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Evaporative cooling in the contact line region Stefurak, Glenn 1995

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EVAPORATIVE COOLING IN THE CONTACT LINE REGIONByGlenn StefurakB. A. Sc. (Mech. Eng. ) University of WaterlooM. A. Sc. (Mech. Eng.) University of WaterlooA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1995© Glenn Stefurak, 1995In presenting this thesis in partial fulfilment of the requirements for an advanced degree atthe University of British Columbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission for extensive copying of thisthesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Mechanical EngineeringThe University of British Columbia2324 Main MallVancouver, CanadaV6T 1Z4Date:/)AbstractEvaporative cooling of small, high power dissipating devices using thin liquid films isbecoming an increasingly important field of research as the requirements for this technology grow. Understanding the mechanisms which cause liquid motion in the ifim andevaporation from the liquid surface is essential to the practical designer attempting tocreate a truly effective cooling scheme.For liquid films of less than 1 pm, previous experiments have confirmed the existenceof an adsorbed layer where no evaporation occurs, plus a region of slightly increasedthickness where limited evaporation takes place. A theory has been proposed for thenon-evaporating region which modifies the thin liquid ifim pressure due to the molecularattraction of the underlying substrate. The pressure adjustment is termed the disjoiningpressure and is thought to be the dominant driving force for liquid motion in this thinfilm regionIn order to test the validity of the disjoining pressure concept for an evaporating film,an experiment was designed which uses a dielectric liquid, FC-72, and a highly polishedsilicon substrate inclined at a 5° angle from the horizontal to create an extended meniscus.A fluorescent light was used in an interferometer to provide increased film profile dataand a specially designed focusing ellipsometer was used to measure film thicknesses in theadsorbed film region. Heat was supplied to the meniscus through a 400 pm wide borondiffused heater within the silicon substrate. Surface temperature and mass evaporationrates were also measured.11It was concluded from the results that the disjoining pressure model which had originally been developed for static non-evaporating thin films is equally applicable to evaporating thin film environments. The model proposes an inverse cubic relationship (1/h3)between the disjoining pressure and the adsorbed thickness. However, values of theHamaker constant used in the relationship, inferred from the experiments were 4-5 timesthe theoretical value.The corresponding heat and mass transfer models which employ the disjoining pressure require further study, as evidenced by their prediction of total heat and mass transfer rates at least one order of magnitude lower than those measured in the experiments.Improvement of the heater design and of the thin film profiling method are two veryimportant areas to be considered for future work in this field.111NomenclatureA Analyzer angle [degrees]A Amplitude of Jones vectorA11 Hamaker constant, vapour-vapour [JJA22 Hamaker constant, liquid-liquid [J]A33 Hamaker constant, solid-solid [J]A12 Hamaker constant, vapour-liquid-solid [J]C Compensator angle [degrees]d Distance difference of reflected waves in medium 0 [m]E Amplitude of the Jones vectorE Jones vectorg Gravitational acceleration [m2/s]h Film height [m]Planck’s constant 6.626 x iO [J.s] (Eq.2.11, 2.12)h Convective heat transfer coefficient [W/m2K]he Evaporative heat transfer coefficient [W/m2K]hfg Heat of vapourization [kJ/kg]h0 Film height at the interline [m]hr Reference film height [m]H Height above pool surfacei Imaginary component of complex numberI Light beam intensityk Boltzmann’s constant 1.380 x 10—23 [J/KjivTherma’ conductivity [W/rn.K]K Curvature [1/rn]The Evaporative mass flux from liquid-vapour interface [kg/m2s]M Molecular weightMe Evaporation rate [kg/s]N Number of molesIndex of refraction of the substraten0 Index of refraction in medium 0n1 Index of refraction in medium 1Ti3 Index of refraction in medium 3P Pressure [N/rn2]Pd Disjoining pressure [N/rn2]Pe External hydrostatic pressure [N/rn2]P2 Pressure in a thin film [N/rn2]P1 Pressure in the liquid [N/rn2]P1, Liquid pressure at a liquid-vapour interface [N/rn2]P Saturation pressure at a liquid-vapour interface [N/rn2]P0 Hydrostatic pressure at the pooi surfaceP0 Pressure in the bulk liquid forming the film [N/rn2]P Saturation vapour pressure of the bulk liquid [N/rn2]P Vapour pressure [N/rn2]PAT Normal pressure perpendicular to the liquid-vapour interface [N/rn2]PT Normal pressure parallel to the liquid-vapour interface [N/rn2]rB Radius of bubbleFresnel reflection coefficient between mediums 0 and 1VFresnel reflection coefficient between mediums 1 and 2R Ratio of electric field componentsR Heater Resistance in Equation 4. 17Z Universal Gas Constants Specific entropy [kJ/kgK]S Entropy [kJ/K]S0 First Stokes parameterS Second Stokes parameter82 Third Stokes parameterS3 Fourth Stokes parametert Time [s]T Temperature [K]T Amplitude of complex compensator transmission ratioTemperature at a liquid-vapour interface [1<]Temperature of the vapour [K]Temperature of the substrate surface [K]u u-velocity [m/s]v Specific volume [m3/kg]V Volume [m31x x-axis position [m]x0 Reference position along x-axis [m]X Argument in determination of film thicknessy y-axis position [m]z z-axis position [m]Greekvia Angle in the camplex polarization ratio amplitude [rad]a2 a for the incident beama for the reflected beam/3 Phase difference of beam travelling in film [rad]-y Film tension [N/miI’ Mass flow rate per unit width [kg/m.s]Evaporation correction factorFca Calculated mass flow rate per unit width [kg/rn.sjMeasured mass flow rate per unit width [kg/m.s]6 Phase in the complex polarization ratio [rad]Phase of the complex compensator ratio [rad]6 of the incident beam6 of the reflected beamox Phase delay on x-axis [radiPhase delay on yaxis [rad]Retardation in the ellipsometer [deg]e Ratio of minor to major ellipse axesDielectric permitivity of a vacuum63 Dielectric permitivity of a liquid8 Azimuth of the ellipse [rad]Angle of incidence in medium 0 [rad]81 Angle of incidence in medium 1 [rad]9 Angle of substrate tilt from the horizontalWavelength [mjWavelength in medium 0 [m]viiWavelength in medium 1 [mJWavelength along a path [m]Dynamic viscosity [kg/msjChemical potential [kJ/kgmole]v Kinematic viscosity [m2/sJ11e Characteristic absorption frequency [1/s]p Ratio of complex polarization ratiosPc Complex ratio of compensator axesDensity in the liquidPv Density in the vapourSurface tension [N/rn]Evaporation coefficientShear stress at the liquid-vapour interface [N/m21Absolute Jones vector phase [rad]Complex polarization ratioAngle in the complex compensator ratio amplitude [radiComplex polarization ratio of the incident beamlbr Complex polarization ratio of the reflected beamw Angular frequency [rad/s]Subscriptsi Incident upon the surfaceI Left hand polarizationp Perpendicular to the plane of incidencer Reflected from the surfacer Right hand polarizationvii’x Component along the x-axisxo Amplitude along the x-axisy Component along the x-axisyo Amplitude along the y-axisx,y axes rotated 45°x,y axes rotated —45°SuperscriptsStokes parameter after the analyzerStokes parameter before the sampleStokes parameter at the laserixTable of ContentsAbstract iiNomenclature ivTable of Contents xList of Tables xivList of Figures xvAcknowledgement xviii1 Introduction 11.1 Background 11.2 Research Motivation and Objectives 62 Literature Review 82.1 General 82.2 Disjoining Pressure 122.3 Experiments with Heat and Mass Transfer in Thin Liquid Films 232.4 Modeling of Heat and Mass Transfer in Thin Liquid Films 282.5 Synopsis 343 Experiment 363.1 General 36x3.2 Ellipsometry 383.2.1 Background 383.2.2 Theory and Operation 393.2.2.1 Methods of Describing the Polarization State of Light. 393.2.2.2 Determining the Stokes Parameters from a Nulling Ellipsometer 443.2.2.3 Determination of Liquid Thickness Using the ComplexPolarization Ratio 463.2.2.4 Fortran Program 503.2.2.5 Focusing Lenses 533.3 Interferometer 533.4 Apparatus . . 563.4.1 Heater 563.4.2 Chamber 583.4.3 Ellipsometer 613.4.4 Stand, Microscope and Camera 643.4.5 Resistance, Power Input and Temperature Measurement 643.4.6 Dielectric Test Liquid 673.5 Procedure 684 Experimental Results and Observations 704.1 Preliminary Operation and Calibration 704.2 Interferometry Photographs 744.3 Liquid Film Profiles 864.4 Adsorbed Thickness 904.5 Evaporation Rate 92xi4.6 Surface Temperature Results 925 Discussion of Results5.1 Experimental Profile Comparison5.2 Unsteady Behaviour5.3 “Lens” Formation5.4 Adsorbed Thickness5.5 Hydrodynamic Model5.6 Heat and Mass Transfer5.7 Experimental Accuracy5.8 Summary of Relevant Experimental6 Conclusions 1327 Future Work 135Appendices 137A Physical Description of the Stokes ParametersA. 1 The First Stokes’ Parameter, S0A.2 The Second Stokes’ Parameter, S1A.3 The Third Stokes’ Parameter, 52A.4 The Fourth Stokes’ Parameter, S3B Stokes Parameter Description Using Ellipse Parameters 141C Complex Polarization Ratio Determination 144D Description of Light Intensity at the Photodiode 146979799103104112122129Observations 130137• 138• 138138139xiiE Determination of Compensator Imperfections 149F Estimate of Hamaker Constant Calculation Errors 151Bibliography 153xliiList of Tables3.1 Measured interference colour thickness values 553.2 Properties of test liquid, FC-72 [2] 675.3 Experimentally calculated Hamaker constants 1115.4 Experimental vs. calculated mass flow rates 125xivList of Figures1.1 Heat flux levels for various microelectronic modules, Chu [1] 21.2 Effectiveness of direct contact cooling methods, [2] 21.3 Surface temperature profiles at a film edge, taken from Hirasawa andHauptmann [3] 41.4 Illustration of an extended meniscus, showing the contact line region. . 52.1 Thin liquid film pressure between two identical phases without overlap 132.2 Thin liquid film pressure between two identical phases with overlap. 142.3 Thin liquid film pressure between two different phases without overlap 162.4 Thin liquid film pressure between two different phases with overlap. 162.5 Disjoining pressure example 1 182.6 Disjoining pressure example 2 192.7 Notation for interfacial mass transfer 313.1 Schematic of experimental design 373.2 Travelling wave sinusoidal components 403.3 Ellipse conventions 423.4 Single reflection 473.5 Multiple reflections 483.6 Transverse plane ellipses for various thicknesses 513.7 General Stokes parameter information for various transverse plane ellipses 523.8 Boron Diffused Silicon Heater 573.9 Schematic of the experimental chamber 59xv3.10 Photograph of the experimental chamber interior 603.11 Schematic of the ellipsometer-chamber setup 633.12 Experimental apparatus without microscope 653.13 Experimental apparatus with microscope 664.1 Experimental calibration of heater temperature vs. electrical resistance 734.2 Schematic description of photographic results 754.3 Photographic result for P=0 mW 764.4 Photographic result for P=4 mW 774.5 Photographic result for P=23 mW 784.6 Photographic result for P=59 mW 794.7 Photographic result for P=114 mW 804.8 Photographic result for P=185 mW 814.9 Liquid film profiles, averaged over 6 readings 874.10 Adsorbed thickness region profiles, averaged over 6 readings 884.11 Adsorbed thickness vs. power input 914.12 Mass Evaporation Rate vs. Power Input 934.13 Heater temperature rise vs. power input 945.1 Input power effect on liquid profiles, [30] 985.2 Normalized liquid profiles 1005.3 Normalized liquid profiles, disjoining pressure region 1015.4 Photograph of lens formation from Cook et al [30] 1035.5 Comparison of adsorbed thickness vs. saturation pressure ratio, [211. . 1055.6 Comparison of non-dimensionalized adsorbed thickness vs saturationpressure ratio, [21] 1075.7 Adsorbed thickness vs. saturation pressure ratio, Equation 2.19 109xvi5.8 Polynomial curves fit to full profiles 1135.9 Profile curvatures at relative positions 1155.10 Profile curvatures as a function of film thickness 1165.11 Evaporative mass flux for the full profile 1175.12 Curve fit to profiles in the disjoining pressure region 1195.13 Disjoining pressure profiles 1205.14 Evaporative mass flux in the disjoining pressure region 1215.15 Heat transfer coefficient for the full profile 1235.16 Heat transfer coefficient in the disjoining pressure region 1245.17 Evaporation coefficient for the full profile 1275.18 Evaporation coefficient in the disjoining pressure region 128xviiAcknowledgementI would sincerely like to thank my supervisor Dr. E.G. Hauptmann for his contributions during my time at U.B.C. Our discussions went far beyond the research topicand his insights and enthusiasm were both appreciated and helpful. I would also like tothank my research committee of Dr. P. Hill, Dr. N. Epstein, and Dr. M. Salcudean fortheir support during our meetings. I would also like to acknowledge the financial supportreceived from NSERC.The construction of the apparatus required expertise far beyond my capabilities. Iowe a large debt of thanks to Dr. M. Parameswaran from S.F.U. for the fabrication of thesilicon heaters which took many trials to obtain a satisfactory result. Mr. Dave Camp,Mr. D. Bysouth, and Mr. T. Besic were very helpful in the design and building of thechamber, stand and the electronics for the experiment. I would also like to thank myfellow students for their friendship and many discussions.I also wish to thank my parents, Wilf and Joan Stefurak, for their encouragementduring all my studies. Hopefully their son is finally finished school. Finally, I wish tothank my wife Laurie without whom this work could never have been finished. Her loveand friendship were very valuable at all times.xviiiChapter 1Introduction1.1 BackgroundAs technology produces new systems and devices which are smaller and more powerfulthan their predecessors, the requirements for effective cooling become greater. Figure 1.1shows the heat flux levels which are produced by modules of various computers. Maintaining a cool and stable operating temperature for these components is becoming agreater challenge with this new technology. The benefits of cool and stable operatingtemperatures are greater microelectronic component reliability and the possibility of future increased power dissipation.Minimizing the resistance to heat transfer between the component to be cooled andthe cooling mechanism is one of the major goals of any practical cooling design. Directcontact cooling is the simplest method of reducing the thermal resistance between a deviceto be cooled and the coolant. Direct contact cooling with fluids may be accomplishedwith the familiar methods of free convection, forced convection and boiling or evaporation.Figure 1.2 shows the effectiveness of these various direct contact cooling methods.Each direct contact method has its own positive and negative features when consideration is given to designing a practical cooling system. Forced or free convection coolinginvolving air is the easiest method if fabrication of the electronics packaging is the mostimportant design parameter. However, due to the low density and the low thermal conductivity of air, the electronic components must be low power and situated in a low1Chapter 1. Introduction 2cG’ 5E0 0 Air Cooled Tecflnology4 •ColdPlateCooledTecflnotogy IBM 3090 TCM 0I COG Cer 203,205IBM 43810) IBM3O8ITCM>‘ ONECLCMFujitsu FACOM M-380I 0 •HitachiS-8100IBM 360 _IBM 370Honeywell DPS-88Vacuum Tubes • • IBM 303355 1960 1965 1970 1975 1980 1985 1990YearFigure 1.1: Heat flux levels for various microelectronic modules, Chu [1].2’E 10lo4 a,oC) . 0)10) lñ’‘ICoCo 0C) C)102 .a) fl C) C> o” > oI oil 0C1O vii oQ)fl II =a)U oii S? 0 00 0 W LL11 U-______Cooling ModeFigure 1.2: Effectiveness of direct contact cooling methods, [2].Chapter 1. Introduction 3packaging density environment to avoid overheating.Forced or free convection using a liquid provides at least one order of magnitudeimprovement in cooling capability due to the increased density and thermal conductivitycompared with air. When using liquid as a coolant in microelectronic packaging, designdifficulties are magnified compared to that for air cooling. Liquid circulation and thesealing of the cooling area are two difficult design considerations which must be overcomefor a truly effective system.The highest level of cooling effectiveness with direct contact cooling methods is thatinvolving phase change. Either boiling or evaporation has been shown to provide thelargest amount of localized cooling. This cooling benefit does not come without designdrawbacks. High component surface temperatures compared with ambient conditions arenormally required to initiate boiling, and even higher surface temperatures may occur ifa dry-out situation develops within the boiling region. These high surface temperaturescan adversely affect microelectronic component performance over a period of time. Also,a practical design limit to the condensate return rate must be overcome in any phasechange cooling application.Evaporative cooling does not require the high surface temperatures that boiling requires to initiate the process, but provides the same cooling benefits that change ofphase supplies. Experimental investigations into film evaporation have demonstratedthese large cooling effects, with particularly effective cooling existing near the film edgewhere the film is thinnest. Figure 1.3 shows the increased cooling levels near the contactline region (area at edge of film where apparent contact angle is measured). Practicaluse of the high cooling rates existing in evaporative ifim cooling has been made with thedesign of small wickless heat pipes and micro-grooved channels to cite just a couple ofexamples. In order to achieve maximum cooling benefit from film evaporation, investigations into the contact line region are required to provide an understanding of the areaChapter 1. Introduction 4Figure 1.3: Surface temperature profiles at a film edge, taken from Hirasawa and Hauptmann [3].where the largest cooling rates occur.Microscopic investigations into the contact line region of a wetting film show a gradually thinning film which eventually reaches a constant thickness, thus an actual zerocontact angle. The constant thickness region is an adsorbed film(no evaporation fromthe interfacial surface) due to the presence of long range molecular forces from the underlying solid surface. The adsorbed layers are typically on the order of iooA thick (lA= 10m), but some layers have been reported as thick as 600A. As the film thickness increases, the effect of the molecular forces is reduced and evaporation occurs. Theboundary between evaporating thin film and adsorbed film is called the interline. Onemethod to produce an elongated contact line region is by situating a liquid pooi at thebottom of a shallow sloping substrate. This design is shown schematically in Figure 1.4.Studies into the contact line region show particular promise for microelectronic cooling. Two major factors, the presence of an optically smooth substrate and a dielectric00$ 908O7o60504030Chapter 1. Introduction 5Figure 1.4: Illustration of an extended meniscus, showing the contact line region.cooling liquid, exist in these applications and create a wide contact line region. Theseconditions are achieved with highly polished silicon wafers and electronic cooling liquids,such as FC-72 (perfluorohexane), available from 3M corporation.Designing evaporative cooling schemes using film thicknesses in the contact line regionrequires a physical understanding of the relevant forces. The question then becomes howto incorporate these forces into fluid dynamic and heat transfer analysis.A concept called “disjoining pressure” has been developed to account for the intermolecular forces important in films with thickness less than 1tm = 106m. Thedisjoining pressure (Pd) is used in equations of motion in the same manner as capillary orsurface tension forces. Due to the nature of the size of the experimental apparatus, nondestructive measurement techniques are required to determine the effect of the pressuresand validate existing theories. Two such thickness measuring techniques are interferometry and ellipsometry. Therefore, creating an extended meniscus with these measuringtechniques should provide valuable insights into the contact line region.Chapter 1. Introduction 61.2 Research Motivation and ObjectivesEvaporative cooling appears to be very promising for small high power devices whichare currently being designed and it appears these devices will be even more complicatedin the future. The practical designer must be concerned with maintaining stable deviceoperating temperatures within a room temperature environment. The most convenientevaporative cooling design will include very little liquid volume and maximum liquid-solidcontact area, resulting in very thin liquid films. Also, packaging will include air mixedwith vapour within the sealed cooling cell. With these expectations, it is necessary tounderstand the important physical factors in thin film cooling.Considering the above factors, it was the general objective of this research to gain afuller understanding of heat and mass transfer processes in the contact line region. Withthis understanding, a detailed evaluation of the existing models which use disjoiningpressure and interfacial mass transfer models in the contact line region.Specifically, the objectives of this research are itemized below.1. To accurately measure the liquid film profiles in the contact line region, particularlyin the region where the disjoining pressure is important, anticipated to be below1pm.2. To measure interline temperature rise above the ambient surroundings.3. To make overall heat and mass transfer measurements in the contact line region.4. To use these experimental measurements to evaluate the analytical models whichpredict the heat and mass transfer in the contact line region with particular attention to the disjoining pressure dominated region.Chapter 1. Introduction 75. To make an overall assessment of the effectiveness of using an evaporating thinliquid film for the cooling of microelectronic devices.The procedure used to obtain the above objectives was to design and construct anexperiment which localized the surface heat flux in the adsorbed film region and also gavean indication of the substrate surface temperature. A practical evaporation - condensation system was designed for a meniscus which exists on a sloping substrate. The filmthickness in the contact line region was measured with greater resolution than in previous studies by the use of a focused ellipsometer beam in conjunction with a fluorescentlight interferometer. This type of interferometer provided detailed profile information atthicknesses less than those which could be obtained using a monochromatic light source,and a spatial resolution not previously used for this application.Chapter 2Literature Review2.1 GeneralWith the miniaturization of technical components, most pronounced being the electronic micro-chip, highly effective dependable cooling systems become increasingly important. Bergies [4] provided a summary of the evolution of cooling for electronic equipment.The summary detailed cooling requirements from vacuum tubes in the 1950’s to ultralarge scale integration chips which have approximately 