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Interaction of particles with an advancing solid/liquid interface Schvezov, Carlos Enrique 1984

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INTERACTION OF PARTICLES WITH AN ADVANCING SOLID/LIQUID INTERFACE By CARLOS ENRIQUE SCH.VEZOV Licenciado en F i s i c a , Universidad Nacional de Rosario(Argentina), 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Metallurgical Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1983 (c) Carlos Enrique Schvezov, 1983  In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t t h e 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 , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . It is understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n permission.  Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date  JdM  U&TH  ii ABSTRACT The i n t e r a c t i o n o f p a r t i c l e s w i t h an a d v a n c i n g s o l i d / l i q u i d i n t e r f a c e has been s t u d i e d  considering  mechanical  f o r c e s and f o r c e s  a r i s i n g from t h e n a t u r e o f t h e p a r t i c l e ,  l i q u i d and s o l i d  materials.  E x p e r i m e n t s were c a r r i e d o u t on l e a d and l e a d containing  i r o n p a r t i c l e s t o d e t e r m i n e whether  were r e j e c t e d  o r encompassed  by an a d v a n c i n g  the p a r t i c l e s interface.  The d i s t r i b u t i o n o f p a r t i c l e s i n t h e s o l i d as a o f the m o r p h o l o g y  alloys  function  o f the i n t e r f a c e was a l s o examined.  was f o u n d t h a t p a r t i c l e s were n o t r e j e c t e d i n t e r f a c e and were p r i m a r i l y  It  by an a d v a n c i n g  segregated into c e l l walls  i n t e r d e n d r i t i c zones f o r a non p l a n a r i n t e r f a c e .  and  For a  p l a n a r i n t e r f a c e t h e p a r t i c l e d i s t r i b u t i o n was u n i f o r m both on a m a c r o s c o p i c and m i c r o s c o p i c l e v e l .  T h i s was a l s o t h e  case f o r s o l i d i f i c a t i o n i n d i f f e r e n t d i r e c t i o n s , v e r t i c a l and  horizontal. A w a t e r model system was examined  to observe d i r e c t l y  how p a r t i c l e s i n t e r a c t w i t h a s o l i d s u r f a c e v e l o c i t i e s and s u r f a c e  conditions.  under  The r e s u l t s  d e m o n s t r a t e d how s p h e r e s s e g r e g a t e to c e l l w a l l s  different  clearly on a  s o l i d c e l l u l a r i n t e r f a c e as a r e s u l t o f m e c h a n i c a l  forces.  iii The p r e s e n t r e s u l t s  a r e compared  to t h e o r i e s f o r  p a r t i c l e i n t e r f a c e i n t e r a c t i o n s ini both metal and non m e t a l l i c systems.  C a l c u l a t i o n s o f L i f s h i t z - V a n der Waals  f o r c e s i n a metal show t h a t an a t t r a c t i v e f o r c e between t h e metal p a r t i c l e and m e t a l s u r f a c e .  exists As a r e s u l t  p a r t i c l e s a r e not r e j e c t e d by an i n t e r f a c e i n a metal s y s t e m i n agreement  w i t h the p r e s e n t r e s u l t s .  The o b s e r v e d  d i s t r i b u t i o n o f p a r t i c l e s i n t h e s o l i d can be a c c o u n t e d f o r on t h e b a s i s o f buoyancy and the i n t e r f a c e  morphology.  forces, convection in the l i q u i d ,  i v  TABLE OF CONTENTS Page Abstract Tabel o f Contents  .  Iv  L i s t of Tables  vii  List of Figures  —  Acknowledgement  —  x  ... —  xix  Chapter I  INTRODUCTION  1  II  REVIEW  4  1.  I n t e r a c t i o n of a P a r t i c l e with a Solid/Liquid  Interface; Definition of  the Problem- ..... — 2.  Comparison  .,-  4  of the Proposed  with Experimental  Theories  Resul t s  a) E f f e c t o f t h e Thermal  13 Conductivity..  19  b) The E f f e c t o f I m p u r i t i e s  22  c) O t h e r E f f e c t s  24  i)  V i s c o s i t y of the Melt  24  i i)  Body F o r c e s  24  iii)  Convection  25  V  Chapter  Page 3. P u s h i n g i n M e t a l s  25  - 4. Summary o f C h a p t e r II  26  m  OBJECTIVES OF PRESENT RESEARCH  IV  EXPERIMENTAL APPARATUS AND  28  PROCEDURE• *  30  1. The M e t a l l i c PLS System a) S e l e c t i o n and P r e p a r a t i o n  30  of Samples..  30  .  b) The A p p a r a t u s and T e c h n i q u e f o r Controlled Solidification  33  i)  S o l i d i f i c a t i o n .....  33  ii)  Metal 1 o g r a p h i c P r e p a r a t i o n  iii)  Counting  .....  and P a r t i c l e S i z e  Distribution  3  c) P r o c e d u r e f o r t h e . C a s t i n g . E x p e r i m e n t . . 2. The Water Model 3.  ^ 38  38  P a r t i c l e I n t e r a c t i o n with a Freezing Water I n t e r f a c e  V  6  3  RESULT AND  42  DISCUSSION  43  1. C o n t r o l l e d S o l i d i f i c a t i o n  43  a) N o n - P l a n a r I n t e r f a c e i)  4 3  V e r t i c a l Growth Pb 1% Sb C e l l u l a r  43  Interface-  Sample V-1 The Real  43  D i s t r i b u t i o n of P a r t i -  c l e s i n the M a t r i x  Chapter  p a  9  e  Pb 2% Sb D e n d r i t i c I n t e r f a c e -  ii) b) P l a n a r  Sample V-2  5 7  H o r i z o n t a l Growth .  7 0  Interface  2. The Water-Nylon  7 6  Sphere Model  a) H o r i z o n t a l Mode  7 9  i)  The Motion of the Nylon Spheres  ii)  Modelling  iii)  Comparison  3. The C a s t i n g  with a S o l i d i f i c a 95  Solidification  100  Experiment  105  ..—  4. Pushing i n Water 5. The L i f s h i t z - V a n der Waals Force VI  7 9  86  t i o n Process b) V e r t i c a l  7 9  SUMMARY AND CONCLUSIONS  REFERENCES  1 22 H  1 29 1 33  APPENDICES I  T h e o r i e s of P a r t i c l e R e j e c t i o n a t a Solid/Liquid Interface  II  C a l c u l a t i o n of w  1 37 167  vi i L I S T OF TABLES Table I  Page Summary of f i n d i n g s f o r the i n t e r a c t i o n p a r t i c l e s w i t h an a d v a n c i n g  of  solid/liquid 7  interface II  (a) V a l u e s of the e x p o n e n t n i n the relation  V=  C/R  c  as a f u n c t i o n  f o r the c r i t i c a l  n  of p a r t i c l e  different theories,  (b)  radius  Raw V-1.  V-l-1  Interface,  V - l - l a , 200  ym from V - l - 2 ; V - l - 3 , 5 mm 200  ym  from  from I n t e r f a c e ;  V-l-2a,  from  Inter-  ym from V - l - 3  .  P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample and % i n the M a t r i x . i n t e r f a c e , .(b) and  V  17  P a r t i c l e S i z e D i s t r i b u t i o n f o r sample  face; V-l-3a, IV  the  V.  V - l - 1 ; V - l - 2 , 25 mm 200  R,for  Experimental  v a l u e s of n and III  velocity  (a) c o u n t s at  c o u n t s b e h i n d the  V-1  the interface,  (c) o v e r a l 1  51  Schwarz-Saltikov c o e f f i c i e n t s for  the  c a l c u l a t i o n of the Real S i z e D i s t r i b u t i o n .. VI  P a r t i c l e S i z e D i s t r i b u t i o n and % of c l e s i n the m a t r i x f o r Sample V-1 on a l l s e c t i o n s )  49  (a) A p p a r e n t  (b) Real D i s t r i b u t i o n  56  parti-  (counts  Distribution 58  v i  Table VII  i  i  Page Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r s e g r e g a t e d and M a t r i x p a r t i c l e s f o r Sample V-2. Section 20Q  \xm  V-2-1 a t t h e i n t e r f a c e ; V - 2 - l a , from V - 2 - 1 ; V-2-2, 2.5 mm from the  i n t e r f ace VIII  64  P a r t i c l e Size D i s t r i b u t i o n f o r segregated and m a t r i x p a r t i c l e s , and % i n t h e m a t r i x f o r Sample V-2.  IX  Counts on a l l s e c t i o n s  ....  65  P a r t i c l e Size D i s t r i b u t i o n f o r segregated and m a t r i x p a r t i c l e s , and % i n t h e m a t r i x f o r Sample V-2.  (a) Apparent  Distribution,  (b) Real D i s t r i b u t i o n X  68  Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r S e g r e g a t e d and m a t r i x p a r t i c l e s f o r Sample H - l . H-l-1 c o u n t s a t t h e i n t e r f a c e ; H - l ^ l a , 200 pm from H-l-1 ; H - l 9 2 , 2.5 mm H - l - 1 ; and H - l - 2 a , 200  XI  from  from H - l - 2  7 2  P a r t i c l e Size D i s t r i b u t i o n f o r segregated and m a t r i x p a r t i c l e s , and % i n t h e m a t r i x f o r Sample H - l . Counts on a l l s e c t i o n s  XII  ....  7  ^  C r i t i c a l V e l o c i t i e s in function of p a r t i c l e radius  predicted  and C i s s e  by t h e t h e o r i e s  of Boiling  (a) and Chernov e t a l . (b)  7 7  ix"  Table XIII  P a  ge  C o r r e l a t i o n t a b l e f o r p a r t i c l e s i z e i n the m e t a l l i c system versus (Equation  XIV  '  frequency  35) and v e r s u s  of r o t a t i o n  p e r i o d of r o t a t i o n .  T h e o r e t i c a l and e x p e r i m e n t a l  88  horizontal  v e l o c i t i e s o f the n y l o n s p h e r e s as a f u n c t i o n of t i l t i n g a n g l e and V XV  p  Terminal  = 1 cm s e c "  f o r w = 0.5  rps  1  90  v e l o c i t y , t r a n s i e n t time, t r a n s i e n t  length versus  radius for iron p a r t i c l e s in 99  l i q u i d lead XVI  P a r t i c l e S i z e D i s t r i b u t i o n f o r the  cast  sample, (a) a t the b o t t o m , (b) 5mm  from  b o t t o m , (c) 15mm  from b o t t o m , (d) 25  from b o t t o m , , ( e ) 30 mm the sample.  mm  from bottom - top o f  In (a) and ( b ) , p a r t i c l e s have  been d i s c r i m i n a t e d as b e i n g s e g r e g a t e d i n t h e d e n d r i t e s ' . In ( c ) , (d) and  or  (e)  t h r e e d i f f e r e n t p o s i t i o n s f o r the s e g r e g a t e d p a r t i c l e s have been c o n s i d e r e d , o f the d e n d r i t e o r t r a p p e d ,  bottom  i n the e u t e c t i c  and a t the top o f the d e n d r i t e XVII  P a r t i c l e Size D i s t r i b u t i o n for heights mm, 25 mm and 30 mm % i n the e u t e c t i c  f o r c a s t sample,  109 15 and 116  LIST OF FIGURES  A p a r t i c l e being  r e j e c t e d by an a d v a n c i n g  s o l i d / l i q u i d interface (schematic). i n t e r f a c e should  exhert  The  a force to acceler-  ate t h e p a r t i c l e and t o w i t h s t a n d t h e drag force Free e n e r g i e s distance  o f t h e PLS s y s t e m v e r s u s  from t h e i n t e r f a c e  (schematic)  i and i ' a r e two i n i t i a l s t a t e s w i t h and  lower energies  final state f.  higher  r e s p e c t i v e l y than t h e  Curves a to d are p o s s i b l e  e f f e c t s of the c l o s e presence of the p a r t i cle at the i n t e r f a c e .  When t h e p a r t i c l e  i s r e p e l l e d ( a - b ) and  attracted  In c u r v e  e  (c-d).  the f r e e energy o f the system  does n o t change u n t i l t h e p a r t i c l e i s i n the s o l i d Changes i n f r e e e n e r g y d u r i n g  the engulf-  ment o f a cube o r s p h e r e o f u n i t a r e a by a s o l i d S.  The s u r f a c e  surface energies  i n t h e b u l k media do n o t change due t o t h e presence of a t h i r d i n t e r f a c e The e f f e c t i v e s u r f a c e e n e r g y o f t h e PLS s y s t e m as a f u n c t i o n o f d i s t a n c e . minimum d i s t a n c e  d  Q  is a  a t which p a r t i c l e and  s o l i d are considered  t o be i n c o n t a c t  Schematic  representation of separation  v e r s u s growth v e l o c i t y f o r v a r i o u s p a r t i c l e s i z e s i n the Uhlmann t h e o r y .  The  upper  b r a n c h are s t a b l e and the l o w e r a r e u n s t a b l e a g a i n s t f l u c t u a t i o n s i n growth v e l o c i t y . The c r i t i c a l v e l o c i t y a t which the p a r t i c l e i s c a p t u r e d i s the maximum v e l o c i t y B o i l i n g and C i s s e scheme o f p u s h i n g F p ( a ) i s the drag f o r c e ; F .',versus the s u r f a c e f o r c e and -r  i s the c o n t a c t r a d i u s  A p a r t i c l e ( l i q u i d , s o l i d o r gas) i n "cont a c t " w i t h the s o l i d .  The d i s j o i n i n g  pres-  s u r e i n a gas b u b b l e as a f u n c t i o n o f s e p a r a t i o n h i s measured by p r e s s i n g the bubble a g a i n s t the s o l i d , T y p i c a l behaviors of the d i s j o i n i n g  pressure  It(h) f o r a gas b u b b l e i n a p o l a r l i q u i d  (1)  and n o n - p o l a r l i q u i d ( 2 ) , as a f u n c t i o n o f the d i s t a n c e from the s o l i d . B r a n c h e s a and 8 give stable f i l m s while y i s unstable  ....  T h r e e media s o l i d (1) - l i q u i d o r gas (3) s o l i d , l i q u i d o r gas (2) l e a d i n g to an important i n t e r a c t i o n at very close d i s t a n c e s ~ 10"^ cm d e s c r i b e d by the L i f s h i t z Ven der Waal t h e o r y  XI  Figure 10  Page R e p r e s e n t a t i o n o f t h e symbols  used i n the  Chernov e t a l . t h e o r y o f p u s h i n g 11  1  Critical for S i 0  162  growth r a t e v e r s u s p a r t i c l e r a d i u s 2  and W p a r t i c l e s .  (a) w i t h bump  r a d i u s o f 5 ym and 10 ym r e s p e c t i v e l y ; and (b) c o p p e r p a r t i c l e s a t two  temperature  g r a d i e n t s and f o r c l e a n e d and non c l e a n e d particles.  The t h r e e d e f i n i t e r e g i o n s c o r -  r e s p o n d t o f o r m u l a s 18 ( a - c ) ( R e f . 23) 12  T h e o r e t i c a l and e x p e r i m e n t a l c r i t i c a l v e l o c i t i e s f o r water.  14  '.  C u r v e s 1 and 2 a r e  f o r s m a l l and b i g p a r t i c l e s r e s p e c t i v e l y g i v e n by the t h e o r y o f Chernov e t a l . 4 c o r r e s p o n d s t o the c r i t i c a l  Curve  velocities  given f o r e l e c t r o s t a t i c i n t e r a c t i o n only (Ref. 13  10)  15  E f f e c t o f the t h e r m a l c o n d u c t i v i t y on the pushing process. (b) k  p  = k  s  (a) kp < k^ , k ;  and ( c ) k  g  p  > k  r  k. s  In (a)  and ( c ) the i s o t h e r m s a r e d i s t o r t e d , heat t r a n s f e r i s enchanced  and i n h i b i t e d r e s p e c -  t i v e l y and so the c a p t u r e o f the p a r t i c l e . .  20  SEM  m i c r o g r a p h o f the ARMCO i r o n  -400  particles  mesh used i n the e x p e r i m e n t s 200X  Diagram o f the e x p e r i m e n t a l s e t up  for  s o l i d i f i c a t i o n of metals ( f o r e x p l a n a t i o n see t e x t ) The  ..  physical  model to s t u d y the m o t i o n o f  p a r t i c l e s at a c e l l u l a r " i n t e r f a c e " . The  (a)  l u c i t e c y l i n d r i c a l container with a  brine  s o l u t i o n , the i n t e r f a c e and  the  nylon  s p h e r e s ; a i s the t i l t i n g a n g l e form horizontal The  f o r the h o r i z o n t a l  c e l l s and  mode,  to b u o y a n c y  c o l l i d i n g w i t h a c e l l t i p , V i s the  resultant tering.  (b)  the p a r t i c l e s i n d e t a i l ,  (c) A s p h e r e w i t h v o l i c i t y V due forces  the  v e l o c i t y immediately a f t e r The  vector V oriented  downward-  i s the v e l o c i t y a f t e r h a l f r e v o l u t i o n respect  scat-  to a s y s t e m f i x e d to the  with  rotating  cells Longitudinal  view of Sample V-1.  quenched l i q u i d and  The  unidirectionally  s o l i d are c l e a r l y d e f i n e d . a p p e a r as l o n g s t r i p s 50X  The  cell  (etched)  grown wall  Transversal  v i e w s o f Sample V - l a t t h e  i n t e r f a c e (a) 100.x;-.,. (b), and ( c ) 500X, (d) SEM-200X; and 12 mm b e h i n d the i n t e r face  (e) 100X.  Most o f the p a r t i c l e s a r e  in the c e l l w a l l s .  In (a.) a t t h e  left a ramification  i s shown.  (d) t h e s m a l l shown.  center  In ( c ) and  p a r t i c l e s i n the matrix are  In (e) most o f t h e c e l l w a l l s  dissolved.  have  The i r o n p a r t i c l e s a r e the w h i t e  spots P a r t i c l e S i z e D i s t r i b u t i o n f o r Sample V - l (cellular interface) and i n t h e m a t r i x .  in the c e l l  boundaries  (a) F o r c o u n t s a t t h e  i n t e r f a c e (b) b e h i n d t h e i n t e r f a c e and ( c ) total counts.  The d i s t r i b u t i o n s a r e s i m i l a r .  O n l y a few p a r t i c l e s a p p e a r i n t h e m a t r i x  ..  % of p a r t i c l e s i n the m a t r i x versus s i z e a t the i n t e r f a c e and b e h i n d t h e i n t e r f a c e . S m a l l p a r t i c l e s a p p e a r more f r e q u e n t l y t h e m a t r i x than l a r g e p a r t i c l e s . i n t h e c u r v e from c o u n t s a t t h e  in  The change interface  and b e h i n d i t , i s a t t r i b u t e d to d i s s o l u t i o n of r a m i f i c a t i o n  i n the m i c r o s t r u c t u r e  (a.) A p p a r e n t D i s t r i b u t i o n and (b) Real D i s t r i b u t i o n o f p a r t i c l e s f o r Sample V-1 (cellular interface).  The shape o f the  h i s t o g r a m does n o t change  substantially  e x c e p t t h a t p a r t i c l e s s m a l l e r than 4 ym are n o t p r e s e n t i n the r e a l  distribution.  In d i a g r a m (b) the volume i s a r b i t r a r y % o f p a r t i c l e s i n the m a t r i x v e r s u s s i z e f o r the a p p a r e n t and r e a l Sample V-1.  distribution.  The e f f e c t f o r s m a l l p a r t i -  c l e s i s remarked L o n g i t u d i n a l view a t t h e i n t e r f a c e  of  Sample V-2 ( d e n d r i t i c i n t e r f a c e ) 100X (etched) (a), ( b ) , (c)  T r a n s v e r s a l views o f Sample  V-2 a t t h e i n t e r f a c e .  The p a r t i c l e s a r e i n  the i n t e r d e n d r i t i c r e g i o n s 500X ( e t c h e d )  ...  Particle size distributions for particles i n the m a t r i x and i n i n t e r d e n d r i t i c f o r Sample V-2. Sample V-1  regions  L e s s p a r t i c l e s than i n  ( c e l l u l a r i n t e r f a c e ) a r e i n the  % of p a r t i c l e s i n the m a t r i x v e r s u s s i z e f o r Sample V-2  (dendritic  interface).  i n Sample V - l the same p a t t e r n a l t h o u g h f o r Sample V-2  As  i s observed  i s less pronounced.,  (a) A p p a r e n t D i s t r i b u t i o n and  (b)  Real  D i s t r i b u t i o n of p a r t i c l e s f o r Sample  V-2  (dendritic interface) Transversal face walls  v i e w s o f Sample H-l at the  inter-  (a) 500X showing p a r t i c l e s i n the and  cell  (b) 200X showing p a r t i c l e s at  edge of the sample i n c o n t a c t w i t h  the  the  container P a r t i c l e Size D i s t r i b u t i o n  f o r Sample H-l  a t the c e l l b o u n d a r i e s and  i n the m a t r i x  ...  % of p a r t i c l e s i n the m a t r i x v e r s u s s i z e  for  Sample H - l .  A random d i s t r i b u t i o n i s  obtained  .  S p h e r e s d i s t r i b u t i o n i n the p h y s i c a l f o r the h o r i z o n t a l  model  mode f o r d i f f e r e n t  rota-  t i o n a l s p e e d s . (a).\w= 0.033 r p s , (b) 0.083 r p s , (c) 0.2  r p s , (d) rps and  (e)  1  rps.  The  p a r t i c l e s p a s s e d from b e i n g c o n c e n t r a t e d a t the w a l l  o f the c y l i n d e r  e x p l a i n e d i n the t e x t . . . . .  to the c e n t e r as  xvi i Figure 32  Page The two c l a s s e s o f p a t t e r n s f o l l o w e d by t h e s p h e r e s i n the model as a f u n c t i o n o f rotation . s p e e d ; (a) t h e s p h e r e s r e a c h t h e w a l l , (b) the s p h e r e s do not r e a c h t h e wal 1 i n one c y c l e  33  35  Nylon spheres t r a p p e d i n the grooves  be-  tween c e l l s f o r t h e h o r i z o n t a l mode. P h o t o g r a p h t a k e n from one end o f the c y l i n d e r , w = 0.5 r p s a = 4 ° . . . 34  92  Nylon spheres i n the grooves i n the v e r t i c a l experiments.  P h o t o g r a p h t a k e n from one end  of the c y l i n d e r ,  v  = 1 cm/sec.  ,  92  P 35  S e m i - q u a l i t a t i v e d e s c r i p t i o n o f the s e g r e g a t i o n of nylon spheres d u r i n g the v e r t i c a l e x p e r i m e n t s w i t h t h e model. face areas are c a l c u l a t e d .  In (a) t h e surIn (b) t h e be-  h a v i o r o f s p h e r e s f o r s t i l l and c o n v e c t i v e l i q u i d a r e compared w i t h t h e e x p e r i m e n t a l observations 36  101 ,  P a r t i c l e s trapped in a well developed d e n d r i t e (a) 200X and (b) 500X.  Particles  a t t h e bottom o f the d e n d r i t e s ( c ) and ( d ) . P a r t i c l e f r e e i n the e u t e c t i c ( e ) .  In (d)  a p a r t i c l e i s p a r t i a l l y t r a p p e d at the top o f t h e d e n d r i t e . In (b) a s m a l l p a r t i c l e appears i n the core of a d e n d r i t e  106  Figure 37  P a r t i c l e Size D i s t r i b u t i o n f o r the cast sample.  (a) At the b o t t o m , (b) 0.5 cm;  from the b o t t o m , ( c ) 1.5 cm from b o t t o m , (d) 2.5 cm  from b o t t o m , (d) 3.0 cm f r o m  bottom 38  D e n s i t y o f p a r t i c l e s v e r s u s h e i g h t i n the c a s t sample  39  view o f t h e t o p p a r t o f t h e c a s t sample. The h i g h c o n c e n t r a t i o n o f p a r t i c l e s t h e r e  /  i s a t t r i b u t e d to f l o t a t i o n .  Solidifica-  t i o n began when the p a r t i c l e s were a l r e a d y there 40  Pushing in water.  (a) Water w i t h Cr  p a r t i c l e s , (b) The i c e - w a t e r i n t e r f a c e a d v a n c i n g to t h e c e n t e r .  ( c ) Dark band o f  p a r t i c l e s pushed by t h e i n t e r f a c e . (d) S i m i l a r to ( c ) but w i t h a h i g h e r d e n s i t y o f p a r t i c l e s which r e s u l t e d i n a w i d e r band  ACKNOWLEDGEMENTS I would l i k e t o thank Dr. F r e d Weinberg f o r h i s a d v i c e and e n c o u r a g e m e n t d u r i n g t h e c o u r s e o f t h i s work. F i n a n c i a l a s s i s t a n c e from t h e N a t i o n a l Council (Canada),  U n i v e r s i d a d Nacnonal  ( A r g e n t i n a ) and F u n d a c i o n  Research  de M i s i o n e s  R o t a r i a n a de Posadas  is : g r a t e f u l l y acknowledged.  (Argentina)  the Memory o f My P a r e n t s  1  Chapter  I  INTRODUCTION The p r e s e n c e o f p a r t i c l e s i n a m a t e r i a l can s t r o n g l y i n f l u e n c e i t s properties;.  