DEPOSITION OF C O L L O I D A L S P H E R E S U N D E R QUIESCENT CONDITIONS by CHAI GEOK TAN B.Sc.Eng., The University of New A THESIS SUBMITTED THE IN P A R T I A L F U L F I L M E N T REQUIREMENTS FOR MASTER Brunswick, 1983 OF THE APPLIED DEGREE OF SCIENCE in i THE FACULTY DEPARTMENT We OF OF GRADUATE CHEMICAL ENGINEERING accept this thesis as conforming to the required THE STUDIES UNIVERSITY OF standard BRITISH C O L U M B I A October 1987 ® CHAI G E O K TAN, 1987 OF In presenting degree at this the thesis in University of partial fulfilment of of department publication this or of thesis for by his or her representatives. Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date October 10, 1987 for an advanced Library shall make it agree that permission for extensive It this thesis for financial gain shall not Chemical that the scholarly purposes may be permission. Department of requirements British Columbia, I agree freely available for reference and study. I further copying the is granted by the understood that head of copying my or be allowed without my written ABSTRACT The phenomenon microscopic surface of deposition species, suspended' in a plays a critical role (or release) liquid, in onto many of fine particles (or from) a natural and or other foreign industrial substrate processes. Traditionally, the analysis of this phenomenon has been conceptually divided into two steps — the role the transport step and the adhesion of the adhesion practical situations interdependent zeta-potential, step on the overall are complicated parameters particle such size by as towards gaining a better understanding of a very deposition simple process are important, was In charged of a large thickness, others. under number of particle Thus, and wall as a first of the phenomenon, an experimental where layer only the random interactions between most nature deposited step study of the particles undertaken. this idealized in a system, and the double colloidal silica medium gravity deposition layer amongst to understand deposition process the presence double and flow, step. Attempts system, a stable spheres specially suspension one micron constructed in diameter, deposition cell, and be deposited permanently onto of monodispersed, negatively suspended in an aqueous were a cationic allowed polymer-coated to settle by glass cover slip. The magnitude of surface potential was altered by adjusting the pH of the suspension using N a O H and HC1, while the electrical double varied by dissolving different predetermined The results showed layer thickness was quantities of KC1 into the suspension. that the trends in the experimental n surface coverages obtained were thickness, increase in accordance 11K, or the particle in the interaction decreased. Furthermore, and $ with zeta energy layer between the particles), (leading the surface quantities of K P O , 3 could differ thicknesses. The results of surface coverages of geometric to an coverage 11K alone was was also the effect performed by into the suspension so that in their respective double obtained in this study the influence exerted by the substrate double layer obtained when both 11K separate series of studies examining and the particles findings, the presence the double $ , increased thickness on surface coverage dissolving different predetermined substrate potential, as to be greater than that when used as the controlling variable. A the in that the extent of surface coverages were changed was found of substrate double expectation layer showed that layer was negligible. Besides these exclusion due to the random nature of the deposition process was also noted, although its effect was difficult to quantify. Besides were made prediction first the systematic experimental to develop which scheme computer could be compared involved non-overlapping two a study of colloidal deposition, attempts simulation models with results measured two-dimensional simple rejection particles were deposited, while the second three-dimensional model to generate deposition where the rolling experimentally. The model scheme of sedimenting where only consisted of a particles over the surfaces of previously-deposited particles as well as the stacking of particles were allowed. Comparison two-dimensional consistently nature model of experimental revealed underpredicted that results for the experimental of the simulation. The trends with those all cases, results due obtained the simulated results to the oversimplifying in the experimentally iii using the obtained results, however, were approximated nature, successful materialize. completed, by the simulated results. Owing to its very complex completion of the three-dimensional model It is expected, however, it will yield predicted that when results such which a simulation model did not is successfully are in better quantitative examining the effects of reaction agreement with those measured experimentally. Besides temperature particles novel the and produced method size range above, the separate types of alcoholic was in which of 0.2 a study solvent used also performed. This on Mm can be produced by temperature and the type of alcoholic solvent used. iv of silica study led to the development dispersed, uniform-sized, spherical to 2.0 the properties silica particles simply varying of a in the the reaction T A B L E OF CONTENTS ABSTRACT ii LIST O F TABLES vii LIST O F FIGURES viii ACKNOWLEDGEMENTS xi 1. I N T R O D U C T I O N 1 2. T H E O R E T I C A L B A C K G R O U N D 15 2.1. The Electrical Double Layer 2.1.1. The Gouy-Chapman Model 2.1.2. The Debye-Huckel Approximation 2.1.3. Surface Charge Density, a 2.1.4. The Electric Double Layer Thickness 11K 2.1.5. The Stern Layer 2.2. Electrokinetic Phenomena 2.2.1. Electrokinetic (Zeta) Potential £ 2.2.2. Determination of Zeta Potential 2.3. Effect of Electrolyte on Surface Potential, Zeta Potential and Layer Thickness 2.4. Particle-Particle and Particle-Surface Interactions 2.4.1. Double Layer Interaction Energies 2.4.2. London-van der Waals Interaction 2.4.3. Other Relevant Forces 2.4.4. Overall Interaction Energy Q 15 15 19 20 21 22 23 25 26 Double 28 29 29 35 47 48 3. E X P E R I M E N T 3.1. Introduction 3.2. Colloidal Particles 3.2.1. Production of Amorphous Silica Spheres 3.2.2. Production of Large Uniform Silica Spheres 3.3. Cleaning of Particles 3.4. The Deposition Cell 3.5. Preparation of Deposition Surface 3.5.1. Production of 2-Vinyl Pyridine/Styrene 3.5.2. Coating of Deposition Surface 3.6. Particle Deposition 3.7. Measurement of Surface Coverage 3.8. Measurement of Zeta Potential 3.8.1. Particle Zeta Potential - Micro-electrophoresis 3.8.2. Substrate Zeta Potential - Electro-osmosis 55 55 55 56 57 66 71 72 73 75 77 79 81 81 85 4. C O M P U T E R S I M U L A T I O N O F R A N D O M P A R T I C L E D E P O S I T I O N 4.1. Two-Dimensional random Particle Deposition Model 4.2. Three-Dimensional Random Particle Deposition Model 4.2.1. One-Sphere Encounter 87 87 89 94 v : 4.2.2. Two-Sphere Encounter 4.2.3. Three-sphere Encounter 98 102 5. R E S U L T S A N D DISCUSSION 5.1. Introduction 5.2. Surface Coverage - Experimental 5.2.1. Effect of Double Layer Thickness 5.2.2. Effect of p H 5.2.3. Effect of Counter-ion Valence Number 5.3. Surface Coverage - Simulation 106 106 108 108 116 121 126 6. C O N C L U S I O N S A N D 142 RECOMMENDATIONS NOMENCLATURE , REFERENCES 146 149 APPENDICES 155 A. Sample Calculations 155 A . l . Double Layer Thickness 155 A.2. Particle Zeta Potential 160 A.3. Wall Zeta Potential 165 A. 4. Errors in Electrophoresis and Electro-osmosis Measurements .. 166 B. Calibrations 171 B. l . Calibrations Made for Electrophoresis Apparatus 171 B . l . l . Inter-electrode Distance and Cell Dimensions ... 172 B.1.2. Eyepiece Graticule and Timing Device 174 B.1.3. Conductivity of Potassium Chloride and Potassium Phosphate as a Function of Concentration ... 174 C. Computer Programmes 179 vi LIST OF TABLES 2.1 Values of Hamaker Constants 3.1 Average diameter (nm) and standard deviation obtained using different temperatures and solvents 5.1 45 of silica 62 Summary of experimental data for the effect of double layer thickness on surface coverage 5.2 Summary 5.3 Summary of experimental data for the effect of K P O „ on surface coverage A.l A.2 spheres of experimental data for the effect of p H 3 on surface 106 coverage 107 concentration 107 Results of chemical analysis showing the types and quantities of ions present in the freshly prepared silica suspension 157 E versus y 164 0 - Run A2 vii LIST OF FIGURES 2.1 2.2 2.3 A schematic representation of a charge surface and together with other parameters as shown The effects of (a)ionic strength and potential and double layer thickness Qualitative sketch of two (0 ) which are possible (b)pH on its double layer, 27 zeta potential, surface 30 general types of overall interaction energy 49 T 2.4 2.5 Effect of Stern (or zeta) potential on overall interaction energy function of separation distance between two charged surfaces as a 52 Effect of double layer thickness {11K) on overall interaction energy as a function of separation distance between two charged surfaces of same polarity 53 2.6 Effect of Hamaker constant A on overall interaction energy as a function of separation distance between two charged surfaces of same polarity 54 3.1 Electron micrograph of silica particles solvent and at a temparature of 0°C 3.2 3.3 3.4 obtained using ethanol as the 60 Normal-probability plot of the cumulative size distribution curve of the particle batch shown in Figure 3.1 61 Average diameters of silica particles temperature and type of solvent used 63 Schametic may diagram showing how a obtained nucleation be separated resulting in a monodisperse as and a a function growth of period sol 65 3.5 Schematic diagram of the cross-flow filter (cross-sectional view) 69 3.6 Schematic diagram of the deposition cell 72 3.7 Schematic diagram of the coating apparatus 76 3.8 Photograph of polymer-coated substrate with a single layer of deposited silica spheres 80 4.1 Schematic flowchart for random sphere deposition 93 4.2 Plan view showing various possible general configurations when a sphere below is encountered by the falling sphere Plan view showing the initial and final positions of the falling sphere when a subordinate sphere is encountered (Case I) 95 4.3 viii 96 4.4 4.5 4.6 4.7 4.8 Plan view showing the initial and final positions of the falling sphere when a subordinate sphere is encountered (Case V) 97 Plan view of various possible general configurations when two spheres below are encountered by the falling sphere 98 Plan view showing the initial and final positions of the falling sphere when two subordinate spheres are encountered (Case I) 99 Plan view showing the initial and final positions of the falling sphere when two subordinate spheres are encountered (Case V) 102 Plan view showing the final resting position of the falling when three subordinate spheres are encountered 103 4.9 Elevation 5.1 Effect of added KC1 concentration on the surface coverage of silica spheres 109 5.2 Surface coverage of silica spheres as function of fca 110 5.3 Particle zeta potential as a function of KC1 concentration 112 5.4 Particle zeta potential as a function of ka 113 5.5 Photographs M view of the falling sphere resting atop sphere two subordinate spheres 105 of deposited spheres at added KC1 concentrations of (a)0 and (b)lxlO"" M 115 5.6 Effect of pH on the surface coverage of silica spheres 5.7 Effect 5.8 5.9 of particle zeta-potential on the surface coverage 117 of silica spheres 118 Comparison of surface coverages obtained as a function of /ca between series A and C 124 Qualitative sketches of the potential distributions in the particle double layer for runs C2 and C4 127 5.10 Plots of surface coverage of silica spheres as a function of /ca 5.11 The total interaction energy as a function of the separation 129 distance between the particles for series A 131 5.12 Surface coverages as a function of /ca for series A 133 5.13 The total interaction energy as a function of the separation distance between the particles for series B ix 136 5.14 The total interaction energy as a function of the separation distance between the particles for series C 137 5.15 Surface coverages as a function of pH for series B 138 5.16 Surface coverages as a function of /ca for series C 139 5.17 Comparison of simulated and experimentally obtained photographs approximately similar values of coverage 140 B.l Solution conductivity as a function of added KC1 B.2 Solution conductivity as a function of added K P 0 B.3 Solution pH 3 as a function of added K P O 3 x at u concentration 4 concentration concentration 176 177 178 ACKNOWLEDGEMENTS The and Dr. author wishes Epstein, for to express their his thanks patience, to his supervisors, Dr. understanding and consistently Bowen wise supervision throughout the duration of the program. Thanks Department of are also due Chemical to the faculty, staff and Engineering who all other personnel of the have in one form of a way or the other contributed to the completion of this project. Finally, throughout the financial course assistance of study in the from the Natural Research Council of Canada is gratefully acknowledged. XI Research Sciences and Assistantship Engineering 1. INTRODUCTION Judging from concluded that great of deal research effort has been devoted towards the phenomenon of deposition (or release) of fine particles or other understanding microscopic a the number of publications in the literature, it can easily be species, suspended surface. This phenomenon processes and a in a plays knowledge of liquid, a onto critical role (or from) a in many foreign substrate natural and industrial it is of paramount importance in the design and phenomenon are: operation of the processes involved. Examples physical packed of applications separation processes bed physiological filtration processes the bloodstream secondary unit from the as as the removal flotation metastasis involving of this suspended of mineral where particles fines; malignant cells are occluding the attachment as and flow of blood of microbes onto establishment chemical and forming the interior where blood cells adhere to injured vessels during injuries; biological other surfaces of intermicrobial thermal and carried in tumor to other locations in the body thrombosis in a biomedical tumors, atherosclerosis where cholesterols are deposited on stabilization such processes froth the primary walls of the arteries, and thereby such and such or fouling of in their search relationships; surfaces; and processes such as for nutrients, industrial deposition processes of sizing materials onto cellulose Fibers during paper making. analysis of the usually accomplished by conceptually dividing the process into two The deposition of 1 colloidal species onto a substrate is steps: INTRODUCTION 1. a transport step in which the particles are transferred from / 2 the bulk of the solution to the interface region, and 2. an adhesion attached step in which the particles to the substrate by in the interface region are overcoming various surface interaction forces which arise at short distances of separation. The forces relevant to these two steps can in turn be classified into three categories: 1. forces related to the motion of the fluid particles relative to the fluid of the particles; the hydrodynamic convection-diffusion effects fall under this 2. external forces such and and forces causing as those the motion of the the Brownian motion drag forces and the magnetic and category; due to electric, gravitational fields; and 3. chemical and substrates and colloidal forces particles which with result molecules from and the interactions of ions in the suspending medium; these include the van der Waals, electrical double layer, Born repulsion and structural forces. The categories region transport which while step is usually are responsible the adhesion controlled by those for bringing the particles to the interfacial step is dominated by the forces category, which decide if a particle will be deposited Thus, it can be clearly in into that the specific role played the overall two actually seen steps deposition this process. the validity in the third onto the substrate. classification is useful in defining by each step and hence each type of force Although is largely conceptual, confirmed in the first two this of the process by some investigators have concept under certain studies of this separation conditions INTRODUCTION / 3 (1,2,3). The chemical transport engineers experimental understood review step and has fluid studies in this for a great paper in this traditionally dynamicists. area variety area been has been popular Accordingly, are abundant of flow a with theoretical and and the subject is well situations. presented field Recently, an by Papavergos excellent and Hedley (4). In the present investigation, the primary understanding particle review objective is to gain a better of the role played by the forces in the adhesion deposition process, and hence this chapter will mainly step of the focus on a of those theoretical and experimental studies that explore the role of colloidal forces on deposition and resuspension. It onto is anticipated that the rate a substrate is critically of deposition of a colloidal species influenced by the physicochemical interactions between the colloidal species and the substrate. Despite the diversity of the applications relevant important listed above, the underlying physical to the deposition of the particles of these interactions and chemical remain are the London-van phenomena the same. der Waals The most and the electrical double layer interactions. Attempts to understand the role of physicochemical phenomena on the deposition process are complicated by the presence of a large number of INTRODUCTION / 4 interdependent parameters and sometimes by the practical difficulties in quantifying some of these parameters sufficiently accurately. In view of the above, many experimental studies have been carried out over the investigate jointly colloidal separately deposition. strength of suspension particles or Examples electrolytes (10), size (12), in and and the of role the and the parameters relevant studied dispersion concentration types of of particles characteristics of on types (5,6,7,8,9), (11), to parameters include medium years and pH of polydispersity of substrates (6,13,14,15), amongst others. Owing to the enormous amount subject, this chapter is not all the the and literature on experimental intended of published to provide a information about comprehensive this review of deposition of colloidal species. Only the theoretical studies which are the most relevant to the present thesis will be briefly discussed. Although there have been some studies of colloidal particles onto spheres in packed do not yield method any used direct for obtaining comparison of inlet systematic study reported in 1966 disc technique suspended in an of by to information and on outlet suspended particle deposition Marshall and investigate the deposition data, particle under Kitchener the the rates which deposition was well-defined of due concentrations. (13), who deposition aqueous medium, onto on beds (16,17,18,19), the results local experimental conducted disc surface the merely The a first conditions introduced carbon to black as a was a rotating particles, function of INTRODUCTION the particle layer. and substrate The surface zeta characteristics of the disc different types of plastics onto was changed concentration by potentials and the thickness through the of accumulated addition the thickness of KC1 particles was rinsing the disc, drying under a microscope. Their it while were to of the double altered that coating suspension. The as a function of time it in a dessicator, and counting results showed by of the double layer the measured / 5 the particles the deposition was greatest when the electrical double layer interaction between particles and solids was attractive. However, particularly when the double layer thickness deposition were was later trends the double layer interaction increased, observed. Similar experiments employing performed particles, used when by Hull and Kitchener a stable colloid a great repulsive, reduction in the same (14) who, instead of monodisperse polystyrene in results, as compared was technique of carbon spheres. Similar with the results of Marshall and Kitchener, were observed. This successful rotating disc and has been technique used by of studying several particle other deposition investigators in the field (20,21,22). Besides the rotating disc collector, collectors of other have also been employed. These include cylindrical proved geometries (23), spherical (24) and parallel-plate channel (25) collectors. plays As mentioned earlier, an important role Ruckenstein, Marmur and deposition in many or adhesion biological co-workers have and long of cells on biomedical been surfaces processes. involved jn INTRODUCTION systematically deposition investigating the effects of cells onto surfaces of various forces on (26,27,28,29). Besides thermal the / 6 static and surface interaction forces, the effect of gravity on cell deposition on either horizontal or inclined surfaces and by others has also been investigated recently by these (30,31). These significant role in the deposition Recently, Tamai, Suzawa results indicated that researchers gravity plays a process. and co-workers (6,15,32) have studied the effects of pH, types and quantities of electrolytes and surface characteristics on both the deposition and rate and the surface cationic latex particles onto various coverage by depositing anionic plastic fibers that exhibit different surface properties. The method involved the immersion of a rectangular (1x2 cm) piece of preweighed plastic known solids concentration. either sodium chloride hydrochloric acid determined through intervals using immersed or or fabric into a 200 ml latex The electrolyte concentration sodium sodium sulfate. hydroxide. turbidity measurements The The withdrawn and rinsed adjusted using was varied using of deposition of the dispersion a spectrophotometer. A t the end of a cloth was was pH rate dispersion of with 12-hour was at regular period, the a particle-free solution of the same composition to remove undeposited free particles. The sample was then dried in air, coated with gold and photographed using a scanning electron microscope. Their concentration results showed that the rate of electrolytes increased. of deposition Different increased electrolytes as the at the same INTRODUCTION / 7 ionic strength increased gave similar results. Increasing pH of the suspension the magnitude of the zeta potentials of both the particles and fibres, thus raising the repulsive energy particle deposition. One authors suggested 500 that orders-of-magnitude barrier and peculiar finding observed energy maxima exceeding surface the greater kT, reducing was observed than predicted rate of that despite potential measurable deposition the the the deposition, still occurred. which is theoretically, must be The many due to heterogeneity. In many experimental studies (e.g., 33,34,35,36) of colloidal deposition involving fluid flow, it is deposition is linear with commonly observed time. However, as rate of deposition decreases asymptotically than For a monolayer coverage is obtained example, Bowen (33) coverage causes attained of incomplete behaviour. The the never first charge of the particles. In an reported exceeded coverage substrate attempt to rate of particle deposition a model based on deposition surface by of each a as initially with time and the In view been predict the effect of in packed beds, Wnek average covered with particles, a new such that, charge when plane surface rationalize this state or coverage on the et al. (37) have proposed particle amount surface possible chemical surface their model, they fixed the becomes covered with this argument. In alters the of "saturated". several to surface surface is maximum this, postulated substrate rate usually much less substrate of change in the the coverage increases, in his thesis that the have the surface when 10%. is that of a that the is created suggested that the density of the entire surface is completely which has the same INTRODUCTION / 8 charge density ^-potential as the particles. Thus, because the (Stern potential) are charge density closely related, the deposition and the of negatively charged particles onto a positively charged substrate effectively decreases the $-potential of the substrate. With $-potential becomes negatively charged. potential energy model, barrier although accumulation it and hence, predicted curve and sufficient a This general gives reduced qualitatively the deposition, rise rate the the to of a shape influence of $-potential, failed to yield good quantitative agreement with the results (33). Furthermore, it is deemed to be repulsive deposition. correct observed substrate This of the the wall experimental fundamentally incorrect for the reason outlined below. Besides may also be energy heterogeneity due to surface geometry of the substrate for deposition first put model aimed through diffusion. a at by by Although the fluid the i.e., due the et particles. al. (27) in a approach to horizontal taken by This their coverage-dependent onto observed behaviour a change a reduction in surface deposited Ruckenstein explaining stagnant heterogeneity, surface and introduced forward mentioned above, the plane this in the area available explanation development rate by of cell is of a deposition sedimentation model was and fundamentally sound, it yields complete surface coverage as time becomes ven' large under all if not all, circumstances. experimental This result will conditions. Thus, to replaced the particle radius by never improve be true upon for this most, model, an effective particle radius on Bowen (33) the recognition of the fact that because of the finite thickness of the electrical double layer INTRODUCTION / 9 surrounding below each which sphere, there exists a two particles cannot surface coverage thus obtained minimum approach each is approached distance of separation other. The resulting final asymptotically and will always be less than complete coverage. It process was further and hence approach recognised there to within exists that, finite any distance because probability (greater than the particles would uncovered surface that twice radius) of a second particle, a correct accounting deposited deposition totally excluded random one particle the effective would particle for the finite size of the therefore lead to the possibility being is a from of some areas of further deposition, and hence an even smaller value of the final surface coverage will be obtained. Based on this coverage reasoning, at which Manifestations Bowen has derived the maximum of this effect on an equation possible geometric further particle representing the exclusion deposition can occur. have been observed experimentally by other investigators (34,35,36,38). However, extremes despite of surface coverage for most predicted by these coverage, runs the two was two relationships Bowen well found below that representing the measured the envelope relationships. Thus, the besides of possible the effects two surface results mentioned above, some other effects must be present. He suggested that an additional contributing factor arises from the fact that the suspension is, when surface, a particle the fluid carried streamlines by a fluid are forced in laminar to move flow over is flowing. That deposits and on a around the I N T R O D U C T I O N / 10 particle. At very particles are relatively bare low portion of the distance becomes over the bare closer these pushed surface coverages, far apart, the few particle particles are to little diameters, wall is influenced by the distribution wall is probably a when other, over the affected. However, when this the distribution the of streamlines of deposited particles; the more away from the wall. Thus, it is hypothesized between deposited deposited of streamlines the presence each randomly the streamlines that, as the particles decreases, the probability of a particle the increased distance to the wall will be greatly reduced, and are spacing diffusing hence there will be a reduction in deposition. Thus, particle it can deposition influenced by thickness, particle others. it will studying very the and and in be simple random seen that surface variety Therefore, behaviour, only a be the coverage of experimental wall order process past to gain a in the necessary a better between simple such size nature of the present number and investigation deposition process and subsequent one, but double flow, is layer among of deposition to begin by conditions, so that the double layer important. is included, it is no random as understanding deposition situations under stagnant randomness use not $-potentials, particle analytically. Several numerical which is parameters interactions between deposited particles are Once relationship longer possible to model the studies have been carried out in the generators deposition or flocculation processes. One of the to help simulate random earliest models in this are_a I N T R O D U C T I O N / 11 was a developed by sediment formed permanently suspension vertically Monte either the the on of y- and bottom The In particle starting simulation simulation, N particles are is allowed previously situation to this not was to on generated points; instead, the path and allowed one by rest is to using one a fall by a into a when it strikes its way down) or its way actually involve cohere represented dropped come that envisaged are dropped particle on were did particles particles which the coordinates studied the volume of spherical dropped particle it contacted were determined by a down. The random motion of x- number the particle resting place of the particle methods of analytic geometry. Similar studies on cluster were performed later by Sutherland (43,44) and, more recently, Meakin (45). Although greater three-dimensional deposition in random have about other generally these include: structure aspects the formed coordinate of focussed system on than the studies published of finding the so-called "jamming limit" this geometry and number deposition, packing density, i.e., the the of physical distributed (i.e. no through successive by contact. in which each generator. The formation settling gravity. model (39,40,41,42), who the randomly first previously and by first under Carlo container Marjorie J. Void the problem of packing (49), have depositing also two-dimensional