500,000 components on a chipof 0.25 cm2 area. The IBM 3090 series chip, for example, produces almost 7 Watts ofpower. This indicates a definite need for reduced thermal resistance between the chipand the ambient environment, i. e. more effective cooling.Many different methods have been reported in the literature. Chu [1] gave a gooddescription of the current direct contact cooling methods being incorporated in electronicspackaging today. The basic methods included free and forced convection, with bothair and liquid, as well as phase change methods involving liquids. Studies into thesemethods are numerous, a select few will be discussed here to indicate the level of coolingeffectiveness for each method.Park and Bergies [5] examined natural convection heat transfer for simulated microelectronic chips. They discovered that heat transfer could be enhanced by changing theconfiguration of the electronic chips, specifically stacking them in vertical arrays. Heattransfer coefficients as high as 600 W/m2K were measured, almost 100% greater than8Chapter 2. Literature Review 9prevailing natural convection theory.Another cooling method which offers higher heat transfer coefficients than convectionis boiling. However, large temperature superheats at the wall are normally required toinitiate boiling unless a surface enhancement, such as an attached porous metal surface,is used to aid in nucleation. Bergles and Kim [6] studied the problem of reducing “temperature overshoots” (wall superheat above saturation required for boiling) in immersioncooling. Heat transfer coefficients in the range of 1,000-10,000 W/mK were recordedwith surface temperature superheats reduced from 30°C to 8°C by generating bubbles inthe vicinity of the heated surface.A variation of immersion cooling which creates thin films along the heat transfersurface was studied by Toni et al [7] using grooved fins and porous tunnels on tubes toaugment heat transfer. Heat transfer coefficients as high as 7,280 W/m2Kwere obtainedin an evaporation mode with superheats as low as 1.2°C. These studies have indicatedthe potential for large cooling rates using change of phase.Evaporating liquid films have been shown to be another method of providing effectivecooling. Many aspects of liquid film evaporation have been studied for films 1 mm thickand less. Fujita and Ueda [8, 9] studied the flow of sub-cooled and saturated liquidfilms down the exterior of an electrically heated stainless steel tube. Wave formationin the liquid films was noted as power was increased with localized film thinning andeventual dry patch formation. For sub-cooled films, the surface tension variations dueto temperature changes along the tube are shown to be important in causing dry patchformation. Local heat transfer coefficients dropped from approximately 2000 W/m2Kto500 W/m2K or less at the formation of a dry patch. For saturated films, boiling heattransfer coefficients in the range of 7000 W/m2K were measured.A very informative study showing the cooling potential for large evaporation rates ofthinning films and subsequent dry-out is given by Orell and Bankoff [10]. Nichrome stripsChapter 2. Literature Review 10embedded in a horizontal plate were used to thin a pooi of liquid. Higher heat transferwas noted as the film thinned but oscillating liquid behaviour of the contact line regionwas found as a dry patch formed. This important fact has subsequently been noted inwetting film evaporation studies and dry-out studies.Sharon and Orell [11] studied heat transfer to flowing laminar liquids in a horizontalplastic channel with a smooth copper bottom. Isothermal and heated cases were considered. Liquid surface temperature measurements were made with a movable 25 4umdiameter thermocouple to include thermocapillary effects in creating a critical heat fluxmodel. Distilled water flowing over the smooth copper channel resulted in thin films,measured to within ±5pm by a micrometer needle probe. Predicted profiles using capillary and thermocapillary forces in the equations of motion provided excellent agreementwith the measured profiles, for ifim thicknesses on the order of 1 mm. Predictions of thecritical heat flux were also very good with this method.Hirasawa et al [12j examined surface tension effects in condensing laminar films formedin various sized small troughs. The most important effect noted was the surface tensionchanges in locally thinning areas of the film, which reduces the resistance to heat transferand, therefore increases the heat transfer coefficient. At the thinnest section of the trough,the films were approximately 100pm thick. Calculations of condensation heat transfercoefficients based on experimentally verified thickness profiles range from 10,000 W/m2Kfor R-113 to 100,000 W/m2K for water.Another area of interest which uses evaporative cooling with liquid films on the orderof 1 mm thick is closed thermosyphons or wickless heat pipes. These heat pipes, generallyon the order of 10 mm in diameter, return condensate by means of gravity or centrifugalforce, and are used in gas turbine blade cooling, electronic component cooling or gas togas heat exchangers. Studies by Imura et al [13], Reed and Tien [14], Chen et al [15]and Negishi and Sawada [16] have examined effects such as critical heat flux, dryout,Chapter 2. Literature Review 11flooding, vapour shear effects, inclination etc. All the above mentioned studies show thedriving forces in liquid films from 1pm to 10mm thickness. These heat pipes contain acomplicated mix of phase change conditions ranging from evaporation to violent boiling,depending upon the design parameters. Studies into liquid film behaviours would benefitthe understanding of the processes in these heat pipes.The above cited papers [8]-[16J show the cooling potential available with evaporatingliquid films but also show the adverse effect dry-out has on temperature. However,the behaviour of the contact line region is usually not considered for liquid films ofapproximately 1 mm thickness. Studies in this region could provide valuable informationon evaporative cooling and dry patch formation.An important feature in films of 1 mm thickness and less is the apparent contactangle of the evaporating liquid with the surface, which is used in hydrodynamic modelsof dry patch formation and stability. The contact angle is termed “apparent” due tothe acknowledged presence of a very thin adsorbed portion of liquid at the interline.Hirasawsa and Hauptmann [3] studied the apparent dynamic contact angle for a flowingrivulet between two heated patches using a colour schlieren method, and noted large heattransfer coefficients at the contact line region with films of the order 1 pm or less. Theypointed to the need for studying heat transfer in the area of very thin films, less than1 pm, to determine heat transfer coefficients and important parameters describing theliquid motion.Bankoff [17] gave a good account of the current state and future needs in thin filmcooling. The previously mentioned macroscopic factors of stability, dry-out, boiling andcontact line motion are mentioned as areas which require further detailed research. Withrespect to molecular forces and adsorbed liquid layers in the contact line region, Bankoffsuggests that this is a “rich area for further research”.Chapter 2. Literature Review 122.2 Disjoining PressureThe concept of disjoining pressure was first proposed for a thin liquid separatingtwo plane, parallel solid surfaces by Derjaguin [18]. Derjaguin found experimentallythat an additional force acting perpendicular to the plane of the surface was requiredto hold the surfaces in equilibrium at a fixed distance of separation. This force (eitherpositive or negative depending on the combination of materials used) was in additionto that calculated from conventional hydrostatic pressure analysis for the liquid layer.The additional force required to maintain this separation distance between the two solidsurfaces showed an increasing significance below 106m separation distances. Dividingthis additional force by the surface area of the solids in contact with the liquid gave ameasure of the disjoining pressure in the thin film. The pressure distribution in the thinlayer was assumed to be uniform through the layer with a pressure jump occurring atboth liquid-solid interfaces. For a plane parallel case, the value of the jump at each faceis identical.The basic definition of disjoining pressure was originally given as (for the case of twosolids separated by a thin liquid film)Pd(h) = Fe — Pi, (2.1)with P2 being the pressure in the thin liquid layer assumed from conventional hydrostaticanalysis and Fe being the measured external pressure on the solids. Any anisotropyof pressure in the liquid was not considered during these initial investigations but wasaddressed by Derjaguin in later work as is discussed below. Derjaguin stressed the pointthat the pressure does not change with distance from the interface but is only a functionof the liquid thickness, h, at each point. He states that the disjoining pressure “is alliedto the concept of phase pressure but not of internal pressure”.Chapter 2. Literature Review 13vapourFigure 2.1: Thin liquid film pressure between two identical phases without overlap.Gibbs interfacial theory [19] treats thermodynamic systems involving thin liquid filmswith interfacial zones as the sum of the bulk phase properties plus the properties of theinterfacial regions between neighbouring phases. This approach assumes there exists asection between interfaces which contains liquid with bulk liquid properties as shown inFigure 2.1, for the symmetric case of a thin film bordered by two identical vapour phases(such as a thin soap film). The interfacial regions, which are anisotropic with respect tointermolecular forces near each interface, are assumed separated by a bulk liquid phase.Figure 2.1 also shows the pressure distribution which exists in a thin film which containsa bulk liquid region separating the two interfacial regions. The normal pressure, N,is everywhere equal to the surrounding normal pressure in the other bulk phases. Thetangential pressure, T, varies in each interfacial zone but returns to an isotropic value(equal to PN) in the bulk zone. The surface tension is a property of the interfacial regionPbulk liquidliquid-vapourinterlacialliquid-vapourinterfacialvapourI .1P.Chapter 2. Literature Review 14liquid-vapourinterfacialoverlappingliquid-vapourinterfacialFigure 2.2: Thin liquid film pressure between two identical phases with overlap.and is defined for this case by the Bakker equation as= f (F — PN) dz = (P — P0) dz, (2.2)where the integration is taken in the direction perpendicular to the interface, z, with thelimits of integration being the region of influence on the tangential pressure. The filmtension for this particular case will be the sum of the two individual surface tensions or+00=—L00 (PT — PN) dz = 2. (2.3)If sufficient thinning of the film occurs such that no region of bulk properties exists, theindividual interfacial zones will overlap and this will significantly change the propertiesof the thin film, causing Gibbs’ approach to be invalid. This situation is shown inFigure 2.2. From the pressure distribution diagrams, it can be noted that the tangentialpressure does not attain an isotropic value anywhere between the two interfacial regions.vapourPI,’0vapourF FChapter 2. Literature Review 15The normal pressure also varies in a complex manner, P, but is approximated by thedisjoining pressure added to the surrounding isotropic pressure, the resultant shown asP2 (P,== Po + Pd). For this symmetric case a relationship between the disjoiningpressure and the film tension may be established using the film tension definition. Writingthe film tension as+00=—L00 (PT—PO—PN+PO)dZ, (2.4)and using the surface tension definition given by Equation 2.2 and the disjoining pressuredefinition, the film tension may be rewritten asy=2o+Pdh, (2.5)where h is the height of the film, assumed to be equal to the integration limits for thethin film.For the case of a wetting thin film bordered by two different phases, namely a vapourphase and a solid phase, the pressure distribution will be quite different from the symmetric case. For a thin film without and with an overlapping interfacial region, the sketch andpressure diagram are given in Figure 2.3 Figure 2.4 respectively. The interfacial zoneswill be distinctly different from each other at the liquid-vapour and liquid-solid interfacesas compared with the identical values for the previous symmetric case. The interfacialtensions will also be much different at each phase boundary. Because the concept of filmtension is not meaningful for the vapour-liquid-solid case, a direct relationship betweenthe film tension and the disjoining pressure, such as Equation 2.5 does not exist for thiscase.From the basic definitions presented, it is apparent that the disjoining pressure andsurface tension are both functions of the height of the thin layer for cases with overlappinginterfacial regions. However, the disjoining pressure concept is concerned solely with theforces acting normal to the interface and may be approximated with intermolecular forceChapter 2. Literature Reviewvapour______bulk liquid___________solid__Figure 2.3: Thin liquid film pressure between two different phases without overlap.16A ApP0 P04liquid-vapour -f—interfacialliquid-solidinterfacialPrvapour liquid-vapourinterfacialoverlappingliquid-solidinterfacialsolidaPrFigure 2.4: Thin liquid film pressure between two different phases with overlap.Chapter 2. Literature Review 17theory while the surface tension is more complex. Therefore, in wetting film analysis,the surface tension is assumed to be the calculated bulk value while the intermoleculareffects are accounted for only in the disjoining pressure.Using the anisotropic pressure description in a thin film, Derjaguin [20] recast thedefinition of disjoining pressure asPd(h) = PlY — P0, (2.6)with P0 being the isotropic pressure in the bulk liquid from which the thin liquid wasformed and PN being the pressure component perpendicular to the interface within thethin liquid layer. The sign convention for the disjoining pressure term in Equation 2.6term is usually chosen such that a negative value will be obtained for a wetting film. Thisis the same sign convention as given in Equation 2.1 with a more defined description ofthe system pressures.For thin film hydrodynamic calculations, the disjoining pressure given by Equation 2.6is added to the existing pressure terms to give the net thin layer pressure. This netpressure is then taken as the isotropic pressure existing in the film under disjoiningpressure theory analysis.For the specific case of a solid-liquid-gas thin film system, two examples illustrate thedisjoining pressure concept. The first is the case of a vapour bubble pressing against asolid surface creating a thin horizontal liquid film between the bubble and the solid, asshown in Figure 2.5. Assuming that the bubble is in thermodynamic equilibrium withthe liquid and maintains a spherical shape except for the flat surface pressed against thesolid, a uniform internal bubble pressure, which is higher than the surrounding liquidpressure by an amount given by the Laplace-Young equation, will create a uniform thinfilm between the bubble and the surface. Assuming mechanical equilibrium, the vapourpressure and the thin liquid pressure should be equal perpendicular to the horizontalChapter 2. Literature Review 18Solid.Liquidthin liquid filmVapour rBFigure 2.5: Disjoining pressure example 1.interface. However, a higher pressure in the thin ifim with respect to the surroundingliquid would result in a flow outwards, toward the bulk liquid. Because this does notoccur, the existence of a pressure jump at the liquid-vapour interface is inferred, indicating the existence of a disjoining pressure. Using the Laplace-Young equation to describeinternal bubble pressure rise, the disjoining pressure is given asPd(h)=PN-Po-—Pj-P=---—— -Ku, (2.7)TBwhere TB 1S the radius of curvature of the bubble and a is the surface tension of theliquid. For an isothermal situation, an experiment varying the bubble size, and thus thecurvature would give disjoining pressure variations with film thickness.A second example is that of a vertical flat plate immersed in a liquid creating ameniscus as shown in Figure 2.6. On a microscopic scale for a wetting liquid, there iszero contact angle and the film is characterized by a plane parrallel adsorbed film atthe top of the extended meniscus. For an equilibrium situation, the pressure from theChapter 2. Literature Review 19/ Adsorbed thin liquid film(assumed zero contact angle)PHVapourLiquidFigure 2.6: Disjoining pressure example 2.horizontal liquid surface may be given asPN = P0—pjgH, (2.8)Po = P0—pgH. (2.9)Where p and Pv represent the liquid and vapour densities respectively. Therefore, thedisjoining pressure is by definitionPd(h) = g(p — pj)H. (2.10)In theoretical modeling, when evaporation causes a gradual thinning of the film, as inan extended meniscus, the basic disjoining pressure definition is used at each incrementalposition in the film to form a gradient of pressure in the liquid. In thin film hydrodynamics, the disjoining pressure is added to the local hydrodynamic pressure and used inthe Navier-Stokes equations to determine liquid velocity and mass flux in the film. TheChapter 2. Literature Review 20film must be a wetting film with slowly varying thickness because the disjoining pressureis assumed to be accurate at each incremental film height. Also, the composition of thefilm must be consistent because the disjoining pressure is generally unknown for mixtures. One inherent assumption that is made when using this disjoining pressure theoryin hydrodynamics is that the pressure is isotropic in the thin liquid film and all pressurecomponents are affected by the disjoining pressure. No separate theoretical descriptionof the pressure parallel to the interface is utilized within the disjoining pressure term.From the definition of disjoining pressure, if the gradient of disjoining pressure withthickness is less than zero,<0, (2.11)the disjoining pressure will aid the stability of the flow causing the liquid to flow towardsthe thinner section resulting in a wetting film. This corresponds to the condition ofmolecular attraction between the liquid and the solid as opposed to repulsion.Using the condition of chemical equilibrium, Derjaguin and Zorin [21] calculated thedisjoining pressure for a one-component thin film in terms of the corresponding saturationpressures. For a single component system, the Gibbs-Duhem relation givesSdT — VdP + Ndii = 0, (2.12)ord,u = —sdT + vdP (2.13)For a constant temperature liquid, 7j, the change in chemical potential may be written()T,= vi, (2.14)For the case of a change of liquid pressure from the thin liquid value, P, to the bulksaturation value of the same liquid,P3,integrating to determine the change in chemicalChapter 2. Literature Review 21potential the relation may be written based on the disjoining pressure definition as(2.15)Using the same analysis for a vapour at a constant temperature, T, the chemical potentialmay be written (a,— 216and integrating between the same pressure conditions gives(7T\ (P=in (2.17)If the vapour and liquid are in equilibrium, the chemical potentials at each state mustbe equal and hence also the change, therefore the relation may be written asPd(h) = (Rt) in (f), (2.18)with P and P2 being the pressure in the thin liquid film and the saturation pressurewhich would occur in the bulk liquid at the same temperature conditions as the thin filmrespectively.Based on experimental data, Derjaguin and Zorin [21] further proposed the relationshipPd(h) = (‘RTvPi) in () (2.19)using the molecular interaction of van der Waals dispersion forces only. This dependencywas also derived theoretically by Frenkel [22], showing the disjoining pressure dependencyof for non-polar liquids using dispersion force interaction. Equation 2.19 was comparedwith experimental data for non-polar liquids and concluded to be in agreement, “albeitby a rather long stretch” [21].A different form of Equation 2.19 was given by Wayner et al [23] and was used ina majority of Wayner’s subsequent contact line studies (work to be described in detailChapter 2. Literature Review 22later in this chapter). The different form of the disjoining pressure equation presentedby Wayner et al [23] gives the potential to calculate the disjoining pressure based ontemperature difference measurements as opposed to pressure.Van der Waals dispersion forces may be taken as the dominant intermolecular forcefor non-polar molecules. The forces are the result of the creation of a temporary dipolemoment in a molecule due to the instantaneous position of the electrons which thenpolarize neighbouring molecules. The attraction is classified as long-range with effectsbeing important at distances of greater than 100)1. The attraction has been showntheoretically to depend inversely on the distance to the sixth power. Frenkel [22] indicatedthat experimentally measured values show effects at distances 20 times larger than thecalculated values.If the molecular separation becomes too large, retardation effects occur due to thereflection of the temporary polarization back to the original site, now possessing a dipoleoriented differently. The effect of this phenomenon is to reduce the attraction betweendistant molecules.The Hamaker constant, A12, is dependent upon the type of liquid and substrate forthe case of a wetting film. Churaev [24] and Wayner [25] proposed methods of calculatingthe Hamaker constant based upon known relationships for the electromagnetic frequencydependent dielectric constants of the materials involved. However, these frequency dependent properties are unknown for the liquid used in this study, FC-72; therefore amore approximate method proposed by Israelachvili [26] may be used. Israelachvili usesa combination of Hamaker constants derived from molecular attraction of identical substances across a vacuum to create an overall system constant. The relationship is givenasA12= (/ — /)(/A — /A), (2.20)Chapter 2. Literature Review 23where the subscripts 2,1, and 0 refer to the solid, liquid and vapour phases respectively.The individual Hamaker constants for the solid and liquid are given in terms of bulkproperties. For a liquid, the constant is— 3 (i_—_E3’2 3hptie (n —A11 — —kT1 I + 3, 2.214 \E+EJ 16/(n?