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Electrostatic manipulation of fluidic interfaces for optical control purposes Kwong, Vincent Hugh 2002

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ELECTROSTATIC MANIPULATION OF FLUIDIC INTERFACES FOR OPTICAL CONTROL PURPOSES by VINCENT HUGH KWONG B.A.Sc, University of British Columbia, 1998  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER 2002 © Vincent Hugh Kwong, 2002  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n .  Department of  pHW<\t.O& AvAJO . { ^ T O l ^ l M V  The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada  Date .  .fiFCJFH&e^  2*6^7-  ABSTRACT  Three original, different, yet complementary optical techniques are presented in this thesis. They can be employed individually or in combination, to yield new methods of optical control. The common theme linking these techniques is the application of patterned electric fields to control the bulk and/or surface properties of fluidic structures, yielding a variety of optical effects. These effects may be useful in a multitude of optical structures, including variable Bragg gratings, reflective displays, optical memory devices, and the like. The first technique, here termed the fluid interface grating, uses a spatially modulated electric field to alter the interfacial pressure between two immiscible fluidic layers having different indices of refraction. The pattern of the electric field determines the amplitude and spatial frequency of the deformation in the interface between the two layers. Since the two fluids have different indices of refraction, this patterned deformation produces a diffractive structure. The second technique, here termed the electrophoretic grating, uses electrophoresis to microscopically control the location of charged, high refractive index particles suspended in a low index perfluorinated hydrocarbon fluid. By controlling the amplitude and spatial frequency of the electric field in the suspension, a spatial periodic change in the effective index of refraction occurs in the bulk of the suspension, forming another new diffractive structure. The third technique uses an effect known as electrowetting to alter the shape, and hence the reflective properties, of tiny water drops on an air-polymer interface. As a result of electrically inducing changes in the surface  energy of the drops, the shape of each drop is deformed as its contact angle is altered. This corresponds to a substantial change in the reflectance of the drop, which could, for example, form the basis for a new type of reflective display. The results presented in this thesis demonstrate the feasibility of these three new techniques to actively control optical properties of surfaces. These three methods can be used separately but there is also potential for combining these techniques. The results are encouraging and are suggestive that further research is warranted.  m  T A B L E OF CONTENTS ii  ABSTRACT  vii  LIST O F T A B L E S LIST O F FIGURES  ,  ACKNOWLEDGEMENTS  viii xi  1  INTRODUCTION  1  2  BACKGROUND WORK  5  2.1  PRIOR WORK INVOLVING ELECTRICAL MODIFICATION OF B U L K PROPERTIES OF  MATERIALS  5  2.2  PRIOR WORK INVOLVING MANIPULATION OF SURFACE PROPERTIES  12  2.3  CONCLUSIONS  17  3  B A C K G R O U N D PHYSICS  18  3.1  DIFFRACTION GRATING  18  3.2  ELECTROPHORESIS  19  3.3  SURFACE TENSION  22  3.4  CONTACT A N G L E  23  3.5  SUPER-HYDROPHOBICITY  27  3.6  ELECTROWETTING  30  3.7  HEMISPHERE REFLECTION  33  3.8  CONCLUSIONS  35  4  FLUID INTERFACE GRATING  4.1 4.2  4.2.1 4.2.2 4.2.3 4.3  ANALYSIS OF OBSERVATIONS  36  MEASUREMENTS AND ANALYSIS OF A SINGLE CONTROL ELECTRODE  39  Materials and Setup Diffraction Intensity and Transmission Time Response MEASUREMENTS AND ANALYSIS OF AN ARRAY OF INDIVIDUAL ELECTRODES  4.3.1  36  Materials and Setup  39 43 50 ..51  51 iv  4.3.2 4.4  5  Diffraction Intensity and Transmission  52  CONCLUSIONS  56  57  ELECTROPHORETIC GRATING 5.1  MEASUREMENTS AND ANALYSIS OF A SINGLE CONTROL ELECTRODE  5.1.1 5.1.2 5.2  58 59  MEASUREMENTS AND ANALYSIS OF AN INDIVIDUAL ELECTRODE ARRAY  5.2.1 5.2.2 5.3  6  63  Materials and Setup Diffraction Intensity and Transmission  63 64  CONCLUSIONS  66  67  REFLECTIVE E L E C T R O WETTING DISPLAY 6.1  MATERIALS AND EXPERIMENTAL SETUP  67  6.2  ELECTROWETTING USING OSCILLATING AND STATIC ELECTRIC FIELDS  72  6.2.1 6.2.2 6.3  Analysis of Observations Measurements and Analysis  72 73  DROP EVAPORATION AND RECONDENSATION  6.3.1 6.3.2 6.4  75  Analysis of Observations Measurements and Analysis  75 77  TIME RESPONSE OF BASE DIAMETER OF A N ELECTRO WETTED DROP  6.4.1 6.4.2 6.5  79 81  TIME RESPONSE FOR TOTAL DROP STABILIZATION  6.6  85  Analysis of Observations Measurements and Analysis  86 87  CONTACT A N G L E OF AN ELECTROWETTED DROP AS A FUNCTION OF APPLIED  VOLTAGE  6.6.1 6.6.2  90  90 91  Analysis of Observations Measurements and Analysis  6.7 REFLECTANCE OF AN ELECTROWETTED DROP AS A FUNCTION OF APPLIED VOLTAGE  6.7.1 6.7.2 6.8  Analysis of Observations Model Results ELECTROWETTING MULTIPLE DROPS  6.8.1 6.8.2 6.9  79  Analysis of Observations Measurements and Analysis  6.5.1 6.5.2  7  57  Materials and Setup Diffraction Intensity and Transmission  19-Drop Hexagonal Pixel Separately Controlled Drops in a 7-Drop Pixel CONCLUSIONS  CONCLUSIONS  :  94  94 95 98  98 99 100  101  v  REFERENCES  104  APPENDIX A: REFLECTION FROM A HEMISPHERE  108  APPENDIX B: CALCULATION OF DEFORMATION OF OIL-AIR INTERFACE GRATING  110  APPENDIX C: MATERIALS LIST  115  APPENDIX D: CALCULATION OF TIME RESPONSE OF OIL-AIR INTERFACE GRATING  118  APPENDIX E: MODEL OF EVAPORATION AND RECONDENSATION OF PURE WATER ONTO SALT WATER  122  APPENDIX F: TIME RESPONSE OF BASE DIAMETER AT VARIOUS VOLTAGES  126  APPENDIX G: APPLIED FORCE DATA AS A FUNCTION OF BASE DIAMETER VELOCITY  131  APPENDIX H: MAXIMUM REFLECTION OF A HEXAGONAL ARRAY OF E L E C T R O W E T T E D DROPS  135  vi  LIST O F T A B L E S TABLE 6.1 THE PREDICTED CONTACT ANGLES AS A FUNCTION OF VOLTAGE BASED ON EQUATION 6.2 85 TABLE 6.2 THE MEASURED CONTACT ANGLE AS A FUNCTION OF VOLTAGE FOR AN ELECTRO WETTED DROP -. 92 TABLE 6.3 THE PREDICTED CONTACT ANGLES AS A FUNCTION OF VOLTAGE BASED ON EQUATION 6.2 92 TABLE C . 1 MATERIALS '. 115 TABLE F . 1 ELECTROWETTED BASE DIAMETER OF A WATER DROP AS A FUNCTION OF TIME AT VARIOUS VOLTAGES AT 2020 H z 130 TABLE G . 1 BASE VELOCITY AND APPLIED FORCE AS A FUNCTION OF TIME AT VARIOUS VOLTAGES AT 2020 H z  133  vii  LIST O F FIGURES FIGURE 2.1 Two PERPENDICULAR POLARIZERS PREVENTING TRANSMISSION OF LIGHT  6  FIGURE 2.2 TRANSMISSIVE STATE OF A NEMATIC LIQUID CRYSTAL CELL FIGURE 2.3 ABSORPTIVE STATE OF NEMATIC LIQUID CRYSTAL CELL FIGURE 2.4 ORIENTATION OF PARTICULATES IN AN ELECTRO-RHEOLOGICAL FLUID  7 8 9  FIGURE 2.5 TRANSMISSIVE AND ABSORPTIVE STATES OF AN ELECTROPHORETIC CONTROLLED LIGHT PIPE SYSTEM 10 FIGURE 2.6 T H E TWO STATES OF A ELECTROPHORETIC REFLECTIVE DISPLAY 11 FIGURE 2.7 T H E TWO STATES OF REFLECTIVE DISPLAY USING ELECTROPHORETIC CONTROL OF TOTAL INTERNAL REFLECTION 12 FIGURE 2.8 CONTROLLING THE SPATIAL FREQUENCY OF A BRAGG GRATING BY COMPRESSION AND TENSION 13 FIGURE 2.9 SCHEMATIC OF A MODULATED ELASTOMERIC DIFFRACTIVE STRUCTURE 14 FIGURE 2.10 INDUCED MOTION OF A LIQUID DROP BY ELECTROWETTING 15 FIGURE 2.11 THE TWO STATES OF AN ELECTROWETTED REFLECTIVE DISPLAY 16 FIGURE 3.1 A MULTIPLE SLIT DIFFRACTION GRATING 18 FIGURE 3.2 FIELD-INDUCED MOTION OF CHARGED ELECTROPHORETIC PARTICLES 19 FIGURE 3.3 THE CONTACT ANGLE OF A LIQUID DROP ON A SOLID SURFACE SURROUNDED BY A GAS OR LIQUID 23 FIGURE 3.4 T H E THREE COMPETING FORCES AT THE INTERFACE OF A LIQUID DROP ON A SURFACE SURROUNDED BY VAPOUR 24 FIGURE 3.5 DIFFERENT ADVANCING AND RECEDING ANGLES OF A WATER DROP DUE TO CONTACT ANGLE HYSTERESIS 26 FIGURE 3.6 COMPARISON OF THE CONTACT ANGLE OF A WATER DROP ON SMOOTH AND ROUGHENED SURFACES 27 FIGURE 3.7 THE PARAMETERS FOR THE ROUGHNESS FACTOR OF CASSIE'S MODEL 29 FIGURE 3.8 DEFORMATION OF A WATER DROP ON A HYDROPHOBIC SURFACE BY ELECTROWETTING 30 FIGURE 3.9 REFLECTION FROM A TRUNCATED SPHERE AND A HEMISPHERE 33 FIGURE 3.10 ANNULUS OF REFLECTION FROM A TRUNCATED SPHERE AND A HEMISPHERE FIGURE 4.1 DEFORMATION OF A LIQUID-AIR INTERFACE BY AN APPLIED ELECTRIC FIELD FIGURE 4.2 APPROXIMATING THE SINUSOIDAL DEFORMATION IN THE FLUID INTERFACE  34 37  GRATING AS A PAIR OF ANGLED STRAIGHT LINES FIGURE 4.3 T H E INTER-DIGITAL ELECTRODE ARRAY FIGURE 4.4 SCHEMATIC OF ETCHING PROCEDURE TO CREATE THE ITO ELECTRODE ARRAY FIGURE 4.5 EXPERIMENTAL SETUP TO OBSERVE THE TRANSMITTED DIFFRACTION INTENSITY DISTRIBUTION FIGURE 4.6 SCHEMATIC OF THE FLUID INTERFACE GRATING FIGURE 4.7 PARAMETERS TO CALCULATE THE ANGULAR SEPARATION OF THE DIFFRACTION PATTERN FIGURE 4.8 THE FLUID-AIR INTERFACE GRATING WITH AND WITHOUT VOLTAGE APPLIED  38 40 41 42 43 44 45  FIGURE 4.9 THE DIFFRACTION PATTERNS CAUSED BY AIR-OIL SURFACE DEFORMATION AND ELECTRODE ARRAY WITH AND WITHOUT PERIODIC APPLIED VOLTAGE 46 FIGURE 4.10 THE LUMINANCE OF DIFFRACTION PATTERNS CAUSED BY AIR-OIL SURFACE DEFORMATION AND ELECTRODE ARRAY WITH AND WITHOUT PERIODIC APPLIED VOLTAGE VS THE POSITION ON THE VIEWING SCREEN 47 FIGURE 4.11 COMPARISON BETWEEN MODEL AND DATA OF ZEROTH ORDER DIFFRACTION AS A FUNCTION OF APPLIED VOLTAGE FOR FLUID INTERFACE GRATING 48 FIGURE 4.12 THE FORMATION OF MACROSCOPIC LENSING SURFACE 49 Vlll  FIGURE 4.13 THE INDIVIDUAL ELECTRODE ARRAY •. 52 FIGURE 4.14 A SCHEMATIC OF ELECTRODE CONNECTIONS 53 FIGURE 4.15 T H E FLUID-AIR INTERFACE GRATING WITH VOLTAGE APPLIED TO VARIOUS ELECTRODES 54 FIGURE 4.16 THE DIFFRACTION PATTERNS CAUSED BY AIR-OIL SURFACE DEFORMATION WITH VOLTAGE APPLIED TO VARIOUS ELECTRODES 54 FIGURE 4.17 T w o FLUID-AIR INTERFACE GRATINGS WITH SIMILAR PERIODICITY BUT DIFFERENT PHASE 55 FIGURE 4.18 T H E DIFFRACTION PATTERNS CAUSED BY TWO FLUID-AIR INTERFACE GRATINGS WITH SIMILAR PERIODICITY BUT DIFFERENT PHASE 56 FIGURE 5.1 SCHEMATIC OF THE ELECTROPHORETIC PARTICLE GRATING .58 FIGURE 5.2 THE ELECTROPHORETIC PARTICLE GRATING WITH AND WITHOUT VOLTAGE APPLIED 59 FIGURE 5.3 T H E DIFFRACTION PATTERNS CAUSED BY ELECTROPHORETIC PARTICLE GRATING AND ELECTRODE ARRAY WITH AND WITHOUT PERIODIC APPLIED VOLTAGE 60 FIGURE 5.4 T H E LUMINANCE OF DIFFRACTION PATTERNS CAUSED BY ELECTROPHORETIC PARTICLE GRATING AND ELECTRODE ARRAY WITH AND WITHOUT PERIODIC APPLIED VOLTAGE 61 FIGURE 5.5 THE SCHEMATIC OF THE ELECTROPHORETIC PARTICLE GRATING MODEL 62 FIGURE 5.6 T H E MODELED RESULTS OF RELATIVE TRANSMISSION OF ZEROTH ORDER DIFFRACTION AS A FUNCTION OF THE THICKNESS OF ELECTROPHORETIC GRATING 62 FIGURE 5.7 SCHEMATIC OF ELECTRODE CONNECTIONS FOR THE ELECTROPHORETIC GRATING.. 64 FIGURE 5.8 THE ELECTROPHORETIC PARTICLE GRATING WITH VOLTAGE APPLIED TO VARIOUS ELECTRODES 65 FIGURE 5.9 THE DIFFRACTION PATTERNS CAUSED BY ELECTROPHORETIC PARTICLE GRATING WITH VOLTAGE APPLIED TO VARIOUS ELECTRODES 66 FIGURE 6.1 DETERMINING THE MAXIMUM SPHERICAL DROP SIZE 69 FIGURE 6.2 A SCHEMATIC OF TWO STATES OF THE ELECTROWETTING SYSTEM 70 FIGURE 6.3 T H E NON ELECTROWETTED AND ELECTROWETTED STATES OF A WATER DROP AND THEIR CORRESPONDING REFLECTIVE ANNULUSES 71 FIGURE 6.4 DEPOSITION OF CHARGE AT EDGE OF DROP DUE TO IONIZATION OF AIR 72 FIGURE 6.5 THE ELECTROWETTED STATES OF A WATER DROP IN AIR AND IN OIL USING 2 7 0 V DC 74 FIGURE 6.6 THE ELECTROWETTED STATES OF A WATER DROP IN AIR AND IN OIL USING 3 8 4 V A C AT 2020 H Z 74 FIGURE 6.7 T H E EVAPORATION AND RECONDENSATION OF DISTILLED WATER DROP DURING THERMAL CYCLING 76 FIGURE 6.8 T H E EVAPORATION AND RECONDENSATION OF DISTILLED WATER ONTO SALT WATER OVER TIME 78 FIGURE 6.9 THE INITIAL AND FINAL STATES OF THE BASE DIAMETER OF AN ELECTROWETTED DROP 79 FIGURE 6.10 THE FORCE DIAGRAM OF AN ELECTROWETTED DROP 80 FIGURE 6.11 A N ELECTROWETTED DROP OVER TIME 82 FIGURE 6.12 BASE DIAMETER OF AN ELECTROWETTED WATER DROP AS A FUNCTION OF TIME USING 384 V AT 2020 H z 83 FIGURE 6.13 T H E INSTANTANEOUS VELOCITY OF BASE DIAMETER OF ELECTROWETTED DROP AS A FUNCTION OF THE INSTANTANEOUS APPLIED FORCE 84 FIGURE 6.14 T H E SURFACE OSCILLATIONS OF A DROP WITH A STABLE BASE OVER TIME IN INCREMENTS OF 2 MS 86  IX  FIGURE 6.15 DROP OSCILLATION STABILIZATION TIME OF A WATER DROP AS A FUNCTION OF FREQUENCY OF INPUT SIGNAL AT 3 8 4 V A C 88 FIGURE 6.16 MEASURING THE CONTACT ANGLE OF AN ELECTROWETTED DROP 91 FIGURE 6.17 COMPARISON OF THE MEASURED AND PREDICTED CONTACT ANGLE OF AN ELECTROWETTED WATER DROP AS A FUNCTION OF APPLIED VOLTAGE 93 FIGURE 6.18 THE DEFINED BEAM DIAMETER FOR TRACEPRO® REFLECTANCE MODEL FIGURE 6.19 THE MODELED REFLECTANCE OF AN ELECTROWETTED WATER DROP AS A FUNCTION OF CONTACT ANGLE FIGURE 6.20 THE MODELED REFLECTANCE OF AN ELECTROWETTED WATER DROP AS A FUNCTION OF VOLTAGE FIGURE 6.21 THE NON-REFLECTIVE AND REFLECTIVE STATES OF A 19-DROP HEXAGONAL ELECTROWETTED PIXEL DISPLAY FIGURE 6.22 SEVERAL REFLECTIVE STATES OF INDIVIDUALLY ELECTROWETTED DROPS IN A DROP PIXEL FIGURE A . 1 CRITICAL ANGLE OF A HEMISPHERE OF WATER FIGURE B . 1 SURFACE DEFORMATION OF TWO FLUIDS IN A CAPACITOR FIGURE B . 2 APPROXIMATING THE SINUSOIDAL DEFORMATION FIGURE D . 1 TIME RESPONSE PARAMETERS FIGURE E . 1 MODEL OF EVAPORATION AND RECONDENSATION OF PURE WATER ONTO SALT WATER FIGURE F . 1 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 384 V FIGURE F.2 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 349 V FIGURE F.3 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 327 V FIGURE F.4 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 301 V FIGURE F.5 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 275 V FIGURE F.6 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 253 V FIGURE F.7 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 237 V FIGURE F.8 BASE DIAMETER VS TIME FOR AN ELECTROWETTED DROP AT 2 2 1 V FIGURE G . 1 GEOMETRIC DROP PARAMETERS FIGURE H . l HEXAGONAL ARRAY OF DROPS  95 96 97 98 799 108 110 113 118 122 126 127 127 128 128 129 129 130 131 135  x  ACKNOWLEDGEMENTS  I would like to express my gratitude to all members of the SSP lab, both past and present; for all of their helpful discussions, insights and lively conversations. Sincere thanks to Andrzej Kotlicki for all of his helpful suggestions and ideas, especially his advice involving the electrical aspects of this research, which was taken much more often than not. Thank you to Andre Marziali for his reading of this thesis and his recommendations. Thank you to my supervisor, Lome Whitehead, for not only providing me with guidance when I needed it most, but also for allowing me the freedom to figure things out on m y own. His enthusiasm and leadership are greatly admired. I am also grateful to NSERC for providing financial support during the course of this research. Finally, I would like to thank my parents and my siblings. Thank you to Mom and Dad for supporting me, even when they didn't understand what I was doing or why I was doing it. And thanks to David, Mona, and Evan, better known to me as David, Jea, and B, for all the laughter.  xi  1  INTRODUCTION  Novel devices, which manipulate optical media to create useful optical effects either by diffraction, reflection, transmission, absorption or a combination thereof have become increasingly common. These optical devices include display technologies used in handheld personal digital organizers, laptop computers, optical switches and gratings in fiber optics systems, and backlit display systems. Consequently, research that may potentially involve the creation of new types of optical devices has increased in demand. The work presented in this thesis deals with three new optical techniques in this field. They involve the electrostatic manipulation of bulk and surface properties of fluidic structures to actively control their optical properties. The first technique, termed the fluid interface diffraction grating, uses an inter-digital array to create a patterned electric field, which in turn is used to produce microscopic changes in inter-facial pressure between two fluidic layers, namely oil and air. As a result of these changes, a spatially modulated deformation occurs between the two fluids. Since the two fluids have different indices of refraction, the patterned deformation forms a diffractive structure. The second technique, termed the electrophoretic diffraction grating, also uses an interdigital array, but instead of two fluids, a suspension, consisting of charged, high refractive index pigment particles and non-pigment counter-ions in a low index dielectric fluid, is placed in a patterned electric field. The location of electrophoretic particles within the fluid is controlled by this electric field, a process known as electrophoresis. By controlling the amplitude and spatial frequency of the patterned electric field, spatially modulated changes in 1  the effective index of refraction occur in the bulk of the suspension, forming another new diffractive structure. The third technique uses an electrostatic effect known as electrowetting, to change the shape of water drops on an air-polymer interface. By electrically inducing changes in the surface energy of the water drop, the contact angle of each drop is reduced, and hence its shape is changed. This, in turn, changes the reflectance characteristics of the drop, which can be used as the basis for a new type of reflective display. To provide background and motivation for the work presented in this thesis, a brief description of prior work in similar areas of research is provided in Chapter 2. Chapter 3 briefly reviews the basic physics of the principal phenomena involved in this research, including the concepts of diffraction gratings, electrophoresis, surface tension of a liquid, contact angle of a drop, super-hydrophobicity of a surface, electrowetting of a drop, and reflection of a hemisphere. Chapter 4 discusses the construction, measurements and analysis of a fluid interface grating. These topics include an estimation of the required electric field and the materials and processes involved in building a test grating. Measurements of diffraction intensity and transmission of such a test grating were performed, and the results, along with the predicted results from a one-parameter model are successfully compared. Observations of the dependence of the diffraction on the spatial frequency of the patterned electric field are presented. Finally, the expected time response of such a grating is discussed based on dimensional analysis arguments and such predictions are compared with experimental results.  