A Mathematical Model for Colloidal Aggregation. Colleen S. O Brien

Transcription

1 A Mathematcal Model for Collodal Aggregaton by Colleen S. O Bren A thess submtted n partal fulfllment of the reurements for the degree of Master of Scence n Chemcal Engneerng Department of Chemcal Engneerng College of Engneerng Unversty of South Florda Maor Professor: Lus H. GarcaRubo, Ph.D. John T. Wolan, Ph.D. Jule P. Harmon, Ph.D. Date of Approval: November, 003 Keywords: Brownan, model, agglomeraton, repulson, partcles, knetcs, populaton Copyrght 003, Colleen S. O Bren

2 DEDICATION I would lke to dedcate ths work to my fancé, Bran. Hs encouragement, patence, and love have enabled me to pursue my passon, and contnue to strve for excellence n all areas of my lfe.

3 ACKNOWLEDGEMENTS I would lke to use ths opportunty to thank everybody wthout whose help ths work would have not been possble. I would lke to thank Dr. GarcaRubo for hs support and gudance over the past several years. I would also lke to thank the Chemcal Engneerng faculty at USF for ther support n my endeavors. I apprecate the help, support, and advce of my workmates at Bausch & Lomb, Santos Vscasllas, Gabrel Salmon, and Don Herber, and also my old workmates at Custom Manufacturng & Engneerng Scott Eler, Tammy Huggns, and Chrs Plonsky.

4 TABLE OF CONTENTS LIST OF TABLES v LIST OF FIGURES LIST OF SYMBOLS v x CHAPTER : INTRODUCTION 3 CHAPTER : AGGREGATION THEORY 6. Introducton 6. Collodal Partcles 6.3 Surface Charge 7.4 GouyChapman Model 8.4. Dffuse Layer Model.4. Electrophoretc Moblty and Zeta Potental 3.5 Aggregaton 4.6 Models for Collson Rates 5.6. Perknetc Collson Mechansm 6.6. Orthoknetc Collson Mechansm Dfferental Settlng Mechansm.7 Comparson of Rates.8 Collson Effcences 4.8. Perknetc Collson Effcences (Stablty Rato) 4.8. Orthoknetc Collson Effcences Hydrodynamc Interacton 7.9 Interpartcle Forces 9

5 .9. Electrostatc Repulsve Forces SphereSphere Interactons 3.9. Van der Waals Hamaker Expressons for Interactng Spheres Other Interpartcle Forces Born Repulson Sterc Interacton 35.0 Populaton Balances 35. Aggregate Structure 39.. Fractal Geometry 40.. Fractal mass scalng 4..3 Packng Densty Coordnaton Numbers Fractals n Partcle Aggregaton 47. Summary 48 CHAPTER 3: MATHEMATICAL MODEL FOR AGGREGATION KINETICS Introducton Interpartcle Forces Van der Waals Attractve Interacton Energy Electrostatc Repulson Interacton Energy Lnearzed PBE Nonlnear PBE Comparson of Lnear and Nonlnear PBE Collson Mechansms Stablty Ratos Perknetc Stablty Ratos Orthoknetc Stablty Ratos Map of Aggregaton Coeffcents Populaton Balance Formulaton Dscretzaton Method Comparson wth Analytcal Solutons SzeIndependent Kernal SzeDependent Kernal Error Levels n Partcle Sze Dstrbuton Mathematcal Model Descrpton 9

6 3.8 Summary 93 CHAPTER 4: EXPERIMENTS CONDUCTED BY MATHEMATICAL MODEL Introducton The Effect of Dscretzng Parameters on the Model The Effect of Temperature and Vscosty on the Model The Effect of Dameter Sze on the Model The Effect of Salt Concentraton on the Model The Effect of Hamaker Constants on the Model The Effect of Fractal Dmenson and Prmary Partcle Szes on the Model The Effect of Surface Potental on the Model The Effect of Shear Rate on the Model The Effect of Smultaneous Perknetc and Orthoknetc Collson Mechansms 4. Summary CHAPTER 5: COMPARISION OF MATHEMATICAL MODEL WITH EXPERIMENTS 3 5. Introducton 3 5. Materals Partcle Sze Dstrbutons Measurement of Surface Potentals Measurement of Aggregaton Phenomenon Expermental Runs Comparson Experment A Experment B Experment C Experment D 43

7 5.7.5 Experment E Experment F Concluson 5 CHAPTER 6: CONCLUSIONS 5 6. Mathematcal Model 5 6. Experments Conducted wth Mathematcal Model Comparson of Mathematcal Model wth Experments Future Work 53 REFERENCES 54 APPENDICES 59 APPENDIX : CRITERION FOR MONOTONIC OR CONCAVE POTENTIAL 60 APPENDIX : BEHAVIOR OF CONCAVE ELECTROSTATIC POTENTIAL AS C VARIES. 6 APPENDIX 3: DERIVATION OF K TERMS FOR ADJUSTABLE DISCRETIZATION OF LITSTER ET AL. 66 APPENDIX 4: PROOF OF VOLUME CORRECTION FACTORS FOR DISCRETIZATION OF LITSTER ET AL. 68 APPENDIX 5: MATLAB CODE FOR MATHEMATICAL MODEL 7 v

8 LIST OF TABLES Table 3.. Table of Aggregaton Events for the Method of Ltster et al. 80 Table 3.. Table of Aggregaton Events for the Method of Ltster et al. (Updated) 85 Table 4. Dscretzng Parameter versus Total Number of Bns 94 Table 4.. Expermental Parameters for Perknetc and Orthoknetc Experments 3 Table 5.. Propertes of Partcle Standards 4 Table 5. Propertes of Partcle Stock Solutons 4 Table 5.3 Summary of the Intal Partcle Sze Dstrbuton for Standards 5 Table 5.4. Expermental Parameters for Experment A 8 Table 5.5. Degree of Aggregaton for Experment A 8 Table 5.6. Expermental Parameters for Experment B 33 Table 5.7. Degree of Aggregaton for Experment B 33 Table 5.8. Expermental Parameters for Experment C 38 Table 5.9. Degree of Aggregaton for Experment C 38 Table 5.0. Expermental Parameters for Experment D 43 Table 5.. Degree of Aggregaton for Experment D 43 Table 5.. Expermental Parameters for Experment E 48 Table 5.3. Degree of Aggregaton for Experment E 48 Table 5.4. Expermental Parameters for Experment F 50 Table 5.5. Degree of Aggregaton for Experment F 50 v