The m a t e r i a l p r o p e r t i e s can be  enhanced as i s t h e c a s e f o r a l l o y h a r d e n i n g p r e c i p i t a t e s , or composite m a t e r i a l s . p r o p e r t i e s can be i m p a i r e d clusions in steel.  with very f i n e  A l t e r n a t i v e l y the  as i n t h e case o f l a r g e i n -  In most c a s e s when t h e p r e s e n c e o f  p a r t i c l e s i n the material i s detrimental  to the p r o p e r t i e s ,  the p a r t i c l e s a r e i n h e r e n t t o t h e f a b r i c a t i o n p r o c e s s and c a n n o t be e l i m i n a t e d .  Accordingly the control of the d e t r i -  mental e f f e c t o f the p a r t i c l e s i s r e l a t e d to the c o n t r o l o f t h e d e n s i t y and d i s t r i b u t i o n o f t h e p a r t i c l e s i n t h e material.  S i m i l a r l y enhanced m a t e r i a l p r o p e r t i e s a r e a l s o  d i r e c t l y r e l a t e d t o t h e p a r t i c l e d e n s i t y and d i s t r i b u t i o n . P r e c i p i t a t i o n from s o l u t i o n , e l e c t r o l y t i c and s o l i d i f i c a t i o n a r e t h r e e common p r o c e s s e s where a s o l i d - l i q u i d i n t e r f a c e i s p r e s e n t .  deposition, in industry  I t i s a t this;  s t a g e t h a t t h e main i n t e r a c t i o n o f t h e p a r t i c l e s ; , which may be p r e s e n t place.  i n the l i q u i d , with the s o l i d w i l l  In t h e c a s e o f w a t e r and many o r g a n i c  take  substances,  2 the s o l i d i n t e r f a c e d u r i n g s o l i d i f i c a t i o n w i l l "push" s m a l l s o l i d p a r t i c l e s ; ahead o f the i n t e r f a c e o v e r distances  large  r e l a t i y e to the d i a m e t e r o f the p a r t i c l e s .  In  m e t a l s i t i s not c l e a r to what e x t e n t p a r t i c l e s ; are j e c t e d by an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e . correlation ti'on and  re-  A  strong  has; been o b s e r v e d between p a r t i c l e d i s t r i b u -  cast structure  ~  in deoxidized steel  the p a r t i c l e s b e i n g c o n c e n t r a t e d p r i m a r i l y dendritic regions.  ingots,  i n the  inter-  T h i s s u g g e s t s the p o s s i b i l i t y t h a t  some p a r t i c l e r e j e c t i o n o c c u r s at the s o l i d - l i q u i d i n t e r f a c e . D i f f e r e n t mechanisms; have been p r o p o s e d to a c c o u n t f o r o b s e r v e d c o n c e n t r a t i o n of p a r t i c l e s i n the regions.  region, pushing  interdendritic  T h e s e range from the p r o p o s a l t h a t the  r e s u l t from homogeneous n u c l e a t i o n w i t h no r e j e c t i o n , at one 32  the  particles  i n the  interdendritic 33 34 extreme ' to p a r t i c l e  d u r i n g s o l i d i f i c a t i o n s i m i l a r to t h a t  observed  i n w a t e r and o r g a n i c m a t e r i a l s ; at the o t h e r e x t r e m e . In t h i s i n v e s t i g a t i o n liquid interface  the i n t e r a c t i o n o f a s o l i d -  in a melt, with s o l i d p a r t i c l e s in  m e l t i s examined to d e t e r m i n e i f p a r t i c l e s a r e by an a d v a n c i n g I n t e r f a c e . to the n u c l e a t i o n  and  No c o n s i d e r a t i o n  no c h e m i c a l r e a c t i o n  rejected  w i l l be  growth, o f s o l i d p a r t i c l e s i n  m e l t ahead o f t h e i n t e r f a c e , and  i t w i l l 6e assumed  occurs between p a r t i c l e and  the given the that  the m e l t .  The r e j e c t i o n i s c o n s i d e r e d from two p o i n t s ; o f view: (.1 ) t h e f o r c e s t h a t o r i g i n a t e a t t h e p a r t i c l e ,  solid-  l i q u i d i n t e r f a c e ; ( 2 ) t h e forces; a r e p u r e l y m e c h a n i c a l due t o t h e r e l a t i v e movement o f t h e s o l i d , l i q u i d  and  p a r t i c l e at the i n t e r f a c e . The c o n d i t i o n s ; f o r p a r t i c l e r e j e c t i o n by an  advanc-  i n g i n t e r f a c e a r e f i r s t r e v i e w e d , f o l l o w e d by a r e v i e w o f t h e t h e o r i e s p r o p o s e d to a c c o u n t f o r t h e s e o b s e r v a t i o n s . T h i s i s f o l l o w e d by a d e s c r i p t i o n . o f t h e e x p e r i m e n t a l p r o c e d u r e and o b s e r v a t i o n s ; i n a metal a l l o y s y s t e m f o r both d i r e c t i o n a l : . ; ' and normal f r e e z i n g c o n d i t i o n s .  The  r e s u l t s ; a r e t h e n compared to t h e o b s e r v a t i o n s , on p a r t i c l e r e j e c t i o n i n w a t e r and t h e t h e o r e t i c a l p r e d i c t i o n s f o r a metal a l l o y  system.  The p r e s e n t r e s u l t s i n d i c a t e t h a t t h e r e i s no r e j e c t i o n o f p a r t i c l e s i n a m e t a l s y s t e m by a s o l i d advancing i n t e r f a c e .  liquid  The o b s e r v e d p a r t i c l e d i s t r i b u t i o n  i n t h e s o l i d can be. accounted  .fbri..by a  m e c h a n i c a l mechanism  f o r the i n t e r a c t i o n of the p a r t i c l e with the i n t e r f a c e . T h i s mechanisjn i s d e v e l o p e d and v e r i f i e d i n a p h y s i c a l model o f t h e s y s t e m .  4  Chapter  II  REVIEW. 1.  Interaction  o f a P a r t i c l e w i t h a Sol i d / L i q u i d  Interface;  D e f i n i t i o n o f t h e Prob!em. Assume a p a r t i c l e i s a t r e s t w i t h r e s p e c t  to the l i q u i d ,  f a r away from a p l a n a r a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e . What then happens when t h e i n t e r f a c e a p p r o a c h e s t h e p a r t i c l e ? Two e x t r e m e s i t u a t i o n s may be c o n s i d e r e d :  either the inter-  f a c e w i l l be c o m p l e t e l y i n d i f f e r e n t t o t h e p r e s e n c e o f t h e p a r t i c l e , o r t h e p a r t i c l e w i l l be r e j e c t e d  by t h e i n t e r f a c e  and move ahead o f t h e f r o n t i n s t e a d y s t a t e . alternative  i s shown s c h e m a t i c a l l y  r e j e c t i o n of the p a r t i c l e requires accelerated  i n F i g . 1.  The s e c o n d The s t e a d y  t h e p a r t i c l e t o be  from z e r o t o t h e i n t e r f a c e v e l o c i t y V.  t h i s v e l o c i t y i s r e a c h e d t h e i n t e r f a c e must  exert  When a  f o r c e on t h e p a r t i c l e i n t h e d i r e c t i o n o f s o l i d i f i c a t i o n to compensate f o r t h e d r a g f o r c e t h a t a r i s e s due t o t h e transport  of f l u i d necessary f o r s o l i d i f i c a t i o n to proceed.  T h i s f o r c e on t h e p a r t i c l e s h o u l d be g e n e r a t e d a t t h e l i q u i d l a y e r between t h e p a r t i c l e and s o l i d t h r o u g h an i n t e r a c t i o n between t h e p a r t i c l e , t h e l i q u i d l a y e r and t h e s o l i d i f y i n g material  ( r e f e r r e d t o as P L S ) . T h i s i n t e r a c t i o n w i l l be  considered later.  5  V L i q u i d (L)  \  \ \  \  \  \  \ \ \  \  S o l i d (S )  F i g u r e 1. A p a r t i c l e b e i n g r e j e c t e d by a S/L i n t e r f a c e . T h e i n t e r f a c e s h o u l d ex e r t a f o r c e to a c c e l e r a t e the p a r t i c l e and w i t h s t a n d the drag f o r c e .  6  Lavalle repelled  i n 1853 n o t e d t h a t f o r e i g n  by a g r o w i n g c r y s t a l .  particles; are  However i t was n o t u n t i l  the n i n e t e e n f i f t y ' s ; t h a t t h e p r o c e s s was: examined i n detail.  The. r e s u l t o f i n v e s t i g a t i o n s ;  of particle  inter-  actions with a s o l i d i f y i n g face reported i n the l i t e r a t u r e to d a t e a r e summarized i n T a b l e I. Pikunov,^ s o l i d i f y i n g several  organic liquids  con-  taining a suspension of particles,found  that the  migration of particles; at the interface  depended on t h e  particular  p a r t i c l e and l i q u i d m a t r i x c o n s i d e r e d .  For  example, s a l o l does n o t pus;h c a r b o n n o r s t a r c h p a r t i c l e s . In d i p h e n y l a m i n e , c a r b o n i s pushed and s t a r c h i s n o t . In azobezene  and b e n z y l , both, s t a r c h  pushed by an a d v a n c i n g  and c a r b o n p a r t i c l e s a r e  interface.  Th;e r e s u l t s , r e p o r t e d above by P i k u n o v i n s a l o l a r e 2 in c o n t r a d i c t i o n  w i t h t h o s e o f Kuo and W i l c o x  who r e -  p o r t e d t h a t c a r b o n p a r t i c l e s ; (Q. 5-2.5ym) a r e pushed by s a l o l . On t h e o t h e r hand the m e l t h o r i z o n t a l l y , and  UliTmann and P i k u n o v s o l i d i f i e d  with., a v e r t i c a l i n t e r f a c e .  Wilcox s o l i d i f i e d v e r t i c a l l y .  s h o u l d make t h e r e j e c t i o n  Solidifying vertically  o f t h e p a r t i c l e more  s i n c e t h e forces; a t t h e i n t e r f a c e  Kuo  difficult  have t o compensate f o r  the drag f o r c e o f t h e p a r t i c l e as w e l l  as the. g r a y i t y  7 TABLE I  EXPERIMENTAL RESULTS ON PUSHING  Particle  Size, pm  Critical Velocity, um/sec  Pushing Observed  Growth Direction and Comments  Ref.  LIQUID: WATER hollow carbon spheres C(grapMte) C(diamond) Si0 Si 2  MgO A1 0 2  3  2°3 Agl fe  glass bead broken glass calcite rutile quartz shale mica silt Ni Zn Sn Cu  w  20  — — 20-50  — —  -  — — 60-120 149-590 149-590 149-590 149-590 149-590 149  — —  -  2.5-62.5 2.5-30  5.5  yes  vertical  >20 >20 0.89-0.66 >20  yes yes yes yes  horizontal horizontal vertical horizontal  23 4  >20 >20 >20 >20  yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes yes  horizontal horizontal horizontal horizontal  4 4 4 4  0.34-0.03 0.2 -0.03 0.2 -0.01 0.33-0.11 0.36-0.12 0.55-0.17 1.39 >20 >20 >20 >20 '5-0.165 3.3-0.18  vertical vertical vertical vertical vertical vertical vertical  23  4 4  21 54 54 54 54 54 54 4 4  horizontal horizontal horizontal horizontal vertical vertical  4 4 23 23  horizontal horizontal vertical horizontal horizontal horizontal horizontal horizontal horizontal vertical horizontal horizontal horizontal vertical vertical horizontal vertical horizontal vertical horizontal horiiontal vertical  4 4 2 1 4 1 1 4 4 2 4 1 1 24 2 4 2 4 24 4 4 24  LIQUID: SALOL Agl C(graphite) C C C(diamond) starch 1icopodium  — —  —  < 0.28 < 0.28 1.94 0 2-2.1 0 0.97-0.55 0.7 2.5 < 1.67 3 1.94-0.83 1.94-0.83 0.35-0.17 < 1.67 0.8 < 1.67 7 0.35-0.17 1 2.3  20-40  0.35-0.17  0.5-2.5 Or5  Fe  — — —  Fe  0.45-1.8  silt 2°3 2°3 MgO 2°3 Cr 0 A1  2  3  wc  5-12 <150  2  Si Si Zn Zn  —  —  --  w  2.5-5  —  Sn Ni Si0  — — —  2  no no yes no yes no yes yes yes no yes yes yes yes no yes no yes yes yes yes yes  TABLE  1  Particle LIQUID:  8  EXPERIMENTAL RESULTS -ON PUSHING (Cont.)  Size, vm  Critical Velocity, um/sec  Pushing Observed  Growth Direction and Comments  Ref.  NAPHTALENE 9.4  carbon  ~  carbon  horizontal,Zonal & rotation 55  yes  vertical  55  carbon  0.5-2.5  3.3-5.8  yes  vertical  7  carbon  0.5-2.5  8.3-10.8  yes  15-20  yes  vertical & stirring 2 horizontal.zonal S rotation 55  6.9  yes  vertical  5  yes  vertical  2  yes ?  vertical  55 29  Cu  --  Cu Cu  8.5-5.5  Ag  .  Fe(Ag coated)  „  several  iron oxide 10-ZOO  acetal  --  acetal nylon  10-240  —  nylon polyestirene polyestirene  ; —  11.1  yes  1.6-7^2 0.49-0.20  yes  vertical  10  yes  horizontal .zonal & rotation  yes  «8  yes  horizontal  yes  horizontal.zonal & rotation  40-10 «8  yes  --  no  «8  no no  --  «8  no  siliconed glass  no  Fe(Ag coated)  82-160  0.42-0.09  yes  Fe and steel (Ag coated) aluminium  92-148  0.56-0.053  yes  0.46-0.0078  yes  LIQUID:  horizontal  2 7 5,6 7  5,6  horizontal.zonal & rotation 7 horizontal 5,6 horizontal.zonal S rotation horizontal  horizontal.zonal rotation  vertical magnetic vertical magnetic vertical magnetic  & field S field. & field  7  5,6  7 29 29 29  THYMOL  2°3 Sn F e  2  yes  horizontal  4 .  4  yes  horizontal  4  6  yes  horizontal  4  6  yes  horizontal  4  8  yes  horizontal  4  8  yes  horizontal  4  9  yes  horizontal  4  10  yes  horizontal  4  12  yes  horizontal  4  16  yes  horizontal  4  — — — — --  Agl Zn MgO Ni C(Diamond) Si C(graphite)  3-5  — —  silt LIQUID:  horizontal.zonal rotation  64-40  teflon teflon  55  ORTHOTERPHENYL __  0.28  no  horizontal  4  C(graphite)  0.28  yes  horizontal  4  MgO  0.5  yes  horizontal  4  silt Si  0.7  yes  horizontal  0.8  4  yes  horizontal  4  yes _  horizontal  4  Agl  Sn  1  TABLE 1  9  EXPERIMENTAL RESULTS ON PUSHING (Cont.)  Particle  S i z e , um  Critical V e l o c i t y , urn/sec  0-5  Pushing Observed  Growth D i r e c t i o n and Comments  Ref.  1.4-1.3  yes  horizontal  4  Ni  —  2.0  yes  horizontal  4  2°3 Zn  — —  2.5  yes  horizontal  4  2.5  yes  horizontal  4  22.2-13.8  yes  horizontal  1  13.9-5.55  yes  horizontal  1  C(diamond)  Fe  LIQUID:  DYPHENYLAMINE  Licopodium pdr.  -— — --  carbon A1 0 2  3  2°3 starch Cr  LIQUID:  16.67-9.72  yes  horizontal  1  13.9-5.55  yes  horizontal  1  0  no  horizontal  1  AZOBENZENE  — — — — —  3.33-1.38  yes  horizontal  1  0.83-0.55  yes  horizontal  1  4.17-8.33  yes  horizontal  1  0.5-11.1  yes  horizontal  1  0.78-8.33  yes  horizontal  1  0.55-0  yes  horizontal  1  0.55-0.28  yes  horizontal  1  19.44-11.11  yes  horizontal  1  2.78-1.39  yes  horizontal  1  2.78-1.39  yes  horizontal  1  38-8  yes  horizontal.zonal & rotation  7  «8  yes  horizontal  32-6  yes  horizontal.zonal & rotation  — —  <<8  yes  — —  «8  tef1 on teflon  —  «8  Licopodium pdr. carbon A1 0  3  Cr 0  3  2  2  Starch LIQUID:  BENZIL  Licopodium pdr. carbon 2°3 Cr 0  A1  2  3  starch  LIQUID:  -— — — —  BIPHENIL 10-200  acetal  —  acetal nylon  10-240  nylon polyestirene polyestirene  — —  t e f l o n covered with s i l i c o n e o i l  LIQUID:  Sn 0 ZnO  0.5-2.5  2  horizontal . horizontal.zonal & rotation  no ?  horizontal  no  horizontal.zonal & rotation  7 5,6 7 5,6 7  no  horizontal  no  horizontal.zonal & rotation  5,6 7  no  vertical  2  yes  casting  1  no  casting  1  CAMPHOR  C LIQUID:  no  5,6  <0.55  COPPER  —  — —  10  force acting  on t h e h e a v i e r p a r t i c l e .  S i n c e p u s h i n g has  been o b s e r v e d w i t h v e r t i c a l s o l i d i f i c a t i o n and n o t f o r horizontal  s o l i d i f i c a t i o n the effect of gravity  used t o a c c o u n t f o r t h e d i f f e r e n c e  c a n n o t be  i n the observed  particle  behavior. The e x p e r i m e n t s o f Uhlmann and Kuo and W i l c o x stitutes  v..examples  where t h e d i r e c t i o n  can a c c o u n t f o r t h e d i f f e r e n c e s Fe  2  0 » S i and Zn p a r t i c l e s 3  Uhlmann w i t h h o r i z o n t a l particles are rejected  con-  of s o l i d i f i c a t i o n  i n the observations with  i n s a l o l as shown i n T a b l e I .  s o l i d i f i c a t i o n found that  these  by s a l o l and Kuo and W i l c o x d i d n o t  observe pushing during v e r t i c a l s o l i d i f i c a t i o n . Returning to the s y s t e m c a r b o n i n s a l o l , p a r t i c l e p u s h i n g i s o b s e r v e d t o depend upon t h e growth v e l o c i t y and p a r t i c l e s i z e . I t w i l l be shown l a t e r t h a t t h e s e two v a r i a b l e s  are related: the  v e l o c i t y a t which a p a r t i c l e ( w h i c h i s i n i t i a l l y pushed by the i n t e r f a c e ) portional  i s t r a p p e d by t h e i n t e r f a c e  i s inversely  pro-  t o t h e r a d i u s o f t h e p a r t i c l e t o some power.  T h i s s u g g e s t s t h a t t h e s i z e o f t h e p a r t i c l e s used by Uhlmann and P i k u n o v may have been l a r g e r t h a n t h e c i r i t c a l r a d i u s f o r p u s h i n g a t t h e growth v e l o c i t i e s t h e y u s e d ; however i t can be shown t h i s was n o t t h e c a s e . Comparing  growth  velocity  f i r s t , P i k u n o v and Uhlmann used v e l o c i t i e s o f I mm/hr, which i s l o w e r t h a n t h a t used by Kuo and W i l c o x o f 7 mm/hr.  The  s l o w e r v e l o c i t i e s p e r m i t l a r g e r p a r t i c l e s t o be pushed which f a v o u r s t h e P i k u n o v p a r t i c l e s f o r p u s h i n g .  Pikunov  11 did not report h i s p a r t i c l e size d i s t r i b u t i o n . Nevertheless, the p a r t i c l e s i z e may be e s t i m a t e d assuming bon p a r t i c l e s were r e j e c t e d by an azobenzene  t h a t t h e same c a r interface at 3  growing r a t e s o f 16 mm/hr.  According to theory  particles of  the o r d e r o f m i c r o n s a r e t r a p p e d a t t h e s e v e l o c i t i e s .  There-  f o r e i t may be assumed t h a t t h i s i s t h e o r d e r o f t h e p a r t i c l e s i z e employed  by P i k u n o v .  Kuo and W i l c o x used c a r b o n  c l e s w i t h s i z e s between 0.5 ym and 2.5 ym and f o u n d o c c u r r e d a t 7 mm/hr f r e e z i n g r a t e s .  partipushing  Pikunov with a p p r o x i -  m a t e l y t h e same s i z e p a r t i c l e s and f r e e z i n g r a t e s o f 1 mm/hr o b s e r v e d no p u s h i n g .  Therefore, the r e l a t i o n v e l o c i t y versus  p a r t i c l e s i z e c a n n o t be t h e r e a s o n f o r t h e d i s c r e p a n c y between t h e r e p o r t e d o b s e r v a t i o n s . A n o t h e r mechanism p r o p o s e d t o a c c o u n t f o r t h e d i f f e r e n c e i s a s s o c i a t e d with t h e r e j e c t i o n o f gases a t t h e moving f r o n t , as r e p o r t e d by Kuo and W i l c o x and o b s e r v e d by P i k u n o v .  The r e j e c t e d gases were a b s o r b e d by c a r b o n  r e s u l t i n g i n the n u c l e a t i o n o f bubbles.  The b u b b l e s may  h e l p t h e p u s h i n g p r o c e s s o r even r e p l a c e i t by a f l o t a t i o n process.  I f t h i s does o c c u r t h e n i t may be c o n c l u d e d t h a t  c a r b o n i s n o t pushed by s a l o l . More examples i n which p a r t i c l e s a r e r e j e c t e d by an a d v a n c i n g i n t e r f a c e i n one s u b s t a n c e b u t n o t i n o t h e r c a n be c i t e d .  Kuo and W i l c o x c o n c l u d e d t h a t c a r b o n i s n o t  12 pushed by camphor b u t i t i s pushed by n a p h t a l e n e .  Uhlmann  showed A g l p a r t i c l e s a r e n o t moved by o r t h o t e r p h e n y 1 s a l o l i n t e r f a c e s , but t h e y a r e d i s p l a c e d by  and  thymol.  One v e r y i n t e r e s t i n g o b s e r v a t i o n i s t h a t i n s a l o l g r a p h i t e i s n o t pushed by diamond ( 0 - 2 y and 3 - 5 y) i s rejected.  T h i s means t h a t t h e c r y s t a l s t r u c t u r e o f t h e  p a r t i c l e i s t h e c r i t i c a l component i n t h i s c a s e m i n i n g w h e t h e r t h e p a r t i c l e i s pushed.  deter-  A n o t h e r example 2  i l l u s t r a t i n g t h a t a s t r u c t u r a l e f f e c t may be s i g n i f i c a n t is the o b s e r v a t i o n i n that d i f f e r e n t c r i t i c a l v e l o c i t i e s a p p l y t o f a c e t e d and n o n - f a c e t e d  i n t e r f a c e s o f t h e same  materi a l . In summary, p a r t i c l e s may o r may n o t be pushed ahead o f an a d v a n c i n g  s o l i d - l i q u i d i n t e r f a c e , depending  on t h e p a r t i c l e , s o l i d and l i q u i d , p o s s i b l y gas e v o l u t i o n and p a r t i c l e and s o l i d s t r u c t u r e .  It i s therefore desir-  a b l e t o p r e d i c t when p u s h i n g w i l l o c c u r i n a g i v e n s y s t e m on t h e b a s i s o f a t h e o r e t i c a l model. proposed  and a r e p r e s e n t e d  Models have been  i n Appendix I.  Formulas 1 to  28; f i g u r e s 2 t o 10; and r e f e r e n c e s 3 t o 22, 59 and 60 are i n t r o d u c e d  there.  13 2.  C o m p a r i s o n o f the P r o p o s e d T h e o r i e s w i t h E x p e r i m e n t a l Results A comparison  critical  o f f o r m u l a 18(.a-c) w i t h the e x p e r i m e n t a l  v e l o c i t i e s f o r a v a r i e t y of p a r t i c l e s in water 23  due t o C i s s e and B o i l i n g  i s shown i n F i g . 11.  In s a l o l  the r e s u l t s were r e p o r t e d t o be s i m i l a r t o t h o s e i n F i g . 11' Chernov and Temlkin 1976 and p l o t t e d  c o l l e c t e d the r e s u l t s a v a i l a b l e  a l l the d a t a as shown i n F i g . 12.  until In  F i g . 12 l i n e s 1 and 2 c o r r e s p o n d t o f o r m u l a s 27 and respectively.  The s c a t t e r i s a l m o s t one o r d e r o f m a g n i t u d e  and may be a t t r i b u t e d ques.  28  to the d i f f e r e n t e x p e r i m e n t a l  Curve (4) c o r r e s p o n d s t o the c r i t i c a l  when t h e r e p u l s i v e  techni-  velocities  force i s of e l e c t r o s t a t i c o r i g i n only  and r e m a r k a b l y i t i s s t r o n g  enough t o s o l e l y a c c o u n t f o r  the r e j e c t i o n o f p a r t i c l e s . An a l t e r n a t i v e theories  method to p r o v e t h e s e  i s to luse a PLS s y s t e m t o d e t e r m i n e the d e p e n d e n c e  of c r i t i c a l obtain  and l e g i t i m a t e  velocity in function  o f p a r t i c l e s i z e ; and t o  n. and C from and r e l a t i o n o f the t y p e V  = c  ...29 R  n  n f o r the s e t o f t h e o r i e s  prescribed  0.5 to 2 as shown i n T a b l e 1 1 ( a ) . f i t t e d experimental c r i t i c a l  here v a r i e s  from  Omenyi and Neumann  velocities for a variety  of  14  F i g u r e 11. C r i t i c a l growth r a t e s v s . p a r t i c l e r a d i u s f o r Si^O and W p a r t i c l e s w i t h bump r a d i u s of 5 fx m and 10 /Urn r e s p e c t i v e l y (a)_; and Cu p a r t i c l e s , c l e a n e d and n o n n - c l e a n e d , f o r two tmperature g r a d i e n t s . ( f r o m r e f . 23).  