UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Hindered settling of charged and uncharged submicron spheres Tackie, Emmanuel Nii 1982

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1982_A7 T33.pdf [ 7.18MB ]
Metadata
JSON: 831-1.0058858.json
JSON-LD: 831-1.0058858-ld.json
RDF/XML (Pretty): 831-1.0058858-rdf.xml
RDF/JSON: 831-1.0058858-rdf.json
Turtle: 831-1.0058858-turtle.txt
N-Triples: 831-1.0058858-rdf-ntriples.txt
Original Record: 831-1.0058858-source.json
Full Text
831-1.0058858-fulltext.txt
Citation
831-1.0058858.ris

Full Text

HINDERED SETTLING OF CHARGED AND UNCHARGED"SUBMICRON -SPHERES by E M M A N U E L N i l T A C K I E B.Sc. (Hons) U n i v e r s i t y o f S c i e n c e and T e c h n o l o g y Kumasi, Ghana, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPL IED SCIENCE •in The F a c u l t y o f G r a d u a t e S t u d i e s D e partment of C h e m i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d The U n i v e r s i t y o f B r i t i s h C o l u m b i a S e p t e m b e r , 1982 0 Emmanuel N i i T a c k i e , 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Chemical: Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) ABSTRACT The e f f e c t o f surface charges on the hindered s e t t l i n g of c o l l o i d a l p a r t i c l e s was studied using chemical ly p rec ip i ta ted mono-dispersed 0.6-micron s i l i c a spheres. S e t t l i n g rates were measured in tubes of 11mm ins ide diameter conta in ing d i s t i l l e d and deionised water that had been adjusted with sodium hydroxide to a l t e r the zeta potent ia ls on.the p a r t i c l e s , with sodium c h l o r i d e to a l t e r the non-dimensional e l e c t r o k i n e t i c radius and with hydrochlor ic ac id to suppress e l e c t r o s t a t i c i n t e r a c t i o n s between the p a r t i c l e s at a pH of about 3, which i s c l o s e to the i s o - e l e c t r i c point of the s i l i c a spheres. The p a r t i c l e s were dispersed by means of a Brinkman homogeniser and the zeta potent ia ls were measured by the method o f m i c r o - e l e c t r o p h o r e s i s . The volume f r a c t i o n c of p a r t i c l e s in the d ispers ions was var ied from 0.005 to 0.26, while the zeta potent ia l and the non-dimensional e l e c t r o k i n e t i c radius were progress ive ly ra ised to as high as -81 m i l l i v o l t s and 240, r e s p e c t i v e l y . The s e t t l i n g rates measured for var ious combinations o f p a r t i c l e concentrat ion c , zeta potent ia l X and e l e c t r o k i n e t i c radius were v i r t u a l l y in a l l cases lower than for the uncharged spheres, thus a t t e s t i n g q u a l i t a t i v e l y to the re tard ing e f f e c t of the Dorn potent ia ls and throwing some l i g h t on anomalous sedimentation r e s u l t s for f i n e p a r t i c l e s in the l i t e r a t u r e . i i i The experimental data for the uncharged spheres were compared with models in the l i t e r a t u r e for sedimentation of f ine p a r t i c l e s and found to f a l l between the theore t ica l p red ic t ion of Batchelor for d i l u t e d ispers ions and that of S tauf fer and Clav in for more concentrated d i s p e r s i o n s . However, comparison of the s e t t l i n g data for the charged spheres with the theory of Levine et a l . on the re tard ing e f f e c t o f the Dorn potent ia ls in concentrated d ispers ions showed t h i s very r e s t r i c t i v e theory to be qui te inadequate to the present da ta . iv TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGMENTS "x Chapter 1 . INTRODUCTION 1 2. BACKGROUND 4 2.1 Hydrodynamic E f f e c t s 4 2.2 E l e c t r i c a l E f f e c t s 9 2.2a Concepts of Double Layer 9 2.2b Mathematical Ana lys is o f the Double Layer Based on the Gouy-Chapman M o d e l . . . ]Q 2.2c L inear ised Poisson-Boltzmann E q u a t i o n . . . 13 2.2d Surface Charge 14 2.2e The Stern Layer 17 2.2f The Concept o f Zeta Potent ia l -E l e c t r o k i n e t i c Phenomena 20 2.2g Stern and Zeta Potent ia l 20 2.2h E f f e c t o f E l e c t r o l y t e on Zeta and Surface Potent ia ls 23 2.2i E l e c t r o k i n e t i c Phenomena 23 2.2j Sedimentaion Potent ia l 29 Page Chapter 3. SOME THEORETICAL CONSIDERATIONS 32 3.1 Sedimentation of M u l t i p a r t i c l e Systems 32 3.1a Hydrodynamic Considerat ions 32 3.1b E l e c t r i c a l Considerat ions 37 3.1c Sedimentation V e l o c i t y .40 4. EXPERIMENT. 45 4.1 Introduct ion 45 4.2 C o l l o i d a l Mater ial 45 4.2a P a r t i c l e s 45 4.2b Preparat ion o f S i l i c a P a r t i c l e s 47 4.2c P a r t i c l e S ize Measurement '$g 4.2d Separat ion of P a r t i c l e s . . . 51 4.3 Equipment and Techniques 51 4.3a S e t t l i n g Apparatus 51 4.3b Instruments 53 4.4 Measurements and Observations 56" 4.4a S e t t l i n g Experiment 56 4.4b Measurement o f Zeta Potent ia l 60 5. RESULTS AND DISCUSSIONS 65 5.1 Introduct ion 65 5.2 S e t t l i n g in Aqueous Media 65 5.3 Hindered S e t t l i n g at N e g l i g i b l e Surface Charge . . 72 5.4 Empir ical Cor re la t ion for Uncharged Spheres 79 5.5 S e t t l i n g of Charged P a r t i c l e s 86 5.6 Development of Cor re la t ions 88 5.7 Richardson and Zaki Plot 94 vi Chapter p a g e 6 . CONCLUSIONS AND RECOMMENDATIONS 98 NOMENCLATURE TOO REFERENCES 105 APPENDIX A. SAMPLE CALCULATIONS 109 Al Rate of Sedimentation 109 A2 Double Layer Thickness 109 A3 P a r t i c l e Zeta Potent ia l 114 B. CALIBRATIONS 119 Bl E lec t rophores is Apparatus 119 Bl .1 Ce l l Dimensions 119 Bl .2 Eyepiece G r a t i c u l e and Timing Device 121 B2 Measurement o f P a r t i c l e Density 122 B3 Measurement o f P a r t i c l e S ize from E lec t ron Micrographs 124 B4 Conduct iv i ty of NaCl So lu t ion with Concent ra t ion . . 125 B5 V i s c o s i t y Measurement 125 C. STATISTICAL AND ERROR ANALYSIS 129 CI Empir ical Cor re la t ion (Uncharged P a r t i c l e s ) T29 C2 Dimensionless Rela t ionships (Charged P a r t i c l e s ) . . . 133 C3 Errors in E lec t rophores is Measurements 135 D. EXPERIMENTAL RESULTS 139 vi i LIST OF TABLES Page TABLE 5.1 Summary o f Experimental data for Charged P a r t i c l e s 66 5.2 Summary of Experimental Data for Uncharged P a r t i c l e s 68 5.3 Experimental Data for Uncharged P a r t i c l e s with U 0 Corrected for V i s c o s i t y E f f e c t s 82 5.4 Comparison o f S e t t l i n g Rates of Charged P a r t i c l e s Obtained from Experimental with the Theory of Levine et a l . (1 976) 87 5.5 Dimensionless Groups 90 Al E Versus Y 0 fo r Run IA 118 Bl: Ce l l Dimensions 122 B2 P a r t i c l e S ize Measurements 125 B3 V i s c o s i t y Measurements Cl Values of C o e f f i c i e n t s Determined by Regressional Ana lys is 132 v i i i LIST OF FIGURES Page FIGURE 2.1 Simple Potent ia l D i s t r i b u t i o n in the Double Layer 15 2.2 Stern-Grahame Model of Double Layer 18 2.3 Double Layer - Structure and Ionic D i s t r i b u t i o n 19 2.4 Concept o f Zeta Potent ia l 21 2.5 E f f e c t of a) Surface P o t e n t i a l , b) Ionic St rength , on Zeta Potent ia l 24 4.1 E lec t ron Micrographs o f S i l i c a P a r t i c l e s . . 50 4.2 S e t t l i n g Apparatus 54 5.1 Comparison of Experimental Results with Pred ic t ions in the L i t e r a t u r e 74 5.2 Comparison o f Experimental Results Re la t ive to Results at Minimum Concentrat ion to Pred ic t ions in the L i t e r a t u r e 76 5.3 Richardson-Zaki Plot of Experimental Data for Uncharged P a r t i c l e s 78 5.4 Comparison of Experimental Results with the Pre-d i c t i o n s o f Batchelor (1972) and Stauf fe r and Clav in (1981) 80 5.5 Comparison of Empir ical Cor re la t ion (Eq.5.4) with the Pred ic t ions o f Batchelor (1972) and Stau f fe r and Clav in (1981) 84 5.6 Comparison o f Empir ica l Cor re la t ions (Eq.5.4a) with the Pred ic t ions o f Batchelor (1972) and Stauf fer and C lav in (1981) . . 85 ix Paje FIGURE 5.7 Scat ter Plot of Predicted V e l o c i t y of Charged P a r t i c l e s (based on equations 5.7 and 5.4) Versus Observed V e l o c i t y 92 5.8 Scat ter Plot of Predicted V e l o c i t y of Charged P a r t i c l e s (based on equations 5 .7 , 5.9 and 5.4a) Versus Observed V e l o c i t y 95 5.9 Richardson-Zaki P lot o f Charged P a r t i c l e s and that of Uncharged P a r t i c l e s Predicted by Equation 5.4 96 A . l P lot of Interface Height Versus Time (Run IA) 110 A. 2 P lot of Inter face Height Versus Time (Tun 1.1) I l l B. l NaCI Concentrat ion Versus Measured Conduct iv i ty of So lu t ion 126 ACKNOWLEDGEMENTS This work has been made poss ib le through the thorough guidance of and inva luable suggestions by Professor Norman E p s t e i n . The ass is tance o f Professor S . Levine in the a n a l y s i s o f var ious mathe-matical problems and Dr. B.D. Bowen in the experimental techniques are very much apprec ia ted . Thanks to a l l the s t a f f and personnel of the Department o f Chemical Engineering who have in one way or the other been a source of encouragement. F i n a l l y I wish to thank the National Science and Engineering Research Council for f i n a n c i a l support . 1 CHAPTER ONE INTRODUCTION The hydrodynamic s o l u t i o n of f l u i d flow r e l a t i v e to a swarm or assemblage of p a r t i c l e s i s very v i t a l to p r a c t i c a l s i t u a t i o n s i n -volv ing p a r t i c u l a t e systems. The complex nature of such systems as a r e s u l t of f ac to rs such as - S o l i d s concentrat ion - Motion of f l u i d and s o l i d s r e l a t i v e to each other - Shape of p a r t i c l e s - E f f e c t of conta in ing wall - Brownian movement - E l e c t r o s t a t i c e f f e c t s has prompted previous workers to resor t to i d e a l i s a t i o n s which enable formulat ion of simple mathematical models that can be handled e a s i l y . In the creeping flow regime, the flow of f l u i d past spher ica l p a r t i c l e s has been t reated a n a l y t i c a l l y by a number of workers (1,2,3) using the c e l l model technique to p red ic t the e f f e c t of concentrat ion on sedimentation ra te . The assumptions or s i m p l i f i c a t i o n s common to a l l these analyses are the neglect o f Brownian motion e f f e c t s and the absence of extraneous forces other than g r a v i t y . 2 However, in ion ic so lu t ions and for very f i n e p a r t i c l e s of s i ze smal ler than about 50 ym, pred ic t ions of the above models disagree s i g n i f i c a n t l y with experimental data (4 ,5 ) . When s o l i d s are brought into contact with a polar medium such as an aqueous s o l u t i o n , a separat ion of charges take place as a consequence o f . . i o n i s a t i o n , ion adsorpt ion or ion d i s s o l u t i o n . The s o l i d s sur face acquires a net charge while the l i q u i d phase acquires a net compensating charge opposi te in s ign to that o f the s o l i d s u r f a c e , rendering the over-a l l system e l e c t r i c a l l y n e u t r a l . This s i t u a t i o n gives r i s e to what is c a l l e d the double l a y e r , one part of which i s r i g i d l y attached to the s o l i d s u r f a c e , the immobile l a y e r , and the o ther , the mobile l a y e r , l o o s e l y a t tached. The shearing of the double layer occurs along a plane between the two layers as a r e s u l t of a r e l a t i v e motion between the two phases. The e l e c t r i c a l potent ia l at t h i s s l i p plane i s known as the zeta p o t e n t i a l . In the sedimentation of s o l i d p a r t i c l e s , the p a r t i c l e s s e t t l e with the r i g i d l y attached immobile layer and the surrounding mobile layer loses i t s symmetry as a r e s u l t of the f l u i d motion past the p a r t i c l e sur face . Consequently an e l e c t r i c a l potent ia l (commonly c a l l e d Dorn Potent ia l ) i s set up which introduces e l e c t r i c a l forces opposing the motion o f the p a r t i c l e s in add i t ion to the c l a s s i c a l hydrodynamic f o r c e s . As a r e s u l t , the charged p a r t i c l e s s e t t l e at a lower v e l o c i t y than t h e i r s e t t l i n g rate when uncharged. Mineral processes such as t h i c k e n i n g , sedimentat ion, storage and t ranspor ta t ion are grea t ly a f fec ted by the act ion of i n t e r f a c i a l e l e c t r i c a l forces because of the s o l i d / l i q u i d r e l a t i v e motion which 3 charac ter ises these opera t ions . Thus a knowledge of the i n t e r f a c i a l e l e c t r i c a l forces which a f f e c t the physical behaviour of f i n e l y d iv ided p a r t i c l e s , such as the s e t t l i n g c h a r a c t e r i s t i c s of mineral suspensions and mineral matter , i s important f o r the e f f e c t i v e e x p l o i t a t i o n of mineral resources and f o r the p o l l u t i o n control of water resources . 4 CHAPTER TWO BACKGROUND 2.1 Hydrodynamic E f f e c t s Numerous theore t i ca l as well as experimental s tudies on the behaviour of p a r t i c u l a t e systems have been reported in the l i t e r a t u r e over the past several y e a r s . E a r l i e s t s tudies (6 ) were conf ined to s i n g l e spher ica l p a r t i c l e s moving in the creeping flow regime and in an unbounded f l u i d , g iv ing the well known Stokes' Law. Subsequent s t u d i e s , which involved the extension o f Stokes' Law to inc lude cases where f l u i d i s bounded by s o l i d wal ls and s o l i d p a r t i c l e s are i r r e g u l a r l y shaped, have been descr ibed in d e t a i l by Happel and Brenner ( 7 ) . Si tua t ions invo lv ing the motion of a swarm of p a r t i c l e s in a f l u i d are very complex. This complexity stems from the in f luence of a number of f ac to rs such as s o l i d s concentrat ion ( i . e . the in ter fe rence on the f l u i d stream around a p a r t i c l e by the presence of neighbouring p a r t i c l e s ) , the e f f e c t of conta in ing vessel w a l l s , Brownian motion, and e l e c t r o s t a t i c e f f e c t s that come into play fo r f i n e p a r t i c l e s . Thus f a r , no complete theore t ica l d e r i v a t i o n has been able to cover a l l the above e f f e c t s . Various inves t iga to rs have therefore approached the problem from various angles . 5 The r e f l e c t i o n technique was employed by several workers (7) f o r var ious p a r t i c l e arrangements. This technique led to numerical r e s u l t s 1/3 which pred ic t that the f i r s t - o r d e r cor rec t ion to U s depends upon C at low Reynolds numbers and which were in f a i r l y good agreement with experiment. For example, the sedimentation v e l o c i t y U Q as obtained by MacNown and L in (8) i s expressed as : _° = ! \2 Oil 11 1/3 L ^ . u i J U s 1+1.6 Q } ' 6 where U g i s the Stokes 1 s e t t l i n g v e l o c i t y and c the p a r t i c l e volumetr ic concent ra t ion . By t r e a t i n g f l u i d - p a r t i c l e suspensions as a continuum ( e . g . as a s i n g l e phase f l u i d ) , other workers such as Hawksley (9) and Vand.) (10) have a lso der ived theor ies based on E i n s t e i n ' s (11) equation for the v i s c o s i t y o f d i l u t e suspensions and the r e l a t i o n s h i p between sedimenta-t i o n rate and v i s c o s i t y o f suspensions. In t h e i r experimental s tudies on monodisperse polystyrene l a t i c e s , Cheng and Schachman (12) obtained an empir ical r e l a t i o n fo r the d i l u t e concentrat ion range of the form U U9- = 1 - 5 - 0 6 c [2.02] s which agreed f a i r l y well with the review of ear ly experimental data by Maude and Whitmore (13) who proposed that (1 [2.03] 6 where & was found to be about 5 for monodisperse, equ i -dens i ty spheres in creeping f low. Another technique which has been employed s u c c e s s f u l l y to estimate the e f f e c t of concentrat ion on the rate of sedimentation is the use of a c e l l model, which involves a modi f ica t ion o f the boundary condi t ions on f l u i d f lowing past a s ing le i s o l a t e d sphere to render them app l i cab le to that of a s i n g l e sphere in an assemblage of spheres. By th is technique a number o f models have been formulated and studied a n a l y t i c a l l y or numer ica l ly . Such models as Happel 's (1) f ree surface model, Kuwabara's (2) zero v o r t i c i t y model and the hexagonal conf igura t ion of Richardson and Zaki (3) are in th is category. In the c e l l model, the en t i re suspension of p a r t i c l e s i s assumed to be made up of i d e n t i c a l c e l l uni ts c o n s i s t i n g o f a p a r t i c l e and a surrounding f l u i d such that the void volume in each c e l l i s i d e n t i c a l to that in the to ta l assemblage. The boundary value problem i s thus reduced to a cons idera t ion of the behaviour of a s i n g l e p a r t i c l e and i t s bounding f l u i d . In the usual s i m p l i f i e d case of a spher ica l p a r t i c l e , a spher ica l c e l l i s assumed such that the r a t i o o f the p a r t i c l e r a d i u s , a , to the c e l l r a d i u s , b, i s given by ( a / b ) 3 = c [2.04] The boundary value problem thus involves a cons idera t ion of the equation of motion in the creeping flow regime ( i . e . neglect of i n e r t i a terms) subject to the appropriate boundary c o n d i t i o n s . The ' f r e e sur face ' and 7 'zero v o r t i c i t y 1 models d i f f e r only in the choice of hydrodynamic boundary cond i t ions at the outer envelope ( i . e . a t r=b). Happel (1) assumed the tangent ia l shear s t ress at the outer surface of the c e l l to be zero while Kuwabara (2 ) , i n s t e a d , argued that the v o r t i c i t y vanishes at the c e l l boundary. The r e s u l t s of both t h e o r i e s , however, give good pred ic t ions o f the r e l a t i v e v e l o c i t y of concentrated suspensions. In the Richardson and Zaki (3) type of c e l l the sedimenting p a r t i c l e s were assumed to be arranged in a hexagonal-type pattern in hor izonta l l a y e r s , in two conf igura t ions which d i f f e r in the v e r t i c a l d is tance between const i tuent p a r t i c l e s . In conf igura t ion I the v e r t i c a l d is tance was assumed to be the same as the hor izontal d is tance between p a r t i c l e s in each l a y e r , and in con f igura t ion II , which agreed with t h e i r experimental data at intermediate concent ra t ions , the p a r t i c l e s were assumed to be touching in the v e r t i c a l d i r e c t i o n . From t h e i r experimental r e s u l t s the r e l a t i v e v e l o c i t y is of the form n where the exponent n, dependent on Reynolds number and d /D , i s equal to 4.65 for n e g l i g i b l e wall e f f e c t and Re g <0.2 (14) . The above studies have been s i m p l i f i e d to a cons idera t ion o f the e f f e c t of p a r t i c l e - f l u i d - p a r t i c l e i n t e r a c t i o n a lone . Several other f a c t o r s , however, a f f e c t the behaviour o f suspensions. When p a r t i c l e s i z e becomes comparable to the mean f ree path of a gas medium, for ins tance , continuum hydrodynamics no longer app l ies and a c o r r e c t i o n to S tokes 1 Law deduced by Cunningham (15) is a p p l i c a b l e . The e f fec t of 8 Brownian Mo ion a lso becomes considerable as the p a r t i c l e s i ze decreases. A comparison of the magnitude of displacement by Brownian movement with that due to g r a v i t a t i o n a l s e t t l i n g is given by Lapple (16) based on the r e l a t i o n s of E i n s t e i n (11). The in f luence o f e l e c t r o s t a t i c forces a lso becomes apprec iable as the p a r t i c l e s i z e decreases and the p a r t i c l e concentrat ion i n c r e a s e s . I t has been argued elsewhere (17) , for i n s t a n c e , that the high values o f the Richardson and Zaki (14) index n obtained exper imental ly by Maude and Whitmore (18) fo r 20-40 ym emery powder in water (n=8), and by Richardson and Meikle (19) fo r 4-7 ym alumina p a r t i c l e s in water (n=10.5), might p a r t i a l l y be due to the e f f e c t of e l e c t r o s t a t i c behaviour at the p a r t i c l e - l i q u i d in te r face in i o n i c media.. . Experimental r e s u l t s o f other workers for sedimentation o f f ine p a r t i c l e s in aqueous media a lso showed some re tardat ion in the se t t ing rate r e l a t i v e to what i s predicted or measured for uncharged p a r t i c l e s o f the same s i z e and d e n s i t y . E l ton (4 ) , experimenting with 2-5 ym _5 carborundum p a r t i c l e s in 10 M K C l , reported values of U /U o of 0 .35, where U Q represents the s e t t l i n g v e l o c i t y of uncharged p a r t i c l e s at the same concentrat ion at which the actual s e t t l i n g v e l o c i t y ' i s " l l . Dul in and El ton (5 ) , working with s i l i c a p a r t i c l e s of about 6 ym s i z e in the same e l e c t r o l y t e medium, obtained values of U/U Q of 0.77. Other workers (20,21,22) have reported s i m i l a r r e s u l t s . In genera l , s o l i d s in contact with an aqueous medium acquire an e l e c t r i c a l charge as a consequence of i o n i s a t i o n , ion adsorpt ion or ion d i s s o l u t i o n . The l i q u i d phase a lso acquires a net compensating charge opposite in s ign to that of the s o l i d s u r f a c e , rendering the overa l l system e l e c t r i c a l l y n e u t r a l . This s i t u a t i o n gives r i s e to what is c a l l e d 9 the e l e c t r i c a l double l a y e r , one part o f which i s r i g i d l y attached to the sol id" surface arid the other l o o s e l y a t t a c h e d . . The shearing o f the double layer occurs along a plane between the two layers as a r e s u l t of r e l a t i v e motion between the two phases. Thus in i o n i s i n g s o l v e n t s , where p a r t i c l e s are charged and are therefore surrounded by e l e c t r i c a l double l a y e r s , the pred ic t ions of the above hydrodynamic models might have to be corrected to include the e f f e c t of the d i f f u s e l a y e r . 2.2 ELECTRICAL EFFECTS 2.2a Concepts of double l ayer The double layer that i s formed as a r e s u l t of separat ion of charges at the s o l i d - l i q u i d in te r face i s regarded as c o n s i s t i n g o f two r e g i o n s , one o f which i s r i g i d l y attached to the s o l i d surface and i s c a l l e d the immobile l a y e r . The other i s the d i f f u s e mobile l a y e r con-s i s t i n g o f compensating ion charges d i s t r i b u t e d in a d i f f u s e manner. In the Gouy-Chapman model (1910)(23), which was an improvement over the capac i to r model o f Helmholtz (1879)(23), a l l the i o n i c charges at the in te r face are assumed to be d i f f u s e l y d i s t r i b u t e d , extending from the surface of the s o l i d which possesses a f i n i t e potent ia l to the bulk of the so lu t ion where the potent ia l i s n i l . The major defect of t h i s model is the assumption of point charges, which neglects the f i n i t e s i zes of the ions and thus the f i n i t e degree of approach to the s o l i d s u r f a c e . This model has been modif ied by Stern (1924)(23) to account for the f i n i t e s i z e of i o n s , and thus the degree o f approach of the ions in the d i f f u s e layer to the i n t e r f a c e i s l i m i t e d by a f i n i t e d i s t a n c e , 6 , wi thin which a very th in immobile layer of counter - ions are s t rongly adsorbed to the s u r f a c e . The d i f f u s e layer thus extends from the d i s t a n c e s , 10 where the Stern plane i s l o c a t e d , to the bulk of the l i q u i d . 2.2b Mathematical a n a l y s i s of the diouble l ayer based on the Gouy-Chapman model (1910) The treatment in standard texts (23,24) is s i m p l i f i e d here. Major Assumption of Model: 1. Surface charge is uniformly d i s t r i b u t e d . 2. The ions in the d i f f u s e layer cons t i tu te point charges, i . e . t h e i r volume i s zero . 