random packing (50). the on two-dimensional random final random maximum (46,47), although been species fraction obtained and literature relationship studies conducted. Some (48), through the monolayer contraction between of one - of and I N T R O D U C T I O N / 12 Feder of and Giaever two-dimensional surface (47,48) have random in an attempt sequential to verify performed the computer deposition of discs simulation onto a planar the results of the experimental study of deposition of ferrition molecules, a visible, nearly spherical, rigid, iron-storage protein found in horse spleen, onto Lexan polycarbonate surfaces. In their experimental studies, although coverage on many physicochemical acknowledged, no attempts and the dependence properties of the carbon of surface surfaces were were made to quantify these properties and only a couple of attempts, in a non-systematic manner, were made to study the effects of variations on deposition. Furthermore, the deposited surface was rinsed and dried before micrographs were These taken. rearrangement in pH procedures and create added electrolytes the possibility of deposited particles. Thus, extremely of resuspension and poor agreement, both in terms of quantity and pattern of surface coverage, between experimental and simulated results were obtained. The only useful conclusion that can be drawn from coverage their study was found is that the computer simulated maximum to be 54.73%, a figure which surface has also been reported by others (51). In study of simulation results first, the present colloidal models which similar two-dimensional investigation, deposition have under been the one approximation with employed the systematic stagnant developed can be compared to besides conditions, to generate those by measured Feder and experimental two predicted computer deposition experimentally. The Giaever, uses a in which disc-like particles are simply placed I N T R O D U C T I O N / 13 at random coordinates on a plane surface. Only those particles that do overlap previously deposited The second model particles are assumed to remain on involves random environment where particles are the deposition surface. When are permitted the deposition allowed they or to become reach that ultimately contact "nested" the surface a the surface. three-dimensional to sediment under gravity towards the to roll over previously deposited surface in not within are vicinity of the surface particles to eventually the other considered particles. to be they contact Only those part of the final deposit. For the both models, the final fraction of the deposited total area particles occupied performed by of the data /ca, by deposited double the reduced the determination layer and thickness the Calculations particles successively reducing enables the where between is determined. total area occupied on particle the of the new substrate (1/K) fraction area diameter. This of surface coverage as is deemed to be original particle radius and are of also manipulation a function of the a by all difference is the reduced particle radius. In Chapter 2 of the present study, background pertaining to the thesis is presented. Section 2.1 layer on theory, relevant theoretical This includes discussions in the initial development of the Gouy-Chapman electrical double the simplifying approximation of double layer thickness, and layer Stern model the by and by Debye-Hiickel, further refinements Grahame. This section on the the definition electrical double is followed by a brief I N T R O D U C T I O N / 14 review (in emergence be Section of 2.2) on electrokinetic $-potential and quantified. In Section the 2.3, a phenomena, techniques by the $-potential and this 2.4 by various in Section surface particularly the interaction electrical its resulting this parameter short qualitative discussion is presented the effects of electrolytes on is followed which with a forces double layer and on double layer thickness, and brief review involved can of the theory in the the London-van of deposition the process, der Waals interaction forces. Chapter 3 the production discusses in detail the experimental and requirements preparation of various experimental of the experimental apparatus, experimental In data. and particular, the techniques a novel such materials, the as design used in the acquisition of method exploiting the effect of temperature control in the preparation of monodisperse silica spheres in the colloidal range is presented Mathematical necessarj' for the 4. The details for the formulation of the computer simulation of particle deposition are presented results of the summary and in detail in Section 3.2.2. experiments are conclusions are presented discussed in Chapter in Chapter 6. algorithm in Chapter 5 while the T H E O R E T I C A L 2. T H E 2.1. E L E C T R I C A L D O U B L E B A C K G R O U N D L A Y E R In general, most substances acquire a surface electric charge when brought into contact with an aqueous medium. This acquisition of an electrical charge occurs either as a result of the ionization of slightly soluble surface groups or as a result of the preferential adsorption of specific ions from solution. This surface charge influences the distribution of nearby ions in the aqueous medium in order to maintain electro-neutrality. As (counter-ions) are attracted towards are repelled away random molecular from over co-ions in maximum surface the the surface (thermal) motion, leads made up of the charged manner consequence, opposite charge and ions of like charge (co-ions) the surface. This, together layer near a the surface. aqueous with of the mixing tendency of to the formation of an electric double surface and a neutralising excess of counter-ions This medium at the interface to zero (bulk of solution). This ions excess such charge that its concentration at distances system is distributed falls diffuse from a removed from the of two charge lashers, a fixed charge on the surface of the solid and a smeared-out sufficiently in a or diffuse layer in the liquid next to the solid, is the Gouy-Chapman model of the electrical double layer. 2.1.1. The Gouy-Chapman Model The mathematical development of this model has been treated in standard texts and hence only a simplified version of the treatment will be given here. - 15 THEORETICAL B A C K G R O U N D / 16 In essence, the model is based on the following assumptions: 1. The surface is assumed to be flat, of infinite extent and uniformly charged. 2. The ions in the diffuse part of the double layer are assumed point 3. to be charges. The solvent is assumed dielectric constant to influence the double layer only through its or permittivity, which is assumed to be constant throughout the diffuse part. 4. The electrolyte is assumed species, both having By to consist of a single cationic and anionic the same charge number z. using the Boltzmann distribution to describe ionic concentration, and equation to relate charge density and electric potential, expressions Poisson's which describe the decay of potential with distance from the charged surface are derived. Let the electric potential be from the surface in the at a flat surface and \p at a distance x electrolyte solution. By applying the Boltzmann distribution, the probability of finding an ion at the position x is: n where n + and n_ = n exp 0 [2.1] kT are the respective numbers unit volume at points where energy is + of positive and negative the potential is \p (i.e. where ions per the electric potential +ze \p and -ze \[/, respectively), z is the valency of the ions, e Q Q electron charge, k the Boltzmann constant, T the absolute temperature Q and the n 0 the identical bulk concentration of both ionic species where the potential \p = 0. The net volume charge density p at points where the potential is \p is THEORETICAL B A C K G R O U N D / 17 related to the ion concentration as follows: p = ze (n 0 - n_) + [2.2] Substituting Equation [2.1] into Equation [2.2] yields -ze \p +ze \p 0 p = ze n 0 Since e - e = 0 exp 0 - exp kT 2sinh(x), therefore Equation [2.3] can be rewritten as p = -2ze n sinh 0 The variation of potential 0 with [2.3] kT [2.4] kT distance from a charged surface is described by Poisson's equation V -]/ 2 = - 4.7TP [2.5] which, for an infinite planar surface, takes the form d</> 2 where 4tfp e is the dielectric constant of the aqueous medium. Combination of Equations the Poisson-Boltzmann equation [2.4] and [2.6] gives rise to what is known as T H E O R E T I C A L B A C K G R O U N D / 18 87rze n d i// 2 0 0 dx • The solution (52) of this expression, with x = 0; and \p — 0, di///dx = 0 when x = °= 2kT yp = - — ze In 0 ze ^ 0 sinh c a [2.7] kT the boundary conditions \p — \p Q when n be written in the form 1 + 7exp(-Kx) 1 - 7exp(-»cx) [2.8] where e x p [ (ze <// )/(2kT) ] - 1 0 7 = 0 exp[(ze ^o)/(2kT)] 0 [2.9] + 1 and 87reon z 2 n K °- 87regN Cz 5 1000ekT where N ^ The can be 0.5 2 A = [2.10] 1OOOekT is Avogadro's constant and C is the concentration of electrolyte. assumption that the potential in the diffuse part of the double layer reasonably well-represented by the solution of the Poisson-Boltzmann equation is generally considered to be valid below electrolyte concentrations of the order of 10" M for 1-1 electrolytes (53). Studies by Krylov 2 concentrations drop in the range of 0.1 - is sharper Gouy-Chapman at theory. these 0.5M concentrations and Levich (54) for seem to indicate that the potential than would Furthermore, one must account be predicted for all relevant such as volume of ions and variation of permittivity, which are neglected above model. by the effects, in the T H E O R E T I C A L B A C K G R O U N D / 19 2.1.2. The Debye-Huckel For 25°C the case where the potential is less than approximately (kT/e = 0 linearized through x + Approximation x /2! + 2 25.6 mV at 25°C), the Poisson-Boltzmann equation a power series expansion of the exponential term ....) in Equation [2.9], and by retaining 25.6 mV at (e can be = only the terms 1 + through first-order, Equation [2.8] reduces to yp = »// exp( -KX ) ro i n 0 This linearization Debye-Hiickel of the Poisson-Boltzmann approximation. Equation equation [2.11] shows that is known as the at low potentials the potential decreases exponentially with distance from the charged surface. However, for very small distances from the charged surface, the potential is likely to be relatively high, so that the Debye-Hiickel approximation becomes inapplicable and the rate of decay of the potential given by Equation [2.8] is predicted to be greater than an exponential rate. The Poisson-Boltzmann equation for a spherical interface takes the form r d</< 2 V i// = 8Trzec.n 2 r 2 where r is the distance from integrated analytically dr dr 0 s inh zeo^ kT [2.12] the centre of the sphere. This expression cannot be without approximation of the exponential terms. Debye-Hiickel approximation is made, then the equation reduces to If the T H E O R E T I C A L B A C K G R O U N D / 20 [2.13] which, on integration (with the boundary conditions, \p = \p 0 at r= a and \p = 0 di///dr = 0 at r = ») gives \p = \p a r 0 exp{-/c(r-a)} [2.14] where a is the particle radius. Unfortunately, limiting for the the Debye-Hiickel treatment of approximation (z\p < many Unapproximated solutions of Equation a computer by Loeb et al. (55). 2.1.3. Surface Charge Density, The density of charge at the two a is often too surface phenomena. numerically with 0 faces be diffuse layer, the solution of unit cross and mV) [2.12] have been obtained condition of electro-neutrality at a phases contains of the a colloid 25 charged interface requires that equal. This implies that when one total charge contained section extending from the in a wall the of the volume element to infinity must contain the same amount of charge, although of opposite sign, as a unit area wall contains. In mathematical form, this statement may be written = - J o p(x) dx of as [2.15] T H E O R E T I C A L B A C K G R O U N D / 21 Combining Equations [2.6] and [2.15] therefore yields [2.16] which upon integration gives OS e As x approaches °° (bulk [2.17] dx 47T 0 of solution), it is envisaged approach zero. Therefore, Equation that both \j/ and di///dx [2.17] becomes eK\jj 0 [2.18] 4JT From Equation o 0 depends composition then a The as both of the the surface medium. If the thus be seen that the surface charge density potential \p and Q K) (through on (i.e. K double layer is compressed the ionic increased), must increase. 0 2.1.4. The to on [2.18], it can the Electric Double L a y e r T h i c k n e s s 11K inverse of K which is defined in Equation thickness of the double layer. This [2.10] is commonly referred is due to the fact that this quantity represents the separation between the plates of a parallel plate capacitor having layer. the same surface A l l distances quantity. within potential and the double charge layer density are as the determined electric relative double to this T H E O R E T I C A L B A C K G R O U N D / 22 However, in the diffuse double layer model, although 11K is still customarily referred to as the double layer thickness, it is a misnomer. This can be the [2.11] which, upon substitution of 1 / K for x, predicts that seen from Equation potential falls to lie of its surface value as opposed to the value of zero expected in the bulk solution. 2.1.5. T h e Stern As Layer stated in Section 2.1.1, the development of the Gouy-Chapman theory of the diffuse layer assumes point charges in the electrolyte medium. However, ions have finite sizes and the finite size of ions will limit the inner boundary of the diffuse approach part of the double the surface When concentrations the electrolyte Gouy-Chapman theory since to within its hydrated adsorbed. The assumption electrolyte layer, that the ions but will break concentration known no volume down as ionic potentials high ion can only becoming specifically is acceptable concentration are very counter-ion at low increases. high, the concentrations at small value for the double layer thickness. In light of this shortcoming, Stern plane have and of an radius without predicts an unacceptably the surface, with an impossibly size of ions hy the centre proposed a model to include the finite considering the double layer to be divided into two parts by a as the Stern radius) from specific ion adsorption surface by electrostatic plane the surface. Within giving located at a distance this plane, a compact a hydrated ion the possibilit}' was layer and van der Waals 6 (about of counter-ions forces strong considered of attached enough to the to overcome T H E O R E T I C A L B A C K G R O U N D / 23 thermal agitation. Grahame Helmholtz through (56) further plane" and the centers an distinguishes "outer Helmholtz of specifically adsorbed within the plane". Stern The ions, which upon adsorption, while the latter corresponds to the Stern Outside Gouy-Chapman's the Stern model. The double layer required measured from used instead plane, the double only layer modifications former potential \p Q "inner dehydrated plane. to be of the analysis described by of the diffuse plane are that x is now than from the wall and that the Stern of the surface an is the plane are usually continues by the introduction of the Stern 5 rather layer potential is as the potential at the inner boundary of the diffuse layer. 2.2. E L E C T R O K I N E T I C P H E N O M E N A Electrokinetic phenomena is the term assigned to the four phenomena that arise when attempts are made to separate the mobile part of the electric double layer from layer phase the charged can be achieved boundary mechanically surface. This of the two parts by directing an externally applied (resulting in a initiating separation a relative relative movement movement between induced electric field and a transport of electricity). electric between of the double field along the the phases) or by the phases (resulting in an T H E O R E T I C A L B A C K G R O U N D / 24 The 1. four electrokinetic phenomena are: Electrophoresis — which involves the movement of a charged surface relative to a stationary electrolyte due to an applied electric field. 2. Electro-osmosis involves — which the movement is complementary of an electrolyte to electrophoresis relative to a and stationary charged surface due to an applied electric field. 3. Streaming refers Potential — to the electric which field is the opposite which is created of electro-osmosis when an and electrolyte is made to flow next to a stationary charged surface. 4. Sedimentation Potential — which is the opposite refers to the electric field which is created of electrophoresis and when charged particles are allowed to sediment relative to a stationary electrolyte. there Specifically, when electrophoresis occurs, several forces are present. Firstly, is the force of the exerted by the applied potential as a consequence charge carried by the particle. Secondly, there is a viscous retarding force due to the flow creeping a of liquid past flow, is given the particle which, for isolated spherical particles in by Stokes' Law. Thirdly, since the diffuse layer net excess of counter-ions, there contains will be a net force acting on the liquid due to the interaction of the ion charges with the electric field. The resulting flow of liquid (electro-osmosis) causes called the electrophoretic a retarding retardation. Fourthly, vicinity of the particle is deformed of its ionic rebuild atmosphere. the ionic force The atmosphere, on the particle. as the particle moves away from attraction between a which process effect is the distribution of ions Coulombic takes a the ions finite relaxation time. Thus in the steady state the charge centre constantly lags behind the centre This time in the the centre tends to called the of the diffuse layer of the particle. Consequently a dipole is formed which results in an electric force that acts on the particle. This force is usually T H E O R E T I C A L B A C K G R O U N D / 25 a retarding one and is called complicated double the relaxation effect. The latter two effects are functions of the zeta potential, the particle size, the thickness of the layer and the valence and mobility of the specific ions in the solution. Electrokinetic (Zeta) Potential $ 2.2.1. When an electrokinetic phenomenon occurs, the potential that is at the surface of shear between the charged surface and the electrolyte solution is called the commonly, electrokinetic usage potential of the Greek parameter assumed or, more letter in the study $ as the symbol. of colloidal to be identical with stability the Stern the zeta This because potential, potential due is an it has been potential, introduced to the important customarily in the Stern theory, which cannot be evaluated experimentally. The assumption of the identity of and £ stems from the following arguments and findings. Although the exact generally conceived location of the shear plane is not known, it is to be located either on (57,58,59) or slightly outside (60) the Stern plane, the latter being the more widely accepted choice. This is because in addition to the immobilized will probably ions in the Stern layer, a certain amount of solvent also be bound Firstly, because moving liquid thereby effective^ to the charged of the microscopic will not penetrate pushing the surface roughness .of all solid the depressions shear plane for the following reasons. but will further away surfaces glide from (61), the across them, the surface. Secondly, there exists an anomalously high viscosity in the vicinity of the surface (62) created by the interaction of polar solvent molecules with the high electric T H E O R E T I C A L B A C K G R O U N D / 26 field strength in this region. This high viscosity effectively moves the shear plane further out into the solution. Furthermore, researchers (63,60,57,64) indicate that the the plane shear is only a few the experimental results distance between the Angstroms. Thus, at of Stern low and various plane and intermediate o electrolyte concentrations, where exceeds the assumption distance between of identical justified. Figure 2.1 11K t the t//g and is a 100A, Stern $ the plane double layer thickness greatly and the shear would, for practical schematic representation of a plane and purposes, appear charged surface the to be and its double layer, together with other relevant parameters mentioned above. 2.2.2. Determination Although of Zeta Potential the zeta electrokinetic techniques technique that is of potential outlined may in be Section determined 2.2, electrophoresis. Theoretical the by any most expressions one of commonly relating the used the zeta potential to the measured electrophoretic mobility under various specific conditions have been established by (68), Booth (69) and Of inclusive the as Smoluchowski only theory of accounts surface conductance, it also can The (66), Hiickel (67), Overbeek Wiersema et al. (70). above, the it not (65), Henry be Wiersema for the et effects al. is conceivably the most of retardation, relaxation used for zeta potentials as high as 150 and mV. main assumptions of this model are: 1. The particle is a rigid, non-conducting uniformly distributed over the surface. sphere with its charge THEORETICAL + - - -I- Figure 2.1 A - + +- H — — B A C K G R O U N D / 27 - schematic representation of a charge surface and its double layer, together with other parameters as shown T H E O R E T I C A L B A C K G R O U N D / 28 2. The electrophoretic behaviour of the particle is not influenced by other particles in the dispersion. 3. Permittivity and viscosity are constant throughout the the double layer, which is described by 4. Only one type each of positive and mobile part of the Gouy-Chapman theory. negative ions are present in the mobile part of the double layer. Wiersema et governs the describes al. simultaneously distribution the fluid solve of electrical motion, and movement of ions in the successive approximations using a the Poisson-Boltzmann potential, the the Navier-Stokes' transport solution. The equation equations equation which solution is obtained by which which control the the method of computer. Their results are given in tabulated form in terms of dimensionless variables. 2.3. EFFECT OF POTENTIAL AND ELECTROLYTE ON POTENTIAL, determining ions will cause in an electrolyte solution, the addition of changes in the magnitude potential which, in turn, produce changes in the zeta potential. For OH ion is a potential determining Thus, addition under solution, a.ny on the conditions of OH" of increase of OH" and zeta ion for silica surfaces in an relatively ions causes (or release of H* surface ZETA DOUBLE LAYER THICKNESS If ionic strength remains constant potential SURFACE high insignificant electrolyte changes aqueous medium. ionic where strength ions increases the specific adsorption of OH" ions from) the surface and potentials. In surface example, the concentrations in the of the of the ions hence changes the magnitude of solutions of low electrolyte concentrations, THEORETICAL addition of any counter-ions ions into unchanged but into the the the Stern solution will layer. As potential at the a affect result, Stern plane the equilibrium the and B A C K G R O U N D / 29 adsorption surface potential remains hence the zeta potential is affected. This addition in effect causes the double layer thickness to be according of to Equation [2.10]. potential, surface potential and Figure The effects pH of and ionic strength reduced on zeta double layer thickness are shown schematically in 2.2. 2.4. PARTICLE-PARTICLE AND PARTICLE-SURFACE INTERACTIONS 2.4.1. Double Layer Interaction Energies When two double layers overlap, for example when two in a polar medium approach each other, an the two diffuse layers occurs. The interacting energy (or surfaces can (72) considerations from electrical electrostatic potential) boundary between conditions All (i) that must be utilizable the the two at solutions for the is described surfaces. exact thus created either force distribution interacting their made to numerical distribution through knowledge of the ec recourse potential energy be . determined interaction energy, 0 jj, is complex. No and electrostatic interaction of the ions in electrostatic a particles dispersed bodies, The by shortcomings have been outlined in Section the of interaction Poisson-Boltzmann 2.1.1 and or free of space charge and analytical expression determination the (71) from the of the calculation solutions or to various by can be given approximations. energy equation (ii) that either a assume whose constant THEORETICAL BACKGROUND / CO fl 1 \ (b) 1 \ 1 \ | shear plane •5 \ \ \ |\ increasing , PH 1 1 1 / / Distance From Charged Surface Figure 2.2 The effects of (a) ionic strength and surface potential and double layer (b) pH thickness on zeta potential, 30 T H E O R E T I C A L B A C K G R O U N D / 31 surface potential or a constant surface charge density boundary condition applies. However, which boundary condition best applies in real situations remains unclear. It has been suggested that if the time taken electrochemical equilibrium between the surface for the re-establishment and the bulk solution is negligible compared to the time taken for a Brownian collision to occur charge be is the result of the adsorption the surface Conversely, collision, charge density will adjust if the equilibrating process then a constant charge and if the surface of potential-determining assumed that the surface potential will remain constant and accordingly takes density of much during ions, then it can during the interaction (see Equation longer than interaction [2.18]). the Brownian is the correct assumption. This problem has been studied by Overbeek (73), who concluded that the rate of double-layer particles true is too fast overlap in a typical Brownian motion encounter between for adsorption situation will, in general, equilibrium lie somewhere to be maintained between constant and that the potential and constant charge density. By adopting Derjaguin's derived approximation an expression energy 4>^\ ^ e = d l the linearized form of the Poisson-Boltzmann (71) at small for the case separations, Hogg of constant-potential equation and et al. (74) have double-layer interaction between two dissimilar spheres, namely ea i a 2 U o i +^ 4(a,+a ) 0 2 ) 2iK iK —_Si_£z_ln( 2 ^Z+i// * oi 62 1+e k H h —„ )+ln(1-e k H 1-e 2 k H ) [2.19] where a, and a their respective 2 are the radii Stern potentials, H of the spherical particles, is the shortest distance \p^ and \p^ are between their Stern T H E O R E T I C A L B A C K G R O U N D / 32 layers, e is the permittivity of the dispersion and K medium is defined in Equation [2.10]. For equal spheres, with a = a = a and i / ^ ='/'g ^§> Equation [2.19] reduces to = 1 2 2 4> " eav> <t> edl £ ln(1+e k ) H [2.20] When one of the particle radii becomes infinite, a situation corresponding to the interaction between energy, 0 ^ a sphere and a plate, the expression for the interaction , reduces to 1+e edl 2 *6i*62 l n U ^)ln(l-e- + ) 2 K H 5 1-e [2.21] For the case where the charge densities at both surfaces remain constant during the interaction, Wiese and Healy (75) give the following expression for the interaction between two dissimilar spherical particles: e e d a i a U 2 5 i +ifr S 2 ) 4(a,+a ) l + J 5 1 fr ln( - " e K H _ -KH )-ln(1- - 2 K e H ) 0 2 [2.22] For the case of equal particle size and equal Stern potentials, Equation [2.22] reduces to: eaxp edl £ ln(1-e' K H ) [2.23] THEORETICAL B A C K G R O U N D / 33 For particle-plate interactions, Equation [2.22] can be rewritten as *edl " 2\p Y j 1+e 61 tZ/ 02 T r Y -KH In " 1-e ( ^ ^ ) l n ( l - e " 2 K ) H [2.24] For constant the mixed potential case, where and a second the interaction is between surface at constant charge one surface at density, K a r et al. (76) obtained for the general particle-particle interaction e d 4(a,+a ) l -tan- (sinh H)J-(^-^ 2\p 1 2 K 5 )ln ( l + e -2KH } 2 [2.25] which, for the case of equal particle size and equal Stern potentials, reduces to ,a-i// *edl e = T a 7T ,2 — ^5 2 - tan" (sinh/cH) 1 [2.26] For particle-plate interaction, Equation [2.25] becomes ea *edl = — 7T 2 ^5i^S2 . 2 tan" (sinhKH) - (^ 2 5 -^ 2 6 )ln(l+e" 2 K H ) [2.27] when the particle surface (surface 1) remains at constant potential and ea *edl = — 2 *6i*52 — 2 - tan" 1 (sinhKH) + ( ^ - ^ 2 ) l n ( 1 + e - 2 K H ) [2.28] when the plane surface (surface 2) remains at constant potential. THEORETICAL At separations large compared to the double B A C K G R O U N D / 34 layer thickness ( K H£ 5) — ic-H such that e « 1, Equations *edl This expression expressions 7 " *6i*«2 = is identical to that obtained linear superposition approximation the [2.21], [2.24], [2.27] and [2.28] all reduce to obtained using [2.29] e by Bell et al. (77), who to arrive at Equation Derjaguin's used the [2.29]. This suggests approximation may be used here to represent the sphere-plate interaction energy at all separation distances. A prediction of the interaction energy based should decrease in an approximately that the range of 0 j e(J should be decreased fashion with general is that 0 jj increasing H and on the above expressions exponential that e( by increasing K (i.e. by increasing electrolyte concentration and/or counter-ion valency). It should calculation be of double noted here layer that all the equations interaction energies derived above are applicable only for the when two double layers are involved. If a third double layer is present, for example, as in the case planar of two substrate, negatively charged these equations spheres will be depositing on a inapplicable. This potential distribution between two spheres situated on a planar positively charged is because the substrate will be distorted by the presence of the additional surface. Except for the deposition that takes approach place on a surface whose macroscopic particles, the presence of a third processes where deposition takes its frequent occurrence double layer double dimensions layer is, in fact, commonplace place on an extended in many deposition processes interaction energy and those hence of the in all surface. However, despite and its obvious effect on the the surface coverage, no published T H E O R E T I C A L B A C K G R O U N D / 35 theory for the potential currently available. Thus, until the equations discussed 2.4.2. London-van In attractive distribution between an d e r Waals layers is of this situation becomes available, Interaction 1873, van der Waals first introduced force present between two gases and liquids. This saturated molecules may Two neutral force the idea of the existence atoms to explain of attraction between of an the behaviour of neutral, chemically originate from any of the following electrical interactions: molecules with permanent dipoles mutually orientate each other in such a way 2. analysis interacting double above will be assumed to be approximately applicable. non-ideal 1. three Dipolar that, on average, attraction results. molecules induce dipoles in other molecules so that attraction results. 