15  F i g u r e 12. Comparison of c r i t i c a l v e l o c i t i e s g i v e n i n the Chernov t h e o r y w i t h the e x p e r i m e n t a l c r i t i c a l v e l o c i t i e s f o r w a t e r . C u r v e s 1 and 2 c o r r e s p o n d to s m a l l and l a r g e p a r t i c l e s . Curve 4 r e p r e s e n t s the c r i t i c a l v e l o c i t i e s when t h e r e i s e l e c t r o s t a t i c i n t e r a c t i o n o n l y (from r e f . 1 0 ) .  16  PLS s y s t e m s , to f o r m u l a s l i k e 29 and f o u n d t h a t n v a r i e s between 0.27  t o 1.51  as shown i n T a b l e 1 1 ( b ) .  In a d d i t i o n ,  i t can be seen t h a t i n a l l the c a s e s n was s m a l l e r i n the range o f p a r t i c l e r a d i u s from 10 t o 100 ym than i n the range 100 t o 200 ym.  T h i s s t r o n g e r dependence f o r  l a r g e r p a r t i c l e s was a l s o e v i d e n t i n the C c o n s t a n t s w h i c h were a l m o s t one o r d e r o f m a g n i t u d e g r e a t e r . T h i s b e h a v i o u r i s o p p o s i t e t o t h a t p r e d i c t e d by the Chernov e t a l . t h e o r y w h i c h f o r b i g g e r p a r t i c l e s g i v e s a s m a l l e r power, 1.0, than f o r s m a l l e r p a r t i c l e s , On t h e o t h e r hand, when c o n s i d e r i n g c o n s t a n t  1.33.  C, the d i v e r -  s i t y . o f v a l u e s f o r d i f f e r e n t p a r t i c l e s i n t h e same m a t r i x , shows the l i m i t a t i o n o f t h e s e t h e o r i e s t h a t g i v e o n l y one c o n s t a n t r e g a r d l e s s o f the n a t u r e o f t h e p a r t i c l e .  In  a d d i t i o n the d i f f e r e n c e i n C c o n s t a n t s f o r t h e d i f f e r e n t s y s t e m s r e i n f o r c e s the s t a t e m e n t t h a t the p u s h i n g  process  depends on an i n t e r a c t i o n where the p r o p e r t i e s o f t h e t h r e e components P-L-S  must be i n v o l v e d .  If other parameters  a r e a n a l y z e d from t h e s c a r c e  e x p e r i m e n t a l work a v a i l a b l e t h e problem seems t o be f a r from s o l v e d and no t h e o r y i s good enough t o e x p l a i n the experimental f i n d i n g s . C i s s e and B o i l i n g r e p o r t e d e x p e r i m e n t a l v a l u e s from  TABLE  lira  VALUES QF n FROM EQUATION 29 FOR THE DIFFERENT THEORIES Theory  Part Size  n  Uhlmann e t a l .  smal 1  0. 5  Chernov et a l .  -^500 ym  1 .0  Chernov et a l .  < 500 ym  1 . 33  B o i l i n g &;;Ci.sse ' s m a l l  1 .5  Uhlmann e t a l .  2.  smal 1  -TABLE I I - b  EXPERIMENTAL VALUES OF n and C FROM EQUATION 29.  Particle  Size  7  10-100 um  100-200 ym  n  C  n  G;  Biphenyl/acetal  0.61  202.1  f.5!  11887.0  Bi p h e n y l / n y l o n  0.54  107.8  1 .49  8210.5  Naphtha!ene/acetal  0.27  124.7  0. 31  143.5  Naphtha!ene/ny1 on  0.42  99.8  0.62  232.5  System  19  which  i t is inferred  that the c r i t i c a l  velocity  increases  f o r l o w e r t e m p e r a t u r e g r a d i e n t s , w h e r e a s no t h e o r y  pre-  d i c t s any r e l a t i o n w i t h t h i s p a r a m e t e r  particles.  On t h e c o n t r a r y C h e r n o v  f o r small  e t a l . p r e d i c t s an o p p o s i t e  behavior f o r large particles  (see Equation 28).  So f a r i n t h e t h e o r i e s p r e s e n t e d t h e c r i t i c a l velocities e.g.  have been o b t a i n e d a s s u m i n g  ideal  conditions  s p h e r i c a l p a r t i c l e s , no s o l u t e i n t h e l i q u i d a n d  same t h e r m a l  c o n d u c t i v i t y f o r p a r t i c l e and s o l i d i f y i n g  m a t e r i a l , e t c . These f a c t o r s w i l l relation  a)  to the p a r t i c l e pushing  E f f e c t o f the Thermal If the thermal  now be c o n s i d e r e d i n process.  Conductivity  conductivity of a particle  h i g h e r t h a n t h a t o f t h e l i q u i d a n d s o l i d , k-j * k capture of the p a r t i c l e through k  s  the following qualitatively  the i n t e r f a c e will  heat flow i n which through  i s enhanced.  be d e f o r m e d  This will  reasoning.  will (Fig.  3  I f k p * k-j  by a n o n - u n i d i r e c t i o n a l  produce  conducted a depression  b e h i n d t h e p a r t i c l e as i t i s shown i n F i g . 13 the p a r t i c l e  the  This i s visualized  heat i s p r e f e r e n t i a l l y  the particle.  g  is  trapping  i n i t . T h e o p p o s i t e c a s e , when kp< k-j , k  g  e n h a n c e p u s h i n g by c r e a t i n g a bump b e h i n d t h e p a r t i c l e 13).  F i g u r e 13. E f f e c t of the thermal c o n d u c t i v i t y on the p u s h i n g p r o c e s s , a) k < , k ; b) k ~ ,k and c ) k > ^ ,k . In a) and.c) ?he isotherms a r e d i s t o r t e l , heat transfer i s i n h i b i t e d and e n h a n c e d , r e s p e c t i v e l y and so the c a p t u r e of the p a r t i c l e . g  P  21  The f a c t t h a t t h e s e e f f e c t s s h o u l d be s t r o n g e r f o r h i g h e r h e a t f l o w s may e x p l a i n the o b s e r v e d dependence o f 23  p u s h i n g on t e m p e r a t u r e g r a d i e n t by Cisse" and  Boiling.  They used Cu p a r t i c l e s which have a much h i g h e r t h e r m a l c o n d u c t i v i t y than water and t h e r e f o r e under a s t e e p g r a d i e n t the c r i t i c a l  v e l o c i t y w i l l be l o w e r .  Neverthe-  l e s s , t h i s i s not i n agreement w i t h t h e o n l y f o r m a l t r e a t ment a v a i l a b l e ^ i n which t h e t h e r m a l e f f e c t s a r e i n c l u d e d to g i v e the f o l l o w i n g e q u a t i o n f o r t h e c r i t i c a l V  c  ^3  =  B 2.  24nR  3  GAS  1 / 4  (l-2x)"  B AS  3 / 4  (1+X)  ...30  3  where x =::(k - k ) / ( 2 k 1  velocity  1  + k ).  I t i s e v i d e n t t h a t t h e above f o r m u l a c o n t r a d i c t s the former a n a l y s i s .  Moreover  for x  1 > i - e . f o r kp>>k-j as  i n the c a s e o f Cu o r any o t h e r m e t a l l i c p a r t i c l e i n w a t e r , the c r i t i c a l  v e l o c i t y tends to zero.  T h i s i s n o t shown  in the C i s s e and B o i l i n g d a t a where the c r i t i c a l f o r Cu and W a r e o f t h e same  velocities  o r d e r as t h o s e f o r S i 0^  p a r t i c l e s d e s p i t e the v e r y d i f f e r e n t t h e r m a l c o n d u c t i v i t y . On the o t h e r hand, t h e s e f i n d i n g s : b r i n g i n t o doubt t h e 25 e x p e r i m e n t a l o b s e r v a t i o n s o f Zubko and c o - w o r k e r s employed  who  m e t a l l i c p a r t i c l e s (W, T a , Mo, Fe, N i , C r ) , 2-3  mm i n d i a m e t e r and m e t a l l i c m a t r i c e s ( Z n , Bi and Sn) t h a t  22 is with s i m i l a r values  of k ,  and  p  k  s>  They v e r i f i e d  t h a t f o r kp/k-j > 1 t h e r e i s c a p t u r e and f o r k^/k^< 1 the p a r t i c l e s are p u s h e d .  Two p;e:c:u;T.i.ar f a c t s must be n o t e d ;  f i r s t , the growth v e l o c i t i e s a t which the were made v a r i e d i n the r a n g e 5-100  observations  cm/hr w h i l e  the  t h e o r i e s p r e d i c t c r i t i c a l v e l o c i t i e s o f the o r d e r _3 10  cm/hr f o r t h e s e p a r t i c l e s .  Cr ( p a r t i c l e s ) - Sn ( m a t r i x )  S e c o n d , f o r the  system  the r e l a t i o n k /k-j  was  p  assumed to be l e s s than 1 (0.07/0.157) and the ments gave r e p u l s i o n w h i l e  of  u s i n g ASM  v a l u e o f kp/k-j i s 76/32.6 >1  t a b l e s the  and a c c o r d i n g l y  have l e d to  trapping.  b)  The  E f f e c t of  The  i n t e r a c t i o n i n the PLS  expericalculated  this  should  Impurities  c a l l y when the 1 i q u i d c o n t a i n  system changes d r a s t i -  impurities.  D e p e n d i n g on  the v a l u e o f the p a r t i t i o n c o e f f i c i e n t t h e r e w i l l  be  a c c u m u l a t i o n or d e p l e t i o n o f s o l u t e i n f r o n t of the a d v a n c ing i n t e r f a c e .  The  p r e s e n c e o f a f o r e i g n body i n f r o n t o f  the i n t e r f a c e w i l l o b s t r u c t t h e d i f f u s i o n o f s o l u t e away from the i n t e r f a c e i n c r e a s i n g the c o n c e n t r a t i o n intermediate  liquid layer.  i n the  As a r e s u l t a d e p r e s s i o n  at  the i n t e r f a c e w i l l a p p e a r f a v o r i n g the c a p t u r e o f a particle.  Tembin  et a l .  introduced  t h i s f a c t o r i n the  theory  23  of Chernov e t a l . and a r r i v e d a t t h e f o l l o w i n g e q u a t i o n for the c r i t i c a l V  velocity  a,1 s DftK  ~T  In  "  30-| n s  -,  DfiK  - l r  1  B AS m C  R ASmC.  0  I n l n A-,-1  .31  In A  where A  3t&. n^DK/B-AS m C,„  D  diffusion  m  liquidus slope  K  partition  coefficient coefficient  concentration of impurity i n the bulk  liquid  It can be seen i n t h i s e x p r e s s i o n t h a t t h e c r i t i c a l v e l o c i t y has a s t r o n g e r d e p e n d e n c e w i t h p a r t i c l e r a d i u s . Whose e x p o n e n t p a s s e s from -4/3 t o -2. occurs at a value of R ~ 2.5xl0~ at c o n s t a n t  Cc  4  This  transition  cm f o r m C / K.: = 1 d e g . ro  , , and t h i s R d e c r e a s e s as m 0^/^:1 i n c r e a s e s .  CO -J  r ;  i  •  On t h e o t h e r hand f o r a g i v e n p a r t i c l e r a d i u s , f o r i n s t a n c e R = 1 ym t h i s change o f b e h a v i o r happens a t a c o n c e n t r a tion of C 10  > 3x10"  2  wt % f o r m = 3 deg/wt % and K =  . At these points the capture process i s  controlled  by t h e d i f f u s i o n o f s o l u t e t o t h e b u l k l i q u i d r a t h e r than the t r a n s p o r t o f l i q u i d t o t h e s o l i d i f i c a t i o n f r o n t o r the d i s j o i n i n g  p r e s s u r e , whose p a r a m e t e r B^ a p p e a r s i n  a weak T!o.g;a<rn"ttunic r e l a t i o n .  24  It i s n e c e s s a r y  to m e n t i o n t h a t t h e r e a r e no  experi-  mental r e s u l t s r e p o r t e d on the e f f e c t o f i m p u r i t y particle c)  on  trapping. Other E f f e c t s i)  V i s c o s i t y o f the m e l t .  A l l the  theories  p r e d i c t lower c r i t i c a l v e l o c i t i e s f o r l i q u i d s with  higher  24  viscosities.  C i s s e and B o i l i n g  found t h i s q u a l i t a t i v e  d e p e n d e n c e c o m p a r i n g the c r i t i c a l r a t e s f o r s a l o l and ii)  Body f o r c e s .  I t i s o b v i o u s t h a t the  water. differ-  ence i n d e n s i t y between t h e p a r t i c l e and the l i q u i d  will  enhance t r a p p i n g d e p e n d i n g on w h e t h e r the r e l a t i o n dp/d-j i s e i t h e r g r e a t e r or l e s s than 1.  The d e n s i t y  difference  w i l l c o n t r i b u t e to s i n k i n g o r f l o a t i n g o f the p a r t i c l e . T h i s e f f e c t w i l l be g r e a t e r f o r l a r g e p a r t i c l e s as -  s i d e r e d by B o i l i n g and  con-  3  Cisse.  The r o l e o f the body f o r c e i s not c l e a r when i t i s 29  p r o d u c e d by a m a g n e t i c f i e l d . ferromagnetic  Cheu and W i l c o x  using^  p a r t i c l e s and inaphithalene: as m a t r i x  t h a t t h i s body f o r c e iinh'ibiit's: c a p t u r e  found  but a t the same time  when t h e p a r t i c l e s were used s e v e r a l t i m e s t h e  critical  v e l o c i t y i n c r e a s e d w i t h the run number s u g g e s t i n g a,, h y s t e r e s i s e f f e c t may  obscure t h e i r  that  observations.  iii)  Convection.  25 I t has n o t been e s t a b l i s h e d i n  what way movement o f t h e m e l t may a f f e c t t h e c a p t u r e 3 28 process.  It is believed '  convection should i n h i b i t  t r a p p i n g o f b i g p a r t i c l e s which may be washed o f f from the i n t e r f a c e ; a t l e a s t f o r p a r t i c l e s l a r g e r than t h e -3 -1 l i m i t l a y e r (10 -10 cm). -3.  Pushing in Metals O b s e r v a t i o n s r e l a t e d to p a r t i c l e p u s h i n g i n an a l l  metal s y s t e m a r e r e l a t i v e l y r a r e .  T h e r e i s no c o n c l u s i v e  evidence i n the l i t e r a t u r e of p a r t i c l e pushing at a metallic solid-liquid Pikunov  1  interface.  i n t r y i n g to f i n d the o r i g i n of the equiaxed  zone i n i n g o t s p e r f o r m e d some e x p e r i m e n t s i n c o p p e r oxygen a l l o y s i n which  " r a d i o a c t i v e Zn^^ or S n  1 1 3  were  i n t r o d u c e d as d e o x i d i z e r . A f t e r s o l i d i f i c a t i o n was comp l e t e samples were a n a l y z e d .  A u t o r a d i o g r a p h s o f t h e sectioned  samples::: d i d n o t show any p r e f e r e n t i a l m a c r o s c o p i c d i s t r i 65 However, Sn; 113 was f o u n d t o be b u t i o n o f Sn 113 o r Zn.*..s e g r e g a t e d i n s t r i p s a l o n g t h e g r a i n b o u n d a r i e s and a l s o 65 i n t h e i n t e r d e n d r i t i c l i q u i d , whereas Zn was f o u n d to be homogeneously d i s t r i b u t e d . T h i s l e d t o t h e c o n c l u s i o n t h a t Sn 113 0 i s pushed w h i l e Zn 65 0 i s n o t . The same 2  11 3 o b s e r v a t i o n on t h e Sn  0 p a r t i c l e s , as w i l l be shown i n  the p r e s e n t i n v e s t i g a t i o n may be due to p h y s i c a l  ,v  26 and  entrapment  not  pushing.  On the o t h e r hand, i n d e o x i d i z e d 33-36 ingots  ~  situated  ingots  or  ESR  i n c l u s i o n s were o b s e r v e d to be p r e f e r e n t i a l l y in i n t e r d e n d r i t i c p o s i t i o n s .  In the l a t t e r  re-  p o r t s the a u t h o r s a s s o c i a t e d t.heise o b s e r v a t i o n s to a homogeneous n u c l e a t i o n  from the i n t e r d e n d r i t i c  With a d i s t i n c t view F r a n k l i n and r i t e s sweep o f f the i n c l u s i o n s , a t t r i b u t e the s e g r e g a t i o n  eutectic.  Evans assumed the i - ' M y e r s  and  dend-  Fleming  o f the i n c l u s i o n between  dendrites  to a p u s h i n g mechanism. „3 B o i l i n g and C i s s e r e p o r t e d  t h a t some Cu  particles  _5  10  cm i n r a d i u s were f o u n d to be c l u s t e r e d  w h i c h Cu i s  i n Pb i n  soluble.  F i n a l l y , the o n l y r e l a t e d work where p a r t i c l e s were.; introduced  i n m e t a l l i c m e l t s to s t u d y t h e i r r e j e c t i o n i s 25  t h a t o f Zub&o e t a l . which has been d i s c u s s e d previous 4.  i n the  section  Summary o f C h a p t e r II The  r e j e c t i o n of p a r t i c l e s by an a d v a n c i n g  l i q u i d i n t e r f a c e has been e x t e n s i v e l y r e t i c a l l y ; ' ! and e x p e r i m e n t a l l y .  studied  solid-  both t h e o -  At the p r e s e n t t i m e  there  27  i s no mechanism a v a i l a b l e t o p r e d i c t when r e j e c t i o n o f particles will  o c c u r by an a d v a n c i n g i n t e r f a c e .  Neverthe-  l e s s , the c a l c u l a t i o n o f o>, the L i f s h i t z - Van d e r Waals', c o n s t a n t , may pushing, w i l l  g i v e a l e g i t i m a t e way t o e s t a b l i s h w h e t h e r  o c c u r under s p e c i f i e d c o n d i t i ons. (see Appendix I ) .  When p u s h i n g o c c u r s s e v e r a l t h e o r i e s have been d e v e l o p ed t o p r e d i c t t h e c r i t i c a l  v e l o c i t y a t which t h e p a r t i c l e s  a r e c a p t u r e d by t h e i n t e r f a c e . However, t h e r e i s no ment among the t h e o r i e s i n t h e d e p e n d e n c e o f t h e v e l o c i t y on the r a d i u s . o f , t h e p a r t i c l e . v e l o c i t y r e l a t i o n s o f t h e form V w i t h n r a n g i n g from 0.5-2. thesettheories  « R~  n  Experimental  agree-  critical  For t h i s  critical  have been o b t a i n e d v e r i f i c a t i o n of  i s c o m p l i c a t e d due t o the p r e s e n c e  of  o t h e r v a r i a b l e s i n t h e system such as t h e r m a l c o n d u c t i v i t y , shape o f t h e p a r t i c l e s , i m p u r i t i e s and c o n v e c t i o n i n the m e l t which a r e i n a r e a l s y s t e m . the c a p t u r e p r o c e s s .  These f a c t o r s can  affect  Some o f t h e s e n o n - i d e a l s i t u a t i o n s  have been s t u d i e d but have not been v e r i f i e d e x p e r i m e n t a l l y . Most o f the work has  been done i n n o n - m e t a l l i c  where the e l e c t r o s t a t i c f o r c e may  be i m p o r t a n t .  The  e l e c t r o s t a t i c f o r c e , which i s not p r e s e n t i n m e t a l s , in i t s e l f be t h e o v e r w h e l m i n g  systems may  f a c t o r in d e t e r m i n i n g whether  p a r t i c l e p u s h i n g may o c c u r i n non m e t a l l i c s y s t e m s .  28  Chapter III OBJECTIVES OF PRESENT RESEARCH The o b j e c t i v e o f t h i s programme i s t o e s t a b l i s h  whether  s o l i d p a r t i c l e s i n an a l l metal s y s t e m a r e r e j e c t e d by an a d v a n c i n g s o l i d - l i q u i d i n t e r f a c e and i f so under what c o n ditions rejection occurs. listed 1)  The p r o c e d u r e t o be f o l l o w e d i s  below. A s u i t a b l e p a r t i c l e , l i q u i d , s o l i d s y s t e m w i l l be  s e l e c t e d and t h e d i s t r i b u t i o n o f p a r t i c l e s ahead o f an advancing s o l i d - l i q u i d i n t e r f a c e determined.  Interfaces  w i t h p l a n a r , c e l l u l a r and d e n d r i t i c m o r p h o l o g i e s w i l l be considered.  S o l i d i f i c a t i o n w i l l be b o t h u n i d i r e c t i o n a l  and n o n d i r e c t i o n a l . 2)  The r e s u l t s from (1) w i l l be c o n s i d e r e d i n terms o f  a p h y s i c a l model o f p a r t i c l e i n t e r a c t i o n s w i t h an a d v a n c i n g i nterface. 3)  A w a t e r and n y l o n s p h e r e model  s i m u l a t i n g the a l l  metal s y s t e m w i l l be examined t o d e t e r m i n e by d i r e c t o b s e r v a t i o n how p a r t i c l e s move a g a i n s t a s o l i d 4)  interface.  The o b s e r v a t i o n s o f p a r t i c l e movement i n t h e a l l  29  metal s y s t e m , w i l l be compared t o t h e o b s e r v a t i o n s i n t h e water and n y l o n s p h e r e model and t o t h e p r e d i c t i o n s o f t h e t h e o r i e s presented i n the previous chapter.  30  C h a p t e r IV EXPERIMENTAL APPARATUS AND 1.  The M e t a l l i c PLS a)  PROCEDURE  System  S e l e c t i o n and P r e p a r a t i o n o f  Samples  In an i d e a l PLS s y s t e m , t h e p a r t i c l e s s h o u l d have the same d e n s i t y and t h e r m a l c o n d u c t i v i t y as t h e m e l t .  In  a d d i t i o n t h e p a r t i c l e s s h o u l d be s p h e r i c a l and must n o t d i s s o l v e i n t h e m e l t but s h o u l d be w e t t e d by t h e m e l t .  A  number o f s y s t e m s were i n v e s t i g a t e d u s i n g e i t h e r Sn, Pb o r Bi as t h e l i q u i d m a t r i x and C r , W, cles.  SnO, Fe o r Cu as p a r t i -  The m a j o r p r o b l e m e n c o u n t e r e d was t h a t i f a p a r t i c l e  had l i m i t e d s o l u b i l i t y i n t h e m e l t , o r was o x i d i z e d , i t was not p o s s i b l e to i n t r o d u c e and keep t h e p a r t i c l e s i n t h e melt during s o l i d i f i c a t i o n . s i l i c o n o i l , Zn C12  a n  d  To overcome o x i d a t i o n e f f e c t s C12 were used as f l u x e s on the  m e l t s u r f a c e s when t h e p a r t i c l e s were added t o t h e m e l t . T h i s was i n i t i a l l y t h o u g h t t o be s u c c e s s f u l i n i n t r o d u c i n g p a r t i c l e s i n t h e m e l t , but s u b s e q u e n t l y i t was f o u n d t h a t the p a r t i c l e s , even w i t h t h e f l u x , were n o t i n t h e m e l t . One s y s t e m , Fe p a r t i c l e s i n a Pb m a t r i x was f o u n d t o be r e a s o n a b l y s a t i s f a c t o r y , i n t h a t t h e p a r t i c l e s were w e t t e d by t h e Pb but were not d i s s o l v e d , and c o u l d be i n t r o d u c e d  31 in the melt.  However the d e n s i t y and thermal  of the Fe p a r t i c l e s matrix.  differed significantly  In s p i t e of these d i f f e r e n c e s  matrix system was adopted as the best  conductivity  from the Pb  the Fe p a r t i c l e alternative.  Master samples of lead of 99.99% and 99.999% were cast particles.  c o n t a i n i n g a high d e n s i t y of ARMCO i r o n The 99.99% p u r i t y  and the higher p u r i t y  Pb  lead was used with  purity spherical alloys  lead when pure lead matrix was used.  The ARMCO powder was screened  and  mesh powder was introduced in the l e a d . SEM micrograph of the s p h e r i c a l  t h e . r-400:  :  F i g . 14 shows a  particles.  Lead was melted in a g r a p h i t e mold 3.5 cm in  diameter  and 6.5 cm h e i g h t , ZnCl^ added to the melt s u r f a c e , and then the Fe p a r t i c l e s . the p a r t i c l e s  The f l u x d e o x i d i z e d the Fe r e s u l t i n g  being wetted by the l e a d .  in  The p a r t i c l e s were  introduced i n t o the melt by vigorous s t i r r i n g of the melt. The amount of p a r t i c l e s  that entered i n t o the lead was  found to depend on the t o t a l  number of p a r t i c l e s  and the  degree of s t i r r i n g of the melt.  Lead antimony a l l o y s tainers  were prepared in graphite  con-  s i m i l a r to pure lead using a part of the master  F i g u r e 14. SEM m i c r o g r a p h of the ARMCO i r o n p a r t i c l e s used i n the e x p e r i m e n t s . 200X.  ,-400  mesh,  33  sample and a d d i n g pure l e a d and pure a n t i m o n y (99.99%) to o b t a i n the d e s i r e d c o n c e n t r a t i o n o f a n t i m o n y and p a r t i c l e s . The m e l t was  then c a s t i n t o a 11 mm  i n s i d e diameter  t u b e , 20 cm l o n g , p a i n t e d w i t h g r a p h i t e .  The m e l t  Vycor was  r a p i d l y s o l i d i f i e d to a v o i d f l o t a t i o n o f the Fe p a r t i c l e s and o b t a i n homogeneous a x i a l d e n s i t y o f p a r t i c l e s . p a r t i c l e d i s t r i b u t i o n was  checked  The  by c u t t i n g the c a s t  samples and o b s e r v i n g the p a r t i c l e d i s t r i b u t i o n w i t h  an  o p t i c a l microscope  Samples  i n the p o l i s h e d s u r f a c e s e c t i o n .  