2  Chapter 5 presents the construction, measurements and analysis of an electrophoretic grating. Topics discussed include the materials and processes involved in building a prototype grating. The diffraction intensity and transmission of the prototype were measured and these results are compared to the predictions of a model analyzed using a diffraction grating software package. Observations of dependence of the diffraction on the spatial frequency of the patterned electric field are also presented.  Chapter 6 describes construction, measurements and analysis of controlled electrowetting to variably change the shape of water drops on a super-hydrophobic surface. The components of the electrowetting system, including electrode materials, insulating coatings, and superhydrophobic surfaces are discussed. As well, studies of the advantages of oscillating instead of static electric fields are described. Also presented is a study of the problems associated with thermal cycling, such as recondensation of water in undesirable locations and changes in the contact angle of the water drop. A solution, based on the relationship between solution concentration and vapour pressure, is presented. A theoretical model, based on measurements of the time response of the base diameter of an electrowetted drop at various voltages, was used to predict the contact angle as a function of voltage. A section dealing with drop stability issues due to coupling of the input voltage to the natural resonant frequency of the water drop is also included. Another theoretical model that predicts the contact angle of the water drop as a function of voltage was derived from previously reported electrowetting work on a smooth surface. The two contact angle models were successfully verified by measuring the contact angle of a water drop at various voltages and comparing the corresponding results. Also, a model of the reflectance of an electrowetted drop as a function of contact angle was produced using Monte Carlo ray tracing software. Based on  3  this reflectance model and one of the contact angle models, a model of the reflectance of the drop as a function of voltage was produced and its predictions are reviewed. Finally, the results of two multiple drop electrowetting systems are presented. Chapter 7 presents overall conclusions based on the study of these three techniques as well as suggestions for improvements and future work in each of these areas individually and, possibly, in combination. While the scope of this research is limited to optical effects, the possibility of non-optical applications of these effects is also considered.  4  2  BACKGROUND WORK  To place the context of this research in perspective, it is helpful to provide some brief details of previous efforts in similar areas of research. The common theme of this thesis and of the previous work described below is electrical modification of the surface or bulk properties of materials to create useful optical effects.  2.1  Prior Work Involving Electrical Modification of Bulk Properties of Materials  A common example of modifying the bulk properties of a material to create a useful optical interaction is the liquid crystal display (LCD). The principle behind an L C D involves the application of an electric field to change liquid crystal molecules from one orientation to another. This re-orientation changes the polarizing nature of the bulk such that the L C D 1  reflects, absorbs or transmits light. While there are several types of liquid crystals, for illustrative purposes it is sufficient to consider the most common type, known as the nematic liquid crystal display. This type of L C D uses long chains of liquid crystal molecules, sandwiched between two polarizing plates whose polarizations are oriented at 90° with respect to each other. When light encounters the first polarizer, about 50% of the energy is absorbed and the remaining transmitted light has purely linear polarization. The polarization direction of this transmitted light is perpendicular to that of the second polarizer, such that this light is completely absorbed there, as depicted in Figure 2.  5  Figure 2.1 Two Perpendicular Polarizers Preventing Transmission of Light  However, the introduction of a layer of nematic liquid crystals between the polarizers allows this situation to be changed by application of an electric field. The alignment of the long molecules comprising the nematic crystals is specified by a unit vector known as the director. The surface of the polarizers can be prepared such that the director is parallel with the polarization vector at the surface of the first polarizer and also with the polarization vector at the surface of the second polarizer. In between, the director undergoes a uniform rotation of 90°, as shown in Figure 2.2.  6  Polarizer 2  Polarizer 1  Figure 2.2 Transmissive State of a Nematic Liquid Crystal Cell  The presence of the liquid crystal material rotates the polarization of the light transmitted through the first polarizer by 90°, such that the polarization matches that of the second polarizer when it is encountered. Thus, light passes through the second polarizer, corresponding to a transmissive state.  However, the application of an electric field between the two polarizers causes the liquid crystal molecules to align themselves parallel to the field, as shown in Figure 2.3. In this arrangement, the polarization of the light remains unchanged by the liquid crystals and is thus absorbed by the second polarizer, corresponding to a black, absorptive state.  7  Polarization Direction  Polarizer 2  Polarizer 1  Figure 2.3 Absorptive State of Nematic Liquid Crystal Cell  Hence, by controlling the electric field, the transmission of liquid crystal displays can be changed. Another example of a device that involves the manipulation of bulk properties of a material is common photographic film. Photographic film is composed of a suspension of silver halide crystals which photo-chemically react when exposed to visible light to form a developable latent image on the film. By developing the film, the bulk characteristics are changed, with the final result being a photographic negative image. Development is achieved by using a developing agent to convert the exposed silver ions into silver metal, with the grains that were exposed to more light developing much more rapidly, while leaving the unexposed silver halides unchanged. The developing process is stopped with a fixing  8  bath, which also dissolves away the unexposed silver halides, yielding a stable photographic negative.  Another example of a material whose bulk properties can be manipulated are rheological solutions whose viscosity, yield stress, and shear modulus, can be altered by the application of an electric or magnetic field within the fluid. '  2 3,4  Most of these solutions, known as  electro-rheological and magneto-rheological fluids, consist of particulates dispersed in an insulating oil but some are composed of a liquid dispersed in oil. By applying an electric or magnetic field through the fluid, previously free flowing particulates within thefluidform fibrillated chains, changing the rheological properties of the solution, as illustrated in Figure 2.4. Devices that use rheologicalfluidsinclude dampers, brakes, clutches, hydraulic valves and shock absorbers.  a) No Electric Field  b) Applied Electric Field  Figure 2.4 Orientation of Particulates in an Electro-Rheological Fluid  9  Electrophoresis uses an electric field to induce motion of charged particles suspended in a fluid medium, which creates changes in the overall bulk properties of the suspension. This can be used as a control mechanism in a light pipe system. As shown in Figure 2.5, a layer 5  of electrophoretic suspension sandwiched between two transparent electrodes surrounds a light pipe. By varying the density of particles near the surface the light pipe, the degree of reflection occurring at the surface of the pipe can be controlled.  \ _ Electrophoretic Particles  a) Transmissive State  b) Absorptive State  Figure 2.5 Transmissive and Absorptive States of an Electrophoretic Controlled Light Pipe System  !()  Electrophoresis can also be used to create a pixilated optical display by switching pigmented 67  particles in and out of view in selected regions of the display area. ' This is illustrated by Figure 2.6.  Apparent Colour is that of the Pigment Particles  <3 Observer Apparent Colour is that of the Suspending Fluid  E Glass  Figure 2.6 The Two States of a Electrophoretic Reflective Display  Another version of an electrophoretic reflective display uses electrophoresis to actively control total internal reflection in a structured layer. The basic concept of this type of 8  display is shown in Figure 2.7. If the particles are away from the prismatic structure, then light is reflected. If the particles are at the surface of the prismatic structure, then light is absorbed.  11  Figure 2.7 The Two States of Reflective Display Using Electrophoretic Control of Total Internal Reflection  In addition to changing the bulk properties of materials, it is also possible to produce interesting optical effects by modifying the various characteristics of surface interfaces. Some examples of previous work in this area are presented below.  2.2  Prior Work Involving Manipulation of Surface Properties  Many different ways of changing surface properties of materials have been explored, but the ones most relevant to the research presented in this thesis are diffraction gratings and electrowetting.  12  If one can control a diffractive grating in terms of its spatial frequency, the index of refraction, or both, then one can control the direction and intensity of diffracted light. Considerable research has been carried out on the principle of physically stretching or compressing a fiber grating in order to control its spatial frequency. ' '  9 10 11  This is a very  simple, yet effective concept in which a Bragg grating fabricated onto a fiber is variably stretched. By varying the degree of stretch, the spatial frequency of the grating can be controlled, as shown in Figure 2.8.  Applied Force  ^ ^ ^ ^ i b e r  Figure 2.8 Controlling the Spatial Frequency of a Bragg Grating by Compression and Tension  A related technique uses controlled heating to change the index of refraction of heat sensitive 12  materials located within a grating.  This method uses highly temperature sensitive polymers  to create a diffractive structure. The structure is heated by passing current through a thin metallic layer that is in contact with the polymeric structure. By varying the current, the 13  amount of heating can be controlled to alter the index of refraction of the diffractive structure. Another method uses periodic encapsulation of air pockets by elastomeric structures to create a diffraction grating.  Electrostatic forces are used to elastically deflect these membranes to  create a change in the width of the air gap, as shown in Figure 2.9.  Elastomeric Diffraction Grating  Elastomeric Membrane  Electrodes Air  a) Undeflected State  b) Deflected State  Figure 2.9 Schematic of a Modulated Elastomeric Diffractive Structure  By changing the width of the air gaps, changes in the effective index of refraction of the grating occur which can be used to vary the intensity of the transmitted diffraction pattern. Another example is a surface acoustic wave grating. One can apply a propagating surface acoustic wave to the surface of a material to create a surface reflection diffraction grating.  14 1 5  14  The amplitude and frequency of the surface wave can be controlled to create a variable diffraction grating by inducing a periodic change in the refractive index of the material.  While it is only necessary to induce microscopic changes in surface properties to produce diffractive effects, macroscopic changes can also produce other kinds of useful optical effects. One method to create macroscopic effects is electrowetting, which involves the application of an electric field to change the contact angle of a liquid on a substrate. If the contact angle can be controlled to a sufficient extent, electrowetting can be used as a means of controlling the motion of fluids on a microscopic scale. The basic principle is shown in Figure 2.10.  Figure 2.10 Induced Motion of a Liquid Drop by Electrowetting  15  This method has been used to dispense, join, divide, and transport micro-drops  1617  and has  also been used to control the amount of fluid in multi-channel micro-structures. By 18  inducing drop motion, electrowetting has been utilized to create liquid micro-motors as 19  well as optical switches in the form of an opaque liquid drop, which acts as a shutter. 20 Electrowetted motion of the fluid is not required to create useful effects; potentially valuable effects can occur by simply changing contact angle of drop by electrowetting. By deforming the surface of a liquid lens, electrowetting can be exploited to create a lens with a variable focal length.  By controlling the amount of spreading of the liquid in contact with the  21  surface, electrowetting can also used to create a pixilated display. ' This concept is shown 22 23  in Figure 2.11.  Observer  Observer  Clear Insulating Fluid  Electro-wetted Drop  Coloured Background Electrodes Insulator a) Observer Sees Mostly Colour of Background  Undeformed Drop  b) Observer Sees Mostly Colour of Drop  Figure 2.11 The Two States of an Electrowetted Reflective Display  16  Depending on whether or not the drop is electrowetted, the viewer will see either the colour of the background or the colour of the drop.  2.3  Conclusions  This body of prior work provided the motivation for new investigation in this area. Before describing the current research in detail, it will be helpful to briefly summarize the basic physics of several key phenomena involved in this work.  17  3  B A C K G R O U N D PHYSICS  Before discussing each of the fluid interface grating, electrophoretic grating and reflective electrowetting display, it is helpful to provide a summary of the relevant background physics pertaining to each of these topics. This brief discussion includes the physics of diffraction gratings, electrophoresis, surface tension, contact angle, super-hydrophobicity, and hemisphere reflection.  3.1  Diffraction Grating  For a multiple slit diffraction grating, the location of the maxima of the diffraction pattern is determined by:  24  (3.1)  dsmd - mA  where m is the order number (...-3, -2, -1, 0, 1, 2, 3...), X is the wavelength of incident light, d is the center to center spacing of adjacent slits and #is the angle of diffracted light, as shown in Figure 3.1.  Grating  A Maximum  Incident Light  dsinB  Screen  Figure 3.1 A Multiple Slit Diffraction Grating 18  Since the angular separation between the adjacent maxima is inversely proportional to the distance between adjacent slits, adjusting this spacing will control the direction of the diffraction peaks.  3.2  Electrophoresis  Electrophoresis is the field-induced motion of charged particles in a fluid. ' 25  26  By applying  an electric field in a fluid containing charged particles, positive particles will tend to move in the direction of the field, and negative particles will tend to move in the opposite direction. This phenomenon is commonly used in biology and chemistry, often to separate mixtures. However, in the scope of this research, electrophoresis is used as method of creating a variable diffraction grating.  Electrodes  Fluid a) No Voltage Applied  b) Voltage of One Polarity Applied  c  ) oltage of Opposite Polarity Applied v  Figure 3.2 Field-Induced Motion of Charged Electrophoretic Particles  19  Consider Figure 3.2, in which an electrophoretic suspension, containing positively charged pigment particles and non-pigment counter-ions in a dielectric fluid, is located between two parallel electrodes. With no voltage applied to the electrodes, the pigment particles are randomly distributed. When voltage is applied, the particles are attracted to the negative electrode. This process can be reversed by simply changing the polarity of the applied voltage. The speed of migration of the particles, v, depends on the electrophoretic mobility of the 07  particles, //, and the magnitude of the electric field, E, as shown by Equation 3.2:" v = fiE  (3.2)  where the electrophoretic mobility, jU, is given by:  (3.3)  ju=—^—  6mja  net  where q is the charge of the particle, v is the viscosity of the fluid and a  net  is the effective  hydrodynamic radius of the particle.  Since the electrophoretic mobility is directly proportional to the charge on the particle, a higher charge yields a higher mobility, which, in turn, leads to a shorter response time. For the electrophoretic solution used in this research, the electrophoretic mobility was approximately lxlO" m -V" -s"', corresponding to a particle charge of about 6.6x 10" C, or 10  approximately 4e.  2  l  19  28  20  Another important parameter of an electrophoretic suspension is the size of the particles. If the particles are small enough such that the settling force, F, multiplied by the settling distance, d, is less than the thermodynamic energy, kT, then it is difficult for the particles to settle out of the fluid.  29  This condition is shown in Equation 3.4 below.  Fd = g[p, - p )vd = ^ng(p p  t  )R d < kT  (3.4)  3  P  p  Where g is the gravitational acceleration, pi is the density of the liquid, p is the density of p  the particles, V is the volume of the particles, d is the settling distance, R is the radius of the particles, k is Boltzmann's constant and T is the temperature of the solution. The density of the perfluorinated liquid used in this thesis was 1900 kg/m . The density and 3  radius of the electrophoretic particles were approximately 1700 kg/m and 200 nm, 3  respectively. The temperature of the solution was 25°C and the settling distance was 0.05 mm. By substituting these values into Equation 3.4, the settling force multiplied by the settling distance was found to be 4x10" J, which is less that the thermodynamic energy of 4xl0" J. Hence, the problem of electrophoretic particles settling out of the fluid was not an 21  issue during the course of this research.  