. No major change was observed in thebehavior of the contact line and the contact line was almost pinned for near 50% ofthe whole evaporation time for all of the suspensions without surfactants. We alsostudied the effect TWEEN R?20 as surfactant on the evaporation of sessile droplets.At low surfactant concentration below 0.0005 wt%, no major change in the ini-85tial contact angle and radius of the sessile droplets was observed. However, weobserved an increase in the pinned-stage of the evaporation process and a lowerreceding contact angle. A fully pinned contact line was observed at surfactant con-centration near 0.0005 wt%.We also investigated the effect of the substrate temperature on the evaporation ofdistilled water and colloidal suspensions with a surfactant concentration of 0.0005wt%. We measured the size of the ring deposition of microspheres after dryingof colloidal suspensions with the surfactant concentration of 0.0005 wt% and weshowed a wider ring at lower substrate temperature. We showed that the main rea-son that causes the wider ring deposition is a result of a stronger thermal Marangoniflow toward the contact line.86Chapter 7Conclusions and future work7.1 ConclusionsThe basic evaporation models of sessile droplets on a flat substrate was discussedand the limitation of the models to predict the evaporation of small sessile dropletswas discussed. The influence of the substrate thermal properties was confirmedthrough sets of experiments of water droplets evaporation on different substrateswith different thermal conductivities. While the basic models predict that the evap-oration rate per unit length of the droplet, i.e. (dV/dt)/R, yields the same value,the model predictions are from the experimental results. A numerical model wasdeveloped to include the thermal effects of the substrate. The numerical modelconfirmed that for thin droplets, substrate cooling has a strong influence on dropletevaporation rate and time. Furthermore, when the substrate is considered non-conductive, the results from our model approach the basic model. A dimensionlessnumber was suggested that relates the influence of thermal properties on the evap-oration process to the droplet size.Furthermore, a model was developed for the evaporation of line droplets withfinite aspect ratio. The complex behavior of the contact line in line droplets andthe existence of both pinned and de-pinned stages simultaneously was discussedthrough a model that solves the most stable shape of the droplet during evaporationusing energy equations. This approach de-coupled the evaporation equations fromthe energy equations. The evaporation flux was found by solving a Laplace equa-87tion in a quasi-steady state for diffusion-controlled evaporation. By knowing theevaporation flux and the geometrical shape of the most stable form of the droplet,the behavior of the contact line at the ends and at the sides was presented. Threedifferent stages for the contact line were distinguished: fully pinned, de-pinnedends and pinned sides, fully de-pinned. Also, the model shows that total evapo-ration along the line droplet per unit area is higher near the ends. By having theshape of the droplet found using our model, the evaporation-induced flow insideline droplets was investigated inside line droplets. A potential flow assumption wasused to find the velocity field. The outward evaporation-induced flow was showninside the line droplets during evaporation.Finally, the effect of surfactant and particles on the contact line behavior was dis-cussed. Sets of experiments were run for colloidal suspensions containing differentparticles (of 0.6 ?m and 1 ?m) at different concentrations. No major effect on thebehavior of the contact line was observed due to the presence of these particlesat different concentrations. However, the surfactant concentration showed to havea major effect on the pinned an de-pinned stages during evaporation. At lowerconcentration of surfactant, below 0.0005 wt%, the duration of the pinned stage in-creased with increasing surfactant concentration. This was mainly due to the lowersurface tension that is the main force for moving the contact line. The effect of thesubstrate temperature was also investigated. The cooling of the substrate showedno effect on the ratio of pinned and de-pinned stage at the absence of surfactant.At the presence of surfactant, cooling the substrate increased the ratio of the dura-tion of the de-pinned to pinned stage. However, despite the longer duration of thede-pinned stage, a larger ring deposition was observed at lower temperature of thesubstrate. This was due to the Maranoni flow at the surface that carries the particleat the surface toward the contact line.The major contributions of this thesis are:? Developing a numerical model for the evaporation of sessile droplets thatincludes the effect of the substrate conductivity based on FLOW-3D subrou-tines88? Showing the strong influence of substrate thermal conductivity on evapora-tion of small