3. The i o n i c d i s t r i b u t i o n in the d i f f u s e layer fol lows the Boitzmann d i s t r i b u t i o n Law. 4. The solvent medium i s s t r u c t u r e l e s s and i t s only e l e c t r i c a l in f luence i s through i t s d i e l e c t r i c constant , which i s independent of the l o c a t i o n in the d i f f u s e l a y e r . Based upon the above assumptions, the potent ia l d i s t r i b u t i o n in the d i f f u s e part of the double layer can be expressed q u a n t i t a t i v e l y through the a p p l i c a t i o n of two basic laws or equations o f p h y s i c s : - The divergence theorem: The surface in tegra l of any vector over a surface s i s equal to the volume in tegra l o f the divergence of the vector over the volume enclosed by the s u r f a c e , i . e . e E-ds = s s " (Ve E) dv [2.06] v - Gauss' theorem: The surface in tegra l of the e l e c t r i c f i e l d vector i s equal to 4ir times the charge enclosed within the s u r f a c e , i . e . e E-ds = 4T:Q s — In the above equat ions, e = d i e l e c t r i c constant of the medium s E_ = e l e c t r i c f i e l d vector Q = e l e c t r i c charge. I f p i s the charge d e n s i t y , then Q = pdv v Combining equations 2.07 and 2.08 e E-ds = 4-rr s — s pdv v and thus from equations 2.06 and 2.09 V-e E_ = 4irp I f the e l e c t r i c potent ia l in the double layer i s T|>, then grad TJJ = E_ i . e . ViJ> = E_ If fo l lows from equations 2.10 and 2.11 that [2.12] which is the Poisson equation for the potential distr ibution in the diffuse layer. A solution of the Poisson equation is dependent on a knowledge of the value of the charge density in the double layer, which can be expressed as where n^  is the number of ions of type i per unit volume at any point in the double layer, e Q the electron charge, and z. the valency of the ions of type i . From Boitzmann's Probability Law the number of ions of species i at any point in the double layer can be obtained from where n. is the number of ions of type i per unit volume in the no neutral bulk of the solution where the potential ^=0, K = Boitzmann's constant and T = absolute temperature. These concentrations s t r i c t l y refer to time-averaged concentrations since the ions are in a constant state of random thermal motion. From equation 2.14, the Poisson equation (2.12) becomes n i = n i o exp^z .e^ /KT) [2.14] 4-rre v 2 ^ = - 0 Vz.n. exp-(z. ip/KT) L I io K io r [2.15] e s 13 which simplifies for a z-z symmetrical electrolyte where P =zeQ(n+ - n_) to 8ne n V 2 ^ = £ ° 0 S1nh(z'e t|«/KT) [2.16] s 2,2c Linearised Poisson-Boltzmann equation Equation 2.16 is called the Poisson-Boltzmann equation. The Debye-HUkel (1923) approximation applies i f z& i|>/KT «1, in which case the equation reduces to 8iTZe n ze ii> = __JL_o . _ o _ [ 2 .17] s k2^ [2.18] wh ere 8 i r z 2 e 2 n „ k2 = £ l K ° 0 [2.19] s" For an infinite plane surface, equation 2.18 can be written as ^ = k 2 ^ [2.20] dx2 where x is the distance away from the plane. A solution of equation 2.20 is * ( x ) = V _ k X [ 2" 2 1 ] 14 For the fol lowing boundary condi t ions: At x = 0, \\b = \pQ As x -»- » , \b -> 0 Equation 2.21 indicates that the potential decreases exponential ly to zero with increasing distance from the surface as i l l u s t r a t e d in Figure 2 .1 . Al so when 1/k (x) [2.22] 2.2d Surface charge For e lec t roneu t ra l i t y , the surface charge on p a r t i c l e , a , must be equal in magnitude and opposite in sign to the spacer-charge in the double l a y e r . . p(x) dx From Poisson's equation (2.12) applied in the x- d i rec t ion . P(x) = e d 2 ib. > 4 ' dx 2 and thus 4TT -*(x) dx dx5 s 4ir (x) dx L2.23] k DISTANCE X Figure 2.1: Simple Potent ia l D i s t r i b u t i o n in the Double Layer 16 A solut ion of th is equation by Overbeek (23) i s £ s d ^(x) a o 4TT dx x=0 for the fol lowing boundary condit ions iji + 0, ^ -> 0 a s x - > » From equation 2.21 , e kifc which i s analogous to that of a para l le l plate capacitor with a plate separation of k - 1 and capacitance, C = a0/i>Q- The quantity k"1 i s therefore general ly referred to as the equivalent thickness of the double layer and thus a decrease in double layer thickness resu l ts in an increase in surface charge or decrease in surface potent ial or both. For an aqueous solut ion of symmetrical z -z e lec t ro ly te where e * 81 at 25°C, equation 2.19 reduces to k = 0.3286 x 10 8 ( c o ) ( z ) 1 / 2 cm" 1 [2.25] where C q = concentration of neutral e lec t ro ly te in gm mo les / l i t r e = 1000 n o / N Q , being the Avrogadro's number. Thus at very high concentrations the double layer thickness i s very smal l . 17 2.2e The Stern layer The Gouy-Chapman model o f the double layer has been found to be q u a n t i t a t i v e l y v a l i d for e l e c t r o l y t e so lu t ions of very small concentra-t i o n s . In s i t u a t i o n s where the concentrat ion and potent ia ls are very h i g h , the theory pred ic ts unacceptably high counter - ion concentrat ions at the s u r f a c e , with an accompanying very small value for the double layer t h i c k n e s s . These shortcomings of the theory stem from the basic assumption upon which the theory was based - the assumption that the ions are point charges which can approach the surface without l i m i t . This assumption can be considered v a l i d in d i l u t e so lu t ions where the double layer th ickness i s la rge compared to the ion s i z e s . Stern (1924) and Grahame (1941 and 1958) took into cons idera t ion the f i n i t e s i z e o f the i o n s , for which the d is tance o f c l o s e s t approach of the centres to the s u r f a c e , <5, i s a lso f i n i t e . Thus the potent ia l appearing in the Gouy-Chapman theory i s a c t u a l l y not the surface potent ia l but rather the potent ia l in the plane of c l o s e s t approach of the counter - ions to the s u r f a c e . This potent ia l i s c a l l e d the stern plane p o t e n t i a l , ^ as shown in Figure 2 .2 . Stern a lso considered the p o s s i b i l i t y of s p e c i f i c ion adsorpt ion g iv ing a compact layer of counter-ions attached to the surface by e l e c t r o s t a t i c and Van der Waals forces s t rong ly enough to overcome thermal a g i t a t i o n . The overa l l double layer i s therefore considered to be d iv ided into three parts (Figure 2 .3 ) : 1. In the immediate v i c i n i t y of the in te r face a ' l a y e r formed by s p e c i f i c a l l y adsorbed ions c a l l e d 'po tent ia l determining' i o n s . They are adsorbed as unhydrated i o n s . Inner Helmholtz Plane Solid Outer Helmholtz Plane Slip Plane Specifically Adsorbed Ion Non-Specifically Adsorbed Ion Figure 2.2: Stern-Grahame Model of Double Layer 19 Figure 2 .3 : Double Layer - Structure and Ionic D i s t r i b u t i o n 20 2. Counter ions of opposite charge to the potent ial determining ions are arranged in the outer Helmholtz layer at a distance 6 , from the in ter face. In th is layer the potential drops to ^ in the stern plane. 3. The d i f fuse layer . Thus any deviat ions from i dea l i t y such as f i n i t e ion volume and spec i f i c adsorption ef fects are assigned to the stern layer and with increasing distance from the surface the double layer becomes more ideal (Figure 2.3) . 2.2f The concept of zeta potent ial - e lec t rok ine t i c phenomena When a force (mechanical or e l e c t r i c a l ) i s applied to the s o l i d -l i q u i d in te r face , re la t i ve motion between the so l id and l i qu i d occurs at a s l i p plane, the potential at which i s ca l led the Zeta Po ten t i a l . The s l ipp ing plane div ides the double layer into an immobile, layer r i g i d l y attached to the surface and a mobile layer that forms the bulk of the solut ion (Figure 2 .4) . The zeta potential determines the repuls ive forces on a pa r t i c le and thus the s t a b i l i t y of the dispersed system. 2.2g Stern and zeta potential The locat ion of the s l i p plane i s not exact ly known. The outer Helmholtz plane (stern layer) of the double layer is the plane beyond which the charge d is t r i bu t ion obeys Poisson-Boltzmann s t a t i s t i c s and the concept of a s l ip -p lane assumes the existence of a f l u i d whose v i scos i t y i s i n f i n i t e near the so l i d surface and changes sharply to the bulk value at the plane of s l i p . The re la t ionsh ip between the 4 ' Figure 2 . 4 : " The Concept of Zeta Potent ia l 22 Stern and zeta po ten t ia ls depends therefore on the l o c a t i o n of the s l i p p lane. Overbeek (23) considers the s l i p plane to be located e i t h e r in the d i f f u s e part of the double layer or in the Stern l a y e r . However i t i s known (25) that the i n t e r a c t i o n of polar solvent molecules such as water with the high e l e c t r i c f i e l d near the surface of the s o l i d r e s u l t s in a boundary layer of s t ructured water with very high v i s c o s i t y . Hence the s l i p - p l a n e l i e s outside t h i s high v i s c o s i t y region fur ther into the s o l u t i o n . Furthermore, the fac t that a l l s o l i d surfaces are m i c r o s c o p i c a l l y rough (26) ind ica tes that during shear , l i q u i d g l ides across depressions on the s u r f a c e , e f f e c t i v e l y moving the s l i p plane away from the s o l i d s u r f a c e . The pos i t ion of the s l i p plane i s thus genera l ly considered to l i e beyond the OHP. Eversole and Boardman (27) have estimated that the d istance between the s o l i d - l i q u i d in te r face and the s l i p - p l a n e for glass-water systems var ies between 8 and 63 angstroms. Hunter and Alexander (28) used the Eversole and Boardman technique and located the s l i p p i n g plane at 10 angstroms from k a o l i n i t e s u r f a c e s . Van Olphen (29) gives a value of 5 angstroms for the th ickness of the stern layer on c l a y m i n e r a l s . Thus i t appears probable that the s l i p -plane i s located approximately 10 angstroms from the sur face whereas the OHP i s at some f r a c t i o n of t h i s d i s t a n c e . For a .s imple Stern-Grahame model of the double l a y e r , the r e l a t i o n s h i p between the s u r f a c e , zeta and stern potent ia ls i s as shown schemat ica l ly in Figure 2 .2 . The zeta potent ia l i s re la ted to both the surface potent ia l IJJ and the stern potent ia l and for c o n t r o l l e d 23 c o n d i t i o n s , measured changes in the zeta potent ia l may be considered to r e f l e c t s i m i l a r changes in these other p o t e n t i a l s . In the case of the Stern potent ia l there i s usua l ly a very c lose r e l a t i o n s h i p between i t s value and that o f the zeta p o t e n t i a l . In fac t i t i s reasonable to suppose that for a l l p r a c t i c a l purposes, ^ = ? (Zeta P o t e n t i a l ) . 2,2h E f f e c t o f e l e c t r o l y t e on zeta and surface po ten t ia ls Changes in the zeta potent ia l w i l l only r e f l e c t changes in the surface potent ia l provided the ion ic strength remains constant and there i s no s p e c i f i c adsorpt ion other than by potent ia l -determin ing i o n s . Th is r e l a t i o n s h i p i s as shown in Figure 2 .5a . For example the OH" ion i s a potent ia l -determin ing ion for s i l i c a s u r f a c e s . Thus under c o n d i -t ions of r e l a t i v e l y high s a l t concentrat ions where the double layer th ickness remains r e l a t i v e l y constant , any increase in the concentra-t ion o f 0H~ ions increases the s p e c i f i c adsorpt ion o f OH" ions on (or re lease of H + ions from) the sur face . As a r e s u l t , the surface potent ia l and hence the ze ta -po ten t ia l both increase in magnitude. The r e l a t i o n s h i p between the ion ic strength and zeta potent ia l i s i l l u s t r a t e d in Figure 2 .5b. Changes in ion ic strength a f f e c t the equ i l ib r ium adsorpt ion of counter - ions in to the stern l a y e r . As a r e s u l t the surface potent ia l remains unchanged but the potent ia l at the outer Helmholtz plane and hence the ^ -poten t ia l i s a f f e c t e d . 2.2i E l e c t r o k i n e t i c phenomena The presence o f the e l e c t r i c a l double layer with i t s associated mobile and immobile layers causes pecu l ia r hydrodynamic e f f e c t s when (a) Figure 2 .5 : E f f e c t of (a) Surface P o t e n t i a l , (b) Ionic Strength on Zeta Potent ia l 25 there i s a r e l a t i v e bulk motion between the two phases. Th is e l e c t r o k i n e t i c phenomenon, which was f i r s t observed by F . F . Reyss (1809) and then by Helmholtz (1877), may a r i s e from an e x t e r n a l l y appl ied e l e c t r i c f i e l d d i rec ted along the phase boundary ( r e s u l t i n g in a r e l a t i v e movement between the phases) or from a mechanical ly i n i t i a t e d r e l a t i v e movement between phases ( r e s u l t i n g in an induced e l e c t r i c f i e l d and a t ranspor t o f e l e c t r i c i t y ) . These two s i t u a t i o n s account for four basic e l e c t r o k i n e t i c phenomena, namely a) E lec t rophores is and b) E l e c t r o - o s m o s i s , both due to the former s i t u a t i o n ; c) Streaming Potent ia l and d) Sedimentation P o t e n t i a l , both due to the l a t t e r ins tance . Other e l e c t r o k i n e t i c e f f e c t s inc lude surface conductance, re laxa t ion e f f e c t s and the e lec t rov iscous e f f e c t . a . E lec t rophores is This is the motion o f small charged p a r t i c l e s r e l a t i v e to a s ta t ionary l i q u i d by an appl ied potent ia l gradient which exerts a force on the p a r t i c l e s . The magnitude of the force and the v e l o c i t y with which a p a r t i c l e moves depends on the p a r t i c l e charge, the p a r t i c l e s i z e and several other f a c t o r s . b. E lec t ro -osmosis This i s complementary to e lec t rophores is and re fe rs to the motion o f l i q u i d through s ta t ionary charged surfaces or porous plugs by an appl ied potent ia l g rad ient . The pressure necessary to counter-balance e lec t ro -osmot ic flow is termed the e lectroosmot ic pressure . 26 c . Streaming Potent ia l This is a converse e f f e c t to e lectro-osmosis• . = V/hen l i q u i d "is forced through an i n s u l a t i n g c a p i l l a r y or porous plug by means o f an appl ied pressure g r a d i e n t , a potent ia l i s set up in the d i r e c t i o n o f f l u i d displacement as a. r e s u l t o f charge displacement caused by the flow of l i q u i d along the s o l i d s u r f a c e . Since the bulk of the l i q u i d contains a net excess of negative (or pos i t i ve ) charge, the flow of charge accompanying the flow of l i q u i d c o n s t i t u t e s an e l e c t r i c current - the convect ive c u r r e n t . However s ince the l i q u i d enter ing the system i s neutral and that leav ing c a r r i e s a net negative (or p o s i t i v e ) charge, an induced potent ia l gradient is set up which exerts a force on the ions in the l i q u i d , se t t ing up an opposing conduction c u r r e n t . At steady s t a t e , when the net current flow i s z e r o , the induced e l e c t r i c f i e l d i s c a l l e d the Streaming P o t e n t i a l . d . Sedimentation Potent ia l This i s a converse e f f e c t to e l e c t r o p h o r e s i s . When f ine p a r t i c l e s suspended in a l i q u i d are subjected to motion by a g r a v i t a -t i o n a l , u l t r a s o n i c or c e n t r i f u g a l f i e l d , a potent ia l gradient i s set up which i s sometimes re fe r red to as a migrat ion potent ia l or a Dorn (1878)(30) p o t e n t i a l , a f t e r the man who f i r s t observed i t . As the sedimenting p a r t i c l e moves downwards, some of the ions in the mobile part of the d i f f u s e layer c lose to i t are sheared 'o f f and l e f t behind as a r e s u l t o f the motion of f l u i d around the p a r t i c l e . An excess o f charge accumulates behind the p a r t i c l e over that ahead of i t , which c o n s t i t u t e s an induced e l e c t r i c f i e l d . A l s o , as a r e s u l t o f the d i s t o r t i o n in the symmetry o f the double l a y e r , the centres 27 o f g r a v i t y o f the p a r t i c l e and the d i f f u s e layer are d i s p l a c e d . Thus the sedimenting p a r t i c l e c o n s t i t u t e s a d i p o l e . Furthermore, the induced f i e l d in turn exerts a force on the p a r t i c l e and on the ions in so lu t ion which acts to retard the p a r t i c l e . Thus the sedimentation v e l o c i t y can d i f f e r apprec iab ly from that expected from pure hydrodynamic c a l c u l a t i o n s . E l e c t r o k i n e t i c phenomena occur only in the mobile part of the double layer .and therefore may be intepreted in terms of the zeta potent ia l or the charge dens i ty at the surface of shear . Theoret ica l expressions r e l a t i n g the zeta potent ia l to the measured e l e c t r o k i n e t i c e f f e c t have been reasonably well es tab l ished fo r a l l the e l e c t r o -k i n e t i c phenomena (23, 24) . Thus the zeta potent ia l may be measured by one of the e l e c t r o k i n e t i c techniques of e lec t roosmos is , streaming potent ia l or e l e c t r o p h o r e s i s . The e lec t rophores is techniques include m i c r o - e l e c t r o p h o r e s i s , which requires measurements of p a r t i c l e m o b i l i t y fo r very d i l u t e suspensions. In g e n e r a l , there are four forces that act on a moving p a r t i c l e under the in f luence of an e l e c t r i c f i e l d . F i r s t and most impor tant ly , there i s the force exerted by the appl ied potent ia l as a consequence of the charge c a r r i e d by the p a r t i c l e . Secondly, there i s a v iscous re ta rd ing force due to the flow o f l i q u i d past the p a r t i c l e . For i s o l a t e d spher ica l p a r t i c l e s in creeping f low, t h i s force i s given by Stokes' Law. The two remaining forces are due to the presence of e l e c t r o l y t e in the s o l u t i o n . Since the d i f f u s e layer contains a net excess of c o u n t e r - i o n s , there w i l l be a net force act ing on the l i q u i d due to the i n t e r a c t i o n o f the ion charges with 28 the e l e c t r i c f i e l d . The r e s u l t i n g flow o f l i q u i d (e lec t ro -osmosis ) causes a re tard ing force on the p a r t i c l e . This e f f e c t i s c a l l e d the e l e c t r o p h o r e t i c r e t a r d a t i o n . A l s o , the d i s t r i b u t i o n o f ions in the v i c i n i t y of the p a r t i c l e i s deformed as the p a r t i c l e moves away from the centre of i t s i o n i c atmosphere. The Coulombic a t t r a c t i o n between the p a r t i c l e charge and the ions tend to rebu i ld the i o n i c atmosphere, a process which takes a f i n i t e time c a l l e d the re laxa t ion t ime. Thus in the steady state the charge centre o f the d i f f u s e layer constant ly lags behind the centre of the p a r t i c l e . Consequently a d ipo le i s formed and thus an e l e c t r i c f o r c e , usua l l y a re tard ing one and c a l l e d the " re laxa t ion e f f e c t " , acts on the p a r t i c l e . These l a t t e r two forces are complicated funct ions of several physical parameters of the suspension such as the zeta p o t e n t i a l , the p a r t i c l e s i z e , the th ickness o f the double layer and the valence and m o b i l i t y o f the s p e c i f i c ions in the s o l u t i o n . In a number of l i m i t i n g c a s e s , simple expressions have been found to be a p p l i c a b l e . The f i r s t of such expressions for a d i l u t e suspension i s that due to Smoluchowski (31) , in which the re ta rda t ion and re laxa t ion e f f e c t s were neglected . A large number of workers, inc lud ing Hukel (32) , Henry (33) Overbeek (34) , Booth (35) , Lyklema and Overbeek (36) and Hunter (37) , have invest iga ted the problem of c o r r e c t l y in te rp re t ing the ro le played by the re ta rda t ion and re laxa t ion e f f e c t s . The re laxa t ion c o r r e c t i o n i s more complex s i n c e , besides being s t rong ly dependent on the r a t i o k a , i t i s a lso inf luenced by such fac tors as the magnitude o f the zeta p o t e n t i a l , the valency and the 29 m o b i l i t y of ions in the s o l u t i o n . For a c a r e f u l l y s p e c i f i e d physical system Wiersema et al . (38) presented a numerical so lu t ion that accounts for the e f f e c t s of r e t a r d a t i o n , re laxa t ion and surface conductance from a cons idera t ion of the Poisson equation and the Navier-Stokes equat ion. In 1974, Levine and Neale (39) extended the theory of e lec t rophores is for a s ing le p a r t i c l e to cover p r a c t i c a l s i t u a t i o n s invo lv ing m u l t i p a r t i c l e systems, for n e g l i g i b l e surface conductance and re laxa t ion e f f e c t s . 2.2j Sedimentation potent ia l Of a l l the e l e c t r o k i n e t i c phenomena, sedimentation potent ia l i s the l e a s t s t u d i e d . The f i r s t theore t ica l estimates o f the magnitude of the e l e c t r i c f i e l d due to sedimentation potent ia l was provided by Smoluchowski (31). He concluded E = — ^ — > k a » l [2.26] 4iTA.y * where E represents the induced e l e c t r i c f i e l d , TJJ i s the potent ia l a at the surface o f a sphere o f radius a with a very th in double layer th ickness 1/k; e , A and u are the d i e l e c t r i c constan t , s p e c i f i c c o n d u c t i v i t y and v i s c o s i t y r e s p e c t i v e l y o f the e l e c t r o l y t e ; p g and are the s o l i d and l i q u i d d e n s i t i e s r e s p e c t i v e l y ; and c i s the volume concentrat ion o f p a r t i c l e s . Smoluchowski a l s o studied the re ta rda t ion o f the s e t t l i n g rate due to the i n t e r a c t i o n o f the induced f i e l d with the double - layer and found that for a s i n g l e charged spher ica l p a r t i c l e , the observed v e l o c i t y U i s re la ted to the v e l o c i t y U s when 30 the p a r t i c l e i s uncharged by the equation k a » l [2.27] 16u 2 a 2 Ay where i s the surface p o t e n t i a l . T i s e l i u s (40) tested t h i s a r e l a t i o n exper imental ly and found agreement with the predicted order Booth (41) proposed a theory from a d i f f e r e n t approach, by consider ing the e l e c t r i c a l d ipo le moment of a s i n g l e p a r t i c l e . He then expressed the e f f e c t i v e d ipo le moment as a power s e r i e s in the charge on the p a r t i c l e surface by a technique of successive approx i -mation. In the zeroth approximation, the f l u i d flow around the spher ica l p a r t i c l e corresponds to Stokes f low. However, in the higher approximat ions, the mathematical complexity increases as a r e s u l t o f i n t e r a c t i o n o f the p a r t i c l e charge with the charge d i s t r i b u t i o n in the d i f f u s e l a y e r , which a l s o a f f e c t s the hydrodynamic flow behaviour around the p a r t i c l e . For very th in double layers t h i s complexity i s e s s e n t i a l l y second-order in nature . From the numerical estimates of Booth's ana lys is presented by Sengupta (42) , the expression fo r the v e l o c i t y of a s ing le i s o l a t e d sphere in a 1-1 e l e c t r o l y t e up to the second-order in surface potent ia l i s where G(ka) , a complicated funct ion of ka , approaches 1 as ka approaches i n f i n i t y . Booth a lso presented an approximate s o l u t i o n of magnitude o f u" s/U. - G(ka) [2.28] 16u- 2a 2Xy 31 for a suspension of p a r t i c l e s in which the double layer thickness i s e i ther much greater or much l e s s than the mean separation,between the 1/3 p a r t i c l e s , i . e . ka « c r e s p e c t i v e l y . However, t h i s s o l u t i o n i s on ly v a l i d for very d i l u t e suspensions s ince Booth neglected the e f f e c t o f hydrodynamic and e l e c t r i c a l i n t e r a c t i o n s in his s i m p l i f i c a t i o n . Other theore t ica l de r iva t ions have been c a r r i e d out to descr ibe the sedimentation p o t e n t i a l . These include those o f Dukhin (43) and o f Derjaguin and Dukhin (44) , who employed the 'Helmhol tz - type 1 double layer s t ruc ture developed by the l a t t e r workers. Recently (1976) Lev ine , Neale and Epste in (45) have published a theore t i ca l approach to the p r e d i c t i o n o f sedimentation potent ia l wi thin a m u l t i p a r t i c l e system employing a geometric c e l l model, which was a lso used by Levine and Neale (39) in t h e i r study of e lec t rophores is in m u l t i p a r t i c l e systems at 0.5 ^_c >_0. This theory i s , however, r e s t r i c t e d to vary small surface po ten t ia ls (^  < 25 mV) in a 1-1 e l e c t r o l y t e for ka > 10, i . e . a — — for a l l but very small p a r t i c l e s with r e l a t i v e l y th ick double l a y e r s . In the l i m i t i n g case o f a s ing le i s o l a t e d p a r t i c l e , the value obtained for the sedimentation v e l o c i t y by t h i s theore t i ca l treatment corresponds to that of Smoluchowski. Amaratunga (46) has measured the potent ia l gradient in a f l u i d i s e d bed o f uniform p a r t i c l e s , which i s equivalent to the streaming potent ia l gradient in a porous medium. He however regarded th is as the sedimentation potent ia l gradient and found agreement with the Smoluchowski (31) r e l a t i o n (equation 2.26) for c <_ 0.25 ± 0 .05. Above t h i s l i m i t , the measured potent ia l gradient f a l l s below the Smoluchowski p r e d i c t i o n , which i s in l i n e with the theore t ica l p red ic t ions of Levine et a l . (45). 32 CHAPTER THREE SOME THEORETICAL CONSIDERATIONS 3.1 Sedimentation of M u l t i p a r t i c l e Systems 3.1a Hydrodynamic cons idera t ions As mentioned in Chapter Two, the c e l l model technique appears to be p a r t i c u l a r l y useful for obta in ing s i g n i f i c a n t numerical r e s u l t s fo> f l u i d flow in concentrated suspensions. Th is model assumes the suspen-sion of p a r t i c l e s to be made up of un i t c e l l s each of radius b, greater than the radius a , of the p a r t i c l e enclosed in the c e l l , such that the s o l i d s volume f r a c t i o n o f the c e l l i s equal to the mean volume f r a c t i o n of s o l i d s in the overa l l suspension. For a suspension of spher ica l p a r t i c l e s , the to ta l volume of suspension V i s given by V = ^ b 3 N [3.1] and the volume V" occupied by the p a r t i c l e s i s a 3 N [3.2] where N i s the number o f p a r t i c l e s . Thus from equations 3.1 and 3 .2 , 33 the volumetr ic concentrat ion of p a r t i c l e s , c , i s c . cf)3 [ 3 - 3 ] The en t i re d isturbance due to each p a r t i c l e i s therefore conf ined to the c e l l o f f l u i d with which i t i s associated and thus the hydrodynamic problem i s reduced to a cons idera t ion of the behaviour of a s ing le p a r t i c l e and i t s bounding envelope. Assuming, for example, that the ind iv idua l p a r t i c l e s are s ta t ionary with the f l u i d moving p a r a l l e l to the z - a x i s of a reference spher ica l coordinate system, the o r i g i n o f which i s located at the centre of a p a r t i c l e , with a uniform v e l o c i t y U, then the f l u i d flow around a t y p i c a l p a r t i c l e i s governed by the Navier-Stokes equat ion. In the absence o f extraneous forces other than g r a v i t y , the Navier-Stokes equation s i m p l i f i e s fo r creeping flow by omission of i n e r t i a terms as VVZU = grad P [3.4] and the c o n t i n u i t y equation for an incompressible f l u i d i s V •• U_ = 0 ' [3.5] where IJ denotes the f l u i d v e l o c i t y v e c t o r , u the f l u i d v i s c o s i t y and P the g r a v i t y - c o r r e c t e d pressure at any point re fe r red to a datum plane. The above equations (3 .4 , 3.5} can then be solved fo r the. f l u i d v e l o c i t y d i s t r i b u t i o n subject to the appropr iate boundary c o n d i t i o n s . 