+nwhere the subscript 3 refers to a vacuum. For a solid, the constant isA22 hptie. (2.22)The dielectric permitivities, , are taken at zero frequency, the refractive indices, n, arevalues in the visible frequency range, and tie is the main absorption frequency.For non-polar liquids, the Hamaker constants calculated by Israelachvili typicallyrange from —0.4x10’9to —4x109J. Israelachvili [26] showed good agreement betweenvalues caluculated using the approximate theory presented above and the more complexanalytical theories (i. e. Churaev [24]), errors typically less than 5%. Comparisons of bothanalytical theory values with the limited experimental values available show differencesas much as 2-3 times the calculated value. Using the approximate method, the Hamakerconstant for FC-72 on a silcon dioxide covered surface is —9.8x1020J. The data for theliquid was obtained from the manufacturer, 3M [2], while the data for the Si02 coveredsilicon surface was obtained from Gregory [27].2.3 Experiments with Heat and Mass Transfer in Thin Liquid FilmsThe theoretical equations for heat and mass transfer are of practical design use ifthey can be verified by direct or indirect experimental observation. Obtaining experimental verification is necessary if thin liquid film analysis is to be used with confidencein analytical and numerical models.Chapter 2. Literature Review 24Derjaguin and Zorin [211 were amongst the first experimenters to attempt to determinedisjoining pressure variation with liquid thickness. Equilibrium adsorbed film thicknesseswere measured, using an ellipsometer, on a superheated glass substrate using polar andnon-polar liquids along with the surface and vapour temperatures. The data were fit tothe equation‘RTp1 1 A12Pd(h)= M ln(Pz/Pj8) —--. (2.23)The Hamaker constant was reported to be in the range of A12 —102ergs (erg =107J) for non-polar liquids. Gee et al [28] measured adsorbed thicknesses of n-alkaneson quartz using ellipsometry for various vapour pressures. Control of the vapour pressure,measured by a transducer, proved difficult although the same disjoining pressure modelas Derjaguin, Equation 2.23 provided good agreement with experimental results. Experiments of this nature have proven the existence of the intermolecular forces described bythe disjoining pressure concept. Both studies used an ellipsometer to measure the filmthicknesses, all below 100 A thickness.Experiments using the disjoining pressure concept to predict equilibrium thin liquidfilm profiles and thin liquid film transport processes have become more frequent the pasttwo decades. Wayner [29]-[32], [35j-[40] has been a leading researcher in this area since1970, conducting experiments with extended menisci developed at the contact line regionof a non-polar liquid and an optically smooth solid.Renk and Wayner [29] used monochromatic light interference through a glass slide toobserve meniscus profiles of ethanol varying with heat flux levels. The profiles were fittedwith a fourth order polynomial from which the profile curvatures were calculated. In anevaporating meniscus, the calculated change of curvature was taken as a measure of thepressure differences in the film (normally associated with capillary forces) which causethe liquid to flow into the evaporating region. The results indicate that the curvatureChapter 2. Literature Review 25increased towards the interline for an evaporating meniscus and the degree of curvatureincreased with increasing heat flux. It appeared that capillary forces were the dominantdriving force for liquid flow into the evaporating region for film thicknesses greater thanapproximately 1000 A. Because the first interference fringe was at 1120 A, no directmeasurement of either the thin film region where disjoining pressure becomes dominantor the adsorbed film region was possible.Cook et al [30] measured meniscus profiles of non-polar decane on silicon in an extended meniscus formed by a 6° tilt of the substrate from horizontal. An improvement ofthe interference technique was achieved through the use of a scanning microphotometer.Again, the meniscus profile curvature was found to increase with increased heat flux, thusshortening the contact line region. This was attributed to the increased capillary forcesrequired to provide more liquid for the higher evaporation rates. Two substantial observations were that the meniscus vibrated at high heat fluxes and “lenses” were presentin the photographic results, concluded to be dust particles or distillation effects of animpure liquid. For this system, the disjoining pressure was assumed to be importantbelow approximately 5000 A. Using capillary and disjoining pressure theory, the profileswere used to calculate a theoretical heat flux due to evaporation which agreed well withthe overall heat flux inferred from twelve 130 m thermocouples attached on the backside of the substrate. The results showed that the evaporation region was estimated tobegin at a liquid thickness of between 7 m and 10 sum.Wayner et al [31] used basically the same setup to study the effects of compositionchanges on the meniscus profiles. Results indicate a more drawn out, shallower meniscus,concluded to be due to distillation within the liquid. This resulted in a small profilecurvature with decreasing purity indicating a different physiochemical process occurringin the contact line region. For the case of 98% hexane and 2% octane, the maximumcurvature occurs at much larger film thicknesses than the tests with a higher percentageChapter 2. Literature Review 26of hexane. This was taken as a clear indication of additional mechanisms contributing toliquid flow. Again, an oscillatory motion of the meniscus and lens formation were notedat higher heat fluxes.Truong and Wayner [32] measured transport properties near the interline directly forfilms less than 100 A. An ellipsometer was used to measure liquid profiles of a decane onsilicon extended meniscus system for varying heat fluxes. Ellipsometer measurements,with a measuring area of the order 200 4um x 200 sum, were taken. Using quadraticregression analysis to fit the data, theoretical analysis based on disjoining pressure theory indicated a heat transfer coefficient of 10,000 W/m2K in the contact line region,specifically near the interline. Again, thermocouple size and spacing on the opposite sideof the substrate made thermal verification difficult in the contact line region, althoughglobal measurements are in good agreement. Adsorbed ifims between 36 A and 55 Awere measured. Beaglehole [33] used a microscopic imaging ellipsometer to examine therate of spreading of a wetting film of siloxane oil on various surfaces. Leger [34] measuredthe thin film ahead of a spreading drop using a focusing ellipsometer and 2-axis scanner.Adsorbed films of a few hundred angstroms or less were found for polydimethylsiloxaneon silicon.Combination of the above interferometry and ellipsometry was used by Truong andWayner [35] to measure equilibrium wetting film profiles of hexane and octane on silicon.Film thicknesses between 200 A - 10,000 A were measured, studying the region whereboth capillary forces and disjoining pressure terms are important. Adsorbed thicknessesof 250 A and 195 A were reported using ellipsometry. Theoretical equilibrium liquidfilm profile calculations using disjoining pressure and capillary forces did not accuratelypredict the film thicknesses in the adsorbed region. However, results in the capillary forceregion (10,000 A) were very good.Sujanani and Wayner [36] also used ellipsometry combined with interferometry toChapter 2. Literature Review 27analyze draining films of non-polar liquids on silicon nitride covered silicon surfaces.Equilibrium thicknesses of approximately 50% less than those predicted by theoreticalcalculations using only van der Waals dispersion forces were found. The changing rateof drainage was linked with surface tension changes due to impurities of liquid or in thecleanliness of the sample. These were taken as very important.Sujanani and Wayner [37], Wayner et al [38], DasGupta et al [39] all used a slightlymodified version of the microphotometer based interferometry combined with ellipsometry to provide increased liquid film profile accuracy. Ellipsometer spot measurements of200 pm x 200 pm were used in the disjoining pressure dominated region. In the capillarydominated meniscus region, a microcomputer was used to digitize the monochromaticmicroscopy image obtained from the meniscus. Having every pixel graduated for lightintensity levels provides resolution for 0.625 pm x 0.625 pm areas in the capillary dominated meniscus region. Sujanani and Wayner [37] used octane on silicon as an extendedmeniscus inside a sealed glass cell which also contained air at atmospheric pressure. Theresults were termed as “near equilibrium” because of the difficulty of achieving stableequilibrium due to the sensitivity of the entire system to small temperature changes. Theadsorbed thickness changed with time, from 50 A to 600 A over 30 minutes, indicating thepossibility of condensation at the interline for small changes in temperature. DasGuptaet al [39] used a meniscus formed at the exit of a circular capillary feeder in a closed cellwith external liquid supply to measure the evaporation rate of heptane on silicon. Fromthe measured thickness profiles, the maximum curvature was found to exist at liquidthicknesses of approximately 1200 A for equilibrium cases and as low as 500 A for evaporating cases. Again, backside thermocouples were used for the temperature profiles fromwhich heat transfer rates in the evaporating region were calculated. Good macroscopicheat transfer agreement was achieved using the thermocouple measurements in a onedimensional conduction model of the silicon substrate as compared with the measuredChapter 2. Literature Review 28evaporation rate. Major observations from all these microcomputer enhanced profilestudies indicated that the meniscus length shortened and adsorbed film layer thicknessdecreased with increasing power until eventually the meniscus displayed an oscillatorybehaviour with respect to position. The adsorbed thickness appeared to approach someasymptotic value as the heat flux increased before oscillation was observed.Sujanani and Wayner [40] used the enhanced microscopy on an inclined silicon platefor an extended meniscus of 1,1,2 - trichlorotrifluoroethane. The measurements indicatethat the capillary dominated region ends at 500 A with the disjoining pressure beingimportant below this thickness. Again, an oscillating meniscus behaviour was observed.Very few experiments have been done with the experimental apparatus approximatingthat of a practical cooling application. Xu and Carey [41] conducted an experimentaland analytical investigation of a copper slab with small V-shaped grooves cut in thesurface (64 pm wide x 190 pm deep x 25.4 mm long). With one end in a methanolpool and the other heated, capillary action drew liquid up the length of the groove whiledisjoining pressure action carried the liquid up the groove sides. Three thermocouplestransverse to the groove length were embedded in the copper at five separate locationsalong the groove. Heat transfer coefficients at the edge of the film were estimated fromthermocouple readings to be as high as 40,000 W/m2K.2.4 Modeling of Heat and Mass Transfer in Thin Liquid FilmsTheoretical analyses of heat and mass transfer in the contact line region which included disjoining pressure theory have been presented in a few different forms but thesetheories still require experimental verification. Ruckenstein and Jam [42], Miller andRuckenstein [43], and Frenkel [22] incorporated the molecular attraction into a bodyforce term for analysis. However, most researchers used the disjoining pressure approachChapter 2. Literature Review 29in their analysis.Combining the disjoining pressure term into the fluid dynamics equations is straightforward. Due to the small evaporation area and the low power inputs to avoid meniscusoscillations, the rate of evaporation is very small. Therefore, the lubrication approximation is used. This approximation is given in numerous sources, such as Probstein[441,=— sinE. (2.24)dy2 dx vUsing boundary conditions of no-slip at the substrate surface and no-shear at the liquid-vapour interface,y = O,u = 0, (2.25)y=h,r=0, (2.26)the resulting equation for the u-velocity in the thin layer is simply=(.jL— sin9) (- — (2.27)While it is not necessarily true for a thin liquid film that the surface tension forces atthe liquid-vapour interface may be neglected, very few experiments have had the capability of measuring any surface tension changes due either to composition or temperaturedifferences, thus no surface tension driven motion is usually considered.Taking the important fluid driving forces in the contact line region as capillary forces,K, disjoining pressure forces, Pd, and gravity, and using a force balance at the liquidvapour interface, the pressure in the liquid may be expressed asPz—P=Pd—uK+pzg(h—y)cosO, (2.28)where the curvature for a very shallow change in height with distance along the substrateis approximated byK = (2.29)Chapter 2. Literature Review 30and the disjoining pressure is given by Equation 2.19. Integrating across the film givesthe mass flow rate (per unit width) ash3 / d3h 3A12 dh . dhF(x) = 0----j- + 6irh4 d + f’iJ sine — cos ). (2.30)For an extended meniscus, the last two terms on the right hand side of Equation 2.30,the gravity force terms, have been shown to be orders of magnitude smaller than the othertwo. Neglecting these two terms results in the equationh3 I d3h 3A12 dh’\F(x) = —+ 6Trh4—) (2.31)The change in the mass flow rate per unit width in the liquid represents mass which isadded or lost from the liquid-vapour surface. This mass evaporation rate per unit surfacearea (as opposed to condensation for this experiment) is given bydl’ o ( 3d4h 2dhd3h’ A12 (ld2h 1 (dh2. (2.32)Theoretical predictions of heat and mass transfer in the contact line region require astatement of the system parameters, most notably the temperature difference between theheated surface and the surroundings. This temperature difference will cause evaporationof the liquid. A sketch showing the model most frequently used for heat and masstransfer analysis is shown in Figure 2.7. Combining the temperature difference and theevaporative mass flux given in Equation 2.32, the heat transfer coefficient for evaporationfrom the liquid-vapour interface may be calculated using the equationh = mehfg (2.33)The most often cited equation used for predicting the amount of interphase masstransfer is that given by Schrage [45]. Schrage’s basic mass transfer equation is derivedfrom the kinetic theory of gases. For an equilibrium condition between a liquid and aChapter 2. Literature Review 31•TFigure 2.7: Notation for interfacial mass transfer.vapour, the number of molecules passing between phases is identical. Therefore, using aMaxwell velocity distribution for gas molecules near the interface, an absolute vapourization rate (w÷) is given based on the number of molecules passing through a planeasIM= PaY 2ir1ZT1V(2.34)The absolute vapourization rate must be modified by a factor which accounts for thefraction of molecules which are reflected at the surface and do not vapourize (). Asimilar analysis may be conducted for the absolute rate of condensation. For a netevaporation or condensation to be occurring, a local movement of the vapour away from ortoward the interface will occur. Accounting for this movement, the absolute condensation(w0_) is given as(Piv’ Iwo— =— I — I I — I 1’cWs+\PsJ \TjJT1plvvapour•pliquidsubstrateT5(2.35)Chapter 2. Literature Review 32Again, correcting for reflection, the net interfacial mass transfer may be written asrAT1me = °elf , —1- — (2.36)v h7I, T2 T2lv VThe evaporation correction factor, F, is a function of the overall velocity of the vapour.For a case where equilibrium exists, the value will be F = 1.0. For an evaporating case,there will be a net movement of vapour away from the liquid-vapour interface. Collier[46] adjusted the interphase mass transfer, Equation 2.36, for a small net motion awayfrom the surface to give the modified interphase equation as12e’/7TPtv Pvme I I 1 — (2.37)\2Ue/ v2tR- T Tlv VThe pressure at the liquid-vapour interface must be adjusted from the normal bulkliquid saturation vapour pressure, P, at the interfacial temperature, lj,,, to account forthe effects of disjoining pressure and curvature. The correction is from Equation 2.23plus the addition of the interfacial pressure difference effects due to the presence of acurved surface(Pd - crK)MP1, = Ph,s exp . (2.38)P1 ‘-1vOne of the least investigated but most important aspects of the interphase masstransfer equation, Equation 2.37, is the evaporation coefficient, Schrage [45] indicated that the theoretical value of 0e 1.0 has not always been consistent with experimental results. The ideal experiment would include knowing the exact temperatureof the liquid surface and the vapour temperature within less than one molecular meanfree path, plus having the condensing surface also within one mean free path from theevaporating surface. When dealing with macroscopic experiments with thin films, theliquid-vapour surface temperature may be approximated using a simple one-dimensionalheat conduction model through the thin film, neglecting the liquid motion in an evaporating film, with corresponding evaporation and natural convection heat transfer fromChapter 2. Literature Review 33the liquid-vapour interface. An order of magnitude calculation for the natural convectionheat transfer coefficient (he) for this particular experimental design gives a coefficient of5W/m2K. The interfacial temperature may then be calculated from the substratesurface temperature, T8, by the equationT1= (k- h)—?i’Iehfg + hT). (2.39)However, the vapour temperature of practical design interest is located within the chamber interior, many orders of magnitude distance greater than the ideal theoretical molecular mean free path. An added difficulty is the presence of any non-condensables. Indications of the uncertainty in the value of e are given by Merte [47], who reportsexperimental results of ae for water from 0.006 to 1.0. Collier [46] reports cr6 for variousliquids from 0.04 to 1.0.Mirzamoghadan and Catton [48] used these basic equations in the theoretical analysisof an extended meniscus on a sloping surface with varying degrees of incline. Maximumheat transfer was predicted for plate tilt angles of between 200 and 30°.Holm and Goplen [49] also used these equations to predict that 80% of the heattransfer for liquid filled capillary grooves in a plate occurred in the capillary force dominated region. Only 8% of the heat transfer occurred in the disjoining pressure dominatedregion.Wayner, in his theoretical investigations [50]-[52] and in his previously describedexperimental investigations, used these basic equations to predict the heat and masstransfer in the contact line region. Wayner’s most frequent method involves equatingEquations 2.31 and 2.37 and adjusting the equation parameters, mainly the temperaturedifference until a best fit with the experimental profiles is achieved. Subsequently calculated interfacial temperature differences, Tk, — T, are typically io-iO K. These arenot measurable experimentally, so verification is difficult.Chapter 2. Literature Review 34The above investigations assume c = 1.0, which could lead to large differences inmass flows and temperature differences if surface contamination or non-condensablesseriously affect the value of2.5 SynopsisHighly effective cooling has been demonstrated using evaporating thin films in wettingfilm configurations. The most obvious use of this cooling technique has been in the areaof micro-electronics. However, many aspects of the transport behaviour and coolingpotential of thin films remain unknown for the practical designer.At the leading edge of a thin wetting film, it has been experimentally proven that athin adsorbed, non-evaporating film is present due to the molecular interaction betweenthe liquid and the solid substrate on which the thin film resides. A theory convertingthe molecular attraction forces into a modification of the liquid pressure has receivedmuch attention in the literature in recent years. This modification, termed “disjoiningpressure”, has been shown to significantly affect equilibrium thin liquid film profilesand liquid transport in evaporating thin films. Analytical predictions of the disjoiningpressure involve two separate theories, one describing the change of disjoining pressurewith film thickness, 1/h3 and the second in the magnitude of the attractive forces betweenthe liquid and the solid. Experimental results have reasonably shown the 1/h3 behaviourto be a good approximation for equilibrium adsorbed films while the Hamaker constant,representing the molecular attraction is difficult to verify. The behaviour of both ofthese theories requires more investigation for evaporating thin liquid films, particularlyin the area of surface temperature effects because only small surface temperature risesare possible before instabilities in the interline region occur. The region of interest beginsas thin liquid films become less than 1 1um.Chapter 2. Literature Review 35Many excellent studies have been reported for thin film evaporation with thin filmthicknesses in the range from 10 pm to 0.1 pm, where the main contribution to fluid dynamic motion is due to capillary forces. However, detailed experimental studies into thedisjoining pressure dominated region are few because of the inherent difficulties in measuring accurate thin film proffies in this region due to various factors such as measuringtechnique and film stability. Previous investigations have shown the apparent accuracyof the fluid dynamics modelling, including the disjoining pressure, by good agreementbetween the predicted and measured profiles. However, independent confirmation of theinterline temperature rise and an accurate measurement of the contact line region masstransfer have not been reported. Therefore, a need exists for detailed information in thedisjoining pressure dominated region to validate fluid dynamic and mass transfer modelsfor use on a macroscopic scale.Particular information is required in the region near the interline with respect to thetemperature difference between the surface and the surroundings so local heat transfercoefficients may be calculated and compared. Closely related to this measurement is aneed for increased liquid film proffle information in the contact line region to validate thedisjoining pressure model. One additional experimental measurement which is lackingin the literature but is very difficult to obtain is the isolation of heat transfer and masstransfer directly into the interline region with the subsequent mass transfer measurementspertaining to this region exclusively.Chapter 3Experiment3.1 GeneralStudying the heat and mass transfer in the contact line region of an evaporatingmenisdus requires an experiment which provides the largest contact line region possibleso measurement techniques can adequately assess the parameters. From the previouslydiscussed modeling equations for a thin liquid film, the thickness profile in the contactline region is a necessary measurement. Also, the heat flux into the region, the surfacetemperature in the contact line region and the ambient surrounding conditions are required to estimate heat and mass transfer. Creating a sealed environment which is in asteady state situation allows the direct measurement of mass transfer if the condensatereturn is calibrated. The entire experiment is shown schematically in Figure 3.1.This experiment uses the method of Truong and Wayner [321 for creating an extendedmeniscus. A highly polished pure silicon wafer is tilted at 5° to the horizontal and adielectric cooling liquid is placed in a pooi at the bottom end. The liquid then formsan extended meniscus on the wafer due to molecular attraction forces. Because theadsorbed thicknesses are anticipated to be of the order of iooA or less, the silicon mustbe highly polished so the surface roughness is less than 100A. This also aids in the profilemeasurement using optical devices due to the high reflectivity of the polished surface.The choice of experimental fluid was determined by the nature of the experimentalinvestigation. The most prominent cooling application for thin film heat transfer is36Chapter • Experjsealed chamber37interferometermeasurementFigure 3.1: Schematic of experimental design.calibratedIdrops /interline ellipsometermeasurementn-type silicon ‘resistance heaterChapter 