9 LIST OF FIGURES Fgure.. The GouyChapman Model 9 Fgure.. Double Layer Structure 0 Fgure.3. Potental Profle for GouyChapman Model 0 Fgure.4. Perknetc Aggregaton 6 Fgure.5. Boundary Condtons for Smoluchowsk s Fast Coagulaton Euaton 7 Fgure.6. Movement of Partcles n Shear Flow 9 Fgure.7. Movement of Partcles n Extensonal Flow 0 Fgure.8. Comparson of Collson Rate Constants for Dfferent Transport Mechansms.3 Fgure.9. Dfference Between Rectlnear and Curvlnear Traectores 5 Fgure.0. Collson Effcency as a Functon of Shear Rate and Partcle Sze 6 Fgure.. Lmtng Stablty Rato W lm, as a Functon of the Hamaker Constant (). 8 Fgure.. Potental Energy Curve for Collodal Interacton. 30 Fgure.3. Populaton Balance Modelng 39 Fgure.4. Arrangement for Dfferent Number of Prmary Partcles 40 Fgure.5. Constructon Process of the Prefractal Koch Curves. 4 Fgure.6. Schematc Illustraton of SelfSmlar Aggregate Structure 43 Fgure.7. 3D Representaton of Several Aggregates wth D f = Fgure.8. Comparson between the Coalescence and Fractal Aggregate Model 45 Fgure 3.. Contact of Two Dssmlar Spheres 50 Fgure 3.. Potental Energy of Interacton for Dfferent Partcle Szes 5 Fgure 3.3. Potental Energy of Attracton for Dfferent Partcle Szes 5 Fgure 3.4. Potental Energy of Attracton for Dfferent Partcle Szes 5 Fgure 3.5. Potental Energy of Repulson for Dfferent Partcle Szes 54 Fgure 3.6. Potental Energy of Repulson for Dfferent Partcle Szes 54 Fgure 3.7. Graphcal Depcton of the Method of Papadopoulos and Cheh 56 Fgure 3.8. Flowsheet for Numercal Integraton of NonLnear PBE 6 Fgure 3.9. PBE Interacton Energy Comparson 63 Fgure 3.0. PBE Interacton Energy Comparson 64 Fgure 3.. PBE Interacton Energy Comparson 65 Fgure 3.. PBE Interacton Energy Comparson 66 Fgure 3.3. PBE Interacton Energy Comparson 67 Fgure 3.4. Perknetc Rate Constants 68 Fgure 3.5. Orthoknetc Rate Constants 69 Fgure 3.6. Dfferental Settlng Rate Constants 70 Fgure 3.7. Stablty Ratos for HHF 7 Fgure 3.8. Stablty Ratos for RKF and Papadopoulos 7 v

10 Fgure 3.9. Stablty Ratos for Orthoknetc Collson Mechansm 73 Fgure 3.0. Stablty Ratos for Orthoknetc Collson Mechansm 74 Fgure 3.. Log of Aggregaton Coeffcents for Perknetc Collson Mechansm 75 Fgure 3.. Log of Aggregaton Coeffcents for Orthoknetc Collson Mechansm 76 Fgure 3.3. Log of Aggregaton Coeffcents for Dfferental Settlng Mechansm 76 Fgure 3.4. Dscrete Bns to Cover a Large Range of Partcle Szes 78 Fgure 3.5. Moments versus I agg 88 Fgure 3.6. Error n the th Moment for SzeDependent Case 90 Fgure 3.7. Error n the th Moment for SzeIndependent Case 9 Fgure 3.8. Flow Dagram of Mathematcal Model 9 Fgure 4.. Comparson of Dscretzng Parameters for Perknetc Collson Mechansm 95 Fgure 4.. Comparson of Dscretzng Parameters for Orthoknetc Mechansm 96 Fgure 4.3. Temperature Comparson 97 Fgure 4.4. Vscosty Comparson 98 Fgure 4.5. Dameter Sze Comparson for Perknetc Aggregaton 99 Fgure 4.6. Dameter Sze Comparson for Orthoknetc Aggregaton 99 Fgure 4.7. Effect of Ionc Strength on Interacton Energy for Clay Mneral Kaolnte 0 Fgure 4.8. Effect of Ionc Strength on Clay Mneral Kaolnte. 0 Fgure 4.9. Ionc Concentraton Comparson for the Model 03 Fgure 4.0. Potental Energy of Interacton for Varous Values of Hamaker Constant 04 Fgure 4.. Hamaker Constant Comparson 05 Fgure 4.. Comparson of Prmary Partcle Sze for Perknetc Collson Mechansm 06 Fgure 4.3. Comparson of Prmary Partcle Sze for Orthoknetc Collson 07 Fgure 4.4. Fractal Dmenson Comparson for Perknetc Collson 07 Fgure 4.5. Fractal Dmenson Comparson for Orthoknetc Collson Mechansm 08 Fgure 4.6. Constant Surface Potental Between Two Plates 09 Fgure 4.7. Scalng Surface Potental Between Two Plates 09 Fgure 4.8. Surface Potental Versus Dameter 0 Fgure 4.9. Comparson of Shear Rates Fgure 4.0. Peclet Number Versus Partcle Dameter Fgure 4.. Log of Aggregaton Coeffcent, G =, Peclet < 4 Fgure 4.. Log of Stablty Ratos, G =, Peclet < 4 Fgure 4.3. Log of Stablty Ratos, G = 70, Peclet < 5 Fgure 4.4. Log of Stablty Ratos, G = 70, Peclet < 5 Fgure 4.5. Perknetc and Orthoknetc Aggregaton 6 Fgure 4.6. Log of Aggregaton Coeffcent for G = 0 and the Peclet number <90 7 Fgure 4.7. Log of Stablty Ratos for G = 0 and the Peclet number <90 7 Fgure 4.8. Log of Aggregaton Coeffcent for G=0 8 Fgure 4.9. Log of Stablty Ratos for G=0 8 Fgure Perknetc and Orthoknetc Aggregaton 9 Fgure 4.3. Log of Perknetc and Orthoknetc Collson Mechansm where G = 0 0 Fgure 4.3. Log of Stablty Ratos for G = 0 0 Fgure Log of Aggregaton Coeffcents for G = 0 Fgure Addton of Perknetc and Orthoknetc Aggregaton Fgure 5.. Zeta Potental for Polystyrene 6 v