34 A general so lu t ion of these set of equations (3 .4 , 3.5) as given by Lamb (47) can be wri t ten in the form U r = U o T" ( £ ) 2 " A 2 - 2 A 3 ( 7 } + 2 A 4 ( r - ) 3 ] C O s Q [ 3 ' 6 ] [ A l l ? > 2 + A 2 + A3<F J + A 4 ( f } 3 J S i n 6 [ 3 - 7 ] u e = U o 5A-,r 2A 3 a ~~2 2~ , a r J cos9 [3.8] where A Q , A-| , A,, and A 3 are constants to be determined from the hydro-dynamical boundary condi t ions adopted. In applying boundary c o n d i t i o n s , Happel (1) envisages the c e l l to possess a ' f r e e ' surface on which the normal v e l o c i t y ( for moving p a r t i c l e and no net movement of f l u i d across c e l l boundary) and tangent ia l s t ress v a n i s h . Thus for a . s t a t i o n a r y p a r t i c l e , U ( r ,6 ) = U 0 ( r , 0 ) = 0 at r = a [3.9] U (r ,6) = - U cose . . r / 9 U r 9U Q U Q \ 1 at r=b [3.10] x r 9 ( r , e ) = - y t e + ^ f - — The cond i t ion in equation 3.9 corresponds to f l u i d adherence without s l ippage on the p a r t i c l e s u r f a c e s . The .condi t ion of no tangent ia l s t r e s s component on the surface o f the outer sphere (equation 3.10) corresponds to vanishing o f the s t ress tensor components x r Q and x ^ , and the cond i t ion that U r = - U c o s Q corresponds to the c o n t i n u i t y o f 35 f l u i d flow across the f l u i d envelope ( for s ta t ionary p a r t i c l e and moving f l u i d ) . A lso because o f symmetry = 0 in the e n t i r e spher ica l s h e l l , a £ r <_b. By using the general so lu t ion o f the creeping flow equations as given by Lamb [47] , Happel obtained the expressions for the v e l o c i t y and pressure of the f l u i d from which the drag force on each const i tuent p a r t i c l e was computed as F D = -4TrpaU o D 1 [3.11] where n = 3+2lf [3.12] 1 2 - 3 y + : Y 5 - Y 6 and y = a / b - By comparison with analogous expressions for a very d i l u t e system where Stokes' Law a p p l i e s , Happel obtained the terminal s e t t l i n g v e l o c i t y of the suspension as i r = f D i C 3 - 1 3 : 0 where u"s i s the v e l o c i t y of an i s o l a t e d sphere in an i n f i n i t e medium. The boundary condi t ions as appl ied by Kuwabara (2) in his treatment are s i m i l a r to the free surface model. However the cond i t ion of no tangent ia l s t ress i s replaced by the cond i t ion that the v o r t i c i t y at the outer spher ica l envelope i s ze ro . This requires that U r ( r , 0 ) = - U cos0 CO = 3U f 9F 3U U r + r 36 r 1 -H- = 0 at r=b [3.14] 36 The values of the constants A ] , A 2 , A 3 , A 4 in Lamb's s o l u t i o n that s a t i s f y the boundary condi t ions (equations 3.9 and 3.14) as appl ied by Kuwabara thus take the form A - I Y I = 5 ( 2 n 3 ) 1 ~ ~ J ' 2 2J A _ _ 15 A = h i l l [3.15] A 3 4J ' A 4 4j where, as be fore , y = a / b . The drag force on each const i tuent p a r t i c l e fol lows as F n = 67ryaU - D 9 C 3- 1 63 D 2 0 c. where D 2 = 5 /J [3.17] and J = 5 -9 Y +5y 3 -Y 6 [3.18] By comparison with Stokes' Law the terminal v e l o c i t y as obtained by Kuwabara is given as U_ / = D 2 [3.19] 0 The major d i f f e rence in the treatment of the two models stems from the choice of the hydrodynamic boundary condi t ions at the outer envelope (r=b) of the c e l l . From equations 3,13 and 3.12 and from 3.19, 3.17 and 3.18 i t i s found that the Kuwabara model shows a higher dependency of r e l a t i v e v e l o c i t y ( U 0 / U s ) on p a r t i c l e concent ra t ion . 37 3.1b E l e c t r i c a l cons idera t ions The e f f e c t of e l e c t r o s t a t i c forces on the f l u i d flow around a p a r t i c l e becomes appreciable as the p a r t i c l e s i z e decreases. G r a v i t a -t iona l force on the other hand becomes less important. For very small p a r t i c l e s o f s i z e s l e s s than about 50 ym, the Navier-Stokes equation s i m p l i f i e d for creeping flow must therefore be modif ied to include the e f f e c t o f e l e c t r o s t a t i c forces due to the existence of double layers on the p a r t i c l e s u r f a c e s . Equation 3.4 i s modif ied to y/72U = grad P + F [3.20] Here F denotes body force due to e l e c t r o s t a t i c e f f e c t s per un i t volume of f l u i d . In the equ i l ib r ium sta te o f zero f l u i d f low, the potent ia l in the d i f f u s e layer ty{r) depends on the d is tance r from the centre o f the p a r t i c l e . I f ib(r) i s small enough to permit the use o f the l i n e a r i s e d Poisson-Boltzmann equation ( i , e , i f the Debye-Huckel approximation app l ies ) then V2<Kr) = k2<Mr) [3.21] which i s the same as equation 2 .18. When the p a r t i c l e i s under the in f luence o f g r a v i t y , the d i f f u s e layer around each p a r t i c l e loses i t s symmetry as a r e s u l t o f the re laxa t ion e f f e c t as descr ibed in Chapter Two. Consequently a perturbat ion potent ia l <f>(r>9) i s set up. Adopting the assumption employed by Henry (33) in h is c l a s s i c a l work on e lec t rophores is that 38 the induced potential, can be taken as simply superimposed on the f i e l d due to the e l e c t r i c a l double l a y e r , the body force F can be expressed as where p i s the charge per uni t volume and grad (<j>+^) the force on uni t charge. Equation 3.20 thus becomes A so lu t ion of equation 3.23 together with the c o n t i n u i t y equation thus requi res a knowledge of ty and <f>. A s o l u t i o n o f the l i n e a r i s e d Poisson-Boltzmann equation as given by Mol ler et al (48) , subject to the boundary condi t ions appropriate to the c e l l model geometry, F = p.grad ($+ty) [3.22] yV2U_ = Vp + pV(<|>+i|0 [3.23] *(a) = * a [3.24] and r=b = 0 [3.25] is ty{r) = ( a / r ^ a [ M 1 e - M r + M 2 e w ] [3.26] where i f [3.27] 39 then M, = [3.28] The cond i t ion in equation 3.25 stems from the a p p l i c a t i o n of Gauss' theorem and the fac t that the system i s e l e c t r i c a l l y n e u t r a l , so that each ind iv idua l c e l l possesses no net charge. The nature of the sedimentation p o t e n t i a l , cj>, i s very complex and the treatment presented by Levine et al (45) i s the only quant i ta t ive theory in the l i t e r a t u r e . In th is theory , they approximated the potent ia l by the s i m p l i f i e d form <t>(r,9) = U[xr + mr" 2 + g(r )J cos9 [3.29] Here, the f i r s t term xr descr ibes the potent ia l due to the g l o b a l l y _2 induced uniform sedimentation f i e l d . The second term mr i s the potent ia l due to m, the d ipo le moment r e s u l t i n g from displacement of charge in the d i f f u s e l a y e r . The l a s t term g(r) i s the potent ia l due to re laxa t ion e f f e c t as a r e s u l t of d i s t o r t i o n in the r a d i a l l y symmetric d i f f u s e l a y e r . Thus i f the l a s t term in <|> i s neg lec ted , <J> becomes r a d i a l l y symmetric and s a t i s f i e s Lap lace 's equat ion , i . e . i f <f>(r,0) = (Mr .e j = U[xr + - 2 ] cose [3.30] r then v 2 cf> 0 (r ,e) = 0 [3.31] 40 3.1c Sedimentation v e l o c i t y There are three forces that act on a sedimenting p a r t i c l e , and the resu l tan t e f f e c t becomes zero at steady s t a t e . Thus f^ denotes the hydrodynamic force due to the f l u i d flow past the p a r t i c l e , f the e l e c t r i c a l force as a consequence of the sedimentation potent ia l and f the bouyancy-corrected force due to g r a v i t y . A so lu t ion of equation 3.23 i s thus required for the evaluat ion o f f^. Levine and Neale (39) gave a general so lu t ion of t h i s problem in the i r theory of e lec t rophores is based on the so lu t ion of Henry (33) for a s i n g l e p a r t i c l e . The r e s u l t s obtained for the v e l o c i t y p r o f i l e may be expressed a s : where H and E denote the hydrodynamic and e l e c t r i c a l cont r ibu t ions r e s p e c t i v e l y . The hydrodynamic parts are of the same form as Lamb's (47) so lu t ion given in equations 3.6 and 3 .7 . I f the port ion of the e l e c t r i c a l con t r ibu t ion which i s due to the e lectrosmot ic e f f e c t s of the ion movement in the e l e c t r o l y t e (corresponding to e lec t rophore t i c re tardat ion) i s neg lec ted , then applying the Kuwabara zero v o r t i c i t y type of boundary c o n d i t i o n s , the hydrodynamic drag on the p a r t i c l e i s given by f H + fe + V° [3.32] [3.24] 41 f H = 2T ra 2 (p cos0 + T n s ine) x sinBde o [3.34] where p = -p + 2y r r r 3r which reduces to f H = 6uyaUD 2 [3.35] The e l e c t r i c a l force on the p a r t i c l e c o n s i s t s of two components. The f i r s t i s the force of the loca l induced f i e l d on the p a r t i c l e charge, The second i s the re tardat ion force which a r i s e s because the centre of the d i f f u s e layer charge lags behind the centre of the p a r t i c l e charge (an e f f e c t analogous to the re laxa t ion e f f e c t in e l e c t r o p h o r e s i s ) . The to ta l e l e c t r i c force in the v e r t i c a l z - d i r e c t i o n as given by Wiersema et al (38) i s a (gradT-kJ ds s [3.36] /•IT = -a2Tra 2 sinede | | -Jo where a = surface charge dens i ty s = surface o f sphere. tn order to evaluate the terms appearing in the s i m p l i f i e d form of <K'r,9), Levine et al adopted three boundary condi t ions and steady state cons idera t ions appropr iate to the c e l l model geometry: 42 1. At the surface of the p a r t i c l e , the rad ia l component of current flow in the d i f f u s e layer vanishes for non-conducting spheres. f e . = 0 1 , e * Sr r=a u 2. For the requirement o f a steady state streaming potent ia l c o n d i t i o n , the net flow o f e l e c t r i c current across any hor izonta l plane by f l u i d convection must be equal and opposite to that due to e l e c t r i c a l conduct ion . 3. As a cond i t ion for c o n t i n u i t y , the r a d i a l component of the l o c a l induced e l e c t r i c f i e l d must be equal to the rad ia l component of the ex te r io r macroscopica l ly uniform induced e l e c t r i c f i e l d . Furthermore, by comparison with the re la ted treatment of e lec t rophores is by Levine and Neale (39) in which the re laxa t ion e f fec ts were neg lec ted , the e l e c t r i c a l force was s i m p l i f i e d as [3.37] where b b 5 I 5 ( r ) dr + 5c c£)'? £Cr) dr A = S ( r ) -2(1-c)Cm-mp) a* and m [3.38] N 2(m-m0) 43 The buoyancy-corrected g rav i ta t iona l force f i s given by f . - ^ T - t P , ^ ' [ 3 - 3 9 ] where p and p 0 are the p a r t i c l e and f l u i d d e n s i t i e s r e s p e c t i v e l y . On s u b s t i t u t i n g for f^, f g and f from equations 3 .35 , 3.37 and 3 .39, equation 3.32 reduces to BTrauUD, - e aE,-A = ^ 9 C P S - - P f r ) g [3.40] 3 The corresponding equation for an uncharged m u l t i p a r t i c l e suspension reads 6TryaU o D 2 = ^ E | l ( p ^ p ^ g [3.41] D iv id ing 3.41 by 3.40 y i e l d s an expression for U Q /U as U e aE. A _P_ = I + s L [3.42] U 1 6%iaUD 9 After s u b s t i t u t i n g for E^ in terms of ka and c , and for A (equation 3 .38) , equation 3.42 reduces to the more general form obtained by Levine et al . (45) i r - = 1 + - H(ka.c) [3.43] U 16T72a2'Xu 44 where H(ka,c) is a complicated funct ion of ka and c shown g r a p h i c a l l y in t h e i r Figure 4 and represented by t h e i r equation 4 .11 . For d i l u t e suspensions, where c+o, and for large ka , H(ka,c) -»- 1; therefore equation 3.43 reduces to that of Smoluchowski (30) fo r a s ing le p a r t i c l e . A l s o , for uncharged p a r t i c l e s , where ty = 0, equation 3.43 c o l l a p s e s to a U = U. o Based on the above assumptions, however, the theory depicted by equation 3.43 i s r e s t r i c t e d to small values of i M ~ 25 mV) and a r e l a t i v e l y large values of ka(>. 10) . Never the less , equation 3.43 ind ica tes that charged p a r t i c l e s s e t t l e slower than uncharged ones. 45 CHAPTER FOUR EXPERIMENT 4.1 Introduct ion The main object o f the experimental work was to inves t iga te the e f f e c t of surface potent ia l ty&, double layer th ickness k"1 and p a r t i c l e concentrat ion on hindered s e t t l i n g in the creeping flow regime, and to tes t how well the c o r r e l a t i o n of Levine et a l . (45) as given by equation 3.43 agrees with experiment. It was therefore des i rab le to carry out the experiments under condi t ions s i m i l a r to those assumed by Levine et a l . 4.2 C o l l o i d a l Mater ia l 4.2a P a r t i c l e s In order to meet the condi t ions implied in the theory , a model c o l l o i d a l suspension of d i s p e r s e d , sub-micron, uni formly s i z e d , spher ica l p a r t i c l e s , in which e l e c t r o s t a t i c e f f ec ts were a p p r e c i a b l e , was r e q u i r e d . A l s o , to f a c i l i t a t e experimental manipu la t ions , a c o l l o i d a l system that i s s tab le over a wide range of e l e c t r o l y t e con-d i t i o n s was p r e f e r a b l e . There are a number o f commercial ly a v a i l a b l e 46 p a r t i c l e s which meet the former requirement o n l y . S i l i c a suspensions are known to be very s table over a wide range of e l e c t r o l y t e condi t ions and the surface chemistry of s i l i c a has been ex tens ive ly studied (49, 50). The most common form i s pyrogen!c or f lame-hydro l ised s i l i c a , which i s produced by the hydro lys is o f s i l i c o n hal ides in a hydrogen-oxygen f lame. However, t h i s method produces p a r t i c l e s i z e s (200 mesh, US-standard s ieves) outs ide the range o f i n t e r e s t . Another method for the production of amorphous s i l i c a i s that o f p r e c i p i t a t i o n by the reac t ion of t e t r a - e s t e r s o f s i l i c i c ac id ( t e t r a - a l k y l s i l i c a t e s ) with water, developed by Stflber (51) . This react ion i s c a r r i e d out in a mutual solvent and i s cata lysed by ac id or base s o l u t i o n s . According to Ae l ion et al . (52) , the hydro lys is of t e t ra -e thy l o r t h o s i l i c a t e (TEOS) fol lows two s teps : S i ( 0 C 2 H 5 ) 4 + 4H2Q + S i ( 0 H ) 4 + 4C 2 H 5 0H S i ( 0 H ) 4 + S i 0 2 + + 2H 20 invo lv ing hydro lys is and dehydrat ion , the f i n a l product o f which i s armophous s i l i c a . In the presence o f hydroxy! ion c a t a l y s t and an excess o f water, both the hydro lys is and the dehydrat ion react ions are f as t and complete. The hydro lys is reac t ion i s r a t e - c o n t r o l l i n g and is f i r s t order with respect to the TEOS concen t ra t ion ; the extent o f reac t ion i s proport ional to the amount o f base present . StOber et al . (51) found that when the mutual solvent i s an alcohol and the c a t a l y s t i s ammonium hydroxide, the reac t ion produces s i l i c a spheres of very narrow s i z e d i s t r i b u t i o n in the s i z e range of 0.05 to 2.0 ym, depending on the type of a lky l group of the s i l i c a t e and on the mutual s o l v e n t . The longer the a lky l c h a i n , the la rger 47 the p a r t i c l e s . This method of product ion , which was ex tens ive ly studied and employed by Bowen (53) , was thus adopted here. 4.2b Preparat ion of s i l i c a p a r t i c l e s The type of reactants and the condi t ions under which the react ions were c a r r i e d out were based p r imar i l y on the experimental conclusions of Bowen (53) . The uni formi ty of the s i l i c a so ls produced by the methods o f Stflber et al . can be explained by the l im i ted s e l f - n u c l e a t i o n model of LaMer and Dinegar (54) , according to which a l l p a r t i c l e nucle i are formed almost simultaneously and fur ther p r e c i p i t a t i o n contr ibutes only to p a r t i c l e growth. It was therefore necessary to maintain a well s t i r r e d reac t ion and to introduce the rea.ctant (TEOS) almost ins tantaneously , so that the nucleat ion period was r e s t r i c t e d in time and separated from the growth p e r i o d . Bowen observed that the maximal s p h e r i c i t y and uni formity of s i z e of p a r t i c l e s depended on the use o f pure reactantsrand the use o f maximum ammonium hydroxide concent ra t ion . Thus a l l the esters (TEOS) used were r e d i s t i l l e d in a s p e c i a l l y constructed vacuum f r a c t i o n a t o r . Th is was a 100 cm long x: 18 mm diameter g lass column packed with 5 mm s i n g l e - t u r n g lass he l i ces (fenske pack ing) , equipped with a magnet ica l ly -operated re f lux head and vacuum-jacketed to prevent heat l o s s e s . Nitrogen dr ied with s i l i c a gel was introduced into the column to maintain i t water-dry p r io r to operat ion o f the column. A l s o , to promote b o i l i n g , the same ni t rogen was bubbled in to the r e b o i l e r through a th in g lass c a p i l l a r y . Each batch of reactant was twice d i s t i l l e d with a re f lux r a t i o of 5:1 and under a vacuum of 740 mm Hg. This produced pure reactants with a b o i l i n g point range of 0.1 C ° . The maximum ammonium concentrat ion 48 was obtained by completely satura t ing the a lcohol -water mixture in the react ion bot t l e p r io r to the add i t ion of the T . E . O . S . Higher water concentrat ions were favourable to obta in ing high ammonium concent ra t ions , s ince ammonia i s far more so lub le in water than in a l c o h o l . Bowen found that the best r e s u l t s could be achieved using TEOS/H^O/alcohol volume r a t i o s of 1:5:25. To produce p a r t i c l e s having mean diameters of about 0.5 ym the system employed was T E O S / ^ O / n - p r o p a n o l . The general sequence of steps fol lowed for the production of a batch of the s i l i c a p a r t i c l e s i s as f o l l o w s : Accurate quant i t i es o f the required alcohol and f r e s h l y d i s t i l l e d water were measured into a reac t ion bot t l e equipped with a motor-operated s t i r r e r that mixed the contents of the b o t t l e . Placed in a constant temperature bath, the bot t le and i t s contents were maintained at 25.0 ± 0.1 °C . The bot t le was a lso provided with a p l e x i g l a s s cap which had a centra l bore , through which the a x i a l l y running s t i r r e r shaf t passed. There were two other openings in the cap . One was connected to t u b i n g , which allowed for the i n j e c t i o n of TEOS, and the other was provided with a screwed-on p l u g , which allowed for the discharge o f product by s u c t i o n . Ammonia from a c y l i n d e r was bubbled in to the water-alcohol mixture through a g lass f r i t u n t i l sa turat ion was ach ieved. The d i s t i l l e d t e t r a - e t h y l - o r t h o s i l i c a t e ( T . E . O . S . ) was then in jec ted at once into the react ion bot t l e from a s y r i n g e . The cap was then f i t t e d on t i g h t l y and the openings c losed to prevent any escape of ammonia. An increas ing opalescence of the mixture , i n d i c a t i n g s t a r t o f the growth p e r i o d , occurred within the f i r s t few minutes a f te r adding 49 the T . E . O . S . The reac t ion was allowed to proceed for about two hours to ensure comple t ion , when samples for e lec t ron microscope examination were taken. Large quant i t i es o f r e a c t a n t s , a lcohol and water e s p e c i a l l y , were required to produce r e l a t i v e l y small quant i t i es of p a r t i c l e s . Thus the s i z e o f the reac t ion bo t t l e depended on the quant i ty of p a r t i c l e s r e q u i r e d . However, too big a react ion b o t t l e made contro l of reac t ion condi t ions by thorough mixing very d i f f i c u l t . The optimum reac t ion bot t l e s i z e se lected was 4.5 l i t r e s and the quant i t i es of reactants used per batch were TEOS : 0.10 l i t r e s D i s t i l l e d & De- ionised water : 0.498 l i t r e s n-Propanol : 2.488 l i t r e s which y ie lded 26.7g of p a r t i c l e s . Three batches .were prepared; in a l l . 4.2c P a r t i c l e s i ze measurements Af te r about two hours of r e a c t i o n , samples of the p a r t i c l e s a l ready in a very high state of d i s p e r s i o n in the reac t ion bot t le were taken for e lec t ron microscope examination. The samples were prepared by p lac ing a drople t of the suspension on a copper t ransmission e lec t ron microscope c a r r i e r g r id covered with a carbon f i l m , using a glass c a p i l l a r y to e f f e c t the t rans fe r from the react ion b o t t l e . The sample was then allowed to evaporate to dryness leav ing the p a r t i c l e s on the f i l m . Four or f i v e samples were prepared in t h i s manner and e lec t ron micrographs [Figure 4.1] o f these samples were obtained using a H i tach i HU 11 A t ransmission e lec t ron microscope, the magni f ica t ion of which was obtained by means o f a germanium shadowed carbon r e p l i c a of 54864 l i n e s / i n . 5 0 x I X 9 0 5 0 Figure '4.1 : E lec t ron Micro-graphs of 3 i l i c a P a r t i c l e s . X 8 3 2 0 X 1 4 4 0 0 51 The average p a r t i c l e s i ze and standard dev ia t ion were then determined by manually measuring about 100 p a r t i c l e s from several random micrographs of the gr id sample. (See Appendix B Sec . 33) . 4.2d Separat ion o f p a r t i c l e s The contents o f the react ion b o t t l e , s t i l l in a h igh ly -d ispersed s t a t e , were sucked into a 500 ml volumetr ic f l ask in batches of about 300 ml using a vacuum suct ion arrangement. Each batch was then f i l t e r e d under vacuum using a 90 mm membrane f i l t e r (0.2 ym, ce lo ta te ) posi t ioned in a 600 ml f r i t t e d glass Buchner f u n n e l . Following Bowen (53) , the s t i l l wet f i l t e r cake was then washed twice in n-propanol in order to remove n o n - v o l a t i l e contaminants and to f a c i l i t a t e the subsequent r e d i s p e r s i o n of the dr ied p a r t i c l e s in water. Thus the wet f i l t e r - c a k e was t rans fe r red to a beaker conta in ing 200 ml of n -propanol . The contents o f the bo t t l e were redispersed by means of a polytron homogeniser. The redispersed suspension was then f i l t e r e d again and dr ied under vacuum for 48 hours at 100°C in a L a b - l i n e Instruments DUO-Vac oven. The add i t ion o f heat in the drying process was to e l iminate the contaminating NH^ ion from the reac t ion v e s s e l , in order to reduce the c o n d u c t i v i t y o f the subsequent dispersed suspension in water. 4.3 Equipment and Techniques 4.3a S e t t l i n g apparatus S e t t l i n g experiments were c a r r i e d out in g lass tubes of 11.0 mm diameter and 400 mm l e n g t h . The s i ze o f the tubes were selected with 52 cons idera t ion to the l i m i t e d quant i ty of p a r t i c l e s (Sec. 4 .2b) a v a i l a b l e fo r the preparat ion o f suspensions of the required concent ra t ions . Larger tube diameters required greater amounts of p a r t i c l e s for suspen-s ions o f the same concen t ra t ion . However smal ler tube diameters introduce the undesirable e f f e c t o f c a p i l l a r i t y and the hydrodynamic e f f e c t due to the w a l l s . The length of the tubes were chosen to al low for a reasonable height o f f a l l of the s u s p e n s i o n - l i q u i d i n t e r f a c e , and at the same time not to i n o r d i n a t e l y increase the e f f e c t i v e volume of the tube which would lead to an excessive demand on the quant i ty of p a r t i c l e s . The bottom end o f the g lass tube columns were sealed with f l a t rubber d i s c s of o r i g i n a l diameter equal to the outs ide diameter of the tubes , using s i l i c o n e g l u e . The v e r t i c a l l y mounted columns were uniform in diameter , so that the s o l i d s volumetr ic concentrat ion could be c a l c u l a t e d by measuring the i n i t i a l height o f suspension, knowing the weight and dens i ty of the p a r t i c l e s . The top ends o f the columns were f i t t e d with rubber bungs greased with s i l i c o n e vacuum seal to pre-vent evaporation from the suspension. Four columns, a l l of the same diameter and l e n g t h , were used to obtain r e s u l t s at d i f f e r e n t concentrat ions at the same t ime. Each tube was mounted on aluminium u-channel provided with t e r r y - c l i p s to al low for easy removal and replacement. The u-channels were a lso held on a support ing frame (car r ied on an angle i ron rack) by two screws ' located at the ends, one o f which ran through a s l o t to -a l low for l eve l adjustments by t i l t i n g and t ighten ing as r e q u i r e d . Thus with the help o f a s p i r i t l e v e l , the v e r t i c a l p o s i t i o n o f each tube was assured by adjust ing the p o s i t i o n of the screw in the s l o t . S u f f i c i e n t space 53 was allowed between tube pos i t ions on the rack to ensure independence o f each. The complete assembly o f the apparatus i s shown in Figure 4 .2 . 