3. Experiment 38micro-electronics. With this application as the main criterion for choosing a liquid, acommercially available electronic cooling liquid which is a dielectric and has low surfacetension was desired. FC-72 from 3M corporation was a logical choice. One additionalfavourable aspect of FC-72 is the ability of the experimental designer to use standardsilicone rubber as a sealant because FC-72 does not dissolve silicone.The basic idea for the experiment was to construct an embedded heater in the siliconwafer to act as a localized heat source and a electrical resistance temperature gauge. Thecontact line region would then be placed as close as possible to the heater. This wouldprovide heat flux and surface temperature measurements locally.The profile measurements were made by a combination of ellipsometry and interferometry. The interferometer used a fluorescent light source since the interference ofthe various colours provided additional thickness details, thus higher resolution than amonochromatic source at thicknesses below approximately 3000A. The ellipsometer contained focusing lenses that reduce the ellipsometer spot measuring size to a width of lessthan 20im. This represents an improvement over standard ellipsometer designs thathave spot widths 10 times this size.The mass transfer measurements were accomplished by counting and calibrating thecondensate return drops. These drops were created by making a preferential condensationface which was cooler than the surrounding chamber interior. Locating the return overthe liquid pooi created an evaporation - condensation situation.3.2 Ellipsometry3.2.1 BackgroundIf we assume that a chamber exists which isolates a thin liquid film in a steady statecondition with its own vapour, an accurate method of measuring liquid thickness to aChapter 3. Experiment 39depth of 1O_8_1O_9 m is by the use of an ellipsometer as described by McCrackin et al[53]. An ellipsometer is a useful tool to measure liquid thickness because it is generallya non-intrusive technique. If the laser power is kept low, a highly reflective surface willabsorb very little energy, a definite requirement for evaporation studies of thin films.The most economical ellipsometer available is the nulling ellipsometer, which is verysimple to operate. If a polarized light beam is thought of as consisting as a combinationof two Cartesian component individual waves, judiciously set combinations of input polarization and retardation will create a linearly polarized light beam after reflection froma sample. This may be verified by having another polarizer crossed with the output beamas observed visually. Analysis of the angular positions of the ellipsometer componentsplus the known sample refractive properties allows a thickness determination. A slightvariation of this nulling ellipsometer method was used in this experiment to ease use andfabrication. Because ellipsometry measures thickness through the change in polarizationstates of a monochromatic coherent light beam, a method of describing the polarizationstate completely upon reflection is required for this method.3.2.2 Theory and Operation3.2.2.1 Methods of Describing the Polarization State of LightIn general, if a propagating light beam exhibits polarization characteristics, the position of the electric field vector in the transverse plane will describe an ellipse with time.Specific cases of linear and circular polarization arise from the general ellipse. A polarizedlight beam may be described in an x—y—zcoordinate system, with z being the direction ofpropagation, by creating two sinusoidal components in the x—z and y—z planes as shownin Figure 3.2. The x and y vector components may be written as2irz=E0cos(wt—----+6),YChapter 3. ExperimentFigure 3.2: Travelling wave sinusoidal components.xE =E0cos(wt—+6).Choosing z = 0 as any transverse plane of interest, the components becomeEliminating time, t,= E0cos(wt+6),= E0 cos(wt + 6).(3.1)(3.2)E E2 2EZEY cos(6 — 6) — 2 6 6‘ 2 , , _sin(— ). (.)-‘--‘zo -‘—‘yo £JxoLJyoEquation 3.3 indicates that, in general, an ellipse will be described in a transverse planefor a coherent monochromatic light beam. Knowing E0,E0,b, 6, completely describesthe ellipse. However, measuring E0,E0,6, 6, directly is not straight forward, therefore an alternative method of describing the elliptic state is required to correlate withmeasurements.Four physical quantities which are measurable in terms of light beam intensity areknown as the Stokes parameters and are described in Clarke and Grainger [54]. The40zChapter 3. Experiment 41parameters areS0 = E02 + E02,S1 = E02 —S2 = 2EXOEO cos(b—S3 = 2E0sin(b—6). (3.4)The four parameters are calculable by measuring the physical quantities listed below anddescribed in Appendix A.S0 = total intensity of light,S1 = difference between x—coordinate intensityand y—coordinate intensity,S2 = difference between +1/4ir coordinate intensityand —1/4?r coordinate intensity,S3 = difference between right-handed circular polarizationintensity and left-handed circular polarization intensity.A more convenient method of describing an effipse is shown in Figure 3.3 . The fourdefining parameters are azimuth, 0, ratio of minor to major axis of the ellipse, tan II, thesign of e, and the absolute ellipse size. The convention for measuring 0 and f as given byHauge et al [55J is 0 positive measured counter clockwise from the x—axis when viewingthe source and e is positive for right-handed polarization. Relating the ellipse descriptionquantities to the measured Stokes parameters, as shown in Appendix B, the relations are= I = E02 + E02,S1 = Scos2ecos20,Chapter 3. ExperimentY42xFigure 3.3: Ellipse conventions.S2 S0cos2Esin2O,S3 S0 sin 2f. (3.5)In subsequent analysis, it is shown that the absolute value of the light beam intensity,S0, is not a necessary factor in thickness and refractive index calculations. Therefore,using Equation 3.5, the ellipse is fully described for our purposes by1 _1S20 = tan -,f = (3.6)along with the sign of €.Knowing the elliptic representation as given by Equation 3.6, this information mustbe converted to a thickness measurement using reflection theory. The thickness theory isbased on complex Fresnel reflection coefficients parallel and perpendicular to the planeof incidence. Therefore, the previously described ellipse parameters must be relatedto complex polarization components of mutually perpendicular planes (choose x and yeChapter 3. Experiment 43components) which may be aligned with the directions parallel and perpendicular to theplane of incidence. Using the Jones vector as phasor representation of a totally polarizedlight wave as given in Clarke and Grainger [54], the x and y components may be writtenasEE = Exoei6E E0e6cos 9 cos — i sin 0 sin e= Ae . (3.7)sin 9 cos + i cos 9 sin €Azzam and Bashara [56] create a complex polarization ratio, ‘&,=== tan crezö, (3.8)which suppresses the absolute amplitude and phase information. This is acceptableprovided the change in polarization is the information of primary interest as it is here.Using Equation 3.7 , the complex polarization ratio for mutually perpendicular phasorcomponents becomes— sin9cosf+isinfcos0‘V— cos0sinf—isinfsin9’— tan0+itanf 39— 1—itan0tanfEquating 3.8 and 3.9 allows determination of c and 6 and thus L’ in terms of the ellipsedescription parameters, as detailed in Appendix C,cos2c = cos2fcos29,tan2ctanb = . . (3.10)sin 20Consequently, measuring the four Stokes parameters and using Equations 3.6 and 3.10results in complete description of the complex polarization ratio, ‘b.Chapter 3. Experiment 443.2.2.2 Determining the Stokes Parameters from a Nulling EllipsometerKnowing the polarization state before and after the interaction of the light beamwith a sample allows the calculation of the change in polarization. From the previoussection, knowing b before and after interaction would suffice for calculating the changein polarization. This requires knowledge of the Stokes parameters immediately beforeand after sample interaction. Providing an input of known polarization may be readilyaccomplished with specific optical components. However, the polarization after samplereflection may assume any polarization state. In order to obtain thickness results, themeasured light intensities must be related directly to the Stokes parameters followingreflection and before any optical components used in the intensity measurement.Hauge [57] describes a method of determining the reflected polarization state usingan ellipsometer in the polarizer—sample--compensator—analyzer (PCSA) setting. The intensity reaching the detector in terms of S0, S1, S2, S3, (the four Stokes parameters afterthe sample but before the compensator) is given byI(C,A,z) = [So+(Scos2C+2sin2C)cos2(A—C)(S2 cos 2C — S1 sin 2C) sin 2(A — C) cosS3sin2(A — C)sinz]. (3.11)Using a quarter wave retarder plate, = 900, the Stokes parameters are obtained withthe following intensity readings I(C, A), C and A measured in degrees counterclockwisefrom the x-axis,S0 = I(0,0)+I(90,90),S1 = 1(0, 0) — 1(90, 90),83 = 1(45,45) — 1(135, 135),84 = 1(0,45) — 1(0, 135). (3.12)Chapter 3. Experiment 45The first three readings may be taken without the compensator in-line because the compensator’s effect is restricted to phase differential. This is assuming a perfect quarter-waveretarder with ideal transmission along both axes.For a compensator in general, the complex ratio of the amplitude and phase of thefast transmission axis compared to the slow transmission axis isp = tane = Te_6’. (3.13)Using this representation of the compensator effects, the intensity at the detector maynow be written as (Appendix D)I(C, A, 6) = So[cos 2’’ cos(2C — 2A) + 1]+ S1 [cos 2C cos(2C — 2A) + cos 2C cos 2’b+ sin 2C sin(2C — 2A) sin cos ‘5]+ S2 [sin 2C cos(2C — 2A) + cos 2’& sin 2C— cos 2C sin(2C — 2A) sin 2’/’ cos L5]+S3[—sin2bsinösin(2C — 2A)J. (3.14)Using the same settings as the previous determination of the Stokes parameters, the fourquantities becomeS0 = [I(O,O)+I(9O,9O)j/[cos2i4’+ 11,S1 = [1(0,0) — I(90,90)j/[cos2 + 1],52 = [1(45,45) — 1(135, 135)],—[1(0, 45) — 1(0, 135)] — 52 sin 2i/ cosS3 — . . . (3.15)sin 2 sinObviously the values of i4’ and ö must be determined in order to obtain the ellipticalstate using the Stokes parameters.Chapter 3. Experiment 46The compensator imperfections may be found by removing the sample and settingthe input linear polarization angle coinciding with the x—axis, as detailed in Appendix E.Measuring the intensity at the detector in the form I(C,A), the compensator values arefound to be1(0,0) — 1(90, 0)cos2= I(0,0)+I(90,0)’ (3.16)1— sin 2b cos 6 — 1(0, 0) — 21(45,0) + 1(90, 0) 3 172 — 1(0,0) + 1(90, 0)This information allows the complete determination of the four Stokes parameters andthus the complex polarization ratio i immediately upon reflection from the sample. Determination of Liquid Thickness Using the Complex PolarizationRatioThe complex polarization ratio which determines the elliptic state of a polarized lightbeam may be written for the incident and reflected beams from a sample. The two ratiosmay be written as= tan= tan (3.18)Experimentally given these ratios, the theoretical description of light reflecting froma thin liquid covered surface must also be presented as the ratio of two electric fieldcomponents for mutually orthogonal planesFor an electromagnetic wave incident upon a solid surface with a thin film present,the reflected component will be a summation of an initial reflected component fromthe film surface plus multiple reflections from the solid—film and film—air interfaces. Aphase difference due to the film and dependent upon the thickness (as well as refractiveChapter 3. Experiment 47Figure 3.4: Single reflection.indices) will be introduced. Figure 3.4 illustrates the phase difference for an ideal caseof one reflection. The phase difference for a reflection is given by2?i-phase difference = -i-— x path difference. (3.19)For the geometry in Figure 3.4 the path difference may be written asIh\path difference = 2 I I — d\cos Oil= [2 (coOi)]— [2h tan O sin O0jmedjumO (3.20)med:umlCombining Equation 3.20 into Equation 3.192r(2h\ 2irphase difference =— I — — (2h tan 8 sin 0)A1 \COSO1j A04ir= A cos i(ni — n0 sin 0 sin Oo). (3.21)Using Snell’s Law,medium 0medium 1elhn0 sin 00 = fl,l Sfl 01 (3.22)Chapter 3. Experiment 48the phase difference becomesFigure 3.5: Multiple reflections.27rh6=—,ç-nicosOimedium 1\/////medium04Khn11 2phase difference = — sin OA cos Oi= n1cos8. (3.23)Therefore, for a beam travelling from the liquid surface to the solid surface, a phasedifference of(3.24)is introduced. Figure 3.5 shows the general case of multiple reflections from an incidentelectromagnetic wave. The amplitude reduction due to multiple reflection and the introduced phase lags due to multiple path differences are shown. The total reflected wavewill consist of all components due to multiple reflection. The ratio of the reflected electricfield component to incident electric field component is= r01 + (1 —r1)rj2e+ (1 —r)(—ro2e4+= roi(1 —rg1)r2e2{i — (roiri2e)+ (roiri2e)+Chapter 3. Experiment 49—+(1 —r1)r2e2iTo1 (1 + ro112e2i)r01 +r1e, (3.25)1 +01122f3n1 cos8 —n0cos01where r01 =nl cos 8 + n0 cos 01n2cos81—n1cos82=fl2 COS 81 + 1 COS 02The preceding analysis is correct for both the components, parallel and perpendicular, tothe plane of incidence. Letting p denote the parallel plane and s denote the perpendicularplane, the reflection coefficients are—2j/3— Ldrp roipl-ri2pe— E2 — 1 + roiprlpeiI-’—2jR —— T013 j r12e 3 26—— 1 + ro1r12eIf the x—plane in the nulling ellipsometer discussion coincides with the plane parallel tothe plane of incidence and the y—plane with the perpendicular plane, the effipsometercomplex polarization ratios and the reflection coefficients are related by- ErpEis— EipErs— Exr—Eyr— E, Exr— Eyr(3.27)Solving Equation 3.27 for the thickness, h, gives (Appendix E)h = [j(4j)_1(n — ng sin2 0)h12 ln a’] A, (3.28)a’= [ — (roiroi825+r12, — TO1sTO1pT12pP —r128p)+ 2— [ (roiroiri2 + r12p — roisroipri2pp —r123p)Chapter 3. Experiment 50—4(rl2rolr123—ri2sroipripp)(roj —r013p) J1/2 j±2(rl2roi8ri2S rl2sTolprl2pp).The real solution to the above quadratic will be the thickness providing the refractiveindices are known. Fortran ProgramThe solution of the complex quadratic equation for the film thickness was written following the general outline of McCrackin [58]. The program solves for a thickness given allthe system inputs including the refractive index of the film. Using the Stokes parameterdetermination of Hauge [57], modifications from the basic nulling ellipsometer input angles must be incorporated so the proper thickness solutions are obtained. This involvessetting the complex polarization ratio angles of and z into the proper quadrant toprovide all possible thickness values as potential solutions. To facilitate an understanding of the proper required quadrants, the full range of thicknesses must be related tothe complex polarization ratio, p, and the Stokes parameters (S0,S1,S2,S3). This wasaccomplished by writing a reverse Fortran program which provided complex polarizationratios given film thickness as an input. Figure 3.6 shows the various transverse planeellipses for given thicknesses. Figure 3.7 generalizes this information into quadrants plusgives the corresponding azimuths, 6, and Stokes parameter signs (+ or-). Note that thefirst Stokes parameter, S0, is always positive as it is a measure of total intensity.Using this information, the Fortran program was written to ensure the solution tothe equations observed the above limitations. Achieving the above restraints requiredwriting corrections into the program for Equations 3.6 and 3.10. Based on the signs ofS, S2, S3, the solutions to the tangent function were maintained in value but placed intothe appropriate quadrant. Another correction which was required was in the solution ofChapter 3. Experiment 510 0 0OA 100A 200A400A 600A 800A1000A 1200A 1200AY0%1200A 1440A 1600AY02000 A 2200 A 2400 A2600 A 2800 A 2880 AFigure 3.6: Transverse plane ellipses for various thicknesses.IChapter 3. Experiment 52s;, s; s;0-90<O<-4590°< L <1800450< çLi < 900x45\ss2:st0-90<6 <-45180°<LS< 270°450< çLi < 90°x045S S s;O<6<4O°<L <90°0°<qJ.<45°N‘‘2’00<0<427O°<L. <3600O°.c/J<45°x450Q Q+ Q+‘245°<6<90°270°< L <?360%%\450 çfj < 90°S1-, S2’ ‘‘345°<e.<90°O0<t<9000 X4545°<qJ<90°\..Yss s;450 e < 00x90°< L < 180°7/’ 0< (J45° Y 7),S1 s;, S-45<0<0180°<Ls <270°o°< LI<45°////450/x/ ////0-45xY/0Figure 3.7: General Stokes parameter information for various transverse plane ellipses.Chapter 3. Experiment 53Equation 3.28. Again, to maintain the full range of thickness values, the solution to Xrequires that the solution angle be measured counterclockwise for increasing thickness. Focusing LensesDue to the extremely small size of the interline region even in an extended meniscus,a standard laser beam size of 5mm diameter is much too wide to be of use. Whenellipsometry is combined with a form of interferometry as was done here, the last visiblefringe is approximately 400A thick. The distance between colours is in the range of 50,umor less so the thickness changes rapidly with distance. Therefore, the smallest possibleellipsometer spot is desirable.Focusing of an ellipsometer beam is not a new technology although application toprofile measurement in the contact line region has not been made previously. Derjaguinand Zorin [21] used a focusing ellipsometer to study disjoining pressure isotherms in 1957.Theories describing the accuracy of focusing the ellipsometer are given in Svitashev et al[59, 60], and Baruskov et al [61, 62]. The conclusions of these studies are that any errorsresult from a small change in the angle of incidence of various parts of the beam due tofocusing. However, the zones converge so the errors tend to cancel and the accuracy ispreserved.3.3 InterferometerAs the meniscus increases in thickness away from the interline, the ellipsometer ceasesto be effective due to the limited range of readings before repetition of the calculatedfilm thickness occurred and the rate of thickness change within the ellipsometer spot sizeresulted in meaningless readings. rTherefore in this experiment, perpendicular fluorescentlighting was used in conjunction with a 30X magnificaation microscope following theChapter 3. Experiment 54method of Plishkin and Conrad [63] to identify the area in which the ellipsometer wasuseful, and also to provide an interference pattern to identify the profile shape. Thethickness colour chart for Si02 was adjusted for index of refraction differences betweenSi02 and FC-72. Table 3.1 lists the values used in the fringe pattern analysis.Using this colour interference pattern provides an increased number of data pointsas the rate of change of thickness with position is very small close to the interline.The very beginning appearance of the colour tan is approximately 400 A. This is animprovement over monochromatic fringes where the first order dark fringe occurs atroughly 1000 A. Uncertainties exist in this method from a combination of the estimatedthickness and subjective interpretation of the colour. Table 3.1 indicates the thicknessat a corresponding colour plus the thickness at a boundary between two colours.With the present combination of liquid and substrate, the ellipsometer cycles itselfat approximately 3800 A. Knowing this, a verification was devised to check the colourthickness pattern given in Table 3.1. Inside a sealed chamber, drops of FC-72 were placedon a silicon wafer under saturated conditions. The drops spread out on the sfficon sothe colour pattern was well defined. Using the ellipsometer and translation stage, theellipsometer spot was positioned over the various colours using a microscope. A seriesof three readings were taken and averaged. The results are also listed in Table 3.1. Therange of measurement variation was less than 10% for each colour. The results showgood agreement between the theoretical and measured values, well within the limits oferror due to colour interpretation and ellipsometer spot positioning.Past 4000 A, it was impossible to accurately position the spot due to the rapidlychanging colour pattern. The bright fringes past this point were pink or purple, therefore,the mid-point of each succeeding red fringe from theory was taken as the thickness ofeach fringe.Chapter 3. Experiment 55Measured Interference Colour ThicknessesFringe No. Colour ( Thickness, A Thickness, A(Theoretical) I (Ellipsometer)Clear-Tan <580 370Tan 580 573Brown 820 853Brown-Blue 1170 1121Blue 1400 1379Blue-Light Blue 1575 1670Light Blue 1870 1962Light Blue-Gold 2330 2360Gold 2567 27601 Purple 3326 3499Blue-Green 3850 4240Yellow 4376 46312 Purple 5368 5213Green 6100 N/AYellow 6700 N/A3 Purple 7400 N/A4 Purple 9740 N/A5 Purple 12080 N/A6 Purple 14420 N/A7 Purple 16760 N/A8 Purple 19100 N/A9 Purple 21440 N/A10 Purple 23780 N/A11 Purple 26120 N/A12 Purple 28460 N/A13 Purple____30800 N/ATable 3.1: Measured interference colour thickness values.Chapter 3. Experiment 563.4 Apparatus3.4.1 HeaterThe desire to simulate a two-dimensional micro-electronic chip was the starting pointfor the heater design. To aid in ellipsometric measurements, a highly polished and optically fiat surface was required. Parameswaran et al [64] have done extensive work inboron doping of silicon surfaces. Dr. Paramaswaran agreed to provide two-dimensionalrectangular heaters created in a piece of polished silicon substrate as per our design. Theheater design is shown in Figure 3.8.Five thin rectangular boron atom heaters were thermally diffused into a 10 cm diameter piece of high resistivity n-type silicon wafer. (40-60 ohm — cm) The heaters were 3cm long with widths varying from 200 um to 600 um. The diffusion process consistedof thermally growing a layer of Si02 in the silicon, then using a precision mask to openrectangles in the oxide to match the heater pattern. The sample was then placed in adiffusion oven with a boron source for 90 minutes, a time determined by trial and errorto be the maximum exposure time which still prevented diffusion through the oxide tounwanted locations. The entire process was then repeated to achieve greater diffusiondepth and density, thus the lowest possible resistance. The wafer was then trimmed to a5 cm x 5 cm square to easily fit the translation stage and chamber dimensions.A problem arose in attempting to make consistent electrical connection to the enlargedboron diffused end squares. Therefore, a thin layer of aluminum was vapor deposited overthe boron squares. Small brass posts were held in contact with the aluminum squaresusing a pair of microscope slide spring clamps. The clamps were attached to wires leadingto a BNC connector fitted in the chamber wall. Care was required once the experimentwas assembled because movement of the clamps caused a small change in the resistancereadings, however, the reading was stable if no apparatus adjustment occurred after theChapter 3. Experiment 57siliconFigure 3.8: Boron Diffused Silicon Heater.Chapter 3. Experiment 58chamber was sealed. Over the silicon wafer and posts, an acrylic cover with a 3.75 cmx 2.5 cm cut-out was fitted over the silicon wafer with the inside edge sealed with asilicone bead. This rectangular space, 6 mm high, was the containment volume for theliquid. The silicon heater fastened in place inside the chamber is shown in Figure 3.10.Five heaters of various widths were constructed in a single silicon substrate to providea range of resistances and insurance in the event of a failure. However, the middleheater produced a stable resistance without failure and was used for the