11 Fgure 5.. Log of Perknetc Collson Mechansm for Experment A 9 Fgure 5.3. Log of Stablty Ratos for Experment A 9 Fgure 5.4. Log of Aggregaton Coeffcent for Experment A 30 Fgure 5.5. Partcle Concentraton versus Tme for Experment A 30 Fgure 5.6. Partcle Concentraton versus Partcle Dameter for Experment A 3 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment A) 3 Fgure 5.8. Log of Perknetc Collson Mechansm for Experment B 34 Fgure 5.9. Log of Stablty Ratos for Experment B 34 Fgure 5.0. Log of Aggregaton Coeffcents for Experment B 35 Fgure 5.. Partcle Concentraton versus Tme for Experment B 35 Fgure 5.. Partcle Concentraton versus Partcle Dameter for Experment B 36 Fgure 5.3. Actual versus Mathematcal Model Experments (Experment B) 37 Fgure 5.4. Log of Orthoknetc Collson Mechansm for Experment C 39 Fgure 5.5. Log of Aggregaton Coeffcent for Experment C 39 Fgure 5.6. Partcle Concentraton versus Tme for Experment C 40 Fgure 5.7. Partcle Concentraton versus Partcle Dameter for Experment C 40 Fgure 5.8. Log of Stablty Ratos for Experment C 4 Fgure 5.9. Actual versus Mathematcal Model Experments (Experment C) 4 Fgure 5.0. Stablty Ratos for Experment D 44 Fgure 5.. Log of Orthoknetc Collson Mechansm for Experment D 44 Fgure 5.. Log of Aggregaton Coeffcents for Experment D 45 Fgure 5.3. Partcle Concentraton versus Tme for Experment D 45 Fgure 5.4. Partcle Concentraton versus Partcle Dameter for Experment D 46 Fgure 5.5. Actual versus Mathematcal Model Experments (Experment D) 47 Fgure 5.6. Actual versus Mathematcal Model Experments (Experment E) 49 Fgure 5.7. Actual versus Mathematcal Model Experments (Experment F) 5 v

12 LIST OF SYMBOLS Alphanumerc Symbols A a a,b a a B C C C,C D D f D e F F g effectve Hamaker constant prmary partcle radus constants n Cauchy euaton actvty coeffcent of on radus of partcle wth sze brth functon concentraton of partcles Constant volume correcton factors n dscretzaton method death functon mass fractal dmenson dffuson coeffcent of partcle wth sze electron charge Faraday Constant growth rate of partcles along the sze axs acceleraton due to gravty x

13 G G Iagg I J k k (l) k f K k n l L l p M m m m ~ average shear rate shear rate degree of aggregaton modfed Bessel functon of the frst knd order one flux of partcle of sze Boltzmann constant absorpton coeffcent for partculate matter as a functon of wavelength fast aggregaton rate constant fracton of successful collsons for each type n dscretzaton procedure rate constant of aggregaton between partcle of sze and knematc vscosty center to center partcle separaton length sze coordnate path length mass of aggregate complex refractve ndex th moment of the partcle sze dstrbuton dmensonless th moment of the partcle sze dstrbuton n n(l,t) n(v,t) n + number concentraton of negatve ons populaton densty functon n length coordnates populaton densty functon n volume coordnates number concentraton of postve ons x

14 n o N 0 n 0 (l) n (l) N D (D,t) N n k N p N v (v,t) P R r number concentraton of ons total ntal number of partcles refractve ndex of medum as a functon of wavelength refractve ndex of the partculate matter as a functon of wavelength number densty functon on a dameter bass number concentraton of partcles n sze range kolmogoroff mcroscale total number of partcles n turbdty euaton number densty functon on a volume bass combned regonal and property space adustable dscretzaton parameter Ideal Gas Constant partcle radus R regon of property space for all x m r g R R S S(P) S A T geometrc rato radus of partcle wth sze collson radus between partcles of sze and arc length n Papadopolous ntegraton technue functon to relate mportant nterval lmts n dscretzaton procedure Surface Area Temperature V regon of dmensonal space x

15 V A ve V f v v o V part V R V T ve r v W x,y,z Xd X ref Z z van der Waals attractve nteracton energy propagaton through property coordnate nteracton energy per unt area of between two parallel plates volume of partcle n n sze range mean ntal partcle volume volume of partcles electrostatc repulsve nteracton energy total nteracton energy average propagaton velocty n all property space dmensonless volume stablty rato between partcles of sze and spatal coordnates separaton between plates dstance between plates correspondng to a partcular soluton total number of dscretzaton bns valence of on Greek Symbols a a m b o collson effcency Me sze parameter sze ndependent aggregaton constant x

16 b e e o e p k l F o aggregaton coeffcent between partcles of sze and alternate volume coordnate delectrc permttvty power nput per unt volume DebyeHuckel parameter alternate length coordnate or wavelength Electrostatc potental ntal aggregate densty, (porosty) y o y r r f r s t t r t(l o ) G,f m n x m aggregate densty of partcle wth radus R Surface potental reduced surface potental densty densty of flud densty of partcle dmensonless aggregaton tme resdence tme turbdty for wavelength l populaton densty functon angles used n Papadopolous ntegraton technue moment generatng functon vscosty property space x

17 A MATHEMATICAL MODEL OF COLLOIDAL AGGREGATION Colleen O Bren ABSTRACT The characterzaton of fne partcles s an area of mmense sgnfcance to many ndustral endeavors. It has been estmated that 70% of all ndustral processes deal wth fne partcles at some pont n the process. A natural phenomenon occurrng n these processes s collodal aggregaton. Ths study examnes aggregaton n collodal systems n order to characterze, examne, and control ths occurrence n ndustral processes. The study of partcle aggregaton has been broken nto many dfferent areas, such as collson mechansms, nteracton energy etc, but a complete model that ntegrates these dfferent aspects has never been fully realzed. A new model s reured to accurately predct the aggregaton behavor of collodal partcles. In ths work, a new model s developed that ntegrates Smoluchowsk knetcs, total nteracton energy between partcles, and stablty ratos for perknetc and orthoknetc collson mechansms. The total partcle nteracton energy necessary for the calculaton of stablty ratos s represented by the summaton of electrostatc and van der Waals

18 nteractons. The electrostatc nteractons are modeled usng DLVO theory, the lnear PossonBoltzmann euaton, and a numercal soluton for the nonlnear Posson Boltzmann Euaton, whle the van der Waals nteractons are represented by Hamaker theory. The mathematcal model s solved usng an adustable dscreton technue, whch s tested aganst a specfc analytc soluton, and yelds an assessment of the error ntrnsc n the dscretzaton method. The bass of the mathematcal model s a populaton balance framework. The model developed n ths study s general n many respects, but could be readly appled to many dfferent aggregaton systems wth mnor modfcaton. A comparson of the mathematcal model wth prevous experments conducted by Scott Fsher (998) s carred out for the perknetc and orthoknetc transportlmted aggregaton regmes. The fractal nature of soldsphere aggregates s consdered when comparng the mathematcal model predctons wth expermental measurements. The prevous experments that are used for comparson utlzed polystyrene partcles rangng from 00nm to 500nm n ntal dameter, several ntal partcle concentratons, and varous strrng rates. Zeta potental measurements are presented n order to set the range of transportlmted aggregaton. An assessment of the results of the mathematcal model wth the expermental results show good agreement for transportlmted aggregaton wthn the perknetc and orthoknetc transportlmted aggregaton, wth average partcle szes rangng from 00nm to well over mm.