To further stabilize the suspension, polymer dispersant molecules were added. One end of these molecules is chemically attracted to the electrophoretic particles and thus attaches to them. The other end is attracted to the fluid and readily dissolves. The result is the dispersant molecules end up surrounding the electrophoretic particles, suspending them within the fluid.  21  3.3  Surface Tension  Surface tension, y is a characteristic of the interface of two immiscible materials, where there is an energy per unit area associated with the interface, measured in J/m . Surface tension can also be though of as the tension force within the surface per unit length perpendicular to the direction of the force, measured in the equivalent units of N/m. Such interfaces tend to change shape, if possible, to minimize their surface area. Such reductions of the interfacial area can be viewed as the tendency of systems to move to a lower energy state. Surface tension arises from inter-molecular forces, wherein the molecules deep within one material experience higher repulsion forces due to their close packing as compared to molecules at the interface of the immiscible materials, which are less dense, leading to a surface that is in tension.  30  The property of surface tension of an interface composed of two  fluids is responsible for formation of liquid drops and soap bubbles.  However, very often a system will consist of three substances, such as a liquid, a gas and a solid surface, which give rise to surface tension phenomena such as meniscuses, capillary action, the absorption of liquids by porous substances and the ability of liquids to wet a surface.  31  For example, in this research, a water drop was placed on a solid surface  surrounded by air. Hence, there are three competing surface tension terms that act to stabilize the shape of the drop. These surface tensions terms are y \, y and y , referring to s  v  sv  the surface tension between the surface and the liquid, the liquid and the vapour, and the surface and the vapour, respectively. The following section considers the physics of this system in more detail.  22  3.4  Contact Angle  As illustrated by Figure 3.3, the contact angle, 0, of a drop is defined as the angle the surface of drop makes with the solid surface that is in contact with the inside of the drop. "  Solid Surface  Figure 3.3 The Contact Angle of a Liquid Drop on a Solid Surface Surrounded by a Gas or Liquid  If the drop is small enough such that gravity has negligible effect and surface tension effects dominate, then the drop is a truncated sphere, since a sphere is the shape with the lowest surface to volume ratio. For such a drop, a higher contact angle means less truncation of the spherical drop. The contact angle, 6, of a smooth solid surface, as well as the three competing surface tension forces at the contact point of the drop, air and surface, are shown in Figure 3.4.  23  I  Solid  Figure 3.4 The Three Competing Forces at the Interface of a Liquid Drop on a Surface Surrounded by Vapour  A force balance of the three terms is shown by Equation 3.5:  r, -{-cos0)+y =y sv  v  sl  (3.5)  where yi , y , and %i are the surface tension co-efficients of the liquid-vapour, sol id-vapour v  sv  and solid-liquid interfaces, respectively. Rearranging the terms leads to Equation 3.6, known as Young's Equation:  cose=  r s v  ~ " r  (3.6)  24  Thus, #can be solved for in terms of the surface tension co-efficients, as shown by Equation 3.7:  6 = cos"  Ysv  Ysi  (3.7)  Experimentally, #and y can be easily measured but y and y \ are difficult to measure. v  sv  s  Hence, Equation 3.6 is usually used, in conjunction with measured #and yj values, to v  calculate the difference between y and %i. Thus far, the maximum contact angle that has sv  been reported for a smooth hydrophobic coating was 119°, which occurred for a water drop in air on a surface of regular aligned closest hexagonal packed -CF3 molecular groups.  33  Based on Young's Equation, the contact angle appears to be a well-defined property that depends solely on the surface tension co-efficients of the three materials. However, in reality, the contact angle is very dependent on the direction of motion of the interfaces. The advancing angle is defined as the contact angle of the drop as the contact area between the drop and the solid surface increases, while the receding angle is the reverse of that situation. Contact angle hysteresis is the term used to describe the difference of these two angles. In practice, to some degree, the receding angle is always less that the advancing angle. This can be easily demonstrated by using a syringe to control the direction of flow of a drop on a surface, as shown in Figure 3.5.  25  Syringe  Syringe  Direction of Flow  e  Direction of Flow  Water Drop  Water Drop  a) Advancing Contact Angle  b) Receding Contact Angle  Figure 3.5 Different Advancing and Receding Angles of a Water Drop due to Contact Angle Hysteresis  As illustrated above, the contact angle of the drop depends on whether the drop is advancing from or retreating to the syringe. This difference in the receding and advancing angles is highly dependent on the properties of the system, including the roughness of the surface and contamination of the surface. Hence, the convention is to record the contact angle as the average of the receding and advancing angles. For example, the highest recorded maximum contact angle for a water drop in air on a smooth surface of -CF molecular groups of 119° is 3  the average of the measured advancing and receding angles of 122° and 116°, respectively.  34  Since the highest contact angle for a water drop on a smooth hydrophobic surface is limited to 119°, if a higher contact angle is desired, super-hydrophobic coatings must be used, as described below.  26  3.5  Super-Hydrophobicity  A super-hydrophobic coating is defined as a surface in which a drop of liquid, usually water, forms a contact angle of 150° or more.  35  A roughened hydrophobic surface exhibits super-  hydrophobic properties by introducing air voids, creating hydrophobic tips on which the drop rests. '  36 37  This decreases the contact area between the surface and the drop, thus increasing  the contact angle of the drop, as shown in Figure 3.6. If the coating is sufficiently hydrophobic, and the surface roughness is also sufficiently large, a super-hydrophobic coating can be formed.  Smooth Surface  a) Contact Angle of Drop on Smooth Surface  Roughened Surface  Trapped Air  b) Contact Angle of Drop on Roughened Surface  Figure 3.6 Comparison of the Contact Angle of a Water Drop on Smooth and Roughened Surfaces  Young's Equation is invalid for super-hydrophobic materials because it does not take into account the roughness of the surface. There are two different models to better estimate the 27  contact angle of a rough surface. One of them, Wenzel's Model, is a simple modification of Young's Equation, as shown by Equation 3.8:  38  cos0  rough  = *cos0  =R » (7  smooth  7  d  (3.8)  )  Yiv  where R is a roughness factor, defined as the ratio of the total area of a rough surface to the apparent surface area of the tips, and is always greater than 1. For example, a water drop on a smooth hydrophobic surface, where R=\, may have a contact angle of  Smooth  = 100°. If  that surface is roughened such that R = 5, then according to Wenzel's Equation, the contact angle of the drop becomes  # ugh r0  = 150°.  Similarly, Cassie's Model also uses the apparent contact area of the super-hydrophobic tips to calculate the contact angle, as shown by Equations 3.9 and 3.10:  39  cosr?  h  = /cos0  s m o o t h  +/-l  (3.9)  where  5> f  =  V ,  M  (  3  '  1  0  )  The parameters a and b, shown in Figure 3.7, are used to calculate the area fraction of the solid surface,/. In other words,/is the ratio of the surface area of the tips to the total surface area covered, which is approximately the inverse of R in Wenzel's Equation.  28  > i t ^ '.  W  a  t  e  r  D  r  ° P ^ '  V^. / / / / / / / / / / / / / / / '  •///j  Super-Hydrophobic Tips  / v  Trapped A i r  Figure 3.7 The Parameters for the Roughness Factor of Cassie's Model  As an example, the case of a water drop on a smooth super-hydrophobic surface with a contact angle of Smooth = 100° is considered once more. According to Cassie's Equation, by roughening the surface such that R = 5 and/= 0.2, the contact angle becomes ^ough = 147°.  As roughness of a transparent hydrophobic coating increases, its super-hydrophobicity increases, but its transparency decreases due to the fact that the rough surface is a source of light scattering. Since visible light is in the range of 400 - 750 nm, if a transparent, superhydrophobic coating is desired, the magnitude of roughness should be on the order of 100 nm or less. While such coatings have been reported , they are not readily available. The 40  41  super-hydrophobic coatings used in this research were not transparent but were translucent,  29  transmitting approximately 90% of incident light. Approximately 95% of that transmitted light was scattered by the rough surface, spreading by about 20°.  3.6  Electrowetting  The contact angle of a drop on a surface can be reduced by applying an electric field between the drop and an insulated electrode beneath the surface. This process, known as 42  electrowetting, is illustrated by Figure 3.8.  Electrode _ Super—p Hydrophobic Coating  a) No Electric Field  b) Electric Field  Figure 3.8 Deformation of a Water Drop on a Hydrophobic Surface by Electrowetting  The application of an electric field adds energy to the system. In order to minimize the total free energy of the system, the drop increases its surface area by reducing its contact angle and spreading onto the surface, a process also known as wetting out. On some surfaces, contact angle hysteresis of the drop prevents the drop from resuming its original shape after the electric field is removed. The degree of hysteresis depends on the type of surface, but it 30  is quite substantial for a super-hydrophobic surface. This is because the air voids inherent in super-hydrophobic coating are filled with liquid during electrowetting, which increases the contact area between the water drop and the surface effectively. However, a slight mechanical disturbance at the surface will dislodge the trapped liquid from the air voids, restoring the original shape of the drop. A model exists which describes the behaviour of an electrowetted drop on smooth hydrophobic surface. The electrowetting equivalent of Young's Equation is shown by 43  Equation 3.11:  Y {v)cos0{v)=y {v)-y {v) lv  sv  (3.11)  sl  Where y (V), y (V) and y i(V) are the three competing surface-tension co-efficients as a v  sv  s  function of applied potential, V. Both y (V) and y (V) are assumed to be independent of v  sv  voltage to a first approximation and are simply equal to y and y , respectively. However, v  sv  %i(V) has contributions from both the chemical potential, which is just y i, and the electrical s  potential,  yi , el  s  as shown by Equation 3.12:  (3.12)  where C is the capacitance of between the drop and the bottom electrode of the electrowetting system. Furthermore, this capacitance depends on only the parameters of the insulating coating, as shown by Equation 3.13:  C=  (3.13)  31  where e and d are the dielectric constant and thickness of the insulating layer, respectively, r  and e is the permittivity of free space. 0  Thus, Equation 3.12 becomes:  r -\^f-  rAy) = r,-^\\dvdv=  (3.14)  sl  By substituting 3.12 into 3.10, the following result is obtained:  (3.15)  y cos6{V) = y - y + lv  sv  sl  2  d  Moreover, by substituting Equation 3.6 (Young's Equation) into Equation 3.15, the following model is obtained:  cos0(v) =  c  2  o s 7,J  0  (  3  .  1  6  )  where 6(V) is the electrowetted contact angle, 0 is the contact angle at zero voltage, £„ is the o  permittivity of free space, e is the dielectric constant of the insulating layer, %, is the liquid r  vapour surface tension coefficient, d is the thickness of the insulating layer, and V is the applied voltage. This model has been demonstrated experimentally. In one reported example, drops of 10" M KNO3 in air were placed on a smooth hydrophobic Teflon™ 4  coating on top of various insulating materials with thicknesses ranging from 6 to 35 urn, and substantial verification of Equation 3.16 was shown.  44  32  3.7  Hemisphere Reflection  A recently noted effect is that a hemisphere can reflect a large portion of incident light in what has been termed a semi-retro-reflective manner.  45  A retro-reflective material reflects  light back to its point of origin, whereas a semi-retro-reflective material reflects the majority of light back towards approximately its point of origin.  A truncated spherical water drop in air with a contact angle of 150° reflects about 3% of incident light back to the viewer. However, if that drop is deformed into a hemisphere, then 4 3 % of the incident light is reflected back towards the viewer. The portions of the hemisphere that yield this reflection lie in an annular region, as shown in Figures 3.9 and 3.10.  a) Reflection from a Sphere  b) Reflection from a Hemisphere  Figure 3.9 Reflection from a Truncated Sphere and a Hemisphere 33  a) No Reflection from Sphere  b) Reflection Annulus from Hemisphere  Figure 3.10 Annulus of Reflection from a Truncated Sphere and a Hemisphere  Equation 3.17 gives the reflectance, R, of a beam of light from a hemisphere of index of refraction nj in a medium of index of refraction n , where R is the observed fractional area of 2  the reflection annulus. The derivation of this equation is located in Appendix A.  R = \-  (3.17) v"i  J  As shown by the above equation, it is necessary for the index of refraction of the hemisphere to be larger than that of the surrounding media for reflection to occur. Also, as index mismatch between the materials increases, the reflection annulus grows, yielding greater  34  reflection. For a water drop in air, with n = n \ ~ 1 and n/ = n 2  a r  ~ 1.33, the reflectance of  water  the hemisphere is 43%.  3.8  Conclusions  Numerous different microscopic and macroscopic effects can be obtained by taking advantage of the phenomena described in this chapter. The scope of the research in this thesis involves the use of these phenomena to create various optical effects. The three candidates of study are the fluid interface grating, the electrophoretic grating and the reflective electrowetting display, which are described in the following three chapters.  35  4  FLUID I N T E R F A C E G R A T I N G  A s mentioned previously, it is possible to control the direction of the diffraction peaks by simply adjusting the spatial frequency of a diffraction grating. The fluid interface grating described in this section accomplishes this by using an inter-digital array, consisting of interleaved electrodes, to produce a spatially modulated electric field which, in turn, induces periodic changes in interfacial electrostatic pressure of an oil-air interface. Since the two fluids have different indices of refraction, the deformation induced by the changes in pressure yields a diffractive structure.  T w o methods of controlling the fluid interface grating were tested. One technique involved using an array of inter-digital electrodes connected together to control the amplitude of the deformation of an oil-air interface with a fixed spatial frequency. The other technique involved using an array of independently connected, inter-digital electrodes to control both the amplitude and the spatial frequency of the deformation of an oil-air interface.  4.1  Analysis of Observations  B y applying an electric field between two fluids, the energy relationships of the two fluids are modified, which introduces a perturbation in the form of a protrusion of the higher dielectric material, as shown in Figure 4.1.  36  (+) Electrode  Z  Substrate — Air - Liquid Substrate <3 Electrode Protrusion  a) No Voltage Applied  b) Voltage Applied  Figure 4.1 Deformation of a Liquid-Air Interface by an Applied Electric Field  Consider Figure 4 . 1 , in which a liquid-air interface is located between two parallel electrodes. With no voltage applied to the electrodes, the fluid minimized its total energy, the sum of the surface energy and the electrostatic potential energy, by assuming a flat surface.  4  However, when voltage was applied, the energy relationships were modified and,  in order to minimize the total energy of the system, an elevated protrusion was formed on the fluid surface. In effect, the liquid was attracted to the region of higher electric field because its dielectric constant was higher than that of air. The height of the protrusion depended on a variety of factors, including the dimensions of the electrodes, the properties of the liquid, such as the dielectric constant and the surface tension co-efficient, and the amount of the applied voltage. An exact solution of the amplitude of this deformation requires numerical  37  simulation, but it was estimated by simple dimensional arguments. By approximating the deformation as two angled, straight lines, as shown in Figure 4.2, and minimizing the total energy of the system, a rough estimate of the magnitude of the deformation was obtained, as shown in Equation 4.1:  '• /  Straight Line Approximation  Sinusoidal Deformation  Figure 4.2 Approximating the Sinusoidal Deformation in the Fluid Interface Grating as a Pair of Angled Straight Lines  V w  £„£„,{£«,-£„)  ~7^Z—1  v~  where a is the height of deformation, V is the applied voltage, £  ( 4 > 1 )  oil  is the permittivity of the  oil, £ is the permittivity of free space, ^is the surface tension coefficient of the oil. d is the a  total thickness of the oil and air, and w is the center-to-center electrode spacing. The details of this derivation are located in Appendix B.  38  By having an array of these inter-digital electrodes, it was possible to create a spatially modulated electric field to control the spatial frequency and height of the deformation of a liquid-gas interface, thus producing a diffractive structure.  