duration of theexperiments.3.4.2 ChamberFigure 3.9 shows a schematic of the chamber with the microscope positioned abovethe optical window located in the chamber top. The translation stage provides relativemotion between the laser and the meniscus position. Figure 3.10 is a photograph ofthe chamber interior without the top in place. The silicon wafer with thin embeddedheaters is visible in the center of the photograph. The microscope clamps which providethe electrical connection to the heater are also evident on top of the acrylic plate thatprovides the liquid containment area. The acrylic piece that extends from the verticalwall in the bottom of the photograph is the return mechanism for the condensate. Thetwo focusing lenses and the tuberculin syringe are also visible.The sample chamber was milled from a solid aluminum block. Two exterior sidewalls were perpendicular to the base with the other two machined at a 20° incline fromperpendicular. The inclined sides are thus perpendicular to the incident and reflectedpaths of the ellipsometer laser. The inner walls are all perpendicular to the base withinterior dimensions of 85 mm x 120 mm x 60 mm.Two 25 mm diameter low birefringent windows were located in the inclined walls toallow laser beam passage into and out of the sealed chamber. A 75 mm diameter windowmicroscope30Xmagnificationlinearlypolarizedfluorescentilluminatorellipticallylightpolarizedlight(generalcase)lowbirefringentwindowstranslationstageCl’Figure3.9:Schematicoftheexperimentalchamber.ctII_____________Figure3.10:Photographoftheexperimentalchamberinterior.CChapter 3. Experiment 61was fixed in the chamber lid for liquid viewing. The windows were Melles Griot models02 WLQ 105 and 02 WBK 007 respectively.In one of the perpendicular outer walls, a 40 mm diameter hole was drilled throughand fitted with a hollow cylindrical insert penetrating 10 mm into the chamber. Athermoelectric cooler, Marlow Industries model M11022T was installed in the insert andconnected to a series of cooling fins via a cooper cylinder. A variable voltage source from0—3.5 Volts was available for the cooler.An Oriel Model 16122 7.5 cm x 8.75 cm translation stage was fastened to the interior chamber floor and connected to an externally mounted Mitutoyo model 15 1-255micrometer head with 0.01 mm graduation. The translation stage surface was removedand replaced with an acrylic cover with a 3.75 cm x 2.5 cm cut-out in the middle. Thisprovided an insulating airspace under the actual heaters so as to approximate an insulated condition. All chamber openings were fitted with 1.8 mm rubber gaskets plus themicrometer shaft was complete with an 0-ring seal fitted in the mounting shoulder. Also,all edges were sealed with silicone and allowed to cure once assembly was completed.3.4.3 EllipsometerA basic nulling type ellipsometer, arranged as Polarizer—Sample—Compensator—Analyzer, (PSCA) was designed to measure the film thickness. In addition, focusing lenseswere used immediately preceding and following the sample to provide the smallest measurement area possible.The basis of the ellipsometer is a 1 mW Melles Griot Model 05 LPL 340-065 polarizedHe—Ne laser. The red laser beam is at a wavelength of 6328 A. A polarized laser waschosen based on Hauge and Dill [65] in order to eliminate intensity variations due toswitching between polarization modes.Using lens focusing theory [66], the focused spot size is inversely proportional to theChapter 3. Experiment 62input beam diameter. Therefore, to minimize the spot size to diffraction limits an expansion of the 0.59 mm diameter standard laser beam output was achieved using a MellesGriot 8—X beam expander Model 09 LBC 003. This resulted in a beam approximately 5mm in diameter.The polarizer and analyzer were 20.6 mm diameter dichroic sheet polarizers withextinction ratios of iO (Melles Griot Model 03 FPG 001). The focusing lenses are 30mm diameter, 100 mm focal length lenses to correspond to the chamber design. A singlelayer MgF2 anti-reflection coating was used on the Melles Griot Model 01 LPX 178/066lenses. The compensator is a 1/4 wave mica retardation plate specifically designed for a6328 A red light laser, model number Melles Griot 02 WRM 015.All the optical components were mounted with optical post holders and posts fastenedto two Melles Griot mini-rail carriers (Model 07 ORM 007). The rail carriers were fixedto an aluminum platform at an angle of 70° from the platform normal axis. This angleprovides a maximum in ellipsometer reading sensitivity according to McCrackin et al[53]. The entire ellipsometer configuration is shown schematically in Figure 3.11.The optic holders for the polarizer, compensator and analyzer were required to rotateto four fixed positions each, thus rotatable mounts were fabricated using fixed MellesGriot component mounts for 30 mm diameter components (and adapters if necessary)and machined aluminum rings. The mounts were scribed to an accuracy of ±0.025°.A silicon photodiode was used as a measuring device for the ellipsometer outputbecause unlike a true nulling ellipsometer, the light intensity reading levels were required.However, the absolute intensity was not required only the relative intensity of the variousreadings. Therefore, absolute detector calibration was not imperative. The detector isan Oriel model 7182 silicon photodiode with 100 mm2 sensitive area. In order to preventany polarization dependent tendencies of the detector, the analyzer output was focussedinto a fibre optic bundle which uses multiple internal reflection inside between 50—200x1:xjYHe-NeLaser4XBeamExpanderPolarizerFocusingLens ChamberAnalyzerFigure3.11:Schematicoftheellipsometer-chambersetup.Chapter 3. Experiment 64bundles to essentially eliminate polarization effects [67]. The fibre optic bundle is anOriel model 77521. The signal was amplified and output to a digital voltmeter.3.4.4 Stand, Microscope and CameraFigure 3.12 shows a photograph of the overall view of the apparatus, including theliquid insertion method through a tuberculin syringe. The thermoelectric cooler fins areevident in the front wall of the chamber. Figure 3.13 shows the apparatus completewith the microscope in position above the chamber. The RTD temperature probe canbe seen in the back wall of the chamber.The ellipsometer and chamber, complete with a space for the microscope, were assembled on a 36” x 12” rectangular aluminum platform with four adjustable legs. Theaccuracy of the adjusting screws was one full turn per one-half degree incline. A levellingbubble was attached to ensure the stand was level in the direction perpendicular to theincline axis.The microscope was an Edmund Scientific Industrial Microscope model A37,659 with30X magnification and a 75 mm working distance. A fluorescent light ring was fittedaround the microscope objective to supply perpendicular fluorescent light. A Yashica 35mm camera was fitted over the microscope eye piece for still photographs.3.4.5 Resistance, Power Input and Temperature MeasurementThe temperature of the vapour inside the chamber was measured with a platinumRTD probe inserted through the chamber wall and secured with a compression fitting.The probe was located about mid-chamber height approximately 1 cm from a side wall.The location was chosen to approximate the bulk vapour temperature while not interfering with the laser beam. The probe and digital readout was an Omega modelPR-13-2-100-1/8-51/2-E with an accuracy of 0.1 °C.3Ct:,Figure3.12:Experimentalapparatuswithoutmicroscope.btj ft‘NFigure3.13:Experimentalapparatuswithmicroscope.03 C,Chapter 3. Experiment 67Liquid Properties of FC-72 and Water at 25°CProperty FC-72 WaterMolecular Weight 340 18Surface Tension (dynes/cm) 12 72Refractive Index 1.252 1.333Vapour Pressure (torr) 232 23.7Solubility of air (mi/lOOmi) 48 1.9Density (g/ml) 1.68 1.0Viscosity (cs) 0.4 0.9Heat of Vapourization (cal/g) 21 540Conductivity (mW/cmC) 0.57 5.86Dielectric Constant (1Hz) 1.76 78Table 3.2: Properties of test liquid, FC-72 [2].The heater power was supplied by a constant current source variable from 4-100mA. The current through and resistance across the heater were measured by 5-1/2 digitmultimeters.3.4.6 Dielectric Test LiquidThe cooling liquid used to create the thin film was 3—M Fluorinert FC—72. Thecomposition of the liquid is C6F14, known as perfiuorohexane. Typical bulk liquid properties are given in Table 3.2. As a comparison the properties of water are also included.Injection into the chamber was accomplished with a 1.0 cc tuberculin syringe with a 4inch long needle fitted through a rubber stopper and inserted through a side wail. The3M representative established the fact that the liquid was tested with a silicone rubberseal for a 6 month period with less than 1% variation in properties.Chapter 3. Experiment 683.5 ProcedureThe pre-testing procedure involved cleaning the chamber and sample as thoroughlyas possible, then assembling and sealing the chamber. The silicon wafer was put intoa dilute HF solution to remove the Si02 layer which was deposited during fabrication.The wafer was then washed with de-ionized water, dried with compressed N2 then furtherdried in an inert gas oven. As it is impossible to keep a small oxide layer from growingwhen exposed to normal atmospheric conditions, it was accepted that a thin oxide layerwas present during testing. The wafer was handled with a lint-free cloth covering thetest surface until the final assembly stage.The chamber and all parts associated with it were washed with alcohol, rinsed withde-ionized water and air dried. Once assembled, before the silicon wafer was placedinside, the interior was rinsed with the test liquid and allowed to dry. The wafer wasthen inserted and the entire chamber sealed. Before liquid was introduced, calibrationreadings were taken with a dry surface. Ellipsometer readings in the vicinity of the heaterwere taken to determine the native oxide layer thickness as well as the surface reflectanceto be used in ellipsometer thickness calculations. Power was introduced to the heater fora short time to check for changes in substrate refractive index with temperature. Thetemperature-resistance calibration of the heater strip was accomplished by wrapping anelectrical resistance heating strip around the chamber and setting various power levels toobtain a range of ambient chamber temperature conditions. Temperatures were limitedto between 20 °C-30 °C as these would sufficiently cover the test range.Finally, liquid was introduced into the test area to a level near the heater with both theheater and interline being in the field of view of the microscope. With the thermoelectriccooler activated, liquid was continually introduced until the return rate maintained theinterline near the heater, a visual measure of steady state. The apparatus was left for aChapter 3. Experiment 69period of 24 hours prior to the beginning of a series of tests. A test was initiated by ofsetting an input current level and monitoring the heater resistance level and the interlineposition relative to the heater. Once a steady state situation was noted for a minimum of2 hours, the microscope and camera were positioned above the heater and a photographtaken. The chamber ambient temperature, heat input, and return rate of condensationdrops were measured as data. After these steps, the ellipsometer spot was located abovethe last visible brown fringe and the micrometer adjusted to find a constant reading inthis region. The reading was then monitored for approximately 5 minutes for any signof changes. After recording the ellipsometer data, power was removed from the heaterand the resistance was monitored until the wafer cooled to equilibrium with the chambertemperature. The resistance change was recorded as a measure of the wafer temperaturerise above ambient.The tests were terminated at a power level which introduced more vapour than couldbe condensed with the cooler alone and condensate appeared on the chamber floor. Oncethe tests were complete the condensate return from the cooler was positioned away fromthe test area and a range of tests repeated. The equilibrium liquid supply was introducedthrough the syringe thus calibrating the return rate as the return drops were countedand timed.Chapter 4Experimental Results and Observations4.1 Preliminary Operation and CalibrationAn initial set of tests was done to verify the accuracy of the ellipsometer. Two siliconwafers with Si02 layers of known thickness, measured by two separate ellipsometers at theMicro-Electronic Research Centre at the University of Alberta (U. of A. ), were measuredwith the ellipsometer with and without the focusing lenses. The measurements withoutthe lenses were accurate to within 0.2% of the values measured by U. of A. With thefocusing lenses in position, the present ellipsometer measured values were slightly higherthan the values without the lenses, but the estimated error remained less than 1%. Theseerrors are small compared with the uncertainty associated with manual positioning of theellipsometer components as is discussed later.Due to the extremely thin nature of the liquid film and the small length of the two-dimensional evaporating region, the most difficult obstacle to overcome was achievementof a steady-state situation. A stable ambient environment for the duration of the experiments was attained by conducting the experiments at night in a photographic dark room.During the experiments, use of the fluorescent microscope ring and laser was kept to theminimum required to obtain the measured results. If left on for any duration of timeother than that required for immediate observations and photographs, it was noticedthat the fluorescent light ring caused a rise in the temperature inside the chamber. Nonoticeable effect on the internal temperature from use of the laser was detected but the70Chapter 4. Experimental Results and Observations 71beam was blocked when not in use as a precaution. With the cell wall thickness currentlyused and no significant changes of the ambient conditions in the room during the night,the vapour temperature in the cell was measured to within ±0.1°C and was very stableduring each test run.Equation 3.28 gives the quadratic solution used to obtain the film thickness from theellipsometer readings. The refractive index of the vapour, liquid and surface are requiredwithin this equation to obtain the thickness. The refractive index of the vapour is takento be 1.0, the liquid taken as that of the bulk liquid, 1.251, but the refractive index of thesilicon wafer with a native surface oxide layer is unknown. Therefore, in order to use theellipsometer, refractive index readings of the bare silicon substrate with native Si02 layerwere required. The readings were taken at the location just below heater No. 3, shownin Figure 3.8, the designed location of the adsorbed film during a test. Five separatereadings were averaged to give a complex refractive index of n = 4.24 — 0.05i. Using thevalue of Palik [87] for pure silicon of n = 3.882 — 0.019i, a native surface oxide thicknesswas calculated to be 18 A, ±5A. The estimate of error was determined by recordingthe change of effipsometer reading with each individual setting of the compensator andanalyzer. The maximum possible error was then calculated for the oxide thickness results.Therefore, the experimental substrate is actually pure silicon with a thin Si02 layer onthe pure silicon surface to be considered.The determination of the surface temperature of the boron heater required a calibration of heater temperature versus heater electrical resistance. The RTD probe usedto measure the vapour temperature was used to measure the ambient temperature inthe chamber while the chamber was heated externally with an electric resistance heatingstrip. The chamber contained no liquid during the calibration, therefore evaporationdid not influence the calibration. Figure 5.8 shows the results of the calibration. TheChapter 4. Experimental Results and Observations 72resistance was measured to within ±0.001 ohms and a linear least squares regression gaveR = 225.466 + 0.539T3, (4.1)with R measured in ohms () and T8 in degrees Kelvin, K. This calibration was repeated after the experiments were completed. The results were similar with a resultingtemperature-resistance curve slope of 0.59, slightly less than a 10% variation from theoriginal. The original calibration was used to calculate temperatures because it was takenbefore any movement or adjustment was made to the experimental setup.Once the heater calibration was completed, the test liquid was introduced into thesealed chamber through the tuberculin syringe. The liquid level was set below the testheater (No. 3 in Figure 3.8) and allowed to evaporate. Liquid input was continued untilisothermal saturation was achieved. The liquid level was then microscopically set so theinterference colour (beige) was just below the test heater. This would situate the adsorbedfilm on the heater. Once the liquid level was set, the thermoelectric cooler voltage was setby trial and error. Initially, 2.0 volts was set across the cooler. Observation through themicroscope showed a large condensation rate at the cooler face for a saturated chambercondition. This resulted in an oscillatory motion of the contact line region without powerhaving been supplied to the heater. Oscillations toward and away from the heater werenoticed plus waves travelling along the interline from one edge of the heater (where theheater and acrylic cover intersect) to the other edge. The voltage was reduced to 0.75volts and this provided a steady interline and low condensation rate at the cooler face.This level was chosen for use during all of the experiments. The power input to the coolerwas not quantified for use in the results.With the cooler level set, the experiments were taken at steady-state. The assumptionof steady-state was not strictly correct as it was noticed that all the liquid would disappearin 7-10 days if the system was allowed to run continuously. Because an entire test would245244243E.2-242ci) o241C240239238237236235en 0HeaterResistancevs.TemperatureCalibration2022242628HeaterTemperature, T(°C)30Figure4.1:Experimentalcalibrationofheatertemperaturevs.electricalresistance.Chapter 4. Experimental Results and Observations 74be completed within 2 nights, this fact was noted but would have to be addressed if anyduration type tests were done in the future.Due to the very specific design of the experiment with respect to the liquid condensation and return path, power input to the heater could not be increased arbitrarilywithout encountering a situation where the thermoelectric cooler face could not condenseat a rate equivalent with the evaporation rate. This would cause condensation on thechamber bottom and walls thus eliminating the condensation-evaporation steady-statesystem, and making the mass transfer measurements unreliable. By trial and error, thisheater power level was found to be anything greater than 0.2 Watts. Therefore, between0 and 0.2 Watts, 6 separate power settings were chosen for each experimental set. Twophotographs, 15 minutes apart, at each power setting were taken during each experimental run and three complete experimental sets done. This resulted in 6 experimentalprofiles for each power setting, a total of 36 individual interference photograph profiles.4.2 Interferometry PhotographsFigure 4.2 shows a sketch which explains the features of the following photographicresults. The photographs are approximately a 40X magnification of the area after photographic enlargement is included. Horizontally in the z direction, each photograph encompasses 16% of the width of the heater. Each photograph is the result of a minimum of 2hours of continuous steadystate operation at the stated power level. The fluorescent lightring was activated and the photograph taken immediately and the light deactivated inorder to avoid any thermal changes to the system while measurements were being taken.The colour fringes correspond to various liquid thicknesses as described in Chapter 3.Figures 4.3 through 4.8 show one typical set of interferometric photographic resultsfor the 6 different heater power settings which comprised an experimental set.00400im4,focussedellipsometerspot0largelenses0• V a) I 0 U) V Cu U) a) C I II 0 0 0. V a.()ufoou:000/Ismalllensesredfringes\photographedareatlj 0 -4 0’Figure4.2:Schematicdescriptionofphotographicresults.Chapter 4. Experimental Results and Observations 76EriDC)CCbJDChapter 4. Experimental Results and Observations 77E4.)JDC)00ci)Chapter 4. Experimental Results and Observations 78(I)0(Chapter 4. Experimental Results and Observations 79njC-)CtCCc1bDChapter 4. Experimental Results and Observations 80Cl)C.)CCs-IChapter 4. Experimental Results and Observations 81Skr.)-02c)CCc)biDChapter 4. Experimental Results and Observations 82Referring to Figure 4.3, a case with no heat flux input, the edge of the beige or browninitial fringe is approximately 400 A, with the thickness reducing towards the heater andeventually the constant thickness adsorbed layer (clear, no interference pattern) whichis located on the heater for each case. The beginning of the adsorbed layer is located ata different distance from the brown edge depending on the heater power level. As canbe seen from Figure 4.3, the fringe pattern is almost parallel to the heater in the rangephotographed. This parallel pattern exists for over 90% of the heater length indicatingthe two-dimensional nature of the experiment. The fringes deviate from the horizontalpattern only at the edges where capillary action draws the meniscus higher. This is notshown in the photographs. Toward the liquid pooi end of the photograph, the fringesdisappear into the liquid pool which becomes 2-3 mm thick at the deepest section, thepooi edge at the acrylic cover.The first 8 or 9 colour bands are easily distinguishable by the change of colours,given in Table 3.1 (i. e. brown to blue to light blue to gold to purple to etc.) andthe determination of film thickness is straight forward. Past these distinguishable colourbands, a series of red (pink) and green closely spaced fringes is evident. The fringe patternbecomes similar to monochromatic interference with the middle of the bright fringe takenas the thickness reference point for position measuring purposes. Due to different colourwavelengths which contribute to constructive and destructive interference, examinationof the bright fringes shows a change from pink to purple as the thickness increases.These minor variations in the bright fringe colour were ignored and the main colour forthe bright fringe was assumed to be red. Therefore the wavelength 6328 A was used todetermine the thickness after the ninth distinguishable colour band, or the first purplefringe (fringe No.1) as shown in Table 3.1.It is observed from the photographs that the evaporating meniscus becomes smallerand the fringe pattern closer together as the power input is increased from 0 - 185 mWChapter 4. Experimental Results and Observations 83in Figures 4.3 - 4.8. The last visible fringe in Figure 4.8 is at a thickness of approximately 30,000 Aor 3 pm. This represents the power setting with the least number ofdistinct visible fringes. In comparison, Figure 4.3 has 6 additional distinguishable fringes.In the photographs, faint horizontal and vertical lines may be observed, these are themicroscopic cross-hairs used for measuring and positioning.Examination of the individual photographs shows some interesting details which maybe qualitatively described by comparing the various pictures. In Figure 4.3, large lensesare visible on the heater edge away from the liquid film. Microscopic impurities areevident in the middle of these lenses. This photograph was taken without power inputto the heater, and the only source of evaporation is due to the condensation of thevapour on the cooler face. This results in a vapour pressure below the saturation leveland causes evaporation from the liquid interface. At the first appearance of interferencefringes, bubbles or lenses appear parallel to the heater and of varied size. These bubblesonly appeared after the cooler was activated and surface evaporation commenced. Thebubbles were always present once evaporation began, regardless of the power input level.Comparing Figure 4.4 to Figure 4.3, it is apparent that the brown colour band isthinner in the former. Between these two tests, no additional liquid was added to thechamber for positioning purposes, however, the heater was activated and with P=4mWof power being input. The edge of the brown fringe is now further away from the heaterand close examination shows that the bright fringe spacing beyond fringe No. 1 is closer.This indicates a higher evaporation rate and accounts for the difference in distance fromthe meniscus edge to the heater. The same bubble pattern is evident in Figure 4.4 as inFigure 4.3. The clarity of Figure 4.4 shows a disturbed fringe pattern near the bubblesat the centre of the photograph. It is apparent that these bubbles disturb the flow intheir immediate vicinity while the bubbles on the left of the photograph do not appearto influence the flow.Chapter 4. Experimental Results and Observations 84Figure 4.5 is the first result where additional liquid was introduced into the chamberto position the adsorbed film directly over the heater. The additional liquid was necessarybecause the increased evaporation rate shortened the meniscus to an extent that the edgeof the colour bands was between 2-3 heater widths from the heater edge. At this powerinput level, P=23mW, a significant compressing of the fringe pattern above fringe No.3 is clearly shown. This is a result of the increased evaporation rate for this case. Thebubbles which exist between the fringes and the pooi are, on average, smaller in sizethan the previous two cases, but one additional large bubble appeared. One obviousfeature of this photograph is the small size of the brown colour band. However, the sizeof the colour bands between brown and the fringe No. 3 does not appear to have changednoticeably.The results from Figure 4.5 which show the decreased fringe spacing above fringeNo. 