were a l s o a n a l y s e d q u a n t i t a t i v e l y u s i n g x - r a y a n a l y s i s i n the SEM.  No t r a c e ; o f Fe was  f o u n d i n the m a t r i x even a f t e r  r e m e l t i n g the same sample. b)  The A p p a r a t u s and T e c h n i q u e s  for Controlled  S o l i d i f i c a t i on i)  Solidification.  o f an a d v a n c i n g necessary  To examine the i n t e r a c t i o n  i n t e r f a c e w i t h p a r t i c l e s i n the m e l t i t i s  t h a t the p a r t i c l e s r e m a i n u n i f o r m l y d i s t r i b u t e d  i n the m e l t d u r i n g s o l i d i f i c a t i o n . Fe (7.87  gr cm  S i n c e the d e n s i t y o f  ) i s a p p r e c i a b l y l e s s than l i q u i d l e a d  3  (10.6  gr cm  ) the p a r t i c l e s w i l l f l o a t to the top o f the  m e l t d u r i n g s o l i d i f i c a t i o n which i s u n s a t i s f a c t o r y .  To  overcome t h i s d i f f i c u l t y a zone r e f i n i n g s o l i d i f i c a t i o n p r o c e s s was  adopted  i n which the l i q u i d zone was  progres-  s i v e l y moved a l o n g the r a p i d l y s o l i d i f i e d rod c o n t a i n i n g the  To g r oph re c o rd er  TC  Chrom eI-Alumel 0  VAC  rMullit e tube - I-5 5 cm I D V y c o r tube - Mem  OJ or V solid  ID  -liquid .i  i  i  i  i  • i  i  i  i  i  i  i  i  heating coils-Chromel 24 '////////////. in su Io t io n - Fiberflax k W W W W W l -shield I T• T• £ f • t I f I M t f l « * f i f «  Sample CO - r otation  Furnace V = hori zo ntg mot ion  V = u p w a r d -» f i x e d motion  cm SCALE  F i g u r e 15. Diagram of the apparatus used f o r the s o l i d i f i c a t i o n of a l l o y s and pure m e t a l s . ( F o r e x p l a n a t i o n see t e x t ) .  hi  co  35 particles.  Two s o l i d i f i c a t i o n modes were u s e d , v e r t i c a l l y  downward and h o r i z o n t a l w i t h r o t a t i o n u s i n g t h e s y s t e m shown s c h e m a t i c a l l y i n F i g . 15.  With t h e s e t e c h n i q u e s t h e  p r o b l e m s due t o buoyancy f o r c e a r e overcome i n t h e f o l l o w i n g ways.  In t h e v e r t i c a l mode t h e f u r n a c e was kept f i x e d  and t h e sample moved  upwards, t h a t i s , t h e s o l i d i f i c a t i o n  f r o n t a d v a n c e d downwards.  As t h e zone moved, m a t e r i a l  m e l t e d a t t h e bottom r e l e a s e d p a r t i c l e s and e n s u r e d t h e continuous face.  presence of p a r t i c l e s at the s o l i d - l i q u i d i n t e r -  In t h e h o r i z o n t a l mode t h e sample was rotated: on --its axis  at constant  angular v e l o c i t y  zontal l y at constant  speed.  and t h e f u r n a c e was/ moved h o r i The'-minimum a n g u l a r v e l o c i t y  employed was e s t a b l i s h e d by t h e time t a k e n f o r t h e b i g g e s t p a r t i c l e t o t r a v e l a d i s t a n c e equal  to the diameter of the  sample.  To c a l c u l a t e t h i s d i s t a n c e t h e c o r r e c t e d 37 v e l o c i t i e s and t r a n s i e n t times were u s e d .  Stokes  The f u r n a c e c o n s i s t e d o f a m u l l i t e tube 1.55 cm I.D. and 3.5 cm l o n g w i t h Chromel 24 w i n d i n g  cemented and i n -  s u l a t e d w i t h f i b e r f l a x . A t h e r m o c o u p l e was p l a c e d i n t h e center o f t h e inner p a r t i n order to c o n t r o l ^ t h e temperature.  To do t h i s , t h e c u r r e n t t h r o u g h  t h e h e a t i n g c o i l was  c a l i b r a t e d so as t o o b t a i n an a d e q u a t e zone which was o f the o r d e r o f 2.5 cm l o n g .  Once t h i s zone was formed t h e  t e m p e r a t u r e was m a i n t a i n e d  constant  by m a n u a l l y a d j u s t i n g  36  t h e c u r r e n t t o a v a l u e t h a t compensated f o r .the heat l o s s e s . The maximum v a r i a t i o n o f t e m p e r a t u r e g r e a t e r than 0.5%.  i n t h e f u r n a c e was  In t h e v e r t i c a l mode t h e  of t h e sample was r e c o r d e d by a t h e r m o c o u p l e the c e n t r e o f t h e r o d . that the temperature  not  temperature positioned in  The measured t e m p e r a t u r e s  showed  v a r i a t i o n s a s s o c i a t e d with convection  i n the m e l t were much g r e a t e r than t h o s e a s s o c i a t e d w i t h c h a n g e s i n the t e m p e r a t u r e  o f the f u r n a c e .  g r a d i e n t i n t h e m e l t was d e t e r m i n e d remained  constant during  The  temperature  t o be 30°C c m ~ \  and  solidification.  Samples i n both s o l i d i f i c a t i o n modes were grown as: few c e n t i m e t e r s and the l i q u i d was q u e n c h e d . was  !  T h i s growth  sometimes r e p e a t e d a l o n g t h e sample i n c r e a s i n g the number  of i n t e r f a c e s and t h e r e f o r e t h e number o f o b s e r v a t i o n s f o r a given s o l i d i f i c a t i o n c o n d i t i o n To change the shape o f t h e i n t e r f a c e from c e l l u l a r t o d e n d r i t i c t h e c o n c e n t r a t i o n o f antimony than c h a n g i n g t h e t e m p e r a t u r e  was  increased rather  g r a d i e n t o r growth v e l o c i t y .  The l a t t e r were m a i n t a i n e d c o n s t a n t i n o r d e r t o  minimize  changes i n c o n v e c t i o n f l o w p a t t e r n s i n the m e l t . ii)  Metal 1 o g r a p h i c P r e p a r a t i o n .  The  samples  were c u t l o n g i t u d i n a l l y and h o r i z o n t a l l y w i t h r e s p e c t to  37  the growth d i r e c t i o n both a t t h e i n t e r f a c e and b e h i n d i t a t various l e v e l s .  Sample p r e p a r a t i o n c o n s i s t e d o f s t a n d a r d  p o l i s h i n g w i t h p a p e r and a l u m i n a s a r y diamond p a s t e 1 ym was u s e d .  up t o 0.05 urn.  When  neces-  The samples were e t c h e d  to r e v e a l t h e m i c r o s t r u c t u r e o f s e g r e g a t i o n .  Two e t c h a n t s  were u s e d , one w i t h 3 p a r t s o f a c e t i c a c i d and 1 p a r t o f hidrogen  p e r o x i d e , and t h e second  etchant containing  molybdic  a c i d , 100 g r ; NH^OH, 140 m l ; H 0 , 240 ml and HN0 , 60 m l . 2  iii)  Counting  3  and P a r t i c l e S i z e D i s t r i b u t i o n . The  d i s t r i b u t i o n o f t h e p a r t i c l e s was determined, by c o u n t i n g t h e p a r t i c l e s with the o p t i c a l microscope  a t 500X m a g n i f i c a t i o n  s c a n n i n g t h e e n t i r e p o l i s h e d sample a r e a .  The a p p a r e n t  s i z e o f t h e p a r t i c l e s was measured by s u p e r i m p o s i n g  a trans-  p a r e n t g r i d , d i v i d e d i!mm;i 11 i m e t e r s , onto t h e p r o j e c t i o n screen o f the microscope.  At t h i s m a g n i f i c a t i o n the biggest  p a r t i c l e (~35 ym) had a s i z e o f 1.75 cm and t h e s m a l l e s t 0.1  cm.  F o r c o u n t i n g p u r p o s e s t h e p a r t i c l e s i z e s were  g r o u p e d i n mm i n t e r v a l s i n 15 p a r t i c l e c l a s s e s . c l a s s e s were l a t e r r e g r o u p e d  These  f o r analysis o f the data.  When s a m p l e s c o n t a i n e d a low p a r t i c l e d e n s i t y on t h e polished surface, i n s u f f i c i e n t to give a s t a t i s t i c a l l y s e n t a t i v e d i s t r i b u t i o n , t h e y were r e s e c t i o n e d and ; p o l i s h e d and c o u n t e d  again.  repre-  38 The  r e a l p a r t i c l e d i s t r i b u t i o n was 38  Schwarz-Salttkov  method  calculated  using  the  which e n a b l e s the number o f p a r t i -  c l e s o f a g i v e n r e a l s i z e to be d e t e r m i n e d u s i n g the p a r e n t d i s t r i b u t i o n o n l y ; t h a t i s , the c a l c u l a t i o n o f  apthe  r e a l number o f p a r t i c l e s of a g i v e n s i z e does not r e q u i r e  the  r e a l number o f p a r t i c l e s of o t h e r s i z e s . c)  P r o c e d u r e f o r the C a s t i n g A 50%  Experiment  l e a d - 50% t i n m e l t w i t h i r o n p a r t i c l e s  p r e p a r e d i n a manner s i m i l a r to t h a t used i n the s o l i d i f i c a t i o n experiments.  The m e l t was  g r a p h i t e mould.  the a l l o y was  s o l i d i f i e d and  A f t e r 2 min the s t r u c t u r e  revealed  d r i t e s . The m e t a l l o g r a p h y was several  controlled  then cast into a completely  f u l l y d e v e l o p e d den-  s i m p l e r b e c a u s e i t was  n e c e s s a r y t o e t c h the sample.  was  C o u n t i n g was  not  performed at  h e i g h t s i n the sample u s i n g the method  described  previously. 2.  The  Water Model  A w a t e r model was cylinder At one  end  17 cm l o n g and  constructed consisting 9.5  cm I.D.  a c e l l u l a r i n t e r f a c e was  w i t h r o u n d e d t i p s p l a c e d 4 mm  closed  of a l u c i t e  a t both e n d s .  s i m u l a t e d w i t h hexagons  a p a r t as shown i n F i g . , 1 6 .  N y l o n p a r t i c l e s 0.375 cm i n d i a m e t e r were u s e d , the d i a m e t e r to c o r r e s p o n d  selecting  to the a v e r a g e p a r t i c l e s i z e  39  I cm  40  F i g u r e 16. The p h y s i c a l model to study the motion of n y l o n spheres at a c e l l u l a r i n t e r f a c e , a) The l u c i t e c y l i n d r i c a l c o n t a i n e r w i t h the b r i n e s o l u t i o n , t h e spheres and t h e " i n t e r f a c e " ; 06 i s the t i l t i n g angle.b) The c e l l t i p s and the spheres i n d e t a i l . c ) A sphere w i t h v e l o c i t y V due to buoyancy f o r c e c o l l i d i n g w i t h a c e l l t i p ; v i s the r e s u l t a n t v e l o c i t y immediately a f t e r c o l l i s i o n ; t h e downward v e c t o r V i s the v e l o c i t y a f t e r h a l f r e v o l u t i o n . T h e v e l o c i t i e s a r e w i t h r e s p e c t to a system o f r e f e r e n c e f i x e d t o the c e l l s .  41  i n t h e r e a l F e - P b s y s t e m (~18 ym) when c o m p a r i n g the  cell  size  A brine adjusting  with  (~100 y m ) .  solution  the density  was u s e d t o s i m u l a t e t h e m e l t .  By  of; theesojuti.o.n by c h a n g i n g t h e s a l t  c o n c e n t r a t i o n » a r a n g e o f p a r t i c l e v e l o c i t y was o b t a i n e d . To s i m u l a t e h o r i z o n t a l cylinder the  was t i l t e d  s o l i d i f i c a t i o n with rotation the  from the h o r i z o n t a l  in-order  to give  s p h e r e s a component o f v e l o c i t y p a r a l l e l t o t h e a x i s  simulating  an a d v a n c i n g i n t e r f a c e .  The h o r i z o n t a l  velocity  was d e t e r m i n e d by m e a s u r i n g t h e t i m e t a k e n by t h e p a r t i c l e s t o move b e t w e e n two m a r k s on t h e t u b e . was c a l c u l a t e d  from t h e h e i g h t and l e n g t h  supporting the cylinder. to t h e s t i l l were r e l e a s e d  The t i l t i n g  angle  o f the frame  The a b s o l u t e v e l o c i t y w i t h  l i q u i d was m e a s u r e d on s e v e r a l  particles  i n t h e l i q u i d by means o f a j - t u b e  in the bottom o f the s o l u t i o n .  respect which  introduced  The motion and d i s t r i b u t i o n  of p a r t i c l e s were o b s e r v e d f o r a range o f f r e q u e n c i e s o f rotation  o f t h e tube and t i l t i n g  angles.  In t h e v e r t i c a l " s o l i d i f i c a t i o n " mode t h e d e n s i t y the  solution  as a f u n c t i o n  was v a r i e d  and t h e d i s t r i b u t i o n o f p a r t i c l e s  o f p a r t i c l e v e l o c i t y was e x a m i n e d .  t e c h n i q u e s were used t o r e l e a s e cylinder,  of  the p a r t i c l e s .  Two In o n e t h e  w i t h a l l t h e p a r t i c l e s a t t h e t o p end o p p o s i t e  42 the c e l l u l a r i n t e r f a c e , was r a p i d l y r o t a t e d t h r o u g h 180 d e g r e e s about a h o r i z o n t a l a x i s .  In t h e s e c o n d p r o c e d u r e t h e  p a r t i c l e s were r e l e a s e d i n t h e s t i l l  l i q u i d , one a t a t i m e ,  with the j-shaped tube. 3.  P a r t i c l e I n t e r a c t i o n w i t h a F r e e z i n g Water I n t e r f a c e A p y r e x t u b e 10 mm i n d i a m e t e r and 25 cm l o n g w i t h one  c l o s e d t e n d was f i l l e d w i t h d i s t i l l e d w a t e r .  Cr p a r t i c l e s  were added t o t h e w a t e r , mixed w e l l and t h e tube then p l a c e d i n a f r e e z i n g b a t h a t -10°C.  The d i s t r i b u t i o n o f t h e Cr  p a r t i c l e s i n t h e w a t e r was o b s e r v e d as t h e w a t e r i n t h e tube solidified.  The d i s t i l l e d w a t e r was d e g a s s e d p r i o r t o use  by b o i l i n g t h e w a t e r and f r e e z i n g s l o w l y . s e l e c t e d as t h e y a r e w e t t e d by w a t e r .  Cr p a r t i c l e s were  43  Chapter RESULTS AND 1.  Controlled a)  { ] % and  2%)  DISCUSSION  Solidification  Non-Planar i)  V  Interface  V e r t i c a l Growth. containing  Two  spherical  lead-antimony a l l o y s  p a r t i c l e s o f i r o n were  grown a t the same v e l o c i t y i n the v e r t i c a l mode.  The  Pb 1% Sn a l l o y had a c e l l u l a r s o l i d l i q u i d i n t e r f a c e the Pb 2% Sb a l l o y s o l i d i f i e d  dendritically.  Pb 1% Sb C e l l u l a r I n t e r f a c e  - Sample  F i g . 17 shows the e t c h e d l o n g i t u d i n a l t h i s sample i n c l u d i n g o f the s a m p l e .  the quenched r e g i o n  section  i n the u p p e r  of part  to the q u e n c h i n g p r o c e s s .  appearance of t r a n s v e r s e s e c t i o n s ing, at d i f f e r e n t distances shown i n F i g . 18. (the white spotes)c e l l u l a r regions.  The  V-1  I t can be c l e a r l y seen t h a t t h e r e i s a s h a r p  change i n m i c r o s t r u c t u r e due  t h a t t h e r e was  and  o f the sample a f t e r  i n F i g . 18 are s i t u a t e d density  negligible  etch-  b e h i n d the quenched i n t e r f a c e  Note t h a t a l m o s t a l l of the The  The  i n the  o f p a r t i c l e s was  interaction  particles low  interso  among them.  r e s u l t s of the measurements o f p a r t i c l e s i z e ,  is  44  F i g u r e 17. L o n g i t u d i n a l view of sample V - l . The quenched l i q u i d and the u n i d i r e c t i o n a l l y grown s o l i d a r e c l e a r l y d e f i n e d by t h e sharp change i n m i c r o s t r u c t u r e . E t c h e d , 5 0 X .  46  47  (e)  F i g u r e 18. T r a n s v e r s a l v i e w s of sample W I a t the i n t e r f a c e (a t o d) and 12 mm from the i n t e r f a c e ( e ) . I n d) the i r o n p a r t i c l e s a r e d a r k e r t h a n t h e m a t r i x . I n a,b,c and e the p a r t i c l e s a r e t h e b r i g h t p s p o t s . M o s t of the p a r t i c l e s a r e i n the c e l l w a l l s . I n a a t t h e c e n t e r l e f t a r a m i f i c a t i o n w i t h a p a r t i c l e i s shown.In c and d t h e s m a l l p a r t i c l e s appear i n the m a t r i x . I n e most of the c e l l w a l l s have d i s s o l v e d . a)100X; b)and c)500X; d)SEM 200X and e ) 1 0 0 X . A l l the sections are etched.  48 d e n s i t y , and d i s t r i b u t i o n on sample V-1 a r e t a b u l a t e d i n Table I I I .  Column V - l - 1 l i s t s t h e number o f p a r t i c l e s on a  p l a n e p e r p e n d i c u l a r t o t h e growth d i r e c t i o n a t t h e quenched interface.  The d i a m e t e r s c o f t h e p a r t i c l e s a r e i n d i c a t e d on  the l e f t , then t h e p a r t i c l e p o s i t i o n as s e g r e g a t e d a t t h e c e l l walls or i n the matrix.  Column V - l - l a  l i s t s t h e number o f  p a r t i c l e s on a p l a n e 200 um b e h i n d t h e p l a n e V-l-2  V-l-1.  Column  i n d i c a t e s t h e p a r t i c l e c o n f i g u r a t i o n on a p l a n e 2.5 mm  b e h i n d t h e i n t e r f a c e and V - l - 2 a 200 ym b e h i n d t h i s p l a n e . Column V - l - 3 i s 5 mm b e h i n d t h e quenched i n t e r f a c e w i t h V - l - 3 a 200 ym b e h i n d  this.  Measurements were n o t made beyond 5 mm b e h i n d t h e i n t e r f a c e as t h e c e l l s b e g i n t o h o m o g e n i z e and some c e l l 39 wal1S;  disappear.  . At  2.5 mm b e h i n d t h e i n t e r f a c e t h e  s o l i d i s c l o s e to the m e l t i n g temperature f o r a p e r i o d of 5 min.  I f s i g n i f i c a n t homogenization  o c c u r r e d t h e number o f  p a r t i c l e s i n t h e m a t r i x s h o u l d i n c r e a s e as c e l l w a l l s d i s appear.  E x a m i n i n g T a b l e I I I t h e t o t a l number o f p a r t i c l e s  i n t h e m a t r i x i n c r e a s e s from 25 t o 42 from V - l - 1 t o V - 1 - 2 a then d r o p s t o 25 i n V - l - 3 a t h e same as t h e i n t e r f a c e . Accordingly the decrease at V-l-2a i s a t t r i b u t e d to s c a t t e r and n o t h o m o g e n i z a t i o n  of the c e l l w a l l s .  G r o u p i n g t h e p a r t i c l e s i n l a r g e r s i z e i n c r e m e n t s and  TABLE I I I  Raw P a r t i c l e S i z e D i s t r i b u t i o n f o r S a m p l e V - l  V-l- 1 S e c t i o n •+ Size : Matrix Segregated um  V-l- 2  V-l-la  V-l-2  V-l-3a  V-lM 3  1  Segregated  Matrix  Segregated  Matrix  Segregated  Matrix  Segregated  Matrix  Segregated  Matrix  4  5  1  4  3  3  1  4  0  3  1  2  1  6  7  4  8  5  8  4  7  2  7  5  5  0  8  16  4  17  3  19  6  10  2  9  4  10  4  10  25  4  17  0  17  6  20  10  17  6  9  5  12  25  5  9  1  15  1  12  4  17  2  7  4  14  23  2  15  0  18  5  23  6  17  4  7  0  16  15  3  21  3  20  3  21  2  18  3  15  1  18  15  0  09  1  7  9  19  4  14  0  13  4  20  17  0  19  3  19  2  25  7  12  0  16  3  22  11  2  05  0  7  0  14  2  5  0  5  1  24  10  0  07  0  3  0  17  1  1  0  11  1  26  5  0  7  0  1  1  7  0  1  0  6  l  28  1  0  4  0  0  0  4  0  1  1  5  0  30  0  0  7  0  0  0  6  1  1  0  6  0  32  0  0  1  0  0  0  1  1  0  0  1  0  175  25  150  19  137  38  190  42  123  26  118  25  TOTAL  50  a d d i n g t h e c o u n t s f o r a number o f s e c t i o n e d p l a n e s g i v e s t h e r e s u l t shown i n T a b l e IV.  The same d a t a p l o t t i n g t h e number  o f p a r t i c l e s a g a i n s t p a r t i c l e s i z e isf'shown i n F i g . 19, a, b and c.  The t.hree.hystograms show s i m i l a r p a r t i c l e  distri-  b u t i o n s w i t h an a v e r a g e o f 11.4% o f t h e t o t a l p a r t i c l e s i n the m a t r i x a t t h e i n t e r f a c e and 18.7% b e h i n d i t due t o homogenization.  I t i s e m p h a s i z e d h e r e t h a t t h e r e l a t i v e number  of p a r t i c l e s i n the matrix i s higher f o r small s i z e s .  This  i s e v i d e n t from T a b l e IV a , b and c and F i g . 20.where t h e p e r c e n t o f p a r t i c l e s i n t h e m a t r i x as a f u n c t i o n o f s i z e i s shown.  In t h i s f i g u r e t h e s o l i d l i n e c o r r e s p o n d s t o t h e  p a r t i c l e s i z e d i s t r i b u t i o n at the interface-vand the broken l i n e t o d i s t r i b u t i o n s b e h i n d t h e quenched i n t e r f a c e .  In both  cases the percent of p a r t i c l e s i n the matrix decreases with s i z e , i n other words, s e g r e g a t i o n a t the c e l l w a l l s i s higher for bigger particles.  T h i s i s an i m p o r t a n t e x p e r i m e n t a l  r e s u l t b e c a u s e i t shows t h a t t h e o b s e r v e d d i s t r i b u t i o n i s due t o phenomenon o t h e r t h a n p u s h i n g .  In t h e r e s u l t s r e p o r t e d i n t h e l i t e r a t u r e i t was observed that there i s a c r i t i c a l  v e l o c i t y at which p a r t i -  c l e s a r e t r a p p e d and t h a t t h i s c r i t i c a l with p a r t i c l e s i z e .  velocity  decreases  T h e r e f o r e , i f p u s h i n g was t h e mechanism  leading to the observed segregation of p a r t i c l e s at the c e l l w a l l s , s m a l l p a r t i c l e s s h o u l d have been  preferentially  TABLE IV  Parti ele Si ze Di s t r i bution and % on Matri x f o r Sample V - l  Size, ym Segregated Matrix  (e) 1 +la+2+2a+ 3+3a  (b) 2+2a+3+3a  (a) 1+lA  Section ->  % Matrix Segregated Matrix  %: Matrix Segregated Matrix  % Matrix  : 24  11  31.4  39  14  26.4  63  27  30.0  7-11  75  11  12.8  111  43  27.9  186  54  22.5  11-15  72  8  10.0  116  26  ' 18.3  188  34  15.3  15-19  60  7  10.4  127  26  \ 17.0  177  33  15.7  19-23  : 52  5  8.8  103  15  12.7  155  20  15.7  23-27  29  0  0  47  5  9.6  76  5  11.4  27-31  12  0  0  23  1  4.7  35  1  6.2  :31-34  1  0  0  2  1  3  1  TOTAL  325  42  11:4  568  131  893  173  3-7  18.7  16.2  80  6  —  cell boundaries in matrix n = 367  0 4  T  CP  -40 + Q.  -£204 1 H  h  19  II  (a)  5  +  27  Size,  fjLm  35  0-  — cell boundaries — in matri x  n = 6 99  «/>  a>  T  - 1004 p  504  i  o 1  1  L.  L,  (b)  19  1  3-  27 S i z e ^ m  35  —  intercellulo r in m a t r i x  150 + o  o QL  o  50 +  i  I  1  I  04 — 4 -  i  II  19  (c)  =4— 27  S ize  35  m  F i g u r e 19. P a r t i c l e s i z e d i s t r i b u t i o n f o r sample V-1 ( c e l l u l a r i n t e r f a c e ) f o r p a r t i c l e s i n the c e l l w a l l s and i n the m a t r i x . a ) A t the i n t e r f a c e . b ) B e h i n d t h e i n t e r f a c e . c ) T o t a l distribution.  54  Q  •  interface  behind  interface  30-  -20 £  10  0  F i g u r e 20. % o f p a r t i c l e s i n the m a t r i x v s . p a r t i c l e s i z e , a t the i n t e r f a c e and behind the i n t e r f a c e . S m a l l p a r t i c l e s a r e more l i k e l y to appear i n the m a t r i x than l a r g e p a r t i c l e s . T h e d i f f e r e n c e o f t h e two c u r v e s i s a t t r i b u t e d to d i s s o l u t i o n of r a m i f i c a t i o n s .  