19 CHAPTER : INTRODUCTION Aggregaton has become ncreasngly sgnfcant durng the last twentyfve years. Industry has become progressvely more nterested n controllng the mcroscopc propertes of partcles, such as composton, shape, surface roughness, surface characterstcs, and porosty. It has been estmated that 70% of all ndustral processes nvolve dealng wth fne partcles at some pont n the process (Bushell 998). Characterzaton of these partcles help us to understand and predct or control ther behavor n many processes. In ndustry, the propertes of partcles determne whether or not a dust s a respraton hazard, whether granular materals wll mx or segregate when agtated, and whether materal n a hopper wll flow n a controllable fashon, behave lke a lud, or not flow at all. Many of the tradtonal partcle characterzaton technues make assumptons about the shape or physcal structure of the partcles beng measured. The extracton of lnearsze parameters from laser scatterng measurements generally assume that the partcles are sphercal. Ths assumpton s normally made because t probably s not mportant to the technue beng used, but often t s because t s a dffcult problem. One class of partcles that has great mportance s aggregates. Almost every partculate system nvolves to a greater or lesser extent some partcles that are aggregates of smaller partcles n the system. Ths may be unmportant for systems such as the handlng of bulk ores, but n processes such as drnkng water fltraton, t s the domnant structure. These aggregates are often wspy, tenuous enttes that are absolutely unlke spheres, plates, or other famlar geometrc forms. The euatons used n ths model are for sphercal partcles, but the sphercal partcles are translated nto ther assocate fractal dmenson at the end of the model. 3

20 The begnnng of the current understandng of collodal aggregaton dates back to the work of Smoluchowsk (97). He dentfed aggregaton as a secondorder process dependent on the concentraton of aggregatng speces. The early work n ths area conssted of measurng and predctng rates of aggregaton based upon sngle euatons. Up untl the mdseventes, the study of aggregaton consdered only the formaton of dstnct aggregates from a monodspersed (unform n shape and sze) ntal partcle system, or the calculaton of rate constants of aggregaton from doublet formaton. The nature and magntude of varous forces actng on the partcles determne the stablty of the suspenson. Generally, n case of a sold dsperson n an aueous medum, the forces actng on a partcle nclude van der Waals forces, electrostatc forces and hydrodynamc forces. Of these, the van der Waals forces are attractve n nature and favor aggregaton. Electrostatc forces exst due to the presence of an electrc doublelayer around the partcles and cause repulson between the partcles. Ths has the effect of opposng aggregaton. Both of these forces are smlar n magntude, and act over comparable dstances from the partcle surface. If the electrc doublelayer repulson domnates, the suspenson remans stable, and does not agglomerate. However, suppressng the electrc doublelayer can destablze a suspenson makng the van der Waals attractve forces domnant, and creatng aggregaton. Ths knd of aggregaton s known as coagulaton. When partcles collde wth each other, not all collsons result n the formaton of agglomerates. Instead, only a fracton of the total collsons lead to the formaton of agglomerates. Ths fracton s known as the collson effcency, and t determnes the overall rate of aggregaton. When a suspenson s sheared, the partculate sze ncreases as aggregaton takes place. However, stresses develop at the same tme due to flud shearng and tend to break up the agglomerates nto a smaller sze. Hence, durng the flow of a suspenson, partcle enlargement and breakage take place smultaneously. Conseuently, agglomerates do not contnue to grow ndefntely n sze durng prolonged shearng; nstead they attan an eulbrum sze. 4

21 Ths paper s a summaton of the work of Lus H. GarcaRubo s collodal partcle students over the last decade. The maor contrbutons are by Esteban MaruezRuelme (994) n the area of the nonlnear PossonBoltzmann euaton to obtan the repulsve nteracton energes between sphercal partcles, and Scott Fsher (998) who researched and mplemented the populaton balance euatons wth the Ltster et al. (995) method of dscretzaton. The three maor sectons of ths paper are () the mathematcal model descrpton, () The nvestgaton of the mathematcal model, and (3) a valdaton of the model aganst a set of aggregaton experments that were performed by Scott Fsher (998). In accomplshng these goals, the work s outlned n the followng matter: Chapter deals wth the background nformaton that s necessary to understand the model. Chapter 3 outlnes the general mathematcal model for aggregaton phenomenon. Chapter 4 deals wth an nvestgaton of the model parameters. Chapter 5 provdes a comparson between the mathematcal model predctons and the measured aggregaton phenomena. Chapter 6 deals wth a general summary and conclusons, as well as future development of the model. 5

22 CHAPTER : AGGREGATION THEORY. Introducton Ths chapter descrbes the necessary background topcs reured to understand the mathematcal model for aggregaton. The frst secton descrbes the basc propertes of collodal partcles. The next few sectons descrbe aggregaton knetcs, collson mechansms, partcle stablty, and aggregate structure. The attractve and repulsve forces, such as electrostatc repulsve and van der Waals attractve forces, wll then be dscussed. Next, a mathematcal ntroducton to the populaton balance model s ncluded. The theory presented n ths chapter ntroduces the basc concepts necessary to understand the mathematcal model.. Collodal Partcles Collods form heterogeneous mxtures that are large enough to scatter lght. Collods usually consst of two phases, or one contnuous phase n whch the other phase s dspersed. These partcles are larger than the sze of molecules, but small enough for the dspersed phase to stay suspended for a long perod of tme. Collodal systems contan at least one or more substances that have at least one dmenson n the range between 0 9 m (0 A) and 0 6 m ( mm) n sze (Hemenz, 986). On the smaller end of ths scale, there are no dstnct boundares between the phases, and the system s consdered a soluton. On the larger end of ths scale, partcles wll begn to fall to the bottom due to gravtatonal force, and the phases are separated. Aggregaton nvolves the assocaton of partcles to form clusters, and depends on two dstnct nfluences: () partcles must move n a way that collsons occur, and () partcles that repel each other are sad to be stable, snce they do not form aggregates. Collods nteract wth each other at 6