3. Attractive forces can universal attractive explained by dipole dipole the between known (1930) and generated around exist forces, London moment electrons also in a polarizes a nearby dispersion are due neutral its nuclei. The neutral induced dipole from atom as B non-polar molecules. These forces, were first to the rapidly fluctuating atom A by the motion electric field of this atom and the returning B then interacts with atom of instantaneous A field of resulting in an attractive force between them. With the exception account for nearly energy of highly interaction of separation with materials, London dispersion all of the van der Waals attraction. London showed of attraction between their distance polar other two atoms and is, to a atoms. varies first Thus, for an inversely forces that the as the sixth power of approximation, independent of the assembly of molecules, dispersion forces are, to a first approximation, additive and the van der Waals interaction THEORETICAL BACKGROUND energy between between two particles can be computed by summing all interparticle molecule pairs. The results of such / 36 the attractions summations predict that the van der Waals interaction energy between collections of molecules decays much less rapidly than that between individual molecules. Using the above assumptions, Hamaker (78) obtained the van der Waals attraction energy, # j , for the interaction of a number of geometrically simple v ( w bodies. For the case of two spherical particles of radii a« and a , separated in 2 vacuum by a shortest distance H, Hamaker derived the following expression for the interaction energy, # j : v c w <fi vdw 1 2 x +xy+x 2 A x +xy+x x +xy+x+y 2 + 2-ln 2 1 x +xy+x+y J 2 [2.30] where x = A a 1+a , a, y = — a2 2 is a constant, known as the Hamaker constant. For the H equal spheres, with a =a =a 1 2 (i.e. x=H/2a), Equation [2.30] takes form 1 <t> vdw 1 2 x(x+2) 1 + (x+1) +2-ln x(x+2) (x+1) 2 [2.31] T H E O R E T I C A L B A C K G R O U N D / 37 If a small interparticle separation is assumed, such that H < a (i.e. x << 1), Equation [2.31] simplifies to Aa *vdw f - ^ " 7^ = 2 32 For the sphere-plate system: <t> , vdw It should be noted H(H+2a) 10 nm) and even bodies whose thicknesses exceed overestimated + ln H of 0 v c j w then molecular calculated from at large distances (H > [2.33] H+2a that the above equations only apply of separation (H < Values 2a(H+a) = -A at very small distances only as a first approximation for dimensions. any of the above equations will be c. 10 nm). This is because if two atoms are further apart than a certain distance, by the time the electric field from one dipole has reached atom and polarised the other, the electron configuration of the first will have changed. This will result in a poor correlation between the two dipoles and the two atoms then experience what is known as "retarded" van der Waals forces. This finite time radiation between the particles required for the propagation results in a weakening of # of electromagnetic j v ( w and hence the attractive force is less than that predicted by the above equations. The effect of retardation has been worked out by Casimir and Polder ret (79), who obtained an expression for the retarded energy, $ of atoms by correlating <t> . and a correction it with the • unretarded energy, a a a a , between a pair T H E O R E T I C A L B A C K G R O U N D / 38 function, f(p), as shown below: *a-a where p = 2TTH/X and *a-a' = f ( P [2.34] } X is the wavelength characteristic of the atom dipole fluctuations. Approximate expressions for f(p) has been obtained by Overbeek (80) as follows: f(p) = 1.01 - 0.14p p < 3 [2.35] 2.45 = Expressions macroscopic for bodies have 2.04 " — P the P P * retarded been van obtained by der Waals Clayfield interactions between et al. (81), who repeated Hamaker's calculation using Equation [2.34], with Equation atom-pair interaction. They derived 3 2 the following [2.35] to represent the expression for the sphere-plate interaction: ^vdw " " 2 A 132-I where (i) I For H = i s p 1 > = p Ca 3 15H (H+2a) 2 2Da (H+a) 3 2 45H (H+2a) 3 3 s p [2.36] T H E O R E T I C A L B A C K G R O U N D / 39 (ii) I For H + 2 a sp c n > p > H = I, z . ' 3 = J 1 12 A ) 2 3 3 H 2 H Hp- B p H 2(a+H)ln — 6 Cp B 6p 5p 6p« 5 C D 20p" 2 30p ! + ap' (2a+H)p' + p' + p i 2 60H p 2 Dp' H(p'-3a) - ap' 3 2 H (!2a-4p') + Hp'(8a-p') 2 360H p 3 (iii) For p > sp D + 3 + 3Hp' + C 3p fl + I2p P P' -In - + — 12 B + 4p + p* ( a - — + A Ua-p' ) + + 2ap' 2 4 H + 2a o = A 4a (p'-a) B + 3 4p 4a A 3 12p A 12 -ln 3p fl B + H+2a H C 6p 3 + C 2 + 3 5p 6p 5 6 D + 20p" 30p 2a(a+H) H(2a+H) B + — 3 ! 2a - (a+H)ln H+2a H THEORETICAL BACKGROUND / 4 0 3X/27T where p = H p' = p A = 1.01 B = 0 . 1 4-2ir/X 2.45/(TT/X) C = D = 2. 1 4 / ( 2 7 r / X ) . 2 For the interaction between two spheres having equal radii a, Clayfield et al. gave the following expression for the retarded van der Waals energy: A ret vdw 4> 132 H+2a [2.37] ss where (i) : ss For p < = J H S1 C (H+2a) ln 30 2 H(H+4a) + 2a H+4a 1 1 + a' H 180 + 2a H (H+2a) 2 H (H+4a) J 2 2 H+2a H+4a + 2a 1 . (H+4a) 1 : H 3 (H+4a) 3 (H+4a) 3 1 2 H : THEORETICAL BACKGROUND / 41 For H+2a > p > H = 1 s2 p (a-H) p' ,3 = I s 1 + ap' 2 + 0 p a(a+H) + ,2 a ,' 3 p' Ha(2a+H)+ 2 p' r 4a(H+a) - H(2a+H) fl H - fl P' 5 p 1 1 p' - 2(2a+H)ln - + H(2a+H)(- - - ) H H p 12 1 1 H(2a+H) 1 1 + 2a 2(a+H)<- - - ) (— - — ) H p H' B P -H 2 2 3 2 p - 2(a+H)p' + (H +4aH+2a )ln H 2 2 T 1 1 + a -p' - H(2a+H)( - - - ) L H p j C p 1 1 H(2a+H) 1 1 In - - 2(a+H)(- - - ) + (— - — ) H H p 2 H p 30 2 f 1 1 1 + 3a -( - - - ) +(a+H)( — . H p H 2 - H(2a+H) 3 2 1 1 1 1 H(2a+H) (- - - )-(a+H) ( — - — ) + H p H p 3 D 60 -4a 1 _) P 2 1 2 ( 1 H 1 ^ 2 p 2 2 2(a+H) ^ 1 1 H p 3 2 1 (_ H 3 1 (— H 3 H(2a+H) 3 1 _ ) p 3 1 - — ) p 3 1 1 • H" p" T H E O R E T I C A L B A C K G R O U N D / 42 (iii) For p > H + 4a S3 A (2a+H)ln 6 H(H+4a) (H+2a) B (H +4aH+2a ) 2 3 (iv) : ss For H + 4a > p > ~ X ln 2 + a 1 — 2 2 + . H 2 (H+2a) 1 + H+2a H+4a 2 - 4a H(H+4a) 2 H + 2a s4 4a p'-2a : = s3 J + 4a a (p'-2a) (2a+H) + 3 - (a-H) - 1 2 + 2a - (p'-2a) (a+H)a 2 6a r H +4aH+2a + - f l 3 2 (p'-2a) a- (p*-2a) 3 (p'-2a) 4a - : - 2 4a(H+a)-H(2a+H) 1 1 Q( - p H+4a H+4a p R )+ 2 2 (p'-2a)« 3 2(2a+H)ln (p'-2a) Ha(2a+H) H + (p'-2a) 1 + H + 4 a - p + R ( - p 1 ( p — 2 (H+4a) 2 1 H+4a 5 ) THEORETICAL BACKGROUND / 43 B (H+4a) -p 2 2 - Q(H+4a-p) 3 H+4a 1 1 ) + a H+4a-p+R( - p H+4a + (H +4aH+2a )ln 2 2 C 30 + 3a H+4a ln( 1 1 R 1 ) - Q( - ) + - ( — p H+4a p Lp H+4a H+- 2 p 2 1 ( H+4a 2 p~ p - Q 3 ( 1 (H+4a) { p (H+4a) - R 1 ) + - ( 3 - 2 1 3 (H+4a) 3 2 1 R 1 1 4 p" (H+4a) (H+4a) ) + - ( — - 3 3 1 2 ~? " 1 3 1 2 1 + 4a 2 R 1 1 -) •) + - (— " (H+4a) 3 p (H+4a) . Q 60 (H+4a) 1 Q 1 - (• 2 - 1 3 ft where Q = 2 (3a + h ) R = (h + a )(h + 4a) A = a = = = 7= 2p — 2A 3p A 4p B C P 3p — — — 2 + 3 — ft B — P 2 B 3p + + 3 4p" C 2p D 5p 5p 4 C 3 D 5 + 5 D 6p 6 Since the frequency of electronic fluctuations 10 ' 3 in atoms is of the order of to 10 ' ° per second, the value of the wavelength, X, may range from 3Q0 THEORETICAL BACKGROUND / 44 o to 3000 A. However, to simplify calculations, Overbeek (81) recommends a value o of 1000 A for X to be applicable for most materials. If the above equations can be considered correct, i.e., the assumption the interactions between individual molecules embedded that in macroscopic bodies are additive, then the remaining problem in calculating the van der Waals interaction between colloidal solids methods are available London-Hamaker evaluated the from who for this microscopic assumed of evaluating calculation. approach, the individual materials involved. (82), is that atomic the Hamaker The first in which constant, of these the A. methods Hamaker Two is the constant polarisabilities and the atomic is densities of The second method is the macroscopic approach of Lifshitz that the attraction force between two interacting bodies was due to a fluctuating electromagnetic field in the gap, arising from all electric and magnetic polarizations within the two surfaces. and the intervening van der Waals interaction energy and the retardation just from calculations properties the medium can be treated In this way, the interacting bodies bulk properties are complex, values between Where the and require materials effects could be determined concerned. However, the the availability of bulk optical/dielectric of the interacting materials over a sufficiently wide frequency range. The various methods dispersion of as continuous phases and hence the of calculating the Hamaker constant, A, from optical data have been reviewed by Gregory (83) and Visser of Hamaker 10" 2 0 a range and constants 10" of values 1 9 for single J. Some is quoted materials examples have been are given for a given material, (84). In general, reported in Table to vary 2.1 (84). it reflects different T H E O R E T I C A L B A C K G R O U N D / 45 methods of calculation within the basic microscopic or macroscopic method. Table 2.1 Values of Hamaker Material A(microscopic) x air, A(macroscopic) 10" J x 2 O 10" J 2 O Water 3.3- 6.4 3.0- 6.1 Ionic crystals 15.8-41.8 5.8-11.8 Metals In Constants 22.1 7.6-15.9 Silica 50.0 8.6 Quartz 11.0-18.6 8.0- 8.8 Hydrocarbons 4.6-10.0 6.3 Polystyrene 6.2-16.8 5.6- 6.4 the presence the of a liquid Hamaker constant constant. For example, dispersion medium A medium, rather must be replaced for the interaction than by in a vacuum an between two effective solids 1 and or in Hamaker 2 in a 3, the effective Hamaker constant is given by A132 where dispersion — Ajj is the Hamaker A 1 2 + constant A - A 3 3 - A 1 3 [2.38] 2 3 for the interaction of material i with material j in a vacuum. If the attractions between unlike mean of the attractions of each A ij phases is taken phase to itself, i.e., if = ( A ii' jj A ) 0 ' 5 to be the geometric T H E O R E T I C A L B A C K G R O U N D / 46 then Equation [2.38] becomes A If the two 132 = interacting ( 11°' S solids are of the same A ~ 33°" A ) 5 ( 22°" A ~ 5 [2.39] 33°' ) A S material, the above expression becomes A 131 A-32 will usually be both greater or both A33 has a value negative, i.e., a However, positive less than an A 3 3 3 3 A J 33 However, criticism. method those of solids attraction between of pair-wise of neighbouring A of A-- interaction A 2 and n and A occurs 2 2 material, them. The interaction are either 2 situation , then between of the same A A where addition due the solids. 1 3 1 is always will be weakest atoms in A and has fallen on strong exerts a force on atom B and will therefore be low. 1 3 1 to Hamaker 1 in A is 1 3 2 medium are chemically similar, since A«« Firstly, it is evident that if atom the presence [2.40] in the unusual will be of similar magnitude and the value of A The B, . der Waals when the solids and the dispersion A ~ A between van for the interaction ^ 11 (attraction), i.e., when intermediate repulsive positive, indicating = is bound to influence 1 in the interaction between A l and B l . Secondly, it is not obvious whether the electric field propagated by an instantaneous dipole ought to take into account the finite dielectric between it and the atoms some distance from properties in the other body. Thirdly, of the material the but another dielectric Lifshitz approach by medium. These difficulties treating the interacting surface that lies it is not clear how addition needs to be modified if the space between bodies A vacuum the free and B the is not air or have been eliminated in bodies and the intervening THEORETICAL BACKGROUND medium This in terms approach properties of their is able bulk to deal and is also able properties, specifically their dielectric / 47 constants. with bodies made of layers of different dielectric to cope with the problem of solids separated by a liquid medium (85). Despite apply the superiority of the Lifshitz approach, it is extremely difficult to in practice. Furthermore, results for many body shapes are not easy to obtain due to the complexity in mathematical treatment of the continuum Even then, the evaluation of the interaction energy still requires theory. a knowledge of the complete optical spectra over all wavelengths for all the materials of interest, of which Equation few are available. In light [2.40] will be used here of the foregoing, the retarded for the calculation of London-van Hamaker der Waals interaction energy. 2.4.3. Other Relevant Forces Besides the above-mentioned electric double layer and van der Waals forces, other forces which are important in the deposition process include 1. Viscous interaction force, which is due to the increased resistance to movement experienced by a particle near a foreign surface. This force reduces the mobility of a particle and hence tends to retard deposition. 2. Born repulsion force, which is a short-range molecular interaction force resulting from the overlap of the electron clouds surrounding individual molecules on the two approaching surfaces. 3. Structural force, which structure (i.e. changes vicinit}' of a surface arises as a result of changes in the local ordering in the solvent of molecules) or interface due to packing constraints. in the THEORETICAL BACKGROUND / 48 2.4.4. Overall The total summation energies Interaction energy of of the electric due to Born Energy interaction double layer and structural between two solids and van der Waals forces could is obtained by energies. Potential be similarly superimposed to provide a more complete picture of the overall interaction, if the expressions were available. Figure 2.3 is a qualitative sketch of two general types of potential energy curves -> which are possible. It should 0, 0 j| -> e( ^vdw ^ constant with 0 }i ec 0 and v c j be noted that as the separation distance w decreasing more ~°° and as H -> rapidly. The two 0 jj and «, both curves H e( show that attraction predominates at very small and very large separations. A t intermediate separations, the effects of the two contributions must detail. In general, the potential energy minima as sketched, although some curve of these considered in more consists of a maximum and two features be may be masked if one contribution greatly exceeds the other. The height of the maximum above \|/ = 0 is the energy barrier. The deeper minimum is due to the repulsion caused small distances when is called by the Born force which predominates minimum falls off more and it is due to the fact that the van der slowly (approximately as 1/H) at large than does the electrical double layer interaction (approximately depth is measured at very the two surfaces come into contact. The more shallow one- the secondary Waals energy is called the primary minimum, and it from 4i = 0 also. Although distances, that is, the secondary minimum attraction separations as e predominates is generally present, it may ). Their at large be quite T H E O R E T I C A L B A C K G R O U N D / 49 O cd cd u cu > ^T2/ ^ o ^ / / 1 / / / — ~~~ ^vdw ^ 7 '/ Separation Distance H Figure 2.3 Qualitative sketch of two general types of overall interaction energy (<j>j) which are possible shallow, especially in view of the effects THEORETICAL BACKGROUND / 50 of on retardation and the medium attraction. When two surface due present or particles approach one to diffusion, either when the pull the or two 0rpj), occurs. On surfaces the prevented. primary This will be much no barrier is primary minimum. As hand, if the to the each net other represented by and thus curve attractive force potential energy flocculation 4> £-j j r L is will a result, flocculation height of the average Brownian thermal (refer or stable energy to curve deposition is with respect to flocculation or deposition. Nevertheless, flocculation or deposition could still occur into the will other repel system, minimum will occur. When shown in curve #«p2), then the is appreciable compared the following barrier is negligible compared to the bodies together into the deposition barrier of the height of the energy of a particle (as another or when a particle approaches a more secondary minimum. However, the reversible than in the primary process in this case minimum case due to the can be finite depth of the energy well. The deduced general from interaction Stern by system interacting of parameters the that potential appear energy-distance in the energy expressions. These parameters are: potentials controlled the the character the of each surface; concentration Hamaker solids and constant, on the double and valence layer of which, depends medium repulsive sign and thickness, the separating them the and attractive magnitude of which electrolyte on curve is (Equation materials of in double layer turn [2.10]); the (Equation [2.39]); and particular pair of boundary conditions assumed during the the two the interaction, THEORETICAL that is, either constant surface potential or constant BACKGROUND surface charge / 51 density, or both. For Stern everywhere by potentials except at very of equal small sign, distances the potential energy of separation is repulsive where it is dominated the London-van der Waals force. In this case, the effect of variations in the value of that on the total interaction potential energy, with the height values of \jj g of the potential energy as would be expected barrier in view will K and A increase of the increase constant, is with increasing in repulsion with this quantity. The effect of variations in the value of K is that as K increases (through increasing either counter-ions), the height barrier disappears the concentration and thickness altogether at very of electrolyte of the energy large values or valence barrier decrease of K. When of the until the the Stern potentials of the two surfaces are of opposite sign, the overall interaction energy is everywhere parameters attractive, regardless mentioned, the Hamaker of the values constant A of on continuous and is the one which controlled because its value is determined by the chemical and K nature A. Of the can be least of the dispersed phases involved. The effect of the variations in the value of A the total interaction energy is that the height of the potential energy barrier decreases values and the depth of the secondary of A. Qualitative sketches interaction energy for both like minimum increases with of the effects of i//g, K, and A and unlike charged Figures 2.4, 2.5 and 2.6, respectively. increasing on the total surfaces are demonstrated in THEORETICAL BACKGROUND / 52 Separation Distance H Figure 2.4 Effect of Stern (or zeta) potential on overall interaction energy as a function of separation distance between two charged surfaces T H E O R E T I C A L B A C K G R O U N D / 53 increasing i/K Separation Distance H Figure 2.5 Effect of double layer thickness (1/K) on overall interaction energy as a function of separation distance between two charged surfaces of same polarity. THEORETICAL BACKGROUND Separation Distance H Figure 2.6 Effect of Hamaker constant A on overall .interaction energy as a function of separation distance between two charged surfaces of same polarity. / 54 3. EXPERIMENT 3.1. INTRODUCTION The of surface charged planar object of the experimental potential and electrical colloidal silica was to investigate the effects double layer thickness surface substrate. The magnitudes of the two surface potentials were altered by the electrical double layer thicknesses by of the suspension while dissolving different pre-determined an coverage of onto varied deposited on gravity the pH spheres work by adjusting were main oppositely-charged quantities of a neutral electrolyte into the suspension. 3.2. C O L L O I D A L In PARTICLES accordance with suspension of dispersed, the conditions implicit in the theory, uniformly-sized, review of the literature showed organic and inorganic materials spherical fact that, besides notably stable over surface chemistry a meeting wide of silica mandatory. (86,87,88,89,90,91) in the size-range the above range has was A that the procedures for the preparation of several are available. In the present study, silica was the particles a model colloidal chosen as the model colloid due to requirements, silica of electrolyte conditions been extensively (93,94). 55 of interest studied suspensions (92) and and well are that the documented E X P E R I M E N T / 56 3.2.1. Production of Amorphous Silica Several particles methods are available (93). The most common hydrogen-oxygen flame. Another that of chemical silicates) with precipitation Spheres for the production method is to hydrolyze of amorphous silica silicon in a halides method for the production of amorphous silica is by reacting tetra-esters water, a technique developed of silicic by Stober (tetra-alkyl et al. (90). This reaction is carried out in a mutual solvent due to the immiscibility silicates. The reaction is catalysed by either acid of water with alkyl acid or base solutions. According to Aelion et al. (95), the overall production takes place via two steps, namely, the hydrolysis of ester to silicic acid (ii) followed by the dehydration (i) of silicic acid forming amorphous silica, i.e., (i) Si(OC H ) (ii) In the 2 Si(OH), the presence hydrolysis hydrolj'sis 5 and reaction a + 4H 0 2 = Si0 2 + = Si(OH) mutual reaction range and + 4C H OH 2 5 2H 0 2 of a hydroxyl ion catalyst and an excess of water, both the dehydration is rate-controlling reactions and are fast is first tetra-ethyl-ortho-silicate (TEOS) concentration. Stober the s order and with silica spheres of 0.05 to 2.0 Mm, of fairly depending narrow respect et al. (90) found solvent is an alcohol and the catalyst is ammonium produces complete. size to the that when hydroxide, the distribution on the type of alkyl The in the size group of the silicate on the mutual solvent. The longer the alkyl chain, the larger the particles. E X P E R I M E N T / 57 3.2.2. Production of Large Uniform Silica Spheres Owing to the inherent limitations in resolution of an optical microscope, it is essential that a model colloid of size in the vicinity Although Um Stober et al. were able to produce range, the method still has two particles (>1.0 non-uniform. silicates nm) which Secondly, to achieve a other than TEOS, range which is micron be silica spheres in the 0.05 notable drawbacks. are occasionally of one used. to 2.0 Firstly, it produces larger non-spherical and often unacceptably of sizes the the only requires tetra-ester use of of alkyl silicic acid commercially available in a pure form. Therefore, in the present investigation, an attempt was made to produce of sizes by reacting only T E O S altered by varying monodisperse and the temperature silica spheres with a suitable range water. In this case, the reaction rate ( 2 0 ° C to 60°C) and - the alcoholic was solvent (methanol, ethanol, n-propanol and n-butanol). The (AnalaR materials grade, BDH), solvents. Distilled Fisher) was maximum ammonia well was vacuum sphericity (99.9%, were as water twice general sequence used and Union as as follows. Methanol, absolute ethanol (Stanchem), deionized just distilled before monodispersity as Carbide) was of steps was n-propanol similar prior use to use. T E O S in reported employed as the by the to that employed and were used (Reagent experiments Bowen n-butanol to grade, ensure (33). Anhydrous catalyst/stabilizer. by as Bowen The (33) and is described below. Measured volumes of alcohol and freshly distilled and deionized water were EXPERIMENT pipetted was into a then 120 ml glass bottle with a teflon-lined screw-on positioned glycol-water in a mixture specially was circulated refrigerated constant temperature reaction temperature constructed plexiglass to within through the attained, water-alcohol anhydrous mixture An ethylene jacket from ±0.1°C of the desired value. The contents of the ammonia through stirrer. After thermal equilibrium gas from a glass a cylinder capillary until was bubbled saturation into the was reached (indicated by no further increase in volume). A t this juncture, the TEOS, had a a bath ( N E S L A B Endocal RTE-9) to maintain the bottle were agitated continuously by a magnetic was cap. The bottle jacket. insulated / 58 which been maintained at the same temperature, was injected into the bottle using glass syringe. The bottle was then capped tightly to prevent ammonia from escaping. For all experiments, the volume ratio of TEOS:water:alcohol was 1:5:25, a ratio found by Bowen (33) to yield optimal results for the TEOS, water and n-propanol system. After an invisible hydrolysis reaction in which mixture The of suddenly precipitate appeared about reaction 15 was consequence, 9 became hours minutes almost cases. indicating reduced as the that of silica temperatures the rate temperature experiments in which Clean glass of 60, 40, 20, 0 average precipitation. but only after a lapse of the was the contents of the reaction bottle were sampled than these having indicated all the onset instantaneously at 60° C at -20°C, considerably for reaction preliminary opalescent signifying silicic acid was formed, the and hydrolysis lowered. As a at 1, 2, 3, 5 and -20°C, sizes were measured respectively, at times shorter that the reactions had already reached completion in capillaries were used to transfer a droplet of the E X P E R I M E N T / 59 suspension allowed to carbon-coated to dry and then locations on the grid The photographs, Research image micrographs using a Hitachi were then were carrier taken HU-llA analyzed analyzer. Between order to determine Figure microscope grids. The at a samples number of random transmission electron microscope. the average 100 automatically by means to 400 were particles of a size and standard deviation of each sample. 3.1 shows an electron micrograph 0°C. Figure sample plotted demonstrate 3.2 in gives the cumulative terms of that the particles of one of the batches size distribution normal-probability are normally curve co-ordinates. and a standard particles differ realized with from deviation of 0.05 of the same The two distributed, reasonably the median size the other by large particle Mm, more batches; 3%. figures spherical and diameter of indicating that only than of larger temperature have a relatively narrow range of sizes. This sample has an average 1.37 |tm Leitz measured in silica particles, obtained in this case using the solvent ethanol and a of were along with those of a calibration standard (54864 lines/in., Ladd Industries), TAS-PLUS electron Similar 5% results generally speaking, of the were the smaller particles were even more monodisperse and spherical. The and runs type number particle of solvent are listed were conditions. average repeated These on showed in Table different excellent considerations, the alcohols used sizes obtained days as a function of 3.1 and plotted as Figure 3.3. Several under otherwise reproducibility. Because identical experimental of water as solvents in the present study to those having low molecular weights temperature and straight hydrocarbon miscibility were restricted chains. Even so, E X P E R I M E N T / 60 it was not possible to employ prevented particle production n-butanol below 0°C. Also, excessive evaporation with methanol and ethanol at temperatures exceeding 40°C. The alcoholic most dominant feature solvent is used, the average inversely with to of Figure size 3.3 of the produced temperature. One possible explanation the saturation concentration temperature. This hypothesis of ammonia, is that, regardless particles varied is that this trend is related which was tested by conducting decreases with increasing two experiments (one with Figure 3.1 Electron micrograph of silica particles obtained using ethanol as the solvent and at a temparature of 0°C of which E X P E R I M E N T / 61 99.9 0.1 1 1.32 ' 1 1.34 1 1 1.36 1 I i 1.38 ' 1.40 • i 1.42 Diameter (um) Figure 3.2 Normal-probability plot of the cumulative the particle batch shown in Figure 3.1 size distribution curve of EXPERIMENT TABLE / 62 3.1 Average diameter (Mm) and standard deviation of silica spheres obtained using different temperatures and solvents SOLVENT T E M P E R A T U R E (°C) Methanol -20 0 1.01±0.04 0.95±0.07 1.87±0.11 Ethanol 1.37±0.05 20 40 0.44±0.02 0.23±0.02 0.44±0.03* 0.21±0.02* 0.54±0.03 0.26±0.02 60 0.55±0.03* 0.53±0.03** 1.67±0.05 n-Propanol n-Butanol 1.05±0.05 0.88±0.03 0.46±0.02 0.27±0.02 0.46±0.03** 0.29±0.02* 0.62±0.02 0.35±0.01 0.25±0.01 0.26±0.02 * replicate experiment ** water-alcohol mixture saturated with ammonia at 40 "C. ethanol and saturated one at 40°C with n-propanol) in which and then cooled to 20 °C the alcohol/water mixtures were before injecting the TEOS. As can be seen from Table 3.1 and Figure 3.3, the results of both tests were virtually the same indicating as those obtained that the ammonia in corresponding concentration solutions near saturated saturation levels at 20°C, does not significantly influence particle size. A more plausible explanation is that the trend is associated with the hydrolysis reaction rate, which was observed to decrease significantly at lower E X P E R I M E N T / 63 Temperature (C) Figure 3.3 Average diameters of silica particles as a function of temperature and type of solvent used. The two filled-in symbols represent cases where the water-alcohol mixture was saturated with at 40 °C before the reaction was carried out at 20°C. ammonia E X P E R I M E N T / 64 temperature. This relationship between particle size and with the earlier observations of Stober et al. (90) reaction rate is in accord and can be interpreted in terms of the limited self-nucleation model of monodisperse particle production proposed by critical LaMer and concentration, rate and the saturation C* , that above Qf below which providing Dinegar (96). The soon as reduces only concentration, nuclei are to form C only after sufficient number the super-saturation growth limited to a very ^ already a particle nucleation nucleation is negligible. model is depicted in Figure 3.4. nuclei begin which model hypothesizes can above which occur proceeds at a an schematic appreciable greater growth will representation of than occur this Thus, as the hydrolysis of T E O S proceeds, silica the silicic acid concentration Si(OH)^ and, present, below particle 0* ^. From a providing that the exceeds growth this C* ^. As a permanently point onwards, self-nucleation period is short initial outburst, the final sol will be monodisperse. If the of nuclei will acid. As at is