55  rejected  instead  of large p a r t i c l e s .  ; !-., The l a r g e  particles  s h o u l d have been f o u n d i n t h e m a t r i x which i s n o t t h e c a s e . The d i f f e r e n c e  i n the percent of p a r t i c l e s in the matrix  on t h e i n t e r f a c e and b e h i n d t h e i n t e r f a c e shown i n F i g . 2 0 , is consistent  with homogenization occurring.  Some o f t h e  l a r g e r p a r t i c l e s , i n i t i a l l y i n t h e c e l l w a l l s , a f t e r homogeniz a t i o n , a r e now c o u n t e d i n t h e m a t r i x i n c r e a s i n g of p a r t i c l e s i n the m a t r i x .  the percent  A higher percentage o f small  p a r t i c l e s ds> i n i t i a l l y i n t h e m a t r i x so t h a t  homogenization  has a s m a l l e r e f f e c t . The  Real D i s t r i b u t i o n o f P a r t i c l e s i n t h e M a t r i x .  Polishing does The  a section  not„ give . t h e f u l l :  containing particle  particle  sections  diameter.  r e a l d i s t r i b u t i o n o f p a r t i c l e s p e r volume i s c a l c u l a t e d  u s i n g t h e S c h w a r z - S a l t i k o v method r e f e r r e d t o  previously.  T h i s method e n a b l e s t h e o b s e r v e d d a t a t o be c o r r e c t e d random p l a n e o f s e c t i o n  of the p a r t i c l e .  S a l t i k o v method g i v e s t h e f o l l o w i n g  f o r the  The S c h w a r z -  f o r m u l a f o r t h e number  of p a r t i c l e s N ( J ) o f s i z e J : v  a  N (i)-a(i N ( J ) = -lii-i v where  m  A = D /k max  =  + l )N (i+.l )- a ( i +2 ) .N . ( i +2 )-... -a(k)H (k) § § a  interval size  D = maximum size :.max k = number o f groups a ( i ) = Schwartz-Saltikov c o e f f i c i e n t s  Table V  Shwartz-Saltikov Coefficients  1  a(l)  J=l J=2 J=3 J=4 J=5 J=6 J=7 J-8  1  o(2)  a(6)  a(7)  a(8)  a(3)  a(4)  0.1547  0.036  0.013  0.0061  0.0033  0.0020  0.009  0.5774  0.1529  0.042  0.0171  0.0087  0.0051  0.0031  0.4472  0.1382  0.0408 : 0.0178  0.0093  0.0057  0.3779  0.1260  0.0386  0.0174  0.0095  0.3333  0.1161  0.0366  0.0168  0.3015  0.1081  0.0346  0.2773  0.1016  a(5).  : 0.2582  57  For t h e d i s t r i b u t i o n o f p a r t i c l e s o b t a i n e d i n t h i s e x p e r i ment D  = 36 um. in d  The i n t e r v a l s i z e r e g r o u p i n g t h e  raw  A  d a t a i s A= 4 ym.  T h e r e f o r e the number o f groups  i s k = 8.  The c o e f f i c i e n t s a ( i ) a r e g i v e n i n T a b l e V. For comparison r e a r r a n g e d because  purposes the apparent d i s t r i b u t i o n i s the S c h w a r z - S a l t i k o v method r e q u i r e s a  d i s t r i b u t i o n of s i z e s beginning with zero. T h i s new g r o u p i n g as w e l l as t h e r e a l  distribution  a r e g i v e n i n T a b l e s VI (a) and (b) r e s p e c t i v e l y and t r a t e d i n F i g . 21 (a) and ( b ) .  illus-  I t can be n o t e d t h a t t h e r e  i s no s i g n i f i c a n t c h a n g e s i n t h e d i s t r i b u t i o n e x c e p t t h a t p a r t i c l e s s m a l l e r than 4 ym a r e not p r e s e n t i n t h i s sample and a l s o t h a t t h e number o f p a r t i c l e s i n t h e m a t r i x s t a n t i a l l y i n c r e a s e s with r e s p e c t to the apparent bution for smaller sizes.  subdistri-  T h i s can be seen i n F i g . 22  where the p e r c e n t o f p a r t i c l e s i n the m a t r i x f o r b o t h distributions  size  is plotted.  Bb 2% Sb D e n d r i t i c I n t e r f a c e - Sample  V-2  A l o n g i t u d i n a l s e c t i o n of a d e n d r i t i c a l 1 y s o l i d i f i e d and quenched sample which has been p o l i s h e d and e t c h e d i s shown i n F i g . 23.  The quenched m a t e r i a l i n t h e upper p a r t  o f t h e f i g u r e has a f i n e r s t r u c t u r e than the l o w e r p a r t as  TABLE VI  Particle Size Distribution and % in the Matrix f o r Sample V - l (a) Apparent Distribution (b) Real Distribution f o r Total Counts. (a) Apparent Distribution Size, ym  Segregated;-;  1 Matrix';••-<.; % Matrix  (b) Real D i s t r i b u t i o n * Segregated;-  . .Matrix  % Matrix  0-4  21  7  25.0  0  0  4-8  123  43  25.9  7.2  3.9  35.0  8-12  190  48  20.2  11.4  3.9  25.4  12-16  213  32  13.1  13.1  1.9  12:6  16-20  185  33  15.1  •12.1  2.5  17.1  20-24  96  7  6.8  5.9  0.4  7.0  24-28  42  3  6.7  2.3  0.2  8.1  28-32  22  0  0.0  1.4  0.0  0.0  *Arbitrary Volume  200-  —  intercellular  —  in m a t r i x  100-  0 0  I  8  16  24  Size  (a)  50-  a  QL  32 fjLm  —  intercellular  —  in m a t r i x  30  10+ 0  0  i  8  r  16 (b)  1  i  i i  1  24  i  Size  32 /JLvr\  F i g u r e 21. a)Apparent and b ) R e a l d i s t r i b u t i o n of p a r t i c l e s f o r sample V-1.The d i s t r i b u t i o n s a r e s i m i l a r except f o r the f a c t t h a t p a r t i c l e s s m a l l e r than L- yO- m a r e not p r e s e n t i n the r e a l d i s t r i b u t i o n . I n b) the volume i s a r b i t r a r y .  60  F i g u r e 22. % of p a r t i c l e s i n the m a t r i x v s . p a r t i c l e s i z e f o r the apparent and r e a l d i s t r i b u t i o n of p a r t i c l e s . T h e e f f e c t f o r s m a l l p a r t i c l e s i s remarked i n the r e a l d i s t r i b u t i o n .  61  expected.  The p a r t i c l e s  interdendritic  region.  are the white spots situated  T r a n s v e r s e s e c t i o n s o f t h e sample  cut a t and behind t h e i n t e r f a c e particles  at i n t e r d e n d r i t i c function  a r e shown i n F i g . 24.  a p p e a r w h i t e and a r e a t i n t e r d e n d r i t i c  The n u m b e r o f p a r t i c l e s  o b s e r v e d on t h e t r a n s v e r s e  i n F i g . 25.  with the corresponding p a r t i c l e  Comparing distribution in the  c a s e i s t h a t t h e a v e r a g e number o f p a r t i c l e s i n  the i n t e r d e n d r i t i c  being segregated to  region.  Applying the Schwarz-Saltikov correction particle sizes  real d i s t r i b u t i o n plotted  are in the  ( F i g . 19) t h e o n l y d i f f e r e n c e  t h e m a t r i x i s l o w e r , more p a r t i c l e s  observed  shows  The t h r e e columns i n T a b l e V I I a r e  combined i n T a b l e V I I I and p l o t t e d  for a c e l l u l a r interface  V-2-1  The t a b l e  a high proportion of the p a r t i c l e s of a l l sizes region.  Section  V - 2 - l a i s 200 um f r o m  and V-2-2 i s 2.5 mm b e h i n d t h e i n t e r f a c e .  dendritic  section  ( s e g r e g a t e d ) and i n t h e m a t r i x as a  V-2-1 i s a t t h e i n t e r f a c e ,  these results  The  positions.  of p a r t i c l e size i s given in Table VII.  interdendritic  in the  for dendritic  growth g i v e s the  g i v e n i n T a b l e IX ( b ) . T h e d a t a i s a l s o  i n F i g . 27j.  Comparing:; F i g . 27 a , t h e a p p a r e n t  d i s t r i b u t i o n , w i t h b, t h e r e a l d i s t r i b u t i o n , two d i s t r i b u t i o n s  to the  are similar.  However w i t h t h i s  ment f o r t h e s i z e i n t e r v a l , t h e d i s t r i b u t i o n does n o t c l e a r l y f o l l o w  shows t h e arrange-  in the;matrix  t h e same p a t t e r n o b s e r v e d i n  62  F i g u r e 23. L o n g i t u d i n a l v i e w o f sample V-2 a t t h e i n t e r f a c e . T h e sharp change i n s t r u c t u r e i s due t o t h e q u e n c h i n g proeess.Etched. 100X.  F i g u r e 24. T r a n s v e r s a l views of sample V-2 ( d e n d r i t i c i n t e r f a c e ) . The b r i g h t c i r c l e s a r e the p a r t i c l e s . T h e p a r t i c l e s a r e i n the i n t e r d e n d r i t i c r e g i o n s , a),b) and c) 5 0 0 X . E t c h e d .  TABLE VII  Raw Particle Size Distribution f o r Segregated and Matrix Particles f o r Sample  1  Section res i z e , um  Segregated  '  V-2-la  V-2-11  Matrix  Segregated  Matrix  V-2-•2 Segregated  Matrix  4 6 8 10 12 14 16 18 20 22 24 26 28 30 32  8 20 31 49 33 44 '. 38 : 24 38 : 19 : 14 9 7 0 1  1 4 1 10 5 5 0. 2 2 0 0 0 0 0 0  8 18 35 52 47 50 45 50 56 32 26 17 9 16 4  1 3 5 13 4 3 3 0 8 7 2 0 0 0 0  7 18 27 47 36 45 34 32 34 19 12 7 3 2 0  0 2 2 10 5 12 7 3 2 1 0 0 0 0 0  TOTAL  334  30  465  49  323  44  65  TABLE VIII  Particle Size D i s t r i b u t i o n and % of P a r t i c l e s in the Matrix for Sample V-2. Counts on a l l sections V-2: (l+la+2) Size, um Segregated  .. Matrix •.:  % Matrix  3-7  79  11  12.2  7-11  241  41  14.5  11-15  255  34  11.76  15-19  223  15  6.3  19-23  198  20  9.17  23-27  85  2  2.3  27-31  37  0  0  31-35  6  0  0  TOTAL  1122  123  9.87  66  300  —  interdendritic  —  in  if 200 o CL  o  matrix Uj= 1247  00-  d  0  i  i  1  1  -!—- =i— II  27  S i z e , /JL m  35  F i g u r e 25. P a r t i c l e s i z e d i s t r i b u t i o n f o r sample W2 (.dendritic i n t e r f a c e ) f o r p a r t i c l e s i n the m a t r i x and i n i n t e r d e n d r i t i c r e g i o n s . L e s s p a r t i c l e s than f o r sample V - l ( c e l l u l a r interface) a r e p r e s e n t i n the m a t r i x .  F i g u r e 26. % of p a r t i c l e s i n the m a t r i x v s . p a r t i c l e s i z e f o r sample V-2 ( d e n d r i t i c i n t e r f a c e ) . T h e same p a t t e r n as i n sample V-1 ( c e l l u l a r i n t e r f a c e ) i s observed a l t h o u g h insample V-2 i s l e s s pronounced.  TABLE IX  Particle Size Distribution for Segregated and Matrix P a r t i c l e s and % in the Matrix, (a) Apparent Distribution  (b) Real Distribution.  Counts on a l l Sections  V-2-(l- la-2)  i  (a) Apparent Distribution Size, utT Segregated..  Matrix % Matrix  1  (b) Real Distribution Segregated.,  Matrix  % Matrix  QT4  23  2  8.0  0  0  -  4^8  149  17  10.2  7.4  0.6  7.9  8-12  264  37  12.3  17.6  2.9  14.1  12-1.6  256  30  10.5  15.3  2.2  12.5  16-20  234  17  6.8  15.4  1.1  6.8  20-24  233  10  7.5  7.6  0.7  9.0  24-28  52  0  0  3.0  0  0  28-32  23  0  0  1.5  0  0  co  69  300  —  interdendriti c  —  in m a t r i x  J200o Q.  H—  ioo4  o d  i  oo  8  1  "L__.  '3-  24 32 Size , /JLm  16  (a)  70+  ^  —  inte r d e n d r i t i c  —  in m a t r i x  50+  o Q.  °  b  30-  10+ 00  ri • i  8  16  (b)  24 32 Size, [JL.™  F i g u r e 27. a)Apparent an b ) R e a l d i s t r i b u t i o n o f p a r t i c l e s f o r sample V - 2 ( d e n d r i t i c i n t e r f a c e ) . P a r t i c l e s s m a l l e r than 4/U. m a r e not p r e s e n t i n the r e a l d i s t r i b u t i o n . I n b) the volume i s a r b i t r a r y .  70  F i g . 26 and  the d i s t r i b u t i o n o f s i z e s i n the m a t r i x a p p e a r s  to be random as i t can be seen i n T a b l e IX (a) and ii) containing  F i g . 28.  at  0.5  Sb mould _3  a t a growth v e l o c i t y o f  17x10  o f t h i s sample a r e shown i n  As b e f o r e most of the p a r t i c l e s a r e f o u n d i n The  number and  Section  H-l-1  um f r o m s e c t i o n  the r e g r o u p e d d a t a  i s a t the i n t e r f a c e , H - l - l a i s  H - l - 1 , H-l-2  i n t e r f a c e and H - l - 2 a i s a t 200  i s a t 2.5  um from H - l - 2 .  o f the d i s t r i b u t i o n i s shown i n F i g . 29. p a r t i c l e s i n the m a t r i x as a f u n c t i o n shown i n F i g . 30.  the  p o s i t i o n of the p a r t i c l e s as a  o f s i z e i s g i v e n i n T a b l e X and  i n T a b l e XI. at 200  rps and  An a l l o y Pb 1.5%  grown i n an h o r i z o n t a l  Transverse sections  cell walls. function  Growth.  Fe p a r t i c l e s was  rotating cm.s"^.  Horizontal  (b).  The  mm  from  the  A histogram percent of  of p a r t i c l e s i z e i s  In t h i s f i g u r e t h e r e i s no c l e a r  trend  o f m a t r i x p a r t i c l e s w i t h p a r t i c l e s i z e , d i f f e r i n g from vertical solidification  results.  Many o f the p a r t i c l e s were f o u n d to be s i t u a t e d outside surface wall.  the  of the sample a d j a c e n t to the  at  the  container  T h i s i s shown i n the l e f t hand s i d e o f F i g . 28  (b)  where a h i g h c o n c e n t r a t i o n o f w h i t e p a r t i c l e s . i s p r e s e n t at the edge o f the  sample.  71  (b)  F i g u r e 28. T r a n s v e r s a l views of sample H - l . a ) A s e c t i o n at the i n t e r f a c e showing p a r t i c l e s i n the c e l l w a l l s . 5 0 0 X . E t c h e d . b ) A s e c t i o n a t the i n t e r f a c e showing p a r t i c l e s near the w a l l of the container.200X.Etched.  TABLE X •Section -> Size, um  Raw Particle Size Distribution f o r Sample H-l. H-l- 1 Segregated Matrix  H-l- la Segregated ! Matrix  H-l- 2 Segregated Matrix  H-l--2a Segregated Matrix  2 4 6 8 10 12 14 16 18 20 22 : 24 26 : 28 : 30  2 4 15 20 12 16 22 22 25 23 11 8 8 3 0  1 5 2 10 6 6 2 4 1 1 3 0 0 .: 0 0  0 7 10 10 20 13 16 15 13 10 4 7 3 3 1  0 2 1 6 2 3 4 6 0 2 1 1 0 0 0  0 4 11 18 23 21 25 25 13 22 10 9 7 5 1  0 0 1 5 4 5 7 3 0 3 2 2 1 : 0 0  1 7 16 16 23 24 28 33 13 23 12 7 12 7 0  TOTAL  191  41  132  28  194  33  222  -  0 1 3 1 5 6 7 0 5 2 1 2 3 0 0 36  73  TABLE XI  P a r t i c l e S i z e D i s t r i b u t i o n f o r S e g r e g a t e d and M a t r i x P a r t i c l e s , and % i n t h e M a t r i x f o r Sample H - l . C o u n t s on a l l S e c t i o n s .  S i z e , ym  . Segregated  Matrix  % in Matrix  3-7  74  14  16  7-11  142  35  19.8  11-15  165  44  21 .0  15-19  159  20  11.2  19-23  115  13  10.15  23-27  61  12  16.4  27-31  20  0  0  31-35  0  0  0  736  138  TOTAL  15.8  74  — interce 11 u I a r —  in m a t r i x n  160+  T  = 87 7  to  ~I20o Q.  80+  r  40+  i 1  0:  m  i i i i 19  +  27 Size ,  m  35  F i g u r e 29. P a r t i c l e s i z e d i s t r i b u t i o n f o r sample H - l f o r p a r t i c l e s at the c e l l w a l l s and p a r t i c l e s i n the m a t r i x .  75  O  E  II  19  27 Size ,  35 fjLrn  F i g u r e 30. % of p a r t i c l e s i n the m a t r i x v s . p a r t i c l e s i z e f o r sample H-l.The d i s t r i b u t i o n i s random.  76  b) - P l a n a r  Interface  S e v e r a l samples o f p u r e l e a d (99.999% p u r i t y ) c o n t a i n i n g f e p a r t i c l e s were gfown v e r t i c a l l y and h o r i z o n tally.  The p r e s e n c e  o f a p l a n a r i n t e r f a c e d u r i n g growth  can be e s t a b l i s h e d u s i n g t h e c r i t e r i o n 6/R-(m C /D)(l-k )k »» o o o f o r c o n s t i t u t i o n a l super c o o l i n g . is the l i q u i d u s slope, C  Q  In t h i s e x p r e s s i o n m  i s the concentration of solute,  D i s t h e d i f f u s i o n c o e f f i c i e n t o f s o l u t e atoms i n t h e m e l t and k  Q  i s t h e p a r t i t i o n c o e f f i c i e n t . I f G/R i s l e s s than  the r i g h t hand p a r t o f t h e e q u a t i o n , no c o n s t i t u t i o n a l s u p e r c o o l i n g i s p r e s e n t and t h e i n t e r f a c e i s p l a n a r .  The  measured t e m p e r a t u r e g r a d i e n t i n t h i s group o f e x p e r i m e n t s was 30°e cm -1 and t h e growth v e l o c i t y was l e s s than 1.5x10 -4 5 cm - 2. s e c cm.s -1 . Under t h e s e c o n d i t i o n s G/R i s 2x10 °C o b t a i n e d f o r l e a d w i t h 10 -2 wt % Sn a l l o y . 40 E x a m i n a t i o n o f the quenched samples showed no a c c u m u l a t i o n  of p a r t i c l e s  ahead o f t h e quenched i n t e r f a c e , and no d i f f e r e n c e i n t h e number o f p a r t i c l e s o f a l l s i z e s was e v i d e n t between t h e grown and quenched m a t e r i a l . horizontal  T h i s a p p l i e d t o both v e r t i c a l and  solidification.  From t h e t h e o r i e s o f p a r t i c l e r e j e c t i o n d e s c r i b e d  pre-  viously, predictions of c r i t i c a l v e l o c i t i e s f o r the t r a n s i t i o n from t r a p p i n g t o r e j e c t i o n a r e g i v e n .  The B o i l i n g  and C i s s e t h e o r y ( f o r m u l a 1 8 ( a ) ) g i v e s f o r l e a d 2 3 V^R J  =  -18 5 -? 4.71x10 '° cm s e c 0  c  ...32  77  TABLE XII 1  C r i t i c a l Velocities Predicted by the Theories C r i t i c a l Velocities , V  R, um  Boiling and Cisse Theory  cm sec" XlO 1  Chernov e t a l .  I  21.67  26.94  5  1.94  3.14  10  0.69  1.25  4  Theory  78 in which v = 33.3  the f o l l o w i n g v a l u e s f o r the c o n s t a n t s were  0.34,  used:  k = 1.38xl0" e r g / d e g " , T = 6 0 0 ° K , <? = -2 41 * o e r g cm , a = 3.5 A , n = 1.67 mPa.s. ( T h i s l a s t 1 6  1  ls  :  Q  two w e r e t a k e n f r o m M e t a l s H a n d b o o k V o l . l i , 9 t h Ed.) Chernov  The  theory gives the f o l l o w i n g e x p r e s s i o n f o r the  critical  v e l o c i t i e s f o r small p a r t i c l e s i nlead (Equation V  C  R  4  /  3  =  1  -  2  5  X  1  0  "  8  C  M  7  /  3  S  E  C  _  • • •  1  3  27).  3  To o b t a i n t h i s e q u a t i o n t h e same v a l u e s o f n and a . Is  -1 4 w e r e u s e d and  was t a k e n a s 10  v e l o c i t i e s versus shown i n T a b l e The a b s e n c e  The  r a d i u s g i v e n by E q u a t i o n s  critical 32 a n d 33 a r e  XII. o f pushing a t growth v e l o c i t i e s p r e d i c t e d  by t h e t h e o r i e s s u g g e s t s of  erg.  t h a t , r e g a r d l e s s o f the e x i s t e n c e  any f o r c e t h a t c a n l e a d t o t h i s k i n d o f r e j e c t i o n ,  segregation o f particles to interstructural governed  by a mechanism i n which  positions i s  t h e b u o y a n c y f o r c e and  c o n v e c t i o n in the l i q u i d are the d r i v i n g f o r c e s . o t h e r hand t h e s e e x p e r i m e n t a l  Ewing  On t h e  r e s u l t s show t h e l i m i t a t i o n o f  these t h e o r i e s in p r e d i c t i n g p a r t i c l e  Note:  the  reported experimental  pushing.  and t h e o r e t i c a l v a l u e s -2 o f a - | w h i c h a r e 55 and 53 e r g cm r e s p e c t i v e l y . The l o w e r v a l u e g i v e n i n r e f 41 was a d o p t e d i n o r d e r t o c a l c u l a t e the lower v e l o c i t i e s . S  79  2.  The M a t e r - N y l o n Sphere Model The p u r p o s e o f s t u d y i n g t h i s model i s t o v i s u a l i z e t h e  motion of the p a r t i c l e s at a c e l l u l a r i n t e r f a c e i n order to d e v e l o p t h e mechanism t h a t i s p r o p o s e d i n t h e n e x t s e c t i o n s . a)  H o r i z o n t a l Mode i)  The M o t i o n o f t h e Nylon S p h e r e s .  The d e n s i t y  of t h e w a t e r was a d j u s t e d by a d d i n g s a l t (NaCT) t o a v a l u e t h a t gave t h e s p h e r e s a t e r m i n a l v v e l o c i t y o f 1 cm sec"''. Comparing t h i s v e l o c i t y w i t h t h e c e l l s i z e , t h i s c o r r e s p o n d s to the s m a l l p a r t i c l e s i n t h e m e t a l l i c s y s t e m which a d i s t a n c e of a c e l l diameter per second.  travel  R e l a t i n g the  s p h e r e t e r m i n a l v e l o c i t y o f 1 cm s e c " ' w i t h t h e d i a m e t e r o f the  c y l i n d e r s e l e c t e d , t h i s corresponds to a p a r t i c l e of  a p p r o x i m a t e l y 16 um i n ^ r a d i u s t r a v e r s i n g t h e d i a m e t e r o f the  t u b e (11 mm)  i n 10 s e c o n d s .  The most i m p o r t a n t  m o d e l l i n g f a c t o r i s the r e l a t i o n of the f i n a l v e l o c i t y of the  spheres  v a l u e o f Vjp  t o the a n g u l a r v e l o c i t y w'. and v a r y i n g w',  For a f i x e d  i t i s p o s s i b l e to o b t a i n a s i t u a -  t i o n i n w h i c h t h e s p h e r e s t r a v e l the d e s i r e d number o f c e l l d i a m e t e r s p e r r e v o l u t i o n and can be compared t o t h e r e a l m e t a l l i c system. the  The f i r s t s t e p i s t o s t u d y t h e m o t i o n o f  spheres in t h i s f l u i d  in r o t a t i o n a l motion.  T h i s was  done i n a r a n g e o f p e r i o d s o f r o t a t i o n between 30 s e c . to 0.5 s e c . p e r c y c l e .  80  After a transient 42 as e x p e c t e d .  period  the l i q u i d r o t a t e d  as a s o l i d  The i n i t i a l t r a n s i e n t was o b s e r v e d t o depend  on the r o t a t i o n speed and i t was f o u n d t o be l o n g e r f o r low v e l o c i t i e s , i . e . the t r a n s i e n t N e g l e c t i n g the c e n t r i p e t a l  f o r c e , the t r a j e c t o r y of the  n y l o n s p h e r e s may be p r e d i c t e d principle.  I f the d e n s i t y  to to!.  time i s p r o p o r t i o n a l  by a p p l y i n g  the  superposition  o f l i q u i d and s p h e r e a r e t h e same,  the t r a j e c t o r y s h o u l d be c i r c u l a r and the v e l o c i t y to t h e v e l o c i t y o f t h e l i q u i d .  identical  In t h i s c a s e the d e n s i t y  the s p h e r e s i s l o w e r t h a n t h a t o f t h e l i q u i d ;  of  therefore  the v e l o c i t y due t o b u o y a n c y f o r c e s has a v e r t i c a l component t h a t d i s t o r t s the c i r c u l a r p a t h o f t h e s p h e r e s .  The  e c c e n t r i c i t y o f t h i s d i s t o r t e d c i r c l e depends on t h e r a t i o of t h e t e r m i n a l  v e l o c i t y to the f r e q u e n c y of r o t a t i o n  -V.j/to'  .  For h i g h w' t h e path i s a l m o s t c i r c u l a r , and f o r low co' t h e p a t h i s e l o n g a t e d i n t h e v e r t i c a l d i r e c t i o n , and w i t h l e s s c u r v a t u r e i n the r e g i o n  i n which t h e v e l o c i t y due t o buoyancy  f o r c e has t h e same d i r e c t i o n a>s t h e t a n g e n t i a l ;  velocity  due t o r o t a t i o n . The t y p i c a l p a t h s and d i s t r i b u t i o n o f s p h e r e s i n t h e c y l i n d e r f o r v a r i o u s v a l u e s o f co' a r e shown i n t h e of p h o t o g r a p h s  i n F i g u r e 31.  sequence  P h o t o g r a p h s a t o e were t a k e n  from one end o f the r o t a t i n g c y l i n d e r ; the w h i t e i n the p h o t o g r a p h s a r e t h e n y l o n s p h e r e s .  