23 an extremely short range, (usually much less than the partcle sze), so that partcles have to approach very close to each other before any sgnfcant nteracton s felt. The nteracton may be attractve (van der Waals) or repulsve (electrostatc repulson, sterc). There are many mportant propertes of collodal systems that are determned drectly or ndrectly by the nteracton forces between partcles. These collodal forces consst of the electrcal double layer, van der Waals, Born, hydraton, and sterc forces. Collodal partcles are domnated by surface propertes. If the surface area to volume, or surface area to mass of a sphercal partcle s looked at, the dependence on the partcle radus s S A /V µ /r. Ths relatonshp shows that as partcles decrease n sze, the surface propertes of the partcle become ncreasngly mportant (Fsher, 998). The measurement of partcle sze s also a defnng property. Optcal mcroscopy reles on vsble lght, whch renders collodal partcles largely nvsble to optcal technues. (Vsble lght lmts measurements to about 0.5 mm). Sedmentaton cannot be used to characterze partcles because the partcles need to be about.5 mm n sze (Fsher, 998)..3 Surface Charge Collodal suspensons usually consst of charged partcles dspersed throughout a contnuous solvent phase. When two phases are n contact, a separaton of charge wll occur whch causes a dfference n electrcal potental. If ths phase separaton s restrcted to a sold nterface wth an aueous electrolyte system, there are several possble mechansms for the separaton of charge: ) a dfference n affnty of ons for the two phases ) onzaton of surface groups 3) physcal restrcton of certan ons to one phase. The frst case of separaton of charge s usually found n metal haldes, calcum carbonate, and metal oxdes. The bestknown example s slver odde (Elmelech et al., 995). When slver odde s n contact wth pure water, slver ons have a tendency to escape from the 7

24 crystal lattce, leavng a crystal wth an excess negatve charge. If the concentraton of slver ons s ncreased, a pont s reached where the hgher escapng tendency of the slver ons s balanced by ther hgher concentraton n soluton, and the sold does not have a net charge (Elmelech et al., 995). For these cases, the surface potental, y 0, can be modeled by the Nerst euaton: Ê RT ˆ y 0 = constant + Á ln a z F Ë (.3.) where R s the gas constant, T s the absolute temperature, F s the Faraday constant, z s the valence, and a s the actvty of the escapng ons (Elmelech et al., 995). An example of phdependent surface charge s the case of metal oxdes. When metal oxdes are n contact wth water, the oxde surfaces become hydroxylated, gvng the possblty of surface onzaton to yeld ether postve or negatve groups. The onzaton of such groups can be wrtten as: S OH + S OH S O _ Where S denotes a sold surface (Elmelech et al., 995). Ths process nvolves the loss of two protons, whch are defned by approprate eulbrum constants. The degree of protonaton depends on the values of these eulbrum constants and the soluton propertes..4 GouyChapman Model If the collodal partcles n soluton are charged, and the soluton s electrcally neutral, the balancng charge s accounted for by an excess number of oppostely charged ons or counterons n soluton adacent to the charged surface and a defct of smlarly charged ons. In ths electrcal double layer, the counterons are dstrbuted accordng to a balance between ther thermal moton and the forces of electrcal attracton. The GouyChapman 8

25 model characterzes ths arrangement of charged speces around the collodal partcle. The electronc double layer s composed of two layers, the nner layer (Stern layer), and the outer layer (dffuse layer). The Stern layer conssts manly of oppostely charged ons adsorbed to the collod surface. The dffuse layer conssts of a mxture of ons extendng some dstance away from the collod. The Shear surface s the surface between the fxed and dffuse layer and defnng the moble porton of the collod. A dagram of the electrc double layer s shown n Fgures. (Anderson et al., 975) and. (Maruez, 994). A dagram of the potental profle for GouyChapman Model n shown n Fgure.3 (Anderson et al., 975). Fgure.. The GouyChapman Model 9

26 Fgure.. Double Layer Structure y 0 Potental Zeta Potental, z y d Potental at DffuseLayer Boundary d k Dstance Fgure.3. Potental Profle for GouyChapman Model 0

27 The Gouy and Chapman model s based on a number of smplfyng assumptons: ) an nfnte, flat mpenetrable nterface ) ons n soluton are pont charges, able to approach rght up to the nterface. 3) the surface charge and potental are unformly smeared out over the surface. 4) the solvent s a unform medum wth propertes (especally permttvty) that are ndependent of dstance from the surface. The relatonshp between charge densty, r (cm 3 ), and potental, y, at any pont s the Posson euaton: r y = (.4.) e where e s the permttvty of the medum. The Boltzmann Dstrbuton gves the dstrbuton of postve and negatve ons away from the partcle surface: n n + = n = n 0 0 Ê zef ˆ expá Ë kt Ê zef ˆ expá Ë kt (.4.) where n + and n are the number of catons and anons per unt volume wth charge +e and e respectvely. N 0 s the number of anons or catons far from the surface where the average electrostatc potental f s zero. T s the absolute temperature and k the Boltzmann constant, and z s the valence of electrons (Elmelech et al., 995).

28 Snce a flat nterface s beng consdered, the PossonBoltzmann expresson wll be used: d f = dx 0 zen e Ê snhá Ë zef ˆ kt (.4.3) The dmensonless parameters are: y = zef/kt and X = kx, where k s gven, for zz electrolytes, by: k e n0z = (.4.4) ekt The DebyeHuckel parameter, k, has the dmensons of recprocal length and plays a very mportant part n the electrcal nteracton between collodal partcles. Substtutng y and X: d y dx = snh y (.4.5) The GouyChapman theory has several shortcomngs. For example, measured capactances at certan nterfaces can be much lower than predcted by theory. Also, counterons concentratons close to charged nterfaces can become unreasonably hgh, even for moderate values of surface potental (Maruez, 994)..4. Dffuse Layer Model The surface charge densty s gven by: egtot s = (.4.6) S + a / K H where e s the protonc charge, G tot the total densty of chargeable stes, S a H demotes the surface actvty of the protons, and K s the dssocaton constant (Behrens et al., 999).

