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Deposition of colloidal spheres under quiescent conditions Tan, Chai Geok 1987

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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. 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(ed.),  Surface  Vol. 8, pp. 3-84, Wiley Interscience (1976)  and  Israelachvili, J.N.,  169-187, Elsevier, Amsterdam (ed.), Polymer  Colloids  and  Surfaces,  Vol. 2, pp.  (1981)  Colloids,  86.  Fitch, R.M.  Plenum  Press, New  York (1971)  87.  Watillon, A. and Dauchot, J., J. Colloid Interface Sci., 27, 507 (1968)  88.  Demchak, R. and Matijevic, E., J. Colloid Interface Sci., 31, 257  89.  Brace, R. and Matijevic, E., J. Inorg. Nucl. Chem., 35, 3691  90.  Stober, W.,  91.  Matijevic, E., Budnik, M.  (1969)  (1973)  Fink, A. and Bonn, E., J. Colloid Interface Sci., 26, 62 (1968) and Meites, L., J. 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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