3 and constant colour band size between this fringe and the heater is very clearlypresented in Figure 4.6. The fringe spacing becomes very small by comparison to theprevious cases. The bubble size appears very similar to Figure 4.5 but the centre bubbleappears to be causing a slight disturbance in the liquid locally, as is evident by the slightbowing of the fringe pattern near this bubble. Again, liquid was introduced to maintainthe adsorbed layer over the heater, therefore, relative positioning may not be considered.Figures 4.7 and 4.8 show the exact same trends as detailed in the description ofFigure 4.6. The fringe spacing at liquid thicknesses greater than those at fringe No. 3becomes very close while the colour bands between this fringe and the heater appear tomaintain the same constant spacing as is evident throughout all the photographs. InFigure 4.7 the local perturbation near the large bubble is clearly shown, but the overallbubble size remains close to the same size as shown in Figures 4.5 and 4.6.The result in Figure 4.8 shows a much different aspect than all the previous photographs. The bubbles between the fringes and the liquid pool appear to be connectedChapter 4. Experimental Results and Observations 85by a wave-like feature. This exact feature was also apparent at higher power input levels,which produced meniscus oscillation and condensation on the chamber floor.The presence and formation of the lenses between the liquid pooi and the fringeswere examined closely. When the chamber was filled with liquid, without the cooleroperating and without power to the heater, the entire fringe pattern was very small, withfringe spacing so close that the resulting photograph could not distinguish the separatefringes. Also, no lenses were present in the liquid. However, once the cooler was activatedresulting in condensation and corresponding evaporation, these lenses appeared as shownin Figure 4.3. Deactivating the cooler would eventually result in the disappearance of thelenses but they would return upon reactivation. The lenses were present for every heaterpower level used and would remain in the photographed location for the duration of eachexperimental run. Disassembling the experiment, cleaning, and re-sealing the chamberbetween experimental sets would not change the appearance or behaviour of the lenses.It was noted that the large lenses near the pooi, at least one of which is present in everyphotograph, would eventually move through the fringes and migrate upwards past theheater and reside above the heater. It would take more than 2 hours for this to occur.The location of the lenses was not consistent in terms of liquid thickness. In the lowerpower range, Figures 4.3 - 4.5, the lenses were centred approximately at the 12th purplefringe while for the higher settings, Figures 4.6-4.8, the centre location was approximatelythe 15th purple fringe.Figures 4.7 and 4.8 show the sensitivity of the entire experimental system to changesin the setup. With the sensitivity of the constant current source feeding power to theheater, there is much greater control at the lower power settings. Increasing the powermay be accomplished slowly at the lower settings; however, at the higher settings theincrease is accomplished within larger increments. This rapid increase causes surges inthe meniscus which produce a wave which propagates up the substrate toward the heaterChapter 4. Experimental Results and Observations 86from the liquid pooi. A small remnant is seen in Figure 4.7 located on the heater whileFigure 4.8 shows a much more noticeable wave. These waves eventually migrate to theopposite side of the heater from the liquid pooi. This resulted in waiting for the meniscusto resume a stable shape before measurements were taken. This same behaviour resultedfrom movement of the translation stage. Once the wave migrated away from the meniscusvisible edge it was assumed to have negligible effect on the steady state evaporatingsystem.4.3 Liquid Film ProfilesCalculation of the liquid film profiles was accomplished using the photographic resultsand the thicknesses of Chapter 3. The colour fringe position was analyzed through a 30Xmagnification microscope on a 0.01 mm graduated translation stage. Each of the 6power settings had 6 photographic results which were all analyzed at the centre of thephotograph. The x-axis positions of each fringe were averaged for each power setting,and the profiles are presented in Figures 4.9 and 4.10.The effect of increasing the power input to the heater is evident in the length of theevaporating meniscus as shown in Figure 4.9. The slope of the profile increases dramatically with increased power input. The relative position of the x-axis, x, was determinedby finding the last visible red fringe on the photograph of the highest power level, Figure 4.8, the result with the least number of visible fringes. This red fringe, number 13,was then taken as the reference position of all the other test cases, with distances weremeasured from this fringe to the interline, x0. For the red fringes, the centre location wastaken for the distance measurement. Once individual colours became distinguishable,the mid-point of the colour was measured along with the boundary between two adjacentcolours. The photographic measurements ended at the visible edge of the beige fringe.en3.5E4-LiquidProfilesatVariousPowerInputs3.0E4E25E4a -oDP=OmWoP=4mW2.0E4VP=23mWP=59mWSOzD0P=ll4mWC1.5E4&P=185mWrJ)E1.0E4-&tD5.0E30.OEOI________________________02505007501000RelativeDistance,x-x0(x108m)Figure4.9:Liquidfilmprofiles,averagedover6readings.4000ThinFilmProfiles(VariousPowerInputs)3500&OO7LL:13000DP=OmWLP=4mW2500&0P=23mWP=59mW&0DvD0P=ll4mW02000&P=185mWo&D1500&Q710&1000&U500&0c’c7LAAAAA0----._cx:xI20040060080010001200RelativePosition,x-x0(x106m)Figure4.10:Adsorbedthicknessregionprofiles,averagedover6readings.Chapter 4. Experimental Results and Observations 89The last point of each data set is the adsorbed thickness and position as measuredwith the ellipsometer. The adsorbed thickness was measured by locating the laser intothe region above the last visible remnants of the beige fringe and moving the translationstage while monitoring the ellipsometer reading until no change was noted. After monitoring the ellipsometer reading in this location for approximately 15 minutes to ensure novariation, the reading was taken. After the reading was noted, the translation stage wasmoved towards the beige fringe and the translation position noted when the ellipsometerreading began to deviate from the equilibrium adsorbed thickness reading. The distancefrom this point to the visible edge of the beige fringe was recorded. This method had to beperformed quickly because translation stage motion caused waves in the meniscus, therefore the distance had to be determined before equilibrium was disturbed. The presentdesign precluded a slow profiling method using the ellipsometer because the meniscuswas highly sensitive to translation stage motion. Although a detailed ellipsometer profilebetween the beige fringe edge and the adsorbed film was desired, (one measurement every10 um along the x-axis) the variation in adsorbed layer position with power input wasstill obtained with this alternate method.An expanded view of the adsorbed layer position from Figure 4.9 is shown in Figure 4.10. The primary reason the curve points are not perfectly smooth is due to thesubjective interpretation of the colours and related thickness as discussed in Chapter 3.The results show that the adsorbed thickness layer begins approximately 350 m fromthe visible beige edge in the case without power to the heater and this same measurement is only 50 um in the highest power case, P=185 mW. Figure 4.9 clearly shows theshortening of the meniscus as the power is increased. The initial slope is much steeperat P=185 mW than the other power levels. Figure 4.10 illustrates the point that belowthe second purple fringe ( 5000A) the profiles are fairly constant in slope. The changenear the adsorbed thickness reflects the widths of the beige fringe in the photographs.Chapter 4. Experimental Results and Observations 904.4 Adsorbed ThicknessWhile the photographic results were limited to 6 power input settings throughout theexperimental range, as the power was increased incrementally through the test range,adsorbed thickness, evaporation rate, and heater temperature were all measured at eachincrement. This resulted in 14 different measurements for each of these three parameters.The 14 measurements were averaged for the 3 experimental runs. The average values forthe adsorbed thickness change with power input are shown in Figure 4.11.Each adsorbed film thickness measurement was made after the laser was determinedto be situated in the adsorbed region using the consistency in ellipsometer readings asan indicator as the ellipsometer spot was situated at the interline visually and subsequently moved toward the adsorbed film region using the micrometer. A minimum of 15minutes was allowed to monitor the reading for any possible variations with time. Thepoints in Figure 4.11 represent an average of the three experimental runs. Variations ofless than 10 % of the value at each condition were noted. For readings in this range ofthe ellipsometer (< 150A), the estimated human error in reproducing the analyzer andcompensator positions for each reading is ±5)t. This estimate was achieved by using oneof the Si02 samples from the University of Alberta as a test sample, and then recording the sensitivity of each ellipsometer component to 1 degree changes in measurementposition (the estimate of repeatability of positioning with the current design). Usingthese sensitivity results in the computer program, the maximum variation in calculatedthickness was computed. The caluculated variation represents approximately 25% of theadsorbed thickness value at the highest power inputs. The two most notable parts ofFigure 4.11 are the lack of variation in adsorbed thickness past 100 mW input powerand the rapid change in steady-state adsorbed thickness below 50 mW input power. Theadsorbed thickness ranged from 138A at P=0 mW to 40)1 at P=185 mW.eliAdsorbedThicknessvs.PowerInput140irE120-o 1 x .0ioo-••0 a).80-0.E I-.•601 o0(I)40••20 0IIIII050100150200PowerInput,P(mW)Figure4.11:Adsorbedthicknessvs.powerinput.Chapter 4. Experimental Results and Observations 924.5 Evaporation RateFigure 4.12 shows the mass evaporation rate as a function of the input power. Themass evaporation measurements involved timing the return rate of liquid drops at steady-state. The drops were timed to within ±0.5seconds, leading to an error of less than 3 %of the calculated evaporation rate.The mass evaporation rate was calculated relative to the background evaporationdetermined from the evaporation result from the P=0 mW case. The evaporation forthis particular case was caused solely by the thermoelectric cooler condensing the vapourand resulting in a vapour space below saturation pressure. As the cooler voltage washeld constant throughout the tests, the evaporation due to the cooler was assumed to beconstant for all the tests. Therefore, this evaporation rate was subtracted from all thecases where the heater was activated. This should provide information on the evaporationcaused only by the heater. The background evaporation comprised 90% of the totalevaporation measured for the P=4 mW case and only 31% for the P=185 mW case. Theassumption was made that the internal chamber conditions were constant throughoutthe test causing the background evaporation to be steady at the pre-test measured level.The net mass transfer results (total minus background) were constant to within less than5% variation when comparing the individual test runs.4.6 Surface Temperature ResultsThe surface temperature change with power input is shown in Figure 4.13. Thetemperature results were averaged over the three experimental sets; however, variationof the temperature readings at each power level was less than 5%.The heater temperature rise was measured by recording the electrical resistance before and after a test, then disconnecting the power supply and monitoring the electrical30-EvaporativeMassRatevs.PowerInput25-.20-I(I)150Ici) >0010-cci> w5IIIull 050100150200Powerlnput,P(mW)Figure4.12:MassEvaporationRatevs.PowerInput.1.5-Ct,5’TemperatureRisevs.PowerInputo I—>Ij1.O-I ba,00. E G) I.I- ci)I-.0.5z.II•I.1IIIIIIII050100150200PowerInput,P(mW)Figure4.13:Heatertemperaturerisevs.powerinput.Chapter 4. Experimental Results and Observations 95resistance until the heater attained thermal equilibrium with the chamber interior. Theelectrical resistance was then recorded again and the difference was taken as a measureof the heater temperature rise above the chamber temperature. Determining when theheater was in thermal equilibrium with the chamber was the most difficult aspect of thereading. However, the results were very consistent over the three experimental runs andit was estimated that the error in the resistance measurement was less than 5% of thereading and much lower at the higher power settings. The esitmate was calculated byrecording the uncertainty in the electrical resistance measurement while monitoring thereading to determine exactly when thermal equilibrium with the chamber was achieved.The one exception to the above error estimate was the temperature reading at the lowestpower input level, 4 mW. The change in electrical resistance was very small and therefore,determing when thermal equilibrium occurred was extremely important. Estimation ofthis error was difficult, however, errors as large as 25% could be possible. The relativechange in the electrical resistance reading for the other power settings was sufficientlylarge so that the determination of thermal equilibrium was not as crucial. These errorscombined with the slight scatter in the calibration data resulted in a temperature riseerror estimates of no greater than 10% of the calculated result and much less at thehigher power levels.The adsorbed thickness, mass evaporation and temperature rise measurements wererepeated on 2 additional experimental runs after the complete experimental set with resuits consistent with the ones shown, within the experimental accuracies. The extra runswere conducted mainly to check the adsorbed thickness results. Surface contaminationwould alter the adsorbed thickness readings dramatically, so the test surface was cleanedwith the test fluid and a lint free cloth before each of the additional runs. Refractiveindex readings of the surface were taken before both tests and the average reading was a20 A layer of Si02, compared with the average reading of 18 A before the tests. This wasChapter 4. Experimental Results and Observations 96an indication that practically no surface contamination was present during the additionaltests.Chapter 5Discussion of Results5.1 Experimental Profile ComparisonComparing experimental extended meniscii thin liquid film profiles is difficult due toof the variety of geometries and materials used in research of interline heat and masstransfer. While silicon is a common substrate among wetting film researchers because ofthe smoothness of the surface and high reflectivity, interline dimensions and experimentalliquids vary depending upon the individual experimental setup. One common factoramong researchers is the use of non-polar liquids leading to theoretical analyses whichuse only London van der Waals dispersion forces as the only significant contributionsto the intermolecular attraction between the liquid and substrate. One experiment thatprovides a useful comparison to the results presented here was that of Cook et al [30].Figure 5.1 shows the present results which indicate the effect of increased input poweron the profile shape. Figure 5.1 also shows the results of Cook, who had a similarexperimental setup but used decane as a working fluid. The effect of increasing theinput power is similar in both experiments. The meniscus length shortens and the slopebecomes steeper in each case. This indicates that an increased evaporation level createsforces within the meniscus, the result of which shortened the length.Further examination of Figure 5.1 indicates the possibility of comparing the currentexperimental film profiles by attempting to eliminate the effect of power input. Normalizing the profile length by dividing the relative position, (x — x0), by the individual971.0E45.0E3I. 03.5E4EffectsofIncreasingPowerInputonFilmProfiles•P=OmW•P=185mW-°—Cooketal,1981P=200mWCooketal,1981P=OmWE2.5E42.0E41.5E40.OEO02505007501000RelativeDistance,x-x0(x106m)Figure5.1:Inputpowereffectonliquidprofiles,[30].00Chapter 5. Discussion of Results 99meniscus length (reference position to the interline (x2 — x0)), the six individual profilesmay be compared. Figure 5.2 shows the normalized profiles plots. It appears that thenormalized profiles are very similar from the thickest portion, 30,000 A, down to approximately 5000 A. This indicates that the meniscus length in this part of the profile isdependent upon the power input almost exclusively. However, below 5000 A the profilesdo not appear similar. Below 5000 A, where the disjoining pressure forces become important, the expanded view profile plots presented in Figure 4.10 are normalized in thesame manner as previously described and shown in Figure 5.3. This Figure illustratesmore clearly the previous observation that the normalized profiles are not similar in thisregion. It can be clearly seen that the profile slopes down to 400 A are almost identicalbut the major differences occur in the region from the 400 A level to the interline. Thiscould indicate that the power input is not the most important factor in determining themeniscus shape in this thickness range. One additional consideration is the fact that thedistance from the edge of the beige fringe to the interline was the singularly most difficultmeasurement to obtain in the experiment. This may account for significant errors in thenormalized profile; a more precise experimental design is needed to accurately examinethis region.5.2 Unsteady BehaviourInitially at very high cooler background voltage settings or at high power settings,meniscus oscillations toward and away from the interline were noted. The oscillationswere comparable to a wave motion on a shoreline when viewed from above. This behaviour was noted in many of Wayner’s experimental works. [[30], [31], [39], [38]]. Waynernoted that the profile shape remained constant (determined by observation of the fringepattern) while the meniscus advanced towards and receded from the interline. These1.0E45.0E3I C3.5E4LiquidThicknessvs.NormalizedDistanceE2.5E40 1 x2.0E4Co Co 0) C 0 I— E U-1.5E4—D--—P=OmW—‘h’-—P=4mW—V-—P=23mW—0-—P=59mW—0——P=ll4mW&P=185mW0.OEO0.000.250.500.75NormalizedDistance,(x-xjl(x-xjFigure5.2:Normalizedliquidprofiles.250020000 0 ci).21500jE E1000I.I.35003000LiquidFilmThicknessvs.NormalizedDistanceL—C----P=0mWP=4mW—V--—P=23mW—0--—P=59mW—0—P=ll4mW&P=185mW500 0 0.500.600.700.800.90NormalizedDistance,(x-xj/(x-xj1.00Figure5.3:Normalizedliquidprofiles,disjoiningpressureregion.Chapter 5. Discussion of Results 102previous experiments used monochromatic light with a first dark fringe of approximately1000A thickness. The oscillations were evident by the position of the leading fringe. Theoscillations were noted but were never quantified in terms of frequency.With the current experimental setup using a fluorescent light ring as a source, avariation of colours beginning at approximately 400)1 thicknesses provided greater detail for the liquid profiles plus added insight into the behaviour of the meniscus duringoscillations. It was observed during oscillations that the leading edge colour would varybetween the colour extremes of tan and light blue over a period of time (plus intermediate colours between). At thicknesses greater that 2000)1 (light blue), no variation ofthe meniscus pattern was noticeable. Upon meniscus advancement towards the heater,the leading edge would appear light blue followed by the clear adsorbed region. During acycle of between 5-10 seconds, the light blue would change to blue, from blue to brown,from brown to tan followed by a return to light blue only. It appears that at high evaporation rates, a smooth supply of liquid into the evaporating zone is not possible so asmall dry-out and rewetting situation is created. Analysis of this periodic condition wasbeyond the scope of this investigation, other than to note the appearance and behaviour.While the meniscus could be approximated as two-dimensional for greater than 90%of width of the silicon substrate, the meniscus did wick further up the substrate, towardsthe heater, at the edges where the substrate joined the acrylic cover and was sealedwith silicone rubber. This appears to cause an interesting action at high evaporationrates. When oscillatory meniscus behaviour appeared at a high power inputs, wavestravelling parallel to the interline were observed. It appears that liquid was drawn intothe evaporating region from the side edges as well as from the liquid pooi below. Thiseffect was not apparent if no oscillations were present.Chapter 5. Discussion of Results 104one order of magnitude in area). Both the previous papers attribute the lenses to thepresence of microscopic dust particles. Their behaviour was similar to that noticed in thisexperiment in that some of the lenses slowly coalesced over time and propagated towardsthe heat source. Because the results indicate that larger lenses near the heater havemicroscopic particles visible while the bubbles between the fringes and the pool have noapparent particles, it is not certain that all the lenses (or bubbles) are necessarily createdby particles. The experiment was disassembled and cleaned on five separate occasions butthe bubbles returned in the same location every time evaporation began upon activationof the heater.One possibility for the bubble formation is the dissolution of air into the liquid FC72, since air is extremely soluble in FC-72, 48% air by volume at 25°C. The dissolvedair may begin to come out of solution as the meniscus begins to thin dramatically anddifferent forces are introduced locally into the meniscus. This would explain the constantpresence of the bubbles in the same location over a period of hours and days. One aspectwhich can be concluded not to cause the bubble formation is nucleation due to boiling.Reeber and Frieser [88] used a smooth silicon substrate with FC-72 as a cooling liquidin superheated plate tests with a superheat of 30°C without producing any nucleation.This superheat is at least 20 times greater than that of the highest substrate superheatproduced in this experiment, therefore no nucleation should appear.5.4 Adsorbed ThicknessIn an effort to test the proposed disjoining pressure relationship, Equation 2.19, themeasured adsorbed thickness was plotted versus the saturation pressure ratio (calculatedfrom measured temperatures), and is shown in Figure 5.5 along with the data of Derjaguinand Zorin [21] for comparison. Derjaguin and Zorin measured adsorption thicknessesE0100‘b x 07t0“.