4.2  Measurements and Analysis of a Single Control Electrode  A fluid interface grating was created by applying a variable electric field with fixed spatial frequency through an oil-air interface. The electric field was produced from an array of inter-digital electrodes connected together to form a common, single control electrode. This section discusses the materials, the results and the analysis of the diffraction intensity and transmission measurements, and the time response associated with this fluid interface grating.  4.2.1  Materials and Setup  The inter-digital electrode array, as shown in Figure 4.3, consisted of two inter-leaved electrodes etched from a transparent indium tin oxide (ITO) layer on a glass substrate. A 47  list of all materials and equipment used, as well as any relevant physical information, is located in Appendix C.  39  © ITO Electrode  Glass / i i i i j  © I T O Electrode  © ITO Electrode Glass © ITO Electrode  (NOTTO SCALE)  Figure 4.3 The Inter-Digital Electrode Array  The etching procedure used to create the array is a well known process. A layer of AZ P4110, a UV-sensitive photo-resist, was spin-coated onto the ITO covered glass slide at 5000 rpm for 45 s. The slide was then baked on a hot plate at 100°C for 10 minutes. A positive mask of the desired electrode array was placed on top of the resist, which was then activated by exposure to a 300W UV source for 100 s, with the unexposed resist under the mask remaining unactivated. Next, the ITO slide was placed into a stirred solution consisting of 75% distilled water and 25% AZ 400IK Developer for 90 s to remove the activated resist, which exposed the unwanted portions of the ITO layer. The slide was then placed in a warm etching bath consisting of 70% distilled water, 25% HC1 and 5% HNO3 for 90 s to etch away the exposed ITO, leaving the desired ITO electrode array. Finally, the slide was cleaned with  40  methanol to remove the remaining UV-cured resist. This procedure is illustrated by Figure 4.4 below.  Mask  Glass a) Exposing Resist to U V Source with Mask In Place  b) Patterning of Resist c) Etching of ITO to After Exposure to U V Form Electrode Array and Developer Bath  d) Removal of Resist to Produce Final Electrode Array  Figure 4.4 Schematic of Etching Procedure to Create the ITO Electrode Array  The individual FTO electrodes were approximately 40 um wide with a center-to-center spacing of 100 p:m and a length of 25 mm. Diffraction was observed using a solid-state green laser with a wavelength of 535 nm and the resulting relative intensity distribution of the transmitted diffraction pattern was projected onto a screen and measured using a CCD array photometer (IQCAM Model II), as shown in Figure 4.5.  41  Diffracted Beams  Grating Laser  CCD Array Photometer  Screen  (NOT TO SCALE)  Figure 4.5 Experimental Setup to Observe the Transmitted Diffraction Intensity Distribution  Figure 4.6 shows the arrangement of the oil layer relative to the electrodes. The fluid used was immersion microscopy oil, which had a dielectric constant of 2.56, a viscosity of 1.25 kg-nf'-s" , and an index of refraction of 1.52. Each of the oil and air layers was 1  approximately 0.175 mm in thickness.  42  Figure 4.6 Schematic of the Fluid Interface Grating  4.2.2  Diffraction Intensity and Transmission  With no voltage applied to the electrodes, the oil-air interface was flat and thus did not contribute to diffraction. In this situation, only the ITO electrodes caused diffraction. A measurement of the angular separation of the ITO diffraction pattern, 9, was obtained by measuring the distance from the grating to screen, /, and the width between the zeroth and first order diffracted beams, w, as shown by Figure 4.7 and Equation 4.2.  43  1st Order Diffracted Beam \  Oth Order Diffracted Beam  Grating  (NOT T O S C A L E )  U Screen  Figure 4.7 Parameters to Calculate the Angular Separation of the Diffraction Pattern tan0 = —  (4.2)  With / = 200 ± 5 cm, and w = 10 ± 2 mm, the angular separation was calculated to be 0.29° ± 0.06°. This was in good agreement with the independent calculation of the angular separation using Equation 3.1 of Section 3.1. In that case, for d = 100 ± 10 urn, m = 1 and A = 535 nm, the angular separation of the peaks was calculated to be 0.31° ± 0.03°.  However, as illustrated by Figure 4.8, when a voltage was applied, a surface deformation with a period of 200 urn was induced, which was twice that of the ITO electrodes.  44  Air  Glass  b) Voltage Applied  a) No Voltage Applied  Figure 4.8 The Fluid-Air Interface Grating With and Without Voltage Applied  This spatially modulated deformation of the surface arose because the profile of the oil-air interface changed to minimize the total energy of the system, creating a new diffractive structure. This diffractive structure yielded diffraction angles at half the angular spacing of the ETO electrodes, a result that was easily distinguishable. This is a key benefit of the fluid interface grating; new diffraction peaks were generated and reversibly removed by controlling the surface of the fluid interface via an electric field. Thus, by using this technique, a new type of variable diffraction grating can be produced.  By estimating e = lxlO" F/m, £ = 2.6£ , y= 0.1 J/m , d = 0.35x10" m, w = 200x10" m, 11  0  2  oil  3  6  ()  and V= 100 V, Equation 4.1 roughly predicted a, the height of the perturbation, to be approximately lxlO" m, which should create a substantial diffraction effect. As expected, it 7  was observed that 100 V did indeed create a surface deformation large enough to cause readily observable diffraction.  45  As mentioned before, the spacing of the diffraction angles of this surface grating was half as large as that of the ITO electrodes. The intensity of the transmitted diffraction pattern with and without the applied electric field was measured using the IQCAM CCD array photometer and the results are shown in Figures 4.9 and 4.10. In this case, the transmission intensity of the 1 order peaks of the ITO diffraction pattern was too weak to be detected by the st  photometer.  a) No Voltage to Electrodes  b) 120 V to Electrodes  Figure 4.9 The Diffraction Patterns Caused by Air-Oil Surface Deformation and Electrode Array With and Without Periodic Applied Voltage  46  Figure 4.10 The Luminance of Diffraction Patterns Caused by A i r - O i l Surface Deformation and Electrode Array With and Without Periodic Applied Voltage vs the Position on the Viewing Screen  The amplitude of the perturbation affects the intensity of the diffraction pattern due to the relative phase shifts involved as the amplitude changes. A simple one-parameter model was considered in which two beams with a relative phase difference that is proportional to the square of the applied voltage are summed to obtain the transmission intensity. If the beams are in phase, then constructive interference occurs and the relative transmission value is unity. If the beams are 180° out of phase, then destructive interference occurs and the relative transmission value is zero. Using this idea, the intensity in such a model has the form: 47  l +  r  T=  cos(kV )^ 2  2  (4.2)  where T is the fractional transmission of the zeroth order peak, Vis the applied voltage, and k is the adjustable free parameter. The successful comparison between this model and the data is shown in Figure 4.11.  1  •  c o ••-* 0.9 u <0  — Model (1 Free Parameter)  0.8  Q  Experimental Data  I-  a>  T3 0.7  o o k_  0.6  N  0.5  a>  0.4 0.3 0.2 0.1  0  20  40  60  80  100  120  140  160  180  200  220  240  Applied Voltage (V)  Figure 4.11 Comparison between Model and Data of Zeroth Order Diffraction as a Function of Applied Voltage for Fluid Interface Grating  Figure 4.11 illustrates the fact that the intensity of the zeroth order decreased as the applied voltage increased. This is due to the fact that, as the applied voltage increased, the amplitude of the surface deformation became a substantial fraction of the laser wavelength, transferring 48  considerable power to the diffraction pattern and substantially reducing the intensity of the zeroth order. Hence, the intensity of the peaks of the oil-air interface grating can be increased or decreased by controlling the applied voltage to the ITO electrodes.  The presence of a macroscopic lensing effect at more than 160 V, as illustrated in Figure 4.12, prevented accurate transmission measurements by blurring the diffraction pattern.  Macroscopic Lensing Surface  Microscopic Deformations  Figure 4.12 The Formation of Macroscopic Lensing Surface  This lensing effect is probably due to imperfections in the LTO electrodes caused by etching inconsistencies. The non-uniform electric field created by these imperfections resulted in macroscopic lensing surfaces, which acted to diverge incoming light. While this effect was always present to some degree, it did not affect the diffraction pattern at voltages less than 160 V. However, as the voltage was increased, this effect noticeably blurred the diffraction pattern.  49  4.2.3  Time Response  It was interesting to consider the equilibration time of the surface perturbation in response to a change in the applied voltage. A complete numerical solution can be used to predict this equilibrium time but no innovative physics would be demonstrated in such an immense exercise. Hence, a rough estimate was deemed appropriate in this case. Simple dimensional arguments were used to estimate the time response, and the result is shown by Equation 4.3 below.  T*  where ris the response time,  v  v  w  y  d  2  (4.3)  and yaxe, the viscosity and surface tension co-efficient of the  oil, respectively, d is the total thickness of the oil and air, and w is the spatial period of the perturbation. The details of this dimensional analysis are located in Appendix D.  By substituting estimated values of w = 200xl0" m, d = 0.175xl0" m, v= 1.25 kg-m's" and 6  3  1  y= 0.1 J/m into Equation 4.3, rwas predicted to be on the order of milliseconds, rwas 2  observed to be in the range 20 to 30 ms, which was not inconsistent with this rough estimate.  According to Equation 4.3, the predicted equilibration time is directly proportional to the viscosity of the liquid. Hence, the response time can be decreased simply by reducing the viscosity of the liquid. However, for sufficiently low values of viscosity, the response would no longer be limited by the viscosity but would instead be limited by inertial effects. Based on simple dimensional arguments, this inertial limited response time was estimated to be:  50  (4.4)  where p and ^are the density and surface tension co-efficient of the oil, respectively, and w is the spatial period of the perturbation. The derivation of Equation 4.4 is also located in Appendix D.  By substituting the estimated values of w = 200x10" m, p = 1000 kg/m , and y= 0.1 J/m 6  3  2  into Equation 4.4, the mass-limited rwas calculated to be roughly 0.5 ms, approximately 100 times shorter than the observed experimental result. Hence, with the oil and spacing used for this grating, it was primarily the viscosity that controlled the response time, and Equation 4.3 was the best estimate.  4.3  Measurements and Analysis of an Array of Individual Electrodes  Active control of the spatial frequency of the field-induced gratings was accomplished by individually controlling each electrode in an inter-digital electrode array. By controlling the amount of applied voltage to specific electrodes, it was possible to control both the angular spacing and intensity of the resultant diffraction pattern. This section discusses the materials, and the results and analysis of the diffraction intensity and transmission measurements for this type of grating.  4.3.1  Materials and Setup  The new inter-digital electrode array, with separate electrodes, is shown in Figure 4.13.  51  Figure 4.13 The Individual Electrode Array  This array was etched from a transparent ITO layer on a glass substrate using the same process described in Section 4.2.1. The individual ITO electrodes were approximately 40 um wide with a center-to-center spacing of 100 p.m. The experimental setup used to measure the transmission intensity of the diffraction pattern was the same as the one shown previously in Figure 4.5.  4.3.2  Diffraction Intensity and Transmission  The oil-air interface grating was filled with a layer of immersion oil and a layer of air, both of thickness 0.175 mm, and the electrodes were connected in 4 sets, as shown in Figure 4.14.  52  Glass ITO  Air —=>—^  ^  ^  ^  ^ ^  —  ^  / / ^ / ^ / ^ O i i - ; ^ l^rT>r?TfTC^rVii^^  ITO Electrodes Glass  Figure 4.14 A Schematic of Electrode Connections  As before, with no voltage applied, only the ITO electrodes themselves diffracted light. With 120 V applied to the 1 and 3 sets of electrodes, a situation similar to the results of Section st  rd  4.2.2, a surface deformation with half the spatial frequency of the electrodes was created, and the angular spacing of the diffraction pattern was halved. When 120 V was applied to the l , sl  2 , and 3 sets of electrodes, the spatial frequency of the deformation became one quarter nd  rd  that of the electrodes, and the angular spacing was halved once more. The surface deformations and the corresponding diffraction patterns of these three situations are shown in Figures 4.15 and 4.16 respectively.  53  Separate ITO Electrodes  a) No Voltage Applied  Glass b) 120 V to 1st and 3rd Sets of Electrodes  c) 120 V to 1st, 2nd, and 3rd Sets of Electrodes  Figure 4.15 The Fluid-Air Interface Grating With Voltage Applied to Various Electrodes  Figure 4.16 The Diffraction Patterns Caused by Air-Oil Surface Deformation With Voltage Applied to Various Electrodes  Applying voltage to only the 1 set of electrodes also created a deformation with one quarter st  the spatial frequency of the electrodes. However, as shown in Figure 4.17, this deformation was not exactly the same as the one created by applying voltage to the 1 , 2 st  n d  and 3 sets of l d  electrodes. Hence, even though the spatial frequencies were the same, there were phase differences between the two deformations, which created differences in the intensity of the diffractive peaks, as shown in Figure 4.18.  Glass  a) 120 V to 1st, 2nd, and 3rd Sets of Electrodes  b) 120 V to 1st Set of Electrodes  Figure 4.17 Two Fluid-Air Interface Gratings With Similar Periodicity but Different Phase  55  a) 120V to 1st, 2nd and 3rd Sets of Electrodes  b) 120V to 1st Set of Electrodes  Figure 4.18 The Diffraction Patterns Caused by Two Fluid-Air Interface Gratings With Similar Periodicity but Different Phase  4.4  Conclusions  Thus, it was possible to create a variable diffraction grating by spatially modulating an electric field to create a periodic deformation in an oil-air interface. The next chapter describes another such method, the electrophoretic grating, which uses the same spatially modulated electric field. However, instead of controlling the surface properties of fluidic interface, this grating controls the electrophoretically induced motion of charged high index particles suspended in a low index fluid.  56  5  ELECTROPHORETIC GRATING  This optical technique used electrophoretic motion of high index, charged pigment particles suspended in a low index fluid to create a diffraction grating. B y applying a patterned electric field produced by an inter-digital array, it was possible to create a tunable diffractive structure by spatially modulating the location of the charged particles to induce substantial periodic changes in the effective index of refraction of the suspension. Similar to the previous fluid interface grating, two methods of controlling the electrophoretic grating were tested. One technique used an array of inter-digital electrodes connected together to modulate the location of the electrophoretic particles to produce and remove a diffractive structure with fixed spatial frequency. The second technique used an array of individually connected inter-digital electrodes to modulate the location of the electrophoretic particles to produce and remove a diffractive structure with variable spatial frequency.  5.1  Measurements and Analysis of a Single Control Electrode  Similar to the single electrode of the fluid interface grating described in Section 4.2, an array of inter-digital electrodes was connected together to create a single common control electrode. This electrode was used to apply a patterned electric field through an electrophoretic suspension to produce the desired diffractive structure. This section discusses the materials, the results and analysis of diffraction intensity and transmission measurements, and a successful transmission model for such an electrophoretic grating.  57  5.1.1  Materials and Setup  This electrophoretic grating used the same inter-leaved transparent I T O electrodes and diffraction measurement setup as in the oil-air interface grating, as shown in Figures 4.3 and 4.5 respectively. However, instead of employing a fluid-air interface, the grating was filled with electrophoretic suspension of thickness 0.05 mm, as shown in Figure 5.1.  Glass ITO Particles f^l  ^  g=3  r^3"E^  ITO Inter-Digital Array Glass  Figure 5.1 Schematic of the Electrophoretic Particle Grating  This colloidal suspension contained positively charged pigment particles with a diameter of approximately 200 nm as well as non-pigment counter-ions. The liquid medium, a perfluorinated hydrocarbon , had an index of refraction of 1.276, while the pigment particles 48  had an estimated index of refraction of about 1.8 at the laser wavelength of 535 n m .  