significantly reaction rate is markedly reduced, it thus seems reasonable number the existence of a spontaneous available. A of nuclei are of C* first be required consequence, since the to relieve same the amount high that a much smaller supersaturation of T E O S is reacted of silicic in each experiment, the average size of the particles must increase as the temperature is diminished. The relationship between is far more complicated. particle size and the different alcohols employed In general, particles produced in ethanol and exhibited the strongest temperature dependence, especially at low elevated the temperatures, the alcohol, but particle size varied directly with the under cooler conditions, this tendency appeared n-propanol temperatures. A t chain length to reverse of itself E X P E R I M E N T / 65 Nucleation Reaction Time Figure 3.4 Schematic diagram showing how a nucleation and a growth period may be separated resulting in a monodisperse except in the but it is case of methanol. probably related to This the complex different behaviour effects is not sol easily temperature explained has on such solution properties as solubility and viscosity. The most important result demonstrated by this study is that, by using a single alkyl alcoholic silicate solvent (commercially (ethanol or available n-propanol), and TEOS) by along simply with varying a short-chain the reaction temperature, it is possible to produce uniform silica spheres over a wide range of E X P E R I M E N T / 66 sizes (0.2 - 2.0 monodisperse, um). with increased. The the The size particles made distribution best results in becoming are obtained when this fashion wider the as are the apparatus relatively temperature is is maintained as clean as possible and reagents of maximum purity are employed. Thus, the size in order to fulfill range of around monodispersed silica was produced by ml) at a the implicit requirement that one micron reacting reaction CLEANING The utilizable TEOS (2 ml) temperature OF of to from with water approximately 1.00 the produced presence silicate, and suspension contaminants are include the mutual hydrolysis of alkyl dispersed which due solution. These in the experiment, a batch of ± 0.04 (10 ml) 0°C. The above in n-propanol (50 number average jum. PARTICLES particles form used model colloid in spheres exhibiting similar qualities as those mentioned particle size obtained in this case was 3.3. be a of silica by of the various alcohol above method contaminants solvent, the are not residing particles in water with in the alcohol produced ammonia. Therefore, in order to yield minimum in a by a highly carry-over the production solution, a suitable cleaning method had the of to be developed. A method that was Bowen (33). In treated silica initially his thesis, particles could pursued rather substantially was Bowen be reported easily a washing redispersed in technique aqueous that due to whereby solution the having E X P E R I M E N T / 67 conductivity for values in the range this technique are described below. A filtered 2 /zmho/cm. The procedures batch of freshly produced almost to dryness and the wet filter cake beaker containing the wash the of 1 - beaker were ultrasonic bath refiltered and then solution and a magnetic stirred and for ten minutes the was dispersion treated after process particles were dried at 100 °C After water hydrosol was was used always as the wash in a the in a vacuum heat during the drying process is required to improve When solution, was stirring bar. The contents of Branson 5200 the contents of the beaker repeated. for 48 hours particles placed in a water-cooled ultrasonically which necessary filtration were step, the oven. The addition of the desorption of ammonia. it was found that the final far less disperse than the initial suspension. However, when two or three washings in either n-propanol or n-butanol were used, the resultant silica suspensions were found to be as well dispersed as the original production suspension. According to Bowen, the reason why in promoting water and dispersion in alcohol, is due drying, the dissolved cementing surfaces rather to the difference the former negligible. It is suggested that silicic alcohol pretreatment is effective being rather in the solubilities substantial as the water in the filter cake acid precipitates on the available the latter is evaporates upon surface, thereby together those particles in contact. Since the solubility of concave is less than than dissolve that of convex upon subsequent surfaces (94), these bridges tend redispersal. Thus, chemically bound together after resuspension. Furthermore, adsorb while of silica in many layers of water. increases the removal of these many particles silica to grow remain silica surfaces strongly Consecutive washing of particles in alcohol greatly adsorbed water silica in alcohols is very small, a negligible layers. Because amount the solubility of of precipitation takes plaee E X P E R I M E N T / 68 during the drying produce an process. Thus, alcohol-treated silica particles can be dried to easily dispersed powder. Although the above method produces silica redispersed, it still suffers from the formation particles which can be easily of small quantities of agglomerates which are undesirable for the deposition experiments. Thus, to avoid this problem, a revised method based on step for the care of removal particle unacceptably in the high values is conductivity was developed. In this method, the drying ions was However, of removing these using clean, distilled and rather of the of the arduous filtrate as to handle this task. cm in screw-on cap. The 55 mm porous height stainless steel annular and 6.2 disc, ring. The ions a the considerable As an filter, which was cm 47 silica desired schematic the filtration and value. This repetition before alternative, a diagram of process, the cross-flow this cleaning constructed of thick-walled top portion consisted of a cylindrical in bottom part housed a 55 polyethylene to the presence of ammonium in ions is to repeat A parts. The 7 with acceptable. plexiglass, was chamber suspensions reaches The of two takes deionized water as the wash solution device is depicted in Figure 3.5. made up elimination it requires solution becomes devised eliminated. This it results of conductivity due way conductivity however, filter of ammonium solution. One the above was agglomeration. dispersion process until the diameter mm mm with an 'O'-ring sealed perforated stainless steel disc, a filter filter membrane membrane, was and sputter-coated a with 55 mm a thin o layer (400 A) of gold-palladium alloy that enabled charged when a potential difference was and the the membrane surface to applied across the membrane at the stainless steel disc at the bottom. The porous disc of polyethylene be top was E X P E R I M E N T / 69 Water Inlet TT LP " 1 Thick- Walled Plexiglass Stainless Steel Annular Ring Alloy-Coated Filter Membrane Porous . Polyethylene Disc Perforated Stainless Steel Disc Water Outlet Figure 3.5 Schematic diagram of the cross-flow filter (cross-sectional view) E X P E R I M E N T / 70 used as a insulator support preventing assembled for the flimsy the two filter charged membrane surfaces from as well as to act as an contacting. The filter was by slipping the top portion into the bottom portion and fastened with six sets of bolts and nuts. The filter was designed to withstand pressures of up to 20 p.s.i.g. The follows. procedures portion of previously A redispersed of general twice using ammonium ions for the cleaning produced of silica silica in the solution. These stirring bar was introduced filtered the filter membrane supply (Anatek Model 50-1.0S). Since charged surface filtered as and silica particles were then magnetic into the chamber to create a tangential motion which across membrane was and transferred into the filter chamber. A swept particles clear of the membrane surface. A the suspension were n-propanol as the wash solution to remove the majority washed with water, dispersed exhibit a negatively suspensions was potential difference was and the stainless steel surface it is known disc by negatively charged a regulated that the silica under the prescribed created power particles will experimental conditions, to maintain a repulsion between the particles and this surface. Distilled and deionized water, which was kept in a specially constructed pressure with a filter membrane been present pressure filtered Seibold 0.2 um in pore size to remove any dust that may in the system. Nitrogen source water conductivity vessel, was forced through a filtration unit equipped to maintain was then (model the pressure allowed of the outflow gas from from L T A ) conductivity cylinder in the vessel to be added the filter meter. a dropwise was The at 10 into continuously cleaning was used have as the p.s.i.g.. This the chamber. The monitored process was with a terminated E X P E R I M E N T / 71 when the desired conductivity value attained. was It was found that for an of 3 u.mho/cm, a washing time of less than 30 minutes initial conductivity value was sufficient to reach a final value of 1 (imho/cm. This However, method in the course managed loosely thus of cleaning, attached the membrane, conducting coating continuous, thus defect would seem ideal for the present it was discovered to deposit on the alloy-coated membrane to be rather through of cleaning may on constraints. Since experiments, mentioned may force the flow particles to the surface; and the it was were filtration may not pursued the loss of particles to the surface with they seemed include which causes particles not be electrically do not repel the particles. Although corrected, cleanings some surface, although Possible surface, patches that easily subsequent above, which surface. the membrane creating be quite to this that purpose. done replaced using by further is very due to time undesirable the tedious revised centrifugation this for in the method solid-liquid separation. 3.4. THE DEPOSITION The deposition CELL of colloidal silica carried out in a specially constructed has mm onto a cylindrical configuration and dimensions 15 mm height. The inner wall sealed with a 0.17 mm a planar substrate was deposition cell. The cell, built of plexiglass, of the cell mechanical entrapment of particles. Both and spheres was cell made I.D., 26 mm very ends were thick glass cover slip 25 mm O.D., and 10 smooth fitted with to minimize an 'O'-ring in diameter. It should E X P E R I M E N T / 72 be noted hence here that only the bottom was coated with firmly in place by wall. The mm, surface acted as the deposition surface plastic. The other was not. The cover slip was an aluminum optical held screwing threaded aluminum end-cups onto the threaded outer cup was designed with a concentric circular opening which enabled the transmission of light through the cell when it was under and microscope. A schematic diagram of the deposition of 15 placed cell is depicted in Figure 3.6. Cylindrical " Plexiglass 25 mm Glass Cover Slip Threaded Aluminum End-Cup 2VP/S Coated 25 mm Glass Cover Slip Figure 3.6 3.5. PREPARATION In DEPOSITION of the deposition cell SURFACE the deposition experiments, circular cover slips 25 0.17 mm were to be in thickness were carried negativelj' charged 4 OF Schematic diagram to 11. The out used under as the conditions deposition where the mm and substrate. The experiments silica stable sols to prevent coagulation. This occurs in the pH iso-electric point of the silica in diameter are range and of about surface is approximately pH 3 arid E X P E R I M E N T / 73 hence the sol is positively counter-ion concentration charged thins below the this double value. layer Above to the pH 11, extent the that high the sol becomes unstable. Unfortunately, in the pH range where the particles are stable, the negatively glass ensure cover adherence slip of will also silica exhibit spheres a onto the charged glass surface surface characteristics of the glass have to be altered to one charge. This was accomplished by coating the clean cover layer of a cationic copolymer of 8 0 % 3.5.1. Production search the copolymer can method has 4 that Utsumi be prepared of Mallinckrodt) and by ml not 20% had to be reported a solution polymerization. A (AnalaR 100 tightly wrapped necessary grade, ml BDH), 4 point inhibitor Carbide), deoxygenated ml H 0) 9 and added with in a to prolong packed bed a foil to exclude of literature whereby styrene this using sealable screw-on light. Both styrene as each contained pyrogallol, 50 anhydrous (99%, injected shelf-life. Nitrogen in a pyrogallol solution (5 g dried of Aldrich) were test tube equipped aluminum ml 2-vinyl pyridine were freshly vacuum distilled prior to use 100 with found. A and are detailed below. and boiling thin modified version of this and higher a very styrene (2VP/S). method cap a the exhibiting a positive available commercially this copolymer had of 2-vinyl pyridine (97%, clean glass syringes into a was was the procedures methanol 15 2VP/S et al. (97) been adopted and ml contact, slip with 2-vinyl pyridine and copolymer method of manufacturing revealed upon of 2-Vinyl Pyridine/Styrene Unfortunately, hence a surface. Thus, to g gas (Union KOH, calcium and sulphate E X P E R I M E N T / 74 (DRIERITE) was allowed to bubble through the mixture via a glass capillary. At the end of one hour, 0.2 through was a glass continued. The the solution no funnel had two days to initiate increased to such an the test tube was tightly at room temperature. The After the glass mass, the solid were dissolved then tube was ml solution was to the breakage then placed in a ml ether (Reagent could be sealed precipitate was was and was carefully was was added the bubbling viscosity of of nitrogen removed stored in a dark down broken into was from the location for a clear golden brown small grade, Fisher) in a removed washed in 400 from pellets the polymer of plastic grade, which Fisher). When filtered to remove glass debris present in the of the test tube. The further away of chloroform (Reagent separatory funnel and continuously fibrous precipitate the extent that the passage contact with the petroleum ether, the polymer which terminated when resultant polymer broken in 500 solution due 500 and was of the test tube. test polymer dissolved, the polymer was reaction this juncture, the glass capillary solid residing at the bottom petroleum the polymerization bubbling of nitrogen longer possible. At solution and g of benzoyl peroxide (AnalaR grade, BDH) purified from by filtered polymer added dropwise well-stirred formed the beaker 2000 solution to 1600 ml was ml of beaker. On a white fibrous precipitate using a glass rod. This repeating the process once. The final ml of fresh petroleum ether and then dried under vacuum for one day. Approximately 15.6 g of 2VP/S were obtained. E X P E R I M E N T / 75 3.5.2. C o a t i n g of Deposition Surface Plastic coated surfaces were prepared by solutions of 2-vinyl pyridine/styrene in chloroform falling level method that electron microscopy. A in Figure 18 cm 3.7. has been to it was of solution. The for preparing bore teflon stop-cock apparatus and uniformity of the coated liquid be plastic of plastic in the effected by very much solution used larger than was 0.5 Revell and solution and having w% 10 its dimensions the final film thickness and be with using the films for cm the to the of I.D. the and glass capillary have been Agar (98). They dimensions that in diameter at the bottom to control the used by can slips support a 4 ft length of 2 mm give similar results to those latter cover (AnalaR grade, BDH) It consists of a thick-walled glass funnel tubing equipped with a 2 mm concentration the schematic diagram of the coating apparatus used is shown in height. Attached discharge employed coating to suggested that film are proportional to the constant of the specimen. which, according designed drainage effective The to Marshall rate. head The of the concentration of (99), yields an o optimal thickness of 200 The cleaned procedures with for hot chromic dichromate) and hour coating to were as acid (concentrated thoroughly were then contacted additional A. washed layers of then fitted to a simple slips were sodium all traces of residual acid. They hour and adsorbed strength of the bond between the plastic film slip was cover sulphuric acid saturated with to remove with methanol for one remove follows. Circular water and dried under vacuum for an in order to improve the glass substrate. The spring clip which gripped the cover the sides of the upper E X P E R I M E N T / 76 10 cm I.D. Cylindrical Glass Funnel < 2 mm I.D. Thick-Walled Glass Capillary Teflon Stopcock (2 mm Bore) Figure 3.7 Schematic diagram of the coating apparatus E X P E R I M E N T / 77 edge of the specimen and the surface on which the Film was to be coated was held vertically in the solution. Care was taken to ensure that the specimen was immersed This to a level where the solution was just beneath the edge of the clip. is necessary otherwise affect to create a the thickness steady and non-disturbed and uniformity of the coated solution in the funnel had come to rest, it was allowed opening When the stop-cock. approximately apparatus Five was then minutes flow washed with the specimen chloroform Film. would When the to drain by carefully Finished, the Film was allowed before which was to dry in situ for removed. to remove The any traces coating of plastic which might alter subsequent solution concentrations or block the capillary tube. 3.6. PARTICLE It DEPOSITION can be anticipated from depends on both the electric the particles. valency The double theory that the extent double layer thickness layer thickness of neutral electrolytes present is dependent upon the p H of the suspension. coverage and the surface potential of is dependent in the system of surface upon while the quantity and the surface potential Thus, in the present investigation, the experiment was divided into two portions. 1. the study of the effect of surface potential on surface coverage of silica spheres, and 2. the study of silica The surface dispersion of the effect of double layer thickness on surface coverage spheres. potential of the particles using either HC1 or N a O H was varied while by altering KC1 was used the p H of the as the neutral electrolyte in achieving various predetermined electrolyte concentrations. E X P E R I M E N T / 78 Although necessary are the experiment for both were outlined below. A spheres similar portion in suspension, eight equal lots When and pH divided into two parts, the procedures and hence only the general sequence (approximately 66.7 mg) of freshly enough process described earlier into was for several (Section experiments, was 3.3). The cleaned suspension and these were placed in eight of steps prepared silica cleaned by the was then divided 50 ml volumetric flasks. the effect of p H on surface coverage was to be studied, distilled, deionized filtered water was used to make up the volume of the suspension and the was adjusted to the desired value using either N a O H hand, when the effect of double studied, a predetermined or HC1. On the other layer thickness on surface coverage amount of KC1 was added to achieve was to be the desired electrolyte concentration and again distilled, deionized and filtered water was used to make up the volume. In either case, both the p H and the conductivity of the suspensions were measured. The ambient temperature of the room recorded. The suspensions were then dispersed for 30 minutes ultrasonic bath, after which they was in a Branson also 5200 were transferred into the deposition cells using clean disposable pipettes. The suspension concentration was such as to provide up to 30 layers of spheres on the deposition surface. Care was taken to ensure that the trapping of air bubbles inside the cell was avoided. This was achieved by filling meniscus the cell was with formed. A the suspension clean cover slip up to the rim at which was then placed carefully meniscus in such a way that the excess suspension was squeezed tightly sealed, air bubble-free cell. The cover slip by screwing the aluminum allowed to settle by gravity cap onto was then held the cell. The particles and be deposited onto a convex over the out, leaving a firmly in place in the cell were the polymer-coated substrate E X P E R I M E N T / 79 for a few by day, although a simple calculation using hours were sufficient. After settling, the rinsing suspension the the suspension with distilled, Stokes' unattached deionized and became almost particle-free, leaving only a substrate surface. This rinsing step possible optical observation was equation was showed that particles were removed filtered water until the single layer of spheres necessary a to ensure that the on best obtained. 3.7. MEASUREMENT OF SURFACE COVERAGE The determination polymer-coated microscope of glass substrate with a lOOx analysis system. The two surface was coverage performed by oil immersion of using objective silica a and particles image These analyser and a Kontron pieces of equipment were "linked" together images were then processed into sharp superimposed onto contrast black a measuring IPS "transferred" to and white 30 Mm by properly account for particles ovelapping of the frame, those touching spheres. after square. To those the touching left and It is assumed the adjustment will the top and right not the number deviate of edges particles significantly from were measuring frame true number before exclusion of particles overlapping will The a cancelling measuring frame was effect. adjusted then counted. For number of while made into complete in the the a the edges eliminated, adjustment, as the inclusion and have had measuring frame was bottom edges were included and that images. frame whose area the present Mm image so that the previously been calibrated. For 30 analysis, the the Zeiss Universal optical particles seen at random positions under the microscope could be the on opposing edges particles within the each cell substrate, the total number -of E X P E R I M E N T / 80 particles at various randomly The total obtained projected by projected determined area of the particles multiplying area of earlier one chosen locations under the microscope were counted. the number particle, the of within particles average using a transmission electron the measuring within diameter the frame frame of which by had microscope. The fractional was the been area occupied by the spheres was then obtained by dividing the total projected area of spheres by the area of the measuring a polymer-coated substrate with a single layer of deposited silica spheres. Figure 3.8 Photograph c frame. Figure 3.8 shows a photograph of of polymer-coated substrate with a single layer of deposited silica spheres E X P E R I M E N T / 81 M E A S U R E M E N T OF Z E T A 3.8. POTENTIAL 3.8.1. Particle Zeta Potential - Micro-electrophoresis The Brothers zeta Mark transparent the of a either particle the suspension, particles. particles, The system and the electrodes of mobility was zeta determined which cylindrical is used microscope determination electrophoretic or cell of $p, apparatus, rectangular microscope. The suspension, within the the II micro-electrophoresis cell electrodes, and particle potential of basically applying for observing (£im/sec.)/(volt/cm), a the potential involves a a Rank pair sample a of of the potential gradient resulting the which a consists of cross-section, for containing for using motion of measurement was obtained of by measuring the time taken for a targeted particle to traverse a prechosen distance (as determined potential by a gradient. In calibrated grid in the general, 20 such about which the average electrophoretic mobility was the numerical their technique results of Wiersema cm of cells, in the in cross-section present and 7.0 measurements. This preference due to the eyepiece) in measurements converted al. (70). A were a known made, after into zeta potential using sample calculation based on is shown in Appendix A. 2. Although the Rank Brothers types et microscope instrument study, a cm in was designed to accommodate both rectangular cell approximately length was employed of the rectangular cell over the fact that the former had the measurements (see Section 3.8.2), which for 0.1 x 1.0 all mobility cylindrical one is advantage of permitting electro-osmotic are taken at the mid-plane between the E X P E R I M E N T / 82 top and from the the bottom of the cell, action of gravity on without the particles during cylindrical configuration were used, the velocity distribution Furthermore, the the the deposition syringe. The clamped be coating applied from sediments resulting mobility measurements. If the sediment would disturb the result (Section in substantial 3.5.2) required more easily to the electro-osmotic measurement for the rectangular errors. electro-osmosis cell than to the one. During in hence plastic measurements could cylindrical and interference eletrophoresis experiments cell was within a measurements, samples were transferred rinsed twice with constant temperature the to the same bath of the cell suspensions used using a clean suspension, refilled maintained at 25.0 glass and ± then 0.2°C, equivalent to the operating temperature of the deposition runs. Once the sample was of a sealed 13 x 20 into x 0.013 in position, blackened platinum mm sheet of bright platinum standard-taper glass fittings, compartments at either end of the cell. The potential gradient length along the •the ends of the cell in order convective currents within blackened electrodes were the of the were After thoroughly rolled into a cylinder placed in mating cell. The glass fittings served each washed a d.c. to seal that might cause unwanted electrophoresis with and electrode electrodes were used to apply to prevent evaporation cell. electrodes, constructed distilled measurement water and stored the in distilled water until required for the next measurement. Direct illumination was provided by a 12 V, 100-watt quartz iodine lamp. E X P E R I M E N T / 83 The an light beam could be focussed adjusting screw on at any position within the cell by means of the illuminating dark-field type, which allowed condenser. The condenser was of the the particles to be seen as points of light against a black background. The movement of the particles microscope employing attached to a a lOx objective and a micrometer along the line normal within stage and hence the cell was observed with lOx eyepiece. The microscope could to, and passing through be focussed a was at any position the centre of, the large face of the cell across its entire smaller width. To register the mobility of the particles, the eyepiece contained a graticule with a square grid. The graticule was calibrated against a "stage micrometer" (see Appendix B, Section B.1.2) and was found to have an average grid spacing of 119 ± 0.09 The a d.c. supply potential gradient was applied using voltage drop between the two electrodes was read from determine the potential measured voltage drop, it was necessary distance, I, which um. a voltmeter. In order to gradient (applied voltage/interelectrode distance) from the can be found to determine the effective interelectrode based on a knowledge conductivity, X, the cross-sectional area of the cell at the plane and the two the measured and the overall resistance between electrodes, R. of the solution of viewing, The A, relationship between these parameters is Z = The procedures RXA r 3 - 1 ] required for measuring these quantities are given in Appendix B, Section B . l . l . The effective interelectrode distance, I, for the rectangular cell used E X P E R I M E N T / 84 in the present study was determined to be 7.085 cm. When a potential difference is applied to the cell, both the particles (electrophoresis) and the solution (electro-osmosis) are set in motion. Since the cell is closed, this electro-osmotic movement, which manifests itself as a constant velocity flow at the four walls of the channel, generates a difference in pressure between the extremities of the cell. As a consequence of this pressure difference, a return flow the down the centre of the channel occurs. Thus, there exists within cell cross-section a region these the fluid velocity is zero. The location of "stationary levels" in a closed rectangular cell can be determined from the solution of the equation in where of hydrodynamic the channel subject to the constant (33). Komagata (100) showed motion describing the steady-state electro-osmotic that for a rectangular slip cell velocity flow at the walls of half-width a and half-thickness b (a/b > 2), and for measurements taken at the centre of the cell width, the stationary levels and are symmetric about the mid-plane of the channel are located at y I" 1 128b _ = + _ + b where y is the distance from [3 7r a 5 the mid-point. experiments, a = 0.4922 cm, b = 0.05605 cm 0.5 For the cell used [3.2] in the present (see Appendix B, Section B.l.l), and hence y/b= ±0.6172. The true electrophoretic velocities of the particles were measured with the microscope focussed at either of the two stationary levels. The stationary levels E X P E R I M E N T / 85 were located by moving the microscope focus a distance of 0.3828b and from the front inside face particle was reversal was instrument. of the cell. To effects, each timed over the same distance in both directions. The necessary field attained by means of a simple eliminate polarization 1.6172b switching circuit provided with the Random errors due to Brownian motion and depth-of-field effects were minimized by measuring the velocity of at least ten different particles. In some experiments, 20 separate determinations were made. The measurements were taken at both stationary levels to assure had been properly located. The average velocity was converted same number of that their positions to an mobility by dividing by the applied potential gradient. 