29 The charge of a latex surface can be related to the soluton propertes wth the socalled dffuse layer model (DLM). Ths model assumes that all the surface charge s located at the soluton nterface, whch s characterzed by the electrostatc surface potental y 0. The proton actvty at the surface s evaluated as: a = a exp( bey ) 0 (.4.7) S H H where ph=log a H, and b =k B T the thermal energy (Behrens et al., 999). These euatons defne a relaton between the surface charge and surface potental. Eulbrum reures the smultaneous fulfllment of a second chargepotental relaton that follows from the dstrbuton of moble ons n the dffuse layer. In a descrpton based upon the Posson Boltzmann euaton for electrolytes, the surface charge densty s of an solated partcle wth radus, R, can be expressed n terms of the surface potental y 0 as: ee0k È s = Ísnh( bey 0 / ) + tanh( bey 0 / 4) be Î kr (.4.8) where ee 0 s the total permttvty of the soluton and / k = ee0 /(N A be I) the Debye length, further nvolvng the onc strength I (N A s Avogadro s number). Wthout the second term on the rght sde, ths s ust the classcal GouyChapman result. The addtonal term was proposed by Loeb, Overbeek, and Wersema, and gves a frst order correcton for the surface curvature, accurate to wthn 5% of the true charge densty for any surface potental whenever the Debye length s smaller than the partcle dameter (Loeb, Overbeek, Wersema, 96). The last three euatons determne the eulbrum value of the surface charge s and potental y 0 at a gven ph (Behrens et al., 000)..4. Electrophoretc Moblty and Zeta Potental For a flat surface n a monovalent electrolyte, the electrostatc potental at a dstance x from the surface s related to the surface potental y 0 va: 3

30 4 y ( x) = arctan h[exp( kx) tanh( bey 0 / 4) (.4.9) be as follows from the ntegraton of the PossonBoltzmann euaton. The zeta potental was computed as the electrostatc potental z = y(x s ) at some dstance x s from the surface, correspondng to the thckness of an mmoble flud layer, adacent to the partcle surface (Behrens et al., 000). The outer end of ths mmoble layer, where the moton of flud relatve to the partcle sets n, s commonly referred to as the surface of shear..5 Aggregaton Three of the most basc propertes of a partculate system are the partcle composton, partcle sze dstrbuton, and partcle shape. As partcles undergo aggregaton, the partcle sze dstrbuton and shape of the partcle can change dramatcally dependng on the fractal nature of the aggregate structure. The foundatons of the rate of aggregaton start from the classc work of Smoluchowsk (97). It s convenent to thnk n terms of a dsperson of ntally dentcal partcles, whch, after a perod of aggregaton, contans aggregates of varous szes and dfferent concentratons. A fundamental assumpton s that aggregaton s a secondorder rate process, n whch the collson s proportonal to the product of concentratons of two colldng speces (Elmelech et al., 995). Threebody collsons are usually gnored n treatments of aggregaton they usually become mportant at very hgh partcle concentratons. The number of collsons occurrng between and partcles n unt tme and unt volume, J, s gven by: J = k n n (.5.) Where k s a second rate order constant, whch depends on a number of factors, such as partcle sze and transport mechansm. In consderng the rate of aggregaton, t must be recognzed that not all collsons may be successful n producng aggregates. The fracton of successful collsons s called the collson effcency and s gven the symbol a. If there s strong repulson between partcles, there wll not be any collsons that gve aggregates and a = 0. When there s no 4

31 sgnfcant net repulson or attracton between partcles, then the collson effcency can approach unty (Elmelech et al., 995). It s normal to assume n aggregaton modelng that the collson rate s ndependent of collod nteractons and depends only on partcle transport. Ths assumpton can be ustfed on the bass of the shortrange nature of nterpartcle forces, whch operate over a range whch s usually much less than the partcle sze, so that the partcles are nearly n contact before these forces come nto play. For the present, t shall be assumed that every collson s effectve n formng an aggregate (collson effcency, a=), so that the aggregaton rate constant s the same as the collson rate constant. The rate of change of concentraton of k fold aggregates, where k = + : dn dt k = = k  + > k = k n n n k  kk k = n (.5.) The frst term on the rghthand sde represents the rate of formaton of k aggregates by collson of any par of aggregates, and, such that + = k. Carryng out the summaton by ths method would mean countng each collson twce and hence the factor _ s ncluded. The second term accounts for the loss of k aggregates by collson, and aggregaton, wth any other aggregates. The terms k and k k are the approprate rate constants (Elmelech et al., 995). The above euaton s for rreversble aggregaton..6 Models for Collson Rates The determnaton of rate constants for aggregaton events s dependent on two factors: () the mechansm by whch partcle collsons occur, and () the presence of nterpartcle nteractons. In consderng the nature of partcle transport, and correspondngly partcle collson, there are three maor mechansms () Brownan moton (Perknetc aggregaton), () flud moton (Orthoknetc aggregaton), and (3) dfferental sedmentaton (Elmelech et al., 995). These mechansms wll be dscussed n the next few sectons. In all cases t s 5

32 assumed that partcles are sphercal and that the collson effcency s unty. Hydrodynamc nteracton wll be neglected n the next few sectons..6. Perknetc Collson Mechansm Small partcles n suspenson can be seen to undergo contnuous random movements called Brownan moton. Ths phenomenon occurs prmarly n partcles 00 nm to 000 nm (Peltomak, 00). Brownan moton s temperature dependent, and becomes ncreasngly mportant when partcles are one mcron or smaller. It s also mportant under condtons of hgh partcle concentraton (greater than 0 g/l), and low or no shear (Ernest, 995). The moton of perknetc aggregaton s shown n Fgure.4 (Ernest, 995). Fgure.4. Perknetc Aggregaton Smoluchowsk derved an expresson for collson freuency n ths case by consderng the dffusve flux of the partcles towards a statonary partcle (Agarwal, 00). Usng Fck s law for the number of partcles J gong through a unt area toward a reference partcle per unt tme: dn J ' = D (.6.) dr where D s the dffuson coeffcent of partcles, N s the number concentraton, and r s the radal dstance from the center of the reference partcle. Smoluchowsk defned a sphercal surface around a reference partcle (Fgure.5) so that any other partcle whose center 6

33 passes through that boundary wll be consdered to collde effectvely and produce coagulaton (Maruez, 994). R r R R Fgure.5. Boundary Condtons for Smoluchowsk s Fast Coagulaton Euaton The number of partcles gong through a sphere of radus r n unt tme s: dn J ' = (4pr ) D (.6.) dr 7