‘0 a) C 0 I—50.a) 0 C’) 025<0-150AdsorbedThicknessvs.SaturationPressureRatio•ExpenmentADerjaguinandZorin,1957:CCI4onglass•DerjaguinandZorin,1957:butylalcoholonglass-1250.9500.9600.9700.9800.990SaturationPressureRatio,P/P.,01.000Figure5.5:Comparisonofadsorbedthicknessvs.saturationpressureratio,[21].CChapter 5. Discussion of Results 106of various polar and non-polar liquids on superheated plates in a static non-evaporatingenvironment. The shape of the curve containing the present experimental data comparesfavourably with the non-polar data, CC14, for values of P/Ps near 1.0, although theabsolute levels are different most notably at the lower saturation pressure ratios. Thiscould be a function of the different material properties used in the two eperiments. Thepolar liquid, butyl alcohol is shown to indicate that the results for all polar liquidsintersected the saturation pressure ratio P/P8 = 1.0 axis while the non-polar fluidsbecame tangent to this axis, as did the present experimental results.Direct comparison of the different experimental results reported above can be achievedby rearranging Equation 2.19 to non-dimensionalize the adsorbed thickness by the remaining terms in the equation which account for the variations in substrates and liquidsamong the experiments. Equation 2.19 may be written as1•) ((Min — \ &r) ‘sjT,pz 5 1hDefining a reference thickness asM 52r6)Tp)Equation 2.19 may be written as1n(-)=(5.3)Evaluating the terms in hr for each particular fluid and substrate in Figure 5.5 (fromtheoretical bulk values), the non-dimensionalized thicknesses are replotted versus thesaturation pressure ratio in Figure 5.6. The results show very good agreement withthe data of Derjaguin and Zorin [21] for the non-polar liquids. This result indicatesthat the adsorbed thickness behaviour of different non-polar liquids is similar when nondimensionalized using Equation 5.3. This supports the use of Equation 2.19 for predictingChapter 5. Discussion of Results 106of various polar and non-polar liquids on superheated plates in a static non-evaporatingenvironment. The shape of the curve containing the present experimental data comparesfavourably with the non-polar data, CC14, for values of P/P8 near 1.0, although theabsolute levels are different most notably at the lower saturation pressure ratios. Thiscould be a function of the different material properties used in the two eperiments. Thepolar liquid, butyl alcohol is shown to indicate that the results for all polar liquidsintersected the saturation pressure ratio P/Ps 1.0 axis while the non-polar fluidsbecame tangent to this axis, as did the present experimental results.Direct comparison of the different experimental results reported above can be achievedby rearranging Equation 2.19 to non-dimensionalize the adsorbed thickness by the remaining terms in the equation which account for the variations in substrates and liquidsamong the experiments. Equation 2.19 may be written as((Min — 6ir) \j?Tpj 5 1Defining a reference thickness asM 52r6)Tp) (.)Equation 2.19 may be written asin () = (h)3 (5.3)Evaluating the terms in hr for each particular fluid and substrate in Figure 5.5 (fromtheoretical bulk values), the non-dimensionalized thicknesses are replotted versus thesaturation pressure ratio in Figure 5.6. The results show very good agreement withthe data of Derjaguin and Zorin [21] for the non-polar liquids. This result indicatesthat the adsorbed thickness behaviour of different non-polar liquids is similar when nondimensionalized using Equation 5.3. This supports the use of Equation 2.19 for predicting•50Non-dimensionalizedAdsorbedThicknessvs.SaturationPressureRatiE0•40ci;• (0) C C)•ExperimentADerjaguinandZorin,1957:CCI4onglass•30DerjaguinandZorin,1957:butylalcoholonglass0 a,20(U C 0 (0 C ci)•101 C 0 zI•I•00.9500.9600.9700.9800.9901.000SaturationPressureRatio,P/PsFigure5.6:Comparisonofnon-dimensionalizedadsorbedthicknessvs.saturationpressureratio,[211.Chapter 5. Discussion of Results 108adsorbed thicknesses given the saturation pressure ratio for any non-polar liquid on asmooth substrate in an evaporating as well as a non-evaporating environment. Nearthe saturation pressure ratio value of 1.0, the agreement among the data is the poorest,however, this constitutes the smallest experimentally measured temperature rise, and issubject to the most uncertainty.The proposed variation of disjoining pressure of 1/hg from Equation 2.19 can beexamined by plotting h0 versus —ln(P/P3)on a log-log scale. Assuming that the valueof hr is constant for each individual experiment, Equation 5.3 may be rewritten asP -h3— in (i-) = -. (5.4)Taking the logarithm of both sides results inlog (—in (i-)) = —3iog(h0)+ 3iog(—h). (5.5)This relationship was fit to the experimental data and is shown in Figure 5.7. Examination of Figure 5.7 shows good agreement with the proposed 1/hg relationship for themajority of the data. This results strongly supports the proposed disjoining pressurerelationship, Equation 2.19. For the data points on either end of the graph, the largestexperimental uncertainties exist creating the possibility of errors in these points.Equation 5.5 was also plotted on Figure 5.7 for the case of the theoretical Hamakerconstant A12 = 1.0x10’9Jused to calculate the non-dimensional thickness, hr. FromFigure 5.7 it is apparent that while the slope of the relationship is well approximatedby the data, the theoretical position of the line as calculated by the Hamaker constantequation, Equation 2.20, does not match the data well. This indicates that the Hamakerconstant equation as presented does not match the experimental results. This equationis an approximation to the more complex theory, as given for example by Wayner [25]. Itis not possible from these experimental results to determine if the difference between theE0 0 x0Co Co a) C C) I a) 0 Cl)AdsorbedThicknessvs.Log.SaturationPressureRatio1 102•ExperimentADerjaguinandZorin,1957:Cd4onglassExperimentalcurvefit,1/h03--Theorteticalcurvefit,1/h031! I. C cc’I101—1o-i0102-LogarithmSaturationPressureRatio,P/PsA101Figure5.7:Adsorbedthicknessvs.saturationpressureratio,Equation2.19.Chapter 5. Discussion of Results 110experimental data and the theoretical calculation is due to the equation approximation orto physical factors, such as surface contamination, surface roughness, or liquid impurities.Calculating the intercept of the line through the experimental points gives a value forA12 =- 4.3x10’9J,approximately four times larger than the theoretical value. However,using the theoretical value of the Hamaker constant results in adsorbed thickness valuesof less than 50% lower than the measured values. This suggests that unless detailedliquid property data is available, using Equation 2.20 to calculate the Hamaker constantwould be useful in practical engineering designs.Assuming that the 1/hg dependency for disjoining pressure is correct as suggestedby theory, the Hamaker constants given in Equation 2.19 were calculated using the adsorbed thickness and saturation pressure ratios which were measured experimentally. Thesaturation pressure ratios were determined from the measured vapour temperature andsubstrate surface temperature. In the adsorbed liquid region, the surface temperaturewas assumed equal to the liquid-vapour surface temperature. The pressures were calculated from the manufacturer’s data on vapour pressure. The vapour pressure relationshipis given byPv= 10(7.6042_1), (5.6)where P is given in torr. The calculated values of the Hamaker constant were subject tolarge errors primarily due to the ellipsometer thickness measurements which are raised tothe third power in the calculation. The best estimate of the random error which occurredin each ellipsometer measurement was ±5)1 as previously discussed in Chapter 4. Errorsdue to temperature uncertainty were much less during each Hamaker constant calculationas the estimate of temperature uncertainty was less than 10% , and significantly lessthan this at the higher power inputs. Accounting for these errors, the calculated valuesof the Hamaker constant are given in Table 5.3. The calculation procedure is detailedChapter 5. Discussion of Results 111Hamaker Constant CalculationsRun h0 P/Ps A12 (data) A12 (err.)10_la m xlO’9J %2 110 0.9994 -1.85 153 100 0.9983 -3.84 134 91 0.9965 -6.05 185 69 0.9960 -3.04 256 66 0.9935 -4.25 257 58 0.9903 -4.37 298 52 0.9860 -4.54 339 50 0.9819 -5.22 3310 45 0.9764 -4.98 3611 40 0.9689 -4.64 4312 42 0.9623 -6.53 4013 42 0.9557 -7.70 4114 40 0.9499 -7.56 44Avg.[_______-4.9Table 5.3: Experimentally calculated Hamaker constants.in Appendix F. Examining the values in Table 5.3 it is apparent that when the errorestimates are taken into acount, a strong case can be presented for the existence ofa constant value of A12. This was confirmed by the results presented graphically inFigure 5.7 where the data fit to the experimental 1/3 slope does not show any obvioustrends. Although averaging over a series of experiments aids in eliminating the randomerror, significant uncertainty in the ellipsometer measurements makes any distinct patternimpossible to establish. The one apparent fact is that the experimentally calculated valueof A12 is approximately 5X larger than the theoretical value. Given the fact that thetheoretical value was calculated from Equation 2.20, which is an approximation to themore detailed theoretical calculations, and surface contaminants and impurities couldChapter 5. Discussion of Results 112affect the experimental results, the result indicates that the given equation is adequatein providing a value to use in evaporating situations.5.5 Hydrodynamic ModelDetermination of the effectiveness of the hydrodynamic model, Equation 2.32 requiresa mathematical description of the liquid film thickness profiles which must be differentiated four times to obtain the evaporative mass flux from the liquid-vapour surface. Inthis experimental setup, there was an insufficient number of data points per profile forthe use of a spline routine. Therefore a polynomial curve-fit of order 6 was chosen tofit the entire experimental profile. The polynomial approximation to the data pointswas very good except in regions close to the interline region where the resulting curveshowed a large gradient. To circumvent this problem, the profile near the interline, in thedisjoining pressure dominated region, was modelled with a separate curve. The resultsof the polynomial curve fit are shown in Figure 5.8.Using the coefficients from the polynomial curve fit in an equation to give the filmthickness as a function of the relative position, Equations 2.31 and 2.32 may be calculatedto give the mass flow rate per unit length through the meniscus, F, and the evaporativemass flux from the liquid-vapour surface, Ihe. An indication of the forces which causefluid motion in the meniscus is shown by the curvature, K, of the liquid film surface. Thecurvature, approximated by Equation 2.29, is shown in Figure 5.9 for the polynomialcurve fit. The Figure shows that the curvature generally increases from the liquid poolregion until a maximum is reached at approximately the same x location from the 3m thickness position, x0. One curve, P=23 mW does not peak as much as expectedin accordance with the other results and is a more even curvature profile. The reasonfor this individual behaviour is not certain. From Figure 5.9 the curve fit limitations at3.Oxl__2.5x104E0 0 !-2.0x104Co C4.1.5x10C) I E Lt1.0x104 5.0x1o0.OxlO°1200ThinFilmProfiles(VariousPowerInputs)—0----—0--—--P=OmWP=4mWP=23mWP=5OmWP=114mWP=185mW02004006008001000RelativeDistance,x-x0(x16m)Figure5.8:Polynomialcurvesfittofullprofiles.I.Chapter 5. Discussion of Results 114either end of the data are immediately apparent. Without the ability to clamp the curveends during the regression analysis, the end information was not sufficient to properlymodel the curves at these locations. This fact becomes more prominent as the profileequation is differentiated.The calculated curvature of the above profiles are replotted in Figure 5.10 as a functionof film thickness. The results show a definite trend of the maximum curvature to belocated at a thinner portion of the film as the input power increased. Also, the value ofthe maximum curvature increased, indicating an greater evaporation rate, for increasedpower input. Due to the need to differentiate the polynomial curves twice to obtainthe curvature, the effect of having a different curve fit to the data was examined usingpolynomial curves of order 4 through 9 and power law curve fits. The size and locationof the maximum curvature did not vary appreciably with any of these other curve fits.The evaporative mass flux described in Equation 2.32 was calculated using the polynomial curve fit and is shown in Figure 5.11. The mass flux is plotted against the liquidfilm thickness. The results show that very little mass flux occurs in the disjoining pressure dominated region, less than 1 um film thickness. The mass flux increased withincreased power input as is expected. The result for P=23 mW follows the same resultas the curvature graph with respect to its totally independent behaviour from the otherexperimental results. These mass flux results indicate that the majority of the evaporating mass is occurring in liquid film thicknesses on the order of microns, outside the zoneof influence of the disjoining pressure.A similar procedure to that previously described for calculating the evaporative massflux for the full profile was used to calculate the flux for the disjoining pressure dominatedregion. One major difference was in the type of profile required to approximate thedata in this region. Polynomial curve fits were not very accurate representations of theexperimental data due to the sharp relative curvature in the data as the profile approaches100E a) C.)600700800ProfileCurvatures—0———P=OmW±P=4mWVP=23mW—0--—P=59mW—0——P=ll4mWP=185mW90 80 70 60 50 40 30 20 10 00100200300400500RelativeDistance,x-x0(x106m)Figure5.9:Profilecurvaturesatrelativepositions.I. CRLCurvaturevs.FilmThickness100 90 80 70 60 50Ezz-0-P=OmWP=4mWP=23mWP=59mWP=114mWP=185mW40 30 20 05.0x1031.0x1041.5x1042.0x104FilmThickness,h(x1010m)2.5x1043.OxllFigure5.10:Profilecurvaturesasafunctionoffilmthickness.0.0080 E0.006U Cl)Cl)Ca (1)0.004> 00.0020.000EvaporativeMassFluxvs.FilmThickness—s-—0-P=0mWP=4mWP=23mWP=59mWP=114mWP=185mWFilmThickness,h(m)2.OE-62.5E-63.OE-6Figure5.11:Evaporativemassfluxforthefullprofile.I.Chapter 5. Discussion of Results 118the interline. A more accurate representation was a power law curve fit of the formh = axb. (5.7)The fitted curves are shown in Figure 5.12. The Figure shows the effect of increasedinput power on the liquid film profile. The major differences between the power settingsoccur in the curvature of the profile below 1000 A.Using the power law profiles in the disjoining pressure region, the disjoining pressureversus the relative distance location is calculated with Equation 2.19. The resultingdisjoining pressure curves are shown in Figure 5.13. The curves are plotted with thereference position of x1 located at the liquid film thickness of 3500 A, estimated to bethe beginning of the disjoining pressure dominated region from the full profile plots.Examining Figure 5.13, this assumption appears valid as the calculated value of thedisjoining pressure is below 1 N/rn2 at this thickness for all input power levels. Therelative size of each disjoining pressure region along with the steepness of each profile inthis region are evident from the graph.Using the second derivative of the disjoining pressure equation and the curvature calculation in Equation 2.32, the evaporative mass flux in the disjoining pressure controlledregion are shown in Figure 5.14. It is evident from the Figure that the evaporativemass flux increased with increasing input power with a large increase occurring betweenP=185 mW and the other power settings. However, the absolute value of the mass fluxis negligible compared with the flux occurring in the film thicknesses greater than 1 turn,as shown in Figure 5.11. Therefore, it appears that the disjoining pressure caused liquidflow in this region but the level of evaporation is very small compared to the region wherethe changes in curvature are responsible for fluid motion, at least 2 orders of magnitudein all cases.cf.Figure5.12:Curvefittoprofilesinthedisjoiningpressureregion.z 0310Co Co 0100) C C 0 :101tn 0 C1oDisjoiningPressurevs.RelativeDistance10010.1—ci----P=4mW—b--—P=23mW—V-—P=59mWD—P=ll4mW—0—P=185mW0100200300RelativeDistance,x-x0(x106m)400500600Figure5.13:Disjoining pressureprofiles.EvaporativeMassFluxvs.FilmThickness1.OE-50 >t7.5E-6LL.Co Co Cu5.OE-60 0. > W2.5E-6O.OEO—0—P=4mW—s--—P=23mW—V-—P=59mWP=ll4mWP=185mW5.OE-81.OE-71.5E-72.OE-72.5E-73.OE-73.5E-7FilmThickness,h(m)Figure5.14:Evaporativemassfluxinthedisjoiningpressureregion.Chapter 5. Discussion of Results 1225.6 Heat and Mass TransferThe evaporative mass fluxes presented above allow the calculation of the interfacialheat transfer coefficient due to evaporation. Using Equation 2.33 the heat transfer coefficients are shown in Figure 5.15 for the full meniscus profiles. It is apparent fromFigure 5.15 that there are very low values of he in film thicknesses below 1.5 ,um. For theinput power levels between 23 and 185 mW, the heat transfer coefficients reach a maximum of between 500 and 1000 W/m2K at a thickness level of 3 /im. The heat transfercoefficients for the P=4 mW case indicate a much greater value. Because this result doesnot follow the others, it appears that the temperature difference measurement of 0.015K for this case could be influencing the result significantly. However, the magnitude ofpossible error in the temperature measurement does not account for the increase in theheat transfer coefficient shown. Therefore, for very low power inputs, significant heattransfer coefficients are possible below 3 sum. For the input power settings above P=23mW, possible large heat transfer coefficients may exist but at film thicknesses greaterthan 3 1um.From Figure 5.15, the heat transfer coefficients in the disjoining pressure dominatedregion are shown to be very small. Figure 5.16 shows the calculated values of he for thefilm thicknesses below 3500 A. Again, the values for P=4 mW are the least accuratedue to the small temperature difference to be measured at this level. The heat transfercoefficients in this area are very low, below 1.0 W/m2Kfor the power levels greater thanP=4 mW. The values are very consistent throughout the range, without much variation.From these results, the available cooling due to evaporation in this region is negligiblecompared with that which was shown for film thicknesses above 1.5 ,um.Using the calculated evaporative mass fluxes for the full profiles, integrating from the-t C (I) 0 ci) 0 0 ci) Cl) C I- Cu zHeatTransferCoefficientvs.FilmThickness500040003000_—__P=4mW—2S-—P=23mW—v---P=59mW—c——P=ll4mW—0—-P=185mW10002.OE-62.5E-63.OE-6FilmThickness,h(m)Figure5.15:Heattransfercoefficientforthefullprofile.0HeatTransferCoefficientsvs.FilmThickness4.