4.3b Instruments For c l e a r v i s u a l i s a t i o n and accurate l o c a t i o n of the p o s i t i o n of the in te r face in the s e t t l i n g tubes, a t r a v e l l i n g microscope was used. This was mounted on a laboratory j i f f y jack placed on the same leve l with and fac ing the s e t t l i n g apparatus. It was necessary to measure the zeta p o t e n t i a l , which was c l o s e l y re la ted to the surface potent ia l of the p a r t i c l e s (See Sec . 2 .2g ) , in order to assess the e f f e c t of changes in surface p o t e n t i a l . This was accomplished by the technique of m i c r o - e l e c t r o p h o r e s i s , which involves the measurement of the ind iv idua l m o b i l i t i e s for a large number of p a r t i c l e s . The Rank Bros . Mark II microe lec t rophores is apparatus was employed for the measurement of such m o b i l i t i e s . This apparatus c o n s i s t s e s s e n t i a l l y of a transparent rectangular or c y l i n d r i c a l c e l l in to which a sample o f the suspension i s p laced , a system of e lectrodes for applying a potent ia l gradient wi th in the suspension, and a micros -cope for the observat ion of p a r t i c l e motion as a consequence of the appl ied f i e l d . Deta i ls o f th is apparatus can be found in Sect ion 4 .4b . The c o n d u c t i v i t y and pH changes of the suspensions in the tubes were measured i n - s i t u . This was done to reduce p a r t i c l e loss through handl ing . In t h i s manner the concentrat ion of p a r t i c l e s in each tube was maintained f a i r l y constant . The c o n d u c t i v i t y was measured with a Stebold c o n d u c t i v i t y meter Model LTA using a s p e c i a l l y constructed c o n d u c t i v i t y m i c r o - c e l l of diameter 8 mm, which was l e s s than those of U — 5 to to- 4 1 Column-Supporting frame 2 Settling column 3 Aluminium U-channel 4 Terry-clips 5 Holding screws 6 Screw slot Figure 4 .2 : S e t t l i n g Apparatus 55 the tubes. The m i c r o - c e l l was constructed by cas t ing two platinum wires (e lect rodes) connected to two cable leads with co ld -curve epoxy r e s i n s , such that the exposed ends o f the platinum wires and the d is tance between them was s u f f i c i e n t to give a c e l l constant wi thin the range s u i t a b l e to the meter. The cab le ends were connected to the meter through banana plugs into sockets provided on the meter. The c e l l constant obtained was 0 .71. The c o n d u c t i v i t y meter was p e r i o d i c a l l y c a l i b r a t e d against standard so lu t ions o f KC1, the c o n d u c t i v i t y o f which was obtained from standard sources (55) . A c a l i b r a t i o n curve r e l a t i n g the concentrat ion o f NaCl in so lu t ion to the measured c o n d u c t i v i t y as given in Appendix B was used to determine the NaCl content o f suspensions at higher e l e c t r o l y t e concent ra t ions . The pH was measured with a Corning Model 101 d i g i t a l electrometer with a Beckman combinat ion, futura s t y l e e lect rode that f i t t e d in to the tubes. The pH meter was a lso p e r i o d i c a l l y c a l i b r a t e d against buf fer so lu t ions of known pH. In order to e f f e c t i v e l y red isperse the dr ied p a r t i c l e s in water at var ious concen t ra t ions , the Brinkmann Polytron homogeniser was used. This i s p r i n c i p a l l y made up of two parts - the basic assembly that c o n s i s t s of a high-speed s i n g l e phase 115V, 60 Hz motor and speed control un i t mounted on a s t u r d y , upr ight stand with weighted enamelled base and holder for beakers; and the generator , which i s a kind o f probe in the form of a m u l t i - s l o t t e d tube with a shaf t connected to the high speed rotor o f the motor to produce frequency shear/shock waves. This shearing act ion i s enhanced by the saw^like c o n f i g u r a t i o n , of the teeth at the circumference of the s ta tor part of the generator . The generator is constructed of 316 s t a i n l e s s s t e e l . 56 The degree o f d ispers ion was known by examining a sample of the suspension using a Wild-Heerbrug Model M20 o i l - immers ion microscope. 4.4 Measurements and Observations 4.4a S e t t l i n g experiment Experiments were c a r r i e d out to determine the rate o f s e t t l i n g of the prepared p a r t i c l e s at var ious concentrat ions and e l e c t r o l y t i c condi t ions from the measurement o f in te r face v e l o c i t i e s . P a r t i c l e suspensions ( c o l l o i d s ) were prepared outs ide the s e t t l i n g tubes in 250 ml t a l l beakers. As mentioned in Sect ion 4 .2b , one o f the major experimental l i m i t a t i o n s was the i n a b i l i t y to prepare large q u a n t i t i e s of p a r t i c l e s without wide p a r t i c l e s i z e d i s t r i b u t i o n . There-fore the quant i ty of suspension prepared in a batch was not more than 50 m l , which was s u f f i c i e n t to f i l l a s e t t l i n g tube to about 5 cm from the top with l i t t l e l e f t over . For var ious p a r t i c l e concent ra t ions , a c a l c u l a t e d quant i ty o f the p a r t i c l e s based on the measured dens i ty of the p a r t i c l e s (see Appendix B Sec'..B2) was weighed into the t a l l c a l i b r a t e d beaker. Freshly d i s t i l l e d and deionised water (using a calgon mixed-bed car t r idge de ion izer ) was added t i l l the content was 50 m l . The beaker and i t s contents was then placed in a water-cooled jacket and dispersed using the Polytron homogenizer. The progress o f the d i s p e r s i o n p rocess , which las ted a few minutes, was fol lowed by measuring the change with time of the c l u s t e r - s i z e d i s t r i b u t i o n using the Wild-Heerbrug microscope. A very high degree of d i s p e r s i o n , u s u a l l y over 99% s i n g l e p a r t i c l e s , was at ta ined in a short t ime. 57 The suspension was then t ransfer red into the s e t t l i n g tube. The i n i t i a l height o f the suspension in the tube was noted. A f te r the experiment, the suspension was c a r e f u l l y f i l t e r e d in a pre-weighed f i l t e r and dr ied to obtain the accurate weight o f p a r t i c l e s . The actual con-cent ra t ion of the suspension was therefore ca lcu la ted from the weight of p a r t i c l e s , suspension i n i t i a l he igh t , dens i ty of p a r t i c l e s and ins ide diameter of tube. The v e r t i c a l alignment of the tube was adjusted using a s p i r i t l e v e l . Conduct iv i ty and pH measurements were made by p lac ing the c o n d u c t i v i t y c e l l (probe) and the pH e lectrode d i r e c t l y into the tube. The temperature of the room was c o n t r o l l e d at about 2 5 ° C , but was how-ever p e r i o d i c a l l y measured with thermometer #802417. The tube was then sealed with a rubber bung greased with s i l i c o n e vacuum seal to prevent evaporat ion . The supernatant l i q u i d - s u s p e n s i o n in te r face pos i t ion was then recorded at time i n t e r v a l s of several hours over periods ranging from 20 to 100 hours, depnding on the c o n c e n t r a t i o n , using a wr is t watch and the t r a v e l l i n g microscope. The in te r face was s u f f i c i e n t l y sharp that i t s pos i t ion could be measured to better than 0.01 mm. Af te r the in te r face had f a l l e n over a substant ia l height ( t y p i c a l l y about 2 cm) at what was estimated to be a constant s e t t l i n g r a t e , the readings were stopped. In some cases the readings were d iscont inued because the i n t e r -face was no longer sharp , due presumably to the e f f e c t o f Browm'an motion. A p lo t of in te r face height versus time was then made and the slope of the best f i t l i n e obtained by regress iona l a n a l y s i s was taken to r e -present the v e l o c i t y of s e t t l i n g (see Appendix A ) . The suspension was, however, allowed to s e t t l e fur ther with no readings recorded for some t ime, to produce s u f f i c i e n t quant i t i es of supernatant l i q u i d (about 7 m l ) . About 5 ml of t h i s c l e a r l i q u i d was withdrawn from the tube 58 using a s y r i n g e . Very small quant i t ies o f the suspension were a lso withdrawn and added to the separated l i q u i d , to provide a very d i l u t e suspension with e l e c t r o l y t i c cond i t ions s i m i l a r to those e x i s t i n g in the tube and su i tab le for p a r t i c l e zeta potent ia l measurement. A f te r th is the d i l u t e suspension was returned to the tube and the e l e c t r o l y t i c condi t ions a l t e r e d . Changes in pH were e f fected by adding e i ther NaOH ( 'cer t i f ied ACS, Fisher) or HC1 ( c e r t i f i e d ACS, Fisher) standard so lu t ions to the suspensions in the tubes by means o f s y r i n g e s . The e l e c t r o l y t e concentrat ions were a lso increased by adding preweighed quant i t i es of NaCI. (reagent grade, ACS). These were i n i t i a l l y d i s s o l v e d in small quant i t i es o f the supernatant l i q u i d and then t rans fer red in to the tube by s y r i n g e s . However, pre-determined changes in pH could not be achieved by adding c a l c u l a t e d quant i t i es of NaOH or HCT . This i s because the OH" ion i s a potent ia l -determin ing ion for s i l i c a (50) and increase in the concentrat ion of 0H~ ions therefore resu l ted in an increase in e i ther the s p e c i f i c adsorpt ion o f OH" ions on , or the re lease of H + ions from, the surfaces of the p a r t i c l e s . Consequently the pH of the suspension was d i f f e r e n t from the pre -deter -mined v a l u e , u s u a l l y lower. The adjustment of the pH in the tubes was therefore p r imar i l y based on previous experience and was e f fec ted with the use of the pH meter. In th is way, the adjustment o f pH las ted cons iderab ly longer than expected, s ince d i s c r e t e add i t ion of NaOH or HC1 had to be made u n t i l the des i red change had been a t t a i n e d . More-over the suspension, which was ag i ta ted a f t e r each add i t ion of NaOH or HC1, was allowed some time to s e t t l e so that subsequent addi t ions were made d i r e c t l y in to the supernatant l i q u i d to avoid l o c a l i s e d high 59 concentrat ions o f HC1 or NaCl that might cause l o c a l i s e d coagu la t ion . A f te r the required change in pH or e l e c t r o l y t e concentrat ion had been e f f e c t e d , the suspension was given a f i n a l manual a g i t a t i o n to r e -d isperse a l l the sediments from the previous run. The pH and con-d u c t i v i t y were then recorded fo l lowing the same procedure as be fore , and the s e t t l i n g experiment repeated. Rep l ica t ion of s e t t l i n g data was found to be very good, usua l ly wi thin a v a r i a t i o n o f about 1%. The double layer thickness was ca lcu la ted from a knowledge of the concentrat ion of e l e c t r o l y t e (counter i o n s ) , which at high e l e c t r o l y t e concentrat ions was obtained from conduc t i v i t y measurements. A sample c a l c u l a t i o n can be found in Appendix B. At high pH v a l u e s , the e l e c t r o l y t e concentrat ion could be evaluated from a knowledge of the pH. It was necessary to determine the rate of s e t t l i n g of the p a r t i c l e s when they are uncharged in order to assess the magnitude o f the e f fec t due to surface charges. The i s o e l e c t r i c point ( i . e . point of zero surface charge) for s i l i c a occurs at pH between 2 and 3. However i t was very d i f f i c u l t to d isperse the p a r t i c l e s at th is pH (which is in agree-ment with the mechanism (50) of adsorpt ion of hydroxyl and hydroxomium ions that determine the sufrace p o t e n t i a l ) . The p a r t i c l e s were therefore d ispersed at pH's between 3 and 3 .5 , depending on the concentrat ion of the suspension, in which case the d ispersed p a r t i c l e s possessed n e g l i g i b l e surface charges which were presumed to have l i t t l e e f f e c t on the sedimentation r a t e . The r e s u l t s obtained at these pH's were thus taken to represent the sedimentation rate of uncharged p a r t i c l e s at various p a r t i c l e concent ra t ions . 60 4.4b Measurement of zeta potent ia l The p a r t i c l e zeta potent ia l was measured by the technique of m i c r o - e l e c t r o p h o r e s i s , which involves the measurement of p a r t i c l e m o b i l i t i e s . M o b i l i t y measurements were made using the Rank Bros . Mark IT e lec t rophores is apparatus, in which a rectangular o p t i c a l glass c e l l was employed in preference to a c y l i n d r i c a l one. The rectangular c e l l possessing c r o s s - s e c t i o n a l dimensions approximately 0.1 x 1.0 cm had the advantage of permit t ing measurements at s ta t ionary l e v e l s , as d iscussed below, without in te r fe rence from sediments r e s u l t i n g from the f a l l o f large p a r t i c l e s in the c e l l . Samples of the d i l u t e suspension prepared with the supernatant l i q u i d r e s u l t i n g from a s e t t l i n g experiment were t rans fer red into the c e l l , which was clamped within a constant temperature bath maintained at 25 ± 0 . 1 ° C . Br ight platinum e l e c t r o d e s , constructed o f 13x26x0.013 mm sheets r o l l e d into c y l i n d e r s and sealed into standard g lass f i t t i n g s , were inser ted into e lectrode compartments at both ends of the c e l l . The g lass f i t t i n g s and the e lect rode compartments mated together such that the ends o f the c e l l were sealed to prevent evaporation that might cause unwanted convect ive currents wi th in the c e l l . D i rec t t ransmitted i l l u m i n a t i o n was provided by a 12V, 100-watt quartz iodine lamp. The l i g h t beam could be focussed through an adjustable s l i t at any point wi thin the c e l l by means of con t ro ls provided on the lamp housing and prov is ion for the r e l a t i v e movement of the i l l u m i n a t i n g o b j e c t i v e . The beam a lso passed through a heat f i l t e r p r io r to enter ing the c e l l to reduce convect ive e f f e c t s . Having brought the appropr iate part of the c e l l under the observing o b j e c t i v e , the observing microscope could now be focussed on var ious marks on the c e l l wall and on the 61 p a r t i c l e s . The movement of p a r t i c l e s could then be observed. The observing microscope with a 2Qx ob jec t ive and lOx eyepiece was attached to a micrometer stage and hence could be focussed within the c e l l at any point along i t s a x i s . To e l iminate any p o s s i b i l i t y of back lash , the micrometer was turned in one d i r e c t i o n only for one determinat ion. The eyepiece o f the microscope a l s o contained a g r a t i c u l e with a square gr id over which the p a r t i c l e s were t imed. The gr id s p a c i n g , which was c a l i b r a t e d against a stage micrometer (see Appendix B, Sec . B1 .2 ) , was found to be 63 .79 ym. The "complex" geometry of the rectangular c e l l (compared to the simpler geometry of the c y l i n d r i c a l c e l l ) required the est imat ion of the potent ia l gradient from measurement of s o l u t i o n c o n d u c t i v i t y , e l e c t r i c a l r es is tance and viewing region c r o s s - s e c t i o n a l a rea . Thus i f R i s the res is tance between the two e lect rodes when the c e l l i s f i l l e d with a s o l u t i o n of known c o n d u c t i v i t y , X, and the c r o s s - s e c t i o n a l area of the c e l l at the plane of viewing i s A , then the i n t e r - e l e c t r o d e d i s t a n c e , I, i s given by A = RXA [4.1] and the potent ia l gradient by V / l , where V is the vol tage drop across the c e l l . Such measurements can be found in Appendix B, Sec. Bl . P a r t i c l e m o b i l i t i e s were recorded with the microscope focussed on the s ta t ionary l e v e l s . The wal ls of the e lec t rophores is c e l l are in general charged in the presence of s o l v e n t ; therefore when a potent ia l d i f f e r e n c e is a p p l i e d , electrq:-osmotic streaming o f the oppos i te ly • charged solvent near the wal ls o c c u r s . Since the c e l l i s c l o s e d , th is 62 flow of solvent bu i lds up a hydrodynamic pressure d i f f e r e n c e between the extremit ies o f the c e l l . For hydrostat ic equ i l ib r ium a reverse flow of solvent ensues in accordance with P o i s e i l l e ' s Law. As a consequence of these opposing f lows , there e x i s t s a region within the c r o s s - s e c t i o n o f the c e l l where the solvent flow i s n i l . Thus the observat ion of the p a r t i c l e v e l o c i t y can best be made a t ' t h e s ta t ionary l e v e l s . The l o c a t i o n of the s ta t ionary leve l can therefore be determined from a cons idera t ion of the hydrodynamic so lu t ion of s teady-s ta te f l u i d flow in the rectangular channel coupled with the d i s t r i b u t i o n of v e l o c i t y due to the e lect ro-osmosis e f f e c t . Komagata (56) der ived a r e l a t i o n -sh ip which ind ica tes symmetry o f the p o s i t i o n of the s ta t ionary leve l about the centre of the channel : = 0.500 - [0.0833 + — h 2 l 4 J 2 ] where s i s the d is tance of the s ta t ionary leve l from the ...inner wall of the channe l , I i s the width and d the th ickness of the channel . Thus the s ta t ionary leve l pos i t ion i s dependent on the r a t i o lid. The thickness of the c e l l was determined by focussing the microscope on imperfect ions on the inner surfaces of the two f l a t s i d e s . The length was determined with a t r a v e l l i n g microscope as descr ibed in Appendix B. The r a t i o Z/d fo r the c e l l used was found to be 10, in which case the r a t i o s /d was evaluated as 0.194. No op t i ca l cor rec t ions were necessary to the p o s i t i o n of s ta t ionary l e v e l s in the f l a t c e l l when c e l l dimensions were measured with d i l u t e suspensions in the c e l l , s ince f r a c t i o n a l d is tances (s /d) were v a l i d even when d was an apparent 63 t h i c k n e s s . The c e l l th ickness was measured each time a new set of m o b i l i t y values was taken. P a r t i c l e v e l o c i t y was determined by measuring the time taken by a number of ind iv idua l p a r t i c l e s to t raverse a chosen number of g r id spaces in the eyepiece g r a t i c u l e with a timer incorporated in the apparatus. Usua l ly 10 separate p a r t i c l e s were timed to obtain a d i s t r i b u t i o n of m o b i l i t i e s . Timing was made in both d i r e c t i o n s to e l iminate the e f f e c t due to p o l a r i s a t i o n . Measurements were a lso c a r r i e d out at both s ta t ionary l e v e l s to ensure t h e i r proper l o c a t i o n . A mean v e l o c i t y was then obtained from the 20 observat ions by c a l c u l a t i n g the mean rec ip roca l t ime. This then was converted to mean m o b i l i t y on d i v i d i n g by the appl ied potent ia l g rad ien t . A f te r each set of measurements, the e lectrodes were cleaned with d i s t i l l e d water and stored in a beaker conta in ing d i s t i l l e d water un t i l t h e i r next use . The e lectrodes were a lso o c c a s i o n a l l y blackened by f i r s t c lean ing them with detergent , water , concentrated n i t r i c ac id and water again and then immersing them in a 3% so lu t ion of c h l o r p l a t i n i c ac id conta in ing ^ 0.02% lead ace ta te . For short periods of about 10 seconds the e lect rodes were made a l t e r n a t i v e l y anodic and cathodic with about 1 •' mA current passing through the s o l u t i o n . The c e l l , when not in use , was a lso cleaned thoroughly with d i s t i l l e d water and stored in more d i s t i l l e d water to c lean out a l l contaminants. The conversion of the measured m o b i l i t i e s to zeta potent ia ls was made using the numerical r e s u l t s of Wiersema et a l . (38) . This method was used because the o r i g i n a l equation u s u a l l y employed to convert m o b i l i t i e s in to zeta p o t e n t i a l s , the Smoluchowski equation (57) , 64 which in i t s dimensional form i s U £ ^ [4.13] 4Trp where U - m o b i l i t y . ^ u s e c e s = d i e l e c t r i c constant Vi = v i s c o s i t y o f the suspending medium, poise X, = zeta p o t e n t i a l , v o l t s i s only s u i t a b l e for cases where ka i s very l a r g e , i . e . for spher ica l p a r t i c l e s which possess r a d i i that are large compared to the double layer t h i c k n e s s . For very small ka , the modif ied equation of Henry (33) is s u i t a b l e for small zeta potent ia l (say < 20 mV). In i t s dimensional form th is equation i s U = ^ [1 + f (ka) ] [4.14] where f(ka) is a c o r r e c t i o n fac tor based on the value of ka. These equations have however been derived by neglect ing the e f f e c t s of re tardat ion and r e l a x a t i o n . As mentioned in Sect ion 2 . 2 j , a number o f workers have invest igated the c o r r e c t i o n s necessary to preserve the v a l i d i t y of the above equations The r e s u l t s of Wiersema et a l . (38) account for the e f f e c t s of r e t a r d a t i o n , re laxa t ion and surface conductance, and apply to s i t u a t i o n s where zeta potent ia l i s very h i g h , as preva i led in some o f the present experimental c o n d i t i o n s . A sample c a l c u l a t i o n based on t h e i r technique i s shown in Appendix A . 65 CHAPTER F IVE RESULTS AND DISCUSSIONS 5.1 Introduct ion The raw experimental d a t a , inc lud ing the s e t t l i n g experiments and the e l e c t r o p h o r e t i c measurements, have been presented in Appendix D. Sample p lots of in te r face height versus time obtained by l i n e a r r e -gressional a n a l y s i s can a lso be found in Figures ( A l , A2 ) , Appendix A. The v e l o c i t y o f sedimentation computed from the slopes o f these p lots (see Sect ion A l , Appendix A) and the condi t ions under which each e x p e r i -ment was r u n , as well as a l l re levant parameters such as suspension volumetr ic c o n c e n t r a t i o n , c , the r a t i o o f p a r t i c l e radius to double layer t h i c k n e s s , ka , and measured zeta potent ia ls have a lso been presented in Table 5 .1 . In t h i s t a b l e , as well as in Table 5 .2 , the v i s c o s i t i e s shown at var ious temperatures were obtained from standard sources (58) for d i s t i l l e d water and adjusted for the e f f e c t of e l e c t r o l y t e con-cent ra t ion by l i n e a r i n t e r p o l a t i o n of the v i s c o s i t y measurements c a r r i e d out at var ious e l e c t r o l y t e concentrat ions (see Appendix B, Sect ion B5) . 5.2 S e t t l i n g in Aqueous Media In order to evaluate the e l e c t r o k i n e t i c re tardat ion of sedimentation TABLE 5.1 Summary of Experimental Data for Charged P a r t i c l e s c T av pH A x 10 6 -5 x 10 3 ka ll x 10 s y x 10 2 Run - °C mhos/cm v o l t s - cm/sec (poise) IA 0.0098 24.80 6.75 8.50 70.64 8.43 2.004 0.8904 IB 0.0098 24.70 5.86 115.0 62.61 27.89 2.013 0.8906 IC 0.0098 25.10 5.52 265.0 44.31 60.74 2.024 0.8915 2A 0.053 24.50 6.39 7.42 71 .28 7.84 1 .512 0.8904 2B 0.053 25.20 5.33 88.33 64.22 26.93 1 .540 0.8906 2C 0. 053 25.10 4.81 260.00 49.32 52.71 1 .640 0.8912 3A 0.1033 25.41 6.22 19.25 61 .01 12.67 0.909 0.8905 3B 0.1033 25.60 4.91 345.0 21 .71 57.45 1 .532 0.8715 4A 0.145 26.10 6.28 13.75 58.44 10.70 0.637 0.8705 4B 0.145 26.40 5.31 260.00 46.24 55.73 0.877 0.8714 1 .1 0.0064 25.20 6.86 10.50 61 .66 9.05 1 .912 0.8904 1 .2 0.0064 24.30 6.86 370.00 10.00 58.54 2.089 0.9121 1 .3 0.0064 23.8 6.86 342.00 35.96 58.54 2.007 0.9121 1 .4 0.0064 24.0 7.12 330.00 56.52 58.54 1 .882 0.9121 1 .5 0.0064 23.5 7.30 320.00 66.02 58.54 1 .850 0.9121 1 .6 0.0064 24.4 9.42 440.0 80.92 58.76 1 .845 0.9124 2.1 0.0537 25.6 5.82 28.80 41 .10 14.92 1.212 0.8706 2.2 0.0537 25.45 5.67 579.00 16.70 59.74 1 .472 0.8915 2.3 0.0537 25.30 5.66 500.00 16.69 59.74 1 .815 0.8915 TABLE 5.1 - continued c Tav pH •X x 10 6 -C x 10 3 ka U x IO5 y x 10 2 Run °C mhos/cm volts - cm/sec (poise) 3.1 0.113 22.50 6.13 34.50 42.39 16.40 0.792 0.9326 3.2 0.113 23.60 6.20 270.0 24.40 47.83 1 .070 0.9118 3.3 0.113 24.10 5.91 250.0 26.33 47.83 1 .218 0.9118 3.4 0.113 24.20 6.09 295.0 31 .34 47.83 1 .225 0.9118 3.5 0.113 24.88 8.51 330.0 54.46 47.83 1 .397 0.8914 4.1 0.170 24.50 6.76 31 .90 46.24 15.80 0.472 0.8905 4.2 0.170 24.30 6.35 283.00 28.25 49.84 0.924 0.9119 4.3 0.170 24.10 7.02 320.00 35.96 49.84 0.994 0.9919 4.4 0.170 24.72 8.69 495.00 61 .65 54.53 1 .025 0.8916 5.1 0.2088 24.12 8.12 34.50 80.92 14.59 0.523 0.9114 5.2 0.2088 24.44 7.48 160.00 41 .10 14.59 0.575 0.9114 5.3 0.2088 24.40 7.51 300.00 32.11 37.13 0.616 0.9116 6.1 0.0102 24.6 5.40 10.90 44.95 7.70 1 .887 0.8904 6.2 0.0102 24.3 12.90 4600.00 65.51 239.9 2.020 0.9168 6.3 0.0102 24.5 11 .34 1060.00 55.23 239.9 2.033 0.9061 6.4 0.0102 24.32 11.13 "950.00 55.23 239.9 1.939 0.9270 6.5 0.0102 24.32 9.94 1080.00 57.80 239.9 2.147 0.9289 7.1 0.2693 24.60 -1.11 30.05 59.09 1 5.35 0.212 0.8911 TABLE 5 . 2 Summary o f E x p e r i m e n t a l Data f o r Uncha rged P a r t i c l e s Run c pH T a v y * 1 0 s U x 10 °C Nm/sec m /sec NI 0 .00514 3 . 2 4 2 4 . 8 4 8 9 2 . 6 2 . 2 2 0 N2 0 .0137 3 . 3 4 2 4 . 6 0 8 9 2 . 5 2 . 1 5 0 N3 0.0251 3 . 3 3 2 4 . 7 2 8 9 2 . 5 1 .990 N4 0 .0502 3 . 3 4 2 4 . 8 6 8 9 2 . 5 1 .840 N5 0 .0799 3 . 1 6 2 4 . 6 6 8 9 2 . 6 1 .670 N6 0 .1085 3 . 3 0 2 4 . 9 7 8 9 2 . 5 1 .430 N7 0 .1550 - 3 . 2 9 2 4 . 8 2 8 9 2 . 5 1 .160 N8 0 .2044 3 . 6 8 2 5 . 2 7 8 9 2 . 5 0 . 6 3 0 N9 0 . 2 5 9 3 . 7 8 2 3 . 7 0 9 1 3 . 0 0 . 