3.8.2. Substrate The was Thus, prior Potential - Electro-osmosis zeta potential of the plastic used to coat the deposition substrate, measured performed Zeta by the method simultaneously with of electro-osmosis. the £p These measurements to the measurements, the retangular cell measurements in the was and with a layer of 2VP/S. The coating procedures were similar to those Section 3.5.2. for creating plastic films on the glass cover slips w were rectangular cleaned $ , cell. coated described in used in the deposition experiments. As mentioned in the previous for the electro-osmotic along the length flow of the which cell. section, the wall zeta-potential is responsible occurs Although when an a potential gradient expression is applied for the electro-osmotic velocity distribution is available in terms of the wall zeta-potential, the motion -of E X P E R I M E N T / 86 the of liquid cannot be measured directly. However, since the electrophoretic the particles velocity well relative to the liquid is everywhere at the stationary levels, then the if the particle location in the cell as electro-osmotic determined by difference. The location for this additional velocity was is minimal. Except for the location electrophoretic particle zeta velocity can be measurement of particle chosen to be the centre of the channel because the gradient of the electro-osmotic velocity profile at this location error constant, relative to the walls is measured at one other as velocity measurements potential. was A t least of the measurement, the procedure for identical 10 is zero and hence, the depth-of-field to particles that were for the timed determination of at the centre of the channel in either direction. For a cell with a/b > 2, the wall zeta-potential, $ , was determined using the relationship derived by Bowen (33) 8TTM * where U 0 U E w [ 1 - (I92b)/(7r a) ] 5 ~ ~e~X [ 1 + ( 3 8 4 b ) / U a ) ] 5 is the true electrophoretic the apparent velocity velocity at the mid-plane, solution dielectric constant, and X the applied U„ - U, (measured u [3.3] at the stationary the solution potential gradient. viscosity, level), e the 4. COMPUTER SIMULATION OF R A N D O M P A R T I C L E DEPOSITION In particles the present onto simulated a study, planar using the while rolling of surface two the rejection second random under different two-dimensional simple deposited the deposition quiescent schemes. particles over first been scheme only non-overlapping scheme consisted of a sedimenting monodispersed spherical conditions has The model where of the involved particles three-dimensional surfaces of numerically a were model where previously-deposited particles as well as the stacking of particles were allowed. 4.1. TWO-DIMENSIONAL R A N D O M P A R T I C L E DEPOSITION M O D E L In this chosen xx 10 The and model, particles y- 0.5 in diameter coordinate positions one units square divided evenly into 100 were the Computer Centre value of either random one of number the prescribed range (i.e., outside the were discarded process was particle particle and and repeated was a new until a dropped, the each of the set generated (RAND). If it was coordinates generated substrate area), the generated of random (x,y) coordinates satisfactory set of (x,y) values was projected distance previously deposited including the lot of the falling particle, was 87 at randomly identical square lots of 1 x generator (x,y) dropped at a time onto a planar surface random position of the centre of the falling particle was UBC a unit in the particles computed. fell 1 unit. using the found that outside the (x,y) coordinates generated. This obtained. When (x,y) plane in the 10 between neighbouring this lots, COMPUTER SIMULATION OF RANDOM If this distance were bigger than case, the particle diameter, since all particles were assumed to be of equal size), it was considered permanently fixed particular were recorded the particles at this position. and the counter generated already in place, then The x and y coordinates of for the number lot was increased by one. The random particle were then and the process continued as before. On carried In this case, no alteration out. The deposition successive number maximum reached value process in the record of particles was continued of failures for the falling of 1000. A t this in this in this trial the other the particle the particle was discarded before the co-ordinates of the next generated. diameter, particle were deposited manner was until the particle to be deposited exceeded a point, it was assumed that the surface had its maximum coverage, i.e., the surface was "saturated". Once saturation had been obtained, the top and right edges of the square the of particles coordinates for the next hand, if any of the projected distances were smaller than then / 88 the sum of the particle radii (or in this i.e., the falling particle did not overlap with this particle PARTICLE DEPOSITION fraction planar of the total surface simulation described model was being followed correctly In maximum order necessary 3.7 substrate surface was carried occupied calculated. This was in Section area the elimination of particles by all the particles process to ensure of particle that out before deposited on the elimination in the the experimental leading to the determination touching procedures of surface coverage was in the simulation. to estimate surface coverage, the influence of double layer it was assumed that each particle thickness on the has an effective COMPUTER SIMULATION radius, a , which g 2(a a) is larger than its actual represents the minimum - e planar surface can approach OF RANDOM P A R T I C L E D E P O S I T I O N / 89 radius, a, such distance to which one another because that the difference two particles deposited on a of double layer repulsion. The distance, (a ~a), cannot be accurately predicted beforehand, but it likely depends e in a monotonic could fashion on the double layer thickness, 1/K. For example, be the distance deposited at which the overall interaction energy particle. Unfortunately, because for under the interaction convenience, that can be refined a energy g when = (a + such circumstances, g between two particles is approximately 2-10 kT; a kT being the average energy of a single Brownian 2(a -a) thermal no theory yet exists it was assumed, for 1/K) in the present analysis. This approximation the appropriate extension to the double layer interaction theory becomes available. Thus, if it is further assumed that the particle with its attendant double layer acts as a solid sphere having radius a , then it is easily shown that the e surface coverage, 8, is related to the maximum coverage, 6 , when Q 1/K = 0 (i.e. as obtained above) by e= " L a a Ka 2 — 0 e J 4.2. T H R E E - D I M E N S I O N A L R A N D O M O = 2 . 1+»ca . PARTICLE DEPOSITION MODEL In this model, as in the case of the two-dimensional model, particles 0.5 unit in diameter positions were one at a time dropped onto at randomly a planar surface chosen 10 x x- and y - coordinate 10 units square divided evenly into 100 identical square lots of 1 x 1 unit. The random (x,y) coordinates COMPUTER SIMULATION of the centre Centre of the falling random number particle generator fell (x,y) found that the value generated. were obtained. When using the U B C the initial assigned values of z coordinate) Computing z coordinate of the to a value of 1000 units. and a new set of random generated until a satisfactory (x,y) coordinates set of (x,y) values was dropped, the projected distance in the (x,y) plane between this particle and each of the previously deposited particles their / 90 (i.e., outside the substrate area), the generated repeated a particle was generated arbitrarily discarded This process was were P A R T I C L E DEPOSITION of either one of the (x,y) coordinates outside the prescribed range coordinates RANDOM (RAND) while centre of the falling particle was If it was OF in the neighbouring lots, including (regardless of the lot of the falling particle, was computed. If this distance were bigger than case, the particle diameter, i.e., considered coordinate particle would were permanently fixed be one recorded, particle and On radius. The the counter smaller than hand, if any the particle diameter, instance where the projected already in place, then and z coordinates in this for the next trial then of this particular lot particle were found to be as before. of the projected distances were the falling sphere and the subordinate one x, y for the particles coordinates and the process continued the other the particles at this position and its final value of the z increased by one. The random then generated of the particle radii (or in this since all particles were assumed to be of equal size), the falling particle did not overlap with it was was the sum it was assumed that a contact between sphere(s) had occurred. If there were only distance was found to be smaller than the COMPUTER SIMULATION particle diameter, then only one OF RANDOM P A R T I C L E D E P O S I T I O N / 91 sphere below had sphere. If, however, more than one been contacted by the falling of the projected distances were found to be smaller than the particle diameter, then a comparison of the z coordinates of all the subordinate spheres in question was performed so that the number of sphere(s) having the largest value of the z coordinate could be determined. These were the spheres Depending on assumed to in continue to descend or be nested within to be subordinate spheres was were encountered within the contacted by the the spheres. It was the a new necessary. Thus, if either simultaneously, the falling subordinate spheres set of random and sphere was three or four spheres assumed to have nested its final resting coordinates determined. coordinates for the next trial increased sphere The by one generated and or two subordinate spheres were encountered, then falling sphere would not be nested but would roll over the surface(s) of the sphere(s). After the falling subordinate sphere(s) and sphere its final distance with all other previously the assumed that, process continued as before. If, however, only one the sphere. nested, simultaneous contact with at least counter for the number of spheres at this lot would then be and falling allowed to roll over the surface(s) of the sphere(s) order for the falling sphere three been the number of subordinate spheres at this largest value of z, the falling sphere would either be and have had would computation be was encountered necessary over the surface(s) position recorded, computation deposited spheres neighbouring lots, based on its new This rolled If no whether additional or not the of the projected same (x,y,z) coordinates, was to check while rolling. at the of z elevation in again carried out. additional sphere(s) were sphere(s) encountered, COMPUTER then its initial and the SIMULATION OF RANDOM P A R T I C L E DEPOSITION / 92 (x,y,z) coordinates were replaced by its final (x,y,z) coordinates fall continued. If, indeed, sphere(s) were encountered, then recorded (x,y,z) coordinates were discarded and a (x,y,z) coordinates for final were the additional position encountered subordinate sphere was allowed position of was reached. falling sphere(s) was to roll the sphere into consideration. first encountered over the by the on calculated For by taking example, when the falling sphere, the falling surface of the Then, based set of new the already this falling subordinate new sphere until its final projected distance between was again carried out but this time only with spheres at the same z coordinate at this new position, then its initial found contacted two (x,y,z) coordinates were fall as well as the new based subordinate spheres spheres that another sphere was newly calculated (x,y,z) coordinates discarded. The as having its neighbouring of the as that of the falling sphere. If it was and a sphere (x,y,z) position, computation sphere the restored while its falling sphere instead of one contacted was and then treated the direction of (x,y,z) coordinates of the falling sphere were recalculated on the algorithm for continued. The mathematical a two-sphere equations coordinates have been formulated and encounter, necessary and for the the falling determination process of new their derivations are described in detail in subsequent sections of this chapter. The deposition process was continued in this manner number of spheres deposited exceeded a maximum value of 1000, the substrate surface was assumed to be "saturated". A used for computing this model is depicted in Figure 4.1. until the total at which stage schematic flow diagram COMPUTER SIMULATION > OF RANDOM P A R T I C L E DEPOSITION / 93 A sphere with random (x,y) is allowed to fall No >^ Can this sphere contact another below as it falls? Yes Identify the number of particle(s) it contacted New (x,y) determined ^ One sphere? ^ — — T w o Yes No spheres? ^ Yes Can this sphere contact another below it while rolling on the first? Yes No Can this sphere contact a third or fourth below while rolling along the first two? Yes Determine its nested (x,y,z) ^ Record and update all relevant information No ^ / Has the number of spheres used exceeded V the maximum allowed value? Yes Stop Figure 4.1 Schematic flowchart for random sphere deposition. Rectangular boxes indicate operations. Oval-ended boxes indicate tests determining the next operation. No COMPUTER SIMULATION Once coverage saturation (i.e., coverages a = a) as fi (i.e., had (a a) - e been well = OF as RANDOM obtained, the 1/K) were the double then P A R T I C L E D E P O S I T I O N / 94 original layer "hard-sphere" surface thickness dependent surface determined according to the methods used for the two-dimensional model described earlier. Thus, it can the two sphere be seen models, differences when its projected that although there are many do exist area in that overlapped instead with the similarities of discarding projected area the between falling of another previously deposited sphere as in the two-dimensional model, the falling sphere in the three-dimensional model sphere(s) it contacted and substrate surface and was was allowed to the process was deposited or roll over continued until was nested case, the final value of the z- the always centre was one sphere radius surface(s) of the it either reached the among spheres. In the former sphere the previously settled (elevation) coordinate of while, in the latter case, z could be equal to (deposited) or greater than (nested) a. It should be noted here that in both models, the simulation did not actually involve the motion particle centre through successive points. Instead, the path and of the final resting place of the particle were determined by methods of analytic geometry. 4.2.1. One-Sphere When the Encounter falling configuration will resemble sphere any encounters a second sphere beneath it, its of the following positions depicted in Figure 4.2. COMPUTER SIMULATION OF RANDOM II I 88 Figure 4.2. Plan view f) IV i n VI V P A R T I C L E DEPOSITION / 95 Vffl v n showing various possible general configurations when a sphere below is encountered by the falling sphere ( Upon C falling sphere; contacting the sphere below, the falling ^ sphere below.) sphere will roll curvature of the sphere it contacted in a direction parallel two centres of the spheres until below. The derivation coordinates is similar the of along the surface to the line joining the its elevation corresponds to that of the sphere algorithms necessary for cases I to IV for determining depicted in Figure 4.2 derivation based on case I will be presented. its new and (x,3-) hence only COMPUTER SIMULATION As initial and Then, the the illustrated in Figure final positions of 4.3, the OF RANDOM let the centres of the falling sphere straight line joining the P A R T I C L E DEPOSITION / 96 centres B, T be B, and sphere below and T F and can be the F, respectively. represented by equation y = mx + c where the slope m y -y T = X and the constant c At x Let DI = 0, y x o = T~ B X y = B T - mx . T c. be the distance between y x o and B, which can be determined as follows: 0.5 DI = U -0) B 2 + (y -y B x o ) COMPUTER SIMULATION When initial the rolling position F OF RANDOM of the falling is reached, the sphere distance has and For identical, cases V p = y F = yxo of B and F the next is equal to the position of the falling sphere is: x (DI+d)/DI B (y -y + B to VIII, where one determination stopped, i.e., when between diameter of the sphere (d). Therefore, the new x P A R T I C L E D E P O S I T I O N / 97 the of the coordinates of the two falling simplified. For example, for case V )(Di d)/Di + x o sphere's shown new coordinates in Figure 4.4 below, the spheres is is greatly new coordinates of of the falling sphere are Xp = yp = y Y,„ T Y_ F = Xg - R Xrp d P \ i / i\ \ N F y z 1 \ 1 Figure 4.4 Plan view showing the initial and final position of the falling sphere when a subordinate sphere is encountered (Case V) (x,y) COMPUTER For SIMULATION number directly it contacted below, i.e., the falling sphere on top of the subordinate sphere, a directional-indicating / 98 was random was generated and the sphere was allowed to roll in this direction. 4.2.2. Two-Sphere When the P A R T I C L E DEPOSITION the special case where the (x,y) coordinates of the falling sphere were identical to those of the sphere sitting OF RANDOM Encounter two spheres are simultaneously encountered by the falling sphere, resultant configuration could resemble any of those shown in Figure 4.5: 1 V 11 VI in IV % VIII vu 99 g) Figure 4.5. Plan view showing various possible general configurations when two spheres below are encountered by the falling sphere ( O falling sphere; ® spheres below.) Again, since the determination of the final sphere is similar for configurations (x,y) coordinates of the falling I to IV, only demonstrate the derivation of the necessary algorithm. case I is used here to COMPUTER SIMULATION OF RANDOM P A R T I C L E D E P O S I T I O N / 99 Consider the configuration illustrated in Figure 4.6. Let the centres of the spheres below be sphere be T B, and F Figure 4.6 and B , 2 the initial Plan view showing final positions of the falling the initial and final positions of the falling subordinate spheres are encountered (Case I) is at the mid-point between B and 1 that X X E = V E = X I ~ B1 X B2 y i y 2 + and and respectively. sphere when two If E and y : B X B E = yB2 Using a linear equation of the form y = mx + c B , then it can be easily 2 seen COMPUTER SIMULATION OF RANDOM PARTICLE DEPOSITION / 100 X the slope of the direction of falling particle motion, m = - B1 X yBr the constant c and thus XQ = = Vg y 2- mx B2 v B 2 E c B m Length between G and I jx^ - X j | (GI ) = 0.5 Length between E and G When position F rolling has (EG) of the been = (y -y ) E falling reached, 2 ( G" I + T sphere has the centres X x ) stopped, i.e., when of the three spheres isosceles triangle. Therefore, the distances between centres F B , 2 will and = d FB = d. 2 Also, the distances between the following points are: 0.5 1 B, and E (B E) 7 = - (y i-y 2 B ) 2 + ( B x B l _ x B 2 ) 2 0.5 F and E (FE) = F and G (FG) (FB ) -(B E) 2 2 = FE + 2 2 EG 0.5 B , and B 2 E and I (B B ) = 1 ( 2 (EI ) = (y y i-y 2 B B 1 "y B 2 B )/2 ) 2 + <x B 1 form and B,, and F equal the sphere diameter, i.e., FB, the -x B 2 ) final an and COMPUTER SIMULATION OF RANDOM P A R T I C L E DEPOSITION / 101 Finally, by the principle of similar triangles, EG EI FG FH FH Thus, = EI FG EG Again, by similar triangles, and EI GI FH GH GH = GI FH EI Therefore, the final x and y positions of the falling sphere are: x y For positions spheres below V to VIII, G new y position, y at the final T = isosceles triangle with B,T GH where one of the (x,y) coordinates of the two of the final (x,y) coordinates of the V, which is illustrated in more detail of the three will illustration. (y position, - = *B2 + F H simplified. Position in Figure 4.7, is used as an Since, F = x is the same, determination falling sphere is greatly The F B 1 ~y B 2 )/2. the centres = B T 2 spheres = d, 0 . 5 therefore the new x position x^, = Xg-^ + (d) -(y -y 2 T B 2 ) form an COMPUTER SIMULATION OF RANDOM Y P A R T I C L E D E P O S I T I O N / 102 B1 Y T X B1 = X B2 Figure 4.7 Plan view' showing the initial and final position of the falling sphere when two subordinate sphere are encountered (Case V) 4.2.3. Three-sphere Encounter When three spheres are simultaneously encountered at the same level, the falling sphere is assumed to be permanent^' nested and hence its final coordinates must be calculated. The determination of the final (x,y,z) (x,y,z) coordinates was based on the recognition of the fact that the distances from its new centre to the centres of the three subordinate spheres are identical. COMPUTER SIMULATION OF RANDOM PARTICLE DEPOSITION / 103 Consider, for example, the configuration shown in Figure 4 . 8 below. Figure 4 . 8 Plan view showing the final resting postion of the falling sphere when three subordinate Let spheres are encountered the centres of the three spheres encountered be A, B of the final resting position of the falling sphere be The x and y coordinates at the mid-point and C, and the centre D. between centres A and B (mpl) are: x m i = P The slope between A m , 1 and ( x A + x B / perpendicular ) 2 a to n y d slope AB m p , = and B, is X m, = - A~ B X ( yA yB / + ) initiating 2 from the mid-point COMPUTER SIMULATION OF RANDOM P A R T I C L E D E P O S I T I O N / 104 Therefore, to describe the x and y relationship, a linear equation of the form y , = n^x, + c, can be written, where the constant X c, A similar linear = y.m p i A~ B X mpi expression can also be written for the line extending perpendicularly from the mid-point of B and C (mp2), as y 2 = m x 2 = Y,mp2 2 + c 2 2 X where c y When corresponding these two lines C X mp2 m p 2 L ^B-yc = (x +x )/2 = (yB yc / mp2 and B : c B + } are extended, 2 they to the centre of the resting position will intersect of the falling sphere enabling the determination of the x and y coordinates of centre D. Since at the point of intersection, x,=x and y ! = y 2 2 therefore, by equating the two linear expressions mentioned above, m x, y and + c, = m x 2 2 + c the x and y coordinates can be determined as c "C , 2 X D = m i -m 2 y D = m,x D at a + c, 2 point D, thus COMPUTER SIMULATION OF RANDOM PARTICLE DEPOSITION The remaining z coordinate can be approximately determined / 105 as follows (refer to Figure 4.9): Figure 4.9 Elevation view of the falling sphere nested atop two subordinate spheres 0.5 1 Length of AM 2 (x -x ) B 2 A + (y -y ) - (AM) B A Length of AD = d o. 5 Therefore, z , z 2 and z 3 Z A + (AD) can be similarly 2 2 calculated, from which be determined as Z D = (z,+z2+z )/3. 3 the final z^ coordinate can 5. R E S U L T S 5.1. AND DISCUSSION INTRODUCTION The summary each experiment such as the ratio of experimental results, including the conditions under which was measured sphere presented in run, as well of particle and radius to double substrate Tables 5.1, 5.2, evaluation of these parameters necessary for these random sphere and as all the relevant experimental zeta potentials, and 5.3. layer $p Sample parameters thickness, /ca, and the and $ respectively, are w calculations leading to the are presented in Appendix A. Computer programs other calculations as well as for the simulation of deposition and the evaluation of the total interaction energies are listed in Appendix C. TABLE 5.1 Summary of experimental data for the effect of double layer thickness on surface coverage Run pH X [KC1] (/xmho/cm) (10" M) 6 Al 5.10 1.83 11K /ca (Mm) 0 0.108 4.63 *w Coverage (mV) (mV) (%) -121 27 33.37 A2 5.24 3.25 1 0.084 5.97 -140 41 35.90 A3 5.19 4.50 10 0.069 7.26 - 99 27 38.14 A4 5.01 9.40 50 0.039 12.70 - 83 25 39.27 A5 5.12 10.50 100 0.029 17.04 - 77 31 39.27 A6 5.06 82.00 1000 0.010 52.18 - 76 12 39.27 106 RESULTS A N D DISCUSSION / 107 T A B L E 5.2 Summary of experimental data for the effect of p H on surface coverage Run X 1/K (umho/cm) (um) pH Ka Coverage *P (mV) (mV) (%) - * BI 2.97 515.00 0.008 62.73 - B2 3.71 123.00 0.014 35.27 -44 9.1 49.39 6.4 44.42 B3 4.64 2.62 0.064 7.86 -62 B4 5.90 2.83 0.066 7.54 -82 21 B5 7.89 3.12 0.059 8.43 -89 -33 B6 8.72 8.60 0.037 12.63 -68 -49 B7 9.51 86.00 0.011 43.84 -75 -18 10.10 388.00 0.005 93.47 -45 -38 B8 35.00 # # # # * Did not determine due to partial coagulation of particles. # Particles did not adhere to the polymer coated surface due to a change in polarity of the surface potential. Hence no determination of coverage was made. T A B L E 5.3 Summary of experimental data for the effect of K^POy Run pH X concentration on surface coverage [K POJ 3 (umho/cm) (10" M) 6 d/K) P (Mm) (mV) t -> w (mV) 6.45 0.026 -75 25 37.00 (fca) p (Mm) Coverage (%) 0 , 0.077 3.20 1 0.067 7.48 0.022 -69 27 40.40 4.05 5 0.058 8.57 0.019 -76 20 44.38 47.12 Cl 5.75 2.40 C2 6.00 C3 6.50 C4 7.03 4.80 10 0.055 9.05 0.018 -76 29 C5 8.50 16.00 50 0.025 20.12 0.008 -76 -50 C6 9.35 35.00 100 0.018 28.46 0.006 -72 -51 C7 10.35 195.00 500 0.008 63.63 0.003 -70 -48 C8 10.70 385.00 1000 0.006 89.99 0.002 -68 -53 # # # # # Particles did not adhere to the polymer coated surface due to a change in polarity of the surface potential. Hence no determination of coverage was made. . RESULTS 5.2. S U R F A C E COVERAGE AND DISCUSSION / 108 - EXPERIMENTAL 5.2.1. Effect of Double L a y e r Thickness In the study of the effect of double layer thickness on surface coverage, a range of values for the double layer thickness of spheres was obtained by adding different pre-determined Experimental data quantities of potassium pertaining to this study chloride into the suspension. have been tabulated in Table 5.1 and the results of average surface coverage of silica spheres as a function of added KC1 is concerned are plotted in Figure 5.1. Since the effect of double layer thickness the study rather than with fundamentally that of the added neutral electrolyte concentration, a plot of the surface coverage versus product of inverse made, as shown double layer in Figure thickness and 5.2, to provide a with sphere more the K a , is therefore radius, direct dimensionless observation of the effect. In Figure 5.1, it can be observed that as the KC1 concentration increases at constant approximately pH, the percent surface coverage of silica spheres also increases. This observation is in accordance with counts. Firstly, as already mentioned counter-ion ( K ) concentration accordance with increases + the Equation .[2.10]. equilibrium surrounding each determined by particle the causes Section the double and is explanable 2.3, layer an surface. As of a counter-ions result, of potential into the surface determining solution) remains unchanged, but the potential at the outer hence the $ -potential is lowered. According increase thickness Secondly, increasing the KC1 adsorption concentration in theory on two in the to decrease in concentration the Stern also layer potential (which is OH" ions in the Helmholtz plane . and to the theorj- discussed in Chapter 2, RESULTS AND DISCUSSION / 109 40.0 39.0 - 38.0 - 37.0 - 36.0 - 35.0 - 34.0 - IS w CO u CD cd o ^ •iH o CO co > o o cu o cd «f-» 33.0 10" 10" 10~ 5 -4 10"' K C 1 Concentration (M) Figure 5.1 Effect of added ,-3 10 KC1 concentration on the surface coverage of silica spheres .-2 10 RESULTS AND DISCUSSION / 110 Figure 5.2 Surface coverage of silica spheres as a function of KSL RESULTS if the double layer thickness one is reduced AND DISCUSSION or the magnitude of the ([-potential of or both of the interacting double layers is decreased, the height of the repulsive energy barrier should inter-particle distance to be decreased also decrease. and, as surface coverage of silica spheres on the planar show zeta how the particle / 111 potential ($p) a and extent These reductions cause the result, increase the substrate. Figures varies as a percent 5.3 and 5.4 function of KC1 concentration and, more specifically, as a function of Ka, respectively. In view of the foregoing, it would thus appear contradictory to observe in Figure 5.2 that the percent approaches a constant surface value coverage does not continue at high K a values. An to increase but immediate explanation for this behaviour is that it is due to the random nature of the deposition process, i.e., However, as obtained may be a result in runs of the A4 inadequate, indisputable. It thus geometric to A6 seem although appears exclusion to suggest the that effect. that the explanation presence some similar of other the geometric factors must be results given above exclusion is present. Upon closer examination, it seems that this observed behaviour can be easily explained through continues becomes consideration to decrease so small of the double at higher compared layer electrolyte to the sphere thickness, in concentration, radius that, although 1 / K its value eventually that its influence on surface coverage becomes negligible. Furthermore, $ seems to approach a constant exerting coverage. no experimental further influence conditions with that a maximum on silica surface spheres surface coverage of only primary controlling variable. Thus, under the value, present as the model colloid, it would 39.27% is attainable when seem 1 / K is the — RESULTS AND DISCUSSION / 112 1 5 0 . 0 1 1 3 5 . 0 h 1 2 0 . 0 T- cd •l-H fl 1 0 5 . 0 0) o O. cd H-> tsi 9 0 . 0 h 7 5 . 0 h 6 0 . 0 10"' Figure 5.3 10"" 10"° 10" -3 4 Potassium Chloride Cone (M) Particle zeta potential as a function of KC1 10 concentration RESULTS AND DISCUSSION / 113 150 60.0 Figure 5.4 Particle zeta potential as a function "of /ca RESULTS AND Since this study coverage, of the double lowest 1x10 M. _3 aimed at investigating the effect of 1/K layer thickness values of 1/K value The were studied. In the studied ranged only from corresponding to an addition of higher KC1 possible because earlier observations added KC1 deposition 0.010 surface experiments, to 0.108 concentration of had shown undesirable and of singly-dispersed therefore was spheres, avoided. At was this concentration to be this concentration the state higher of end the this value absorption was of found ambient secondly, to the presence to be carbon experimentally dioxide gas into the of a multitude of ions normally on the called for a coagulation of unobtainable not exceeding the due, deionized was 1/K scale, M (pure although the lowest theoretical counter-ion concentration possible is 1x10" water), with approximately this limit would result in coagulation of the spheres. Since the study suspension um, concentrations into the suspension brink of the critical coagulation concentration, so that any stable on 114 it would thus have been most desirable if as wide a range as possible however, the the was DISCUSSION / firstly, water, to and associated with sea salt in the solution, probably because of the proximity of the experimental site to the sea (see Table A.l). For coverage of silica spheres the range of 1/K on values studied, the extent of surface the planar substrate was found to be in the range 33.37-39.27%. Figure 5.5 different added the actual demonstrates KC1 visually the density of deposited concentrations. It should micrographs of the original be noted deposited here spheres that these particles, but are at two are not instead photographs taken from the video screen of the image analyser which explain the poor resolutions observed. It can be seen in the photographs that the partiele RESULTS AND • • •^ » o * A_ A 0 — - * 1 «^ J•**T • * o ° ° o o • ^ ° DISCUSSION / 115 ^ 4 - o o o o o _ , added KC1 concentrations of (a) 0 M o o and (b) 1x10"" M RESULTS AND DISCUSSION / 116 deposition is indeed random. Furthermore, it can also be observed that the effect of geometric exclusion is definitely present, although its impact on overall surface coverage is difficult to quantify, since some sparsely deposited than density within the same others. photograph also be attributed to surface deposition site due Although areas on the photographs are more this observed (i.e., at a fixed experimental condition) heterogeneity, of charge-sites magnitude of the coverage since on the the substrate results obtained indicate otherwise, it is highly obtained if surface heterogeneity may e.g. the presence of locally-favorable to variations in micro-roughness distribution difference in the packing and non-uniformity in the surface, the consistency in the at higher unlikely that values such of /ca seems to consistency would be were present. 