particles  F i v e speeds o f  81  rotation rps  a r e shown u = 0.033 r p s ( a ) , 0.083 r p s ( b ) , 0.2  ( c ) , 0.5 r p s ( d ) a n d 1 r p s ( e ) .  At t h e l o w e s t r o t a t i o n a l  s p e e d 0.033 r p s t h e  v e l o c i t y of the l i q u i d at the cylinder order of the f l o t a t i o n v e l o c i t y . positioned  at the radius  - 1  = 1 cm s e c  terminal  .  t h e l i q u i d moved a t t h e t u b e w a l l  h a v i n g t h e same t e r m i n a l  - 1  = 3.45 cm s e c . - 1  velocity  That i s , at that  in lead  a t a speed An i r o n  particle  s h o u l d be 177 ym i n  f r e q u e n c y o f 0. 5 r p s t i h n s ' k i n d  movement d o e s n o t e x i s t i n t h e m e t a l l i c p a r t i c l e s used were t e n t i m e s  vertical  to the  s p e e d o f 0.5 r p s u s e d i n t h e s o l i d i f i c a -  V.p-| = 6.28x1.1 cmx0.5 s e e  As  respect  - 6.28x4.75  2Trrcu'  1  At a r o t a t i o n  radius.  is V^=  = 0.995 cm s e c " , v e r y c l o s e  v e l o c i t y of the spheres  tion of lead  As a r e s u l t t h e s p h e r e s a r e  t h i s i s shown i n F i g . 3 1 ( a ) .  f l u i d v e l o c i t y at the wall  cm 0.033 s e c  i s of the  and e s s e n t i a l l y a t r e s t w i t h  to t h e system o f r e f e r e n c e , The  radius  tangential  of  system because the  smaller.  t h e f r e q u e n c y w' i n c r e a s e s  the time allowed f o r  displacement decreases.  T h e s p h e r e s do n o t h a v e  time to reach the c y l i n d e r  walls  i n each r o t a t i o n  spheres remain i n the l i q u i d moving i n l o o p s . at a r o t a t i o n a l  and the  This  happens  s p e e d o f 0.5 r p s , t h e d i s t r i b u t i o n o f  82  83  84  F i g u r e 31. D i s t r i b u t i o n of spheres i n the p h y s i c a l model f o r t h e h o r i z o n t a l mode a t d i f f e r e n t r o t a t i o n speeds. a ) 0 , 0 3 3 r p s . b ) 0 . 0 8 3 r p s c ) 0 . 2 r p s , d ) 0 . 5 r p s . e ) l r p s . T h e p a r t i c l e s f o r low speeds a r e c o n c e n t r a t e d i n the w a l l , a n d f o r h i g h speeds they a r e c o n c e n t r a t e d i n the c e n t e r of the c o n t a i n e r as e x p l a i n e d i n t h e t e x t .  85  (b) F i g u r e 32. The two c l a s s e s of p a t t e r n s f o l l o w e d by the spheres i n the model as a f u n c t i o n of r o t a t i o n speed.In a) the spheres r e a c h the w a l l of the c o n t a i n e r and i n b) the spheres do not r e a c h  86  s p h e r e s i n t h e l i q u i d i s shown i n F i g . 3 1 ( d ) .  At v a l u e s o f  to' g r e a t e r than 0.5 r p s t h e c e n t r i p e t a l f o r c e s become i m p o r t a n t and t h e s p h e r e s t e n d t o c o n c e n t r a t e a l o n g t h e c y l i n d e r a x i s as shown i n f i g . 31(e) f o r to = 1 r p s . The s p h e r e s moving  i n t h e l i q u i d due t o r o t a t i o n and  g r a v i t y f o r c e s may be s e p a r a t e d i n t o two g r o u p s , one i n which the s p h e r e s r e a c h t h e c y l i n d e r w a l l i n one and a s e c o n d where t h e s p h e r e s do n o t . the  rotation,  In t h e f i r s t  group,  l o o p d i a m e t e r t a k e n by t h e s p h e r e s t e n d s t o expand w i t h  to' as shown s c h e m a t i c a l l y i n F i g . 3 2 ( a ) ; i n t h e s e c o n d group the l o o p d i a m e t e r s h r i n k s as shown i n F i g . 3 2 ( b ) .  The  l o o p s were o b s e r v e d t o change a l i t t l e w i t h f u r t h e r r o t a t i o n so t h a t i n some c a s e s s p h e r e s may t o u c h t h e ' w a l l even a t h i g h v a l u e s o f to' . ii) the  Modelling.  The p a r a m e t e r employed  t o model  s y s t e m i s t h e number o f c e l l s t r a v e l l e d by a p a r t i c l e  in the r e a l s y s t e m and by a s p h e r e i n t h e model.  This  q u a n t i t y i s ' e x p r e s s e d as N = —f C. to V  where <o  =  N • = -ff C . V  . . .32  r  ID  i s t h e t e r m i n a l v e l o c i t y , C i s t h e c e l l s i z e and  i s the f r e q u e n c y o f r o t a t i o n .  to a model v a r i a b l e .  The dashed terms  refer  In t h e a l l metal s y s t e m G and  f i x e d , C ~ 100 ym and to = 0.5 r p s .  In t h e model V  f  to a r e and C  87  are f i x e d , V  "  = 1 cm s e c  _i  i  and C ~ 2 cm.  = C.or; ' _ 10 -2 cm x 0.5 r p s f • >' ' f ' CO: 2 cm y 0.25 x TO" f v  This gives 1 cm.sec -1 co  ...33  2  =  v  CO  U s i n g the f o r m u l a f o r t e r m i n a l v e l o c i t y g i v e n by t h e Stokes  Law =  2a^MP  .78xl0" a 4  =  3  ...34  2  9-u where V.J- i s g i v e n i n cm s e c " ' and a ( t h e p a r t i c l e r a d i u s ) i n ym.  In t h i s c a l c u l a t i o n t h e f o l l o w i n g v a l u e s were used = 10.6 gr.cm - 3 , p = 7.7 gr cm - 3 , n = 1.67 m P a . s e c and g = -2 981 cm s e c . The S t o k e s t e r m i n a l v e l o c i t y g i v e s a c c u r a t e 42 p  v a l u e s when t h e R e y n o l d s number i s l e s s than one. R e y n o l d s number i s g i v e n by P*e =?-  p^dU/n,  where  i s the  d e n s i t y of the l i q u i d , d i s the p a r t i c l e diameter, t e r m i n a l v e l o c i t y and n i s the v i s c o s i t y .  The U i s the  S u b s t i t u t i n g the  a p p r o p r i a t e v a l u e s f o r l e a d g i v e s Re = 3.4x10  4  f o r t h e smal-  l e s t , p a r t i c l e . o f y.2 ym r a d i u s .and;Re .= 2.57x10"' f o r t h e l a r g e s t o f 17.5 ym r a d i u s p r e s e n t i n t h e m e l t . C o m b i n i n g E q u a t i o n s 34 w i t h 33 g i v e s 6.61 w  w i t h a i n um to  V  ... 35 a i s given i n r p s . Using t h i s equation the =  —2~  88  TABLE X I I I  C o r r e l a t i o n Table f o r P a r t i c l e Size i n the M e t a l l i c System V e r s u s F r e q u e n c y o f R o t a t i o n "11 i n t h e Model. a, ym  u>' =  6.61/a (rps)  2  i  T., s e c  2  1 .65  4  4.13-10"  1  2.42  6  1.84x10"  1  5.44  8  1.03xl0  - 1  10  6.6xl0~  2  15  2.94xl0"  2  34.0  17  2.29-10"  2  43.1  0.6  9.68 15.1  89  c o r r e l a t i o n of the p a r t i c l e  s i z e in the m e t a l l i c  system  i  with oi  is  l i s t e d in Table XI T I .  In the t a b l e ,  it  is  evident that the range of f r e q u e n c i e s from 0.5 to 0.033 rps used to study the motion of spheres covers the range of particle  s i z e in the metal system between 2 ym and 17 ym.  However, due to the wall e f f e c t  i t was necessary to  restrict  the observations in the model to those f r e q u e n c i e s f o r which the type of loops shown in F i g . 32(b)  are o b t a i n e d , that  f o r f r e q u e n c i e s higher than 0.2 r p s .  In the m e t a l l i c  t h i s corresponds to p a r t i c l e s In t h i s  respect, this  because outside t h i s of c e l l  system  s m a l l e r than 6 m in r a d i u s . u  i s the most i n t e r e s t i n g range of s i z e s range l a r g e r p a r t i c l e s  diameters per c y c l e are more l i k e l y  among the c e l l  is  travelling  tens  to be entrapped  tips.  For a given frequency <D  the axis of the c y l i n d e r was  t i l t e d at d i f f e r e n t angles to the h o r i z o n t a l in order to vary the p a r t i c l e  velocity  in the a x i a l  simulate an advancing f r o n t .  This a x i a l  d i r e c t i o n and component i s  given by V , = V' ph f where a i s the t i l t i n g values of v e l o c i t y ,  sin a angle.  . . . 36 Table XIV  gives i  the  as a f u n c t i o n of a f o r V^. = i  Note, from Table XIV,  _i  c m  sec" .  that the measured values are always  smaller than those c a l c u l a t e d using Equation 36.  Sometimes  90  TABLE XIV  T h e o r e t i c a l and E x p e r i m e n t a l  1  the Model f o r V  0  Velocities  o f T i l t i n g AngTe f o r oo = 0.5 s e c .oh  as a F u n c t i o n  a  Horizontal  = 1 cm/sec = p a r t i c l e  ph cm/mi n V  ( t h )  V (exp) cm/min p h  4..Q  4.2  1.4-1.0  2.6  2.7  1.14-0.9  1 .1  1 .2  0.75-0.69  velocity.  91  the measured values  are l e s s than h a l f  the c a l c u l a t e d  This may be accounted f o r by the large s i z e of the in the model.  In t h i s  case the average  spheres f o r a r o t a t i o n velocity  period i s  velocity  l e s s than the  due to a large t r a n s i e n t  spheres  of  the  terminal  period.  In the observations where the c y l i n d e r axis was greater  than 1° above the h o r i z o n t a l the spheres  the " i n t e r f a c e "  were observed to enter  between the c e l l s .  A typical  values.  result  i n t o the  tilted  reaching grooves  is shown in F i g . 33  where most of the nylon spheres are found " s e g r e g a t e d " and even p i l e d up in i n t e r c e l l u l a r  positions.  For t i l t  smaller than 1° the spheres tended to come out of channels with f u r t h e r  angles  the  r o t a t i o n and move to another  channel  temporarily.  From the observations and the a n a l y s i s the mechanism leading to i n t e r c e l l u l a r formulated with the a i d of F i g . respect  to a system of reference  interface"  position,  face"  t r a p p i n g can be and ( c ) .  f i x e d to a  With  "cellular  the spheres oscxllTate v e r t i c a l l y with a frequency  equal to the r o t a t i o n a l  fixed.  16(b)  of the model  frequency c o ,  and with an average  projected onto the " i n t e r f a c e " ,  that is  In a d d i t i o n the spheres move towards the at a constant average  axial  velocity.  nearly "inter-  The d i r e c t i o n  92  F i g u r e 33. The n y l o n s p h e r e s i n the grooves between the c e l l s  i n the  h o r i z o n t a l mode. Photograph taken from the o p p o s i t e s i d e of t h e "interface".  oJ  =0.5 r p s ;  = 4°.  F i g u r e 34. N y l o n spheres i n the grooves between the " c e l l s " i n t h e v e r t i c a l mode.Photograph t a k e n from the o p p o s i t e end t o the " i n t e r f a c e " , v = 1 cm.sec !. -  93  o f the a x i a l v e l o c i t y may  i n t e r s e c t the " i n t e r f a c e " at three  d i s t i n c t r e g i o n s : t h e g r o o v e s between t h e c e l l s , the f l a t portion  a t t h e t o p o f the c e l l , and t h e c u r v e d s u r f a c e j o i n -  i n g t h e f l a t r e g i o n to t h e g r o o v e s .  The i n t e r a c t i o n  of  the s p h e r e s w i t h t h e i n t e r f a c e depends on t h e a r e a s o f the surface i t approaches.  As a s p h e r e a p p r o a c h e s t h e s u r f a c e  i t has a v e r t i c a l v e l o c i t y V which changes t o a s m a l l e r v a l u e v i m m e d i a t e l y a f t e r an i n e l a s t i c c o l 1 i s i o n , as shown i n Fig. 16(c).  In t h e l e s s f a v o u r a b l e s i t u a t i o n t h e s p h e r e  r e a c h e s t h e v e l o c i t y V upward i m m e d i a t e l y .  In any c a s e due  to t h i s c o l l i s i o n the s p h e r e v e l o c i t y has d e c r e a s e d and t h e p a r t i c l e i s u n a b l e t o move o v e r t h e normal a m p l i t u d e o f oscillation.  In t h e n e x t s t e p , and due t o t h e r o t a t i o n  the c y l i n d e r , a f t e r half a r e v o l u t i o n  the v e l o c i t y w i t h r e s p e c t  to t h e c e l l t i p i s downward as shown by t h e v e c t o r downward i n F i g . 1 6 ( e ) .  of  oriented  In b o t h c a s e s , w i t h V up o r down, .  t h e r e i s a component o f v e l o c i t y p e r p e n d i c u l a r t o the " i n t e r f a c e " t h a t s i m u l a t e s t h e " i n t e r f a c e " movement. drives  the s p h e r e s i n t o t h e g r o o v e s .  scattered  This velocity  On t h e o t h e r hand the  v e l o c i t y v a l s o has a component w h i c h i s i n t h e  opposite direction. the g r o o v e s .  T h i s tends to d r i v e the spheres out of  When t h i s o u t w a r d component i s s m a l l e r than  the i n w a r d v e l o c i t y component, t h e s p h e r e w i l l t e n d t o e n t e r i n t o t h e g r o o v e s by r e p e a t i n g t h e same p r o c e s s each tion.  revolu-  94 The d i r e c t i o n and magnitude o f t h e v e l o c i t y v depends on many f a c t o r s .  T h r e e f a c t o r s a r e t h e most r e l e v a n t f o r  the model and t h e a l l metal s y s t e m .  The s c a t t e r i n g  d i r e c t i o n i s always a f u n c t i o n o f b o t h , t h e i n c i d e n t d i r e c t i o n and t h e p o s i t i o n on t h e t i p a t which occurs.  collision  The magnitude o f v i s a f u n c t i o n o f t h e i n e l a s t i c  or r e s t i t u t i o n c o e f f i c i e n t .  A f t e r c o l l i s i o n the magnitude  ->-  o f V a l s o d e c r e a s e s by the damping  due t o t h e v i s c o u s f o r c e .  These w i l l be c o n s i d e r e d i n more d e t a i l when t h e r e s u l t s o f t h e w a t e r model a r e compared  t o the metal  system.  In t h e w a t e r m o d e l , f o r s m a l l a n g l e s o f t i l t , t h e ->  component o f V t h a t compensates v i s small.  t h e outward component o f  In a d d i t i o n , as t h e p a r t i c l e i s e n t e r i n g i n t o  a g r o o v e by s u c c e s s i v e c o l l i s i o n s , t h e i n c i d e n t a n g l e , measured w i t h r e s p e c t t o t h e normal t o t h e t i p s u r f a c e a t the c o l l i s i o n p o i n t , d e c r e a s e s f r o m 90 d e g r e e s a t t h e f l a t surface to zero i n the groove.  T h e r e i s a r e g i o n i n between  where t h e c o l l i s i o n a n g l e i s 45 d e g r e e s .  At t h i s point  the s c a t t e r e d v e l o c i t y v has a maximum component i n t h e outward d i r e c t i o n .  T h e r e f o r e , when a s p h e r e r e a c h e s a  p o s i t i o n i n which t h e outward component o f v i s g r e a t e r than t h e i n w a r d o f V t h i s s p h e r e comes o u t a g a i n .  This i s  what i s o b s e r v e d f o r low t i l t i n g a n g l e s i n t h e w a t e r  model.  If the s t r i k i n g d i r e c t i o n c o i n c i d e s with a f l a t region  95  on t h e c e l l , a p a r t i c l e can s t i l l  reach a curved p o s i t i o n  or a g r o o v e because t h e a m p l i t u d e o f o s c i l l a t i o n i s o f the  <-ovder  of the c e l l  the  m e t a l s y s t e m f o r t h e s m a l l e s t Fe p a r t i c l e . iii)  d i a m e t e r i n t h e w a t e r model and i n  Comparison w i t h a S o l i d i f i c a t i o n P r o c e s s .  When c o m p a r i n g t h e model w i t h t h e r e a l s i t u a t i o n i n t h e m e t a l l i c s y s t e m t h e main n e g a t i v e a s p e c t o f t h e mechanism d e r i v e d f r o m t h e model the  i s t h a t i t has n o t c o n s i d e r e d t h a t  v e l o c i t y o f t h e i n t e r f a c e i n t h e model and i n t h e r e a l  s i t u a t i o n are d i f f e r e n t .  In t h e model t h e "growth"  v e l o c i t y i s t e n t i m e s f a s t e r than i n t h e m e t a l l i c a l l o y s . T h i s means t h a t i n t h e m e t a l s y s t e m t h e d r i v i n g f o r c e p u s h i n g t h e p a r t i c l e s i n t o t h e g r o o v e s i s much s m a l l e r a n d , at f i r s t , i t may be t h o u g h t i t i s u n a b l e t o compensate any outward component o f s c a t t e r e d v e l o c i t y v. the  However i n  r e a l p r o c e s s t h i s v e l o c i t y s h o u l d be s m a l l e r b e c a u s e  of t h e f o l l o w i n g t h r e e f a c t o r s (1) t h e p r e s e n c e o f d r a g f o r c e ( E q u a t i o n 1) p u s h i n g t h e p a r t i c l e a g a i n s t t h e i n t e r f a c e , ( 2 ) t h e r e s t i t u t i o n c o e f f i c i e n t i n t h e metal s y s t e m i s s m a l l e r t h a n i n t h e model the  and ( 3 ) i n t h e m e t a l  p a r t i c l e s a r e s e v e r a l o r d e r s o f magnitude  t h e r e f o r e t h e damping  system  s m a l l e r and  f o r c e i s much more i m p o r t a n t .  A q u a n t i t a t i v e e v a l u a t i o n o f the e f f e c t o f the drag force i s d i f f i c u l t  t o o b t a i n because i t i s i m p o r t a n t o n l y  96 d u r i n g t h e c o l l i s i o n t i m e when t h e p a r t i c l e i s v e r y to t h e i n t e r f a c e .  close  The r e s t i t u t i o n c o e f f i c i e n t f o r b o t h  model and metal s y s t e m has been d e t e r m i n e d u s i n g v /V , • n n where v  n  and  a r e t h e components o f r e s u l t a n t  and  incident  p a r t i c l e v e l o c i t y normal t o t h e s u r f a c e o n t o which t h e p a r t i cle collides.  The r e s u l t s o f t h e c a l c u l a t i o n s ,  detail b e l o w , g i v e s v / V n  given i n  = 0.88 f o r n y l o n s p h e r e s i m p i n g -  n  i n g on a l u c i t e s u r f a c e .  The c o r r e s p o n d i n g v a l u e f o r i r o n -2  spheres s t r i k i n g a lead surface i s / V V  n  n  -10  .  In l e a d  t h e r e f o r e t h e p a r t i c l e l o s e s 99% o f i t s v e l o c i t y when s t r i k i n g t h e l e a d s u r f a c e , whereas t h e n y l o n s p h e r e o n l y 12% a g a i n s t l u c i t e .  loses  The n e a r l y p l a s t i c b e h a v i o r o f  p a r t i c l e s i n l e a d d u r i n g c o l l i s i o n i s s i m i l a r t o t h e beh a v i o r o f l i q u i d p a r t i c l e s - c o l l i d i n g w i t h any s o l i d s u r f a c e . The  r e s t i t u t i o n c o e f f i c i e n t was c a l c u l a t e d  ing the energy l o s t a f t e r c o l l i s i o n .  by measur-  In f r e e f a l l i n g f r o m  a h e i g h t h^ a p a r t i c l e o f mass m g a i n s v e l o c i t y V . the c o n s e r v a t i o n o f e n e r g y , t h e s e q u a n t i t i e s in the following  Using  are related  way 1 m V2 2 n  mgh, • 1 =  T  ... 37  After c o l l i d i n g perpendicular to a planar i n t e r f a c e the resultant  velocity  to p o t e n t i a l  i s v , the k i n e t i c energy transforms n  and t h e p a r t i c l e r e a c h e s a maximum h e i g h t h  9  97 l e s s than h 1  -  A g a i n , by c o n s e r v a t i o n o f e n e r g y mgh  2  =  j  . . .38  m  D i v i d i n g E q u a t i o n s 37 and 38 and t a k i n g s q u a r e r o o t i t is obtained that  . . .39 With t h i s e x p r e s s i o n t h e r e s t i t u t i o n c o e f f i c i e n t c a n be d e t e r m i n e d by m e a s u r i n g t h e h e i g h t s h  2  a p a r t i c l e or sphere  r e b o u n d s a f t e r s t r i k i n g a s u r f a c e o r t h o g o n a l l y . Rebound h e i g h t s were measured f o r b o t h n y l o n s p h e r e s s t r i k i n g and Fe p a r t i c l e s s t r i k i n g l e a d .  ;  lucite  The n y l o n s p h e r e s were  d r o p p e d from h e i g h t s o f 10 and 20 cm a g a i n s t l u c i t e . F o r t h e c a s e o f i r o n a g a i n s t l e a d , s t e e l s p h e r e s 0.32, 0.4 and 0.48 cm were d r o p p e d from t h e same h e i g h t s as t h e n y l o n s p h e r e s against a polished lead surface. h /h^ 2  In t h i s c a s e t h e v a l u e o f  a p p e a r e d t o be i n d e p e n d e n t o f s p h e r e d i a m e t e r and  i n i t i a l height. The damping f o r c e on t h e Fe p a r t i c l e s r e s u l t s i n the p a r t i c l e s s t o p p i n g much f a s t e r and i n a s h o r t e r d i s tance than the nylon spheres.  T h i s d i s t a n c e and t i m e f o r  i r o n p a r t i c l e s i n l e a d c a n be a n a l y s e d i n t h e f o l l o w i n g way. A f t e r a c o l l i s i o n has o c c u r r e d t h e d y n a m i c s o f t h e  98  p a r t i c l e i s g o v e r n e d by the m ^  =  equation  6TTTI av"  ...  —  40  where n i s the v i s c o s i t y , a i s the p a r t i c l e r a d i u s , m i s 3 HP  the p a r t i c l e mass = 4 / 3 The  where p i s the p a r t i c l e  r i g h t hand s i d e o f the e q u a t i o n i s the S t o k e s  drag v a l i d f o r s m a l l p a r t i c l e s . v e l i c i t y v is horizontal act in t h i s  and  viscous  I t i s assumed t h a t  therefore  density the  g r a v i t y does  not  direction.  E q u a t i o n 40 has the s i m p l e  _ i 2 v  =  v.e  where v. i s the i n i t i a l  2  solution t  ...41  pa  v e l o c i t y and  the  quantity  2  T = TT -— 9  i s a c h a r a c t e r i s t i c t i m e at which the  velocity  n  d e c r e a s e s to 1/e  = 36.7%  Integrating  of i t s i n i t i a l  a g a i n E q u a t i o n 40,  value.  the d i s t a n c e  travelled  by the p a r t i c l e a f t e r c o l l i s i o n on t h i s T i s X  = x  v. §i 9  (1 n  h  e  In the l e s s f a v o u r a b l e s i t u a t i o n the s c a t t e r i n g p l e t e l y e l a s t i c and  . . .42 i s com-  v^ i s the v e l o c i t y j u s t b e f o r e c o l -  l i s i o n ; i . e . the t e r m i n a l  v e l o c i t y g i v e n by E q u a t i o n  34.  TABLE XV  Terminal Velocity, Transient Time, Transient Length f o r Fe P a r t i c l e s in Liquid Lead Assuming Stokes Drag Force Applies. i  1  Radius (pi)  V (cm/sec) =  3.78xl0 a -4  f  2  T(sec)  =  1.0246xl0" a 6  2  X  = 2.45xl0" a 10  T  4.1xl0"  0.39 A  2  : T.51xl0"  4  6.06xl0"  3  1.64xl0~  6  1.36xl0"  2  3.7xl0"  5  31.7 A°  8  2.42x10"  2  6.5xl0"  5  100.4 A°  10  3.78xl0"  2  l.OxlO"  4  245.2 A°  12  5.45xlQ  -2  1.5xl0"  4  508.4 A  14  7.42xl0"  2  2.0xl0"  4  16  9.69xl0~  2  2.6xl0"  4  0.16 ym  18  1.23X10"  3.3xlQ"  4  0.26 ym  1  3  6  5  0  6.27 A°  941:9  0  A°  4  100  Substituting  this  velocity  in Equation 42 gives 4  X  Calculating X  81  T  9  • n.2:  e  u  '* '  J  4  3  from Equation 43 f o r l i q u i d lead and iron  t  _2  p a r t i c l e s with g = 981 cm see , n = 1.67 mPa.s, p = 7.7 -3 -3 gr cm > P-J 10-. 3. gir cm , t h i s distance i s =  X  = 2.45xl0~ a 1 0  ...44  4  T  with a . i n ym X  i s given in cm.  This f u n c t i o n i s tabu-  T  lated  in Table XV together with the y v a l u e s .  for a particle  For  of radius a = 4 ym,' the' t r a n s i e n t  instance time and  _ o  length are  X'  results  may be concluded that once a p a r t i c l e  it  =  6.3x10"  i n t o the i n t e r c e l l u l a r previously  it  cm and x = 16.4 yisec.  