3 5 0 P r e s u m a b l y t h e r e was a s m a l l c h a r g e on t h e s e s p h e r e s . 69 ve loc i t y and to compare the resul ts with the predict ion of Levine et a l . (45), i t was necessary to investigate independently the ef fect of each of the relevant parameters discussed in Chapter Three. Hence the ef fect of changes in surface po ten t i a l , which is c lose l y related to changes in zeta po ten t i a l , was studied by a l te r ing the suspension pH and the effect of changes in double layer thickness was also investigated by varying the concentration of neutral e lec t ro ly te through the addit ion of NaCl. It was, however, very d i f f i c u l t to maintain an absolutely constant counter-ion concentration while varying the suspension pH by the addit ion of NaOH. Thus for a good approxima-t ion of such condi t ions, the ef fect of a l t e r ing suspension pH was investigated at high sa l t concentrations, in which case changes in pH by the addit ion of NaOH had r e l a t i v e l y small e f fect on the double layer thickness. In Table 5.1, the alphanumeric series of runs (IA through 4B) shows the effect of a l te r ing the double layer thickness at various par t i c l e concentrations. In runs IB and 2B, measured quantit ies of NaCl were dissolved in the suspension to increase the counter-ion concentration 4 -5 to 7.851 x 10 moles/ l i t re from an i n i t i a l value of 7.170 x 10 _5 moles/ l i t re in the case of IB and from 6.162 x 10 moles/ l i t re to -4 7.317 x 10 moles/ l i t re in the case of 2B. There was l i t t l e or no change in the rate of sedimentation as a resu l t of these e lec t ro ly te concentration changes; thus larger concentration changes were effected as demonstrated in runs IC and 2C. In these runs the counter-ion concentration was increased to 3.722 x 10 " 3 moles/ l i t re and 2.802 x 10~ 3 moles/1itre 70 r e s p e c t i v e l y , which resu l ted in s l i g h t increase in the s e t t l i n g v e l o c i t i e s . These condi t ions were repeated for higher p a r t i c l e con-c e n t r a t i o n , as ind icated in runs 3B and 4B. From the experience of the previous r u n s , the neutral e l e c t r o l y t e concentrat ion changes in these _3 runs were made very l a r g e , i . e . up to 3.329 x 10 m o l e s / l i t r e and _3 3.134 x 10 m o l e s / l i t r e , r e s p e c t i v e l y , from i n i t i a l values of 1.621 x 10" 4 m o l e s / l i t r e in 3A and 1.155 x 10" 4 m o l e s / l i t r e in 4A. Increases in the s e t t l i n g rate were accord ing ly s u b s t a n t i a l . Further increase in counter - ion concentrat ion was not made for fear of coagula-t ion . The above r e s u l t s were not unexpected s ince an increase in the counter - ion (Na +) concentrat ion decreases the double layer thickness (1/k) as given by equation 2 .19 , and increases the equ i l ib r ium adsorpt ion o f counter - ions into the Stern l a y e r , which subsequently lowers the zeta potent ia l as a lready explained in Sect ion 2 .2h. Conse-quent ly , p a r t i c l e - p a r t i c l e e l e c t r i c a l i n t e r a c t i o n i s g rea t ly reduced and the p a r t i c l e s s e t t l e f a s t e r . In order to confirm the above c o n c l u s i o n s , these more concentrated e l e c t r o l y t e condi t ions were repeated in the second runs of each of the run se r ies 1, 2 , 3 and 4 at s l i g h t l y d i f f e r e n t p a r t i c l e concent ra t ions . The r e s u l t s obtained are ind icated in runs 1.2, 2 . 2 , 3.2 and 4 .2 . Again the concentrat ions of counter - ions were increased sharply to values of 3.457 x 1 0 " 3 , 3.50 x 1 0 " 3 , 2.308 x 10" 3 and 2.506 x 10" 3 m o l e s / l i t r e r e s p e c t i v e l y . Increases in s e t t l i n g rate were accord ing ly observed, with the greatest increase occurr ing in Run 4.2 where the p a r t i c l e concentra -t ion was h ighes t . 71 At these high counter - ion concent ra t ions , subsequent runs in each of the se r ies 1 , 2 , 3 and 4 were made with the aim of i n v e s t i g a t i n g the e f f e c t of changes in p a r t i c l e zeta potent ia l on s e t t l i n g rate by the add i t ion of NaOH. As explained in Sect ion 2 .2h , increases in OH" ion concentrat ion r e s u l t s in increased adsorpt ion of OH" ions on the p a r t i c l e s u r f a c e , thus increas ing the surface potent ia l as well as the zeta p o t e n t i a l . This phenomenon expla ins the observat ions made in the i n i t i a l pH adjustments, i . e . in runs 1.3, 2 . 3 , 3.3 and 4 . 3 . In these runs c a l c u l a t e d quant i t i es of NaOH from a pre-determined pH leve l were not s u f f i c i e n t to e f f e c t the necessary pH changes. Therefore in sub-sequent r u n s , higher 0H~ ion concentrat ion changes, e f fected by the add i t ion of a r b i t r a r y but reasonable quant i t i es of NaOH s o l u t i o n s , were necessary for any s i g n i f i c a n t changes in pH. This d i f f i c u l t y was pronounced at higher p a r t i c l e c o n c e n t r a t i o n s , where a v a i l a b l e surface area for OH" ion adsorpt ion was a lso h igh . Measured changes in s e t t l i n g rate with changes in pH were very s m a l l , almost n e g l i g i b l e , i r r e s p e c t i v e of the concentrat ion of p a r t i c l e s . Run ser ies 5 was c a r r i e d out with a view to i n v e s t i g a t i n g the e f f e c t of changes in p a r t i c l e zeta potent ia l at very large double layer th ickness (small ka) and very high p a r t i c l e concent ra t ion . There were again on ly small changes in s e t t l i n g r a t e , as ind icated by runs 5.1 and 5.2 . In run 5 .3 , ka was i n c r e a s e d , but the change was not s u f f i c i e n t to in f luence the s e t t l i n g rate apprec iab ly , j us t as in the cases of runs IB and 2B descr ibed e a r l i e r . In run se r ies 6, the pH of the suspension was increased sharply to a very high v a l u e , as in run 6 .2 , by the add i t ion of a very con-centrated NaOH s o l u t i o n . There was no marked increase in zeta 72 p o t e n t i a l , which i s cons is ten t with theory , and the change in s e t t l i n g rate was s l i g h t and p o s i t i v e , presumably because ka increased markedly. In subsequent runs , zeta potent ia l was adjusted by the add i t ion of HC1 so lu t ion (runs 6.3 through 6.5) and increase in s e t t l i n g rate was minimal . The highest p a r t i c l e concentrat ion e f f e c t invest igated was that presented in Run 7 .1 . Higher p a r t i c l e concentrat ions could not be invest igated because of inadequacy of t ime, i n s u f f i c i e n t quant i ty of p a r t i c l e s and d i s p e r s i o n d i f f i c u l t i e s associated with higher concentra-t i o n s . In Run ser ies 5 and 6 the p a r t i c l e s of batch III (Table B2 -Appendix B) with smaller average diameter were used. Therefore the measured v e l o c i t i e s were corrected to correspond to the p a r t i c l e diameter in a l l other runs through the r a t i o of the square o f the p a r t i c l e diameters. In Table 5.1 the recorded v e l o c i t i e s are the corrected ones. 5.3 Hindered S e t t l i n g at N e g l i g i b l e Surface Charge In order to assess the reduct ion in s e t t l i n g rate as a r e s u l t of surface charges, or of sedimentation potent ia l for that matter , i t was necessary to compare measured v e l o c i t i e s in aqueous media under condi t ions out l ined above to s i t u a t i o n s where the surface charge i s z e r o . I n i t i a l l y , comparisons with the pred ic t ions o f Kuwabara's theory of hindered s e t t l i n g of uncharged p a r t i c l e s was thought to be adequate, s i n c e , as explained in Chapter Two, the assumptions inherent in t h i s theory were adopted by Levine et al . (45) in t h e i r cons idera t ion of the e f f e c t o f sedimentation p o t e n t i a l . However, the r e s u l t s of such 73 comparisons showed e f f e c t s of surface charge on the rate of sedimentation, e n t i r e l y contrary to expec ta t ion , i n d i c a t i n g that the Kuwabara model does not adequately represent the sedimentation of uncharged spheres at the r e l a t i v e l y low p a r t i c l e concentrat ions s t u d i e d . A set o f s e t t l i n g experiments were therefore c a r r i e d out at var ious p a r t i c l e concentrat ions that covered the concentrat ion ranges for the previous experiments with charged p a r t i c l e s , in an attempt to obtain a hindered s e t t l i n g v e l o c i t y , U Q , at n e g l i g i b l e surface charge. These r e s u l t s are presented in the run s e r i e s 'N ' shown in Appendix D. The technique employed here has been explained in Sect ion 4.4a and the ca lcu la ted r e s u l t s are shown in Table 5.2. In a previous attempt to measure the zeta potential of-suspensions prepared under these c o n d i t i o n s , the p a r t i c l e movement in the e l e c t r o p h o r e t i c c e l l was so small that the measurements were abandoned. This was not unexpected, s ince the surface charge on such p a r t i c l e s was very minute. The double layer thickness was, however, estimated from a knowledge of the pre-dominantly hydrogen ion concen t ra t ion , and presented in Appendix D as wel 1. It was of i n t e r e s t to compare the r e s u l t s obtained with the pred ic t ions o f the Kuwabara model and o f other models in the l i t e r a t u r e . As shown in Figure 5 . 1 , agreement with the Kuwabara and Happel models was poor, at l e a s t wi thin the range of concentrat ions i n v e s t i g a t e d . This was not unexpected s ince i t confirmed the observat ions mentioned above with regard to the s e t t l i n g of charged p a r t i c l e s . The experimental, data lay well above the s e t t l i n g rate predicted by these theore t ica l models, as given by equations 3.13 and Figure 5 .1: Comparison of Experimental Results with Pred ic t ions in the L i te ra tu re 75 3.12 in the case o f the Happel model and by equations 3 .19 , 3.17 and 3.18 in the case of the Kumabara. On the other hand, comparison with the Richardson and Zciki (14) r e s u l t s as depicted by the i r empir ical equation 2.05 with n.= 4.65 p lot ted on the same f igure (5.1) ind icated higher v e l o c i t i e s than the experimental points and a good degree of agreement at intermediate concent ra t ions . The Stokes ' v e l o c i t y appearing in the above theore t i ca l p red ic t ions was determined from the p a r t i c l e diameter of 0.606 ym and measured densi ty of 3 2.172 g/cm . One s i g n i f i c a n t feature o f th is comparison i s the c l o s e s i m i l a r i t y in the slopes of the Richardson and Zaki curve and that generated from the experimental d a t a , wi thin the working p a r t i c l e concent ra t ions . In order to more e f f e c t i v e l y compare the in f luence o f p a r t i c l e concentrat ion in the theore t i ca l and experimental r e s u l t s , a p lo t was made o f sedimentation rate U o at concentrat ions wi th in the experimental range normalized with respect to the sedimentation rate at the minimum experimental concen t ra t ion . Th is i s shown in Figure 5 .2 , from which the good c o r r e l a t i o n with^the Richardson and Zaki empir ica l equation i s ev ident , e s p e c i a l l y at low concent ra t ions . On the same f i g u r e , the r e s u l t s of Cheng and Schachman as summarized by t h e i r empir ical equation 2.02 shows a per fect f i t with the experimental data in the d i l u t e range fo r c <_ 0.08. Both equat ions , however, show some degree of dev ia t ion from the experimental data at higher p a r t i c l e concentra -t i o n s , the Richardson and Zaki equation showing a l e s s e r d e v i a t i o n . Comparison with the McNown and Lin c o r r e l a t i o n (equation 2.10) showed l i t t l e agreement at intermediate p a r t i c l e concentrat ion ranges. Figure 5.2: Comparison of Experimental Results Re la t ive to Results at Minimum Concentration to Pred ic t ions in the L i t e r a t u r e , 77 In view of the f o r g o i n g , a pre l iminary Richardson and Zaki type of p l o t was generated from the experimental data with a view to est imat ing the Richardson and Zaki index, n , from the r e s u l t s . F igure 5.3 gives an index of 4.10 f o r the d i l u t e range, with a higher index of 8.70 at higher concent ra t ions . The higher index of 8.70 might be viewed to be in T ine .wi th previous experiments (18,19) fo r which the high values of n have been a t t r ibu ted to the e l e c t r o k i n e t i c e f f e c t (17). Th is is because i t became necessary to carry out the d ispers ion process f o r the l a s t two data points (c = 0.2044 and 0.259) at values o f pH very near 4 to f a c i l i t a t e d ispers ion (see Appendix D). Thus s e t t l i n g experiments c a r r i e d out at these concentrat ions were presumably a f fec ted by t h e i r surface charge. The p a r t i c l e zeta potent ia l f o r these two runs was subsequently measured to be -42 mV. Quite recent ly (August 1981) S tau f fe r and C l a v i n (59) genera l ised B a t c h e l o r ' s theory (60) of sedimentation in d i l u t e d ispers ions to the case of concentrated d i s p e r s i o n s , using the p a i r c o r r e l a t i o n funct ion of a hard sphere system. The i r r e s u l t s were represented by A pre l iminary comparison of th is equation with the experimental data obtained in t h i s work, as well as with the Batchelor model, [5.1] U r- = 1 - 6.55c [5.2] O Uncharged spheres, pH~3 • Charged spheres, pH~3-7 •692 0-75 0-80 0-85 0-90 0-95 1-1- C Figure 5.3: Richardson-Zaki Plot of Experimental Data for Uncharged'Part ic les 7 9 is shown in Figure 5.4 where almost perfect agreement with the data in the dilute range (c<0.08) was exhibited in the case of the Stauffer and Clavin correlat ion. The foregoing-comparisons thus give some indication of the.• r e l l a b i l i t y of the experimental data. 5.4 Empirical Correlation for Uncharged Spheres In view of the foregoing, i t was decided to f i t the experimental data to an empirical equation of the polynomial form generally adopted in the l i terature . U 2 m jp- = 1 + b-jC + b 2c + . . . + bmc o [5.3] where m is an integer = 1 , 2, 3 etc. U = Stokes velocity s U = sedimentation velocity of uncharged spheres at concentration c and b = constant, m In order to do th is , the method of multiple regressional analysis was applied to the experimental data for m=l, 2 and 3 with the aid of the computer programme STPREG of the UBC sta t is t ica l TRP package, details of which are presented in Appendix C, Section C l ) . To account for the effect of sl ight variations in viscosity due to the introduction of electrolyte and to small changes in temperature, a l l measured sett l ing velocit ies were standardised to the condition of Figure 5.4 Comparison of Experimental Results with the Pred ic t ions of Batchelor (1972) and Stauf fer and Clavin (1981) 81 pure water at 2 5 ° C , at which the v i s c o s i t y i s 0.8904 c e n t i p o i s e (58) . In the v iscous flow regime, temperature changes and e l e c t r o l y t e concentrat ions such as were encountered in the experiments (see Table B3) have n e g l i g i b l e e f fec t on the re levant physical proper t ies o f the system other than v i s c o s i t y . Thus from considera t ions o f Stokes 1 Law, fo r a given suspension, Uy = constant where U i s the s e t t l i n g rate at any temperature and e l e c t r o l y t e con-cent ra t ion at which the v i s c o s i t y of the s e t t l i n g medium i s y . The standardised s e t t l i n g rate at 25°C i s then given by Uy 25 y 2 5 where y 2 g is the v i s c o s i t y of pure water at 2 5 ° C . In Table 5.3 a l l the reported s e t t l i n g rates have been standardised in the above manner. From the regress iona l a n a l y s i s , the best empir ical equation that f i t t e d the experimental data for c •< 0.16 was U 1 + 4.44c - 3.10C2 + 89 .50c 3 L D ' 4 J s with a value of U g = 2.2640 x 10" 7 m/s obtained by ex t rapola t ion of the regressed equation (see Appendix C , Sect . CI) and a mul t ip le 2 c o r r e l a t i o n c o e f f i c i e n t , R , o f 0.9979. The l a s t two experimental data in Table 5.3 ( for 0 0 . 1 6 ) , for which the surface charge was found 82 TABLE 5.3 Experimental Data for Uncharged Par t i c les with U Corrected for V iscos i ty Effects P H T av y x i o 6 U o X 1 0 ? g r e e t e d ) ' Run - "C Nm/sec m/sec m/sec NI 0.00514 3.25 24.84 892.60 2.220 2.225 N2 0.0137 3.34 24.60 892.50 2.150 2.155 N3 0.0251 3.33 24.72 892.50 1.990 1.995 N4 0.0502 3.34 24.86 892.50 1.840 1.844 N5 0.0799 3.16 24.66 892.60 1.670 1.674 N6 0.1085 3.30 24.97 892.50 1.430 1.433 N7 0.1550 3.29 24.82 892.50 1.160 1.163 N8 0.2044 3.68 25.27 892.50 0,630 0.632 N9 0.259 3.78 23.70 913.00 0.350 0.359 83 to be apprec iable (see Sect ion 5 .4 ) , were not included in t h i s a n a l y s i s . As shown in Figure 5 .5 , equation 5.4 at a l l concentrat ions f a l l s above the t h e o r e t i c a l p red ic t ion of Batchelor (60) , equation 5 .2 , for d i l u t e d ispers ions and at O 0 . 0 7 , f a l l s below that of S tauf fe r and C lav in (59) , equation 5.1. At c<0.07, however, equations 5.1 and 5.4 ind ica te almost per fect agreement. From the value of U s , taking the average p a r t i c l e diameter as 0.606 ym and using Stokes' Law at a 3 temperature o f 2 5 ° C , the p a r t i c l e dens i ty was obtained as 2.005 g/cm . 3 This compares f a i r l y well (8% var ia t ion ) with the value of 2.172 g/cm obtained by experimental measurements. For m=l, the b e s t - f i t empir ical equation takes the form s with a value o f U"s = 2.369 x 10" 7 m/s and a p a r t i c l e dens i ty from Stokes' Law of 2.052 g /cm 3 (dev ia t ion o f 5.5% from 2.172 g/cm 3 ) . - The c o e f f i c i e n t 5.78 i s almost, i den t i ca l with the 5.80 proposed for hard spheres by Reed and Anderson (70). I f the Batchelor equation (5.2) i s in terpreted as r ~ = T-TO5T= 1 - 6- 5 5 C [ 5- 2 A ] fo l lowing the ana lys is o f Burgers (61) , then agreement with equation 5.4a i s good. This i s i l l u s t r a t e d in Figure 5 .6 . From the value o f 0"s in equation 5 .4a, taking the measured dens i ty of the p a r t i c l e s , the value obtained from Stokes' Law for the p a r t i c l e s i z e d-> was 0.574 ym, which compares f a i r l y well ( v a r i a t i o n of 5%) with the value of 0.606 ym obtained by e lec t ron microscopy as descr ibed in Appendix B3, Figure 5.5: Comparison of Empir ical Cor re la t ion (Eq.5.4) with the Pred ic t ions o f Batchelor (1972) and Stauf fer and Clav in (1981) 85 # Charged spheres 0 0 4 iJ08 012 016 0^0" 0 2 4 C Figure 5.6: Comparison of Empir ical Core la t ions (Eq. 5.4a) with the Pred ic t ions of Batchelor (1972) and Stauf fer and Clav in (1981). Points d i f f e r from those on Figure 5.5 due to d i f fe rence in value of II 86 Thus the er rors in s i z e measurement and in determining the most representat ive p a r t i c l e s i z e could account, at l e a s t in p a r t , for the discrepancy in the measured and c a l c u l a t e d p a r t i c l e d e n s i t i e s mentioned above. 5.5 S e t t l i n g Rates of Charged P a r t i c l e s The reduct ion in s e t t l i n g rate as a r e s u l t of the existence of surface charges was then evaluated by comparing the rate at substant ia l surface charge to that at n e g l i g i b l e surface charge for the same p a r t i c l e concent ra t ion . In Table 5 .4 , t h i s reduct ion i s represented as ( U Q / U ) E , where U i s the s e t t l i n g rate at any e l e c t r o l y t e condi t ion and at the same p a r t i c l e concentrat ion as that ind icated by U o , which was computed from equation (5 .4 ) . The subscr ip t E denotes experimental r e s u l t s . In the alphanumeric runs (Table 5.1) in which the double layer thickness was decreased from run to r u n , the general trend was a decrease i n U ^ / U . In the run se r ies 1, 2 , 3 and 4 , t h i s phenomenon was again demonstrated in the f i r s t two runs of each s e r i e s . Subsequent runs in each of the above se r ies in which zeta potent ia l was var ied showed l i t t l e e f f e c t For a comparison with the theory of Levine et a l . , i t was necessary to evaluate U 0 /U from the pred ic t ions of the theory at the experimental c o n d i t i o n s . Thus, expressing equation 3.43 in the most convenient form, we have £ s a . H(ka,c) 16TT 2 a 2Xu B ¥ 2 a . H(ka,c) = 1 + [5.5] 87 TABLE 5.4 Comparison of S e t t l i n g Rates of Charged P a r t i c l e s  Obtained from Experiment with the Theory of  Levine et a l . (1 976] Run c "' Beta PSI KA' ; ; H T , (V U ) E (uo/u)T IA .0098 0.326 .2.7521 ! • , .8;43 . 2.39 ' 0.42100 ! 1 .08285 1 .015 IB .0098 0.363 2.4401 27.89 38.70 0.77200 1.07801 1 .002 IC .0098 0.543 1 .7246 60.74 164.84 0.79200' 1 .07215 1 .000 2A .0530 0.322 2.7798 7.84 5.13 0.77800 1 .20760 1 .032 2B .0530 0.321 2.4986 26.93 67.23 1.29400 1 .18564 1 .004 2C .0530 0.417" 1 .9195 52.71 204.99 1.45200 1 .11335 1 .001 3A .1033 0.326 2.3721 12.67 55.43 1 .71200 1.63398 1 .020 3B .1033 0.383 0.8435 57.45 -369.29 2.36800 0.96951 1 .000 4A .1450 0.335 2.2669 10.70 61 .18 2.48100 1 .91949 1.037 4B .1450 0.481 1 .7919 55.73 793.05 3.45900 •::1.39420 1.-002 1 .1 .0064 0.305 2.3990 9.05 7.07 0.41800 1 .15150 1 .,009 1 .2 .0064 0.351 0.3902 58.54 3455.59 0.79200 1 .05393 1 .000 1 .3 .0064 0.379 1.4057 58.54 444.22 0.79200 1.09699 1 .000 1 .4 .0064 0.393 2.2079 58.54 303.85 0.79200 1 .16985 1 .000 1 .5 .0064 0.404 2.5834 58.54 241.67 0.79200 1 .1 9009 1 .001 1 .6 .0064 0.298 3.1568 58.76 225.02 0.79200 1 .19331 1 .001 2.1 .0537 0.310 1.5969 14.92 141.57 1.06800 1 .50237 1 .004 2.2 .0537 0.241 0.6492 59.74 8328.35 1.38400 1.23701 1 .000 2.3 .0537 0.279 0.6491 59.74 98.36 1.38400 1 .00324 1 .000 3.1 .1130 0.286 1 .6643 16.40 270.71 2.06300 1 .79633 1 .006 3.2 .1130 0.320 0.9545 47.83 2578.03 2.65900 1 .32838 1 .000 3.3 .1130 0.347 1.0282 47.83 1049.12 2.65900 1 .16806 '1.000 3.4 .1130 0.288 1 .2235 47.38 839.10 2.65900 1 .16138 1 .001 3.5 .1130 0.270 2.1212 47.83 34.63 2.65900 1 .01839 1 .001 4.1 .1700 0.304 1.8033 15.80 322.49 3.35200 2.27851 1 .013 4.2 .1700 0.333 1 .1025 49.84 1006.34 4.28000 1 .16392 1 .001 4.3 .1700 0.294 1.4043 49.84 351 .10 4.28000 1 .08195 1 .001 4.4 .1700 0.234 2.4025 54.53 108.53 4.24900 1.04923 1 .002 6.1 .0102 0.212 1 .7524 7.70 13.50 0.39800 1 .14806 1 .004 6.2 .0102 0.472 2.5565 239.90 1352.01 1.05300 1.07247 1 .000 6.3 .0102 0.208 2.1539 239.90 392.21 0.03500 1.06561 1 .000 6.4 .0102 0.226 2.1552 239.90 642.77 1.05300 1 .11727 1 .000 6.5 .0102 0.198 2.555 239.90 51 .46 1.05300 1.00903 1 .000 5.1 .2088 0.234 3.1598 14.59 60.21 5.27600 1 .66044 1.058 5.2 .2088 0.505 1 .6032 14.59 836.34 5.27600 1 .51028 1.003 5.3 .2088 0.174 1 .2527 37.13 2063.49 5.50200 1 .40976 1 .001 7.1 .2693 0.305 2.3037 15.35 271 .97 17.70500 2.87100 1 .122 88 where £ = ( f c J ^ K ) 4 A v and r . =. ex/kJ a o are dimensionless parameters depending on the bulk e lec t ro ly te proper-t i es and the surface po ten t ia l , and e c the permi t iv i ty of the medium. The theoret ica l se t t l ing ra t io ( U 0 / U ) T was then evaluated from equation 5.5 using the calculated values for 3, V and H(ka,c) (Table 5 .4) , T denoting theoret ical r esu l t s . Although the predicted values for (U 0 /U) j followed general ly the same qua l i ta t i ve trend with changes in e lec t ro ly te propert ies as those computed from the experimental r esu l t s , quant i tat ive agreement was poor. This was not unexpected since only 5 out of the 36 runs sa t i s f i ed the cond i t ion , f <_ 1, to which the theory of Levine et al i s res t r i c ted (see Chapter Two). The exper i -mental resu l ts also showed larger values of UQ/U for higher par t i c le concentrat ion, which i s in qua l i ta t i ve agreement with the theoret ical predict ions since for a constant value of the parameter ka, the value of the factor H(ka,c) increased with c . This trend i s espec ia l l y marked in runs IA, 2A, 3A and 4A as well as runs 1.2, 2 .2 , 3.2 and 4 .2 . As a resu l t of the poor quant i tat ive agreement for reasons mentioned above, an attempt was made to f i t the experimental data to an empirical co r re la t i on . 5.6 Development of Correlat ions The method of cor re la t ion by dimensional analysis was therefore adopted. From e lec t rok ine t i c theory, the var iables that might have any s ign i f i can t ef fect on the rate of sedimentation of charged par t i c les are the ve loc i ty of the par t i c les when uncharged, U Q ; the 89 zeta po ten t i a l , t, ; the pa r t i c l e concentration, c ; the par t i c l e rad ius , a ; the e l e c t r i c a l double layer th ickness, ; the solut ion v i s c o s i t y , y; the solut ion e l e c t r i c a l conduct iv i ty , X; and the solut ion e l e c t r i c a l pe rm i t t i v i t y , e c ; i . e . U = U(U o , e c , 5, X, y , a, k, c ) . since these variables can be represented by only four fundamental dimensions, namely mass, length, time and current , one obtains from the Rayleigh theorem a re l a t ion among f i ve dimensionless groups: U U e r 2 e U U U [U ya C ' k a ' Xa ; L 5 - b J The values of these dimensionless groups as obtained from the experimental data are presented in Table 5.5 as VEL, ZETP, CONC, KA and COND respect ive ly . The form of the functional re la t ionsh ip between them was then determined empir ica l ly from the experimental data. The method of least squares was adopted with the aid of s t a t i s t i c a l sub-routines from the UBC Computer Centre, the de ta i l s of which are explained in Appendix C. These include the SPSS ( S t a t i s t i c a l Package for Social Sciences),TRP (Triangular Regression Package) and BMDP: PAR (Derivative-Free Nonlinear Regression). I n i t i a l estimates showed that the dimensionless groups r e -presenting the surface potential and the e lec t ro ly te conduct iv i ty were inseparable. Therefore these two groups were combined as c x c o _ c y X a Xya' 90 TABLE 5.5 Dimensionless Groups Run VEL ZETP CONC KA COND x 108 IA 1.0828552 59269.984 0.0098" 8.43 58.591633 IB 1.0780144 46550.285 0.0098 27 .899 4.3306816 IC 1.0721550 23291.641 0.00098 ,60:74 1 .8793525 2A 1.2075977 71723.563 0.053 "7.840- 56.475000 2B 1.1856413 58206.227 0.053 26.93 4.7440803 2C 1.1133461 34306.992 0.053 52.71 1 .6117102 3A 1.6339808 64586.637 0.1033 12.67 17.707868 3B 0.96951008 8356.5586 0.1033 57.45 0.98804769 4A 1.9194899 73639.813 0.145 10.70 ' 20.408368 4B 1.3942013 46055.223 0.145. 