5.2.2. Effect of pH The study experimental are listed data in Table and other 5.2 of particle are plotted in Figure pertaining surface to this coverage of 5.6. In the experiments, was varied (using either HC1 or NaOH), the magnitude $-potential changed as well. Thus, it would seem more appropriate to refer to the study as the effect of particle the information and the results of average silica spheres as a function of pH when the suspension pH relevant surface coverage made in Figure of silica spheres. A $-potential rather than that of pH on separate plot reflecting 5.7. In both of the figures, it can be noted coverage is inversely proportional to pH (or $p) in that as pH the as in the case surface concentration, coverage decreases. the explanation Again, this fact is that the surface (or $ ) increases, p of changing counter-ion for these results also seems to follow directly from RESULTS AND DISCUSSION / RESULTS AND DISCUSSION / 118 52.0 32.0 • 40.0 1 ' 1 i i I i i I 55.0 70.0 85.0 -Zeta Potential (mV) i i I 100.0 Figure 5.7 Effect of particle zeta-potential on the surface coverage of silica spheres RESULTS the theory silica, discussed in Chapter 2. Since increasing the concentration specific adsorption of O H " As a result, OH" of O H " AND is a potential determining ions in the solution increases the + potential and hence the $-potential increases in magnitude. For instance, changing the p H from in p ion for ions on (or release of H i o n s from) the silica surface. the surface the magnitude of $ DISCUSSION / 119 to increase from -44 to -82 mV. of the particle 3.71 to 5.90 caused As was observed earlier Figure 2.4, increasing the magnitude of the ^-potential of one or both of two similarly charged surfaces under otherwise of the interaction energy barrier constant and, presumably, separation between deposited particles. It should concern here during the is the study the course suspension agents Thus, into of the effect changed the suspension. the surface As coverage the minimum distance of be noted that, although of pH of altering the suspension inevitably conditions increases the height (or $ ) on surface pH, the counter-ion the main coverage, concentration of as well due to the addition of pH-changing a results result, the double obtained layer thickness changed. are actually due to the combined effect of zeta potential and double layer thickness, rather than that of $-potential alone. The obtained 1/K observation between decreases, 39.27% made series A surface and remains constant surface coverage continues without thus explains the difference in the results and series B. In series A, it can be noted coverage however, it can be seen 49.4% above that increases despite under but reaches further decreases comparable £ p maximum value of in 1/K. In series B, conditions to increase with decreasing approaching a constant a that as (in terms of 1/K), and reaches a value of value. The only difference between the two RESULTS AND series 1/K A and B is decreased, reducing smaller the is that further value absolute obtained $ becomes approximately p in the latter, barrier $ and p continues thereby constant to decrease, presumably as thereby resulting in a of inter-particle distance. Thus, it seems reasonable to conclude that difference of approximately in £ suspension, whereas the energy between reduction in the former, DISCUSSION / 120 the two series is due primarily from p 1 0 % in the maximum -82 to -44 mV, as opposed to changing surface coverage to the effect of $ , i.e., a p and that the counter-ion by altering the p H concentration alone, of the a wider range of surface coverage can be attained. The present maximum surface experimental conditions maximum value would also is stable. This mentioned above is less 54.73% (47,48,51) obtained appear contention than is supported but close under the to be in the vicinity by the fact to the maximum of the the colloidal that the value theoretical value of for the random deposition of non-overlapping discs on of the nature of the experiments, spheres adhere permanently to the substrate fulfil the required pyridine/styrene of coating experimental (2VP/S) surface. Nevertheless, a 5 0 % obtained infinite flat surface. Because to of almost attainable under all experimental conditions, provided suspension an coverage value was necessary that the surface upon contact. Thus, in order conditions, prepared it was and a cationic coated over copolymer the glass of 2-vinyl deposition the copolymer of 2VP/S may not have been the best choice material as it became negatively charged under alkaline conditions. As consequence of the double layer repulsion between the similarly-charged partiele RESULTS AND and substrate, particle that had DISCUSSION / 121 already reached the polymer coated surface were found not to adhere to the surface but to be in a continuous state of Brownian motion it, such just above that they were readily resuspended when the deposition cell was inverted. Thus, in these cases, no determination of the surface coverage carried 5.2 could be for runs B5 to B8 out. Details of these observations are given in Table inclusively. It can also be noted from that no result is given for case B l , where the suspension p H vicinity of the iso-electric point of silica). This is because it was observed the polymer coated coagulation. Under meaningful measured and for levels was that although was this surface, these they were conditions, hence case the spheres adhered found the to extraction the too slow to be measurable, electrophoretic was permanently be in of 2.97 during the a $p mobility (in the experiment, upon contacting state results excluded. Further, values of because the same table of would and $ at the w suggesting further that the pH partial not be were not stationary was near the iso-electric point of silica. 5.2.3. Effect o f Counter-ion V a l e n c e N u m b e r As mentioned two double two equally planar spheres charge earlier in Chapter layers are present, as would be the case in the present study when negatively charged spheres substrate, it is anticipated will on 2, under the conditions where more than be distorted by the particles and the are that the presence deposited potential of this on a positively distribution additional charged between the surface. Since the the substrate is opposite, it is very likely that the resulting double layer interaction energy between the two equally charged spheres RESULTS AND will be smaller than that predicted by Equation DISCUSSION / 122 [2.20] or any other similar equations given in Chapter 2. In view surface of the above, it was thus of interest to observe what the final coverage substrate was would be sufficiently when the suppressed double to exert layer thickness of the planar negligible influence on the double layer interaction energy between the deposited spheres. The easiest way this reduction in the substrate double layer thickness could be by which brought about would be to add to the suspension greater quantities of KC1 than were added in series A. However, since both the potassium addition of KC1 inevitably results in an and chlorine ions are univalent, the equal reduction thickness of the spheres, rendering the experiment this phenomenon, an electrolyte with a in the double meaningless. greater anion (which layer Thus, to avoid is the counter-ion for the positively charged substrate) valence number than that of the cation had to be used so that the double layer thickness of the substrate would be affected differently Equation than [2.10], proportional (K PO ), 3 a chosen to that of the where the the spheres. double counter-ion The layer valence rationale behind thickness is seen number whose anion (PO^~) valence number z. stems to be Potassium for this from inversely tri-phosphate is thrice that of its cation, for the present study. The experimental results are tabulated this series was of runs in Table 5.3 and the values of average surface coverages obtained as a function of /ca are plotted in Figure 5.8. Along with this plot, the surface coverages obtained as a function of /ca when are also plotted to provide a direct comparison KC1 was used of the results. as the electrolyte RESULTS AND DISCUSSION Figure 5.8 reveals that, instead of approaching a constant value case where using the K PO 3 the electrolyte was KC1, the values as the electrolyte a experimental Furthermore, continue range is small) with the values of surface which are greater than those of surface to increase as coverages obtained obtained as in the coverages /ca increases no sign of approaching obtained (although a constant in the present when KC1 was used / 123 value. series, as the electrolyte at similar /ca values, contradict what would have been predicted. That is, since the charges layer on the particles interaction energy and the substrate between the two are opposite, spheres the resulting should be reduced double in the presence of this additional surface and the thicker the substrate double layer, the greater this reduction will be. Thus, under the present circumstance where the substrate double layer thickness has been greatly reduced (although a reduction in the substrate double layer thickness was qualitatively deduced, an accurate quantitative calculation of l//c for the substrate was not possible since the values of the dissociation of HPO ' and deposited surface for the incomplete to H PO 2 anticipated constants 2 that its effect spheres coverage were li on not obtainable the would accordingly would be reduction double be attained. layer reduced " 3 to H P O ~ 2 in the literature), it was interaction energy between smaller value of and However, of P O since hence the a opposite trend was actually obtained, it is apparent that the observed results cannot be attributed to a reduction can at least spheres, by in the substrate be concluded double that its effect on the surface if any, is negligible. In fact, a similar observation Hamai et al. (6) in their study latex layer thickness. Or, more particles onto anionic conservatively, it coverage was of silica earlier reported of the static deposition of cationic polystyrene synthetic fibers, in which the use of NaCl as R E S U L T S A N D D I S C U S S I O N / 124 50.0 30.0 1 — —'— — —'—'—'— — —>—'— — —'—'—I 1 3.0 1 1 7.0 1 1 11.0 1 1 15.0 19.0 KR Figure 5.8 Comparison of surface coverages obtained as a function of /ca between series A and C RESULTS AND compared with Na SO 2 as the neutral electrolyte f l DISCUSSION / 125 showed no difference in the results obtained. Since the surface coverage reduction in the substrate double responsible for the observed results of series layer C cannot thickness, some behaviour. A be attributed to a other closer examination layer thickness on surface coverage hence no repetition will be made in series C, the measured values of $ to indicate the observed £ p p 5.3 shows that for all the runs are approximately constant, which would to be a non-contributing factor. Thus, in order behaviour and the role of £ p constant values of $ well be fortuitous. Further, it can be observed concentration of K P O 3 a to explain in the present study, it is necessary to begin by pointing out that the approximately may was discussed earlier here. The role of £ , however, may not be so obvious in the present study since Table seem very well be none the particle double layer thickness and the particle zeta potential. The role of particle double and must be of the experimental data in Table 5.3 reveals that these contributing factors may other than factors increases, the suspension p H in Table observed p 5.3, that as the increases too. As discussed earlier, this increase in pH results in a corresponding increase in the magnitude of the surface potential. However, due to a greater amount of added K P O 3 the suspension double run (c.f. runs C2 and C4) and hence a further reduction layer thickness, the range of potential distribution in the double C4 will be larger than that in run C2, resulting f l to in the layer in in a cross-over of the potential distibution curves. If, fortuitously, this cross-over point were located near the shear observed. plane, Figure this 5.9 would serves explain the approximately to illustrate this constant hypothesis. Thus, values of $ since smaller p RESULTS values of $ repulsive result p interaction spheres that in series C energy will can be deposited compared be AND to series A lower for series per unit area DISCUSSION / 126 for a C and given fca, the number of the substrate surface the of will be larger. 5.3. S U R F A C E In COVERAGE the two-dimensional fraction simulation rejection of the total process random area value particle the average occupied simulated obtained on using five exceeded thickness number the obtained simple for the particles on the planar independent the assumption 1000, and is infinitely it represents small. of particles is inadequate the case simulations, was that the deposition been reported were then that a series of new that the assumed since, by allowing an extremely large number surface coverage of 54.73% has final surface coverage obtained. Based performed using listed in Table made for the experimental the suspension the method described on this value, in Chapter 4 so surface coverages as a function of K a could be predicted. These results are plotted in Figure series A seen double (47,51). Thus, in view of the foregoing, this latter value of 54.73% was used as the maximum calculations where the particle Clearly, it can be (10 ) of particles to be deposited, a maximum and using value by all the deposited coverage, based was deposition was complete when the succesive number of failures for a particle to be deposited layer of model, surface, i.e., the surface 0.495±0.006. This - SIMULATION pH 5.10, along 5.1. It should be noted results obtained were changing with the experimental results of here that no comparison was when both the double layer thickness (i.e., as in series B and C). This -is RESULTS AND DISCUSSION / 127 Distance Prom The Charged Surface (um) Figure 5.9 Qualitative sketches of the potential distributions in the particle double layer for runs C2 and C4 RESULTS AND because under these situations, a comparison between simulated surface coverages based effect of changes DISCUSSION / 128 on tea is both in the magnitude inappropriate and experimental and inadequate, since the of $p is not accounted for in the present simulation model. It can experimentally be seen obtained in results Figure 5.10 that are qualitatively results, poor quantitative agreement is obtained consistently model overpredicting the measured reveals that the overpredicted although well described trend by can in the the simulated between the two, with coverages. A results the the latter closer examination likely be of the attributed to the unrealistic use of the double layer thickness, 1/K, as the basis for the calculation of the simulated the double which layer thickness interaction considerable g = represents the distance the potential drops to approximately repulsive a surface coverages. Since, as mentioned earlier between two from in Section 2.1.4, the particle surface at 1/e (^37%) of its surface value, the neighbouring deposited particles will still at this degree of double layer overlap. Hence, the assumption (a + 1/K) likely underpredicts the effective radius surface coverages which, in all cases, exceed the values leading measured be that to simulated experimentally. Therefore, in order to obtain a meaningful quantitative comparison between theory and experiment, a more appropriate basis for the calculation of simulated surface coverages has to be chosen. One such basis is the distance of separation between two deposited particles (8.232 x 2 1 10" J energy. The reason at which at 25°C), why this the overall i.e. equivalent parameter interaction to their is considered energy is equal to 2kT combined average thermal an appropriate basis for comparison is that, if each particle has an average Brownian energy of lkT, two RESULTS AND DISCUSSION / 129 R E S U L T S AND such particles will presumably, on the repulsive approach energy is account that the exceeds the effects average, be twice this interaction of 1/K changing An $ . The unrealistic prediction that the surface coverage was In order to employ this new additional benefit calculation and 130 unable to move into regions where value. energy DISCUSSION / automatically previous basis independent of of this takes into yielded the $. p basis, it becomes necessary to obtain for, each run in the series, the separation distance (H) between deposited particles at which the equals combined 2kT. determined In general, directly parameters as electric straightforward the since shown double value it is in layer and implicitly Equations substitution of of the this der Waals separation dependent [2.20] values van and distance upon a [2.40]. of relevant technique was graph depicting the is shown in Figure be other of into the a two bisection root-finding overall interaction energy obtained a function of the separation distance between the particles for all the runs in series A of instead parameters to obtain the required separation distance, a simple energy cannot number Thus, equations used. A interaction as approriate 5.11. Once the values for the separation distance are obtained, the ratios of the circular area occupied particle and its double calculation of these of surface run Al, determined coverages the by particle layer in the 0.8935 (diameter (diameter ratios enables separation to be the the = distance between um Figure 2a) 2a+ H) determination two-dimensional from = are 5.11. then by the calculated. The of corresponding simulation the to that occupied model. For spheres Based when on <p^ new values example, in = 2kT this value, the is ratio RESULTS AND Figure 5.11 The total interaction energy as a function between the particles for series A DISCUSSION / 131 of the separation distance RESULTS AND mentioned value above, which is [7r(0.5) of the surface coverage 2 condition where H = 0.8935 um +0.8935/2) ], equals 2 ]/[7r(0.5 obtained in the is 15.26% (0.279 x 54.73%). Similar As again The under calculations results are plotted in can be seen in the figure, the well described by the trend in the experimental results simulation. However, with the use the separation distance when the total interaction energy equals 2kT for underpredicts comparison experimental of in the results as thickness was the experimental model which, greatly over by present opposed used as downward their simulation model, it is expected Another too 5.12 large. Equations now overpredictions obtained more due nature continuing surface coverages a which of the to roll over progress, as was that higher basis limitations overlap particle of as the basis when the the double for of the choosing deposition an two-dimensional previously-deposited process. ones, If the the surfaces of those below before planned packing in the densities three-dimensional and hence higher will result. possible cause of the low predicted surface coverages is that the separation distance corresponding The are that this underprediction reasonable to the spheres overlapping particles were allowed simulation basis. It is suspected is largely the the to the when discarding simplifies case, the results effective radius is used Figure the together with the experimentally obtained results for comparison. qualitatively layer 132 0.279. Hence, the two-dimensional model were carried out for other runs in the same series A. Figure 5.12, DISCUSSION / overall [2.20] and interaction energies [2.40], which apply were to ^ calculated = on 2kT the only to the interaction of two shown on may be basis of particles RESULTS AND DISCUSSION / 133 45.0 10.0 I ' 0.0 ' ' I 4.0 i ' ' I 8.0 ' ' ' I ' 12.0 Figure 5.12 Surface coverages as a function ' ' I ' 16.0 ' ' ' 20.0 of K a for series A R E S U L T S AND DISCUSSION / 134 in an infinite medium. Unfortunately, as has been pointed out before, there is no theory available at the present time which describes either the van double layer interaction energies for two spheres located surface. Although the limited number of experimental suggest that the underlying surface has strongly suspected positively-charged that, may two particles be spheres in an surface coverages energies (for two interaction predicted values. at = 2kT based on shown energy coverages the large a double potential distribution between the two that and at is further evidence distances 10 and 5.14, and 20 kT computations separation corresponding (i.e., in Figure quantitative the 5.12. separation largest As distance agreement interaction distances were then hence surface coverages based on the distance the basis of to can is with energy interaction 20 kT be seen, as were reduced), the the measured chosen, to underpredict of the inadequacy of the for series B layer compensate for this effect, the the measured two-dimensional model simulation. of separation distances corresponding were performed respectively). These double infinite medium) equal to 5, 10 and the need for an improved three-dimensional Similar repulsive words, the two-dimensional simulation model generally continues results. This layer the approximately separation better in series C the reduced. In other increased in even results obtained flat thicknesses, way for comparison is are infinite is likely smaller than that predicted on spheres in an However, such infinite medium. To also calculated and the in significantly to <f> corresponding least an little effect on the surface coverage, it is substrate will distort the negatively-charged interaction at on der Waals or and C used to ^ (refer to Figures to estimate = 2, 5, 5.13 and equivalent radii two-dimensional model saturation limit. R E S U L T S AND These predicted results, plotted with their comparison, are shown as a function of pH function of K a for series C able to predict the DISCUSSION / experimental for series B 135 counterparts (Figure 5.15) and for as (Figure 5.16). Again, even though the simulation correct qualitative trends, it always gave a was underestimates of the experimentally measured surface coverages. As improved 2kT expected, the quantitative agreement if separation distances corresponding between experiment larger than were employed. the measured surface coverages are poor quantitative agreement is obtained the two-dimensional Figure of and to interaction energies Based on the above findings, it can be concluded in theory deposited that, although qualitatively described by because of the the simulation, oversimplifying nature of model. 5.17 shows a particles computer-generated obtained using the depiction of the two-dimensional photograph (taken from the image anatyser monitor) of an substrate. The the trends size of the plotted spheres the coverage determined has experimentally. It can model seen orientation along with a experimental deposition been reduced be final by in the a/a to simulate g figure that both photographs show a high degree of randomness in the particles deposited. In the case of the three-dimensional simulation program is partially functioning, successful completion a tangible value for the surface coverage when model, although the of the simulation leading to 1/K approaches zero did not RESULTS AND DISCUSSION / 136 80.0 Separation Distance H (nm) Figure 5.13 The total interaction energy as a function distance between the particles for series B of the separation RESULTS AND DISCUSSION / 137 Separation Distance H (um) Figure 5 14 The total interaction energy as a function distance between the particles for series C of the separation RESULTS AND DISCUSSION / 138 55.0 ST 20.0 ' ' 3.0 1 1 1 ' ' 4.0 1 1 1 1 ' ' ' 5.0 1 1 ' ' 6.0 1 7.0 PH Figure 5.15 Surface coverages as a function of pH for series B RESULTS AND DISCUSSION / 139 RESULTS AND DISCUSSION / 140 O O » <fr O • o w o** o o O • O O o o _ ck O © © ° 0 o o J o O o J o O o » 0 o o o o 7 o o v 0 4 o o , o_ ° o ° o °o ° oo_°o o o o o • o o o o o o o o* o -°o o °- o " ° "> i ^ ^ . o ooo o ^ o ^ o <^ 0 0 A Figure 5.17 Comparison of simulated and experimentally obtained photographs at approximately similar values of coverage. RESULTS materialize and, due, firstly, secondly, program. to to the very the exclusion Unfortunately, due to a complex of -some lack nature conditions AND DISCUSSION / 141 of the simulation process deemed in the essential of time, further debugging of, and the inclusion of more generalized conditions in, the program were not carried out. 6. C O N C L U S I O N S The double effects of the random layer thickness gravitationally allowing deposited and AND nature onto of the deposition process, the surface colloidal potential on spheres spherical, uniform-size, colloidal medium, to settle RECOMMENDATIONS cationic have silica been the surface glass coverage of experimentally spheres, potymer-coated the electrical suspended cover studied by in an aqueous slips under stagnant conditions. Unfortunately, colloidal silica spheres the techniques were unable size range (> 1/xm) where they Thus, before developing this requirement length revealed to be by simply of alcoholic the surface varying coverage decreases. expectation, total since repulsive interaction the a study aimed at out. It was found, after the reaction temperature colloidal an and/or spheres in could be produced. coverage This an increase particles in colloidal silica spheres which met solvent, spherical, uniform-size, experiments, the experimental is dependent potential (£p) and the double layer thickness surface uniform get underway, carried the conditions of the present that available for the production of produce for producing had the size range of 0.2 - 2.0 um Under to reliably investigation could investigation, that chain were could be easilj' observed in an optical microscope. a satisfactory technique important extensive the the primary that observation in either energy upon both 142 the particle zeta (1/K); as 1/K or $p increases, the is found to be in accordance 1/K or £p causes between results a pair an increase of particles and in with the hence - a CONCLUSIONS AND reduction in the surface coverage range of surface coverages greater than that when A thickness on surface predetermined obtained. Furthermore, obtained when both 1/K alone was used separate series of experiments coverage RECOMMENDATIONS examining was quantities of K P O „ 1/K and $ as the primary that the were changing performed by was controlling variable. the effect of substrate double also into 3 it was found / 143 dissolving layer different the suspension. Because of the different valences of the resulting cations and anions, the double layer thicknesses of the substrate and the particle are affected differently. The surface coverages obtained in these experiments showed that any influence manifest itself in the results. Besides these exclusion due to the random nature the maximum In factor findings, the presence of the deposition although its effect was difficult to quantify for exerted by this addition were predicted deposition consisted over also noted, due to the unavailability of a value to the systematic made to develop results experimental two which computer could be study of colloidal simulation compared non-overlapping particles were models with deposited, while deposition, to generate those measured rejection model the second scheme of a three-dimensional model where the rolling of sedimenting particles the surfaces of previously-deposited particles particles of geometric process was experimentally. The first scheme involved a simple two-dimensional only not surface coverage without interaction effects. attempts where did were allowed. Comparison using the two-dimensional consistently underpredict oversimplifying nature of experimental as well results as the stacking of with those obtained model revealed that for all cases, the simulated results those of the of the deposition experimental process results simulation. The due trends to the in the C O N C L U S I O N S AND experimental various results, however, were reasons, involved model, such and time which is as the very constraints, more approximated complex successful realistic by the nature of completion than materialize. It is speculated that the RECOMMENDATIONS the simulations. Owing to the of simulation the process three-dimensional two-dimensional surface coverages / 144 model, provided by did the former model will likely be greater than those obtained using the two-dimensional and hence the three-dimensional model will quantitative agreement with those measured that, in a future study, further debugging model be undertaken coverage may so that be carried correctly results which out, the reflect the differently charged and tangible refinement of the three-dimensional value for the simulated comparison raw of the simulated and series was narrow due at the maximum simulated data experimental experimental results obtained must be conditions studied. However, between two A manipulated this to manipulation spheres situated on a substrate, which is unavailable at the present time. Thus, it is also suggested that the derivation of this interaction theory be coverage in better experimentally. Thus, it is suggested requires a knowledge of the interaction energy In are model be obtained. Before a meaningful can a yield not of the investigated, present the study, range of where 1/K the used effect was of undertaken. 