34 The dffuson coeffcent of a sphercal partcle s gven by the StokesEnsten euaton: D kt = (.6.3) 6 a m p where k s the Boltzmann s constant, T s the absolute temperature, a the partcle radus, and m the vscosty of the suspendng flud (Elmelech et al., 995). Smoluchowsk (97) calculated the rate of dffuson of sphercal partcles of type to a fxed sphere : J = 4pR D n (.6.4) Where D s the dffuson coeffcent of partcles of type and n s ther concentraton n the bulk expresson. Ths collson radus can be consdered the centertocenter dstance at whch a contact takes place. Ths s smply the sum of the partcle rad: R =a + a. (Elmelech et al., 995). In realty, the central sphere s not fxed, but s tself subect to Brownan dffuson. If the concentraton of partcles s n, then the number of I collsons occurrng n the unt volume per unt tme s: The rate constant for aggregaton s now: J = 4pR D n n (.6.5) k kt ( a + a ) = (.6.6) 3 m a a Ths euaton gves the rate constant for perknetc collsons. For partcles of approxmately eual sze, the collson rate constant becomes almost ndependent of partcle sze. 8

35 .6. Orthoknetc Collson Mechansm Collsons brought about by Brownan moton do not usually lead to the rapd formaton of large aggregates. Partcle transport brought about by flud moton can gve an enormous ncrease n the rate of nterpartcle collsons, and aggregaton brought about n ths way s known as orthoknetc collson. Ths type of collson becomes relevant between and 0 mcrons (Peltomak, 00). Smoluchowsk (97) also was the frst to study the rate of orthoknetc aggregaton due to unform lamnar shear forces. A dagram of the movement of partcles n shear flow s shown n Fgure.6 (Agarwal, 00). Z u dz b G=du/dz a Start Vew End Vew Fgure.6. Movement of Partcles n Shear Flow A unform lamnar shear feld s one n whch the flud velocty vares lnearly n only one drecton, perpendcular to the drecton of flow. Smoluchowsk assumed that partcles would flow n straght lnes and collde wth partcles movng on dfferent streamlnes accordng to ther relatve poston. The collson freuency depends on the szes of the partcles and on the velocty gradent or shear rate G (Elmelech et al., 995). By consderng a fxed central sphere of radus a and flowng partcles of radus a, t can be assumed that those movng partcles on streamlnes that brng ther centers wthn a dstance (a + a ). The collson freuency can then be calculated by consderng the flux of 9

36 partcles through a cylnder of radus R, the axs whch passes through the fxed sphere. The total flux towards the center partcle, J s ust twce that n one half of the cylnder and s gven by: J R 3 4Gn z ( R z ) dz = 4Gn R 0 = Ú (.6.7) The total number of collsons occurrng between and partcles n unt volume and unt tme s then smply: 4 J + 3 The rate constant for orthoknetc collsons between and partcles s: 3 = nn G( a a ) (.6.8) 4 k = G( a a ) (.6.9) Ths euaton shows that the rate s proportonal to the cube of the collson radus, whch has a maor effect on aggregate growth rate. As aggregaton proceeds and aggregate sze ncreases, the chance of capture becomes greater. The other maor flow feld of nterest s extensonal flow, whch s shown n Fgure.7 (Agarwal, 00). a Fgure.7. Movement of Partcles n Extensonal Flow 0

37 In ths case the collson freuency s gven by: J 6p 3 3 = g ext a N (.6.0) where g ext s the stran rate..6.3 Dfferental Settlng Mechansm Another mportant collson mechansm arses whenever partcles of dfferent szes and densty are settlng from a suspenson. Partcles of dfferent dameters settle at dfferent veloctes causng the faster movng partcles to collde wth slower movng partcles leadng to aggregaton. Ths type of collson mechansm usually becomes relevant at partcles of 000 mcrons n sze and larger. By balancng the forces of gravty, buoyancy and drag, the sedmentaton velocty of a partcle of radus a and densty r s n a medum of densty r s gven by Stokes euaton (Agarwal, 00): v g ( r s r) a = (.6.) 9 h The relatve velocty between two partcles of dameters a and a would be u = v v. The rate of N partcles through a cylndrcal cross secton of (a + a ) s gven by: dn dt = N p ( a + a ) ( v v ) (.6.) Usng the last two euatons, the resultng collson freuency, for partcles of eual densty s: Ê pg ˆ J ( ) ( ) 3 = Á r s r nn a + a ( a a Ë 9m ) (.6.3) where g s the acceleraton due to gravty, r s s the densty of the partcles and r s the densty of the flud.

38 .7 Comparson of Rates A summary of the collson mechansms descrbed n the prevous sectons are summarzed below: Perknetc: k = kt ( a + a ) 3 m a a Orthoknetc: 4 k = G( a + a 3 ) 3 Ê Dfferental Settlng: pg ˆ k ( )( ) 3 = Á r s r a + a ( a + a ) Ë 9m Usually t s assumed that the three mechansms of nterpartcle collsons are ndependent and when they operate smultaneously the aggregaton rates are addtve: J total =J Br +J Sedmentaton +J shear (.7.) The relatve magntudes of each contrbuton depend on the characterstcs of the suspenson and flow condtons. If the denstes of the partcles and the dspersng medum are nearly the same, contrbuton due to sedmentaton can be neglected. Other factors that lmt the effect of sedmentaton are hgh vscosty of the dspersng medum and the relatvely small sze of partcles (Bushell, 998). For a comparson of rates, t s convent to take one partcle of fxed sze and compute the varous rate constants as a functon of the sze of the second partcle. To compare the collson freuences due to shear flow that wth due to Brownan moton, ther rato s characterzed by the Peclet number (Agarwal, 00). If Pe >>, shear flow domnates, but f Pe<<, Brownan moton wll domnate. 3 4hga Pe = (.7.) k T B

39 As an example, one partcle s taken to have a dameter of mm and the other dameter vares between 0.0 and 0 mm. The shear rate s assumed to be 50 s and the densty of the partcles g/cm 3. All other values are approprate for aueous dspersons at 5 C. 0 Comparson of Collson Mechansm Rates] 0 m [ t n] a 3 t /s s n o C e t a R Perknetc Orthoknetc Sedmentaton Dameter of Partcle [mcron] Fgure.8. Comparson of Collson Rate Constants for Dfferent Transport Mechansms. As shown n Fgure.8 (Fsher, 998), t s clear that the perknetc mechansm gves the hghest collson rates for partcles less than 0.6 mm n dameter, but for larger partcles orthoknetc collsons and dfferental settlng become more mportant. As the sze of the second partcle becomes greater than a few mcrospheres, the collson rate due to sedmentaton ncreases sharply and becomes comparable to the shearnduced rate (Elmelech et al., 995). 3