55.73 1.0792892 1 .1 1 .1514997 44509.863 0.0064 9.05 48.122524 1 .2 1 .0539331 1142.8552 0.0064 58.54 1 .3656393 1 .3 1.0969944 14778.527 0.0064 58.54 1 .4774461 1 .4 1 .1698551 36508.676 0.0064 58.54 1 .5311713 1 .5 1 .1900911 49813.012 0.0064 58.54 1 .5790203 1 .6 1 .1933155 74810.063 0.0064 58.76 1 .1483785 2.1 1.5023737 24455.145 0.0537 14.92 14.510226 2.2 1.2370090 3942.9097 0.0537 59.74 0.72175190 2.3 1 .0032377 3938.1892 0.0537 59.74 0.83578868 3.1 1 .7963333" 31081 .711 0.113 16.40 9.4640882 3.2 1 .3283806 10533.043 0.113 47.83' 1 .2093007 3.3 1 .1680593 12265.242 0.113 47.83 1 .3060450 3.4 1 .1613846 17376.887 0.113 47.38 1 .1068177 3.5 1.0183935 53673.047 0.113 47.83 0.98942756 4.1 2.2785139 51238.242 0.170 15.80 : 7.7372874 4.2 1 .16391 56 18675.906 0.170 49.84 0.87215390 4.3 1.0819502 30261.082 0.170 49.84 0.77131084 4.4 1 .0492277 90968.000 0.170 54.53 0.49862514 6.1 1 .1480579 24039.457 0.0102 7.70- 45.613695 6.2 1 .0724678 49589.633 0.0102 239.90 0.10808461 6.3 1 .0656099 35663.527 0.0102 239.90 0.46904596 6.4 1 .1172695 34859.465 0.0102 239.90 0.52335665 6.5 1 .0090284 38101.074 0.0102 239.90 0.46036028 5.1 1 .6604424 189872.81 0.2088 14.59 5.7768673 5.2 1 .5102806 48981 .848 0.2088. 14.59 1 .2456372 5.3 1 .4097586 29890.762 0.2088 37.13 : 0.66433969 7.1 2.8710051 147746.69 0.2693 15.35 4.6562302 91 to give a dimensionless group that was o r i g i n a l l y der ived from e l e c t r o k i n e t i c theory by Smoluchowski (29). The remaining four parameters were then c o r r e l a t e d . The best c o r r e l a t i o n that f i t t e d the experimental data was found to be U_ e 2 ? 2 e 2 ? 2 P 2 where the constants P-| to P^ were evaluated as P 1 = 7.73 P 2 = 0.334 P 3 = -540.9 P 4 = 1.245 with a mul t ip le c o r r e l a t i o n c o e f f i c i e n t of 0.818 and standard er ror of 0.166. Figure 5.7 shows a sca t te r p lo t o f U predicted from equations 5.7 and 5.4 versus U observed. Run 7.1 was not included in the above ana lys is because equation 5.4 did not apply to the value of c in t h i s run. It can be observed that equation 5.7 has been constructed to s a t i s f y the cond i t ion that when % -* 0, i . e . when there i s no surface X charge, U=UQ. A lso as both c -* 0 and ka -> °°, the r e l a t i o n reduces to the Smoluchowoski equation (2.27) for a s i n g l e charged p a r t i c l e , assuming 5 = ty . Combining equations 5.7 and 5 .4 , the s e t t l i n g rate of charged and uncharged p a r t i c l e s normalised with respect to the Stokes v e l o c i t y i s obtained a s : 92 Figure 5.7: Sca t te r P lo t of Pred ic ted V e l o c i t y of C h a r g e d . P a r t i c l e s (based on Eqs . 5.7 and 5.4) Versus Observed V e l o c i t y 93 U = (1 + 4.44c - 3 .10c 2 + 89 .50c 3 ) ' j - g g-j U s 1 + z + 7 . 7 3 ( z c ) 0 ' 3 3 4 - 5 4 0 . ' 2 4 5 where In a recent experimental study by Buscal l et a l . (62) , the e f f e c t of i o n i c concentrat ion on s e t t l i n g rate in d i l u t e suspensions of la tex p a r t i c l e s was found to be minimal , provided f l o c c u l a t i o n does not occur . This i s cons is ten t with observat ions made in the present work (see Sect ion 5 .2 ) . Busca l l et a l . found a l s o that at a r e l a t i v e l y high i o n i c concent ra t ion , where e l e c t r o s t a t i c i n t e r a c t i o n was suppressed the r e l a t i v e s e t t l i n g rate U Q / U S was s u b s t a n t i a l l y higher than that predicted by Ba tche lo r , which is cons is ten t with the present work (Figure 5.5) as well as with the data of Cheng and Schachman (12). In another s tudy, Reed and Anderson (63) t h e o r e t i c a l l y predicted a s i g n i f i c a n t e f f e c t o f i o n i c concentrat ion on U / U s , e s p e c i a l l y at very high i o n i c s t rength . At the same i o n i c concent ra t ion , t h e i r p red ic t ions of s e t t l i n g v e l o c i t y agreed with the data o f Cheng and Schachman. The i o n i c concentrat ion e f f e c t on U /U s was, however, observed in the present work only at higher p a r t i c l e concentrat ions (see Sect ion 5.2) and r e l a t i v e l y low i o n i c s t rengths . As predicted by equation 5 .8 , at high i o n i c strength where ka i s very l a r g e , only the z-group and the p a r t i c l e concentrat ion show any e f f e c t on U / U s . 94 In another c o r r e l a t i o n , the constants in equation 5.7 were evaluated using U Q as predicted by equation 5.4a. The values obtained are P 1 = 8.85 P 9 = 0.320 1 [5.9] P 3 = -348.97 P 4 = 1 . .H7 With a mul t ip le c o r r e l a t i o n c o e f f i c i e n t of 0.83 and a standard error of 0.203. These values are very c lose to those obtained e a r l i e r using equation 5.4 to pred ic t U Q . A sca t te r p lot of U predicted from equation 5.7 with the above values of the constants versus those obtained exper imental ly is shown in Figure 5.8. 5.7 Richardson and Zaki P lot A Richardson-Zaki type of p lot was made with a view to determining the e f f e c t of surface charge on the Richardson-Zaki index n. In order to do t h i s , a l l the measured v e l o c i t i e s for the charged p a r t i c l e s were standardised for pure water at 25°C and plot ted on a log sca le against the p o r o s i t y . This p lo t i s shown in Figure 5 . 9 , in which the s o l i d l i n e represents the s e t t l i n g rate of uncharged p a r t i c l e s predicted by equation 5.4. It can be observed from t h i s p lo t that the v e l o c i t i e s of the charged p a r t i c l e s , as expected, fa l l , well below the s o l i d l i n e . The l i n e of slope 7.1 represents the s e t t l i n g rate for a set of runs at near ly neutral pH and otherwise s i m i l a r e l e c t r o l y t i c condi t ions at o 95 6 10 14 18 22 OBSERVED VELOCITY, (M/S)X108 Figure 5.8: Scat ter P lo t of Predicted V e l o c i t y of Charged P a r t i c l e s (based on equations 5 .7 , 5.9 and 5.4a) Versus Observed V e l o c i t y 96 80 60 40 20 CO O X E O Data points for charged spheres CD Multiple points, number shown ® Run series with <=60±.2mV, *a=-|2±3, pH=67±05 Empirical equation (5-4) for uncharged spheres 10 O Slope=7.1 0-7245 075 0-80 0-85 1 - C 090 0-95 1-00 Figure 5.9: Richardson-Zaki Plot of Charged P a r t i c l e s and that of Uncharged P a r t i c l e s Predicted by Equation 5.4 97 var ious concent ra t ions , as i n d i c a t e d . The s l o p e , which is a s i g n i f i c a n t departure from the Richardson-Zaki index of 4 .65 , a value c l o s e to the slope of the uncharged p a r t i c l e l i n e at higher p a r t i c l e concentrat ions, demonstrates the re tard ing e f f e c t of surface charge on the s e t t l i n g rate o f f i n e p a r t i c l e s in a typ ica l sedimentation s i t u a t i o n . The higher value o f the index can therefore be sa id to be s o l e l y due to the e f f e c t of surface charge in t h i s c a s e , s ince p a r t i c l e shape e f f e c t s (17) were e l iminated by the use of spher ica l p a r t i c l e s . It appears therefore that both decreasing s p h e r i c i t y (17) and surface charges can contr ibute to increas ing the Richardson-Zaki index. The shape e f f e c t would apply to both r e l a t i v e l y large and r e l a t i v e l y small p a r t i c l e s , while the e l e c t r o k i n e t i c e f f e c t would in f luence the s e t t l i n g v e l o c i t y of f ine p a r t i c l e s o n l y . 98 CHAPTER S IX CONCLUSIONS AND RECOMMENDATIONS The e f f e c t of surface charge on the hindered s e t t l i n g r a t e of f i n e l y d i v i d e d s p h e r i c a l p a r t i c l e s has been experimentally s t u d i e d . This study has shown an appreciable reduction i n the s e t t l i n g r a t e which i s be l i e v e d to be a r e s u l t of the e l e c t r i c a l p o t e n t i a l that i s set up by v i r t u e of the presence of charges on the p a r t i c l e s u r f a c e . The extent of the reduction has been found to depend on the zeta p o t e n t i a l , on the e l e c t r o k i n e t i c r a d i u s ka, and on the p a r t i c l e volumetric c o n c e n t r a t i o n . The value of the Richardson-Zaki index, n, may be increased as a r e s u l t o f t h i s e f f e c t to as much as 7.1. The s e t t l i n g r a t e o f the charged p a r t i c l e s can be well represented by the e m p i r i c a l equation Un r 2 r 2 c 2 r 2 0-334 ° = 1 + 7.73 • c) u a 2Ay a Ay _ 540.9 ( 4 * 1 . 1 ) 1 - 2 4 5 a 2Ay ka 99 which a lso gives a quant i ta t ive comparison between the s e t t l i n g rate of the uncharged and the charged p a r t i c l e s . This r e l a t i o n agrees with the r e l a t i o n of Smoluchowski (30) at the l i m i t i n g condi t ions when c -> 0 and ka •> °°. However, agreement with the only re levant theore t ica l treatment in the l i t e r a t u r e , that of Levine et a l . (45) , i s found to be inadequate. I t i s suggested that more data be c o l l e c t e d fo r var ious e l e c t r o -l y t i c condi t ions at higher p a r t i c l e concent ra t ions . From the r e s u l t s of the present work and from theore t ica l c o n s i d e r a t i o n s , i t i s bel ieved that the e f f e c t of surface charge w i l l be very s i g n i f i c a n t ' a t higher p a r t i c l e concent ra t ion . The large quant i t ies of p a r t i c l e s required for t h i s purpose could be prepared in several batches of small q u a n t i t i e s , for which the react ion condi t ions could be well c o n t r o l l e d to ensure a narrow s i ze d i s t r i b u t i o n . A l s o , for e f f e c t i v e red ispers ion of the p a r t i c l e s at such high concent ra t ions , i t i s recommended that very d i l u t e d ispers ions are f i r s t made and then adjusted to the required concentrat ions by uncontro l led s e t t l i n g and subsequent withdrawal of supernatant l i q u i d . 100 NOMENCLATURE (The uni ts shown are the most commonly used) a p a r t i c l e radius cm A c r o s s - s e c t i o n a l area of e l e c t r o -phoresis f l a t c e l l at the plane 2 of viewing cm A , A , , A ? , A o constants as used in equations 0 1 c 6 3 .6 , 3.7 and 3.8 b radius of hydrodynamic c e l l envelope cm c p a r t i c l e volumetr ic concentrat ion C capacitance C/V C Q neutral e l e c t r o l y t e concentrat ion g / m o l e s / l i t r e d thickness of e lec t rophores is f l a t c e l l cm d p a r t i c l e diameter cm D diameter of conta in ing vessel cm D-j,D2 c o r r e c t i o n fac to rs to stokes v e l o c i t y as used in equations 3.12 and 3.17 e - elementary charge = 4.80298 x 1 0 " 1 0 c m 3 / 2 g J s s " 1 E appl ied e l e c t r i c f i e l d V/cm E* induced e l e c t r i c f i e l d with V/cm E, induced e l e c t r i c f i e l d with neglect of re laxa t ion e f f e c t as used in equation 3.37 V/cm f = f ( k a ) , funct ion of ka as used in equation 4.14 F body force due to e l e c t r o s t a t i c e f f ec ts as used in equation 3.20 dynes Resultant force due to hydrodynamic e f f ec ts used in equation 3.32 dynes resu l tan t fo rce due to e l e c t r i c a l e f f ec ts used in equation 3.32 dynes bouyancy-corrected gravi ta t iona l force used in equation 3.32 dynes drag force dynes a c c e l e r a t i o n due to g rav i ty ? = 980.98 cm/sec g ( r ) , potent ia l due to re laxa t ion e f f e c t used in equation 3.29 V G(ka) , funct ion of ka as used in equation 2.28 H ( k a , c ) , funct ion of ka as used in equation 3.43 Boltzmann constant = 1.38054 x 1 0 " 1 6 erg ° K _ 1 i n te re lec t rode d is tance f o r e l e c t r o -phoresis c e l l cm width of e lec t rophores is f l a t c e l l cm constant used in equation 3.27 d ipo le moment dyne/cm d ipo le moment f o r a s i n g l e i s o l a t e d sphere dyne/cm constants as used in equation 3.28 Richardson-Zaki index number of ions of type i in the double 1 ayer number of ions of type i in the bulk of s o l u t i o n Avrogadro's number parameter def ined by equation 3.38 number of p a r t i c l e s 102 P f l u i d pressure re fe r red to a datum 2 plane dynes/cm P normal force per un i t area on the 2 surface dynes/cm Q e l e c t r i c charge Coloumbs R mul t ip le regress iona l c o e f f i c i e n t R e l e c t r i c a l r es is tance ohms Re Reynolds number based on Stokes v e l o c i t y T absolute temperature °K 2 U mob i l i t y as used in equation 4.13 cm / (V ) (sec) U average f l u i d v e l o c i t y cm/sec U average s e t t l i n g rate of charged p a r t i c l e s cm/sec U average s e t t l i n g rate of uncharged o r p a r t i c l e s cm/sec U s Stokes v e l o c i t y of a s i n g l e i s o l a t e d p a r t i c l e cm/sec U rad ia l component of f l u i d instantaneous v e l o c i t y cm/sec U tangent ia l component of f l u i d instantaneous v e l o c i t y cm/sec U <f)-component of f l u i d instantaneous • .: * v e l o c i t y cm/sec V voltage drop across e l e c t r o p h o r e t i c c e l l V 3 V tota l volume of suspension cm 3 Vp to ta l volume occupied by p a r t i c l e s cm x d is tance of any point in the double layer from the p a r t i c l e surface cm X l o c a l l y induced potent ia l as used in equation 3.29 v o l t s 103 z dimensionless group = e 2 5 2 / a 2 X y z valence of ion of type i B dimensionless group = (e c kTK) / e 2Xy Y r a t i o of p a r t i c l e radius to c e l l radius <5 th ickness of Stern layer cm e voidage f r a c t i o n of suspension e d i e l e c t r i c constant , e (H ?0) at 25°C = s 78.30 s c e p e r m i t t i v i t y C/V cm £ zeta potent ia l mV k inverse double layer th ickness cm"^ X conduct iv i t y . mho/cm A parameter def ined by equation 3.38 y f l u i d v i s c o s i t y , y H 2 0 at 25°C = 0.008904(58) g/cm-sec r | ( r ) , funct ion of r def ined by equation - : . . -3.38 P f l u i d d e n s i t y , P h 2 0 at 25°C = 0.99708 g/cm 3 p charge dens i ty C/cm p ,p f l u i d and p a r t i c l e densi ty r e s p e c t i v e l y „ as used in equation 2.26 g/cm 2 aQ sur face charge dens i ty C/cm <j) <j>(r,e), sedimentation potent ia l V ty ty{x) = ty{r), potent ia l d i s t r i b u t i o n in the double layer V IJJ ^(a)> potent ia l at surface of p a r t i c l e V a ty Stern potent ia l V ¥ , dimensionless surface potent ia l = e 5 / kT a ¥ ^+<j>, to ta l potent ia l as used in equation 3.36 V T re v o r t i c i t y tangent ia l shear s t ress on the surface of a sphere A B B R E V I A T I O N S V E L Z E T P C O N C KA C O N D P S I S U B S C R I P T S H E E T dimensionless group = U Q / U dimensionless group = E C C 2 / U 0 y a c dimensionless e l e c t r o k i n e t i c radius = ka dimensionless group = £ C U 0 A a Y , dimensionless surface potent hydrodynamic e f fec ts e l e c t r i c a l e f f ec ts experimental r e s u l t s theore t ica l r e s u l t s 105 REFERENCES 1. Happel , J . , A . I . C h . E . Journal 4 , 197 (1958). 2. Kuwabara, S . , J . Physics S o c , Japan 1_4, 527 (1959). 3. R ichardson, J . F . , Z a k i , W.N. , Chem. Eng. S c i . , 3_, 65 (1954) 4. E l t o n , G . A . H . , Proc. Roy. Soc. London A197, 568 (1952). 5. Dul i n , C I . , E l t o n , G . A . H . , J . Chem. S o c , 286 (1952). 6. Stokes, G.G. Trans. Cambridge P h i l . S o c , 9_, I I , 51 (1851). 7. Happel , J . , Brenner, H . , Low Reynolds Number Hydrodynamics, P r e n t i c e - H a l l (1965). 8. McNown, J . S . , L i n , P . N . , "Proc . Second Midwestern Conf. F l u i d . Mechanics", No. 105, Iowa State U n i v e r s i t y , 1952. 9. Hawksley, P.G.W. Some Aspects of F l u i d Flow, Inst . Phys. London. A r n o l d , 1951. 10. Vand, V . J . . P h y s , and C o l l o i d Chem 52, 277 (1948). 11. E i n s t e i n , A. The Theory of Brownian Movement. New York: Dover, 1956. 12. Cheng, P . Y . , Schachman, H.K. J . Polymer Science 1_6, 19 (1955). 13. Maude, A . D . , Whitmore, R.L. B r i t . J . A p p l . Physics 9_, 447 (1958). 14. Richardson, J . F . , Z a k i , W.N. , Trans. Inst . Chem. Engrs. 32, 35 (1959). 15. Cunningham, E. Proc. Roy. Soc. (London). A83, 357 (1910). 16. Lapple , C . E . in F l u i d and P a r t i c l e Mechanics. U n i v e r s i t y of Delaware, 1954. 17. Chong, Y . S . , Ratkowsky, D . A . , E p s t e i n , N. Powder Technology, 23 55 (1979). 18. Maude, A .D . and Whitmore, R.L . Trans. Inst . Chem. Eng. 36 ,^ 269 (1958). 19. Richardson, J . F . , Me ik le , R. Trans. Inst . Chem. Eng. 39,348 ( 1 9 6 1 ) . 20. P a v l i k , R .E . and Sansone, E .B. Powder Tech. 8, 159 (1973). 106 21. Sansome, E .B . and C i v i c , T .M. Powder T e c h . , 1_2, 11 (1975). 22. Dav ies , L . , Do l l imore , D . , and Sharp, T . H . Powder T e c h . , 1_3 123 (1976). 23. Overbeek , j . j h . G. in Kruyt , H.R. (ed) , C o l l o i d S c i e n c e , V o l . 1 , Chapt. IV E l s e v i e r , Amsterdam (1952). 24. Davies , J . T . and R i d e a l , E.K. I n t e r f a c i a l Phenomena, 2nd E d . , Academic P r e s s , New York (1963). 25. Dukhin, S . S . and Der jaguin, B.V. Surface C o l l o i d S c i . , 7_, 1 (1974). 26. Bikermann, T . T . Surface Chemistry, Academic P r e s s , New York (1958). 27. E v e r s o l e , W.G. and Boardman, W.W. J . Chem. P h y s . , 9 , 789 (1941). 28. Hunter, R . J . and Alexander , A . E . J . C o l l o i d S c i . , 18_, 820 (1963). 29. Van Olphen, H. An Introduct ion to Clay C o l l o i d Chemistry, I n t e r s c i e n c e , New York (1963). 30. Dorn, N. Ann. Physik 3_, 20, 1878. 31. Smoluchwoski, M. in "Handbuch den E l e c t r i z i t a t und des Magnestismus" (L . Graetz , E d . ) , V o l . 2, L e i p z i g , 1914. 32. HUkel , E. "Phyzik Z . " 25_ 204 (1924). 33. Henry, D.C. Proc. Roy. Soc . (London), A133, 106 (1931). 34. Overbeek ,J . Th . G. Kollvidchem B e i t h , 54 287 (1943). 35. Booth, F. Proc. Roy. Soc. (London), A203, 514 (1950). 36. Lyklema, T . Overbeek, j . T h . G . ; J . C o l l o i d S c i . 1_6, 501 (1961 ) . 37. Hunter, R . J . T . C o l l o i d Inter face S c i . , 22, 231 (1966). 38. Wiersema, P . H . , Loeb, A . L . , and Overbeek, J . Th . G . , J . . C o l l o i d Inter face S c i . 22 ,^ 78 (1966). 39. L e v i n e , S . and Neale , G.H. J . C o l l o i d Inter face S c i . , 47, 520 (1974). 40. T i s e l u i s , K o l l o i d Z . , 59, 306 (1932). 41. Booth, F. J . Chem Phys. 22, 1956 (1954). 42. Sengupta, M.J . C o l l o i d Inter face S c i . , 26, 240 (1968). 107. 43. Dukhin, S . S . in "Research in Surface F o r c e s " , (B.V. Derjaguim, Ed) V o l . 2 , p. 54; Consultants Bureau, New York, 1971. 44. Derjaguim, B.V. and Dukhin, S . S . in "Research in Surface Forces" (B .C . Derjaguim, Ed.) V o l . 3 , p. 306; Consultants Bureau, New York, 1971 . 45. Lev ine , S . , Neale , G and E p s t e i n , N. J . C o l l o i d Inter face S c i . 57, 424 (1 976). 46. Amaratunga, L. Ph.D. T h e s i s , Dept. of Mining and Mineral E n g . , U n i v e r s i t y of Birmingham (1978). 47. Lamb, H. Hydrodynamics, 6th Ed . Cambridge; Cambridge U n i v e r s i t y P r e s s , 1932. 48. M o l l e r , W . T . H . , Van Os, G . A . J . , and Overbeek, J . M . G . Trans . Faraday Soc. 57 325 (1961). 49. I l e r , R.K. The C o l l o i d Chemistry o f S i l i c a and S i l i c a t e s , Cornel l U n i v e r s i t y P r e s s , I thaca, New York (1955). 50. I l e r , R.K. Surface C o l l o i d S c i . , 6 1 (1973). 51. S tober , W. , F ink , A . and Bohn E. J . C o l l o i d Interface S c i . , 26^  62 (1968). 52. Aelron,, R. , L o e b e l , A . , and E i r i c h , F. J . Am. Chem. S o c , 72^ 5705 (1950). 53. Bowen, B.D. Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, Vancouver 1978. 54. La Mer, V .K . and Dinegar, R. J . Am. Chem. Soc. 72, 4847 (1950). 55. Hodgeman, C D . (ed) Handbook o f Chemistry and P h y s i c s , 44th E d . , p. 2691, Chemical Rubber Publ ishing C o . , C leve land , Ohio (1962). 56. Komagata, S . Researches E l e c t r o t e c h . L a b . , Tokyo, Comm. No. 348 (1933). 57. Smoluchowski, M. Von, B u l l . Acad. S c i . Cracov ie , 1903, 182 (1903). 58. Hodgema-n, C D . (Ed) Handbook o f Chemistry and Physics 57th E d . P. F21, Chemical Rubber Publ ish ing Co. C leve land , Ohio (1976). 59. S t a u f f e r , D. and C l a v i n , P. Le Journal de Physique - Le t te rs 42, No. 15, L-353, 1981 . 60. Ba t c he lo r , G.K. J . F lu id Mech. 52 ,^ 245 (1972). 108 61. Burgers , J . M . Proc. Acad. S c i . Amsterdam, 45, 126 (1942). 62. Buscal l , R. , Goodwin, J . W . , O t t e w i l l , R . H . , Tadros, Th . F . , J . C o l l o i d and Interface S c i . , 85 , 78 (1982). 63. Reed, C C . and Anderson, J . L . J . AIChE J . 26_, 816 (1980). 64. Moore, W- .J . , -Phys ica l Chemistry, 3rd E d . , Prent ice H a l l , Englewood C l i f f s , N . J . (1962). 65. Hodgeman, C D . (ed) . Handbook of Chemistry and P h y s i c s , 56th E d . , p. B-137, Chemical Rubber Publ ish ing Co. C leve land , Ohio (1976). 66. Per ry , J . H . (ed) Chemical Eng. Handbook, 5th E d . McGraw-Hill New York (1973). 67. Guttman, I., W i l l i s , S . S . Introductory Engineering S t a t i s t i c s , John Wiley and Sons, New York (1965). 68. M i c k l e y , H . S . , Sherwood, T . K . and Reed, C E . Appl ied Mathematics in Chemical Engineer ing , 2nd E d . McGraw-Hi l l , New York (1957). 69. Pearson, E . S . and H a r t l e y , H.O. Biometrika Tables for S t a t i s t i c i a n s , V o l . I, 2nd Ed . Cambridge U n i v e r s i t y P r e s s , Cambridge (1958). 70. Reed, C C , and Anderson, J . L . in " C o l l o i d and Interface S c i e n c e " , (Mi l ton Kerker , ed.) V o l . 4 , p.501; Academic Press I n c . , New York, 1976. APPENDIX A SAMPLE CALCULATIONS Al Rate of Sedimentation A p lo t of i n te r face height versus time was made f o r each set of experiments and the best l i n e was obtained by l i n e a r regress ional f i t of the experimental data . The slope of th is l i n e represented the v e l o c i t y o f sedimentation in the constant s e t t l i n g rate range. Figures Al and A2 are sample p lots fo r runs IA and 1.1 r e s p e c t i v e l y . A2 Double Layer Thickness The double layer thickness was ca lcu la ted from equation 2.19 which at 25°C and f o r univa lent counter - ions reduces to (38) K = 0.3286 x 10 8 x Ch Al where.Gr.is the counter - ion concentrat ion in m o l e s / £ . A knowledge of C was therefore v i t a l to the c a l c u l a t i o n of K. The nature of the i o n i c content of i n i t i a l suspensions was very complex and i t s est imat ion was based on the measured conduct iv i t y and c e r t a i n assumptions. In the case of. f r e s h l y prepared suspension i t was assumed (64) that the cont r ibu t ions of the ind iv idua l species to the overa l l conduct iv i t y through t h e i r m o b i l i t i e s was a d d i t i v e , in which case (64) Figure Al : P lot of Interface Height Versus Time (Run i n T I M E ( H R S ) Figure A2: Plot of Interface Height Versus Time (Run 1 . 1 ) i 112 N o C i l Z i e o l U i A h. 1000 where X i s conduct iv i t y of the so lu t ion (mho/cm). N Q i s Avogadro's number, C the concentrat ion of i o n i c species ( m o l e s / l i t r e ) , Z.. the valence of the i o n , e 0 . the charge of an e lec t ron (Coulomb) and U. the mob i l i t y of the ion (cm / v o l t - s e c ) . In cases such as run 2.1 where the p a r t i c l e s had been treated with NaCI in a prev ious ly discarded run , the i o n i c concentrat ion was assumed to be predominantly NaCI and the NaCI concent ra t ion -conduct iv i ty p l o t (see Appendix B) was used to estimate the i o n i c concent ra t ion . In other runs the weights of added e l e c t r o l y t e s were known and therefore the i o n i c concentrat ion could be estimated from t h i s information and a knowledge o f the suspension volume, the conduct iv i t y and/or the pH of the suspension. In run IA, f o r i n s t a n c e , the suspension was f r e s h l y prepared and the l i k e l y i o n i c species present were H + and OH" from the complete d i s s o c i a t i o n of water, and NH^, HS^O^" and HCOg from the incomplete desorpt ion of NH^, d i s s o l u t i o n of s i l i c a and absorpt ion of CO2, r e s p e c t i v e l y . Since pH = 6 .75, [H + ] = I O ' 6 ' 7 5 = 0.178 x 10" 6 Therefore [OH"] = 1 0 " 1 4 / [ H + ] = 0.056 x 1 0 " 6 , -14 s ince the d i s s o c i a t i o n constant o f water is about 10 A l s o , to maintain e l e c t r o n e u t r a l i t y , [H + ] + [NHj] = [HS.Og] + [HCO3] + [ 0 H " ] ' 113 Assuming the t ransport proper t ies of HS^O^ are equivalent to those of HCO3 (53) , s ince values for HS^O^ could not be found in the l i t e r a t u r e . then [H + ] + [NHj] = [HCO3] + [OH"] A3 where [HCO3] now represents the combined concentrat ion o f HS..O3 and HCO3. From equations A2 and A3, the two unknown concentrat ions [NH^] and [HCOj]could be computed. Thus i f [NHj] = y x 10" 6 mo les /1 , then [HCO3] = (y + 0.178 - 0.0.056) x 10" 6 and s u b s t i t u t i n g into equation A2, x = 6.023 x 10 2 3 ^x 1.6023 x I O " 1 9 [ n J 7 8 % + . + Q Q S 6 ^ + yU N H+ + (y + 0.178 - 0 . 