The c o l l i s i o n stops the p a r t i c l e  b)  Vertical In t h i s  almost i n s t a n t l y  the small solution.  are  important than in h o r i z o n t a l  Changing the d e n s i t y of the s o l u t i o n by  l i m i t was very density  and at  mode of " s o l i d i f i c a t i o n " the r e s u l t s  of the nylon spheres were  in the range from 1 cm s e c "  This lower  to come out a g a i n .  Solidification  a d d i t i o n of NaP.lt the v e l o c i t i e s varied  for it  site.  more obvious but not l e s s "solidification".  moves  band by the mechanism d e s c r i b e d  i s very d i f f i c u l t  the same c o l l i s i o n  From these  difficult  difference  Any casual  1  to 0.05 cm s e c " . 1  to adjust because of  between nylon spheres and  p e r t u r b a t i o n d i s t u r b e d the system,  (bj  Particle  Velocity  F i g u r e 3 5 . S e m i - q u a l i t a t i v e d e s c r i p t i o n of the s e g r e g a t i o n o f n y l o n s p h e r e s f o r the v e r t i c a l e x p e r i m e n t s w i t h the m o d e l . I n a) the r e l a t i o n i n a r e a s i s c a l c u l a t e d . I n b) the b e h a v i o u r o f the s p h e r e s f o r s t i l l a n d c o n v e c t i v e l i q u i d are compared w i t h the e x p e r i m e n t a l observations.  102  t h e f l o w p a t t e r n made t h e m e a s u r e m e n t o f t h e v e l o c i t y d i f ficult.  A typical  experiments grooves. 35.  d i s t r i b u t i o n of spheres  i s shown i n F i g . 34.  obtained i n these  The s p h e r e s  are fn the 1  The r e s u l t s a r e q u a l i t a t i v e l y p r e s e n t e d  To u n d e r s t a n d  them,two o p p o s i n g  For a l i q u i d a t r e s t i t i s r e a s o n a b l e  inFig.  s i t u a t i o n s are analysed. t o assume t h a t t h e  p a r t i c l e d i s t r i b u t i o n , d i s c r i m i n a t i n g o n l y between the i n t e r c e l l u l a r grooves  and c e l l  faces, will  be d i c t a t e d by  the r e l a t i o n between t h e groove and c u r v e d p a r t o f t h e t o p s u r f a c e a r e a s , and p a r t i c l e approaching  face surface areas. a f l a t area w i l l  coming to a curve p a r t o f a c e l l  dimensions still  stop there while  or a channel  end up a t t h i s p o i n t on t h e s u r f a c e . faced area of the c e l l  In o t h e r w o r d s , a  finally  The p r o p o r t i o n o f f l a t  to t o t a l area, according to the  o f t h e m o d e l , i s 0.37.  T h i s means t h a t i n a  l i q u i d the percent of p a r t i c l e s found  s h o u l d be 3 7 % .  will  those  T h i s was o b s e r v e d  i n the c e l l s  e x p e r i m e n t a l l y by r e l e a s -  ing p a r t i c l e s at the bottom of the c y l i n d e r .  The i m p o r t a n c e  o f c o n v e c t i o n was d e m o n s t r a t e d by i n -  t r o d u c i n g a very slow c o n v e c t i v e motion i n the l i q u i d a small temperature  g r a d i e n t when t h e s p h e r e s  against the l u c i t e c e l l placed the spheres  surface.  with  were r e s t i n g  The s m a l l c u r r e n t  dis-  to p o s i t i o n s where t h e g r a v i t y f o r c e  c o u l d move t h e s p h e r e s  i n t o the grooves.  The r e s u l t s o f  103  a d e t a i l e d s t u d y o f the e f f e c t o f c o n v e c t i o n on the t i o n o f the n y l o n s p h e r e s  i n the model c a n n o t be e x t r a -  p o l a t e d t o the metal system f o r two main r e a s o n s . r e a s o n , which may  segregaOne  be t h e most i m p o r t a n t , i s r e l a t e d to the  s i z e o f t h e p a r t i c l e s and the b o u n d a r y l a y e r . -1 of the b o u n d a r y l a y e r i s o f the o r d e r o f 10  The  thickness -3  to 10  cm  which i s a l w a y s s m a l l e r than t h e s p h e r e s i z e i n the model. The s p h e r e s w i l l a l w a y s be a f f e c t e d by c o n v e c t i o n i n the liquid.  In the metal system the p a r t i c l e s a r e o f the o r d e r  of the t h i c k n e s s o f the b o u n d a r y l a y e r . f l o w c o n d i t i o n s t h e r e may by c o n v e c t i o n .  D e p e n d i n g on the  be p a r t i c l e s t h a t a r e n o t a f f e c t e d  The e s t i m a t i o n o f the t h i c k n e s s o f the  b o u n d a r y l a y e r f o r the metal s y s t e m  is d i f f i c u l t .  e x p r e s s i o n f o r t h i s l e n g t h i s g i v e n as 6^ = 1.7 for a boundary l a y e r developed  The  (^p)'  2  42  o v e r a f l a t s u r f a c e or a body  i n a f l u i d o f k i n e m a t i c v i s c o s i t y v moving at v e l o c i t y U. - 3  F o r l e a d v = 1.57x10  cgs  u n i t s and U may  from the p e r i o d i c v a r i a t i o n i n t e m p e r a t u r e the v e r t i c a l growths as - 1 cm s e c " ' . f a c e i t may  be  estimated  recorded  during  For a p l a n a r i n t e r -  be assumed t h a t t h e b o u n d a r y l a y e r d e v e l o p s  a l o n g the d i a m e t e r o f the mould (~1 cm); t h i s g i v e s a maximum v a l u e o f 680 ym f o r the t h i c k n e s s w i t h i n w h i c h the p a r t i c l e s a r e n o t a f f e c t e d by c o n v e c t i o n . i n t e r f a c e t h e b o u n d a r y l a y e r may the c e l l d i a m e t e r X~10  cm,  For a c e l l u l a r  be assumed to d e v e l o p  i n t h i s c a s e S, = 68  ym.  over  104 However, t h i s t h i c k n e s s may be s m a l l e r b e c a u s e t h e c e l l t i p i s c u r v e d and as a r e s u l t t h e f l a t p a r t o f t h e t i p i s s m a l l e r than 10  cm.  In any e v e n t i t i s e x p e c t e d  that  some p a r t i c l e s w i l l be a f f e c t e d by c o n v e c t i o n w h i l e w i l l not  depending  others  on t h e p a r t i c l e s i z e and i t s p o s i t i o n  in the t i p . The s e c o n d r e a s o n p r e v e n t i n g a d i r e c t c o m p a r i s o n the model t o t h e metal  system  of  i s the d i f f e r e n c e i n the  P r a n d t l number between t h e two s y s t e m s .  For a given d r i v i n g  f o r c e f o r t h e r m a l c o n v e c t i o n t h e f l o w o b t a i n e d i n each  case  differs s u b s t a n t i a l l y . ^ " ^ With t h e t e c h n i q u e o f t u r n i n g t h e c y l i n d e r w i t h t h e spheres  i n i t i a l l y a t the t o p , the d i s t r i b u t i o n o f spheres  as a f u n c t i o n o f v e l o c i t y i s g i v e n by t h e f u l l l i n e i n Fig. 35(b). still  The t r a n s i t i o n e x h i b i t e d i n t h i s c u r v e ,  from  t o c o n v e c t i v e , o c c u r s a t v e r y low r a t e s o f a p p r o x i -  m a t e l y 0.08-0.1 cm s e c " . 1  At t h i s speed t h e time  taken  by t h e s p h e r e s t o r e a c h t h e o t h e r end ( t h e " i n t e r f a c e " ) was l o n g enough f o r c o n v e c t i o n t o decay i n t h e l i q u i d t o a minimum due t o v i s c o u s f o r c e s . From t h e r e s u l t s i n t h e metal s y s t e m and t h e o b s e r v a t i o n s made i n t h e m o d e l , t h e mechanism f o r t h e s e g r e g a t i o n  105  of p a r t i c l e s i n t h e metal s y s t e m i s p r o p o s e d t o o p e r a t e i n the f o l l o w i n g dendritic  way:  interface.  p a r t i c l e s s t r i k e the c e l l u l a r or I f the s t r i k i n g d i r e c t i o n  coincides  w i t h t h e c u r v e d p a r t o f a t i p t h e p a r t i c l e moves into the i n t e r s t r u c t u r a l force  further  p o s i t i o n as f a r as t h e g r a v i t y  i s not e q u i l i b r a t e d .  I f the s t r i k i n g d i r e c t i o n  coin-  cides with the face of the t i p the p a r t i c l e w i l l  stop  there.  larger  T h i s s t o p may be t e m p o r a r y f o r p a r t i c l e s  than t h e t h i c k n e s s o f t h e b o u n d a r y l a y e r . the p a r t i c l e s a r e d r a g g e d t o p o s i t i o n s forces  In t h i s c a s e  on which t h e g r a v i t y  c o u l d move t h e p a r t i c l e s t o i n t e r s t r u c t u r a l  positions.  Small p a r t i c l e s a r e 1 i k e l y t r a p p e d where t h e y s t r i k e t h e interface  because they a r e not a f f e c t e d  a l s o b e c a u s e o f t h e i r low m o b i l i t y .  by c o n v e c t i o n and  T h i s mechanism f o r  the s e g r e g a t i o n o f p a r t i c l e s s h o u l d be more e f f e c t i v e when the i n t e r f a c e previously  has s h a r p e r t i p s l i k e d e n d r i t e s .  T h i s was n o t e d  when t h e a v e r a g e number o f p a r t i c l e s i n t h e  m a t r i x d r o p p e d from 16.7 t o 10% by c h a n g i n g t h e i n t e r f a c e shape f r o m c e l l u l a r t o d e n d r i t i c . 3.  The ^ C a s t i n g  Experiment  The c a s t i n g  experiment produced very well  dendrites. dendritic  developed  The d e n d r i t e s were l e a d r i c h and t h e i n t e r region  contained lead-tin eutectic.  were s e g r e g a t e d p r i m a r i l y as shown i n F i g . 3 6 ( a ) .  i n the i n t e r d e n d r i t i c  The p a r t i c l e s regions  The s i z e and p o s i t i o n o f t h e  (b)  107  (d)  108  F i g u r e 36. C a s t i n g e x p e r i m e n t , a) and b) P a r t i c l e s t r a p p e d i n w e l l d e v e l o p e d d e n d r i t e s . 2 0 0 X and 500X r e s p . c) and d ) P a r t i c l e s a t the bottom p a r t of the d e n d r i t e s . 5 0 0 X . d)A p a r t i c l e a t t h e top o f t h e d e n d r i t e . 5 0 0 X . And e ) a p a r t i c l e moving f r e e i n the e u t e c t i c . 5 0 0 X .  109 Table XVI  P a r t i c l e Size Distribution f o r Cast Sample. (a) Bottom, (b) 5 mm form the bottom, \. (c) 15 mm'from bottom, (d)-25 mm from bottom, (e) top of sample.  (a) Size, u  Bottom  Segregated  (b) 5 mm'from Bottom Matrix  Matrix  Segregated i  4  3  i !  2  2  6  11  2 \  13  2  8  23  1  22  1  10  37  1  23  1  12  23  0  i  29  2  14  36  0 !  42  2  16  15  0 i  35  0  25  0  j  1'  i  18  19  o  20  29  0  29  0  22  16  0 :  13  0  24  14  0 ;  16  0  26  8  0  12  0  28  3  0  3  0  30  4  0  3  0  32  1  0  0  0  242  6  267  10  TOTAL Density  45 mm  % i n matrix 2.4%  :  50 mm 3 / 6 %  no  TABLE XVI (cont.) /  (c) Size,v  Bottom or Trapped Eutectic  Top  Total 1  Matrix  Total  4  0  0  0  0  6  5  1  1  7  8  6  5  1  12  12  10  21  2  2  25  25  12  26  2  3  31  31  14  20  11  0  31  31  16  18  8  2  28  28  18  22  4  1  28  V  2-  19  1  4  23  23  22  .8  5  1  14  14  24  <9  1  0  10  10  26  5  1  0  6  ';6  28  5  0  0  5  5  30  2  0  0  2  2  32  4  0  0  4  4  170  41  !4  225  TOTAL % % % %  15'..' mm from'Bottom  bottom or trapped on eutectic on top dendrite  75.2% 18.1% 6.2% 0.4%  0 1  8  1  226  density 41 mm -2  TABLE XVI (cont.)  Ill (d) 25 mm from Bottom  Size,^  Bottom or Trapped  Eutectic  4  0  1  1  6  13  2  15  8  8  3  11  10  18  2  20  12  8  4  2  14  14  11  3  1  15  16  10  1  0  11  18  6  1  0  7  20  15  4  1  20  22  5  2  7  24  6  1  7  26  5  0  5  28  3  0  3  30  1  1  2  0 109  Q 25  0 138  32 TOTAL  Top  4  Total 1  Dend  0  % bottom or trapped  79.0%  % eutectic  18.1%  density  "I" top  2.9%  25 mm"  % dendrite  0%  2  112 TABLE XVI (cont.) (e)  3 mm from Bottom - Top of Sample Eutectic  Top  4  Bottom or Trapped 0  0  0  0  0  6  5  1  0  6  6  8  5  1  0  6  ID  14  4  0  18  12  13  7  0  20  14  19  2  0  21  21  16  10  3  2  15  15  18  9  1  1  11  11  20  10  3  2  15  15  22  7  1  0  8  8  24  6  2  1  9  9  26  5  2  0  7  7  28  3  0  0  3  3  30  2  0  0  2  2  32  1  0  0  1  1  TOTAL  109  27  6  142  144  S i ze j  u  % bottom or trapped  75.7%  % eutectic  18.7% 4.2%  % top % dendrite  1.4%  Total T Matrix  density _ -2 26 mm c  1  Total  7 18  1  21  113  — total  604  n  T  = 248  n/A = 4 5 m to OJ  40-  O Q. O  2 0-  0:  II  j j q  to o  o Q-  o  o'  19  27 Size,  35 jJLm  —  604  total in e u t e c t i c  n  =226  T T  CO  -2  n/A = 41 mm  ^40 + o CL  o  20 +  d  r" I  i  i—  i_,  19  1  i  27 S i z e , yU,m  35  (c)  CO  —  u 40+  - - in e u t e c t i c  QJ  total n = 138 _ n/A = 2 6 mm T  O CL  2  o20 +  l — S 0:  n  r-n 19  I 27  S i z e , jM m (d)  ,  35  115  —  £40+  totol  in eutectic  n = 144 T  -2 n/A = 2 6 mm 1  o  CL  o  20-  d I  04=4  1 L  19  27 35 Size , / m m  (e)  F i g u r e 37. P a r t i c l e s i z e d i s t r i b u t i o n f o r t h e c a s t sample, a) and b) t o t a l number o f p a r t i c l e s a t t h e bottom o f t h e sample and 0.5 cm from t h e bottom. c ) , d ) and e) t o t a l namber o f p a r t i c l e s and i n t h e e u t e c t i c a t 1.5; 2.5; and 3.0 cm from t h e bottom of t h e sample respectively.  116 TABLE XVII  P a r t i c l e Size Distribution f o r Heights 15 mm, 25 mm and 30 mm f o r Bottom or Trapped P a r t i c l e s and Particles in the Eutectic. Bottom or . Trapped  Eutectic  3-7  30  5  14.3  7-11  93  17  15.4  11-15  133  29  17.9  15-19  99  18  15.4  19-23  87  16  15.5  23-27  44  7  13.7  27-31  17  1  16.7  31-35  5  0  TOTAL  508  93  Size, y  % i n Eutectic  15.5  117 p a r t i c l e s were measured at the bottom o f the s o l i d and 0.5,  1.5,  2.5  and 3 cm f r o m the b o t t o m ,  b e i n g c l o s e to the top o f the s a m p l e . regions  at  the l a s t p o s i t i o n In the f i r s t  the p a r t i c l e s were s e p a r a t e d i n t o two  two  position  g r o u p s , t h o s e i n the d e n d r i t i c c o r e or b r a n c h and t h o s e i n the e u t e c t i c . The ( b ) . The  r e s u l t s are g i v e n  i n T a b l e s XVI  (a)  r e s u l t s i n the t a b l e s show t h a t o n l y a few  and  parti-  c l e s are p r e s e n t i n the d e n d r i t i c m a t r i x , s p e c i f i c a l l y and  3.6%  at the bottom arid at 0.5  cm above the  bottom.  E x a m i n i n g the p a r t i c l e s i n the i n t e r d e n d r i t i c r e g i o n  in  more d e t a i l , the p a r t i c l e p o s i t i o n s were s u b d i v i d e d t h r e e g r o u p s t a k i n g c a r e to n o t e the v e r t i c a l  arms and  (a) P a r t i c l e s t r a p p e d between the  the d e n d r i t e s ,  (c) P a r t i c l e s i n e u t e c t i c  w i t h the d e n d r i t e s  regions  (Fig. 36(e)).  These  eutectic  D i v i d i n g the number of p a r t i c l e s o b s e r v e d  i n the i n t e r d e n d r i t i c r e g i o n  into these three  the r e s u l t s shown i n T a b l e s XVI corresponding  of  some of them p a r t i a l l y e n g u l f e d as shown  p a r t i c l e s were f r e e to move p r e v i o u s to the solidification.  as shown i n  (b) P a r t i c l e s t h a t were at the top  in photograph 36(d). not i n c o n t a c t  polished  dendrite  i n the bottom p a r t o f the d e n d r i t e s  F i g . 36 ( a - d ) .  into  positions  o f the p a r t i c l e s , as w e l l as the p o s i t i o n s on the faces:  2.4%  (c) , (d) and  groups (e).  gives The  d i s t r i b u t i o n of p a r t i c l e s as a f u n c t i o n  s i z e from the t a b l e s are p l o t t e d i n F i g u r e s  37  (a-e).  of  118  The r e s u l t s i n d i c a t e s t h a t the p a r t i c l e s e n t r a p p e d the d e n d r i t e arms were caught s o l i d i f i c a t i o n , F i g . 36(a).  t h e r e a t an e a r l y s t a g e  by of  The p a r t i c l e s t h a t managed t o  s t a y i n the l i q u i d moved w i t h i t , due t o c o n v e c t i o n ,  and  w i t h r e s p e c t t o i t , due t o buoyancy f o r c e s . Some o f the p a r t i c l e s are s t o p p e d  by the d e n d r i t e s and remain i n t h i s  p o s i t i o n , i n the bottom o f the d e n d r i t e .  Other  particles,  a b o u t 15% o f the t o t a l , can s u r v i v e and c o n t i n u e ; t o move w i t h i n the e u t e c t i c l i q u i d . a d e n d r i t e arm i t may  Once a p a r t i c l e i s s t o p p e d  be d e t a c h e d  by  from the d e n d r i t e by the  c o n v e c t i v e l i q u i d and move t o a new p o s i t i o n . T h e r e are s e v e r a l s o u r c e s f o r c o n v e c t i o n : convection  i s c r e a t e d when p o u r i n g the m e l t .  v e c t i o n a r i s e s by n o n - s t a b l e temperature  forced Natural  d e n s i t y g r a d i e n t s due  g r a d i e n t s and to the s o l u t e g r a d i e n t  con-  to built  up, as s o l u t e i s r e j e c t e d d u r i n g growth o f a d e n d r i t e . A n o t h e r c a u s e o f c o n v e c t i o n may  be the d i s p l a c e m e n t  l i q u i d as d e n d r i t e s (more d e n s e ) f a l l caused  down.  of  T h i s may  have  n e g l i g i b l e c o n v e c t i o n s i n c e i f a l a r g e number o f  d e n d r i t e s f a l l t o t h e bottom d u r i n g s o l i d i f i c a t i o n , top of the i n g o t s h o u l d m a i n l y be o f e u t e c t i c w h i c h was  not o b s e r v e d  in the present c a s t i n g .  composition Moreover,  i f d e n d r i t e s f a l l i n g to the bottom o f the c a s t i n g important  the  was  the number o f p a r t i c l e s s h o u l d be h i g h e r at  to c QJ  Q  o-l— 0  1  10  1  20  1—  30  Height , mm  F i g u r e 38. D e n s i t y  of p a r t i c l e s v s . h e i g h t  i n the c a s t sample.  120 the bottom o f the sample. t h a t i s t h e r e was  S u r p r i s i n g l y , t h i s was  observed,  a l a r g e r number o f p a r t i c l e s i n the bottom  h a l f o f the c a s t i n g than i n the u p p e r h a l f , as shown i n F i g . 38.  However, t h i s d e p l e t i o n can be a t t r i b u t e d to  f l o t a t i o n o f p a r t i c l e s from t h i s r e g i o n t o the top r a t h e r than the f a l l i n g o f d e n d r i t e s .  F i g . 39 i s a view o f the  top p a r t o f the i n g o t showing a v e r y h i g h d e n s i t y o f particles. completely  F l o t a t i o n took p l a c e when the sample was  still  l i q u i d s i n c e t h i s top l i q u i d s o l i d i f i e d when  the p a r t i c l e s were a l r e a d y  t h e r e -as can be  from F i g . 39.  o f 15.5%  The p r e s e n c e  concluded  o f the t o t a l  parti-  c l e s i n the e u t e c t i c r e g i o n s i s c o n c l u s i v e p r o o f t h a t the l i q u i d was moving.  I f t h i s was not the s i t u a t i o n a l l  the p a r t i c l e s s h o u l d have been i n the bottom p a r t o f the dendrites. The phenomenon o f p u s h i n g factor  cannot  be c o n s i d e r e d as the  l e a d i n g to t h i s s e g r e g a t i o n b e c a u s e o f the f o l l o w -  i n g f a c t s : (1) the random d i s t r i b u t i o n o f s i z e s i n the e u t e c t i c ; (2) t h e s m a l l amount (4.5%) o f the p a r t i c l e s i n the t o p o f the d e n d r i t e s and (3) the p r e s e n c e s m a l l p a r t i c l e s i n the d e n d r i t e s ( F i g . 3 6 ( b ) ) .  o f a few The  second  f a c t i n d i c a t e s t h a t the c a p t u r e i s s t r o n g enough t o compens a t e the b u o y a n c y f o r c e . The most p r o b a b l e t h i r d o b s e r v a t i o n may  c a u s e f o r the  be the low m o b i l i t y o f the s m a l l  p a r t i c l e s w i t h r e s p e c t to the l i q u i d w h i c h e n h a n c e d t h e i r  121  F i g u r e 39. Photograph of the top p a r t of the c a s t sample. The h i g h c o n c e n t r a t i o n of p a r t i c l e s i s a t t r i b u t e d to f l o t a t i o n . T h i s p a r t o f the sample s o l i d i f i e d when the p a r t i c l e s were already there.  122  e n g u l f m e n t by the g r o w i n g d e n d r i t e s . d e n d r i t e s may 4.  have n u c l e a t e d on the  the  particles.  P u s h i n g i n Mater When w a t e r c o n t a i n i n g  Cr p a r t i c l e s was  b e h a v i o r of the p a r t i c l e s was that observed previously F i g . 40(a)  c o m p l e t e l y d i f f e r e n t than frozen  The  is given in Fig.  40.  shows the i n i t i a l q u a s i - s u s p e n s i o n o f Cr p a r t i -  c l e s in water.  The  u n i f o r m g r e y a p p e a r a n c e means t h a t  p a r t i c l e s were d i s t r i b u t e d F i g . 40(b)  homogeneously i n the  the  melt.  c l e a r l y shows the a d v a n c i n g i n t e r f a c e by a d i f -  f e r e n c e i n g r e y between s o l i d and density  s o l i d i f i e d the  for p a r t i c l e s in lead.  r e s u l t s f o r w a t e r which was  had  In a d d i t i o n  l i q u i d due  o f p a r t i c l e s i n the l i q u i d .  t o the  higher  A f t e r the e n t i r e  sample  s o l i d i f i e d the p a r t i c l e s pushed by the i n t e r f a c e  were  i n a dark band on the c e n t e r as shown i n F i g . 4 0 ( c ) .  The  r e s u l t of a s i m i l a r e x p e r i m e n t but w i t h a h i g h e r d e n s i t y p a r t i c l e s i s shown i n F i g . 40(d) band i n the The  which shows a w i d e r dark  center.  f a c t t h a t the i n t e r f a c e moved h o r i z o n t a l l y  the p a r t i c l e s are f a l l i n g down e n h a n c e d p u s h i n g .  and Similar  o b s e r v a t i o n s o f p a r t i c l e p u s h i n g have n e v e r been o b s e r v e d in m e t a l l i c  s y s t e m s under s i m i l a r c o n d i t i o n s  low s o l i d i f i c a t i o n r a t e s .  and a t v e r y  T h e r e f o r e the w a t e r  freezing  of  (a)  (b)  F i g u r e 40. P u s h i n g i n water, a) The l i q u i d w i t h the s u s p e n s i o n of Cr p a r t i c l e s , b) The i c e - w a t e r i n t e r f a c e a d v a n c i n g t o the c e n t e r , c) Dark band of p a r t i c l e s pushed to the c e n t e r by the advancing f r o n t , d) S i m i l a r to c) but w i t h a h i g h e r d e n s i t y of p a r t i c l e s i n the l i q u i d which r e s u l t s i n a wider band.  124  experiments c l e a r l y indicate that  particles  in water behave e n t i r e l y d i f f e r e n t l y than p a r t i c l e s in a metal system.  A s o l i d i f y i n g water i n t e r f a c e  rejects  p a r t i c l e s , a m e t a l l i c i n t e r f a c e does not.  5.  