49  58  5.1.2  Diffraction Intensity and Transmission  With zero voltage applied to the electrodes, the individual free-floating particles caused negligible diffraction. In this situation, there was no substantial index of refraction changes present in the bulk of the electrophoretic solution and only the inter-digital ITO electrodes diffracted light. However, as illustrated by Figure 5.2, when voltage was applied, the high index electrophoretic particles were attracted to every second electrode, causing substantial diffraction effects due to spatially modulated differences in the index of refraction in the bulk of the suspension.  — Glass -ITO ParticlesITO Array  + la -  BI  + m -  BI  + ia -  BI  +  Glass  a) No Voltage Applied  b) Voltage Applied  Figure 5.2 The Electrophoretic Particle Grating With and Without Voltage Applied  As illustrated above, this induced electrophoretic diffraction grating had a period of 200 urn, which was twice that of the ITO electrodes. This led to the expectation that the diffraction angles of the new grating should be half that of the electrodes. Figures 5.3 and 5.4 show the  59  diffraction pattern and its intensity profile created by the electrophoretic particle grating for an applied voltage of 45 V, corresponding to an electric field of approximately 10 V/m. In 6  the zero-field case, unlike the oil-air interface grating, the IQCAM photometer was able to detect the 1 order diffraction spots formed by the array of ITO electrodes. This effect was st  attributed to the larger refractive index differences between the ITO and the perfluorinated hydrocarbon compared to the ITO and the microscopy oil. As predicted, when the voltage was applied, the electrophoretic particle grating yielded diffraction angles at half the angular spacing as those caused by the electrode array. After the voltage was removed, it was observed that a faint electrophoretic diffraction pattern remained, due to the fact that some particles remained trapped on the ITO electrodes.  1st Order due to Particles Remaining on ITO Electrodes  1st Order From ITO Electrodes  I  —Zeroth Order—  a) No Voltage Applied  1st Order From .Electrophoretic Particle Grating  b) 45 Volts Applied  Figure 5.3 The Diffraction Patterns Caused by Electrophoretic Particle Grating and Electrode Array With and Without Periodic Applied Voltage  60  Figure 5.4 The Luminance of Diffraction Patterns Caused by Electrophoretic Particle Grating and Electrode Array With and Without Periodic Applied Voltage  As with the fluid interface grating, this result is the key feature of the electrophoretic grating. New diffraction peaks were created and reversibly removed simply by controlling the location of the electrophoretic particles via an electric field. Thus, this technique can be used to create a new type of variable diffraction grating. Using G-Solver®, a diffraction grating software package , a diffraction model was developed to predict the relative zeroth order 50  transmission as a function of the thickness of the field-induced electrophoretic grating. In this two dimensional model, the electrophoretic particle grating formed by the pigment particles was approximated as a solid slit grating with variable thickness, t. The transmission 61  of the zeroth order peak was then calculated as a function of t. A schematic of the model and its results are shown in Figures 5.5 and 5.6, respectively.  Air(n=l) Glass (n=1.5) ITO (n=1.96) Liquid Medium (n=1.27) Particles (n=1.8) ITO (n=1.96) Glass (n=1.5) lOOum  Air(n=l)  200um (Not to Scale)  Figure 5.5 The Schematic of the Electrophoretic Particle Grating Model  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  Thickness of Electrophoretic Grating (Normalized w.r.t. Laser Wavelength)  Figure 5.6 The Modeled Results of Relative Transmission of Zeroth Order Diffraction as a Function of the Thickness of Electrophoretic Grating 62  As illustrated by Figure 5.6, an applied voltage of 45 V created an electrophoretic grating with fractional zeroth order transmission value of about 0.87 ± 0.09. When this result was compared with the model, it was apparent that an applied voltage of 45 V corresponded to a grating thickness of approximately 160 ± 50 nm. Since the diameter of one particle was approximately 200 nm, the effective thickness of the electrophoretic grating was roughly one single layer of pigment particles.  5.2  Measurements and Analysis of an Individual Electrode Array  As in the case of the fluid interface grating, active control of the spatial frequency of the field-induced electrophoretic grating was achieved by individually controlling each electrode. By applying voltage to specific electrodes, it was possible to control the angular spacing and intensity of the resultant diffraction pattern. This section deals with the materials, and the measured diffraction intensity and transmission of this type of electrophoretic grating.  5.2.1  Materials and Setup  The inter-digital array used in this case was exactly the same as the one used previously for the fluid-interface grating, as shown in Figure 4.13. Similar to the first electrophoretic grating, this grating was filled with the same electrophoretic suspension of thickness 0.05 mm. The electrodes were connected in sets of 4, as shown in Figure 5.7.  63  Glass ITO Particles rn  m  r n ran m  m  m  r n - m  rxirn  nn  ITO Inter-Digital Array Glass  Figure 5.7 Schematic of Electrode Connections for the Electrophoretic Grating  The experimental setup used to measure the intensity and transmission of the diffraction pattern through the grating was the same used for the fluid interface grating, as shown in Figure 4.5.  5.2.2  Diffraction Intensity and Transmission  Using the same procedure used for the multiple electrode oil-air interface grating, it was possible to vary the spatial frequency of the electrophoretic diffractive structure by controlling the amount of applied voltage to specific electrodes. With no voltage applied to the electrodes, only the electrodes themselves diffracted light. Applying voltage to the 1  st  and 3 sets of electrodes created an electrophoretic diffractive structure with a spatial r d  frequency half that of the ITO electrodes. A s such, the corresponding angular spacing of this electrophoretic grating was half that of the ITO electrodes. When voltage was applied to the  64  1 , 2 and 3 sets of electrodes, the spatial frequency of the electrophoretic diffractive st  nd  rd  structure became one quarter that of the electrodes, and the angular spacing was halved once more. This also occurred when voltage was applied only to the 1 set of electrodes. st  However, even though the spatial frequency of the last two situations was the same, the location of the particles was not the same, causing a phase difference that varied the intensity of the diffracted peaks. These four electrophoretic structures and their corresponding diffraction patterns are shown in Figures 5.8 and 5.9, respectively.  Glass ITO Particles Separate ITO Electrodes  a) No Voltage Applied  Glass  b) 60 Vto 1st Set of Electrodes  c) 60 Vto 1st and 3rd Sets of Electrodes d) 60 V to 1st, 2nd, and 3rd Sets of Electrodes  Figure 5.8 The Electrophoretic Particle Grating With Voltage Applied to Various Electrodes  65  a) No Voltage Applied  •  *  |'# *  if Zeroth Order  *  b) 60 V Applied to 1st Set of Electrodes  1  1  Zeroth Order  if  W'  c) 60 V Applied to 1st and 3rd Sets of Electrodes  *  i  d) 60 V Applied to 1st, 2nd and 3rd Sets of Electrodes  Figure 5.9 The Diffraction Patterns Caused by Electrophoretic Particle Grating With Voltage Applied to Various Electrodes  5.3  Conclusions  The electrophoretic grating was similar to the oil-air interface grating in that, by applying a patterned electric field through the electrophoretic solution, it was possible to create a diffractive structure of selectable spatial frequency with the only limit being the physical dimensions of the electrodes themselves. However, there are also macroscopic optical effects that can be obtained by means other than microscopic gratings. One such effect is the electrostatic control of the shape of water drop on a super-hydrophobic surface using the concept of electrowetting, as discussed in the next chapter.  66  6  R E F L E C T I V E E L E C T R O W E T T I N G DISPLAY  It is possible to use the reflectance characteristics of an array of variably electrowetted water drops on a super-hydrophobic surface as a basis for a new type of semi-retro-reflective display. As mentioned previously in Section 3.7, a truncated water sphere in air with a contact angle of 150° reflects very little (~ 3%) of incident light, whereas a hemisphere reflects 43% of incident light back towards the viewer in a semi-retro-reflective manner. Thus, a reflective display can be created by using the concept of electrowetting to change the shape of an array of drops from truncated spheres to hemispheres and back. This is the key difference between the electrowetting display described in this section and the prior electrowetting display work; the prior work uses electrowetting to change the surface area of coloured drops ' while this research uses electrowetting to change the reflective properties 51  52  of the drops themselves.  This chapter describes the experiments performed to determine the required properties of a reflective electrowetting display. Topics include the appropriate materials, the prevention of charge deposition, the stabilization of the system during thermal cycling, the time response of the system, the electrowetted contact angles and their corresponding reflectance, both as a function of applied voltage, and the results of two multiple drop electrowetting systems.  6.1  Materials and Experimental Setup  Several hydrophobic and super-hydrophobic materials were obtained to determine the one most suited for an electrowetting display system. The candidate materials were Teflon AF 1600® from Dupont , CRC 6000® from Boyd Coatings Research Company Ltd. , 53  54  67  Fluorothane® from Cytonix , and HIREC 100® from NTT Technologies . A table of these 55  56  materials and their relevant properties is located in Appendix C.  Teflon AF 1600® had a contact angle of 105°, which was too low for the purposes of an electrowetting display. Boyd CRC 6000® had contact angle of 140°, but was not transparent and so could not be used for display purposes. Fluorothane® had a high contact angle of 140° - 150°, was translucent and slightly yellowish and so was acceptable. However, the equipment needed to properly create a uniform coating was not readily available. HIREC 100® had a contact angle of 140° - 150°, was translucent and white. Additionally, NTT Technologies uniformly coated this material onto the surface of the test cells. Hence, HIREC 100® was chosen as the best available material because of its high contact angle, relative transparency and uniformity. However, the best choice of material for an electrowetting display would be a transparent super-hydrophobic surface. As mentioned previously, these transparent materials have been reported but, unfortunately, are not readily available.  57  The contact angle of a water drop on a smooth and roughened HIREC 100 surface was measured to be 110° and 140° - 150°, respectively. From Cassie's Equation (Section 3.5, Equation 3.9), the roughness factor,/, was calculated to be 0.40 - 0.45. In other words, approximately 40 - 45% of the rough surface consisted of tips that were in contact with the drop.  The effect of gravity determined the maximum size of a spherical drop. This maximum size was easily obtained experimentally, as shown in Figure 6.2.  68  Figure 6.1 Determining the Maximum Spherical Drop Size  The 1 three drops of Figure 6.2 are spherical and were approximately 1 mm, 1.5 mm and 2 st  mm in diameter, respectively. The last drop, approximately 2.5 mm in diameter, is noticeably non-spherical, as it was deforming under its own weight. Thus, to eliminate the influence of gravity, the diameter of all subsequent electrowetting drops were kept to 2 mm or less.  The electrowetting substrate was an ITO covered glass slide, with the ITO acting as the bottom electrode during the electrowetting process. The ITO layer was insulated with a 10 um layer of Parylene C , a material with a high dielectric strength (270MV/m). Parylene C 58  is commonly used in the electronics industry as an insulator and has a dielectric constant of 3.15. Next, the Parylene C was coated with a layer of HIREC 100® 18 um in thickness. While the dielectric constant of HIREC 100® was approximately 2 - 3, it was also assumed to be 3.15 during calculations for convenience. Using tweezers, HIREC 100® was carefully 69  removed to expose a layer of Parylene C approximately 1 mm in diameter. The purpose of this Parylene C "hole" was to anchor the water drop in place. As mentioned previously in Section 3.6, when a drop was electrowetted on a super-hydrophobic surface, contact angle hysteresis prevented the drop from returning to its original state. A slight mechanical disturbance eliminated this hysteretic effect, restoring the original shape of the drop. However, the disturbance displaced the drop from its original location due to the low surface energy between the drop and the super-hydrophobic coating. By placing the drop in the Parylene C hole, the drop was locked in place due to the high surface energy between the drop and the Parylene C. Only the center of the base of the drop was in contact with the Parylene C. Since the outer diameter of the base of the drop remained in contact with the HIREC 100®, the drop retained its high contact angle, as shown in Figure 6.1.  Gold Wire Electrode  a) Non-Electro-wetted State  b) Electro-wetted State  Figure 6.2 A Schematic of Two States of the Electrowetting System  70  The coated slide was placed on a vibration table, which was the disturbance mechanism used to eliminate contact angle hysteresis. A gold wire, approximately 50 urn in diameter, acted as the electrode connecting the drop to the voltage source. Gold was chosen due to its inertness to prevent oxidation, which would contaminate the water drop, altering the surfacetension co-efficients and reducing the contact angle of the drop. The thinnest possible wire was used to minimize any surface deformation of the water drop due to surface tension effects between the wire and the drop. A digital still camera and a digital slow motion video camera, capable of taking pictures every 2 ms, were used to record various electrowetted states of a water drop. A sample picture of a water drop on HIREC 100® and its electrowetting equivalent are shown below in Figure 6.3.  a) Non-Electrowetted State  b) Electro-wetted State  Figure 6.3 The Non Electrowetted and Electrowetted States of a Water Drop and their Corresponding Reflective Annuluses  The remaining sections of this chapter deal with further testing and analysis of various properties of similar electrowetted drops.  71  6.2  Electrowetting Using Oscillating and Static Electric Fields  A comparison between electrowetting a water drop using an oscillating and a static electric field was performed. This was accomplished by using A C and D C voltage sources to create oscillating and static electric fields, respectively. A n oscillating electric field was determined to be much more effective in an electrowetted application for a variety of reasons.  6.2.1  Analysis of Observations  It was observed that the contact angle of a water drop did not decrease as much when electrowetting with D C voltage instead of the equivalent R M S A C voltage. It was surmised that this effect is caused by deposition of charges at the edge of the drop due to air ionization, as shown in Figure 6.4.  Figure 6.4 Deposition of Charge at Edge of Drop due to Ionization of Air The presence of charges at the contact edge of the drop would decrease the electric field in this region, reducing the effective electrowetting force on the drop. This would decrease the  72  electrowetting effect on the drop such that the contact angle would not be reduced as much. Furthermore, it was postulated that this effect would be reduced when using AC voltage because the switching of the polarity would prevent the accumulation of charges at the contact edge.  6.2.2  Measurements and Analysis  To test this theory, a drop of water was electrowetted in both air and microscopy oil using an applied DC voltage of 270 V and an applied AC voltage of 384V at 2020 Hz . Since the oil has a higher dielectric strength than air (approximately 20 MV/m and 3 MV/m, respectively), any charge deposition that may occur in air will be suppressed by the oil. Hence, in the DC situation, the contact angle of the electrowetted drop should be greater in air than in oil. However, if there is no charge deposition due to air ionization, then the contact angle of the electrowetted drop will be similar in air and oil. In the AC situation, due to the lack of air ionization, the contact angle of the electrowetted drop should be similar in air and oil. As shown in the DC case of Figure 6.5, the contact angle of the water drop was substantially larger in air than in oil. As shown in the AC case of Figure 6.6, the contact angles of a water drop in air and in oil were very similar. The slight differences were attributed to the different dielectric constants and surface tension co-efficients of the two fluids.  73  a) Electrowetted Water Drop in Air  b) Electrowetted Water Drop in Oil  Figure 6.5 The Electrowetted States of a Water Drop in Air and in Oil Using 270V DC  a) Electrowetted Water Drop in Air  b) Electrowetted Water Drop in Oil  Figure 6.6 The Electrowetted States of a Water Drop in Air and in Oil Using 384V A C at 2020 Hz  74  Hence, the results of these two measurements verify the hypothesis. Unfortunately, oil cannot be used as the medium in which a water drop wets out because the index of refraction of oil is larger than that of water. As mentioned previously in Section 3.7, this would cause little light to be reflected from the electrowetted hemisphere. It was also observed that the contact angle of an electrowetted water drop did not decrease as much at low AC frequencies in the range of 1 - 20 Hz as compared to 50 Hz and higher. This was due to the fact that, as the frequency decreased, the system approached the DC case and there was some charge deposition at the contact edge. Low frequency AC was also not desirable due to its coupling to several resonant frequencies of the drop, which is discussed in Section 6.5. Thus, the simplest solution to avoid these issues was to use an oscillating AC voltage supply, with frequencies of 1000 Hz and higher, to create a reflective electrowetted water drop in air.  6.3  D r o p Evaporation and Recondensation  For an electrowetting system to be practical, the small water drops of the system must be able to withstand temperature changes in the environment. This section provides a discussion of the observations and corresponding solutions to the problems encountered during thermal cycling of an array of distilled water drops.  