0 5 6 ) 0 ^ - ] x 10" 6 A4 4 3 where the ind iv idua l m o b i l i t i e s are (64) UH+ = 36.30 x 10" 4 U Q H -= 20.50 x 10~ 4 U N H , = 7 . . 6 x l 0 - 4 U u r n - = 4.6 x 10" 4 A5 HCO3 6 6 + Thus for a c o n d u c t i v i t y of 8.50 x 10" mhos/cm, 10" xy = [NH^] can be solved from equations A4 and A5 as 1T4 [NHj] = 71.529 x 10' The tota l counter - ion concentrat ion C i s obtained as C = [HNj] + [H + ] = 71.529 x 10" 6 + 0.178 x I O - 6 = 71.707 x 10" 6 moles/1 and from equation A l , k = 0.3286 x 10 8 x (71.707 x -\0~6)h = 2.783 x 10 5 cm" 1 -4 For a p a r t i c l e r a d i u s , a , of 0.303 x 10 cm, ka = 2.783 x 0.303 x 10 5 x I O - 4 = 8.43. A3 P a r t i c l e Zeta Potent ia l As mentioned in sec t ion 4 .4b , the conversion of the measured p a r t i c l e mob i l i t y was based on the numerical tabula t ion of Wiersema et a l . (38) that accounted f o r the r e t a r d a t i o n , re laxa t ion and surface conductance e f f e c t s . The i r r e s u l t s were reported in terms of dimension-l e s s v a r i a b l e s ; thus the dimensionless e l e c t r o p h o r e t i c m o b i l i t y , E = -U [A6] e Is. I the dimensionless double layer t h i c k n e s s , q Q = ka [A7] 115 the dimensionless zeta p o t e n t i a l , and the dimensionless m o b i l i t i e s o f the p o s i t i v e and negative ions in so lu t ion Ne T , 1-where y = v i s c o s i t y o f the s o l u t i o n , po ise . e Q = the un i t of e l e c t r o s t a t i c change (s ta t coulumbs) e = the d i e l e c t r i c constant of so lu t ion K = the Boitzmann constant ( e r g s / ° K ) T = the absolute temperature ( ° K ) U = the e l e c t r o p h o r e t i c m o b i l i t y (cm / v o l t - s e c ) C = the p a r t i c l e zeta potent ia l (mv) k = the rec ip roca l double layer thickness (cm~^) a = the p a r t i c l e radius (cm) N = Avogadro's constant Z + = the va lenc ies of the ions X° = the l i m i t i n g conductances of the ions in so lu t ion (cm /ohm-equiv) . The r e l a t i o n s h i p s between the dimensionless var iab les were tabulated in the form E = E ( y Q , q Q , m ±) [AlO] Thus fo r known value of q Q , E and y Q are obtained from graphical i n t e r p o l a t i o n of Wiersema's (38) Table I. A l s o , the in terpo la ted values of E are corrected to correspond to the actual m + and m_ from t h e i r m+ = 0.184 as E' = E + (m + - 0.184) | £ + (nr - 0.184) | | [ A l l ] where is in terpo la ted from t h e i r (38) Table III. The value of y Q corresponding to the experimental value of E ' , E , i s then extracted from a f i n a l p lo t of E' versus y . As in the case o f Run IA, the average rec ip roca l e lec t rophore t ic time (see Appendix D) was evaluated as 0.36445 s e c " ^ . _3 E lec t rophore t i c v e l o c i t y = 2 x 63.79 x 10 x '0.36445 cm/sec = 4.649 x 10" 3 cm/sec where the 1-gr id spacing = 63.79 x 10~ 4 cm (Appendix B) _3 E lec t rophore t i c m o b i l i t y U = 3 , 7 8 2 * 1 0 x 7.4603 = 4.955 x IO" 4 c m 2 / v o l t - s e c where the in te re lec t rode d is tance = 7.4603 cm and appl ied voltage =70 .0 vo l ts From equation A6, E = 0.7503;x 10 4 x U at 25°C ex = 0.7503 x 10 4 x 4.955 x 10" 4 = 3.72 Also q Q = ka = 8.43 (Sect ion A2) and from A8 and A9 , y 0 = ?/25.69 [A12] at 25°C m+ = at 25°C for univalent e l e c t r o l y t e s . [A13] K The dimensionless m o b i l i t i e s m + and m are then c a l c u l a t e d from the numerical averages o f x° and \ ° _ . . The ind iv idua l l i m i t i n g i o n i c m o b i l i t i e s are (64) H + = 349.82 cm 2 /ohm-equiv. + 2 NH^ = 73.4 cm /ohm-equiv. OH" = 198.0 cm 2 /ohm-equiv. 2 HC0 3 = 44.43 cm /ohm-equiv. at 25°C and t h e i r corresponding m + values from A13 a re : H + = 0.0368 NH* = 0.175 OH" = 0.0649 H C O 3 = 0.289 Subsequently, from the concentrat ions obtained in Sect ion A2, the number average volumes are : 7.152 x 1 0 " 6 C n T 7 r , 0.178 x 10" 6 Y n mfift m+ . = 7.1707-x::10: 6 x 0 J 7 5 + 7.170 x 10-6 x 0.0368 = 0.1747 0.056 x 1 0 " 6 , „ n n , A Q . 7.1651 x 10" 6 „ n ? 8 q m - = 7.1707 x IO" 6 X ° - 0 6 4 9 + 7.1707 x 10-6 x ° - 2 8 9 = 0.2888 Values fo r E are obtained by graphical i n t e r p o l a t i o n of Wiersema's Table I (38) at q Q = 8.43 (see Sect ion A2) for var ious y Q . A l s o , values for 8E /8 . and 8E /8 are obtained for the same values of m+ m-118 y Q by graphical e x t r a p o l a t i o n . E is then corrected to E' through equation A l l as tabulated in Table A l . TABLE Al E Versus Y„ - Run IA o *o E 9 E / 3 m + 9 E / 3 m - E' 1 1.36 -0.02 -0.04 1 .359 2 2.60 -0.07 -0.23 2.596 3 3.60 -0.14 -0.60 3.594 4 4.26 -0.18 -1.44 4.255 5 4.50 -0.21 -2.55 4.499 A p lo t of E 1 versus y Q i s then made from which the value fo r y Q corresponding to the experimental dimensionless m o b i l i t y , E , i s C A read o f f as 2.75. The zeta potent ia l o f the p a r t i c l e s in t h i s run is thus c a l c u l a t e d from equation A12 as 5 = 25.69 x 2.75 = 70.64 mV which is in f a c t negat ive , as ind icated by the d i r e c t i o n of the p a r t i c l e s in r e l a t i o n to the appl ied voltage during the m o b i l i t y measurements. 119 APPENDIX B CALIBRATIONS BI C a l i b r a t i o n s Made for E lec t rophores is Apparatus BI .1 Ce l l dimensions The i n t e r - e l e c t r o d e d is tance in the micro - e lec tophore t ic c e l l was required to evaluate the potent ia l gradient appl ied for m o b i l i t y measurements. In view of the complex nature of the c e l l t h i s could not be measured d i r e c t l y . It was therefore obtained from measure-ment o f s o l u t i o n c o n d u c t i v i t y , res is tance across the e l e c t r o d e s , vol tage drop and c r o s s - s e c t i o n a l area of viewing r e g i o n . I f R and A are the e l e c t r i c a l r es is tance and conduc t i v i t y r e s p e c t i v e l y o f a so lu t ion placed in the c e l l , then the e f f e c t i v e i n t e r - e l e c t r o d e d is tance l i s given by £ = RxA, where A i s the c r o s s - s e c t i o n a l area o f the viewing reg ion . The thickness of the c e l l was measured using the micrometer focussing adjustment of the e lec t rophores is microscope. Two f l a t c e l l s were a v a i l a b l e , one as a spare . The c e l l was clamped i n the water bath o f the e lec t rophores is apparatus in the same manner as in pro-m o b i l i t y measurements . The c e l l was however empty. The microscope was focussed at both the fa r and near inner surfaces that determine the th ickness . A l l measurements were made at the centre of the c e l l in the viewing p lane, and several measurements were taken for an average va lue . 120 The width of the rectangular sect ion of the c e l l was a lso measured by means of a t r a v e l l i n g microscope. The c e l l was removed from the bath and placed in an upr ight pos i t ion on a hor izonta l l e v e l . The c e l l was then f i l l e d with pottassium per manganite (KMnO^), the d i s t i n c t co lour o f which f a c i l i t a t e d an accurate view of the boundaries o f the c e l l . Several measurements were a lso taken in th is case and averaged. The area o f the view region was therefore obtained from the product o f the thickness and width o f the c e l l . Average values of such measurements are ind icated in Table B l . The product Rx in the expression fo r the i n t e r - e l e c t r o d e d is tance (see above) was a lso determined by measuring the res is tance across a so lu t ion of known/conductivi ty placed i n : t h e c e l l with the e lectrodes in p o s i t i o n . A standard so lu t ion (0.1N) of KC1 was prepared by c a r e f u l l y weighing 74.555 g of KC1 and d i s s o l v i n g in one l i t r e of d i s t i l l e d and deionised water. The 0.1N so lu t ion was then obtained by d i l u t i o n of the IN so lu t ion with d i s t i l l e d and deionised water. A quant i ty of th is 0.1N so lu t ion was placed in the c e l l to f i l l i t . The e lectrodes were put in to pos i t ion and the c e l l clamped in the e l e c t r o -phoresis water bath. The water bath and i t s contents were allowed to stand for 12 hours to ensure thermal e q u i l i b r i u m . A f te r t h i s time the temperature o f "the bath (and i t s contents) were measured, and the res is tance across the c e l l was measured by means of a Beckman Model 16B2 A . C . c o n d u c t i v i t y bridge operat ing at 1000h z, to avoid p o l a r i s a t i o n . The c o n d u c t i v i t y o f the so lu t ion at the equ i l ib ra ted temperature (24°C) was then obtained from a standard tabu la t ion given by Hodgeman (55). Table Bl gives the values obtained from these measurements. 121 TABLE BI Ce l l Dimensions Ce l l #1 Ce l l # 2 C e l l width , cm 1.0086 ± 0.0038 1.0160 ± 0.0015 Cel l t h i c k n e s s , cm 0.1022 ± 0.0037 0.1030 ± 0.0010 2 Area o f plane o f view, cm 0.1031 0.1046 X mmhos/cm 12.64 12.64 R ohms 5440 5640 Intere lect rode d is tance U ) Rx .cm-1 68.762 71.289 £ cm 7.0894 7.4603 BI.2 Eyepiece g r a t i c u l e and t iming device The g r a t i c u l e spacing in the eyepiece of the e lec t rophores is microscope over which the p a r t i c l e s were timed during m o b i l i t y measurements required c a l i b r a t i o n . This was done by observing a "stage micrometer" (a s l i d e engraved with 1 mm d iv ided into 100 equa l "par ts ) . The "stage micrometer" was held in a v e r t i c a l p o s i t i o n in the water bath maintained at 25°C with clamps. Several measurements were taken and averaged from which one g r a t i c u l e spacing was found to be 63.79 ± 0.053 ym. The e l e c t r i c a l timer which was incorporated in the apparatus was a lso c a l i b r a t e d against a Heuer stop watch. The two timing devices were allowed to run together for short times (1-2 minutes) on several occas ions . It was observed that the timer on the apparatus recorded 122 times that were quite c o n s i s t e n t . However the r a t i o of the two records , stopwatch to t imer , was always constant and equal to 5/6. Therefore a l l measured times for e l e c t r o p h o r e t i c v e l o c i t i e s were corrected by m u l t i p l y i n g by a fac tor of 5/6. B2 Measurement of P a r t i c l e Density P a r t i c l e d e n s i t y , which was v i t a l for the determination o f p a r t i c l e volumetr ic concentrat ion arid the evaluat ion of the Stokes v e l o c i t y , was determined by the use o f conventional s p e c i f i c g rav i ty b o t t l e s . In a previous attempt, dr ied and c a r e f u l l y weighed p a r t i c l e samples were placed in a s p e c i f i c g rav i t y b o t t l e and then f i l l e d with d i s t i l l e d water. The volume occupied by the p a r t i c l e s was obtained from the d i f f e r e n c e in volume of bot t le and volume of added water. Values of d e n s i t i e s obtained by th is method were far l e s s than the 2.17-2.20 reported in the l i t e r a t u r e (65). This was because the p a r t i c l e s were not in a dispersed s ta te in the bo t t l e and therefore were not thoroughly wetted. The above method was therefore modif ied in the fo l lowing manner. The volume o f the s p e c i f i c g rav i t y bo t t l e was found by determining the volume o f d i s t i l l e d water that f i l l e d i t at 25°C based on the den-s i t y of pure water at 25°C (66). The emptied bot t l e was then f i l l e d with a h ighly dispersed suspension of the p a r t i c l e s and maintained at a temperature of 25°C in a water bath. The weight o f the bo t t l e and i t s contents were recorded. The suspension was then c a r e f u l l y poured into a pre-weighed p e t r i - d i s h and dr ied in an oven. The s p e c i f i c g rav i t y bo t t l e bearing t races o f the p a r t i c l e s was a lso d r i e d . Both the p e t r i - d i s h and the s p e c i f i c g rav i ty bo t t l e were weighed a f te r dry ing in order to determine the quant i ty of dr ied p a r t i c l e s . The 1 2 3 d i f f e r e n c e between the weight of the p a r t i c l e s and the weight o f suspension gave the weight of water in which the p a r t i c l e s were d i s -persed. The volume of th is quant i ty o f water was therefore obtained from a knowledge o f the water dens i ty . The volume of the s p e c i f i c g rav i ty bo t t l e in excess of t h i s represents the volume occupied by the p a r t i c l e s . A sample c a l c u l a t i o n i s presented below. A s i m i l a r method was t r i e d out in which the water was replaced by propanol , the densi ty of which was determined by the use of a Westphal balance. Results obtained by t h i s method were s l i g h t l y higher than the values of s i l i c a dens i ty in the expected range. The high v o l a t i l i t y of propanol and the high moisture adsorpt ion by the dr ied p a r t i c l e s a f t e r treatment with propanol apparent ly a f fec ted the measurements. The r e s u l t s obtained for the case of d ispers ion in water were therefore adopted, s ince the measured d e n s i t i e s were found to' be wi thin the reported range and because sedimentation experiments were to be c a r r i e d out in water. The average value o f p a r t i c l e dens i ty obtained 3 from several measurements was 2 . 1 7 2 ± 0 . 0 0 4 g/cm . Sample C a l c u l a t i o n Wt. of empty b o t t l e , a Wt. o f bo t t l e and water, b Wt. of water, (b-a) = c .'. volume o f bo t t l e c / 0 . 9 9 7 0 7 = d Wt. of bo t t l e and suspension, e Wt. of empty d i s h , g Wt. of suspension, (e-a) = f Wt. o f d ish and dr ied p a r t i c l e s , h Wt. o f p a r t i c l e s in d i s h , (h-g) = i Wt. of bo t t l e and t races of p a r t i c l e s , j Wt. of t races in b o t t l e , ( j -a ) = k-1 8 . 4 2 0 9 4 g 4 5 . 9 4 1 1 0 2 7 . 5 2 0 1 6 2 7 . 6 0 1 0 3 4 6 . 7 0 6 0 ' 4 6 . 4 3 1 9 6 2 8 . 2 8 5 0 6 4 7 . 8 3 5 0 0 1 . 4 0 3 0 4 1 8 v 4 3 0 0 0 0 . 0 0 9 0 6 124 Sample C a l c u l a t i o n - continued .". to ta l wt. o f p a r t i c l e s (i+k) = £ Wt. of water with suspension ( f - £ ) = m .*. volume o f water with suspension, m/o.99707=n .'. volume occupied by p a r t i c l e s , (d-n) = 0 .'. Density of p a r t i c l e s , l/o 26.95193 26.87296 1.41310 0.64910 2.17547 B3 Measurement o f P a r t i c l e S ize from Elect ron Micrographs P a r t i c l e s i zes were measured d i r e c t l y from the e lec t ron micrographs produced using a "micrometer s tage" , [Bausch and Lomb Inc. ] attached to a 7 x eyepiece. The micrographs were placed on a glass top table i l luminated from below for c l a r i t y . The "micrometer stage" was engraved with l i n e s 0.1mm apar t . Several random measurements were made form each micrograph, from which the average and standard dev ia t ion were obta ined. Measure-ments obtained th is way were then converted to actual p a r t i c l e s i z e by d i v i d i n g by the magnifying fac tor employed. Three batches of p a r t i c l e s were prepared in a l l . The f i r s t two batches were combined to provide a s u f f i c i e n t quant i ty to cover the higher p a r t i c l e concent ra t ions . This combined batch was used for Run s e r i e s 1-4 and 7, and the l a s t batch for Run ser ies 5 and 6. E lec t ron micrographs o f the three batches are shown in Figure 4.1 and the r e s u l t s of s i z e measurement presented in Table B2. 125 TABLE B2 P a r t i c l e S ize Measurement Batch Number Average . Diameter (ym) I 0.592 + 0.0178 IT 0.620 + 0.013 I arid II 0.606 + 0.022 III 0.519 + 0.021 5.4 Conduct iv i ty of NaCI with Concentrat ion At high e l e c t r o l y t e concentrat ions and in other spec ia l cases in the experimental runs , the e l e c t r o l y t e c o n d u c t i v i t y was assumed to be mainly due to the presence o f NaCI. A c a l i b r a t i o n curve fo r the c o n d u c t i v i t y versus concentrat ion of NaCI was convenient for the est imat ion o f the e l e c t r o l y t e concen t ra t ion . Various NaCI so lu t ions of known concentrat ions were therefore prepared and the i r temperature-corrected c o n d u c t i v i t i e s were measured with the S iebold Conduct iv i ty meter, using the same spec ia l c o n d u c t i v i t y c e l l that was cons t ruc ted . The r e s u l t s obtained are presented in Figure B I . B5 V i s c o s i t y Measurement Although the s e t t l i n g experiments were conducted at temperatures c l o s e to 2 5 ° C , the e f f e c t of small v a r i a t i o n s in temperature were not over looked. A l s o , v i s c o s i t y changes due to the in t roduct ion of NaCl CONCENTRATION (moles/litre) Figure Bl : NaCl Concentration Versus Measured Conduct iv i ty of So lu t ion 127: e l e c t r o l y t e s , though very s m a l l , were a lso accounted f o r . Thus a set of v i s c o s i t y measurements were c a r r i e d out at 25°C for extreme e l e c t r o l y t e concentrat ions that covered the range of e l e c t r o l y t i c so lu t ions encountered in the main experimental s t u d i e s . The kinematic v i s c o s i t i e s o f the so lu t ions were determined by the use of the connon-Fenske viscometer o f s i z e 200 (ASTM). The viscometer was immersed in a constant temperature kinematic v i s c o s i t y bath maintained at a uniform temperature of 25 ± 0 . 0 1 ° C . The e f f lux time of the tes t so lu t ion was measured with an e l e c t r o n i c d i g i t a l s-top-watch graduated in d i v i s i o n s of 0.01 sec . The viscometer was prev ious ly c a l i b r a t e d with d i s t i l l e d water, the v i s c o s i t y of which was known (58) . In order to convert the measured kinematic v i s c o s i t i e s into absolute v i s c o s i t i e s , the d e n s i t i e s of the tes t so lu t ions were i n i t i a l l y measured with a Westphal balance which had been c a l i b r a t e d with d i s t i l l e d water at the same working temperature. The r e s u l t s are shown in Table B3. A l l v i s c o s i t i e s were therefore corrected to al low for the e f f e c t of e l e c t r o l y t e concentrat ion at the reported temperatures by l i n e a r i n t e r p o l a t i o n of these r e s u l t s , s ince the measured v i s c o s i t y charges due to the in t roduct ion of e l e c t r o l y t e s within the l i m i t s encountered in these experiments were very s m a l l . TABLE B3 V i s c o s i t y Measurements Sampl e Descr ipt ion Density (gm/cc) Average:ef f lux time (sec) Time r a t i o with respect to S a m p l e ! -V i s c o s i t y . (cp) 1 D i s t i l l e d water 0.9989 284.67 1 0,8904 2 NaCI, 4.0 mmoles / l i t re 0.9990 285.03 0.9987 0.8916 3 Sample 2 ra ised to pH 12.37 0.9990 285.22 0.9981 0.8922 4 HC1; pH 2.02 1.0009 285.22 • 0.9981 0.8939 5 NaOH; pH 13.02 1 .0010 285.92 ' 0.9956 , 0.8962 6 Sample 5 reduced to pH 11.40 1 .0054 287.78 0.9892 0.9060 129 A P P E N D I X C .STATISTICAL AND ERROR ANALYSIS Cl Empir ical Cor re la t ion (Uncharged P a r t i c l e s ) The mul t ip le regressiorial ana lys is which was adopted in the development of the empir ical c o r r e l a t i o n s i s a s t a t i s t i c a l technique that employs the method of l e a s t squares to pred ic t from experimental data the r e l a t i o n s h i p of a dependent v a r i a b l e , Y, to the X m independent va r iab les in an equation of the form: b^ + b,x, + — b x [C5 ] o 1 1 mm with in tercept b Q , and c o e f f i c i e n t s b-j, b 2 , . . . , b m . The STPREG (Stepwise Regressional Ana lys is ) rout ine of the UBC S t a t i s t i c a l TRP (Tr iangular Regressional Package) Programme, which was used in the computations, t reat the ' n ' experimental data made up of the dependent and i t s corresponding independent v a r i a b l e s , p lac ing them in a matrix of the form Y x l — Xm Y n X l n Xnm The programme then proceeds to the est imat ion of the parameters 130 b Q , b-|, bm by the standard method of least squares using an array of the means, standard deviat ions and cor re la t ion matrix of the data .set . In order to select the best equation for predict ing the value of the dependent va r i ab le , the programme also returns values for s t a t i s t i c a l indicators such as the standard error of est imate, 2 residual variance and the mul t ip le cor re la t ion c o e f f i c i e n t , R . The 2 R , which i s always between zero and un i ty , gives the best ind icat ion of how well the regression equation f i t s the experimental data. It i s simply the square of the simple cor re la t ion of Y and Y. Thus IT = I (Y. - Y)(Y. - Y) L1=1 I (Y 1 - Y ) 2 I (Y - Y ) ' i=l 1 i=T where Y. = the i th observation value of the dependent var iab le Y.j = the i t h predicted value Y = the mean of the dependent var iab le Y = the mean of the predicted values. 2 The c loser R i s to un i ty , the better the regressional f i t . The experimental data for the uncharged par t ic les were then' . f i t t ed to functions of the form of equation C5. The se t t l i ng rate UQ at various pa r t i c le concentrations was normalized with respect to the se t t l i ng ra te , U r , at the minimum par t i c le concentration invest igated (c = 0.00514), and the reciprocal of th is ra t io correlated with par t i c le concentration in the form U T T ~ " = b + b,c + U o 1 0 .b c m m [C6] 131 11 m where in t h i s case the dependent v a r i a b l e i s _r , with c , U m = 1, 2 , 3, e t c . as the independent v a r i a b l e s . For c=0, UQ= U s > Therefore equation C6 reduces to I T = b o W s from which U s could be est imated. Subs t i tu t ing U^ , from C7 into C6 U b - V 9 - = b + b n c + b c m [C8] U o 1 m o which s i m p l i f i e s to ir =1 + trc + ± T T - C C 9 ] O O 0 and U o U 1 + b, c + ...+b c" s 1 m [CIO] o o Regressional ana lys is of equation C6 was then c a r r i e d out for var ious values o f m. The r e s u l t s are shown in Table Cl . The r e s u l t s for m=3 gave the best f i t , with R = 0.9979. Thus putt ing t h i s regressional equation in the form of equation CIO, the f i n a l empir ical equation becomes TABLE Cl Values of Coe f f i c i en ts Determined by Regressional Ana lys is standard 2 m b Q bj b 2 b 3 b 1 / b Q b 2 / b Q b 3 / b Q e r ror R 1 0.9393 5.428 - - 5.78 - - 0.05161 0.9793 2 0.9968 3.06T 18.223 - 3.071 18.28 - 0.02333 0.9966 3 0.9827 4.362 -3.046 - 88.002 4.44 3.010 89.55 0.02485 0.9979 133 and U"s = Ur/'b0 (equation C7) = 2 ' 2 2 0 . 9 8 2 7 " 7 m / S = 2 ' 2 6 4 * 1 0 " 7 m / s -C2. Dimensionless R e l a t i o n s h i p s (Charged P a r t i c l e s ) The determination of the f u n c t i o n a l r e l a t i o n s h i p between the dimensionless groups obtained by the Rayleigh method (see Section 5.6) fo r the charged p a r t i c l e s was extremely d i f f i c u l t , since there were a l a r g e number of p o s s i b l e r e l a t i o n s h i p s . An i n i t i a l t r i a l i nvolved the power-product equation of the form / e - C 2 \ a ' i / e u A a 2 a 3 ai* where K, a i a 2 a 3 on».are constants to be determined. By transforma-t i o n i n t o a l i n e a r equation of the form C5, with the a i d of s t a t i s t i c a l computer programmes such as the SPSS ( S t a t i s t i c a l Package f o r S o c i a l Sciences) or TRP ( T r i a n g u l a r Regressional Package), the constants were evaluated from the experimental data. The value for a i was estimated to be negative, which d i d not agree with theory, according to which U increases; with increase i n 5 , the zeta p o t e n t i a l . An i n s p e c t i o n of the experimental data revealed the opposing i n f l u e n c e on U by the c o n d u c t i v i t y even at an increased zeta p o t e n t i a l . The two dimensionless groups c o n t a i n i n g the zeta p o t e n t i a l and the c o n d u c t i v i t y were t h e r e f o r e combined to give, another dimensionless group, ' : j 2 ^ , that y i e l d e d a p o s i t i v e exponent upon a n a l y s i s . However, the R2 value 134 obtained from the regress ional ana lys is was very poor, namely 0.294. Moreover, t h i s form of the funct ional r e l a t i o n s h i p did not s a t i s f y the l i m i t i n g theore t ica l cond i t ion that as c->-o and ka-»-«> the r e l a t i o n -ship reduces to that of Smoluchowski for a s i n g l e p a r t i c l e . The power-product funct ional r e l a t i o n s h i p was therefore abandoned. Other a lgebra ic funct ional r e l a t i o n s h i p s were cor re la ted with the experimental da ta . The f i n a l s e l e c t i o n of the form of r e l a t i o n s h i p used was made so that the theore t ica l l i m i t i n g condi t ions were s a t i s f i e d and so that the number of b e s t - f i t constants did not exceed that required by the Raleigh theory. Inc lus ion of more adjustable constants would a r t i f i c i a l l y reduce the standard error of estimate and 2 increase the c o e f f i c i e n t o f determinat ion, R , which are the s t a t i s t i c a l ind ica to rs for the best regress ional f i t . The two r e l a t i o n s that s a t i s f i e d the above condi t ions were and such non- l inear equations could not be transformed into the l i n e a r form of equation C5; therefore the BMDP:PAR ( s t a t i s t i c a l programme for non- l inear functions), was used for the determination of the parameters . Values for the dimensionless groups computed from the experimental r e s u l t s are shown in Table 5 .5 . The programme 135 returned the values o f the constants PI through P4 for the case where the res idua l sum of squares was smal lest by the method of 2 i t e r a t i o n s . The R and the standard e r ror o f estimates were then evaluated from the experimental r e s u l t s in a separate computation, using the values of the returned constants . The best estimates obtained for the constants o f equation C14 were P 1 = 7.74 P 3 = 0.334 P 2 =-540.90 P 4 = 1 .245 2 with standard error of estimates o f 0.16 and R of 0.818. A sca t te r p lo t of predicted v e l o c i t y versus measured v e l o c i t y was then made, as shown in Figure 5 .7 . A l s o , a sca t te r p lo t of predicted absolute v e l o c i t y versus observed absolute v e l o c i t y is shown in Figure 5.8 for the case o f m = 1. Both p l o t s , however, d i s p l a y only moderate c o r r e l a t i o n of the experimental da ta . C3. Er rors in E lec t rophores is Measurements Wide ranges of e lec t rophore t i c times measured (see Appendix DO for p a r t i c l e s in one set of measurements. This might be due to a number of inherent problems associa ted with such measurements. How-ever , some problems such as jou le heating and p o l a r i z a t i o n were e l iminated through proper experimental technique, as explained in Sect ion 4 .4b. Er rors due to Brownian motion and d e p t h - o f - f i e l d are d i f f i c u l t to e l iminate and therefore c o n s t i t u t e the primary source of e r r o r . 