droplets and finding a critical radius of droplets below whichthe substrate conductivity becomes substantial.? Developing a new model based using an energy method to study the evapo-ration of line droplets where pinned and de-pinned segments of the contactline exist simultaneously? Finding the outward evaporation-induced flow inside line droplets using apotential flow assumption? Showing the effect of surfactants and particles on the pinned and de-pinnedstages during the evaporation of sessile droplets of colloidal suspensions? Investigating the effect of temperature on the pinned and de-pinned stagesduring the evaporation of sessile droplets of colloidal suspensions? Studying the effect of temperature on the deposition of particles from sessiledroplets of colloidal suspensions and showing the strong influence of theMaranoni flow at the surface on the deposition.7.2 Future workIn evaporation of small droplet, the technique that was used in the numerical ap-proach to include the thermal effects was to assume a uniform yet separate phasefor air as a void. Although this is a better approximation than assuming a uniformtemperature across the fluid-gas interface, the temperature discontinuity at the in-terface is not physically realistic. Also, the values of the vapor concentration andthe diffusion coefficient change with the temperature gradient across the droplet.While for small droplets, the temperature difference is small and this gradient doesnot produce significant error, this effect should be included for larger droplets. Asa future task, this error can be fixed by using the two fluid model with phase changeand a non-condensable gas. This allows the saturation properties in a cell to be afunction of the local vapor concentration, which is affected by both phase changeand diffusion. However, this requires a new method for fixing the contact line dur-ing the pinned stage. The technique that was used here to fix the contact line by89defining a dummy substance in the void will no longer work. 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Here we showthe geometrical calculations of the spherical cap for point droplets that were usedin the previous chapters.A.2 Spherical cap calculationsFor a point droplet with a circular footprint radius of R and the equilibrium contactangle of ? as shown in Figure A.1, we can find the radius of the spherical cap a asa =Rsin? . (A.1)If we write a based on hm and R as:a2 = (a?hm)2 +R2 (A.2)98 ? Figure A.1: Schematic of a sessile droplet on a substratethen we will finda =h2m +R22hm. (A.3)We can also find ? in terms of R and hm by rewriting Equation A.1 as:sin? = 2Rhmh2m +R2, (A.4)or? = arcsin(2Rhmh2m +R2). (A.5)The height of the droplet, hm, can be written in terms of a and ? asa?hm = acos? , (A.6)and thus:hm = a(1? cos?) . (A.7)99By replacing a from Equation A.1 into Equation A.7, we can write hm in terms ofR and ? as:hm =Rsin? (1? cos?)=Rsin?(1?1?2sin2(?2))=R2sin(?2)cos(?2) ?2sin 2(?2), (A.8)to findhm = R tan(?2). (A.9)The cross sectional area of the droplet, A, can be calculated by subtracting thetriangle area of 2A2 from a minor circular sector seen by 2? as:A =2?2pi pia2?2A2= ?a2?2 ? 12?Rtan? ?R=?R2sin2 ??R2tan? (A.10)The volume of the droplet,V , can be found by calculating the volume of the spher-ical segment as:V =hm?0pir2 dhr=hm?02pi(a2? (hr +a?hm)2) dhr=hm?0pi(2ahm +2hmhr?2ahr?h2r ?h2m)dhr= pi(2ahmhr +hmh2r ?ah2r ?h3r3?h2mhr)????hm0=pi3h2m (3a?hm) . (A.11)100By inserting the Equation A.7 into Equation A.11, we can write the volume of thedroplet asV =pi3a2 (1? cos?)2 (3a?a(1? cos?))=pi3a3 (1? cos?)2 (2+ cos?) , (A.12)orV =pi3(Rsin?)3(1? cos?)2 (2+ cos?)=piR33sin3 ?(cos3 ? ?3cos? +2). (A.13)in terms of R and ? . Also, from Equation A.11 and Equation A.3, we can find thevolume of the droplet as a function of R and hm:V =pi3hm (3ahm?hm)=pi3hm(32(R2 +h2m)?h2m)=pi6hm(3R2 +h2m). (A.14)A.3 Sectional parameters across the dropletThe sectional area of the spherical cap is shown in Figure A.2. To find the sectionheight hr, we can writea2 = (a?hm +hr)2 + r2, (A.15)similar to Equation A.2 and find hr as:hr =?a2? r2?hm?a, (A.16)orhr =?(Rsin?)2? r2?Rtan(?2) ?Rsin? , (A.17)101AA(a)Section A-A ?(b) cross-sectional viewTop view(c) top-viewFigure A.2: Sectional parameters of a droplet in a sessile droplet.in terms of R, ? and r. We can also write the section height in terms of ?1 as:hr =rtan(?1)+hm?a, (A.18)where ?1 is equal to?1 = arcsin( ra). (A.19)Finally we can find the sectional curvature of the droplet (Section A-A in FigureA.2b). The sectional area is the segment of a circle with the radius of R? that canbe calculated versus R and r from Figure A.2c as:R? =?R2? r2. (A.20)102Hence, we can find the sectional contact angle using the Equation A.8 as:? ? = 2arctan(hrR?)= 2arctan(hm?a+ rtan(?1)?R2? r2)= 2arctan(hm?a+ rtan(arcsin( ra))?R2? r2). (A.21)And eventually, we can find the radius of sectional curvature using Equation A.1as:a? =R?sin? ?=R?sin(2arctan(hm?a+ rtan(arcsin( ra ))?R2?r2)) . (A.22)103