6.3.1  Analysis of Observations  Since small water drops evaporate quickly in open air, in any real device the drops would have to be enclosed in a sealed cell. A sealed cell containing a distilled water drop on a super-hydrophobic surface was heated to approximately 50°C from room temperature and 75  then cooled back down. The size of the drop decreased as the water vapour increased until thermal equilibrium was reached. However, two problems were observed during cool down, Firstly, the water vapour did not return back to the drop as condensation formed everywhere within the cell. Secondly, the water vapour also recondensed within the air voids of the super-hydrophobic surface. This effectively decreased the hydrophobicity of the surface, changing the shape of the remaining small drop by substantially reducing its contact angle. Thus, not only was the drop reduced in size after thermal cycling, but its contact angle was also changed, as depicted in Figure 6.7.  a) Initial State  b) Vapourization  c) Recondensation  Figure 6.7 The Evaporation and Recondensation of Distilled Water Drop During Thermal Cycling  A possible solution to these problems was proposed, based on the use of saline solution, for example, N a C l in distilled water, instead of just distilled water. It was reasoned that the lower vapour pressure of a saline solution would prevent pure water drops from forming within the cell during the thermal cycling process. The salinity of a salt-water solution is defined as the amount of salt in parts per thousand of distilled water; a higher salinity 76  corresponds to a lower vapour pressure. By enclosing a salt water drop and heating it, the drop should reduce in diameter as water vapour forms until equilibrium is reached, similar to the distilled water case. However, upon cooling, droplets of pure water should not form due to the higher vapour pressure of pure water. Moreover, the water vapour should preferentially condense onto the salt water because of its lower vapour pressure. A simple model was generated to roughly predict the time required to completely recondense a drop of pure water onto a salt-water drop at 25°C. This model is summarized by Equation 6.1:  t = —^-r  (6.1)  an Ay/ dx  where t is the time required for the pure water to evaporate and recondense onto the salt water, N is the number of atoms of pure water to be evaporated and recondensed, A is the cross-sectional flux area,  y/is  the diffusion constant, and  dn/dx  is the vapour concentration  gradient. Using this model, the recondensation time of a drop of distilled water into a drop of salt water is estimated to be on the order of 20 hours. The details of the model, as well as the estimated recondensation time, are described in Appendix E.  6.3.2  Measurements and Analysis  To verify the above hypothesis, 9 drops of water, approximately 2 mm in diameter, were placed in an enclosed cell, with 4 of them composed of distilled water and the remaining 5 composed of salt water with a salinity of 50. If the theory holds true, then, over time, the distilled water should evaporate completely, and only recondense onto the salt-water drops. 77  As shown by Figure 6.8, approximately half of the distilled water did recondense onto the salt-water drops and nowhere else in 12 hours.  Figure 6.8 The Evaporation and Recondensation of Distilled Water onto Salt Water over Time  Next, a salt-water drop was electrowetted and its shape was compared to that of an electrowetted distilled water drop. There were no observable differences between the contact angles of the two systems. Thus, using salt-water solution instead of distilled water eliminated the thermal cycling problems by controlling the location of recondensation in enclosed cells.  78  6.4  Time Response of Base Diameter of A n Electrowetted D r o p  The velocity of the base of an electrowetted drop was used to develop a model that predicted the contact angle of an electrowetted water drop as a function of applied voltage. This section describes the measurements and analysis of the time response of the base of an electrowetted water drop.  6.4.1  Analysis of Observations  The time response of the base diameter of an electrowetted water drop was defined as the total time from when drop first began to electrowet to when the base diameter did not change anymore, as shown in Figure 6.9 below.  '///////////'/, -/^//////////,  V/  Initial Base Diameter  a) Initial State of Drop  •///////// .////////  ///// //// //////'. "////////////////•  Final Base Diameter  b) Final State of Drop  Figure 6.9 The Initial and Final States of the Base Diameter of an Electrowetted Drop  Even though the base diameter may be stable, the overall drop may not be due to surface oscillations, which are discussed in detail in Section 6.5. However, determining the time  79  response of the base diameter is still valuable in that it can used, in conjunction with the instantaneous force on the contact edge of the drop, to determine the final contact angle of the drop at various applied voltages as well as the minimum required electrowetting voltage.  The net force per unit length, F, at the edge of the water drop is given by Equation 6.2:  F=-y, cos0 + (y -y ,) + v  sv  s  d  (6.2)  where y , y , and %i are the surface tension co-efficients of the water-air, the surface-air, and v  sv  the surface-water interfaces, respectively, 6 is the contact angle, £„ is the permittivity of free space, e and d are the effective dielectric constant and total thickness of the superr  hydrophobic and insulating coatings, respectively, and V is the applied voltage. Each of the terms of Equation 6.2, including the final term, which is the electrowetting force, F , is ewet  shown in Figure 6.10.  Solid Surface ewet  Figure 6.10 The Force Diagram of an Electrowetted Drop  80  If the base diameter as a function of time at a specific electrowetting voltage is known, two parameters can be calculated. Firstly, the instantaneous velocity, v, as a function of time, r, can be determined by taking the derivative of the base diameter with respect to time. Secondly, the contact angle of the drop, 6, can be geometrically calculated as a function of time, by realizing the volume of the drop must be conserved and 6„ = 140°. B y substituting these calculated contact angles and the specified electrowetting voltage into Equation 6.2, the instantaneous force, F, can be determined as a function of time. Hence, it is possible to obtain the relationship between the instantaneous velocity of the base diameter of the drop and the instantaneous force on the contact edge of the drop. From this relationship, the final force on the contact edge of the drop at zero base velocity, F , can be determined. B y a  substituting F in Equation 6.2, the final contact angle can be calculated for various applied 0  voltages. The required minimum electrowetting voltage can also be calculated by substituting F , and & into Equation 6.2. a  6.4.2  0  Measurements and Analysis  Water drops were electrowetted at various A C voltages, all at a frequency of 2020 H z . A digital slow-motion video camera was used to capture frames every 2 ms. In all cases, the base of the drop reached equilibrium within 8 - 10 ms. A sample of the electrowetted drop at various times is shown in Figure 6.11.  81  a) Initial State  b) 2 ms  c) 4 ms  d) 6 ms  Figure 6.11 An Electrowetted Drop over Time  As illustrated above, the surface of the drop did not maintain its spherical shape during the electrowetting process but instead deformed due to the inertial effects of the drop. Also, the surface of the drop did not reach equilibrium at the same time as the base diameter; it continued to oscillate. These oscillations, caused by both a beat phenomenon and the frequency of the applied voltage, are discussed in detail in Section 6.5.  The base diameter of the drop in each frame was measured, and from this, graphs of base diameter as a function of time for various voltages were generated. A sample graph with measurement errors of ±5% is shown in Figure 6.12 with the rest of the graphs located in Appendix F. The instantaneous velocity was obtained by taking the derivative of the fitted cubic equation. The cubic equations at the various voltages are tabulated in Appendix F as well.  82  Figure 6.12 Base Diameter of an Electrowetted Water Drop as a Function of Time Using 384 V at 2020 Hz  Using the process described at the end of Section 6.4.1, the relationship between the instantaneous velocity of the base diameter of the drop and the instantaneous force on contact edge of the drop was determined at 8 different voltages, ranging from 221 to 384 V, and the results are displayed in Figure 6.13. The numerical data, with ±10% errors in both the instantaneous velocity and instantaneous force, as well as a sample calculation, are located in Appendix G.  83  0.4  0.35  0.00  0.01  Fo 0.02  0.03  0.04  0.05  0.06  0.07  0.08  Instantaneous Applied Force, F (N/m)  Figure 6.13 The Instantaneous Velocity of Base Diameter of Electrowetted Drop as a Function of the Instantaneous Applied Force  It was observed that regardless of the magnitude of the applied voltage, there was slight logarithmic behaviour as the drop neared zero velocity. Hence, the above results were fitted to a quadratic equation. From this, the force at zero base velocity, F,„ was found to be 0.01597 N/m. The final contact angle of the drop at the 8 different voltages was calculated by substituting F and the 8 values of V into Equation 6.2. The results are shown in Table 0  6.1.  84  Table 6.1 The Predicted Contact Angles as a Function of Voltage Based on Equation 6.2 Applied Voltage at  Predicted Contact  2020Hz AC (Volts)  Angle (Degrees)  (±5°) 384  89  349  99  327  105  301  112  275  118  253  123  237  127  221  130  The minimum electrowetting voltage was determined by substituting F = 0.01597 N/m, and 0  0 = 0 = 140° into Equation 6.2. The result of V o  m m  = 180 V, with an error of ±10%, was in  good agreement with the observed fact that electrowetting did not begin until approximately 200 V.  Thus, by measuring the time response of base diameter, it was possible to derive a model that predicted the final contact angle as a function of applied voltage as well as the minimum voltage necessary for electrowetting to occur. This model was verified based on the contact angle measurements described in Section 6.6, but first, a section summarizing the surface oscillations of an electrowetted drop is presented.  6.5  Time Response for Total Drop Stabilization  It was observed that, even if the base diameter of the drop was stable, the overall shape of the drop remained unstable due to surface oscillations, which are undesirable for a reflective  85  display. This section describes the methods used to observe and minimize the effects associated with these surface oscillations.  6.5.1  Analysis of Observations  The time response for total drop stabilization was defined as time interval from when the drop began to electrowet to when the surface oscillations of the drop reached steady state. This time was always longer than the base stabilization time. A sample of a drop with a stable base and oscillating surface is shown in Figure 6.14.  1  2  3  6  7  8  4  9  5  10  Figure 6.14 The Surface Oscillations of a Drop with a Stable Base over Time in Increments of 2 ms  The two factors that contributed to surface oscillations were coupling of the input frequency to the natural frequencies of the drop, and the input frequency itself. It was observed that, if the input frequency was low enough, at 50 Hz or less, then the total drop stabilization time  was on the order of seconds or greater to damp out. It was surmised that these surface oscillations at low frequency were due to a beat phenomenon caused by the frequency of the input voltage signal being near one of several natural resonant frequencies of the water drop itself. If this is true, then moving the frequency of the voltage signal away from natural resonant frequency of drop should decrease this beat phenomenon, reducing the total drop stabilization time. Eventually, if the input frequency is far enough away from the resonant modes of the drop, then the beat phenomenon should quickly damp out. A t this point, the only surface oscillations present should be the steady state oscillations with a frequency that matches that of the applied signal. If this input frequency is high enough, these steady state oscillations should be effectively eliminated, as they should fluctuate too rapidly to be detected by the human eye.  6.5.2  Measurements and Analysis  It was possible to determine the total drop stabilization time of an electrowetted drop as function of input frequency by observing the images of the drop captured every 2 ms. The results shown in Figure 6.15 clearly demonstrate that the total drop stabilization time decreased as the frequency of input voltage increased.  87  300  250  c o  200 •  V)  c o  o 100  50 •  250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 4500 4750 5000  Frequency (Hz)  Figure 6.15 Drop Oscillation Stabilization Time of a Water Drop as a Function of Frequency of Input Signal at 384V AC  It was observed that a total drop stabilization time was on the order of seconds for an input frequency in the range of 20 - 40 H z . In the range of 1 - 10 H z , the drop never reached equilibrium. If this is due to the beat phenomenon, then there should be several resonant frequencies modes of the water drop in this frequency range. Several resonant frequencies of the water drop were discovered in the 10 - 45 H z range by coupling the vibration of a speaker onto the super-hydrophobic surface and observing the frequencies at which the drop began to oscillate. Hence, to eliminate this beat phenomenon, the frequency range of the applied A C  88  voltage was set at 50 Hz and higher. In this range, the only oscillations present were the steady state ones which match the frequency of the applied signal.  However, since these steady state oscillations changed the shape of the drop, the reflectance of the drop was also affected. If the input frequency was high enough to eliminate the beat phenomenon but still low enough for the steady state oscillations to be detected by the human eye, then the reflection changes due to these oscillations were visually noticeable, in the form of a pulsating reflection annulus that changed shape. Hence, the applied voltage must be considerably higher than 50 Hz, not only to rapidly damp out the beat oscillations, but to also reach a steady state oscillating mode that is undetectable to the human eye. Based on these requirements, all the measurements in this chapter used an input frequency of 2020 Hz. Hence, from these observations it was concluded that a low frequency input signal was undesirable. Using a high frequency input signal of 2020 Hz eliminated the problems associated with both the beat phenomenon and the steady state oscillations, reducing the total drop stabilization time to an order of 10 ms.  89  6.6  Contact Angle of an Electrowetted D r o p as a Function of A p p l i e d Voltage  It was possible to develop another model to predict the relationship between the contact angle and input voltage by modifying Equation 3.16. This model and the model presented in Section 6.4 were validated by measuring the contact angle as a function of applied voltage.  6.6.1  Analysis of Observations  It is reasonable to expect a roughened surface to create more friction-like effects. Hence, a rough surface should obstruct the electrowetting effect more as compared to a smooth surface, resulting in a higher contact angle. The magnitude of surface roughness is also expected to affect the degree of electrowetting. A rougher surface would hinder the electrowetting process more, leading to a higher contact angle. Since the electrowetting model summarized by Equation 3.16 does not take into account surface roughness, the expectation that this model would predict a lower contact angle than the experimental result is not unreasonable.  Analogous to Wenzel's Equation (Equation 3.8), which modified Young's Equation (Equation 3.6) to account for surface roughness, a new electrowetting model was similarly derived to account for surface roughness by introducing a roughness factor, R , into Equation a  3.16, as shown by Equation 6.3.  cos 6{V) = cos 6 + - R„ " '' £  0  2  £  V  7d  (6.3)  lv  90  The roughness factor R , is different from the roughness factor R, in Wenzel's Equation. 0  Since surface roughness is expected to reduce the electrowetting effect, R should always be t)  less than 1. By adjusting this roughness parameter, it should be possible to obtain a model that fits the contact angle data.  6.6.2  Measurements and Analysis  The digital still camera was used to take several pictures of a single drop electrowetted at various voltages. From these pictures, the contact angle of the drop was measured several times and averaged, producing a standard deviation of ±7°. This technique is shown in Figure 6.16, and the results are listed in Table 6.2.  Figure 6.16 Measuring the Contact Angle of an Electrowetted Drop  91  Table 6.2 The Measured Contact Angle as a Function of Voltage for an Electrowetted Drop Applied Voltage at  Measured Contact  2020Hz A C (Volts)  Angle (Degrees)  (±7°) 402  98  370  104  338  110  300  116  269  124  238  130  207  137  The predicted results of Equation 6.2, as summarized by Table 6.1, are presented once more in Table 6.3 for comparison.  