136 In order to determine the er ror l i m i t s , i t w i l l be assumed that the measured e lec t rophore t i c times are approximately normally d i s t r i b u t e d and the 95% confidence l i m i t s based on the student t -d i s t r i b u t i o n w i l l be adopted to pred ic t the c loseness of the sample mean to the populat ion mean. Thus i f the measured e lec t rophore t ic m o b i l i t y i s M E , the 95% conf idence in terva l fo r Mr- i s (67) where t^ ^ , 0.025 is the t - d i s t r i b u t i o n with N-1 degrees o f freedom at the 95% leve l of p r o b a b i l i t y , N i s the sample s i ze and S m the sample standard d e v i a t i o n . The e lec t rophore t ic m o b i l i t y i s a funct ion of the measured e lec t rophore t i c t ime; thus the m o b i l i t y var iance is a lso a funct ion of the var iance of the e lec t rophore t ic t imes. According to Mickley et a l . (68) , i f y = f ( x 1 , x 2 — x n ) then CL = M E ± tj N-1 ' 0.025) "CNT; [C15] [CI 6] Thus s ince [C17] b Et We have by equation C16, assuming i = l , y=M E , x ^ t , e r x i = s t , and S =S , that y nT [C18] where Dg = grtd spacing E = e l e c t r i c f i e l d appl ied t = mean e lec t rophore t ic time and S t = standard dev ia t ion ' .o f t . . given by _ ? 2 N U i - t ) ' = v- ! t i=i N" 1 Therefore from equations C15 and C17, together with a tab le of t d i s t r i b u t i o n (69 ) , the range of m o b i l i t i e s are estimated and conse-quently the errors in the p a r t i c l e zeta potent ia ls are c a l c u l a t e d . Considering Run IA N = 44 t = 2.691 sec S t= 0.1634 D = 2 x 63.79 x 10" 4 cm 9 E = 9.383 v/cm = 70.0/7.4603 (Sec.A3 and Appd.B) Therefore from equation C18 _4 c 2 x 63.79 x 10 cm „ n 9.383 x (2.691 ) : = 3.067 x 10" 5 c m 2 / v o l t - s e c From T a b l e s , t , 0.025 = 2.021 4.3 Thus from equation C15 the conf idence l i m i t s are Mp ± 2.021 x 3.067 x 10" 5 M E ± 0.9313 x 10 rt-2 •5 and using equation A6, the confidence l i m i t s on E are 138 E ± 0.9313 x 10" 5 x 0.7503 x 10 4 i . e . E ± 0.0698 From sec t ion A3 , E was found to be 3.72 the range of E then becomes E = 3.65 and E = 3.79. The corresponding Y values obtained from a p lot of E 1 versus Y Q are 2.70 and 2.80 r e s p e c t i v e l y . It fol lows from equation Al2 that the confidence l i m i t s on the p a r t i c l e zeta potent ia l are 69.36 and 71.93. i .e. 70.6 ± 1.3 mV. APPENDIX D EXPERIMENTAL DATA The s e t t l i n g r e s u l t s obtained for a l l the runs are tabulated in th is s e c t i o n . Also the e lec t rophore t ic times measured for the determination of p a r t i c l e zeta po ten t ia ls are a lso presented. The condi t ions under which each run was conducted inc lud ing a l l re levant parameters such as p a r t i c l e concent ra t ion , pH of s o l u t i o n , s o l u t i o n ion ic c o n c e n t r a t i o n , p a r t i c l e s i z e , so lu t ion measured i o n i c con-d u c t i v i t y and measured zeta potent ia l can a lso be found in th is s e c t i o n . Abbreviat ions used in th is sect ion are def ined as f o l l o w s : C p a r t i c l e vo lumetr ic -concent ra t ion pH pH of s o l u t i o n ka e l e c t r o k i n e t i c radius TEMP average temperature °c y v i s c o s i t y of s o l u t i o n cp D^y Average p a r t i c l e s i z e ym COND c o n d u c t i v i t y o f so lu t ion ymhos/cm IONIC CONC i o n i c concentrat ion mmoles / l i t re 5 p a r t i c l e zeta potent ia l mV DI N e g l i g i b l e Surface Charge RUN NI C = 0.00514, pH = 3.24, TEMP = 24.84 ka = 23.88 y = 0.8926 = 0.606 Height of Interface (mm) Time (hrs) 96.72 2.00 93.35 6.08 89.45 10.83 87.81 13.41 78.73 24.41 RUN N2 C = 0.0137 pH = 3.34, TEMP = 24.60 k a = 2 1 . 2 8 y = 0.8925 =0.606 Height of Interface (mm) Time (hrs) 85.50 3.33 83.38 5.91 79.27 11.16 70.50 22.66 RUN N3 C = 0.0251 pH = 3.33 TEMP = 24.72 ka = 21.53 :y = 0.8925 D * 0.606 M V Height of Interface (mm) Time (hrs) 86.90 3.92 84.27 6.59 81.78 10.50 74.60 20.75 RUN N4 C = •-0.0502 pH = 3.34 TEMP =. 24.86 ka = 21.29 y = 0.8925 DV = 0.606 AV Height of Interface (mm) Time (hrs) 79.79 4.17 73.26 14.25 70.00 19.25 86.44 24.25 RUN N.5 C = 0.0799 pH< = 3.16 TEMP = 24.66 ka = 26.21 y = 0.8926 = 0 . 6 0 6 Height of Interface (mm) Time (hrs) 91 .39 0.00 90.20 3.08 88.78 6.41 85.48 10.33 79.22 20.91 RUN N6 C = o 1085 pN = 3.30 TEMP = 24 .'97 ka = 22.29 y = 0.8925 = .0.606 Height of Interface (mm) 85.04 83.83 78.65 77.48 75.94 Time (hrs) 6.08 8.41 18.41 20.83 23.75 RUN N7 C = 0.155 pH = 3.29 TEMP = 24.82 ka = 22.56 y = 0.8925 = 0.606 Height of Interface (mm) Time (hrs) 76.27 0.00 76.75 3.33 73.50 5.75 69.31 16.42 68.0 19.50 RUN N8 C = 0.2044 pH = 3.68 TEMP = 25.27 ka = 14.39 y = 0.8925 D y^ = 0.606 Height o f Interface (mm) Time (hrs) 90.33 0.00 89.42 3.83 88.42 7.91 86.04 18.49 85.60 20.91 RUN N9 C = 0.259 pH = 3.76 TEMP = 23.70 ka = 13.13 y = 0.9130 DAy = 0.606 Height of Inter face (mm) Time (hrs) 98.80 11.75 97.12 23.50 95.75 35.58 94.20 47.75 D2 SUBSTANTIAL SURFACE CHARGE RUN IA C = 0.0098 5 = 70.64 ± 1.3 y = 0.8904 pH = 6.75 COND = 8.5 TEMP = 24.80 IONIC CONC = 0.072 n A = 0.606 Height of Interface (mm) 99.58 97.98 95.74 89.49 86.26 83.50 82.24 Time (hrs) 0.00 3.92 6.67 15.25 19.17 23.00 25.00. ELECTROPHORETIC TIME (Sec) Timing Distance = 2 Grid Spacing Upper Stat ionary Level Lower Stat ionary Level 3.26 3.24 3.24 3.48 3.24 3.26 2.26 3.54 3.18 3.28 3.28 3.56 3.14 3.14 3.14 3.'44 3.20 3.36 3.36 3.32 3.26 3.18 3.18 3.26 3.26 3.20 3.20 3.36 3.20 3.26 3.26 3.40 3.16 3.12 3.12 3.36 2.90 3.08 3.08 3.20 3.14 3.40 3.40 3.26 RUN IB C = 0.0098 C = 62.61 (+ 0.84 - 3.53) Vi = 0.8906 pH = 5.86 COND = 115.0 TEMP = 24.70 IONIC CONC = 0.785 =0.606 AV Height of Interface (mm) Time (hrs) 95.48 2.48 93.60 5.09 91.04 7.51 85.06 16.18 82.00 21.10 ELECTROPHORETIC TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 3.16 2.96 3.22 3.16 3.06 3.00 2.98 3.00 3.06 3.08 3.06 3.00 2.96 3.10 3.02 3.00 3.18 3.16 3.10 3.24 3.00 3.14 3.20 3.30 3.06 2.98 3.14 3.00 3.14 3.16 3.04 3.12 3.12 3.20 3.18 3.10 3.06 3.08 3.00 3.08 3.10 3.14 RUN i e C = 0.0098 5 =. 44.31 ± 1 .93 U = 0.8915 pH = 5.52 COND = 265.0 TEMP - 25.10 IONIC CONC = 3.722 D = 0.606 AV Height of Interface (mm) Time (hrs) 102.04 0.0 99.20 3.00 97.15 6.17 89.70 16.25 87.19 20.17 ELECTROPHORETIC TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 4.26 5.10 5.02 4.34 4.28 5.02 5.48 6.24 4.80 4.34 6.04 5.64 6.66 4.34 6.06 5.06 4.22 4.'52 6.10 5.42 4.10 4.46 4.68 4.82 4.50 4.20 4.96 4.74 4.46 4.58 5.12 4.86 5.76 5.50 4.92 4.96 5.68 6.64 5.28 4.38 5.66 6.58 5.50 4.44 5.98 6.54 5.32 4.74 6.34 5.76 6.08 5.86 RUN 2A C - = 0.053 5 =71 .28 ± 3.22 y • = 0.9804 PH = 6.39 COND = 7.42 TEMP = 24.50 IONIC CONC = 0.062 D.„ = 0.606 Height of Interface (mm) Time (hrs) 100.50 2.75 98.62 6.58 97.70 8.58 96.00 10.08 91.69 19.16 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Stat ionary Level 4.56 4.98 4.08 4.42 4.22 5.34 4.68 4.52 4.58 5.22 4.26 4.47 4.62 5.08 4.44 4.46 4.62 4.92 4.20 4.30 4.24 5.26 4.26 4.62 4.36 5.34 3.98 4.44 4.40 5.36 4.44 4.98 3.90 4.98 4.52 4.80 4.12 4.66 4.18 4.54 4.20 4.84 4.14 4.24 4.18 5.22 3.94 4.24 4.20 4.90 3.92 4.12 4.46 4.70 3.98 4.18 3.96 4.82 4.10 4.38 4.18 4.76 RUN 2B c = 0.053 = 64.22 y = 0.8906 PH = 5.33 COND = 88.33 TEMP = 25.2 IONIC CONC = 0.7317 DAV = 0.606 Height of Interface (mm) 93.11 91 .74 86.95 84.18 82.40 Time (hrs) 3.92 6.34 15.01 19.93 23.26 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 4.60 4.46 3.62 3.66 4.22 3.92 3.62 3.56 3.60 4.00 4.18 3.88 4.40 4.26 4.36 4.14 4.04 3.92 4.80 4.32 4.38 4.54 4.20 4.20 4.28 4.10 4.54 4.44 4.08 3.94 4.40 4.30 3.98 3.74 3.96 3.88 4.44 4.26 3.80 3.96 148 RUN 2C C = 0.053 x,- = 49.32 (+ 2.06 - 0.51 ) u = 0.8912 PH = 4.81 COND = 260.0 TEMP =25.1 IONIC CONC = 2.802 = 0.606 Height of Interface (mm) Time (hrs) 96.26 0.00 93.34 6.00 86.95 16.00 84.71 19.92 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 7.74 6.52 9.78 8.84 8.54 7.64 8.42 7.48 9.02 7.44 8.24 7.58 7.82 7.16 9.48 8.32 8.60 7.42 9.10 8.02 Lower Sta t ionary Level 6.62 5.86 6.92 6.44 7.36 6.40 7.80 6.62 7.74 6.52 7.30 7.96 7.68 8.20 7.68 8.06 9.04 8.36 8.32 8.40 8.66 8.00 RUN 3A C = 0.1033 u = 0.8905 COND = 19.25 IONIC COND = 0.162 Height of Interface (mm) 96.93 95.90 94.34 93.66 89.80 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Stat ionary Level 4.24 4.18 5.02 5.00 4.42 4.34 4.62 4.76 4.04 4.12 4.64 5.10 4.12 3.98 4.88 4.90 4.58 4.26 4.88 4.84 4.56 4.58 4.86 4.76 3.98 4.22 5.04 4.92 4.30 4.48 5.04 4.92 4.22 4.30 4.74 4.98 4.24 4.36 4.98 5.16 4.20 4.22 4.88 4.94 4.26 4.22 4.78 4.74 4.50 4.40 4.86 4.94 4.38 4.28 5.08 4.90 4 ..T2" 4.32 4.88 4.94 4.22 4.38 5.18 5.36 4.06 3.92 5.00 5.06 4.26 4.26 4.86 5.04 4.42 4.48 PH TEMP DAV 61 .01 ± 1 .93 6.22 25.41 0.606 Time (hrs) 0.00 5.00 7.92 11 .75 22.08 RUN 3B C = 0.1033 y = 0.8715 COND = 345.0 IONIC CONC = 3.329 Height of Interface (mm) 99.70 96.44 (91.03 88.74 86.48 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Stationary Level Lower Stationary Level 10.20 8.80 8.70 8.62 10.24 10.66 8.08 7.76 9.40 8.74 7.92 8.44 9.80 9.30 8.16 7.84 7.82 7.60 9.06 9.30 9.32 10.48 8.08 8.46 8.54 8.88 8.00 7.46 9.12 9.36 8.88 9.60 9.62 9.02 7.72 6.38 9.54 9.78 10.80 8.88 c = PH TEMP DAV 21 .71 ± 2.49 4.91 25.60 0.606 Time (hrs) 0.00 6.08 15.75 19.83 24.08 \ 15i: RUN 4A C = 0.145 = 0.8705 COND = 13.75 IONIC CONC = 0.116 Height' df Interface (mm) 86.940 86.25 84.96 82.32 80.97 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Stat ionary Level 5.30 4.94 4.62 4.70 5.08 5.10 5.02 4.92 5.32 5.28 4.98 4.52 5.00 4.82 5.06 5.04 4.88 5.02 4.60 4.78 5.18 4.92 4.86 4.78 5.02 5.22 4.74 4.42 5.22 4.92 4.94 4.74 5.18 5.10 5.30 4.88 5.36 5.22 4.78 4.96 5.12 5.00 4.40 4.40 5.24 5.02 4.84 4.96 5.42 5.34 4.90 5.12 5.12 5.02 4.80 4.96 5.24 5.02 5.14 5.02 5.34 5.32 5.02 5.12 5.24 5.12 5.14 4.98 5.26 4.84 4.80 4.98 5.32 5.28 X, = 58 .44 : PH = 6 .28 TEMP = 26 .10 DAV = 0 .606 Time (hrs) 0.00 7.92 11 .75 22.08 27.50 RUN 4B C = 0.145 5 46.24 ± 1.29 y = 0.8714 PH = 5.31 COND = 260.0 TEMP = 26.40 IONIC CONC = 3.134 = 0.606 Height of Interface (mm) 90.30 89.03 86.11 84.07 83.11 Time (hrs) 0.00 6.00 15.58 1.9.66 23.91 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 5.72 5.14 5.12 5.00 5.58 5.50 5.68 5.28 5.10 4.86 5.08 5.00 4.92 4.82 4.78 4.78 5.12 4.70 5.32 4.98 4.18 4.56 4.38 4.58 Lower Stat ionary Level 4.36 4.70 4.54 5.00 4.42 4.80 4.48 4.58 4.58 4.70 4.34 4.40 4.26 4.20 4.38 4.44 4.46 4.22 4.54 4.42 RUN 1 .1 C = 0.0064 £ = 61 .66 ± 1 .29 Vi = 0.8904 PH = 6.86 COND = 10.50 TEMP = 25.20 IONIC CONC = 0.083 D f t v = 0.606 Height o f Inter face (mm) 75.00 73.36 71.77 64.85 6T.65 55.20 Time (hrs) 0.00 4.92 7.59 17.42 21 .59 29.42 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 5 .28 4.98 5 .00 4.96 4 .98 4.88 5 .02 4.88 4 .54 4.52 5 .00 4.72 5 .00 4.80 4 .82 5.02 5 .02 4.92 5 .10 4.80 5 .12 4.96 Lower Sta t ionary Level 4.60 4.54 4.78 4.54 4.64 4.50 4.56 4.70 4.58 4.82 4.38 - 4.30 4.62 4.64 4.48 4.68 4.46 4.58 4.. 2 6 4.34 RUN 1 .2 C = 0.0064 5 = 10.00 ± 0.18 y = 0.9121 PH = 6.86 COND = 370.0 TEMP = 24.30 IONIC CONC = 3.457 D f l W = 0.606 Height of Interface (mm) Time (hrs) 91-62 0.00 87.36 5.33 83.24 10.33 74.90 22.33 71-60 26.33 67.39 31.83 ELECTROPHORESIS TIME (Sec) Timing Distance = 1 Grid Spacing Upper Sta t ionary Level 9 .52 12.60 12 .00 12.60 10 .10 13.24 10 .12 13.06 15 .42 _ 13 .60 13.84 11 .26 11 .22 12 .64 13.00 10 .42 11 .42 Lower Sta t ionary Level 9.82 13.68 11 .60 12.70 12 .77 13.60 12 .88 11 .65 12 .77 12.52 12 .33 11 .25 13 .10 10.14 11 .60 10.23 11 .33 155 RUN 1.3 C = 0.0064 u = 0.9121 COND = 342.0 IONIC CONC = 3.457 •5 PH TEMP DAV 35.96 (+ 1 .80-1 .28) 6.86 23.80 0.606 Height o f Interface (mm) 50.58 ' 87.20 85.15 78.60 75.21 73.15 Time (hrs) 0.00 4.75 7.17 17.09 20.92 24.08 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 5.52 5.94 5.94 5.98 6.92 7.62 5.46 6.00 5.54 5.86 5.54 5.58 6.14 6.62 5.92 6.52 6.40 6.38 Lower S ta t ionary Level 5.98 6.34 5.64 5.68 7.86 7.10 6.48 6.36 6.04 6.20 5.74 5.78 7.12 7.40 5.64 5.12 4.98 5.02 RUN 1.4 C = 0.0064 u = 0.9121 COND = 330.0 IONIC CONC = 3.457 Height o f Interface (mm) 94.23 92.82 89.81 87.23 80.00 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 4.46 4.18 3.60 3.68 4.14 4.14 4.28 . 4.12 3.76 3.86 4.24 4.30 4.96 4.98 4.04 4.12 4.28 4.00 3.56 3.62 4.26 4.20 3.82 4.34 3.82 3.96 3.66 3.40 3.64 3.60 3.90 3.96 3.92 3.98 4.18 3.90 4.60 4.60 4.24 4.04 PH TEMP DAV 56.52 (+ 1.79-1.29) 7.12 24.0 0.606 Time (hrs) 0.00 5, 9, 11. 21 , 25 25 92 92 RUN 1.5 C = 0.0064 Vt = 0.9121 COND = 320.0 IONIC CONC = 3.457 Height o f Interface (mm) 67.50 65.59 64.05 60.48 58.82 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 3.36 3.40 3.74 3.94 3.62 3.22 3.60 4.16 3.30 3.66 4.16 4.40 3.22 3.36 3.90 4.24 3.54 3.58 3.60 4.16 2.88 3.16 3.46 3.66 3.32 3.36 3.76 3.96 3.38 3.56 3.36 3.74 3.24 3.60 3.56 3.84 3.24 3.42 3.90 3.92 3.38 3.76 5 PH TEMP DAV 66.02 (+ 2.05-2.80) 7.30 23.50 0.606 Time (hrs) 0.00 3.33 6.33 10.83 13.16 RUN 1.6 C = 0.0064 C". = 80.92 (+ 2.57-3.85) y = 0.9124 PH = 9.42 COND = 440 TEMP = 24.4 IONIC CONC = 3.457 D = 0.606 Height of Inter face (mm) Time (hrs) 79.75 0.00 78.48 3.25 76.31 6.83 73.94 10.00 71.29 12.67 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 2,44 2.82 3.18 3.08 3.02 3.18 2.74 3.00 2.70 3.24 2.94 3.02 3.00 3.32 2.98 3.76 2.76 3.14 2.76 3.30 3.08 3.34 Lower S ta t ionary Level 2.46 2.44 2.58 3.10 2.56 3.36 3.26 4.04 3.18 2.72 3.64 3.38 3.18 3.48 2.96 3.38 3.10 3.02 2.88 2.84 2.96 3.32 2.78 . 3.02 2.78 2.92 2.72 3.02 RUN 2-.1 C = 0.0537 S = 41.10 ± 1.29 Vi = 0.8706 PH = 5.82 COND = 28.80 TEMP = 25.6 IONIC CONC = 0.225 Q = 0.606 Height o f Inter face (mm) Time (hrs) 94.00 0.00 91.57 6.08 89.54 9.50 88.17 12.67 84.45 22.17 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 6.84 6.90 5.84 6.12 6.22 6.28 6.42 6.26 6.52 6.52 6.66 6.32 6.72 6.36 6.94 6.56 6.06 5.60 6.10 6.12 Lower Sta t ionary Level 5.76 5.26 7.06 7.30 6.16 6.14 6.22 6.22 5.60 5.58 7.06 7.50 6.70 7.14 5.92 5.88 6.44 6.66 5.74 5.96 RUN 2.2 C = 0.0537 £ = 16.70 (+ 1.56-0.52) y = 0.8915 PH = 5.67 COND = 579 TEMP = 25.45 IONIC CONC = 3 . 5 0 DAV = ° ' 6 ° 6 Height o f Interface (mm) Time (hrs) 95.79 0.00 93.16 3.67 90.97 6.67 88.70 13.59 82.67 23.67 ELECTROPHORESIS TIME (Sec) Timing Distance = 1 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 7.76 7.26 6.20 6.36 7.96 7.12 6.36 6.68 5.66 5.42 5.92 5.78 7.46 7.02 6.08 6.42 5.50 5.30 5.74 5.82 6.66 6.82 6.12 5.84 5.26 4.90 5.68 6.42 6.38 6.06 6.16 6.04 5.64 5.62 5.12 4.80 5.48 5.18 5.34 4.72 RUN 2.3 C = 0.0537 y = 0.8915 COND = 500.0 IONIC CONC = 3.50 Height o f Interface (mm) 97.42 93.20 91 .63 89.95 82.72 ELECTROPHORESIS TIME (Sec) Timing Distance = il Grid Upper Sta t ionary Level 6.14 6 .40 6.82 7 .28 8.36 7 .34 6.82 6 .42 8.06 8 .82 6.98 7 .02 7.54 7 .58 6.70 8 .06 6.20 7 .10 6.02 6 .90 6.28 6 .06 7.37 8 .26 5 16.69 PH = 5.66 TEMP = 25.30 DAV = 0.606 Time (hrs) 0.00 5.33 9.33 12.00 22.00 Lower S ta t ionary Level 7.02 9.02 6.44 7.28 6.60 6.10 6.62 8.06 7.20 6.94 6.64 6.22 6.32 6.42 6.96 7.19 6.28 7.22 5.98 6.48 6.96 6.54 6.94 5.96 RUN 3 . 1 C = 0 . 1 1 3 5 = 4 2 . 3 9 ( + 1 . 2 8 - 2 . 0 6 ) y = 0 . 9 3 2 6 p H = 6 . 1 3 C O N D = 3 4 . 5 0 -TEMP = 2 2 . 5 0 I O N I C C O N C = 0 . 2 7 1 D = 0 . 6 0 6 Height of Interface (mm) Time (hrs) 9 8 . 2 6 0 . 0 0 9 7 . 5 0 4 . 8 3 9 6 . 3 3 8 . 2 5 9 4 . 6 3 1 1 . 4 2 9 2 . 6 2 2 0 . 9 2 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 6 . 5 4 5 . 9 8 5 . 0 8 5 . 0 8 7 . 1 8 6 . 9 2 6 . 3 4 6 . 0 2 6 . 9 6 6 . 8 8 5 . 4 4 5 . 0 0 5 . 3 6 5 . 1 8 6 . 0 8 6 . 0 4 6 . 2 2 5 . 7 2 5 . 8 6 5 . 7 0 7 . 5 8 6 . 9 2 6 . 4 2 6 . 7 4 6 . 6 2 6 . 8 8 5 . 6 6 5 . 9 8 6 . 3 6 6 . 1 2 5 . 8 6 5 . 9 0 6 . 0 8 5 . 5 8 5 . 7 2 5 . 8 4 5 . 8 8 5 / 3 4 5 . 5 8 5 . 8 4 4 . 9 8 4 . 8 8 6 . 5 4 6 . 6 4 6 . 6 6 6 . 7 0 5 . 8 6 5 . 7 2 RUN 3.2 C = 0.113 5 » 24.40 (+ 1.29-0.05) u = 0.9118 PH = 6.20 COND = 270 TEMP = 23.60 IONIC CONC = 2.308 D f t v = 0.606 Height o f Inter face (mm) Time (hrs) 91.41 • 0.00 88.77 6.33 86.22 13.16 82.30 23.33 80.53 28.08 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 7.74 8.70 9.04 6.14 6.94 6.40 8.98 8.68 8.46 8.42 6.26 7.52 7.56 7.58 9.28 9.20 7.96 8.66 9.18 8.52 7.50 7.40 6.28 8.04 8.46 7.38 8.54 9.02 6.24 7.36 7.26 9.26 6.36 6.76 7.20 7.88 7.96 RUN 3.3 C = 0.113 y = 0.9118 COND = 250 IONIC CONC = 2.308 Height o f Interface (mm) 101.83 100.39 97.94 96.95 92.47 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 10.08 10.02 8.46 7.68 10.00 10.92 9.30 8.52 10.46 10.62 8.26 8.08 9.58 10.26 10.74 11 .18 7.48 7.78 7.74 7.36 8.98 7.82 7.94 8.18 9.38 9.26 7.72 7.98 8.50 9.72 9.10 8.30 9.52 9.00 8.06 8.68 8.38 9.20 9.50 8.94 8.90 8.00 8.44 8.68 PH = TEMP = DAV = 26.33 (+ 1 .06-0.92) 5.91 24.10 0.606 Time (hrs) 0.00 5.25 9.25 11 .92 21 .92 RUN 3.4 C = 0.113 y = 0.9118 COND = 295.0 IONIC CONC = 2.308 t, = 31.34 (+ 0.77-0.52) PH = 6.09 TEMP = 24.20 0.606 D AV Height o f Interface (mm) Time (hrs) 94.25 93.27 91 .88 90.25 88.88 84.74 0.00 3.42 6.42 9.67 12.42 22.17 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 7.72 7.96 7.62 7.30 8.02 7.70 7.92 8.46 8.24 7.64 8.16 8.32 8.90 8.84 8.86 8.34 8.48 8.40 8.76 8.56 9.92 9.52 7.76 7.96 8.00 9.14 7.38 7.36 8.48 9.14 9.36 8.64 8.16 8.04 7 .82 7.60 RUN 3.5 C = 0.113 jr. = 54.46 (+ 0.77-0.52) u = 0.8914 PH = 8.51 COND = 330.0 TEMP = 24.88 IONIC CONC = 2.308 n = 0.606 Height o f Interface (mm) Time (hrs) 85.82 0.00 84.65 5.00 82.00 9.0 79.65 13.75 74.00 24.33 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 5.12 5.26 4.46 4.42 4.78 4.54 4.64 4.34 4.58 4.54 5.18 5.06 4.68 4.64 4.82 5.14 5.00 5.12 4.04 4.44 4.38 4.56 Lower S ta t ionary Level 4.72 4.66 4.20 4.18 4.26 4.30 5.18 4.90 4.38 4.20 4.00 4.10 4.62 4.76 5.42 5.24 4.44 4.34 4.36 4.58 RUN 4.1 C = 0.170 n = 46.24 (+ 1 .76-1 .32) y = 0.8905 PH = 6.76 COND = 31.90 TEMP = 24.50 IONIC CONC = 0.252 D = 0.606 Height o f Inter face (mm) 91.00 89.62 88.44 87.74 87.00 85.17 Time (hrs) 13.08 22.75 29.75 34.08 37.08 47.66 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 5.32 4.82 6.20 5.84 5.52 -5.16 5.38 5.68 -5.48 5.64 4.98 -4.64 4.66 5.84 5.78 5.08 -5.14 4.90 4.78 5.10 5.56 5.28 5.44 5.28 Lower S ta t ionary Level 4.62 5.54 4.72 4.46 5.98 5.84 4.98 4.94 4.84 -4.84 4.92 5.16 4.96 6.02 6.48 5.02 5.30 5.04 4.86 RUN 4.2 C = 0.170 y = 0.9119 COND = 283.0 IONIC CONC = 2.506 Height o f Interface (mm) 96.91 96.24 94.80 93.32 89.97 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 7.44 7.40 8.62 8.86 8.08 8.10 8.40 8.28 7.64 7.68 7.74 7.70 7.82 7.70 8.22 7.88 7.00 7.34 8.46 7.98 7.96 7.48 8.40 8.24 7.82 7.86 7.14 7.82 8.44 8.14 7.94 7.92 7.00 7.50 7.52 7.46 7.50 7.56 8.14 8.12 PH TEMP DAV 28.25 (+ 1 .29-0.77) 6.35 24.30 0.606 Time (hrs) 0.00 5.25 9.25 11.92 21 .92 RUN 4.3 C = t = 35.96 (+ 0.77-1.80) P = 0.9119 PH = 7 > o 2 COND = 320.0 T E M P = 2 4 10 IONIC CONC = 2.506 D ^ = Q c]Q6 Height of Interface (mm) Time (hrs) 97.77 4. ^ 95.50 11.50 93.78 16.17 89.85 26.75 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Stationary Level 6.98 6.62 6.46 6.34 5.64 5.42 6.14 5.90 6.16 5.66 6.04 6.04 7.34 7.58 6.94 7.12 5.54 5.56 6.28 5.98 7.76 7.06 Lower Stationary Level 6.74 6 .70 6.02 6 .69 5.92 6 .24 6.46 6 .30 6.78 7 .20 6.88 7 .50 6.76 7 .22 6.42 6 .32 6.02 6 .22 6.36 6 .14 RUN 4.4 C = 0.170 Vt = 0.8916 COND = 495.0 IONIC CONC = 2.999 £ = 61 .65 (+. 2.57-0.04) PH =8.69 TEMP = 24.72 0.606 D AV Height of Interface (mm) 79.45 77.65 76.50 72.70 70.98 Time (hrs) 5.33 10.25 13.25 23.25 28.50 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Stationary Level Lower Stationary Level 3.56 3.52 4.04 3.68 3.66 3.62 3.44 3.32 3.72 3.62 3.96 3.82 3.62 3.72 3.84 3.74 3.20 3.26 3.52 3.42 4.20 3.82 4.04 3.84 3.82 3.78 3.68 3.58 3.74 3.54 3.88 3.02 3.58 3.46 3.96 3.86 3.92 3.92 3.64 3.40 3.90 3.80 3.64 3.40 3.30 3.48 3.28 3.14 4.08 4.00 RUN 5.-1 C = 0.2088 % = 80.92 (+ 2.36-1.79) y = 0.9114 PH = 8.12 COND = 34.50 TEMP = 24.44 IONIC CONC =0 .293 n = 0.519 Height of Interface (mm) 97.50 96.40 94.97 93.64 92.80 91.38 Time (hrs) 0.00 10.50 21.08 29.25 35.25 45.42 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 3.50 3.54 3.74 3.50 3.42 3.46 3.22 3.28 3.80 3.82 3.60 3.48 3.54 3.58 3.26 3.46 3.56 3.52 3.58 3.86 Lower S ta t ionary Level 3.40 3.48 3.24 3.44 3.46 3.44 3.46 3.60 3.34 3.24 3.80 3.70 3.22 3.30 3.12 3.06 3.20 3.24 3.32 3.14 RUN 5.2 C = 0.2088 c = 41.10 (+ 2.00-1.29) y = 0.9114 p H = 7.48 COND = 160.0 TEMP = 24.44 IONIC CONC = 0.293 D = 0.519 Height o f Interface (mm) 90.25 88.59 86.91 85.50 83.40 81.67 Time (hrs) 0.00 11.92 22.00 31.00 46.42 56.59 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower Sta t ionary Level 6.94 7.02 5.88 5.72 6.98 7.56 6.02 5.82 7.18 6.68 5.68 5.38 8.40 8.52 5.02 5.12 6.74 7.20 5.16 5.24 7.76 8.00 5.38 5.42 6.76 6.86 5.48 5.86 7.42 7.44 6.12 5.94 7.82 7.54 6.18 5.72 6.70 6.94 5.08 5.76 5.48 5.24 RUN 5.3 c 0.2088 = 32 .11 V 0.9116 PH = = 7. 51 COND 300.0 TEMP = = 24 .40 IONIC CONC = 1.903 DAV = = 0. 519 Height o f Inter face (mm) 85.82 ' 84.23 81.60 80.17 78.17 Time (hrs) 13.50 25.25 38.17 48.84 62.09 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 7.14 6.72 6.90 6.26 6.14 5.86 7.00 6.48 6.42 5.86 8.54 7.60 8.98 8.36 9.22 8.18 8.26 9.34 7.38 6.54 Lower S ta t ionary Level 6.90 6.14 6.42 5.60 6.36 6.12 7.14 7.26 7.36 7.50 6.14 6.70 8.30 8.36 6.98 6.36 7.22 6.50 7.36 6.44 RUN 6.1 C = 0.0102 .g = 44.95 (+ 1.51-1 .29) y = 0.8904 P H = 5.40 COND = 10.90 TEMP = 24.60 IONIC CONC = 0.082 n = 0.519 Height of Inter face (mm) 96.29 94.00 92.19 90.33 85.15 Time (hrs) 0.00 5.33 10.00 13.00 22.50 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 5.46 5.54 5.58 5.86 5.78 5.34 5.36 5.64 5.64 5.86 5.44 5.68 5.68 5.68 5.40 5.42 5.56 5.50 5.68 5.58 5.92 5.82 6.06 5.90 5.98 5.92 5.60 5.80 5.42 5.32 5.40 5.48 5.40 5.66 5.42 5.34 5.88 5.84 RUN 6.2 C = 0.0102 C = 65.51 + ± 1.29 y = 0.9168 p H = 12.90 COND = 4600.00 TEMP = 2 4 - 3 0 IONIC CONC = 79.51 D = 0.519 Height o f Interface (mm) 94.18 93.29 91.09 89.50 84.75 80.50 Time (hrs) 0.00 4.08 7.75 10.58 19.83 26.33 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 3.30 3.12 3.14 2.98 3.40 3.30 3.48 3.30 3.34 3.02 3.50 3.10 3.44 3.02 3.48 3.14 3.42 3.32 3.70 3.14 Lower S ta t ionary Level 3.86 3.32 3.72 3.32 3.74 3.24 3.60 3.12 3.54 3.06 3.68 3.16 3.48 2.94 3.80 3.40 3.80 3.48 3.32 3.04 3.50 3.12 RUN 6.3 C = 0.0102 y = 0.9061 COND = 1060.0 IONIC CONC = 79.51 5 = 55.23 i 2.57 PH = 11.34 TEMP = 24.50 D A V = 0.519 Height o f Interface (mm) Time (hrs) 97.38 0.00 95.30 5.00 91.70 12.42 84.50 24.25 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 4.44 4.46 4.78 3.64 4.00 5.04 4.45 3.77 4.71 3.53 4.66 3.46 5.02 3.94 4.26 3.98 4.80 3.69 5.02 3.95 4.77 3.91 Lower S ta t ionary Level 4.83 3.65 4.77 3.65 5.20 3.36 4.45 4.09 3.66 3.34 3.87 3.61 5.00 3.60 5.06 3.92 4.04 3.88 4.46 3.78 4.83 3.72 RUN 6.4 C = 0.0102 - 5 = 55.23 + 2.57 y = 0.9270 p H = 11.13 COND = 950.0 TEMP = 24.32 .. . IONIC CONC = 7 9 . 5 1 n = 0.519 Height o f Interface (mm) 81.50 79.84 76.09 69.98 65.16 Time (hrs) 0.00 4.33 10.16 23.83 31.83 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 4.28 3.52 4.46 3.78 4.74 3.50 4.76 3.90 4.68 3.44 4.83 3.72 5.04 4.00 5.02 3.94 4.26 3.98 4.44 4.46 Lower S ta t ionary Level 4.78 3.64 5.04 3.94 4.84 3.64 5.00 3.60 5.22 3.34 4.42 4.12 4.20 3.56 3.64 3.36 3.90 3.58 4.06 3.86 RUN 6.5 C = 0.0102 . 5 = 59.60 (+ 1.29-2.05) U = 0.9289 p H = 9.94 COND = 1080.00 TEMP = 24.32 IONIC CONC = 79.51 n = 0.519 Height of Inter face (mm) Time (hrs) 84.02 6.00 81.75 11.50 75.00 24.50 71.36 30.50 68.25 35.08 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level Lower S ta t ionary Level 5.00 4.24 3.14 3.16 4.02 4.16 3.46 3.40 3.96 4.28 3.46 3.38 4.12 4.00 3.72 3.76 4.16 3.88 3.28 3.34 4.06 3.94 3.58 3.42 4.06 4.08 3.66 3.58 3.88 3.76 3.42 3.50 4.14 4.16 3.40 3.32 3.96 4.16 3.24 3.06 3.86 3.86 3.20 3.20 3.52 3.56 3.20 3.20 3.88 3.70 3.22 3.08 4.06 3.96 4.30 3.70 RUN 7.1 C = 0.2693 5 = 62.61 (+ 0.84-3.53) y = 0.8911 p H = 7.27 COND = 30.05 JEW = 24.60 IONIC CONC = 0.2377 D = 0.606 Height of Inter face (mm) 79.24 78.28 76.50 74.59 72.21 Time (hrs) 8.42 23.83 51.08 75.16 99.16 ELECTROPHORESIS TIME (Sec) Timing Distance = 2 Grid Spacing Upper Sta t ionary Level 4.74 4.84 5.00 4.78 4.66 5.04 4.68 5.18 4.50 4.60 4.70 4.72 3.80 3.78 4.30 4.12 4.34 4.36 4.88 4.82 Lower S ta t ionary Level 4.84 4.56 3.78 3.82 4.16 3.98 4.52 4.60 3.64 3.64 4.98 4.76 4.96 5.20 5.12 4.92 4.74 5.18 4.96 4.60 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0058858/manifest

Comment

Related Items