1/K on surface unfortunately rather to several reasons discussed earlier. While not too much can be done lower end of the range (since further reductions in the double layer thickness through the addition of more electrolyte into the suspension would result in particle coagulation), expansion in the upper range of 1/K is presumably CONCLUSIONS AND RECOMMENDATIONS / 145 attainable if the experiments were to be carried out in a controlled environment (e.g., in an inert chamber ions would be absorbed where only nitrogen is present) so that no unwanted into the suspension. Thus, it is suggested that in a future study, experiments be carried out in the above-mentioned manner so that a far greater extended (lower) range of measured surface coverages can be obtained. range would provide a more comprehensive test for any This future simulation models and, hopefully, would allow a clarification of the separate roles played by 1/K Besides extended surface and £ on the coverage-dependent deposition the above, it is also that deposition study should be to include situations where the suspension is flowing. The difference in coverages experimental obtained conditions significance under must motion of the suspension. particles. suggested process. static and flowing, be attributable to new In this wa}', it would of hydrodynamic exclusion on but otherwise effects introduced identical, by the be possible to investigate the the surface coverage of depositing NOMENCLATURE Only those symbols used in the main text of the thesis are defined Those in the Appendices are defined in the particular section here. in which they occur. The units given are those which are most frequently used. a half-width of flat electrophoresis cell cm a the radius of spherical particle cm A — Hamaker constant J — cross-sectional area of flat electrophoresis cell at the plane of viewing A,B,C,D constants Ajjk Hamaker constant for the interaction of material i with used in Equation cm [2.36] — material k in a medium j J b half thickness of flat electrophoresis cell C concentration of electrolytes in the solution e cm moles/litre electronic charge (4.80324 x 1 0 " ) esu 10 Q f = f(p), fitted correction factor used in Equation [2.34] — H distance of separation between two solid surfaces — I parameter used in Equation [2.36] — parameter used in Equation [2.37] — bp I g s k Boltzmann constant (1.380662 x 1 0 " ) I interelectrode distance for electrophoresis cell 23 n Q bulk ionic concentration n + numbers of positive ions per unit volume J7K cm ions/cm 146 3 ions/cm 3 / 147 n_ numbers of negative ions per unit volume Avogadro's constant (6.02276 x 1 0 ) molecules/mole 23 p = r distance from the centre of a sphere R interelectrode resistance of electrophoresis cell T absolute temperature U-g true electrophoretic mobility of particles U apparent electrophoretic mobility of particles at the 0 27TH/X, parameter used in Equation ions/cm [2.34] — cm ohm K mid-plane cm /volt-sec 2 cm /volt-sec 2 V voltage drop across electrophoresis cell x distance in the double layer from a charged surface X applied potential gradient y distance normal to median plane of flat electrophoresis V cm V/cm cell cm z valency of counter-ions — a,P,y parameters used in Equation 5 distance from the solid surface to Stern plane e dielectric constant of the dispersion medium — $ electrokinetic (zeta) potential mV 0 partical surface coverage — K inverse electric double layer thickness cm 1/K electric double layer thickness cm [2.37] — cm o X — wavelength of atom dipole fluctuations — X? A electrical conductivity of electrolyte solution umho/cm 7 limiting conductances of the ions in solution cm /ohm-equ7v 2 / 148 coul/cm p volume charge density p,p' parameters used in Equation [2.36] o surface charge density <j> interaction energy between two solids \p electric potential at a distance x from the charged i// Q cm coul/cm J surface mV electric potential at the charged surface mV \p g Stern mV potential poise (JL viscosity of the solution SUBSCRIPTS a-a atom pair van der Waals interaction edl electrical double layer interaction p particle sp interaction between a particle and a planar ss interaction between two particles T total vdw van der Waals interaction w wall SUPERSCRIPTS ret retarded a constant charge density i// constant potential surface 2 REFERENCES 1. Ruckenstein, E. and Prieve, D.C, J. Chem. Soc. Faraday II, 69, 1522 (1973) 2. Spielman, L.A. and Friedlander, S.K., J. Colloid Interface Sci., 46, 22 (1974) 3. Rajagopalan, R. and Tien, C , Canadian J. Chem. 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Colloid Interface Sci., 61, 302 (1977) 92. Depasse, J. and Watillon, A., J. Colloid Interface Sci., 33, 430 (1970) 93. Her, R.K., The Ithaca, N.Y. 94. Her, R.K., Colloid Chemistry of Silica and (1955) Surface Colloid Sci., 6, 1 (1973) Silicates, Cornell Univ. Press, / 154 95. Aelion, R., Loebel, A. and Eirich, F. J. Am. 96. LaMer, V.K. and Dinegar, R., J. Am. Chem. Soc, 72, 4847 (1950) 97. Utsumi, I., Ida, T., Takahashi, S. and Sugimoto, N., J. Pharm. Sci., 592 Chem. Soc, 72, 5705 (1950) 50, (1961) 98. Revell, R.S.M. and Agar, A.W., Brit. J. Appl. Phys., 6, 23 (1955) 99. Marshall, J.K., Ph.D. Thesis, University 100. Komagata, S., Researches Electrotech. 101. Moore, W.J., Physical Chemistry , : of London (1964) Lab. Tokyo, Comm. no. 348 (1933) Third Edition, Prentice Hall, Englewood Cliffs, N.J. (1962) 102. Scheaffer, R.I., and McClave, J.T., Statistics for Engineers, First Edition., Duxbury Press, Boston (1982) 103. Mickley, Chemical H.S., Sherwood, Engineering, T.K., and Reed, C.E., Applied 2nd Ed., McGraw-Hill, New Mathematics York (1957) in APPENDICES A. S A M P L E CALCULATIONS A.l. Double Layer Thickness The At inverse double layer thickness = 4.8030 x 10" ° = 6.0226 x 10 2 3 e = 78.54 k = 1 .3805 x 10" 1 6 T = 298.15 N 0 A for predominantly where Equation [2.10]. C is the predominantly measuring K [2.10] reduces to K cm" = counter-ion of ionic of those the 0.3286 x 10 concentration species ions added concentration-conductivity conductivities, content of suspensions however, due 5 1 [ knowledge of C the addition of KC1, was assumed the concentration the suspension curve to 0 the given fact in can that the N a O H or 1 ] was made HC1, to be be determined by using the and Appendix very complex, exact determination 155 A high solution conductivities, where the present of C - in moles/litre. A For and calibration was x 8 been adjusted by conductivity solution erg/K univalent counter-ions, Equation electrolyte concentration has concentration e.s.u. 1 therefore vital to the calculation of K. the calculated using 25 °C, e and was B.1.3. nature of C of For the up low ionic is impossible. / Hence only pH and of the approximate ion concentrations conductivity data on individual species additive. In that case the to the can be basis of the calculated from assumption overall conductivity that the through = L e the charge of an 0 contributions their mobilities was N^ilziedUi i 1000 [ A where X is the conductivity of the solution (umho/cm), the concentration suspension (101) X C- the 156 - 2 ] the Avogadro number, of ionic species (moles/litre), zj the valence of ionic species i , electron (Coulomb) and Uj the mobility of the ionic species i (cm /volt-sec). 2 Consider, for conductivity of the example run A2, suspension were 5.24 and case, the suspension was adjusted. A the of ions from also 1x10" M KC1 6 the complete present, 2 + and K . + revealed the measured the pH a these was by anions. For of the suspension the them far the were presence HSjO" constituted + = 10" ' " 5 2 = 5.754 x 10~ 6 this of H not before and + HCO" from ionic species insignificant most abundant, followed reflected in the run and of other the determination in question, since =5.24, [H ] and of the suspension was a multitude majority quantities average mobilities of cations and pH 2 great found that Na* Thus, that, besides of CO , pH /xmho/cm respectively. In dissociation of water, and absorption although proportions. It was Ca 3.25 freshly prepared and the dissolution of silica and were where chemical analysis (see Table A.l) performed on addition OH" in moles/litre by of / 157 TABLE A.1 Results of chemical analysis showing the types and quantities of ions present in the freshly prepared silica suspension. Element Quantity (< mg/1) Boron B 0.01 Berylluim Be 0.001 Cadmium Cd 0.002 Cobalt Co 0.005 Chromium Cr 0.002 Copper Cu 0.01 Mercury Hg 0.05 Molybdenum Mo 0.01 Nickel Ni 0.005 Lead Pb 0.02 Antimony Sb 0.05 Selenium Se 0.05 Thorium Th 0.05 Vanadium V 0.002 Zinc Zn 0.004 Aluminum Al 0.04 Barium Ba 0.001 Calcium Ca 0.26 Iron Fe 0.003 Potassium K 0.2 Lithium Li 0.05 Magnesium Mg 0.05 Manganese Mn 0.001 Sodium Na 0.78 Phosphorus P 0.1 Silicon Si 0.1 Strontium Sr 0.001 Titanium Ti 0.001 Zirconium Zr 0.005 / 158 Therefore, [ O H " ] = 1 0 " " / [ H ] 1 = 0.001 7 3 8 X 1 0 " moles/litre, + 6 since the dissociation constant of water is about 10" . 1ft Also, in order to maintain electroneutrality, [H ] + [K Mother + = cations] + [ H S j O j ] + [HC03] + [OH"] + [ C l " ] + [other anions] It can be noted from the literature that the ionic mobility [ A 3 ] values for most types of ions are similar. Thus, to simplify calculations, most cations and anions were respectively lumped together, giving [ H ] + [ K ] + [ o t h e r cations] = [OH"]+[Cl"]+[other + + anions] [A. 4] From and Equations [A.2] and [A.4], the two unknown concentrations, [other [other anions], could then be computed. Thus if [other cations] = y then [other anions] = (y+5. 754+1 - 1-0 . 00 1 738 = (y+5.7522) and cations] x 1 0 " moles/litre, 6 x ) x 10" 6 1 0 " moles/litre 6 substituting these expressions into Equation [A.2], 6.023 x 10 2 3 x 19 1.602 x 10 5.754U + H 10 1.0U K + 1.0U + 0.001738U Q H - + : C 1 - + y U c a t i o n + (y + 5.7522)U a n i on x 10' [A. 5] / 159 where the individual mobilities are (101) = 36.30 x 10"" cm /volt-sec u - 2 H U Thus = 20.50 x 10"" OH" u K + u c l = 7.61 x 10"" - = 7.91 4 cation = 5.614 x 10" 4 U (average) anion = 6.255 x 10~ 4 u (average) for a conductivity value solved from Equations y x 10" x 10" of 3.25 x 10" mho/cm, [other cations] can be 6 [A. 5] as 6 = [other cations] = 6.432 x 10" moles/litre 6 The total counter-ion concentration C is then obtained as C = [other cations] + [H ] + [K ] + + = (6.432 + 5.754 + 1.0) x 10" 6 = 13.186 x 10" moles/litre 6 and from Equation [A.l], K = 0.3286 x 10 = 1.193 x 10 For a particle radius, a, of 0.50 5 8 x (13.186 x 10" )°6 cm" 1 x 10"" cm, >ca = 1 . 193 x 0.50 x 10 = 5.97 s x 10" ft S / A.2. Particle Zeta Potential As mentioned in electrophoretic mobility Wiersema al. (70) et conductance effects. variables and Section 3.8.1, the conversion zeta potential was based to that Their accounted results were tabulated for were of on the the retardation, reported in relaxation terms 0 E = = 67rue surface dimensionless by 0 0.7503 x the dimensionless double layer thickness, at 25°C 10* • [A.7] by = [A. 8] the dimensionless zeta potential, by Yo e$ 0 kT at 25°C and of ekT Qo 0 of and tables [A.6] where E, the dimensionless electrophoretic mobility, is given y , numerical m+) q r 0 0 measured particle in the form of E = E(y , q , 160 m_|_, the dimensionless mobilities of the positive solution, by N ekT A Z ± and negative [A.9] ions in / 161 12.86 X with Mu e = the unit of electrostatic charge = the particle zeta potential *P e a t 2 5 ° [A.10] C the viscosity of the solution (poise) 7ZZ o — - (stat-coulombs) (mV) the dielectric constant of the solution k the Boltzmann constant (ergs/K) = the absolute temperature T (K) = the electrophoretic mobility (cm /Volt-sec) 2 U E K - the reciprocal double layer thickness (cm" ) a • the particle radius = Avogadro's constant - the valence of cations and N Z A ± 1 (cm) the limiting conductances Because of the form explicitly value of q , 0 E 0.184, the actual m + and 3' 0 m_ values of E, are obtained by since this table interpolated and anions in solution of the Wiersema tables, a solution for y for known reference. Furthermore, = of cations and /ohm-equiv). (cm obtained anions values q , 0 and m must be = E + values. This can be accomplished (m -0.l84) be given results obtained for m + to the using the equation 3E + + 9m . Instead, for a corrected to correspond 9E E' cannot graphical interpolation of Table I in the gives only the of E + 0 + (m_-0.l84) r A 3m_ n l '_ / 162 where E' is the corrected interpolated the from Table experimental value mobility and 3E/9m and + III of the reference. The value of E ' , E e x , is then 3E/9m_ of y extracted from 0 must be corresponding to a final plot of E ' versus y . 0 Consider the case of Run A2, where the concentration of various ionic species in the solution as well as the double layer thickness have been estimated in Section A . l . The average electrophoretic time = 3.85 sec 1-grid spacing = The = 7.085 interelectrode distance Applied voltage 119.0 X 10~" cm (Appendix B.1.2) cm = 5 1 . 0 volts Therefore, electrophoretic mobility U E (0.0119cm)(7.085cm) = = (51V)(3.85sec) 4.2939 x 10"" cm /Volt-sec 2 Substituting this value into Equation [A.7], E e x = 0.7503 x 10* x U = 0.7503 x 10" x 4 . 2 9 3 9 x 10"" = 3.22 From Section A . l , q The dimensionless averages present. Q = £ at 25°C /ca = 5.97 mobilities m of the individual + and m_ limiting ionic can be calculated mobilities, X+ from and X° the numerical of the ions / 163 At 25°C, the X° H values for the various ionic species are (101): = 3 4 9 . 8 2 cm /ohm-equiv + 2 OH" = 1 9 8 . 0 0 cm /ohm-equiv K + = 7 3 . 5 2 c m /ohm-equiv Cl" = 7 6 . 3 4 cm /ohm-equiv 2 2 Other c a t i o n s = 5 4 . 2 1 6 (average) cm /ohm-equiv Other anions and their corresponding m H = 6 0 . 3 7 6 (average) cm /ohm-equiv 2 values from equation [A. 10] are: + + = 0.0368 OH" = 0.0649 K + = 0.1749 Cl" = 0.1685 Other c a t i o n s = 0.2372 (average) Other anions Subsequently, from average values of m the + = 0.2130 concentrations obtained 0.0368 6 13.186X10" and + 13.186x10' 6.43x10" = in Section A . l , the number are: 5.754x10" m. = (average) 0.145 0.2372 6 1 .0x10' 13.186x10 - e 0.1749 / 164 0.00174x10 m_ = -6 13.186x10 12.185X10" 13.186x10" = From graphical 1 .0x10 0.0649 + 6 0.213 6 0.2096 Table I of Wiersema et al. (70), values interpolation at q 0 = 5.97, m = + of E Values m + at q of E and 0 = 5.97 for the same are then 9E/9m + values determined above. These + and 9E/3m_ are by graphical extrapolation. 0 using equation data [A. 11] together with the are tabulated in Table A.2. TABLE E E VERSUS y 0 A.2 - 9E/9m + by Q values of y corrected to E ' are obtained 0.184 for various values of y . Then, from Table III of the same reference, values of 9E/9m obtained 0. 1685 13.186x10' Run A2 9E/9m_ E' 1 1.1506 -0.0855 -0.0166 1.1627 2 2.1216 -0.6524 -0.5156 2.1338 3 2.8034 -0.2566 -0.7733 2.7936 4 3.1444 -0.3161 -1.3804 3.1214 5 3.2272 -0.3755 -1.9875 3.1910 / 165 The E' corresponding Equation versus value y of y curve 0 is then obtained 0 from plotted. Since this plot is 3 [A.9], the zeta potential of the particles used r 0 E = g x = 3.22, the 5.443. Thus, from in run A2 is calculated as S = 25.69 x p = which is in fact negative, as 139.8 5.443 mV indicated by the direction of the particles in relation to the applied voltage during the mobility measurements. A.3. W a l l Zeta Potential The wall electrophoresis zeta potential can be calculated using cell dimensions, and other information used Equation [3.3], the in the calculation of particle zeta potential. For Run A2, electrophoretic time measured at mid-plane = 2.63 sec Therefore, apparent mobility U n = (0.0119cm)(7.085cm) (51V)(2.63sec) = Electrophoretic mobility U From E found 6.286x1 0"" = 0.4922 Cell half-thickness, b = 0 . 0 5 6 0 5 At 25°C, p. = 0 . 0 0 8 9 3 7 poise e = 78.54 2 in Section A.2 = 4 . 2 9 3 9 x 1 0 " " cm /volt-sec 2 Section B . l . l , Cell half-width, a cm /volt-sec cm cm / 166 Thus, substituting the above values into Equation [3.3], yields the wall zeta potential $ A.4. Errors i n Electrophoresis Although electro-osmosis, converting = +41.4 w mV. and Electro-osmosis Measurements there are many potential sources of error in electrophoresis and such as errors due to locating the measuring levels, errors due to time to electrophoretic mobility, etc., it is thought source of error in the present measurements may measuring the electrophoretic time resulting from that the primary be attributed to the errors in the effects of Brownian motion and the finite depth-of-field of the observing system. Many of the potential errors can be whereas largely eliminated the errors eliminate. Thus, resulting particle through proper due to Brownian the problem and wall here design motion of the measuring and depth-of-field is to determine $ -potentials which arise the error when a apparatus, are difficult to limits number on the of time measurements are taken at each of the prescribed levels. In order to determine the error that the measured electrophoretic limits, it will be necessary to assume times are approximately normally distributed, so that the 9 5 % confidence limits based on the student t-distribution will be adopted to predict the closeness of the sample mean to the population mean. Thus, if the measured electrophoretic mobility obtained at the stationary / 167 level is U , the 9 5 % confidence interval for U E E U where 95% ± <t -l'°- N p (t^j _ 0.025) is the t-distribution probability that U represents E limits shown, N is the sample Since the electrophoretic time, the £ mobility variance will electrophoretic times. According and N -l [A.12] degrees £ electrophoretic a^ function also be a of freedom mobility the sample E is a to Mickley "T^T" x the true size mobility > 0 2 5 with 1; E is (102) E function within standard of the measured of the et al. (103), if y for a deviation. electrophoretic variance = the of the f(x ,x ,....,x ), 1 2 n then the standard deviation of y will be: 0.5 .1 (dy / d x , ) i=i -of.. i 2 x In the present case, the stationary level is given measured [A. 13] electrophoretic mobility obtained at the by / U where E [A. 14] = I = graticule spacing X = applied potential gradient = mean electrophoretic time measured at the stationary level t E Therefore, from Equation [A.13], assuming n = l , y = U , E —O- x ^ t g , 0" x 1 0 " y = O j j , the standard deviation of U p E o , and E fc , can be written as U E I a UE = „Xt 2 E * a t ^E [A. 15] / 168 inhere o~- is the standard deviation of t N Thus, from the above given by the expression: E E ( t equations, t r together (102), the range of mobilities are estimated E ) 2 with a table and consequently the errors particle zeta potentials are calculated. Consider, for example, Run where N t £ = of t-distribution in the A2 25 = 4.62 sec E a- - H 0.1247 sec l = 0.0119 cm X = 7.1983 V/cm Therefore from Equation [A. 15], (0.0119 a UE = cm)(0.1247 sec) ( 7 . 1 9 8 3 V/cm)(4.62 sec) 9.658 x 10~ 6 cm /volt-sec. 2 From t-distribution tables (102), ( 1 „ , 0 . 0 2 5) = 2 Thus, from 2 2.064 Equation [A. 12] the 9 5 % confidence limits are 9.658 U P ± 2.064 x E x 10~ — = U_ ± 3 . 9 8 7 3 (25) 6 0 5 x 10~ 6 E and using Equation [A.7], the 9 5 % confidence limits on E are E From ± 3.987 x 10~ 6 x 0.7503 Appendix A, Section A.2, E was found x 10" = E ± to be 3.22. Thus, the range of E becomes 3.1901 < E < 0.0299 3.2499 / 169 The Table corresponding A.2 for E ' respectively. confidence -150.22 mV The values 0 values It follows limits y of 3.1901 from on the particle limits error limits slight the plot of E ' versus [A.9] that $-potential are no longer value because the E ' versus y with Equation from and 3.2499 respectively. In other words, confidence The obtained $ p are 4.9878 the lower for Run = A2 y 0 in and 5.8476 and upper 9 5 % are -128.14 and -139.8 (+11.66, -10.42) mV. symmetrically positioned about the mean curves are not linear. 0 on the wall $-potential, $ , can be similarly w variations. As mentioned in Section 3.8.2, $ evaluated as determined w by a combined electrophoresis-electro-osmosis experiment is given by 8TTM [1 - (I92b)/(7r a)] [1 + (384b)/(7r a) ] 5 U $ w " "TF = 2.087 where $ dielectric x 10 5 5 (U - UQ) E is the wall potential (mV), u the fluid w constant, X the applied potential E " at U 0 25°C viscosity (poise), ratio ( = 0.1139), U g the electrophoretic mobility measured level (cm /sec-volt), and UQ e the fluid gradient (V/cm), b/a the thickness to width 2 [A. 17] the electrophoretic mobility at the stationary measured at the mid-plane (cm /sec-volt). Thus, if it is assumed normally (102) distributed, then that the U £ the 9 5 % confidence and UQ values limits on $ w were approximately can be obtained as $ = 2.087 w x l0 (t ,0.025) 5 M where 1 (1-P ) 2 M P N - 1 £ U = U N E VN E ^UE /N )-(a 'UO and CJJ Xt - 1 ) 2 E 0 u o VN ) E a- 2 t, as defined in Equation [A. 15]. E Consider, for example, the same run (Run A2) where N E = 25 N Q = 21 t 0 ot = 3 . 1 6 sec = 0.1045 sec I = 0.0119 cm X = 7.1983 V/cm 0 'UE 9.658 x 10 cm /Volt-sec. 2 Using Equation [A.21], (0.0119 'UO cm)(0.1045 s e c ) (7.1983 V / c m ) ( 3 . 1 6 = 1.73 x 1 0" 5 sec) cm A'olt-sec 2 2 / 171 Using Equation [A.20], (9.658X10" ) /25 6 P = 2 [ (9.658x1 0 " ) / 2 5 ] + [ (1 . 73x 1 0" ) /21 ] 6 2 5 2 = 0.2075 Using Equation [A.19], 1 (0.2075) M 24 2 (0.7925) 2 20 or M = 30.12 When M $ w = 30.12, 0^,0.025) = 1.96 (102). Thus, the 9 5 % confidence limits on may be found, using Equation [A.18], as (9.658x10" ) /25 " 6 A$ w = ± 2.087 x 10 = ± 1.73 Therefore, the lower 5 x 1.96 x 1 0 - 5 (1 .73X10" ) /21 ) 5 2 mV. and upper 9 5 % confidence limits on the wall Run A2 are 39.67 and 43.13 mV respectively. B. CALIBRATIONS B.l. Calibrations 2 Made for Electrophoresis Apparatus ^-potential for / 172 B.l.l. Inter-electrode Distance and Cell Dimensions A was knowledge of the inter-electrode distance required to evaluate the applied potential in the micro-electrophoretic gradient during cell electrophoretic mobility measurements. In view of the complex geometry of the whole cell, this distance cannot be measured measurement based across R directly. It can, on a knowledge however, be obtained are the electrical indirect of the solution conductivity, the resistance the electrodes, and the cross-sectional area and X from resistance of the viewing region. Thus, if and the conductivity respectively of a solution placed in the cell, then the effective inter-electrode distance I is given by I = RXA where A obtained is the cross-sectional area from of the viewing the product of the thickness region. and width This area can be of the cell, which were determined as follows. The thickness adjustment of the cell of the electrophoresis was measured using microscope. The cell the micrometer was first clamped water bath of the electrophoresis apparatus in the same manner measurements. The microscope was then adjusted near inner thickness walls of the cell of the cell made at the centre were made before The width could to focus and the micrometer be determined focussing readings in the as in mobility on both the far taken and so that the by difference. All measurements were of the cell in the viewing plane, and several measurements an average value of 0.1121 cm was obtained. of the rectangular section of the cell was also measured, by means of a bath and filled travelling placed with facilitated microscope. In in an upright potassium an this case, the position on permanganate accurate measurements were taken a was removed horizontal level. The (KMnO ), the u viewing of and average value the cell the distinct boundaries was of / 173 from the cell was colour the determined then of which cell. Several to be 0.9844 cm. The was resistance determined R by in the above expression measuring the for the resistance across inter-electrode a solution conductivity placed in the cell with the electrodes in position. A of 0.1M prepared KC1 it in one was by litre of distilled and solution was then used to rinse and were placed in position and The at water bath and fill up small quantity and dissolving of the prepared the electrophoresis cell. The cell clamped in the known electrophoresis electrodes water bath. its contents were then allowed to reach thermal equilibrium 25°C, after which the a Beckman Model 16B2 The the water. A of standard solution carefully weighing 7.4555g of KC1 deionised distance conductivity resistance A.C. X of measured using a Seibold L T A across the conductivity bridge the same cell was measured by operating at 1000Hz. solution at the same means of temperature was conductivity meter. In this case, the interelectrode distance was determined to be 7.085 cm. I 174 B.l.2. Eyepiece Graticule The and Timing graticule spacing in the eyepiece which the particles were timed a Device of the electrophoresis microscope over during mobility measurements was calibrated using "stage micrometer", which is a glass slide engraved for 1 mm divisions. The "stage bath maintaineed micrometer" at 25 °C with was held in a verticle The calibrated run electrical timer position clamps. Several measurements averaged to obtain a graticule spacing of 119.0 ± which was 0.9 incorporated for short times to 5/6. Therefore were taken and was times also were allowed to (1-2 minutes) on several occasions. It was all measured mm in the water in the apparatus that the ratio of the two records, stopwatch to timer, was equal 0.01 um. against a Heuer stop watch. The two timing devices together with observed always constant and for electrophoretic velocities were corrected by multiplying by a factor of 5/6. B.l.3. of Conductivity of Potassium Chloride and Potassium Phosphate as a Function Concentration In solution mainly curve the experiments, pH was for high not adjusted, electrolyte the solution conductivity due to the presence of added electrolyte for the solution conductivity versus concentration in the calculation was and when the assumed (KC1 or K P O ) . A 3 various electrolyte in the solution was therefore prepared solution concentrations known to be calibration f l amounts of added to facilitate the determination of of double layer thickness. The results / 175 obtained are presented in Figures In the B . l and B.2 for KC1 and KjPO,,, respectively. the case of potassium phosphate, as the amount of added K PO 3 f l increases, p H of the solution increases. Thus, a separate plot of the solution p H as a function of K P O , 3 concentration was also made (Figure B.3). / 176 Potassium Chloride Concentration (M) Figure B . l Solution Conductivity as a function of added KC1 concentration / Figure B.2 Solution Conductivity as a function of added K P O 3 f l 177 concentration / Figure B.3 Solution pH as a function of added K P O 3 f l concentration 178 COMPUTER PROGRAMMES Two-Ot mens tonal Random Particle Deposition Mo<3e 1 IMPLICIT R£AC8(A-H,0-Z ) DIMENSION X(1O0.100).Y(100.100).Z<100.100>. 1 TX(100).TY(100).T2(100).TTX(1O0).TTY(100).TTZ( lOO). 2 N(100),NZ1(20).NZ2(20).DGL(10), 3 AREA(20),THICK(20) INTEGER FLAGXJ.FLAGX2.FLAGX3. FLAGYL FLAG Y2.FLAGY3. FLAG2A.FLAG2B ICOUNT a 0 EPS-0.01 ZMIN»0.25 DIA-0.50 RAD-DIA/2.0 RAD1=3.0*RA0/4.0 IN03-0 NP"0 ID»0 IID-0 00 10 I"1.100 N< I )-0 TX(I)«0.0 TY(IJ-0.0 TZ(I)-0.0 TTX(IJ-0.0 TTY(I)-0.0 TTZ<I)-0.0 IF{I.GT.20)G0 TO 10 NZ 1 ( I ) "0 NZ2( I )'0 DO 5 J- 1 , 100 X(I,d)"0.0 Y(I.d)-0.0 Z(I,J)-0.0 CONTINUE CONTINUE C Random nuraoer generator C Ini t i a l i ze 10000 XX=PRANO(0.) I-IRAND(O) YY«PRAND(0.) J-IRAND(O) C C Generate XX»FPRAND(1.> I»IRAND(20) XX«XX-I C Y Y"FPRANO( 1. ) J-IRANO(20) YY-YY'J C WRITE(6.4321)XX.YY C4321 FORMAT(2X.F7.2,2X.F7.2 > C IF((XX ) .GE.( 10.0+RAD) .OR. (YY).GE.( 10.0+RAD) .OR. XX.LE.RAD .OR. YY.LE.RAO)G0 TO lOOOO C 4444 ICOUNT"ICOUNT+1 IF(ID.EO.1001)G0 TO 88888 C Compute the distance between centres (OBC) in the (X.Y) plane C C (I) Lower level XTEMP-XX YTEMP-YY I F(XTEMP, LT .O.0)XTEMP-0.0 IF(XT EMP .GE .10.0)XTEMP=9.5 IF(YTEMP, LT .O.0)YTEMP=0.0 IF(YTEMP, G£ .10.0)YTEMP-9.5 I * XTEMP d»YTEMP JM1-J-1 JP1-J+1 NP-0 C (a) Check If the particle is at the lowest level C C IF(J.EG.0)G0 TO 123 C LL-JM1- 10+I LLP2-LL+2 C C (D) Check if the particle Is at the left and r i ght edges C - c . 112 IF(I.E0.0)G0 TO 112 IF(I.E0.9)G0 TO 1.13 GO TO 114 LL»LL+1 GO TO 114 LLP2-LLP2-1 1 13 C 114 00 20 LP1=LL.LLP2 C C (c) If no particle at a certain slot, do not compute IF(N(LP1).EQ.O)GO TO 20 NNT T »N(LP 1) DO 30 LP2-1.NNTT DBC'DSQRT((XX-X(LP1,LP2))•"2+<YY-Y(LP 1.LP2))•-2) IF(DBC.LT.DIA)G0 TO ICOOI CONTINUE CONTINUE C (II) Ma in leval C -- c 123 ML"J-10+I MLP2-ML+2 C C (a) Check rf the particle is at the left and C - c 124 right edges IF<I.EG.O)G0 TO 124 IF(I.EQ.9)G0 TO 125 GO TO 126 Ml_ = ML+1 GO TO 126 MLP2-MLP2-1 125 C 126 DO 40 LP1-ML.MLP2 C C (b) If no particle at a certain slot, do not compute C -- c C IF(N(LP1).EO.O)GO TO 40 NNTT SN(LP 1) DO 50 LP2-1,NNTT DBC-OSQRT((XX-X(LP1,LP2))-•2+(YY-Y(LP 1.LP2))**2) IF(DBC.LT.DIA)GO TO 10001 CONTINUE CONTINUE 50 40 C C (III) Upper level c C C (a) Check if the particle is at the highest Y level C C IF(J.E09)G0 TO 90O0O C NL=JP1«10+I NLP2=NL*2 C C (b) Check if the particle is at the left or right edges c C 236 IF(I £Q.O)GO TO 236 I F(I . EO.9 )G0 TO 237 GO TO 238 NL=NL+1 GO TO 238 NLP2*NLP2-1 237 C 238 00 60 LP1=NL.NLP2 C C (c) If no particle at a certain slot, do not compute C c C IF(N(LP1).EQ.OJGO TO 60 NNTT »N(LP 1) DO 70 LP2-1,NNTT DBC*DSQRT(<XX-X(LP1. LP2>) •-2+(YY-Y<LP 1.LP2) )**2) IF(DBC.LT.OIA)GO TO 10001 CONTINUE CONTINUE C Recording the permanent X.Y.Z values for the settled particle 90000 XTEMP-XX YTEMP-YY IF(XTEMP.LT.O.O)XTEMP=0.0 IF(XTEMP.GE.10.0)XTEMP=9.5 IF(YT£MP.LT.O.O>YTEMP=0.0 IF(YTEMP.GE.10.0)YTEMP=9.5 I-XTEMP J=> YTEMP 1 BOX»(J*10)+I*1 C N(IBOX )-N(IB0X)+1 NUM=N(IB0X) C X(IBOX,NUM)«XX Y<IBOX.NUM) • Y Y C GO TO lOOOO C 10001 IF(XO.GT.O)GO TO 10002 10OO4 ID>1 IID-ICOUNT GO TO 10000 C 10002 IF((11D*1).EO.ICOUNT)G0 TO 100O3 GO TO 10004 10003 ID-ID-M IID=ICOUNT GO TO 10000 C 88888 SUMN=0. DO 8765 LOOP =1.100 SUMN=SUMN+N(LOOP) 8765 CONTINUE C AREA( 1 l-SUMN-3. 14 15926-DIA«DI A/4.0 THICK( J )•100O-O 00 8766 LOOP 1-2.20 DIA1=DIA-0.02*DIA*(L00P1-1) AREA{LOOP 1 )=SUMN-3. 14 1 5926 *DI A 1 -DI A 1 /4.0 THICK(LOOP1)-DIA1/(DIA-0IA1) 8766 CONTINUE C WRITE(6.5436) 5436 F0RMAT(9X,'KA'.13X,'% AREA'//) DO 8767 LLP2-1.20 WRITE(6.98765)THICK{LLP2).AREA(LLP2) 8767 CONTINUE 98765 F0RMAT(3X.F10.2.7X.F10.2/) CALL PLOT(0. C 220 210 200 CALL CALL CALL CALL N0RAW-RAD/0.02 DO 2 O 0 LP20-1,100 NIB0X*N(LP20) IF(NIBQX.EQ.O)GO TO 200 0 0 2 1 0 L P 2 1 - 1 .NIBOX 0 0 2 2 0 LP22-1.NORAW PX*X(LP20.LP21) PY-Y(LP20.LP21) R-RAD-(LP22-1)»0.02 CALL PLDT(PX.PY.3) CALL PCIRC(PX,PY,R,0) CONTINUE CONTINUE CONTINUE CALL PLQTND C 99999 PLOT(O.,10..2) PLOT(10..10..2) PL0T(10..0.,2) PL0T(0.,0.,2) STOP ENO Tnree-Diroensiona 1 Rorxloni Particle Deposition Model IMPLICIT REAL •8(A-H.0-2) DIMENSION X( 10O,150).Y(10O, 150),Z( 10O,150), 1 TX(150).TY(ISO).TZ(150).TTX(150).TTY(150).TTZ(150), 2 N(1OO).NZ1(2O).NZ2(20).0GL(10). 