Table 6.3 The Predicted Contact Angles as a Function of Voltage Based on Equation 6.2 Applied Voltage at  Predicted Contact  2020Hz A C (Volts)  Angle (Degrees)  (±5°) 384  89  349  99  327  105  301  112  275  118  253  123  237  127  221  130  The measured results of Table 6.2, along with the predicted results of the three electrowetting models summarized by Equations 3.16, 6.2 and 6.3 are plotted and compared in Figure 6.17. 92  Figure 6.17 Comparison of the Measured and Predicted Contact Angle of an Electrowetted Water Drop as a Function of Applied Voltage  As shown above, Equation 3.16 predicted lower contact angles compared to the measured data, as expected. The model summarized by Equation 6.3 was fitted to the measured data within the error of the measurements by adjusting R , to be 0.65. The model summarized by (  Equation 6.2 was also within the error of the measurements. Hence, the models based on Equations 6.2 and 6.3 were verified by the contact angle measurements. Since the relationship between contact angle and applied voltage was now established, it was a simple matter to determine the corresponding relationship between drop reflectance and applied voltage.  93  6.7  Reflectance of an Electrowetted Drop as a Function of Applied Voltage  The reflectance of a water drop as a function of applied voltage was of much interest because of the potential of creating grey-scale effects by varying the reflectance of the drop. This section describes the model used to calculate the reflectance of the drop as a function of the applied voltage.  6.7.1  Analysis of Observations  TracePro®, a commercial ray tracing software package , was used to predict the change in 59  reflectance of an electrowetted water drop. TracePro® uses the Monte Carlo technique to probabilistically ray trace situations to within known statistical errors. The reflectance model developed was not complicated. The contact angle of a water drop was changed from 80° to 145° in 5° increments, while keeping the volume of the drop constant. In each case, the incident light on the drop was ray-traced to calculate the magnitude of the reflectance. The diameter of incident light beam was kept constant at the diameter of the hemispherical drop, as shown by Figure 6.18.  94  / / V /  ///////// /  I  /  /  /  /  /  /  /  \/ / / / / / / /  Beam Diameter -  a) Initial State  /  /  /  /  / /  / /  /  /  /  /  /  / / / / /  '  /  Beam Diameter -  b) Electro-wetted State  Figure 6.18 The Defined Beam Diameter for TracePro Reflectance Model  The reflectance of the drop was defined as the ratio of reflected light over the total incident light, which corresponds to the ratio of area of the reflected annulus to the area of the incident light beam. The model only accounted for reflections from the drop; it did not account for any partial reflections from other materials, such as the glass substrate. It also neglected the diffusing effects of the rough super-hydrophobic surface.  6.7.2  Model Results  The results of the TracePro® model are shown in Figure 6.19.  95  45  o-l 70  1  !  !  1  80  90  100  110  !  j  j  i  120  130  140  150  Contact Angle (Degrees)  Figure 6.19 The Modeled Reflectance of an Electrowetted Water Drop as a Function of Contact Angle  As shown above, the maximum reflectance value of the water drop occurred when the drop was in the shape of a hemisphere. The maximum value of 43% is exactly in agreement with the predicted result of Equation 3.17, as summarized in Appendix A. Another interesting result of the reflectance model was the fact that a contact angle of 130° minimized the reflectance of the drop to approximately 3%. Hence, the non-electrowetted contact angle of 140° - 150° for a drop on HIREC 100® was unnecessarily high. Possible advantages of a lower initial contact angle of 130° include less contact angle hysteresis and/or lower applied voltages to create an electrowetted hemisphere.  96  By combining the results of Table 6.2 and Figure 6.19, the relationship between reflectance and applied voltage was obtained, as shown in Figure 6.20.  o-l—I—! 200  225  1—! 250  1—! 275  !  1  !  1  !  1  !  300  325  350  375  400  425  450  , 475  : 500  Applied Voltage (Sine Wave at 2020Hz) (V)  Figure 6.20 The Modeled Reflectance of an Electrowetted Water Drop as a Function of Voltage  As shown above, the reflectance of a water drop can be varied from 3% to 43% simply by controlling the applied voltage from 230 V to 440 V. Thus, by electrowetting multiple drops together at various voltages, it is possible to create a pixilated display capable of grey-scale effects. The results of two electrowetting multiple drop systems are presented in the next section.  97  6.8  Electrowetting Multiple Drops  A large reflective pixel can be achieved by controllably electrowetting an array of water drops. Two prototypes of a multiple drop electrowetting system are presented in this section, one of which uses a common electrode for an array of drops, the other of which uses separate electrodes for each drop.  6.8.1  19-Drop Hexagonal Pixel  A reflective electrowetting pixel composed of 19 drops in a hexagonal close packed structure was built to demonstrate the potential of an electrowetting display. The hexagonal structure was used to maximize the reflectance per unit area. The materials for this prototype were the same as the ones used in the previous experiments except much stiffer inert tungsten wire replaced the gold wires for ease of assembly. The drops were not encapsulated in this prototype display. The electrowetting result of this 19-drop array is shown in Figure 6.21.  DO *> O O  a) Non-Ele ctrowette d State  b) Ele ctrerwette d State  Figure 6.21 The Non-Reflective and Reflective States of a 19-Drop Hexagonal Electrowetted Pixel Display  98  Although one drop did not wet down due to electrode damage, this prototype clearly demonstrated the concept of an electrowetting display. The reflection of a hexagonal array of drops can be maximized by packing the drops as close as possible. The maximum reflectance of an ideal close packed hexagonal structure of electrowetted hemispherical water drops in air is 38%, which is quite close to the maximum reflectance of a single hemisphere at 43%. The details of this calculation are located in Appendix H. Obviously if the drops were to be individually encapsulated, the packing fraction of the hexagonal array would be reduced and the reflectance would drop accordingly.  6.8.2  Separately Controlled Drops in a 7-Drop Pixel  It was also possible to individually control each drop within a large pixel. If the drops are small enough, then separately electrowetting these droplets into fully reflective or fully transmissive states will create grey-scale effects. An individually controlled array of 7 drops is shown in Figure 6.22 below.  a) State 1  b) State 2  c) State 3  d) State 4  Figure 6.22 Several Reflective States of Individually Electrowetted Drops in a 7Drop Pixel  99  This prototype clearly demonstrated the ability to change the reflective state of a multiple drop electrowetting system by controlling the applied voltage to each of the drops.  6.9  Conclusions  It was verified by the experiments described in this section that, by using the properties of super-hydrophobic coatings and the techniques of electrowetting, it is possible to produce a reflective electrowetting display capable of displaying grey-scale effects. From the experiments performed, it was determined that the most useful electrowetting system employs a super-hydrophobic surface to create a low reflective state, uses an AC voltage supply in the KHz range to prevent charge deposition and drop oscillations, and exploits the properties of salinity to withstand thermal cycling. Also, the system should be provided with a slight mechanical disturbance to reduce contact angle hysteresis. Two contact angle models and one reflectance model were developed to predict the behaviour of an electrowetting system. Using the experimental and modeled results, two prototypes were built to demonstrate the concept of a reflective electrowetting display.  100  7  CONCLUSIONS  Three new separate, yet complementary, optical techniques, specifically the fluid interface diffraction grating, the electrophoretic diffraction grating, and the electrowetting reflective display, have been investigated. The common theme of these three techniques is the spatial modulation of applied electric fields to control the bulk and surface properties of fluidic structures to yield optical effects. A new type of tunable diffraction grating, the fluid interface grating, was produced by spatially modulating the deformation of an oil-air interface using a patterned electric field produced by an array of inter-digital electrodes. Since the two fluids had different indices of refraction, the deformation produced a diffractive structure. By controlling the amplitude and spatial frequency of this deformation, many different diffractive structures were be created and reversibly removed. A model was developed using simple dimensional arguments to predict the amplitude of the deformation as a function of applied voltage. Additionally, another model was developed using simple dimensional arguments to predict the response time of the deformation.  Another new type of tunable diffraction grating, the electrophoretic diffraction grating, was created by applying a patterned electric field, produced by an array of inter-digital electrodes, through an electrophoretic solution consisting of high index charged pigment particles in a low index fluid. By controlling the location of the high index particles in the low index fluid, substantial spatially modulated changes in the index of refraction occurred in the bulk of the suspension, producing a new diffractive structure. Many different diffractive structures were formed and removed by controlling the pattern and magnitude of the applied electric field. 101  Using diffraction grating software, a model was developed to predict the effective thickness of the electrophoretic diffraction grating as a function of applied voltage.  Further research pertaining to these two techniques includes the use of increasingly finer gratings with individual control electrodes to pursue potential applications in the areas of wavelength division multiplexing and active controllable holography. Not only will the use of finer gratings increase the angular distribution of the diffraction patterns, but it will also substantially decrease the time response of the fluid interface grating. Research into suitable materials may yield higher index of refraction changes for both types of gratings as well as further reductions in time response. Another possible area of future work involves the creation of a diffraction grating with graded spatial frequencies.  The third new technique used the electrowetting effect to change the shape, and hence the reflectance characteristics, of tiny water drops on an air-polymer interface. From the experiments performed, it was determined that this electrowetting effect can be used as a basis for a new type of reflective, grey-scale display. This reflective display would require the use of a super-hydrophobic surface to create a low reflective state as well as the use of an A C voltage source in the K H z range to eliminate charge deposition and surface oscillations of the drop. To eliminate thermal cycling properties, a saline solution, whose vapour pressure is lower than that of water, should be used to form the water drops. The system should also be provided with a slight mechanical disturbance to eliminate contact angle hysteresis. A s well, two models were developed to predict the contact angle of a water drop on an air-polymer surface as a function of applied voltage. These models, verified by contact angle measurements, were then used, in conjunction with Monte Carlo ray-tracing software, to develop a third model to predict the reflectance of a water drop as a function of applied 102  voltage. It was determined that for the experimental electrowetting system, an applied electrowetting voltage ranging from 230 V to 440 V changed contact angle of a water drop from 140° to 90°, which caused a change in reflectance from 3% to 43%. Based on all of these results, two prototypes were built to demonstrate the concept of a reflective electrowetting display. Future developments pertaining to this new reflective display technique may include research of new materials, such as higher index liquids to increase reflectance, thinner insulating layers with higher breakdown strength to reduce voltage requirements, and superhydrophobic coatings with less roughness to increase transparency and reduce contact angle hysteresis. In short, the ability to manipulate optical information by applying patterned electric fields to fluidic structures to change their bulk or surface properties has been demonstrated. While the research presented in this thesis tested the three techniques separately, there is also potential for combining these techniques. The results are encouraging and are suggestive that further research in each of the three techniques is warranted. The research has been oriented towards creating images, but the overall scope of the research is not limited to optical manipulation. 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A., 'Electrically Tunable Efficient Broad-Band Fiber Filter', IEEE Photonics Technology Letters, 1999, vol. 11, p445-447.  104  13  Gabathuler, W., and Lukosz, W., 'Electro-Nanomechanically Wavelength-Tunable Integrated-Optical Bragg Reflectors', Optics Communications, 1997, vol. 135, pp385-393.  14  Pinnow, D.A., 'Guidelines for the Selection of Acoustooptic Materials', IEEE Journal of Quantum Electronics, April 1970, Vol. QE-6, No. 4.  15  Clark, A. J., 'A Variable Spacing Diffraction Grating Created with Elastomeric Surface Waves', June 1997, Thesis.  16  Shenderov, A.D., Pollack, M.G., Fair, R.B., 'Electrowetting-Based Actuation of Liquid Droplets for Microfluidic Applications', Applied Physics Letters, Sept 11 2000, Vol. 77, No. ll,ppl725-26.  17  www.nanolytics.com. Prins, M.W.J., Welters, W.J.J., Weekamp, J.W., 'Fluid Control in Multichannel Structures by Electrocapillary Pressure', Science, 12 January 2001, Vol. 291, pp277-80.  19  Lee, J., Kim, CJ, 'Surface-Tension-Driven Microactuation Based on Continuous Electrowetting', Journal of Microelectromechanical Systems, June 2000, Vol. 9, No. 2, pp171-80.  2 0  Jackel, J.L., Hackwood, S., Beni, G., 'Electrowetting Optical Switch', Applied Physics Letters, 1 Jane 1982, Vol. 40, No.l, pp4-6.  2 1  Berge, B., Peseux, J., 'Variable Focal Lens Controlled by an External Voltage: An Application of Electrowetting', The European Physical Journal E, Oct 2000, Vol. 3, No. 2, ppl59-63.  2 2  Beni, G., Tenan, M.A., 'Dynamics of Electrowetting Displays', Journal of Applied Physics, October 1981, Vol. 52, No. 10, pp6011-15.  2 3  Sheridon, N. K., "Electrocapillary Color Display Sheet", US Patent 5757345, 26 May 1998.  2 4  Halliday, D., Resnick, R., Walker, J., Fundamentals of Physics. 1993, pi088.  2 5  ibid 10. Hayward, R. C , Saville, D. A., Aksay, I. A., 'Electrophoretic Assembly of Colloidal Crystals with Optically Tunable Micropatterns', Letters to Nature, 2000, Vol. 404, pp5659.  105  2 7  Mossman, M. A., 'Spectral Control of Total Internal Reflection for Novel Information Displays', April 2002, Thesis, pp31-2.  2 8  ibid 27, p51.  2 9  ibid 27.  3 0  White, F.M., Fluid Mechanics. 1986, p30.  31  Issacs, A., A Concise Dictionary of Physics, 1990, p274.  3 2  ibid 24, p32.  3 3  Nishino, T., Meguro, M., Katsuhiko, N., Matsushita, M., Ueda, Y., 'The Lowest Surface Free Energy Based on - C F Alignment', Langmuir, 1999, Vol. 15, pp4321-4323. 3  3 4  ibid.  3 5  ibid 33.  3 6  ibid.  3 7  Herminghaus, S., 'Roughness-Induced Non-Wetting', Europhysics Letters, 15 Oct 2000, Vol. 52. No. 2, pp 165-170.  3 8  ibid 33.  3 9  ibid.  4 0  ibid.  4 1  Nakajima, A., Hasimoto, K., Watanabe, T., 'Transparent Superhydrophobic Thin Films with Self-Cleaning Properties', Langmuir, 25 June 2000, Vol. 16, No. 17, pp7044-47.  42  Verheijen, H.J.J., Prins, M.W.J., 'Reversible Electrowetting and Trapping of Charge: Model and Experiments', Langmuir, 28 Sept 1999, Vol. 15, No. 20, pp 6616-20.  4 3  ibid.  4 4  ibid.  4 5  Whitehead, L.A., "Wide Viewing Angle Reflective Display", Patent Application No. 10/086349, filed 4 March 2002.  4 6  Burdon, R.S., Surface Tension and the Spreading of Liquids, 1949, ppl5-6, 65-41.  106  www.delta-technologies.com. 4 8  www.mmm.com, keyword: fluorinert.  4 9  ibid 2 7 , p l 0 7 .  5 0  www.g-solver.com.  5 1  ibid 19.  5 2  ibid 20.  5 3  5 4  www.dupont.com/teflon/af. www.boydcoatings.com/crc6000.html.  5 5  www.cytonix.com/fluorothane.html.  5 6  www.ntt-at.com/products_e/hirec.  5 7  ibid 34. www.paratronix.com.  5 9  www.lambdares.com.  APPENDIX A: R E F L E C T I O N F R O M A H E M I S P H E R E  The critical angle at which total internal reflection (TIR) of light entering a hemisphere occurs is shown in Figure A . l below. The relevant parameters include the'index of refraction of the hemisphere, n , the index of refraction of the surrounding media, n , the critical angle t  2  at which T I R occurs, 6 , the radius of the hemisphere, R , and the critical radius at which T I R C  occurs,  0  R ritC  \ n,2  R  0  Ret crit  Observer  Figure A . l Critical Angle of a Hemisphere of Water  According to Snell's L a w , the critical angle for total internal reflection is:  (A.l)  108  One can also calculate the critical angle for a hemisphere drop of radius R„ as:  sinr? -  R  c  r  (A.2)  "  R..  This leads to:  cri,  R  In other words, for all R < R  it,  cr  ~  (A.3)  o  R  light passes through the hemisphere. For R > R , light is cri!  reflected back towards the viewer.  The reflection in percent is defined as the ratio of the total reflected area to the area of the hemisphere base and is equal to:  %* = * * - - f » » 7JR. (  2  2  2 )  (A.4)  =l\  n  \ J  For the electrowetting water drop system,rc = n \r ~ ' and n = n 2  a  t  water  ~ 1.33, leading to %R  43%.  