—C—P=4mW—--—P=23mW—V-—P=59mW—0--—P=ll4mW—0——P=185mWI I1.5 1.0 0.5 5.OE-81.OE-71.5E-72.OE-72.5E-73.OE-7FilmThickness,h(m)Figure5.16:Heattransfercoefficientinthedisjoiningpressureregion.Chapter 5. Discussion of Results 125[__Mass Flow Rate ComparisonsPower(measured) (calculated)mW kg/s x108 kg/s x1080 9.6 0.144 11 0.1323 14 0.02159 19 0.35114 27 0.60185 39 2.6Table 5.4: Experimental vs. calculated mass flow rates.interline to the reference position of 3 sum, Equation 2.31 (multiplied by the experimental width) gives the total mass evaporation in each input power setting. These valueswere compared with the experimentally measured values and are shown Table 5.4. It isimmediately obvious from the results presented that the calculated mass flow rate intothe region of the experiment is at least 1 order of magnitude less than the measuredvalue. This indicates that another substantial driving force might exist in this area toaccount for the flow. However, examining the presented results closely indicates that theevaporative mass flux is increasing substantially in the region of the data that is on theouter limit of the current measurement capabilities. This region exists at approximatelya thickness of between 2-3 ,um. Therefore, when comparing the calculated mass flowswith the actual measured values, it appears that the majority of the evaporative massflux must occur outside the present measurable region to account for this substantialdifferenceTheoretical prediction of the interfacial mass transfer involves the use of Equation 2.37.Chapter 5. Discussion of Results 126Corrections to the pressure and temperature in this mass transfer model for the specific case of thin evaporating liquid films are given in Equations 2.38 and 2.39. Usingthe evaporative mass flux calculated with Equation 2.32, the evaporation coefficients inEquation 2.37 may be calculated and compared with the ideal theoretical value of 1.0.The calculated values of the evaporation coefficient, a, are shown in Figure 5.17. Itis apparent that the value of e is not constant over the range of the meniscus profile.In the region where the evaporative mass flux was significant, greater than 1.5 1um, theevaporation coefficient ranged between 0.001 and 0.0025 for input power levels aboveP=4 mW. Again, the result for P=4 mW is significantly different from the other results,reflecting the previous differences in all calculations involving the measured temperaturedifference. These values of o differ from the ideal value of 1.0 by almost 3 orders ofmagnitude but results similar to these have been reported [46]. The extremely smallvalues of cr shown in liquid thicknesses below 1.5 um are indicative of the very smallamounts of evaporation occurring in this region. A more detailed graph of this regionis shown in Figure 5.18. The values of the evaporation coefficient are reasonably constant in the disjoining pressure dominated region, but the overall values are 3 orders ofmagnitude less than those occurring in the thicker film region dominated by capillaryforces (greater than 1.5 tm), and 6 orders of magnitude less than the theoretical values.The most obvious reason for this large discrepancy is in the use of Equation 2.37. Thederivation was made for interfacial pressure and temperature differences on the order ofmolecular distances, whereas the current design attempts to use a form of this analysison a macroscopic scale. The driving temperature and pressure differences in the experiment are those required for a practical design and the results reflect this desire. Theevaporation coefficient when used in the present context must account for macroscopiceffects such as non-condensables and temperature and pressure gradients, the effects ofwhich are shown to be substantial and must be accounted for.7.5E-3b C 0 a,5.OE-3W2.5E-3CEvaporationCoefficientvs.FilmThickness—C—P=4mW—s-—P=23mW—V-—P=59mWDP=ll4mW0P=185mW1.4E-91.5E-6FilmThickness,h(m)2.OE-62.5E-63.OE-6-4Figure5.17:Evaporationcoefficientforthefullprofile.iFEvaporationCoefficientvs.FilmThickness4.5E-64.OE-63.5E-63.OE-60 C2.5E-6Cu 02.OE-6> Ui1.5E-61.OE-65.OE-71.6E-13 O.OEO—0——P=4mW—Lx-—P=23mW—V--P=59mW—0—-P=ll4mW—0——P=185mW5.OE-81.OE-71.5E-72.OE-72.5E-73.OE-7FilmThickness,h(m)Figure5.18:Evaporationcoefficientinthedisjoiningpressureregion.00Chapter 5. Discussion of Results 129There is a basis for expecting evaporation coefficients of this low a magnitude from the3M corporation product manual [2]. When air is present, the FC-72 vapour is heavierthan air and does not diffuse easily through air. Using the interfacial mass transferequation, Equation 2.37, with the measured evaporation loss at room temperature [2], acalculated evaporation coefficient of approximately 1.0x105 is obtained. This indicatesan extremely large influence on the evaporation rate due to the presence of air, and thisis verified by the present experimental results.5.7 Experimental AccuracyThe previously listed results were all subject to the limitations of the current experimental setup and procedure. The most obvious type of error which occurred wasrandom error (operator error or unanticipated exterior factors) and was easily controlledby repeating the experiments numerous times and evaluating the repeatability of theresults and to average the results in order to eliminate the random errors. As mentioned,all the results were averaged over a minimum of six full experimental tests plus somepartial tests which did not include thin film profile photographs. All the results obtainedwere consistent and repeatable. This was assumed to effectively reduce the random errorcomponent to a negligible value.The other type of error was the accuracy limitations of the experimental equipment.The most significant error was in the profile region greater than 1tm where the interference colour bands in the experimental photographs were not precisely defined by amathematical expression. Due to this fact, the calculation of evaporative mass flows inthis region using the change in curvature as representative of the flow rate had the possibility of large errors after having the resulting least squares curve fit differentiated fourtimes. The change in curvature among the profiles shows the effect of increased powerChapter 5. Discussion of Results 130input but definite conclusions about the actual numerical comparison of mass flow ratescould not be drawn.The remaining experimental measurements were all determined to be very accurate.The mass flow rate was measured with a maximum error of less than 3%. The surfacetemperature rise had a maximum error of less than 5%. The only other measured quantitywas the adsorbed thickness. The ellipsometer measurements contained two sources oferror, one of which was random and the other systematic. The random error involvedthe manual setting of the analyzer and compensator angles. Tests on this repeatabilitywith samples of known thickness indicated an error of no more than 2% and usuallylower. The other error involved the determination of the substrate index of refractionwhich would affect the absolute level of the effipsometer reading. Cleaning of the siliconwafer was done before each test, however, after cleaning there was some variation inthe refractive index deterimination, sufficient to cause an uncertainty of 18A ± 5)1 inthe Si02 thickness measurement. This error was systematic because the same refractiveindex measurement was used for each test case, thus resulting in no effect on the adsorbedthickness readings relative to each other. Therefore, the relative temperature effects onA12 contain maximum possible errors of 10%.5.8 Summary of Relevant Experimental ObservationsDuring the course of this research, various interesting and not always explainableobservations were made that have already been discussed and will be summarized here.1. The most obvious experimental phenomena was the continued presence of the bubbles in the liquid between the extended meniscus and the liquid pool. Unlikeprevious researchers, the majority of the bubbles did not appear to be caused bydust particles and did not behave as reported in the literature. The opinion ofChapter 5. Discussion of Results 131the author was that they are dissolved air bubbles which cannot remain dissolvedas the film thins to less than approximately 1tm due to the changes in the filmproperties.2. Another interesting result which has been reported in the literature numerous timesbut has yet to be satisfactorily recorded or analytically explained is the usteadyoscillatory behaviour of the interline as the evaporation rate is increased to a significant level. It appears as a dry-out and rewetting cycle which has a frequencyunrelated to any time dependent process measured in the experiment.These two observations require investigation separate from the current work becauseof the potential impact on the use of thin liquid film evaporative cooling.Chapter 6ConclusionsWith reference to the original objectives of an increased understanding of the heatand mass transfer processes occurring in the contact line region, the following conclusionswere drawn from this experimental work.1. The concept of a disjoining pressure in thin liquid films (typically less than 1 micron), originally developed to describe equilibrium thin films (non-evaporating),appears to be applicable to non-equilibrium films also, based upon experimentalresults. The support for this conclusion comes from:(a) Measured values of adsorbed liquid film thickness for an evaporating, nonpolar, extended meniscus on a silicon substrate compared favourably with theresults of data for a non-evaporating experiment, Figure 5.6. For non-polarliquids on a smooth substrate under either static or evaporating conditions,when the adsorbed thickness is non-dimensionalized by the reference thicknessgiven by Equation 5.1,M (61)ri67r)7zTp)the non-dimensional thickness becomes a unique function of the saturationpressure ratio.132Chapter 6. Conclusions 133(b) The inverse cubic relationship, proposed for equilibrium thin film conditions,between the disjoining pressure and the adsorbed thickness Equation 2.19,flTp1 P A12 1Mln-=---. (6.2)was shown (Figure 5.7) to also hold for the case of a non-polar liquid on asmooth substrate in an evaporating environment.2. Despite the strong support for using the disjoining pressure concept in evaporatingthin films, the Hamaker constant, A12, inferred from averaging experimental measurements of a non-polar evaporating liquid on a silicon substrate was 4-5 timeshigher than predicted by the theoretical equation of Israelachvili [261, Equation 2.20A12=(JA- /- ./Ai), (6.3)However, the adsorbed thickness predicted by this equation was within 50% of themeasured value and therefore could be useful in an engineering design calculation.More detailed property data are needed for evaluating the Hamaker constant usingmore complex Hamaker constant equations.3. While the overall concept of disjoining pressure seems to be valid as noted in conclusion 1, above, hydrodynamic and heat transfer models using disjoining pressure topredict the mass flow in a thin film as well as evaporating mass flux from the liquidsurface and the total heat transfer rate require further study. No firm conclusionscan be drawn regarding the hydrodynamic and heat transfer model validity owingto the poor comparison between predicted and measured values, as cited below:(a) The hydrodynamic model based on thin film lubrication theory, Equation 2.32dr o / 3d4h 2dhd3h\ A12 1 ld2h 1(6.4)Chapter 6. Conclusions 134which uses the disjoining pressure and changes in curvature as driving forcesfor liquid into the evaporating region, predicts overall evaporating mass flowrates at least one order of magnitude less than measured results.(b) The evaporation coefficient in interfacial mass transfer theory, as presented inEquation 2.372Ue’ F7iPiv Pvm=( )V , (6.5)2Ue 27rR. TLv Vwas calculated to be approximately 1.0x106 (see Figure 5.15) in the disjoiningpressure dominated region of an evaporating non-polar liquid when the reference conditions for mass transfer are taken to be the macroscopic temperatureand pressure. This is significantly less than previously published values.Chapter 7Future WorkThe current experimental design has been shown to be very useful in examining thedisjoining pressure controlled region of a thin film. However, additional experimentalinformation that would lead to analytical predictions which verify overall measurementsof mass evaporation and heat transfer in this region requires some specialized adjustmentsto the present design.1. First, and most important is the heater design. The concept is quite viable but alower electrical resistance would be beneficial so an electrical controller, such as aconstant temperature hot wire anemometer, could be used. Also, a thinner heater,in the range of 20 — 30gm, is required to restrict the heater surface to the disjoining pressure region exclusively plus allow some freedom of heater movement withinthis region. These two requirements indicate a need for much higher boron deposition rates or complimentary ion implantation. If this technology is insufficient,a different method such as etching and vapour depositing a pure metal would benecessary. Ideally, the substrate would also be of a higher thermal insulation thansilicon to restrict the heat flow to the heater-liquid interface. One additional criterion would be the addition of more robust electrical lead connections to the heaterto prevent resistance change with movement. A good path for future investigatorswould be convenient access to a microelectronic fabrication location and also accessto a precision machining centre. These two are required to easily construct a rangeof devices based on ideas drawn from the current work.135Chapter 7. Future Work 1362. One item that could be incorporated into the present design that would significantlyenhance these results would be the incorporation of a more detailed profiling mechanism which incorporated all the features which create a very detailed profile. First,remove the scanning stage from inside the chamber and fix the chamber positionwhile moving the ellipsometer past the meniscus region. This will eliminate all thevibration which disturbs the meniscus profile in the current design. Second, have atwo light system in the interferometer, one a fluorescent light for accurate fringe details below 3000 A, and another monochromatic light for exact fringe details as themeniscus thickens. Third, incorporate a more powerful microscope and a digitizedimage similar to that described in Wayner’s works (i. e. [38]).3. Elimination of the air inside the experimental chamber and repeating the experimental conditions described here would satisfy the question as to the origin of thebubbles.4. Another modification to the present experimental design which would be extremelyuseful would be the automating of the ellipsometer so that measurements could betaken on the order of 1 second while scanning the meniscus.5. One last recommendation for further work in this area is the combining of the resistance of the heater with that of the probe so the change in the respective resistancesis measured immediately, automating the temperature difference measurement.Appendix APhysical Description of the Stokes ParametersDescribing the X—Y components of a transverse electric wave propogating in the Z—direction of an X—Y—Z co-ordinate system as27rz=E0cos(wt—-----+&)= cos(wt__4+6) (A.1)Choosing an arbitrary transverse plane as z = 0, the components become= E0cos(wt + o)= cos(wt+6) (A.2)Suppressing the time variance and representing the electric vector in phasor notation,(the Jones vector)E0e6= (A.3)Examining the Stokes’ parameter definitions to determine the physical meaning of thedefinitions, the four parameters are— i’2 t’200—c1,0 + rfy0S1 = E0—ES2 = 2EXOEYO cos(6—S3 = (A.4)137Appendix A. Physical Description of the Stokes Parameters 138A.1 The First Stokes’ Parameter, S0The intensity of an electromagnetic wave is proportional to the amplitude of the wave.I—Se = 1E02= + EYOI2 (A.5)Therefore, S0 is the total intensity of the wave.A.2 The Second Stokes’ Parameter, S1The intensity of a component is also proportional to the amplitude of the componentwave.S = EXO — IE02 (A.6)Therefore, S1 is the X—axis intensity minus the Y—axis intensity.A.3 The Third Stokes’ Parameter, S2The phasor components for an axis notation of —r from the previous X—Y positionis given by using an axis rotation is given by using an axis rotation matrix.E_114,,.= =R(—1/47r)E—cos(—1/4ir) sin(—1/47r)—sin(—1/4ir) cos(—1/47r) E1 E-E2 E+EAppendix A. Physical Description of the Stokes Parameters 139The intensity of a component beam given in phasor notation is I = IE012 = E*E. Usingthis defintion for the rotated axes,11/41r— 1—1/4r = + Ey)*(Ex + E) — — Ey)*(Ex — E)= + +E0e)— —EyoeiY)(Exoe2ö—E0e6u)= ExoEyoe_öz +E0e6—6w)= 2EXOEYO cos(6 — c5) (A.8)Therefore, S2 is the differecne between the 1/4ir — axis intensity (measusred counterclockwise from X—axis) and the —1/4ir — axis intensity.A.4 The Fourth Stokes’ Parameter, S3The phasor components for left and right circular polarizations involves transformingthe component phasors asE1 1 1 iEi,r = —Er 2 —i E—1 E+iE2 E,—iEThe intensity of a component beam given in phasor notation is I = 1E02 = E*E. Usingthis definition for the left and right polarizations,Jr — Ii (E — iEy)*(Ex — iE) — (E + iEy)*(Ex + iE)= (Exoe6 +iEy0eiöv)(Eiô — iE0e)—(Exoe6z— + iE0e”)Appendix A. Physical Description of the Stokes Parameters 140=iEzoE0e—iE0e6r)= 2E0Esin(6—(A.10)Therefore, S3 is the difference between right circular polarization intensity and left circular polarization intensity.Appendix BStokes Parameter Description Using Ellipse ParametersThe concise Jones vector representation of a polarized monochromatic wave is givenbyE E e’E== so (B.1)In general, the above Cartesian components describe an effipse in the transverse plane.A more convenient elliptical description is show in Figure 3.3. The unknowns of azimuth,8, and ratio of minor to major axis, tan el, are related to the Cartesian components byAzzam and Bashara [56].cos 8 cos e — i sin 0 sin eE= —Ae° (B.2)E sin8cose+icos8sin8To relate the ellipse parameters of 8 and e to the Stokes parameters, we start with theStokes parameter definitions in Cartesian terms,S0 =S =S2 = 2EZOEO cos(6—ö)83 = 2E500sin(—6) (B.3)Using the expanded forms of the Stokes parameters given in Appendix A, the followingtransformations are shown.S0 = E0 + E0 = EEX +141Appendix B. Stokes Parameter Description Using Ellipse Parameters 142= [Ae°(cos0cose — isinosine)1* [Ae(cos8cosE — isin0sinf)]+ [AeiO (sin 0 cos e + i cos 0 sin f)] [Ae° (sin 8 cos + i cos 0 sin )]= [Ae_i0(cos8cosf + isin0sinf)] {Ae°(cos0cosf — isin0sinf)]+ {Ae° (sin 8 cos — i cos 8 sin f)] [Ae° (sin 8 cos + i cos 0 sin e)]= A2(cos2 0 cos2 + sin2 8 sin2 €) + A2(sin2 0 cos2 + cos2 0 sin2 e)= A2 (B.4)== [Ae2°(cos0cos — isin8sine)]* [Ae6(cos0cosf — isinosine)j— [AeiO(sin 8 cos € + i cos 8 sin )]* [Ae° (sin 0 cos € + i cos 0 sin= [Ae8(cos & cos € + i sin 0 sin €)] [AeiO(cos 8 cos € — i sin 8 sin— [Ae_i0 (sin 0 cos € — i cos 0 sin €)] {Ae20 (sin 0 cos € + i cos 8 sin= A2(cos2 8 cos2 e + sin2 0 sin2 €) — A2(sin2 0 cos2 € + cos2 0 sin2 €)= A2cos2€cos20 (B.5)82 = 2E0Ecos(b—&)(Ex+Ey)*(Ex+Ey)_ (Ex_Ey)*(Ex_Ey)= [Aez0[cosocos€+sinecos€+i(cososin€_sinosin€)]j= x {Ae° [cos0cos€ + sin8cos€ — i(cos0sin€ — sin0sine)]]= — [Ae8 [cos0cos€ — sin0sin€ — i(cos8sin€ +sinosin€)]j= x [Ae [cos0cos€ — sin0sin€ + i(cos0sin€ + sinosin€)J]= 2A {cos6cos2€sin8 _cos0sin2€sin8]Appendix B. Stokes Parameter Description Using Ellipse Parameters 143= A2 cos 2e sin 20 (B.6)S3 ==— iEy)*(Ex— iE)—+ iEy)*(Ex + iE)= [Ae [cos 0 cos e + cos 0 sin — i(sin 0 sin + sin 0 cos f)j]= x [Ae[cos0cose+cos0sinf+i(sin0sinE+sin0cosE)]]= — [Ae° [cosOcosf — cos0sin€+i(sin0cos€ — sin0sine)]]= x [Ae[cos0cosE — cos0sinf — i(sin0cosE — sin0sine)]]= 2A {cos20cosfsinf+sin20sinfcose]= A2sin2f (B.7)Therefore, the Stokes parameters in terms of the elliptical parameters A, 0, E areS0=A2S1 = A2cos2ecos20S2 = A2cos2Esin2OS3 = A2sin2E (B.8)Appendix CComplex Polarization Ratio DeterminationThe Cartesian X and Y components written in phasor notation may be related toellipse description parameters 6, (azimuth), tan II, (ratio minor to major axis), A, (overallamplitude), q, (overall phase), by Hauge et al. [55] asE EE= X = 0eE0e6cos 6 cos — i sinS sin= Ae’ (C.1)sin 6 cos + i cos 6 sin eThe complex polarization ratio defined by Azzam and Bashara [56] is(C.2)Equating equations C.1 and C.2,sin6cos+isinEcos6tanoe = (C.3)cos S sin E — i sin E sin 0Expanding both sides of equation C.3sin 6 cos 6 cos2 e — sin2 e cos 0 sin 6tanc(cos-f-zsin6) =. 2 2cos26cose+sin csm 0cos2 6 sin c cos + sin2 0 cos e sin c+1 2 2 (C.4)cos0cosf+sln sin 6Therefore, equating the real and imaginary components of equation C.4sinS cos 0 cos2 — sin2 E cos 6 sin 6tanocos6 = 2 2 (C.5)cos26cosf+sln sin 6c0sinfcosf+sinOcoscsinftancsin6 = 2 2 (C.6)cos2Ocosf+sln sin 6144Appendix C. Complex Polarization Ratio Determination 145Solving equation C.6 for 6,cos2 0 sin cos + sin2 0 cos e sin etanö=sin0cos0cose— in €cos0sin0— cosfsine— sin 0 cos 0(cos2 e — sin2 )— 1/2 sin 2E— 1/2sin20cos2f= tan2f (C.7)sin 20Solving equation C.6 for a,tan2 a = tan2 a cos2 6 + tan2 a sin2 62 •2 . 2— sm0cosOcos e — sin fcos0sin0— cos29cos2e+sinesincos0sinfcosf+sincosesine 2cos20cosf+sln sin 0— cos2 sin2 0 + cos2 Osin2 C 8— cos20cos2f+sinesinBut,1 — tan2 acos2a =1 + tan2 a= cos2 ecos2 0 + sin2 Esin2 0— cos2 sin2 0— cos2 Osin2= (cos2 0 — sin2 0)(cos2 e — sin2 f)= cos29cos2f (C.9)Summarizing, the complex polarization parameters aretan2ftanö= sin 20(C.1o)cos2a = cos20cos2f (C.11)Appendix DDescription of Light Intensity at the PhotodiodeIn order to calculate the change in polarization state due to the sample, the Stokesparameters before and after the sample must be known. Before the sample, the light beamis linearly polarized without any other devices present to affect the polarization. However,after the sample the Stokes parmeters must be calculated from intensity measurementsat the detector. Converting the intensity measurements to Stokes parameters is notsufficient becasuse the Stokes parameters at the detector must be related to the Stokesparameters immediately following the sample. Using Mueller matrices to describe theeffects of the compensator and analyzer (see Hauge [68] and Clarke and Granger [54]) theStokes vectors may be related. The analyzer is assumed to be perfect, (high extinctionratio) while the compensator cannot be quantified as easily. Providing the exact ratio ofthe fast axis to slow axis of the compensator may be measured, a complex representationof the compensator may be given asPc = tan = Tceöc (D.1)(for a prefect quarter-wave retarder T = 1, = 45°, ô = 900)146Appendix D. Description of Light Intensity at the Photodiode 147Using the Mueller matrices, the Stokes vectors are related byS1 1100 1 0 0 0S— 1 1 1 0 0 0 cos2(A—C) sin2(A—C) 02 0 0 0 0 0 —sin2(A—C) cos2(A—C) 0S!3 0000 0 0 0STOKES CONVERSION OF STOKESANALYZERDETECTOR TO ANALYZER F.O.R.1 cos2b 0 0cos2& 1 0 0x0 0 sin 2’’ cos t5 sin 2& sin 60 0 — sin 2’z/’ sin sin 2’ cosRETARDER1 0 0 0 So0 cos2C sin2C 0 Sx (D.2)0 —sin2C cos2C 0 S20 0 0 1 S3CONVERSION OF STOKES STOKESTO ANALYZER F.O.R. SAMPLE EXITTherefore, the intensity at the detector is given by equation 3.112S = S0 [cos 21’ cos(2C — 2A) + 1]+ S1 [cos 2C cos(2C — 2A) + cos 2G cos+ sin 2C sin(2C — 2A) sin 2’1’ cos+ S2 [sin 2C cos(2C — 2A) + cos 2’ sin 2C— cos 2C sin(2C — 2A) sin 2b cosAppendix D. Description of Light Intensity at the Photodiode 148+ S3 [— sin 2’& sin 5 sin(2C — 2A)]Appendix EDetermination of Compensator ImperfectionsThe compensator imperfectins written in the ratio of fast axis to slow axis,Pc = tan bce_i6 (E.1)must be determined to accurately calculate the Stokes parameters.If the polarizer axis is set along the X—axis (P = 00), the Stokes vector before thesample is given by1 1 0 0 S”1 1 1 0 0 sr=- (E.2)s’ 2 0 0 0 0 s”S!3’ 0 0 0 0 Si”STOKES STOKES ©POLARIZERBEFORE SAMPLE LASERTherefore, S’ = 0, S’ = 0, and S = S’1’ in this configuration.From equation 3.11, the intensity readings for the three compensator—analyzer combinations (C,A) = (00,00), (450,00), (900,00) are21(0,0) = So[cos2& + 1] +S1[cos2’& + 1] (E.3)21(45, 0) = S0 + S1 [sin 2’b cos ‘5] + S2 [cos 2’b] + S3 [— sin 2’/’ sin &J (E.4)21(90,0) = S0[— cos2& + 1] + S1[1 — cos2’&j (E.5)149Appendix E. Determination of Compensator Imperfections 150Solving for ?,b and ‘Sc,21(0, 0) — 21(90, 0)cos21(0,0) + 21(90,0) (E.6)and1 — sin 2’ cos_____________— 2S[cos2] —2S3[—sin2sin6] = 21(0,0) — 41(45,0) + 21(90,0)1 + 21(0, 0) + 21(90, 0)(E.7)For a system without a sample the Stokes parameters before and after the sample areidentical (ie S = S”) and using the results from the P = 00 configuration b may beevaluated by1 — sin 2’i/’ cos b — 1(0, 0) — 21(45, 0) + 1(90, 0) (E.8)2 — 1(0, 0 + 1(90, 0)Appendix FEstimate of Hamaker Constant Calculation ErrorsThe estimate of the error arising in the Hamaker constant calculations presentedin Table 5.3 was made using estimated values of the uncertainty in the experimentalmeasurements reported in Chapter 4. The Hamaker constant was calculated from theexperimental results by rearranging Equation 2.19 to giveA12 = 6irh (7?Tt) in (f-). (F.1)Assuming that the vapour temperature, T, and the liquid density, P1, were not subject toany experimental uncertainty (density varies negligibly over the entire range of temperatures considered), all the experimental uncertainty was concentrated in two areas, thepressure ratio, P/Ps, and the adsorbed thickness, h. The pressure ratio is determinedby manufacturer’s data given the liquid temperature, as given in Equation 5.6.p = 1o(7.6042_). (F.2)Rearranging Equation F.2 into exponential form the ratio P/P5 may be written asP Z2ZT (F.3)Substituting Equation F.3 into Equation F.1, A12 is given asA12 = (6ir)(3597) (7TvPt) (- — ) h. (F.4)Combining all the constant values into one term, K, Equation F.4 may be written asA 1312 — — J L0\A5 .LI151Appendix F. 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