40 .8 Collson Effcences In the collsons mechansms that were dscussed so far, t was assumed that all partcle collsons are successful n producng aggregates. In realty, ths s not the case, and a reduced collson effcency must be factored n. All that s needed to factor n ths rate s to nclude the collson effcency, a, nto the rate expressons. There remans the problem of assgnng a value to a, and ths wll be dscussed n the next few subsectons..8. Perknetc Collson Effcences (Stablty Rato) The effect of repulsve collodal nteractons on perknetc aggregaton s to gve a reducton n rate. In ths approach, a stablty rato, W, s used and s expressed as W = /a (the recprocal of the collson effcency). The stablty rato s smply the rato of the aggregaton rate n the absence of collodal nteractons. For cases where only van der Waals attracton and electrcal repulson need to be consdered, there s a energy barrer when the partcles approach the partcles. The stablty rato can be calculated by treatng the problem as one of dffuson n a force feld: W Ú 0 exp( ft / kt) = du ( u + ) (.8.) where f T s the total nteracton at a partcle separaton dstance d, and u s a functon of d and partcle sze. For eual partcles, u = d/a. It s also been cted as (Maruez, 63): ÊV ( l) ˆ dl W = + Ú Á ( R R ) exp (.8.) ( R + ) Ë kbt l R where V(l) s the total nteracton energy between partcles. To evaluate W, the ntegral n the last euaton has to be evaluated numercally, usng approprate expressons for the electrcal and van der Waals nteractons. Due to the exponental term, most of the contrbuton to the ntegral comes from a regon close to the maxmum. 4

41 .8. Orthoknetc Collson Effcences For collsons of nonbrownan partcles (> mcron), the Fuchs concept of dffuson n a force feld s not approprate and the relatve moton of partcles nduced by flud shear, or external gravty have to be consdered. It has been observed that aggregaton of otherwse stable collods can sometmes be acheved by the applcaton of suffcently hgh shear. For a gven suspenson, the collson effcency for Brownan aggregaton could be very dfferent from that of orthoknetc collsons. Smoluchowsk s theory makes the assumpton that partcles travel n straght traectores along streamlnes (streamlnes are not dsturbed by the presence of partcles. However, due to van der Waals, electrc doublelayer and hydrodynamc nteractons, partcle traectores devate from a straght lne as partcles approach each other. The dfference between rectlnear and curvlnear traectores s shown n Fgure.9 (Agarwal, 00). Fgure.9. Dfference Between Rectlnear and Curvlnear Traectores 5

42 The net velocty of a partcle s gven by the sum of velocty felds, whch are ndependent and superposable (Zechner and Schowalter, 977): U = u flow +u c (.8.3) The velocty feld u flow s due to the hydrodynamc flow whle u c s due to the presence of collodal forces. It should be noted that traectory analyss s vald for calculatng the collson effcency of doublet formaton resultng from two prmary partcles. No theory s avalable for the calculaton of the collson effcency of aggregated grow from doublets to trplets and larger aggregated because of the complexty of the hydrodynamcs nvolved. The collson effcency s shown as a functon of shear rate n Fgure.0 (Agarwal, 00). Partcle Dameter of Colldng Partcles, mcrons + + Collson Effcency Shear Rate (/s) Fgure.0. Collson Effcency as a Functon of Shear Rate and Partcle Sze 6

43 If the electrc double layer s completely suppressed by addng an electrolyte, numercal calculatons result n an easy expresson for collson effcency (Van de Ven and Mason, 977): 0.8 Ê A ˆ a = KÁ (.8.4) 3 Ë 36phga where K s a constant whose value s close to unty. Potann (99) suggested the followng expresson for aggregaton collson effcency: Ê ˆ ÁÊ d ˆ a ª ( ln( / )) Á (.8.5) d a ËË a 3 / where d s the dameter of the aggregate. Ths expresson shows that collson effcency decreases wth ncreasng d wth respect to prmary partcle sze a (Potann, 99). The collson effcency for orthoknetc collsons cannot be adeuately dscussed wthout reference to hydrodynamc nteracton..8.. Hydrodynamc Interacton The Smoluchowsk approach to aggregaton knetcs takes no account of the effect of the vscosty on the suspendng medum. The hydrodynamc or vscous effects can have a great effect on the aggregaton rates. As partcles approach very close t becomes ncreasngly dffcult for the lud between them to dran out of the gap, whch slows the aggregaton process. For orthoknetc and perknetc aggregaton, ths resstance wll prevent partcle contact completely unless a rapdly ncreasng attractve force such as van der Waals nteractons brngs the partcles together (Spelman, 978). The combned effect of van der Waals and hydrodynamc nteractons on the lmtng stablty rato of sphercal partcles n water s shown n Fgure. (Elmelech, 995). For comparson, the fnely dotted lne also shows the result n the absence of hydrodynamc nteracton. 7

44 3 Lmtng Stablty Rato Hamaker Constant/ 0 J Fgure.. Lmtng Stablty Rato W lm, as a Functon of the Hamaker Constant () It has been suggested that the collodal forces between agglomerates are determned by a couple of prmary partcles whereas hydrodynamc forces are euvalent to forces one would expect between two partcles of the sze of complete aggregates (Agarwal, 00). Thus, as the aggregate sze ncreases, hydrodynamc forces ncrease much more rapdly than collodal forces resultng n a much lower aggregate collson effcency than the prmary partcle collson effcency (Agarwal, 00). From the basc rate of aggregaton and the orthoknetc euaton, the orthoknetc rate of aggregaton becomes (Agarwal, 00): dn dt = an b (.8.6) 8

45 If the volume fracton of partcles f s assumed to reman constant then at any nstant, the number concentraton of partcles can be related to the partcle sze by 00). The last euaton can be ntegrated to obtan: 4 p b 3 N (Agarwal, 3 ln N N 0 4ayf = t (.8.7) p where N 0 s the number concentraton at tme t=0 and N s the number concentraton at any tme t. Ths euaton forms the bass for expermentally determnng the collson effcency. By followng the number concentraton wth respect to tme, the last euaton gves a straght lne durng the early stages of aggregaton. From the slope of ths lne, the expermental value of the collson effcency can be determned (Agarwal, 00)..9 Interpartcle Forces In collodal systems, there are three basc types of ntermolecular forces actng between molecules: () Van der Waals forces, () Electrostatc forces, and (3) sterc hndrance. The combnatons of these forces control the type and rate of coagulaton n partculate systems. Fgure. (Agarwal, 00) shows the potental energy curve for collodal nteracton. 9