109  APPENDIX B: C A L C U L A T I O N O F D E F O R M A T I O N O F OIL-AIR INTERFACE GRATING The rough estimation of the magnitude of surface deformation of two fluids, namely oil and air, in a capacitor was based on minimizing the total energy of the system. Assumptions and approximations included neglecting the fringing electrical field effects, providing a rough estimate of the effective dielectric constant of the two fluids and approximating the sinusoidal perturbation as a triangular deformation. The parameters of interest in this calculation included the permittivity of free space, £ , the permittivity of the oil, £•„,/. the 0  surface tension co-efficient of the oil in air, y the thickness of the capacitor, d, the width of the electrode, w, the length of the electrode, z, the height of the perturbation, a, and the applied voltage, V.  Figure B . l Surface Deformation of Two Fluids in a Capacitor  110  The first step involved calculating the capacitive energy of the system in the following manner. The effective dielectric constant of the oil and air layers between two capacitor plates was estimated as:  «*="3  — —V a  a  2  <  B  J  )  2  +  £  £ o  , oil  such that when a = d/2, corresponding to the situation in which the only fluid between the capacitor plates is o i l , £ jf = £ a. When a = d/2, in which the only fluid between the plates is e  0  air, £ ff = £ i = £ . e  a r  0  The above equation was simplified, such that:  C, (l + C a ) 2  where  C, = " "" 2£  8  (  .  (B.3)  jAi^A  .  Ci  (B 4)  B y assuming a was very small, which was a valid assumption since a was only required to be on order of 100 nm to produce diffractive effects, the above equation was expanded by using:  [l + x) = l + rx+ ^ r  ] )  x - . . . . ~ \ + rx 2  (for small x)  (B.5)  Equation B.2 becomes:  (B.6)  «C,(l-C a)  e  £  2  ff  The capacitance of the system was:  (B.7)  - eff - eff  L  £  £  d  d  The capacitive energy of system per unit length was:  "cap  '  c_  (B.8) d  where  2wV  2  C = 3  C= 4  1  £ £,  o oil \ o+ o„ J  d  4wV d  f  £  2  (B.9)  £  £ £ {£ -£„,,) (£ £ J 0  ml  0  0+  (B.10)  t  The second step in estimating a involved calculating the surface energy of the system in the following manner. A s mentioned previously, the sinusoidal deformation was approximated as two angled, straight lines as shown below:  112  Figure B.2 Approximating the Sinusoidal Deformation  The length of the two angled straight lines defining the surface deformation was:  r  L,  ?  \  =2L = 2^4a + w = 2w — - - + 1 2  2  V  Using the approximation shown in Equation B.5, L  TOT  l / 2  (B.ll)  J  w  became:  4a  z  1+v  2  = 2w + w  (B.12)  w  J  The surface energy of the system was simply:  u*»rf  =  r-  A  =  r-L ,-z t0  (B.13)  and the surface energy per unit length was:  1 13  E^L z  =  r  .  L  u  n  =  2  r  .  w  +  ^  (B.13)  w  To minimize the total energy of the system for a stable configuration, the derivative of the sum of Equations B.8 and B.13 was taken with respect to a, and set to zero. From this, the magnitude of a was roughly estimated to be:  Cw  4w -V 2  A  2  =  a~  Sy  0  nil  0  ml  ;  Sd y 2  V W £„£„,-,(g 0l7  e £ {e -e ) —  (e +£ J 0  0  =  ;  dy 2  -e ) 0  —  (e s )  2  0+  oil  i.o.l'*)  Phone: 613-729-0614 Fluorinated Hydrocarbon  HIREC 100  3M  Product Name: Fluorinert ™ FC-75  Website: www.3m.com  Index of refraction: 1.276  Phone: 888-364-3577 NTT Advanced Technology Corporation Shinjuku Mitsui Bldg. 2-1-1, Nishi-shinjuku, Shinjuku-ku, Tokyo, 163-0431,Japan  Super-hydrophobic Coating Contact Angle: 140-150° Colour: Translucent White Dielectric Constant: 2-3  Website: www.ntt-at.com  Fluorothane  Kumiko Sudo Email: kumiko@ibh.ntt-at.co.jp Cytonix Corporation 8000 Virginia Manor Road, Beltsville, MD USA, 20705  Super-hydrophobic Coating Contact Angle: 140-150° Colour: Translucent Yellow  Website: www.cytonix.com Jim Brown Phone: 301-470-6267 Email: emailbox@cytonix.com CRC 6000  Boyd Coatings Research Co. 51 Parmenter Road, Hudson, MA 01749 USA  Super-hydrophobic Coating Contact Angle: 130-140° Colour: Opaque White  Website: www.boydcoatings.com Teflon AF1600  Phone: Dupont978-562-7561  Hydrophobic Coating  Website: www.dupont.com  Contact Angle: 105° Colour: Clear and Transparent 116  Parylene C  Paratronix Inc., 129 Bank Street, Attleboro, MA USA 02703-1775.  Dielectric Constant at 60Hz: 3.15 Dielectric Strength: 270MV/m Index of Refraction: 1.649  David Scherma/Joe Monaghan Phone: 508-222-8979 Website: www.paratronix.com Digital Still Camera  Nikon  Product Name: Nikon Coolpix 995 Digital Camera  Website: www.nikon.com Resolution: 3.34 megapixels Digital Zoom: 2.5X Optical Zoom: 4X Macrolens for Digital Camera  Digital Slow Motion Camera  AAA Camera 113 West 17th Street New York, NY USA 10011  Product Name: Raynox SuperMacroLens  Phone: 212-242-5800  Magnification: 2.5X  www.aaacameras.com Redlake MASD, Inc. 11633 Sorrento Valley Road San Diego, CA USA 92121-1010  Model: MSN-500  Frame Rate: 500 frames per second Resolution: 320 x 240 pixels  Phone: 800-462-4307 Website: www.redlake.com  1 17  APPENDIX D: C A L C U L A T I O N O F T I M E RESPONSE O F OIL-AIR INTERFACE GRATING This appendix describes in detail the steps involved in estimating the time response of an oil air grating in the two limits of finite and zero viscosity. The first estimation dealt with the time response with a finite viscosity. The second estimation dealt with the scenario involving sufficiently low viscosity in which the time response was limited by the inertia of the drop. The important parameters of the oil included the viscosity, v, the surface tension co-efficient, y, the density, p, the thickness and width, d and w, respectively, the volume, V, and the velocity and acceleration, v and a, respectively. The height of the deformation was approximated to be A, the wavelength of incident light. Some of these parameters are shown in Figure D. 1 below.  r  Electrode"  d/2  X  y  Electrode X  w  d/2  -  1  Figure D . l Time Response Parameters  118  The viscosity limited time response,  was roughly estimated in the following manner.  Using a combination of fluid mechanics and dimensional analysis, two estimates of the acceleration of the deformation were obtained:  .  p y  a  5 « AV  =  L A A-V  =  (D.i)  and  ,  dv 2  p-a  v- —  =  n  „.  (D.2)  dy  From the two above equations, the following relationship was obtained:  y-A  dv 2  —7  =  dy  2  :—  (D.3)  I  A-V  This result was integrated from y = 0 to y = A, the height of the deformation, to obtain the velocity, v:  v= ^  (D.4)  V v Using this equation,  T{,i  SCOUS  can be determined in the following manner:  T co«s vlS  A  = -  v  vV yA  = -  v y  T  = - • L  _ _.  /T  (D.5)  To determine the relationship between the length parameter, L, and the width and thickness of the fluid, w and d, the logical dependence of  on w and d was taken into account. If  d is increased, the horizontal cross-sectional area of flux is increased. Hence, the vertical 119  flow must increase to maintain flux consistency with this increase in area. The time for flow stabilization to occur will decrease due to this increase in vertical flow. Thus,  Tj V  Saius  decreases as d increases. As w increases, more time is required for the flow to stabilize. Thus,  Viscous  increases as w increases. From this, the length parameter, L was predicted to be:  w  2  (D.6)  L = —  d such that  v ^viscous  w  y d  2  i  *^)  As the viscosity approaches zero, the time response must be limited by inertial effects. Using dimensional analysis, the inertial limited time response,  T na mer  was roughly estimated in the  following way:  E ~ m-v - p-V • 2  ' \  d "  2  (D.8)  inertia J  E = y- A  (D.9)  From the above two equations, the relationship for Ti  nertia  7  was found to be:  (D.10)  Similar to the previous situation, Tj j is proportional to w and inversely proportional to d. nert  a  120  To maximize  Tj ia, nert  the length parameter, L, was assumed proportional to w, since any  additional dimensionless parameter of w/d was less than unity for the given dimensions of this grating. Thus, the inertial limited time response was found to be:  ^ = ^ w  (D.11)  3  Substituting y = 0.1 J/m , v = 1.25 kg-nf'-s" , J = 0.35xl0" m, w = 200xl0" m,p= 2  1  kg/m , the viscosity limited response time, T J  3  the inertial limited response time,  tmertia,  1000  , was estimated to be on order of 1 ms, and  3  V  6  SCOUS  was estimated to be roughly 0.5 ms. Thus, it was  viscosity effects, not the inertial effects, which dominate the time response of the fluid interface grating.  121  APPENDIX E: M O D E L O F E V A P O R A T I O N A N D RECONDENSATION OF PURE WATER ONTO SALT W A T E R This appendix summarizes the steps involved in estimating the time required from a drop of pure water to evaporate and recondense onto a drop of salt water in an enclosed system. In this model, the water drops were approximated as two flat reservoirs of fixed thickness t = t t  2  = 1 mm, as shown in Figure E.l below. The temperature of the system, T, was set at 25°C and the salinity of the salt water, S, defined as the amount of salt in parts per thousand of distilled water, was fixed at 50. The vapour gap thickness,  t  , was also set at  gap  1 mm, and the  dimensions of the container was fixed to be w = / = 3 mm. Other assumptions included a linear concentration gradient, dn/dx, and the treatment of the salt and pure water vapours as ideal gases.  fffly, ^  ^  Salt Water ^  ^  ^  /  ^  Salt Water Vapour  i l  i i i i l  I  l  l  Pure Water Vapour  MESH U  w  t. —1  tl-  Figure E . l Model of Evaporation and Recondensation of Pure Water onto Salt Water 122  The vapour pressure of pure water at room temperature (25°C) is 3.16508 kPa. The vapour pressure of salt water is related to that of pure water by the following equation:  ^water '  ^ s a l t water  e  X  0.018  P  (E.l) B  m" J  where Imi, is the total molality of the dissolved species, 0is the osmotic coefficient of salt water, and m° is a constant equal to 1 mol/kg. For salt water of salinity, S:  Y B _ 31.998 5 Y m ° ~ 10 -1.005-5 m  (E.2)  3  fV B  ^ = 0.90799-0.08992 V  + 0.18458-  2m°  v  J  \  B  2m°  )  5 B  -0.07395V  B  -0.00221-  2m°  \  J  (E.3)  2m" )  Substituting S = 50 into the above equations, the vapour pressure of salt water at room temperature was found to 3.07820 kPa.  The ideal gas law is used to calculate n  wa  ter vapour  and n i sa  t  water vapour,  the atomic concentrations  of pure water and salt water respectively:  N_ _P_ V~ kT  (E.4)  123  3.16508X10 !  watervapour  3  g  x  1 0  "23 .  2 9  g_  „ „ atoms 7.6926x10  3  1  '  5  ^  3.07820X10 ~ 1.38 X 10" • 298.15 ~ '  atoms  3  "sal, water vapour  2 3  23  1T1  3  A linear concentration gradient was assumed and calculated to be the following:  dn _ dx  _ (7.6926 - 7.48 14) X 1 Q  "watervapour ~ "salt water vapour  IO"  t  gap  3  23 =  1  '  ]  1  6  x  l  Q  2 5 atOHIS  m  4  The flux of water molecules going from pure water to salt water, J, in a t o m s - m ' V , is calculated using the following equation:  J = <p— dx  (E.5)  S  9  I  where <p, the diffusivity constant, is <p= 2.6x10" m -s" . 7 c a n also be calculated by using the following equation:  N j  _  wate^  (  E  i  6  )  A-t where N  water  is the number of water atoms in the pure water reservoir, A is the cross-sectional  area in which diffusion occurs and is equal to l-w, and t is the total time required for complete diffusion of the pure water. The relationship for the time for complete diffusion to occur was found by combining Equations E.5 and E.6:  t=  N  xez— A-0 dx  (E.7)  124  For the model, the total number of atoms of pure water was calculated to be:  ^  -Avagadro's Number  _ p-l-w-t  l  1000- 3 x l 0 " ) - K T - 6 . 0 2 X 1 0 = i 'z 18.0153X10 3  N  mur  (E 8)  Molecular Weight of Water  wa,er  3  2 3  . _ , = 3.007 x 10-° atoms i n  0  -3  which led to an estimation of t:  3.007 x l O t =r= 60856s = 17 hours (3xlO ) -2.6xl0~ -2.1116x10" 2 0  :  2  - 3  5  125  A P P E N D I X F: T I M E R E S P O N S E O F B A S E D I A M E T E R A T V A R I O U S VOLTAGES This appendix shows the graphs of the base diameter as a function of time at various A C voltages at a frequency of 2020 H z . A cubic fit of the data and 5% error bars are included in each figure as well. The equations of the cubic fits are summarized at the end of this appendix.  h 1  2.5  E 1.5  TO  a  a> V)  n ffi  0.5  8  10  12  14  16  Time (ms)  Figure F . l Base Diameter vs Time for an Electrowetted Drop at 384 V  126  0  2  10  4  12  14  16  T i m e (ms)  Figure F.2 Base Diameter vs Time for an Electrowetted Drop at 349 V  0  2  4  6  10  12  14  16  16  20  T i m e (ms)  Figure F.3 Base Diameter vs Time for an Electrowetted Drop at 327 V 127  10  12  14  16  F i g u r e F.4 Base Diameter vs T i m e for an Electrowetted D r o p at 301 V  10  12  14  16  Time (ms)  F i g u r e F.5 Base Diameter vs T i m e for Electrowetted D r o p at 2 7 5 V 128  2.5  T i m e (ms)  Figure F.6 Base Diameter vs Time for an Electrowetted Drop at 253 V  i  1  T i m e (ms)  Figure F.7 Base Diameter vs Time for an Electrowetted Drop at 237 V 129  0  2  4  6  8  10  12  Time (ms)  Figure F.8 Base Diameter vs Time for an Electrowetted Drop at 221 V  Table F . l Electrowetted Base Diameter of a Water Drop as a Function of Time at Various Voltages at 2020 Hz Voltage (V)  Cubic Equation of Base Diameter, d (mm) as a Function of Time, t (s)  R  384  d = 7.73xl0"V - 2.95xl0"V + 3.52x10"'-? + 1.43  0.996  349  d = 7.89xl0" -? - 2.56xl0" -? + 2.78x10"'-? + 1.53  0.984  327  d = 4.36xl0"V - 1.76xlO"V + 2.26x10"'-? + 1.56  0.974  301  d = 3.42xl0" -? - 1.40xl0" -? + 1.86x10"'-? + 1.50  0.994  275  d= 1.58xlO"V-8.53xlO-V+ 1.31x10"'-?+ 1.59  0.996  253  d = 3.00xlO"V-9.94xlO-V+ 1.09x10"'-?+ 1.66  0.996  237  d = 5.79xl0"V - 5.11x10" V  + 7.49xl0" -?+ 1.50  0.982  221  d = 2.31xlO"V - 6.60xlO"V + 6.82xl0" -?+ 1.50  0.979  4  3  4  2  3  2  2  2  2  2  2  130  APPENDIX G: APPLIED F O R C E D A T A AS A F U N C T I O N O F B A S E DIAMETER VELOCITY This appendix tabulates the applied force on a water drop as a function of the velocity of the base diameter of the drop. A sample calculation is also shown.  B y taking the derivative of the cubic fits of the base diameter as a function of time for various voltages at 2020 H z , as shown in Table F . l , the instantaneous base diameter velocity, v, was determined at any time, t. A l s o , the base diameter,  db , ase  determined at t, and from  this, the contact angle, 6, was calculated by conserving the volume of the drop.  For example, at t = 2 ms with an applied voltage, V, of 349 V , v was determined from the cubic fit to be 0.184 m/s. A t t = 2 ms, dbase = 2.0 mm. 6 was determined in the following manner. The initial volume of the drop was calculated using the initial contact angle, 0= 6-„ and the initial base diameter,  dbase = d , t  as shown by Figure G . l and Equations G . 1 - G.4.  Figure G . l Geometric Drop Parameters 131  base  R= e-  COS  \  (G.l)  2 • cos 0-  2;  (G.2)  V„„. .=-nR sphere  3  r  f  =*R> 2-3sin e - -  V  3  cap  V,  =V  drop  Thus, for dbase =  dj  sphere  + sin  2y  V  0-v  (G.3) 2j  j  -V  (GA)  cap  =1.5 mm, and # = 140°, Vd  = 6.402 mm .The volume of the drop must  wp  be conserved and, since the base diameter was known to be 2 mm at t = 2 ms, Equations G. 1 - G.4 were used to calculate  0 = 6  ewet  =  124°.  Next, the net force on the drop, F, in N/m, was calculated using Equation 6.2, repeated below:  1£ £ V  2  F = -y  cos0 + (y - y ) +  lv  sv  sl  (G.5)  ~^ — 2 d LJ  where y = 0.073 N/m, £ = 8.85x10"" F/m, £ = 2.6 and d = 2.8xl0" m. 6  v  0  r  Young's Equation (Equation 3.2) was used to calculate (y - y j) as shown below: sv  y  s v  -y  s l  =y -cos0 lv  i  s  =0.073 cos 140" = -0.05592IN/m  132  For V= 349 V , and 6= 124°, F was found to be 4 . 6 x l 0 " N/m. Hence, the applied force on 2  the drop and the corresponding velocity of the base diameter at a specific time was now known for a certain voltage. The rest of the data is shown in Tables G . l below.  Table G . l Base Velocity and Applied Force as a Function of Time at Various Voltages Applied  Instantaneous  Instantaneous  Voltage  Time  Applied Force  Base Velocity  (V)  (ms)  (N/m)  (m/s)  384  0  0.074  0.352  384  2  0.055  0.243  384  4  0.037  0.153  384  6  0.025  0.081  384  8  0.020  0.028  349  0  0.060  0.278  349  2  0.046  0.185  349  4  0.034  0.110  349  6  0.027  0.055  349  8  0.025  0.019  327  0  0.052  0.226  327  2  0.040  0.161  327  4  0.030  0.106  327  6  0.023  0.062  327  8  0.018  0.028  327  10  0.018  0.005  301  0  0.045  0.186  301  2  0.036  0.134  301  4  0.028  0.090  301  6  0.022  0.054  301  8  0.019  0.027  301  10  0.019  0.008  275  0  0.038  0.131  275  2  0.032  0.099  275  4  0.027  0.070  275  6  0.023  0.046  133  275  8  0.021  0.025  275  10  0.020  0.008  253  0  0.032  0.108  253  2  0.027  0.072  253  4  0.024  0.043  253  6  0.023  0.021  253  8  0.023  0.007  237  0  0.028  0.074  237  2  0.024  0.055  237  4  0.022  0.036  237  6  0.020  0.019  237  8  0.019  0.004  221  0  0.024  0.068  221  2  0.021  0.044  221  4  0.019  0.026  221  6  0.018  0.014  221  8  0.018  0.007  221  10  0.017  0.005  APPENDIX H: M A X I M U M R E F L E C T I O N O F A H E X A G O N A L A R R A Y O F E L E C T R O W E T T E D DROPS It was very straightforward to calculate the maximum reflection for an array of close packed hemispheres. A hexagonal array maximizes the packing fraction. Three hemispheres of radius, R, can be packed into a hexagon of side length, 2R, as shown in Figure H . 1 below.  R  Figure H . l Hexagonal Array of Drops  Determining the reflection the drops arrayed in such a manner was simply a matter of calculating the area of the reflection annuluses over the area of the hexagon. Using Equation A . 4 of Appendix A , the area of the three reflection annuluses was found to be:  135  \2 \  (  1V  (H.1)  2 ,"1  J )  The area of a n-sided polygon of length b is:  cos —  1  \ )  (H.2)  \  sin —  J  For the above hexagon, n = 6 and b - 2R, which led to:  (H.3)  A =6fiR  2  ha  Thus, the reflectance of a close packed hexagonal array of drops was found to be:  2A  %R= " Ari  (H.4)  gs  V i J n  For water drops in air, nj = n  water  ~ 1.33, n = n j ~ 1, which led to %R = 38%. 2  a  r  136  

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