3 XXX(150).YYY(150),ZZZ(150). 4 XPX(150).YPY(150) INTEGER FLAGX1.FLAGX2.FLAGX 3,FLAGY1.FLAGY2.FLAGY3. FLAG2A,FLAG2B I COUNT-0 EPS-0.010 EPS1-0.0O5 ZMIN-O.250 DIA-0.50 RAD-0IA/2.0 N0RAWRA0/0.015 IND3«0 NP-0 DO 10 I « 1 . l O O N(I )»0 TX(I)-0.0 TY(I)»0.0 TZ(I)-0.0 TTX(I)-0.0 TTY(I)«0.0 TTZ(I)"0.0 DO 5 J»1,100 X(I.J)«0.0 Y(I,0)"0.0 Z(I.J)»0.0 CONTINUE IF(I.GT.20)G0 TO 10 N2l(I)»0 NZ2(I)»0 CONTINUE C Random number generator C Ini t ialize 10000 XX-PRAND{0.) I»IRAND(O) Y Y"PRAND(O. ) JMRAND(O) C C Generate XX-FPRAND(1.) I*IRAND(25) XX-XX-I C YY-FPRANO(1.) J>IRAN0(25) YY-YY-J C C 4444 C C IF(<XX+RAD).GT.10.0 .OR. (YY+RAD).GT.10.0 .OR. XX.LT.RAD .OR. YY.LT.RA0)G0 TO 1OO0O 22*100.0 ICHEQ1«0 ICHEQ2-0 ICOUNT aICOUNT*1 IFUCOUNT.EQ. 1)G0 TO 90OOO IFtlCOUNT.EO.150O)G0 TO 88888 »«.«-......«»««•««•.--».»»«"..-«:.*"».= •«««-*»*""»«»«•««" = « C Compute the distance between centres (DBC) in the (X,Y) plane C C (I ) Lower level 5555 IF(ZZ.LE.ZMIN)GO TO 556G GO TO 5577 556G IF(ICHEQ1.£0.0 .AND. ICHE02.EO.0)G0 TO 90000 5577 XTEMP-XX YTEMP-YY IF(XTEMP.LT.O.O)XTEMP="0.0 If(XTEMP.GE.10.0)XTEMP=9.5 IF(YTEMP.LT.O.0)YTEMP*0.0 IF(YTEMP.GE. 10.0) YTEMP = 9 . 5 I-XTEMP J"YTEMP JM1=J-1 OP 1-0+1 NP'O C C (a) Check if the particle is at the lowest level C IF{J.EQ.0)G0 TO 123 LL=JM1*10+I LLP2-LL+2 C C (b) Check if the particle is at the left or right eages 112 If(I.E0.0)G0 TQ 112 IF(I.EQ.9)G0 TO 113 GO TO 114 LL'LL+1 GO T D 114 LLP2«LLP2-1 113 C 114 00 20 LP1«LL.LLP2 C C (c) If no particle at a certain slot, do not compute !F(N( LP1 ) . EQ.OGO TQ 20 NNTT »N(LP 1) ICL-0 00 30 LP2-1.NNTT IF(ICHE01-EQ.1 .OR. ICHEQ2.EQ.1)G0 TO 2176 GO TO 2177 2176 IF(ZZ.EG.Z(LP1.LP2))G0 TO 2178 GO TO 30 2177 IF((ZZ-O.O5).LE.Z(LP1.LP2))G0 TO 30 2178 ICL-ICL+1 ICLM-ICL-1 XXX(ICL)-X<LP1,LP2) YYY(ICL)«Y(LP1.LP2) ZZZ(ICL)=Z(LP1.LP2) IF(ICL.EQ.1)G0 TO 21787 DO 44 LP44-1.ICLM IF(DABS(XXX(LP44>-XXX{ICL)).LE.EPS .AND. DABS{YYY(LP44)-YYY(ICL ) ) .LE.EPS .AND. • 0A8S(ZZZ(LP44)-ZZZ<ICL)).LE.EPS)G0 TO 30 44 CONTINUE 21787 DBC"DSQRT((XX-X(LP1,LP2))"«2+(YY-Y(LPt.LP2))««2) IF((DBC+EPS1).LT.DIA)G0 TO 111 GO TO 30 111 NP-NP+1 TX(NP)«X(LPl.LP2) TY(NP)*Y(LPl.LP2) TZ(NP)«Z(LPl,LP2) 30 CONTINUE 20 CONTINUE C C (II) Main level C c 123 ML-J*10+I MLP2-ML+2 C C (a) Check If the particle is at the left and right edges C C IFU ,EQ.O)GO TO 124 IF(I.E0.9)G0 TO 125 GO TO 126 124 ML-ML+1 GO TO 126 125 NLP2"MLP2-1 C 126 00 40 LP 1"ML.MLP2 C C (b) If no particle at a certain slot, do not compute c c C 2276 2277 2278 55 22787 222 IF(N(LP1).EQ.O)G0 TO 40 NNTT-N(LPt) ICM-0 DO 50 LP2«1,NNTT IF (ICHEQ1 . EQ . 1 .OR. ICHEQ2.EQ.DG0 TO 2276 GO TO 2277 IF(ZZ.EQ.Z(LP1.LP2))G0 TO 2278 GO TO 50 IF((ZZ-0.0S).LE.Z<LP1.LP2))G0 TO 50 ICMMCM+1 ICMM=ICM- 1 XXX(ICM)«X(LP1,LP2) YYY(ICM)-Y(LP1.LP2) ZZZ(ICM)-Z(LP1.LP2) IF(ICM.EQ.1)G0 TO 22787 DO 55 LP55«1.ICMM IF(OABS(XXX(LP55)-XXX(ICM)).LE.EPS .AND. DABS{YYY(LP55)-YYY(ICM)).LE.EPS .AND. DABS(ZZZ(LP55)-ZZZ(ICM)).L£.EPS)G0 TO 50 CONTINUE 08C»DSQRT<(XX-X(LP1.LP2))* *2+(Y Y-Y(LP 1,LP2) ) " 2 ) I F((DBC+EPS1).LT.DI A)G0 TO 222 GO TO 50 NP=NP+1 TX(NP)«X(LP1,LP2) TY(NP)»Y(LP1,LP2) TZ(NP)=2(LP1.LP2) CONTINUE CONTINUE 50 40 C C (III) Upper level C c C (a) Check if the particle is at the highest Y level C C IF{J.EQ.9)G0 TO 234 C NL-JP1-10+I NLP2-NL+2 C C (b) Check if the particle is at the left or right edges C 236 IF(I . EQ.OGO TO 236 I F(I .EQ.9)G0 TO 237 GO TO 238 NL »NL+1 GO TO 238 NLP2»NLP2-1 237 C 238 DO GO LP1-NL.NLP2 C C (c) If no particle at a certain slot, do not compute C - c C 2376 2377 2378 IF(N(LP1).EQ.O)GO TO 60 NNTT »N(LP 1 ) ICN=0 00 70 LP2a1.NNTT IF(ICHEQ1.EQ.1 .OR. ICHEQ2.EO.1)G0 TO 2376 GO TO 2377 IF(ZZ.EQ.Z(LP1.LP2))G0 TO 2378 GO TO 70 IF({ZZ-0.05).LE.Z(LP 1 ,LP2) )G0 TO 70 ICN=ICN+1 ICNM-ICN-1 XXX(ICN)-X(LP1.LP2) YYY(ICN)-Y(LP 1.LP2) ' • 66 23787 333 ZZZ(ICN)-Z(LP1.LP2) I F U C N . E O . D G O TO 23787 DO 6 6 L P 6 6 - I . I C N * I F ( 0 A B S < X X X ( L P 6 6 ) - X X X < I C N ) ) . L E . E P S .UNO. OABS(YYY(LP66)-YYY(ICN)).LE.EPS .ANO. D A B S ( Z Z Z ( L P 6 6 ) - Z Z Z ( I C N ) ) . L E . E P S ) G O TO 7 0 CONTINUE O B C - D S O B T ( ( X X - X U P 1 . L P 2 ) ) • ' 2 * ( YY - Y ( L P 1 . L P 2 ) ) • • 2 ) I F ( ( 0 B C . E P S 1 ) . L T . 0 I A ) G 0 TO 3 3 3 CO TO 7 0 NP-NP*1 TX(NP)"X(LP1.LP2) TY(NP).Y(LPt.LP2) 70 60 TZ<NP)-Z(LP1.LP2) CONTINUE CONTINUE C Oeterratne partfcle(s) In the highest Z level (Zl) C (a) Check If there is O or 1 particle below the f a l l i n g sphere C 234 1912 1918 I F ( N P . E 0 . O ) G O TO GO TO 1 9 1 8 IF<ICME01.EO.1 .0 GO T O 9 C O O O IFlNP.EO.O . A N O . IF(NP.E0.0 . A N O . IFtNP.EO.1 . A N O . IFINP.EO.1 . A N O . IF(NP.E0.2 . A N O . IF(NP.E0.2 . ANO . IF(NP.E0.3 - A N O . IF (NP.EO.1)G0 TO GO TO 346 I C H E 0 2 . E O . D G O TO ICHE01.EO.1)G0 ICHE02.EO.1)G0 ICHE01 . EO. 1 )G0 ICHE02.E0.1)G0 ICHE01 . EO. 1 )G0 I C H E 0 2 . E 0 . 1)G0 ICHE01-EO.1JGO 334 TO TO TO TO TO TO TO 1912 1916 1917 1919 1920 1922 1923 1924 ICHEOfO GO TO SS55 ICHE02-0 GO TO S555 ICHEOI'O TTX(1)»TX(1) TTY(1)-TY< 1 ) TTZ<1)-TZ(1) TTX(2)-XB0T TT Y(2)•YBOT TTZ(2)-ZB0T GO TO 1444 ICHE02«0 TTX(1)-TX( 1) TTY(1)"TY( 1) TTZ(1)-TZ(1 ) TTX(2)"XB0T1 TTY(2)-YB0T1 TTZ(2)-ZB0T1 TTX(3)-XB0T2 TTY(3)'YB0T2 TTZ(3)"ZB0T2 GO TO 892 ICHE01 T T X ( 1 ) TX( 1 ) T T Y ( 1) " T Y ( 1 ) T T Z O ) TZ( 1 I TTX(2) TX(2) TTY(2) TY(2) TTZ(2) TZ(2) T T X ( 3 ) XBOT T T Y ( 3 ) YBOT T T Z O ) ZBOT GO T O 8 9 2 ICHE02 TTX(1) •TX(1) T T Y ( 1 )• T Y ( 1 ) TTZ(1) •TZ(1) TTX(2) •TX(2> TTY(2) -TY(2) TTZ(2) •TZ(2) T T X ( 3 ) •XBOT1 T T Y ( 3 ) •YBOT1 T T Z O ) •ZBOT1 TTX(4) •XB0T2 TTY(4 ) -YB0T2 T T Z ( 4 ) ZBDT2 GO T O ICHE01T T X ( 1 )• TX( 1 ) T T Y ( 1 ) • TY( 1 ) T T Z ( 1 )• T 2 ( 1 ) TTX(2)' TX(2) T T Y ( 2 )• T Y ( 2 ) TT2(2>- TZ(2) TTXO)' TXO) TT Y { 3 ) • T Y ( 3 ) TTZO)' TZO) T T X ( 4 ) . XBOT T T Y ( 4 ) YBOT T T Z ( 4 ) ' ZBOT GO TO I 1 C XBOT.TXl 1 ) YBOT-T»( I ) ZBOT.TZ( 1 ) C CALL ONES! DIA, XX, XBOT. YY. YBOT. ZZ , ZSQ T , XT0P2 . YT0P2 ) 334 335 XX*XT0P2 YY«YT0P2 ZZ-ZBOT 1CHEQ1-1 GO TO 5 5 S 5 / 185 C (0) Find 21MAX C c 346 C NPZt»0 NPZ2-0 IND-0 Z1MAX»TZ< 1 ) 00 80 LP3-2.NP IF(TZ(LP3).GT.Z1MAX)Z1MAX=TZtLP3) CONTINUE 80 C C (c) Find number of particles at Z1MAX (NPZl) C C 00 90 LP4»1.NP IF(TZ(LP4).EQ.Z1MAX)G0 TO 444 GO TO 90 444 NPZ1-NPZ1+1 NZKNPZ1 )=LP4 90 CONTINUE C C (d) Check if there is only 1 particle at level ZI c c C 445 C - IF(NPZ1.EO.1 )G0 TO 445 GO TO 446 NTZ1-NZ1(NPZ1 ) XB0T-TX(NTZ1) YB0T«TY(NTZ1) ZB0T-TZ(NTZ1) GO TO 335 C Determine the next highest 2 level (Z2) C C (a) If a l l the particles are at ZI level, then skip checking at Z2 C C 446 IF(NPZ1.EQ.NP)GO TO 778 C C (D) Find Z2MAX 555 100 C 666 DO 100 LP5=1.NP IF(TZ(LP5).LT.Z1MAX)G0 TO 555 GO TO 100 Z2MAX=TZ(LP5) GO TO 666 CONTINUE 00 110 LP6 =1,NP IF(TZ(LP6).GT.Z2MAX .ANO. TZ<LP6).LT.Z1MAX)Z2MAX»TZ<LP6> 1 10 CONTINUE C C (c) Find number of particles at Z2MAX 777 120 00 120 LP7*1.NP IF(TZ(LP7).EO.Z2MAXJG0 TO 777 GO TO 120 NPZ2»NPZ2+1 NZ2(NPZ2)«LP7 CONTINUE C Determine the distance between the two Z levels c C ZDIFF-Z1MAX-Z2MAX C C Assign the particles at ZI level to a temporary array (TT/X/Y/Z) C C 778 DO 130 LP13»1,NPZ1 NT-NZKLP13) TTX(LP13)-TX(NT) TTY<LP13)-TY(NT) TTZ(LP13)-TZ(NT) CONTINUE 130 C C Check if the number of particles at Z1 is greater than 2 c c IF(NPZ1.E0-3JG0 TO 892 IF(NPZ1.E0-4)G0 TO 888 C For 2 spheres oe1ow, oetermlne which sphere has a bigger Y C The Digger Y will be assigned the position of 1 1444 C 989 C 999 C 1010 FLAG2A=0 FLAG26-0" I F(TTY( 1).GT.TTY(2 ) )FLAG2A-1 IF(FLAG2A.EO. ' )G0 TO 999 IF(TTY(1).EQ.TTY(2))G0 TO 1010 XBOT 1»TTX(2 ) XB0T2«TTX( 1 ) YBOT1"TTY(2 ) YB0T2=TTY( 1 ) ZB0T1=TTZ(2) ZB0T2-TTZ( 1 ) GO TO 1111 XBOT 1=>TTX( 1 ) X80T2"TTX(2) YBOT1»TTY(1) YB0T2=TTY{2) ZB0T1-TTZ(1} ZB0T2"TTZ(2) GO TO 1111 IF(TTX(1).GT.TTX(2)JFLAG2B-1 IF{FLAG2B.EO.1)G0 TO 999 GO TO 989 C For the 2 spheres below, determine if the f a l l i n g particle C is actually touching both of them C ----- c 1111 DBC 1 - DSQRT( ( XX-XBOT 1 ) " * 2* ( YY-YBOT 1 ) " 2 ) DBC2-DSQRT((XX-XBOT2)*'2*(YY-YBOT2)**2) IF(DBC1.E0.DBC2)6O TO 1222 AVG-(DBC1+0BC2)/2.0 C " C C 1222 CALL TW0S(AVG,XX.XBOT1.XB0T2.YY.YBOT1.YB0T2.ZBOT1.ZB0T2. XT0P2,YT0P2) XX-XT0P2 YY-YT0P2 CALL TWOS(D£A.XX,XBOT1,XB0T2.YY.YBOT1.Y80T2,ZBOT1.ZBOT2, • XTQP2,YT0P2) C C Assign XT0P2 and YT0P2 as XX and YY and repeat C XX-XT0P2 YY-YT0P2 ZZ-ZB0T1 ICHEQ2-1 C 892 897 893 c GO TO SS5S IF(IN03.E0.1)G0 TO 897 GO TO 893 IN03-0 GO TO 896 CALL THR£ES(DIA,XX.TTX<1).TTX(2).TTX(3).YY.TTY(1).TTY(2).TTY(3). 1 ZZ.TTZ(1 ).TTZ(2).TTZ(3).XT0P2.YT0P2.HEIGHT) XX-XT0P2 YY-YT0P2 ZZ-TTZ(1)+HEIGHT C Check if the f a l l i n g particle will be permanently rested c ........................................................ C FLAGX1-0 FLAGX2-0 FLAGX3-0 FLAGY1=0 FLAGY2-0 FLAGY3-0 C 160 C DO 160 LP16-1,NPZ1 IF(XX.GT.TTX(LP16))FLAGX1»1 IF(XX.LT.TTX(LP16))FLAGX2-1 IF(YY.GT.TTY{LP16))FLAGY1-1 IF(YY.LT.TTY(LP16))F LAGY2- 1 CONTINUE IF(FLAGX1.EQ. 1 .AND. FLAGX2.EO. 1 JFLAGX3-1 IF(FLAGY1.EQ.1 .ANO. FLAGY2.EO.1JFLAGY3-1 IF(FLAGX3.£Q.1 .AND. FLAGY3.EQ.1)GQ TO 90000 IN03-1 C 896 C C 1334 GO TO 5555 DGL(1)-OSQRT((TTX(1)-TTX(2))''2+(TTY(1)-TTY(2 ) ) • "2 ) DGL(2 ) -DSQRT((TTX( 1 )-TTX(3 ) )--2*(TTY( 1)-TTY(3))••2J DGL(3)-0SQRT((TTX(2)-TTX(3))"2+(TTY(2)-TTY(3))"2) IF(DGL(1 ) .GT.DGL(2 ) .ANO. DGL< 1 >.GT.DGL(3))G0 TO 1444 IF(DGL(2).GT.DGL(1) .AND. DGL(2).GT.OGL(3))G0 TO 1334 TTX( 1 )-TTX(2) TTY(i)-TTY(2) TTZ(1)-TTZ(2) TTX(2)-TTX(3) TTY<2)»TTY(3) TTZ<2)-TTZ(3) GO TO 1444 TTX(2)«TTX(3) TTY(2)-TTY{3) TTZ(2)-TTZ(3) GO TO 1444 Determine the biggest Y pos11ion for the case of more than 3 particles at the highest Z level (21) Y81G-TTY( 1 ) 00 140 LP14-2.NPZ1 IF(TTY(LP14).GT.Y8IG)YBIG-TTY(LP14) CONTINUE DO 150 LP15-1.NPZ1 IF(TTY(LP15).EQ.YBIG)NP0S»LP15 CONTINUE IF(NP0S.£Q.1 )G0 TO 889 XB0T2-TTX( 1 ) YB0T2-TTY( 1 ) ZB0T2-TTZC 1 ) IF(NP0S.EQ.2)G0 TO 890 XB0T3«TTX<2 ) Y80T3«TTY(2) ZB0T3"TTZ<2) IF(NP0S.EQ.3)G0 TO 891 XS0T4-TTX(3 ) YB0T4-TTY(3 ) Z80T4=TTZ(3) XB0T1-TTX(4) YBOT 1-TTY(4) ZB0T1-TTZ(4> GO TO 894 XBOT 1 -TTX( 1 ) YBOT 1 »TTY{ 1 ) ZBOT 1«TTZ< 1 ) XB0T2-TTX(2) V80r2-TTV<2) ZBOT2-TTZ(2) C XB0T3-TTX(3) YB0T3»TTY(3) ZBOT3*TTZ(3) C XB0T4-TTX(4) YB0T4*TTY(4) ZB0T4-TTZ(4) GO TO 894 C 890 XB0T1»TTX(2) YBOT 1 »TTY(2 ) ZB0T1»TTZ(2) C XB0T3-TTX(3) YBDT3-TTY(3) ZB0T3-TTZ(3) C XB0T4-TTX(4) YB0T4 »TTY(4 ) ZB0T4-TTZ(4) GO TO 894 C 891 XB0T1»TTX(3) YBOT1-TTY(3 ) ZB0T1-TTZO) ' C C 894 C 895 C XB0T4»TTX(4) Y80T4»TTY(4) Z80T4-TTZ(4) GO TO 894 • CALL FOURS(DIA,XX,XBOT1,XBOT2,XBOT3,XBOT4.YY.YBOT1.Y80T2, Y80T3.YB0T4,XB0T2.YB0T2.HEIGHT) XX-XT0P2 YY-YT0P2 ZZ-TTZ(1)•HEIGHT C Recording the permanent X.Y.Z values for the settled particle c .».....».«.««*...«...-............<..............«**>..«...... C 90000 XTEMP-XX YTEMP•YY IF(XTEMP.LT.O.O)XTEMP-0-0 IF(XTEMP.GE.10.0)XT£MP-9.5 IF(YTEMP.LT.0.0)YTEMP»00 IF(YTEMP.GE.10.0)YTEMP-9.5 I-XTEMP J-YTEMP IB0X»(J*1O)+I+1 C ISIG-0 NUN«N(IBOX) IF(NUN.EQ.O)G0 TO 6547 00 300 LP30-1.NUN RES-DS0RT((XX-X(IBOX,LP3O))**2+(YY-Y(IBOX.LP3O))*"2) IF(RES.LT.DIA)ISIG-1 300 CONTINUE IF(ISIG.EG.1)G0 TO 10OOO 6547 N(IBOX)»N(IBOX)*1 NUM-N(IBOX) C X( I BOX,NUM)-XX Y( IBOX,NUM)=>YY C IF(ICOUNT.EO.1 .OR. NP.EO.O)GO TO 90001 Z(IBOX.NUM)*ZZ GO TO 1OOO0 C 90001 Z(IBOX,NUM)«ZMIN GO TO 10OO0 C 888S8 CALL PLOT(0..0..3) CALL PL0T(0..IO..2) CALL PL0T(10..10..2) CALL PL0T(10..0..2) CALL PLOT(0..0..2) C DO 200 LP20-1.100 NIB0X=N(LP20) IF(NIB0X.E0.0)G0 TO 200 IC-0 00 210 LP21"1.NIBOX IF(Z(LP20,LP21).NE.ZMIN)G0 TO 210 ICMC+1 ICM-IC-1 XPX(IC)»X(LP20,LP21) YPY(IC)-Y(LP20,LP21) IF<IC.EO.1)G0 TO 434 INP-0 DO 546 L546-1.ICM RE5"DS0RT((XPX(L546)-XPX(IC))•*2+ (YPY(L546)-YPY(IC))««2) IF(RES.LT.DIA)INP-1 546 CONTINUE IF(INP.EO.1)G0 TO 210 434 00 220 LP22-1.NDRAW PX-XPX(IC) PY»YPY(IC) R»RAD-(LP22-1)*0.02 CALL PLOT{PX.PY.3) CALL PCIRC(PX.PY.R.O) 220 CONTINUE 2 10 CONTINUE 200 CONTINUE CALL PLOTND C 99999 STOP END C C c C C • * SUBROUTINES • * c C 100 C C 40 C C 45 C C C C C C 20 C 10 C C 30 C 50 C 70 C 60 C 80 C 999 C C C SUBROUTINE ONEStDIA,XTOP1,XBOT,YTOP1.YBOT,ZZ,ZBOT,XTOP2,YTQP2) IMPLICIT REAL •8(A-H.0-Z) DELX-XTOP1-X80T DELY"YTOP 1 - YBOT IF(DELX.E0.O .AND. DELY.EQ-0)GO TO 40 GO TO 45 I-IRAND(O) I«IRAN0(4) IF(I.EO.1)XT0P1"XTOP1+0.1 IF(I-EO.2)YTOP1-YTOP1+0.1 IF(I.EO.3)XTOP1»XTOP1-0.1 IF(I.EO.4)YTOP1"YTOP1-0.1 GO TO lOO IF(OELX.EO.O)GO TO 50 IF(DELY.E0.O)GO TO 60 SLOPE"DELY/DELX C*YTOP1-SLOPE"XTOP1 IF(SL0PE.GT.O)GO TO 10 XYO--C/SLOPE DI »DSQRT ((YTOP 1 - YBOT) " 2 * ( XTOP 1 -XBOT ) * *2 ) 0=OSORT(YBOT«*2+(XBOT-XYQ)*«2) IF(DELX.LT.0)GO TO 20 XT0P2-XY0+((0-0IA)'(XB0T-XY0)/D) YT0P2-(0-0IA)«YT0Pl/D GO TO 999 XT0P2-XY0+((DIA+0)«(XT0P1-XY0)/(DI+D)) YT0P2=((DIA+D)*YT0P1)/(DI+D) GO TO 999 YXO=C D"DSQRT(XB0T«*2+( YB0T-YX0)"2 ) IF(DELX.LT.O)GO TO 30 XT0P2»(XBDT-(D+DIA))/D YT0P2-{(YB0T-YX0)*(D*DIA)/D)+YX0 GO TO 999 ' XT0P2=XB0T"(0-0IA)/D YT0P2»((YBOT-YXO)"(D-DIA)/0)+YXO GO TO 999 IF(YTOP 1 .GT.YBOT)G0 TO 70 XT0P2-XT0P1 YT0P2*YB0T-0IA GO TO 999 XT0P2-XT0P1 YT0P2-YBOT*DIA GO TO 999 IF(XTOP1.GT.XBOT)G0 TO 80 YT0P2-YT0P1 XT0P2-XBOT-OIA GO TO 999 YT0P2-YT0P1 XT0P2-XB0T+DIA RETURN END SUBROUTINE TWOS(DIA.XTOP1.XBOT1.XBOT2.YTOP1.YBOT 1 .YB0T2. ZBOT1,ZB0T2,XT0P2.YT0P2) IMPLICIT REAL •8(A-H.0-Z) C C Common equations for a l l cases C EPS-0.005 C IF(OABS(XBOT 1-XB0T2).LE.EPS•OR.OABS(YBOT1-YBOT2).LE.EPS)G0 TO 40 C SLOPE"-(XBOT1-XBOT2 )/< YBOT1-YBOT2) XE»(XB0T1+XB0T2)/2.0 YE»(YB0T1+YB0T2)/2.0 C-YE-SLOPE-XE C IF(YTOP1.LT.YBOT 1 )GO TO 10 C C Equations for Cases 1 anO 2 C XI-XE YI-YB0T2 XYB2=(YB0T2-C)/SL0PE XIL-0ABS(XYB2-XI ) XEL-DSQRTUYE-YI )**2 + (XYB2-XI )-«2) ADL"DIA ABL "DI A 8EL»DSORT((YBOT1-YB0T2)"2+(XBOT1-XB0T2)'«2)/2.O 0IF"ABL*-2-BEL"2 IF(0IF.LT.0.0)DIF-0.0 AEL»DSORT(DIF ) XAL»AEL+XEL BDL=OSORT((YB0T1-YB0T2)*"2+(X80T1-XB0T2)-*2) EIL"O.5»(YB0T1-YB0T2) AA1-EIL*XAL/XEL A1XL«XIL*AA1/EIL C YTOP2"YBOT2*AA1 C IF(XTOP1.GT.XBOT1)G0 TO 20 XTOP2-XY82-A1XL GO TO 999 C 20 XTOP2"XYB2+A1XL GO TO 999 C C Equations for Cases 3 and 4 C 10 YXB2-C+SL0PE-XB0T2 B1B2-DSQRT((XBOT1-XBOT2)"*2+(YBOT1-YB0T2 ) **2 ) EB2-B1B2/2.0 FB1-DABS(XBOT1-XB0T2 ) EI-FB1/2 .0 C XI-XB0T2 Y I •¥E BlT2=DIA B2T2-DIA C DIF-B1T2--2-EB2*-2 IF(01F.LT.0.0)OIF«0.0 ET2-DS0RT(0IF) EYXB2»0SQRT( ( YXB2-YI )«»2+EI "2) T2YXB2-ET2+EYXB2 C C New Y for Case 3 and 4 C YT0P2*YXB2-((YXB2-YI)•T2YXB2/EYXB2) C IF(XTOP1.LT.XB0T2)G0 TO 30 C C New X for Case 3 C XT0P2«((XE-XI)"T2YXB2/EYXB2)+XB0T2 GO TO 999 C C New X for case 4 C 30 XT0P2»XB0T2-((XE-XI)*T2YXB2/EYXB2) GO TO 999 C 40 IF(DABS(XB0T1-XB0T2).LE.EPS)G0 TO 50 C SO XT0P2-(XB0T1+XB0T2)/2.0 DIF-DIA"2-(XB0T1-XT0P2)**2 IF(DIF.LT.0.0)DIF-0.0 YL"DSQRT(DIF) IF(YT0P1.GT.Y80T1)G0 TO 100 IF(YTOP1.EQ.YBOT1)G0 TO 110 120 Y T0P2"YT0P1-YL GO TO 999 C 100 YT0P2-YT0P1+YL GO TO 999 C 110 I-IRAND(O) I=IRAN0(2) IF(I.EQ.1>G0 TO 100 GO TO 120 C 50 YT0P2•(YBOT1+YB0T2)/2.0 DIF=0IA»«2-(YB0T1-YT0P2)*-2 ]F(DIF.LT.O-0)OIF-0.0 XL-DSQRT(OIF) I F(XTOP1.GT.XBOT1)G0 TO 60 IF(XT0P1.E0.XB0T1)GO TO 70 C 80 C 60 C 70 C 999 C c C C C 10 C 20 C 30 40 C 50 C 60 70 C 80 XT0P2-XB0T1-XL GO TO 999 XTOP2*XBOT1+XL GO TO 999 I*IRAND(0) I•I RAND(2) IF(I.EO.1)G0 TO 60 GO 70 SO RETURN END SUBROUTINE THREES(0IA,XTOP1,XBOT1.XB0T2,XB0T3,YTOP1.YBOT1, 1 YBOT2,Y80T3,ZTOP1.ZBOT1,ZB0T2,ZBOT3.XT0P2.YT0P2,HEIGHT) IMPLICIT REAL •8(A-H,0-Z) DIMENSION AD(3),HT(3),XB0T(3),YB0T(3) EPS-0.005 ID12-0 ID13-0 ID23-0 IF(0ABS(YB0T1-YB0T2).LE.EPS)G0 TO 10 GO TO 20 ID12»1 GO TO 30 S 1 2"-(XBOT1-XBOT2)/(YBOT1-YBOT2) XMP12-(X80T1+XB0T2)/2.0 YMP12»(YBOT1+YBOT2)/2.0 C12«YMP12-(S12*XMP12) IF(DABS(YB0T2-YB0T3).LE.EPS)G0 TO 40 GO TO 50 ID23-1 GO TO 60 S23--(XB0T2-XB0T3)/(YB0T2-YB0T3) XMP23-(X80T2+XB0T3)/2.0 YMP23=(YB0T2+YB0T3)/2.0 C2 3=YMP23-(S23-XMP23) IF(DABS(YB0T1-YB0T3).LE.EPS)G0 TO 70 GO TO 80 ID13-1 GO TO 90 S13--(XBOT1-XBOT3)/(YBOT1-YBOT3) XMP13"(XBOT1+XBQT3)/2.0 YMP13-(YBOT1+YBOT3)/2.0 C13-7MP13-(S13"XMP13) IF(1013.EQ.1)G0 TO 1O0 IF(1012.EO-1)G0 TO 110 IFU023.EQ. 1)G0 TO 120 / 191 C 100 C 110 C 120 C 888 C IF(S12.EQ.S23 .ANO. ID13.NE.DG0 TO 110 XT0P2-(C23-C12)/(S12-S23) V TOP 2 »(512•XT0P2) *C 12 GO TO 888 IHS13.EQ.S23 .ANO. 1012.NE.DGO TO 120 XT0P2«(C23-C13)/(S13-S23) YT0P2-(S13-XT0P2)+C13 GO TO 888 XT0P2-<C12-C13)/(S13-S12) YT0P2-(S13'XT0P2)+C13 XBOTI D-XB0T1 XB0T(2)-XB0T2 XB0T(3)-X60T3 YB0T( 1 ) "YBOT t YB0T(2)"YB0T2 YBOT(3)»YBOT3 SUM«0.0 DO 2345 L-1 .3 AD(L)-(XTOP2-XBOT(L))--2+(YT0P2-YBOT(L))**2 DIFF-DIA«"2-A0(L) IF(DIFF.LT.O)DIFF-0 MT(L)«DS0RT(DIFF) SUM«SUM+HT(L) 2345 CONTINUE C HEIGHT-SUM/3.0 C RETURN END C C SUBROUTINE FOURStDIA.XTOP1,XBOT1.XBOT2.XBOT3.XB0T4, 1 YTOP1.YBOT t,YB0T2.YBOT3.YB0T4. 2 XT0P2.YT0P2.HEIGHT) C IMPLICIT REAL *8<A-H.0-Z) DIMENSION XB0T(4) .YB0T(4),HT(4).DBC(3,A).AO(4) C EPS-0.005 I04A-0 ID4B<0 C XBOTC1)"XBOT1 XBOT(2)-XBOT2 XB0T(3)-XB0T3 X60T(4)»XB0T4 Y80T( 1 ) -YBOT 1 YB0T(2)"YBOT2 YB0T(3)*Y80T3 YB0T(4)-YB0T4 C C Compute the distance between centres(DBC) for the 4 bottom C spheres and also determine the largest DBC C 06IG=0.0 NL-1 DO 10 L1-1.3 NL-NL+1 DO 20 L2-NL.4 DBC(L1.L2)"DS0RT((XB0T(L1)-XB0T(L2))'*2+ 1 (YB0T(L1>-YB0T(L2))-*2) IF(DBC(L1,L2) .GT,DBIG)GO TO 10OO GO TD 20 1O0O 0BIG-0SC(L1,L2) ID1-L1 I02-L2 20 CONTINUE 10 CONTINUE C C Determine the next largest DBC C ........=».. C DBIG2-0.0 NN- t 00 30 L3"1,3 NN-NN+1 00 40 L4=NN.4 IF(DBC(L3.L4).GT.DBIG2 .ANO. DBC(L3,L4).LT.DBIG)G0 TO 200 GO TO 40 200 0BIG2»DBC(L3,L4) ID3-L3 ID4-L4 40 CONTINUE 30 CONTINUE C IF(OABS(YBOT(ID1)-YBOT(ID2)).LE.EPS)GO TO SO GO TO 70 60 ID4A-1 GO TO 80 C 70 S1*-(XB0T(ID1)-XB0T(ID2))/(YB0T(ID1)-YB0T(ID2)) 80 XBOTI1-XB0T(101 ) XBOTI2-XBOTUD2) YBOTI1-YB0T(101) YB0TI2"YBOT(ID2) XMP1•(XBOTIl+XBOTI2)/2 .0 YMP1•(YBOTI 1+YBOTI 2)/2.0 C 1 F( ID4A.EO- 1 )G0 TO 90 C1»YMP1-(S1*XMP1) C 90 IF{DABS(YBOT(ID3)-YB0T(104)).LE.EPS)G0 TO 100 GO TO 110 100 ID4B-1 GO TO 120 C 1 »0 S2--(XBOT(ID3 ) -X80T(104))/(Y80T(ID3) -Y80T<104)) 120 XB0TI3-X80T(ID3) X80TI4-X80TUD4) YB0TI3-YB0T(103) Y80TI4-YB0T{104) XMP2-(XB0TI3+XBOTI4)/2.0 C 130 C C 140 C 150 YMP2-(YB0TI3+YB0TI4 )/2 .0 IF(ID46.EQ.1)G0 TO 130 C2-YMP2-(S2"XMP2) IF(I04A.EO.1)G0 TO 140 IF( ID4B.E0 . l)GO TO 150 XT0P2-(C2-C1)/(S1-S2) YT0P2-(S1-XT0P2)+C1 GO TO 868 YT0P2-(S2"XMP1>*C2 XT0P2*XMP1 GO TO 688 YT0P2»(S1"XMP2)+C1 X T0P2"XMP2 C C 3 . . , . . . B . . . B a . . - . - « = . e . . . . . e C Determine the 2 position of the settled particle C .-........-...**-*........».*......*--.....*«... c 888 50 C C SUM-0.0 DO 50 L5-1.4 AD(L5)»(XT0P2-XB0T(L5))-*2+(YT0P2-YB0T(L5)) DIFF"DIA*'2-AD(L5) IF(0IFF.LT.O)DIFF»O MT<L5)-DS0RT(DIFF) SUM-SUM+HT(L5) CONTINUE MEIGHT-SUM/4.0 RETURN END C C C C C C C C C C C C C C c of total potent** energy for the Interaction between two spheres in an queous medium. Calculation Equations from thesis used tn the calculation: ( i ) [2.19] for electrical double layer interaction ( t i ) 12.40] for van der Waals Interaction Uni ts used: Length cm Energy ergs Temp • K Potent i a I statvolt C Parameters to be changed: PSI1,PSI2,KAPPA C IMPLICIT BEAL-e(A-H.O-Z) REAL *8 KAPPA.TOTAL(80),EDL(80).VDW(80).HH{80).KT COMMON/BLKA/PSI1.PSI2,KAPPA EXTERNAL FUNC1,FUNC2 C BOLT2-1.38066244D-16 TEMP-298. IS KT«BOLTZ*TEMP STATV-3.33566830-6 C PSI1--75.83-STATV PSI2"-75.83*STATV KAPPA- 181818 . 1818 C HI-1.00-5 H-HI DO 10 LP 1-1.80 HH(LP1)-H*1.D4 £DL(LP1 )-FUNCKH) VDW(LP1)-FUNC2(H) TOTAL(LP 1 )-(EDL(LP1)+V0W(LPl))/KT H«H+(HI/5. ) 10 CONTINUE C WRITE(6.2000JKAPPA.PSI',PSI2 2000 F0RMAT(/2X.'KAPPA -'.613.6.* CM'/2X.'PSI1 - ' , £ 1 3 . 6 , ' STATV'/ • 2X,'PSI2 - ' . E 1 3 . 6 . ' STATV//6X. 'H(UM) ' . 10X, 'EOL(KT) ' P ,7X.'VDW(KT)'.8X.'TOTAL(KT)'/) C DO 20 LP2-1.80 WRITE(6.1OO0)HH(LP2),E0L(LP2),V0W(LP2).TOTAL(LP2) 10O0 F0RMAT(/2X,E13.6.2X.E13.6,2X.E13.6.2X.E13.6) 20 CONTINUE C STOP END C C ....................... C electric double layer Interaction energy C C DOUBLE PRECISION FUNCTION FUNCl(H) C IMPLICIT REAL*8(A-H,0-Z) COMMON/BLKA/PSI1.PSI2,KAPPA REAL'S KAPPA C EPS-78.54 A-5.D-5 B-5.D-5 C TEl-(EPS-A*S"(<PSI1-'2>*<PSI2"2)))/(4'(A+B)) TE2-(2'PSI1"PSI2)/((PSI1**2)+<PSI2••2)) TE3-DL0G((1+DEXP(-KAPPA*H))/(1-DEXP(-KAPPA*H))) TE4»DL0G(1-DEXP(-2"KAPPA*H)) C FUNC1«T£1*((TE2*TE3)-t-TE4) C RETURN END C C .............. . =C............. S 3 C van der Waals interaction energy c c C C C C C C C C ................................ DOUBLE PRECISION FUNCTION FUNC2(H) IMPLICIT R£AL*8(A-H,0-Z) REAL *8 LAMBDA,IS 1,IS2,IS3,IS4.IS5,ISS.HAMAKE HAMAKE-0.63920-13 .BOLTZ-1.38050-16 TEMP-298.15 A-5.0-5 B-5.D-5 LAMBDA-1.0-5 RH0-0.477464829*LAMBDA AA«1.01 BB-O.879645943/LAMB0A CC =0.38992961'LAMBDA DD-0.051673803'LAMBDA*LAMB0A RHOP-RHO-H ALPHA-(AA/(2*RH0**2))-(BB/RHO)-{CC/( 3*RH0**3)) +(00/(4.*RH0**4 BETA 3(-2*AA/(3*RH0**3)) + (BB/(RHO* *2)) + (CC/(2-RHO* *4)) O -(2*DO/(5*RH0**S)) GAMMA-{AA/(4-RH0**4))-(BB/(3•RHO * * 3))-(CC/{5 *RHO* *5)) » +(DD/<6*RH0«*6)) Q-2-(2*B+A+H) R«(H+B)*(H+2*A+2*B) H2A2B-H+2*A+2*B IF(RH0.GT.(H+2«A))GO TO 1000 C C Calculate IS1 C T1-DLDG(((H*2*A)«(H+2*B))/(H"(H+2*A+2*B))) T2-(A+8)'(( 1 ./(H+2*A*-2*B) )-( 1 ./H) ) T3»(A-B)*(( 1./(H+2*6) )-(1./(H+2*A ) ) ) T4-(A-B>-(( 1 ./(H-H) l * l l . / I H * ! ' » l " I W l . / I H * l ' « l " l l + <1./<H+2-A+2-B)"2>) T5«<1./H)-( 1./(H+2 - A)1 -(1./(H+2-B))»<)./(H+2-A+2-8)) T6»(A+Bt-(( l . / I W i ' W l ] " ] ) - ! I./(H-H)I) T7>< A-B)-( < I ./(H+2-B1--2 )-( 1 ./(H+2 • A ) • -2 ) ) T8-2-A-B-((1./(H-H-H))+<1./(H+2-A)•-3)•(1./(H+2-8)--3) O • ( 1 ./(H+2-A+2-BI--3)) • C C IS1-((CC/30.)-<T1+T2*T3+T4))-((D0/180.)•(T5+T6+T7+TB)) IF(RH0.GT.H)G0 70 3000 ISS-IS1 GO TO 9999 C C Calculate IS2 C 3000 T9-ALPHA-RH0P-RH0P-(A-RH0P/3. ) T10»BETA-((RH0P"RH0P-A-(B+H)) + ((RH0P-'3)•(2-A-B-H)/3. ) • -((HH0P'-4)/4.)) T11«RHOP-RHOP-H-A-(2«8+H) T12-(RHaP-"3)-((4-A*(H+8))-(H-(2*B+H)))/3. T13-((RH0P--4)-(A-B-H)/2.)-((RHOP-"S)/S.) T14-RH0P-(2-(A+B+H)'DL0G(RH0/H)) T15=H-(2-A*H)"((1./H)-(1./RHO)) T1G-2-( A + H W (1./H)-(1./RHO)) T17»(H-l2-A+H)/2.)•((1./(H-H))-(1./(RHO•RHO))) T18-(((RHO-RHO)-(H-H))/2.)-2-(A+H)-RHOP T19«((H-H)+2-H-<A»B)+2-A-S)-DL0G(RH0/H) T20-B-(-RHOP-(H-(2-A+H)-<(1./H)-( 1./RHO)))) T2 1-DL0G(RHO/H)-(2-(A«H)-( ( 1 ./H)-( 1 ./RHO))) T22-(H-(2-A+H)/2.)•((!./(H*H))-(1./(RHO'RHO))) T23--((1./H)-<1./RHO))«((A+H)-<<1./<H-H))-(1./(RHO-RHO)))) T24.(H-(2-A»H)/3. )•( ( 1 ./(H-H-H))-( 1 ./(RH0--3) ) ) T25>((1./H)-(1./RHO))-(A+H)-(<1./(H-H))-(!./(RHO'RHO))) T2G-(H-(2-A*H)/3. )•( ( 1 ./(H-H-H)) - ( 1 ./(RHO--3 ))) T27-O.SO-((1./(H-H))-(1./(RHO-RHO))) T28»(2-(A+H)/3.)•((1./(H--3))-(1./(RHO--3))) T29-(H-(2-A+H)/4.)•((1./(H--4))-(1./(RHO••4))) C IS2=IS1*T9+T10*GAMMA-(T11+T12+T13) » +(AA/12.)-(T14+T15+(2-B-(T16-T17))) • -(BB/3.)-<T18+T19+T20)-(CC/30.)-(T21+T22 » *(3"B)-(T23-T24)) • +(00/6O.)-(T2S»T26-(4-B)-(T27-T26+T29)) C ISS-1S2 GO TO 9999 C C Calculate IS3 C T30«(4-(A--3)/3)-(ALPHA+BETA-(A+B+H)) T31-(4-A-A-A-GAMMA/3)-((H-H+2"H-(A+B)+A-B)•(6-A-A/5)) T32»4-A-2-( A+B+H )-DLOG((H+2-A)/H) T33-2-A-B-((1./H)*<1./(H+2-A)I) T34.(H-H+2-H-(A+B)+2-A-B)-DL0G((H+2-AJ/H) T35-2-A-(A+2-6-H) T36-DL0G((H+2'A)/H)-A-((1./H)+<1./(H+2-A))) T37.(- 1./H).<1./(H+2-A)) T38-A-((1./(H-H))*(1./((H*2'A)--2))) T39-(1./H)-(1./(H+2•A)) T40-A-((l./(H-H>) + (l./((H+2-A)"2))) T41-(-1./<H-H)) + (1./((H+2-A)"2)) T42-2-A"((1./(H-'3))+< 1 ./((H+2-A)•-3)) ) C • • IS3-IS1+T30+T31 + UAA/12.) •(T32+T33))-((BB/3.)-<T34-T35)) -((CC/30.)-(T36+B-(T37+T38>)) *((00/180.)-(T39-T40*B-(T4 1+T42))) C C Calculate IS4 C 1000 T43-( A+B+H )-DLOG( (H' (H+2-A+2-B))/ ((H+2- A ) • (H+2-B )) ) T44.A-B"((1./H)+(1./(H+2-A))•(1./(H+2-B))•(1./(H+2-A+2-B))) T45-(H-H+2-H-(A+B)+2-A-B)-0L0G(((H+2-A ) • (H+2-8) ) e /(H-(H+2-A+2-B)))-4-A-B C IS4=«(AA/6.)-(T43+T44)-(8B'T45/3.) C IF(RH0.LT.H2A2B)G0 TO 2000 ISS-IS4 GO TO 9999 C C Calculate IS5 C 20OO T46-ALPHA•(((4-A--3J/3.)-((RHOP-2-B)•-2)•(A-((RH0P-2-B)/3.))) T47-((4.•(A--3) )/3.)-(A+B+H)+(((RHOP-2-B)*-4)/4. ) T48»(((RHOP-2-B)--3)/3.)•(2•A-B-H)-(((RH0P-2-B)•-2)•A-(B*H)) T49-((4."(A--3))/3.)•((H-H)*(2-H-(A+8))•(2-A-B) +(6-A-A/5)) T50"((RHOP-2-B)--2)'A-H-(2-B+H) T51-(((RH0P-2-B)--3)/3.)*((4*A*(H+B))-(H"(2-B+H))) T52M(((RH0P-2-B1--4)/2.)•( A-B-H ))•((( RH0P-2-B )•-5 )/5.) T53-( -2 . • (A+B+H) *DLOG(H2A2B/RH0)) +H2A2B-RH0 e »R-((1,/RHO)-(1./H2A2B)) T54-2-B-(-0*((l./RHO)-(1./H2A2B)) • +0.50-R-< <1./(RHO--2))-<1./(H2A2B--2) )) I T55=(<(H2A2B--2)-(RHO--2))/2)-<0-(H+2•A+2-B-RHO)) T56-((H-H)+2-H-(A+S)+2-A-B)-DL0G(H2A28/RH0) T57-B-(H2A2B-RH0+R-((1./RHO)-(1./H2A2B))) T58-DL0G(H2A2B/RHO)-0-((1./RHO)-(1./H2A2B)) T59-0.5-R"((1./(RH0--2))-(1./(H2A2B--2))) T60-3-BM(1./RHO)-(1./H2A2B)-0.5-0-((1./(RHO--2)) «• -( 1 ./(H2A28--2)) >+0.333-R- (( 1 ./(RHO--3)) • -<l./(H2A2B--3)))) T61-(1./RHO)-(1./H2A2B)-O.5-0'((1./(RHO--2))-(1./(H2A2B--2))) • +0.333-R-((1./(RHO--3))-(I./(H2A28•-3))) T62-4-B-(0.5'((1 . / ( R H 0 " 2 ) ) - ( 1./(H2A28--2))) » -0.333-0-((1./(RHO--3))-(1./(H2A2B-•3))) » +0.25-R-I( 1./(RHO--4))-(1./(H2A2B--4) )) ) C IS5-IS4+T46+8ETA-(T47-T48)+GAMMA-(T49-T50-T51-T52) » -(AA/12. )•< TS3+T54)-0.333-B-(T55+T56 + T57) • -(CC/30.]-(T58+T59+T60)+<00/60.)-(T61+T62) C I5S-IS5 C 9999 FUNC2-(-HAMAKE-ISS)/(A+B+H) C RETURN END / 195 C C C C C C C C 111 100 ELECTRIC DOUBLE LAYER AND KAPPA-A CALCULATION REAL KA.MINUS.MPLUS.MMINUS.K.KAPPA REAL MH.MOH.MK.MCL.MPLUS.MHINUS UH=36.3E-4 UOH=20.SE-4 UNH4=7.6E-4 UHC03M.6E-4 UCL-7.91E-4 UK«7.61E-4 UNA-5.19E-4 UPLUS-5.614E-4 UMINUS-6.255E-4 MH=.03676 MOH-.06495 MK".1749 MCL-.1685 MPLUS*.2372 MMINUS-.213 A=0.50 AVQ=6.023E23 EL"1.6023E-19 WRITE(6.100) FORMAT(1X,'INPUT PH VALUE : FORMAT XX.XX'/ 1 IX.'TO QUIT. TYPE 0.00'/) REA0(5.1000)PH lOOO FORMAT(G5.2) IF(PH.EO.O.O)GO TO 999 C WRITE(6,101) 101 FORMAT<1X.'INPUT CONDUCTIVITY VALUE : FORMAT O.OOOXXXX'/) READ(5.2OOO)C0N0 2000 F0RMAT(G13.8) C WRITE(6,105) 105 FORMAT(IX,'INPUT KCL CONCENTRATION : FORMAT O.XXXXXXX'/) READ(5.2005)K 2005 F0RMAT(G13.8) CL=K C H-10**(-PH) OH- 10" ( - 14 .0)/H PLUS-((lOOO.O-COND/tAVO-EL))-(H*UH+0H»U0H+(H+K-CL-OH) 1 •UMINUS+K*UK+CL'UCL))/(UPLUS+UMINUS) C IF(PLUS.LT.O.O)PLUS=0.0 CQNC*PLUS+H+K MINUS»CONC-OH-CL C KAPPA-O.3286E8'(C0NC*»O.5) E0L-100O0.0/KAPPA KA=KAPPA*A/10000.0 C XMP-< (H«MH) + (K*MK)*(PLUS*MPLUSn/CnNr. XMI«<(OH-MOH)+<CL-MCL)+(MINUS"MMINUS))/C0NC C WRITE(6.102)CONC.H.K.PLUS.OH,CL.MINUS.EDL.KA,XMP.XMI 102 FORMAT(IX,'CONC -'.G13.6// 1 'H •'.G13.6/ 2 'K •'.G13.6/ 3 'CATIONS •'.G13.6// 4 'OH -'.G13.6/ 5 * CL •'.G13.6/ 6 'ANIONS -'.G13.6// 7 'THE ELECTRIC DOUBLE LAYER THICKNESS - ' . F 9 . 3 . ' UM'/ 8 'KA .G13.6// 9 'M+ •'.G13.6/ 1 'M- .G13.6/) C GO TO 111 C 999 STOP END
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Deposition of colloidal spheres under quiescent conditions Tan, Chai Geok 1987
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Title | Deposition of colloidal spheres under quiescent conditions |
Creator |
Tan, Chai Geok |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | The phenomenon of deposition (or release) of fine particles or other microscopic species, suspended in a liquid, onto (or from) a foreign substrate surface plays a critical role in many natural and industrial processes. Traditionally, the analysis of this phenomenon has been conceptually divided into two steps — the transport step and the adhesion step. Attempts to understand the role of the adhesion step on the overall deposition process under most practical situations are complicated by the presence of a large number of interdependent parameters such as double layer thickness, particle and wall zeta-potential, particle size and flow, amongst others. Thus, as a first step towards gaining a better understanding of the phenomenon, an experimental study of a very simple deposition system, where only the random nature of the deposition process and the double layer interactions between deposited particles are important, was undertaken. In this idealized system, a stable suspension of monodispersed, negatively charged colloidal silica spheres one micron in diameter, suspended in an aqueous medium in a specially constructed deposition cell, were allowed to settle by gravity and be deposited permanently onto a cationic polymer-coated glass cover slip. The magnitude of surface potential was altered by adjusting the pH of the suspension using NaOH and HC1, while the electrical double layer thickness was varied by dissolving different predetermined quantities of KC1 into the suspension. The results showed that the trends in the experimental surface coverages obtained were in accordance with expectation in that as the double layer thickness, 1/К, or the particle zeta potential, ζ⍴’ , increased (leading to an increase in the interaction energy between the particles), the surface coverage decreased. Furthermore, the extent of surface coverages obtained when both 1/К and ζ⍴ were changed was found to be greater than that when 1/К alone was used as the controlling variable. A separate series of studies examining the effect of substrate double layer thickness on surface coverage was also performed by dissolving different predetermined quantities of K₃PO₄, into the suspension so that the substrate and the particles could differ in their respective double layer thicknesses. The results of surface coverages obtained in this study showed that the influence exerted by the substrate double layer was negligible. Besides these findings, the presence of geometric exclusion due to the random nature of the deposition process was also noted, although its effect was difficult to quantify. Besides the systematic experimental study of colloidal deposition, attempts were made to develop two computer simulation models to generate deposition prediction which could be compared with results measured experimentally. The first scheme involved a two-dimensional simple rejection model where only non-overlapping particles were deposited, while the second scheme consisted of a three-dimensional model where the rolling of sedimenting particles over the surfaces of previously-deposited particles as well as the stacking of particles were allowed. Comparison of experimental results with those obtained using the two-dimensional model revealed that for all cases, the simulated results consistently underpredicted the experimental results due to the oversimplifying nature of the simulation. The trends in the experimentally obtained results, however, were approximated by the simulated results. Owing to its very complex nature, successful completion of the three-dimensional model simulation did not materialize. It is expected, however, that when such a model is successfully completed, it will yield predicted results which are in better quantitative agreement with those measured experimentally. Besides the above, a separate study examining the effects of reaction temperature and the types of alcoholic solvent used on the properties of silica particles produced was also performed. This study led to the development of a novel method in which dispersed, uniform-sized, spherical silica particles in the size range of 0.2 to 2.0 µm can be produced by simply varying the reaction temperature and the type of alcoholic solvent used. |
Subject |
Colloids |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0058839 |
URI | http://hdl.handle.net/2429/26744 |
Degree |
Master of Applied Science - MASc |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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