The  L i f s h i t z - V a n der Waals Force  In p r e v i o u s the  segregation  sections  i t has  been d e m o n s t r a t e d  of p a r t i c l e s at a non-planar m e t a l l i c s o l i d -  liquid  i n t e r f a c e i s due  forces  and  convection  to a mechanism i n which the i n the l i q u i d are the d r i v i n g  responsible  for particle segregation,  mechanism.  M o r e o v e r , t h e p u s h i n g m e c h a n i s m has  t o be a b s e n t f o r t h e two  main t h e o r i e s  rather  Fe-Pb system employed.  The  search  this apparent contradiction  However,  t h a t can This  be r e s p o n s i b l e  t a n c e s the s i g n of t h i s f o r c e by e x p r e s s i o n  In o r d e r  physical  of force  Walls force.  At s h o r t  i s d e t e r m i n e d by t h e  disconstant  19.  t o c a l c u l a t e to,  d i e l e c t r i c constant  rejection  f o r the r e j e c t i o n of p a r t i c l e s .  i s the L i f s h i t z - V a n - d e r  to g i v e n  the  samples  f o r an e x p l a n a t i o n  l e d to the only  forces  b e e n shown  p r e s e n t e d i n C h a p t e r II p r e d i c t e d  front.  buoyancy  than a pushing  of p a r t i c l e s at growth v e l o c i t i e s at which the have a p l a n a r  that  are used.  the Drude f o r m u l a e f o r  the  These f o r m u l a e have been 43-4  demonstrated to f i t r e a s o n a b l y well  f o r many l i q u i d  metals  125 46  and p a r t i c u l a r l y f o r l e a d .  The Drude e x p r e s s i o n s f o r t h e  r e a l and i m a g i n a r y p a r t o f t h e c o n s t a n t s a r e 1  £  - T I  _  2 2 4-rrne T 2 2 m(l+oo x )  11  £  2 _ 4Trne T . 22 mco(1 +o) T )  (a) where  A  —  K  •• • ^v  (b)  n - number o f v a l e n c e e l e c t r o n s p e r u n i t volume e = e l e c t r o n charge m = e l e c t r o n mass T = r e l a x a t i o n time i n d e p e n d e n t o f f r e q u e n c y .  To f i t e x p e r i m e n t a l measurements t h e c a r r i e r d e n s i t y n i s u s u a l l y r e p l a c e d by an e f f e c t i v e d e n s i t y . 48  The same f o r m u l a e a r e used f o r s o l i d s d i f f e r e n c e that the product  COT~1,  with  co  49  '  with the  i n the v i s i b l e  f r e q u e n c i e s f o r l i q u i d and i n t h e I.R. f o r s o l i d s . In a d d i t i o n , f o r s o l i d m e t a l s t h e e f f e c t i v e e l e c t r o n mass m* s h o u l d be u s e d . The v a l u e co g i v e n by p  2  co _  p  =  4irne  2  — —  m  i s t h e pi asma f r e q u e n c y of t h e metal:. metal becomes t r a n s p a r e n t t o r a d i a t i o n .  ...  » 46 c  At t h i s value the  126 S u b s t i t u t i n g t h i s v a l u e o f tOp and r e p l a c i n g to f o r H , where .5 i s t h e pure i m a g i n a r y  part of the frequency, the  constant i s 2 2 T  CO  e(ic-)  =  1 +  -  ...47  p 2  z  1 +£  T  w h i c h f o r h i g h f r e q u e n c i e s , such t h a t  £x>>  1 E q u a t i o n 45  t a k e s t h e form to e(U)  =  2  -4-  1 +  ...48  U s i n g t h e same f o r m u l a f o r t h e t h r e e media and e v a l u a t i n g —to t h r o u g h t h e i n t e g r a l 19 t h e f o l l o w i n g expression i s obtained:  _  pl3 2  A  p23 2 " 8  r 223 ~ .2r i 3 ,,2 P  P  co  to  =  Ato„  where  2  Ato A  1 0  . . P1J  Ato„oo  —  to  2 . PI  =  .IT  -2  u  p23 "plSp23 — U  c  -  to  2  "' p ^  ,  AQ  .  ...49  2 . PJ  and 2 tO  ..  PIJ  _  1 ,.  -  TT  I  CO  2  . •  pi  2  ?  .  + C 0 . 1  p;r  The f r a c t i o n , o r s e c o n d f a c t o r i n f o r m u l a positive.  49 i s a l w a y s  T h e r e f o r e t h e s i g n o f to i s g i v e n by t h e f i r s t  f a c t o r i n A. f p r the system  U s i n g t h e v a l u e s o f u>p. g i v e n i n A p p e n d i x I , F e - P b ( s ) - P b ( l )„ to i s g r e a t e r than z e r o , i . e .  ii = 1. 90x1 0  15  sec"  1  127 which g i v e s f o r the c o n s t a n t Bg the v a l u e B  = 2.54xl0"  3  1 4  erg.  T h i s i s o f the o r d e r o f the v a l u e s which a p p e a r i n the literature/ ' 1 0  experimental  1 7  '  1 8  o b t a i n e d t h r o u g h c a l c u l a t i o n s or  measurements.  However, the most i m p o r t a n t r e s u l t i s t h a t t h e s e c o n s t a n t s co and Bg a r e p o s i t i v e ; i . e . the L i f s h i t z - V a n der Waals f o r c e i s a t t r a c t i v e .  T h i s , t h e r e f o r e , i s in agree-  ment w i t h the e x p e r i m e n t a l o b s e r v a t i o n which showed no s i g n o f p u s h i n g a t v e r y low s o l i d i f i c a t i o n r a t e s w i t h a planar i n t e r f a c e . I t i s n e c e s s a r y to n o t e t h a t the v a l u e s o f  plasma  f r e q u e n c i e s f o r the s o l i d s were c a l c u l a t e d u s i n g d a t a o b t a i n e d a t room t e m p e r a t u r e  because  t h e r e i s no d a t a a v a i l -  a b l e f o r l e a d a t the m e l t i n g p o i n t nor f o r i r o n a t the same t e m p e r a t u r e .  However  for Ni, a transition  l i k e i r o n , the a b s o r p t i o n s p e c t r u m  changes o n l y s l i g h t l y 51  on h e a t i n g t h r o u g h t h e X u r i e p o i n t , the same b e h a v i o r f o r i r o n .  metal  which may  Consequently  on i t s o p t i c a l p r o p e r t i e s a r e e x p e c t e d .  suggest  s l i g h t changes The  principal  problem t o e v a l u a t e the i n t e r a c t i o n a t a s o l i d - l i q u i d  inter-  f a c e r e s i d e s i n the f a c t t h a t the o p t i c a l p r o p e r t i e s o f the  128  s o l i d at the melting point are not known. ported values and s o l i d Co.  of the o p t i c a l  This o i s r e l a t e d  of the d i e l e c t r i c a (to)  The values  conductivity  Miller a ('to-)  52  re-  for liquid  to the imaginary  part  constant by  =-(to/4Tr)  e",/t°)  of a f o r s o l i d and l i q u i d Co at the melting  point are within  the experimental  data at room temperature  error,  f o r the same material  conclude a convergence of the o p t i c a l melting p o i n t . peaks t y p i c a l  ."However the f a c t s of a metal  but the lack of enable us to  p r o p e r t i e s at the  that the absorption  in the s o l i d state are l e s s 46  pronounced and broader as the temperature that these peaks are not present  increases  and  in the l i q u i d s t a t e ,  sug-  gest that the o p t i c a l  p r o p e r t i e s converge to a common value  at the melting p o i n t .  In such cases the L i f s h i t z - V a n der  Walls should be negligible-' at. a s o l i d i f y i n g  interface.  129  C h a p t e r VI SUMMARY AND  CONCLUSIONS  Lead-antimony a l l o y s containing  i r o n p a r t i c l e s were  s o l i d i f i e d u s i n g a zone r e f i n i n g t e c h n i q u e o p e r a t i n g i n two  modes, v e r t i c a l and  horizontal  with r o t a t i o n .  At a  n o n - p l a n a r i n t e r f a c e the p a r t i c l e s were s e g r e g a t e d to interstructural  positions  l i k e c e l l w a l l s or  r e g i o n s f o r b o t h s o l i d i f i c a t i o n modes.  interdendritic  When the  moved v e r t i c a l l y downward l a r g e r p a r t i c l e s were e n t i a l l y segregated. f o r the d e n d r i t i c  The  prefer-  a v e r a g e s e g r e g a t i o n was  higher  i n t e r f a c e compared to a c e l l u l a r  w i t h a l m o s t 90% of the p a r t i c l e s i n the zones.  interface  surface  interdendritic  For a c e l l u l a r i n t e r f a c e a d v a n c i n g  horizontally  the d e g r e e o f s e g r e g a t i o n o f p a r t i c l e s was  s i m i l a r to  v e r t i c a l mode but the s i z e d i s t r i b u t i o n was  random.  the  When pure l e a d w i t h i r o n p a r t i c l e s were s o l i d i f i e d with a planar i n t e r f a c e  i n both modes no a c c u m u l a t i o n of  p a r t i c l e s or d i f f e r e n c e  in density  and  s o l i d was  pushing.  The  between quenched l i q u i d  f o u n d i n d i c a t i n g no p a r t i c l e s e g r e g a t i o n or B o i l i n g and  Cisse  1  t h e o r y as w e l l as Chernov  et a l . t h e o r y p r e d i c t p u s h i n g at the growth" v e l o c i t i e s employed.  130  With the h e l p o f a p h y s i c a l were simulated,,a mechanism has  model i n which both modes been d e v e l o p e d to a c c o u n t  f o r the s e g r e g a t i o n o b s e r v e d i n the m e t a l l i c a l l o y s .  The  d r i v i n g f o r c e f o r the p a r t i c l e s e g r e g a t i o n i s p r i n c i p a l l y the b u o y a n c y f o r c e a c t i n g on the p a r t i c l e .  In the v e r t i c a l  c a s e , l a r g e r p a r t i c l e s ha v ring more mo me rit um c o l 1 i de w i t h c e l l c e n t r e s or d e n d r i t i c t i p s and final positions.  are d e f l e c t e d  For p a r t i c l e s l a r g e r than the  of the .boundary l a y e r the p r e s e n c e o f n a t u r a l h e l p s them to f i n d t h e i r way In the h o r i z o n t a l  into their thickness  convection  to i n t e r s t r u c t u r a l  positions.  mode, due t o r o t a t i o n and  buoyancy  forces^ t h e p a r t i c l e s sweep the i n t e r f a c e u n t i l t h e y t r a p p e d i n the i n t e r c e l l u l a r g r o o v e s . p a r t i c l e s travel distances sions  per r e v o l u t i o n  structural  regions.  interstructural forces  The  small  which e x p o s e d them t o the  are  iron  o f the o r d e r o f the c e l l  dimen-  inter-  Once the p a r t i c l e g e t s i n one  r e g i o n s i t i s shown t h a t due  the  to t h e  of these viscous  the p a r t i c l e o f a v e r a g e s i z e i s s t o p p e d a l m o s t  i n s t a n t l y , and c a n n o t come out i n t o the l i q u i d and move further. S e g r e g a t i o n was  also observed in a casting  experiment.  Most o f the p a r t i c l e s were t r a p p e d i n the b r a n c h e s o f  the  d e n d r i t e s i n an e a r l y s t a g e o f the s o l i d i f i c a t i o n p r o c e s s .  131 T r a p p i n g by d e n d r i t e  b r a n c h e s c o u l d r e s u l t as the p a r t i c l e s  floated, towards the s u r f a c e .  In a c a s t i n g 15.5%  of  the  p a r t i c l e s managed to remain i n the l i q u i d due to buoyancy f o r c e s and c o n v e c t i o n a few s m a l l suggesting  i n the i n t e r d e n d r i t i c l i q u i d .  p a r t i c l e s were f o u n d i n the d e n d r i t i c t h a t t h e i r e n g u l f m e n t was  m o b i l i t y or the d e n d r i t e s  may  Only  cores  e n h a n c e d by t h e i r  have n u c l e a t e d  low  on them.  D i f f e r i n g markedly from the m e t a l l i c system, pushing o f Cr p a r t i c l e s was  observed in water.  To s o l v e the a p p a r e n t c o n t r a d i c t i o n between the observations  where no r e j e c t i o n was  detected  present  in m e t a l l i c  s y s t e m s , and the t h e o r e t i c a l p r e d i c t i o n s t h a t r e j e c t i o n t a k e s p l a c e , the L i f s h i t z - Van p a r t i c l e s was  der Waals f o r c e between s o l i d and  c a l c u l a t e d u s i n g the Drude f o r m u l a e f o r the  d i e l e c t r i c constants.  T h i s f o r c e i s shown t o be a t t r a c t i v e  w i t h a m a g n i t u d e s i m i l a r to t h a t r e p o r t e d  i n the l i t e r a t u r e .  Cone!us i o n s : For the Fe-Pb m e t a l l i c s y s t e m a d o p t e d i n t h i s i n v e s t i g a tion: (1)  Segregation  of p a r t i c l e s a t a n o n - p l a n a r i n t e r f a c e  i s due t o n a t u r a l c o n v e c t i o n  i n the l i q u i d and  b u o y a n c y f o r c e s a c t i n g on the p a r t i c l e s .  (2)  No e x p e r i m e n t a l e v i d e n c e o f p u s h i n g o f p a r t i c l e s by an a d v a n c i n g s o l i d - 1 i q u i d i n t e r f a c e was  (3)  The L i f s h i t z - V a n  observed.  d e r Waals f o r c e between p a r t i c l e  and s o l i d has been c a l c u l a t e d t o be a t t r a c t i v e i n t h i s system.  A c c o r d i n g l y , p a r t i c l e s s h o u l d not be  r e j e c t e d by an a d v a n c i n g i n t e r f a c e i n t h i s system as is observed.  133 REFERENCES 1.  M. V. P i k u n o v . 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" P r i n c i p l e s of the Theory of S o l i d s 2nd Ed. Cambridge U n i v . P r e s s , Cambri dge , ( 1 972••).  49.  F. A b e l e s . O p t i c a l P r o p e r t i e s o f M e t a l s i n " O p t i c a l P r o p e r t i e s o f S o l i d s " . Ed. by F. A b e l e s .  50.  C. K i t t e . l . " I n t r o d u c c i o n a l a F i s i c a d e l E s t a d o S o l i d o " . Ed. R e v e r t e S.A. B a r c e l o n a ( 1 9 6 5 ) .  Press.  London  (1977).  136 51.  G.P. S h i g a and M. P e l l s . 2, 2, (1969) p.1847.  J . Phys. C. ( P r o c . Phys. S o c ) ,  52.  O.C. M i l l e r . P h i l . Mag., 20, (1969) p p . I l l 5-1132.  53.  A.E. C o r t e . J . o f Geophys. R e s e a r c h . , 67, 3 (1962) pp.1085-1Q90. ~~  54.  V.H.S. Kuo and W.R. (1973) pp.375-377.  55.  R.H. Ewing.  56.  M.J. S t e w a r t and F. W e i n b e r g . J . o f C r y s t a l 1_2 , ( 1 972) pp. 21 7-227.  57.  M.J. S t e w a r t and F. W e i n b e r g . j_2, ( 1972) pp.228-238.  58.  0. D r i e d g e r , A.W. Neumann and P . J . S e l l . Z.U.Z. P o l y m e r e , 201, (1965) pp.52-57.  59.  I b i d . 2_04, (1965) pp.101-105.  W i l c o x . Sep. S c i e n c e , 8, 37,  P h i l . Mag., 2_5, (1972) p. 779.  J. of Crystal  Growth Growth Kolloid-  137  Appendix I THEORIES OF PARTICLE REJECTION AT A SOLID-LIQUID INTERFACE  a)  Conditions f o r Pushing  138  It was seen above^ t h e e x i s t e n c e o f a s t e a d y s t a t e r e j e c t i o n o f a p a r t i c l e i n a PLS system depends on the system t h r o u g h stituents.  the i n t e r a c t i o n o f the three  con-  The r e s u l t o f t h e i n t e r a c t i o n s h o u l d be r e -  f l e c t e d i n s p e c i f i c c o n t r i b u t i o n s t o t h e t o t a l energy o f the system.  The knowledge and a n a l y s i s o f t h i s  inter-  a c t i o n and t h e e n e r g i e s i n v o l v e d s h o u l d e n a b l e p r e d i c t i o n s to b,e Fig.  majdeoaSiQto  when t r a p p i n g o r p u s h i n g w i l l  occur.  2 shows some o f t h e s i t u a t i o n s t h a t c a n be i m a g i n e d .  In t h e f i g u r e t h e f r e e e n e r g i e s o f t h e PLS system as a f u n c t i o n o f t h e d i s t a n c e from t h e S-L i n t e r f a c e a r e schematically represented.  I t i s assumed t h a t t h e r e i s  e q u i l i b r i u m and t h e phase t r a n s f o r m a t i o n i s n o t t a k e n i n t o consideration. cle  F and F.. and F^ , a r e t h e e n e r g y o f t h e p a r t i f  i n t h e s o l i d and i n t h e l i q u i d r e s p e c t i v e l y , f a r from  the i n t e r f a c e .  F. , and F.. r e p r e s e n t two i n i t i a l  energies  which a r e h i g h e r and lower than t h e e n e r g y o f t h e f i n a l s t a t e , F^. As t h e p a r t i c l e a p p r o a c h e s  t h e i n t e r f a c e an  i n t e r a c t i o n among t h e t h r e e c o n s t i t u e n t s d e v e l o p s .  This  i n t e r a c t i o n s h o u l d be m a n i f e s t e d t h r o u g h a f o r c e o f a s t i l l unknown n a t u r e .  T h i s f o r c e i n p r i n c i p l e s h o u l d be i n -  d e p e n d e n t o f t h e e n e r g i e s o f t h e i n i t i a l and f i n a l s t a t e s where t h e i n t e r a c t i o n i s n e g l i g i b l e . M o r e o v e r , i t may be a t t r a c t i v e , r e p u l s i v e o r f i r s t a t t r a c t i v e and then r e p u l s i v e o r any o t h e r c o m b i n a t i o n .  As a r e s u l t o f t h i s i n -  t e r a c t i o n , c u r v e s a t o e may be c o n s i d e r e d p o s s i b l e , a) and b) a r e t h e r e s u l t o f t h e p r e s e n c e o f a r e p u l s i v e f o r c e  139  F i g u r e 2. F r e e e n e r g i e s of the PLS system v s . d i s t a n c e from t h e i n t e r f a c e ( s c h e m a t i c ) . i ' and i a r e two i n i t i a l s t a t e s w i t h h i g h e r and lower energy than the f i n a l s t a t e f . Curves a to e a r e t h e p o s s i b l e e f f e c t s of the c l o s e presence of the p a r t i c l e i n t h e f r o n t . In a and b the p a r t i c l e i s r e p e l l e d by the s o l i d , a n d i n cand d the p a r t i c l e i s a t t r a c t e d by the s o l i d . I n case e t h e r e i s no i n t e r a c t i o n between p a r t i c l e and s o l i d .  b e t w e e n p a r t i c l e and  i n t e r f a c e . To move a p a r t i c l e  with,  e n e r g i e s ; F^ » o r F | t o t h e i n t e r f a c e r e q u i r e s ; a w o r k t o d o n e on t o t h e p a r t i c l e t o i n c r e a s e i t s e n e r g y .  be  Curves, c )  and d) i l l u s t r a t e tfie. o p p o s i t e c a s e i n which, t h e f o r c e i s ; attractive.  In t h i s c a s e /moving t h e p a r t i c l e t o t h e  face r e l e a s e s energy, energy  r e d u c i n g F- a n d . F - , .  value in the s o l i d .  From t h e p o i n t o f v i e w o f p u s h i n g lead to the steady r e j e c t i o n  should lead to trapping. to a s s e s s . final  Cases  c a s e s ( a ) and  s t a t e has a h i g h e r e n e r g y  not o c c u r i n t h i s case.  a c t i o n i n the system  (b)  of a p a r t i c l e . . C a s e (c) ( d ) and  (e) are  In c a s e ( d ) t h e f o r c e i s a t t r a c t i v e  t h i s c o n d i t i o n e x i s t s on a r e a l may  In c a s e ( e ' ) t h e  i s c o n s t a n t i n t h e l i q u i d and i n c r e a s e s l i n e a r l y t o  the f i n a l  may  inter-  difficult but  than the i n i t i a l PLS s y s t e m  the  one.  If  p u s h i n g may  In c a s e ( e ) t h e r e i s no  or inter-  when t h e p a r t i c l e i s i n t h e l i q u i d  in the s o l i d the energy  i n c r e a s e s l i n e a r l y to the  and,  final  v a l u e Ff-.  O n c e more i t i s r e c a l l e d  t h a t a r e p u l s i v e f o r c e is;  n e c e s s a r y not o n l y to a c c e l e r a t e the p a r t i c l e from zero to b u t a 1 so t o w i t h s t a n d t h e d r a g f o r c e  t h e growth, v e l o c i t y , which  o r i g i n a t e s from the motion  particle during s o l i d i f i c a t i o n .  of f l u i d r e l a t i v e to the In t h i s c a s e t h e  d r a g f o r c e , (F = 6,frryR v w h e r e n. i s t h e v i s c o s i t y , n  Stokes R>  n  is  141  the p a r t i c l e radius; and y i s t h e v e l o c i t y o f t h e l i q u i d ) does, n o t a p p l y  b e c a u s e o f t h e p r e s e n c e o f an i n t e r f a c e  i m m e d i a t e l y b e h i n d the. p a r t i c l e .  This, i n t e r f a c e e f f e c t i v e l y  4  acts; as a s i n k . t e d the v a l u e given  Carrier  o f the drag  by t h e f o l l o w i n g F = 6mR  where d The  g  introduced  this, f a c t and c a l c u l a -  force f o r a planar  s i n k which is,  expression 2 b  i s the distance  v/d  ...(1)  s  between p a r t i c T e : a n d  interface.  form o f t h i s f o r c e , that tends to i n f i n i t e  as t h e d i s t a n c e  d  $  values  g o e s t o z e r o , may l e a d t o t h e c o n c l u s i o n  t h a t a PLS s y s t e m . w i t h a ( d ) - o r ( e ) - l i k e e n e r g y  diagram  can n e v e r push a p a r t i c l e s i n c e t h e r e p u l s i v e f o r c e  begins  t o be i m p o r t a n t  captured  when t h e p a r t i c l e has a l r e a d y  been  by t h e s o l i d . 5  Neuman a n d c o - w o r k e r s fication and  assumed t h a t a t very  rates only the thermodynamic aspect  any o t h e r  is  low s o l i d i important,  c o n t r i b u t i o n such, a s t h a t d u e t o d r a g  i s n e g l i g i b l e i n comparison to the p o t e n t i a l d r i v i n g  forces, force  for migration.  T h i s d r i v i n g f o r c e w a s a s s u m e d t o he t h e  surface  o f t h e PLS system.  tension  free, e n e r g i e s  Computing the changes i n  through the d i f f e r e n t steps of engulfing of  a p a r t i c l e [ s e e F i g . 3) t h e n e t c h a n g e i n f r e e e n e r g y , f o r a p a r t i c l e o f u n i t a.rea x a n .be...given by ;  142 A G  where a  net  a  ps; " p l  •••  a  C2i  and a - | a r e th.e i n t e r f a c i a l e n e r g i e s , o f P-S and p  P-L r e s p e c t i v e l y . Th..is. change is; i n d e p e n d e n t pens, i n between t h e i n i t i a l  o f what hap-  and f i n a l s t a t e s ; .  I t i s assumed  t h a t t h e i n t e r f a c i a l e n e r g i e s ; do n o t change w i t h t h e p r e sence o f the p a r t i c l e very c l o s e t o the i n t e r f a c e .  In t h i s  c a s e t h e r e i s no c h a n g e i n t h e f r e e e n e r g y o f t h e system when t h e p a r t i c l e i s i n t h e l i q u i d . to i n c r e a s e when t h e low; e n e r g y  The f r e e e n e r g y  begins  s u r f a c e a r e a between p a r t i c l e  and l i q u i d i s r e p l a c e d by a h i g h e n e r g y s u r f a c e a r e a between p a r t i c l e and s o l i d , t h a t i s when t h e p a r t i c l e i s e n g u l f e d by the s o l i d . sector i s  C o n s i d e r i n g that the surface area o f a s p h e r i c a l 2/3TR  h,- where  R  i s t h e r a d i u s o f t h e s p h e r e and  h. i s t h e h e i g h t o f th.e spherical sect'or :(that is,thecd'ep'thtthe )  p a r t i c l e has p e n e t r a t e d i n t o t h e s o l i d ) , t h e v a r i a t i o n i n f r e e energy  i s p r o p o r t i o n a l t o h. When h = 2 R , i . e . t h e  whole p a r t i c l e i s i n t h e s o l i d , t h e f r e e e n e r g y r e a c h e s i t s maximum v a l u e . depths  T h i s maximum v a l u e does n o t change f o r f u r t h e r  i n t o the s o l i d .  In t h i s c a s e t h e v a r i a t i o n i n f r e e  e n e r g y o f t h e PLS system w i t h d i s t a n c e from t h e S-L i n t e r f a c e may be r e p r e s e n t e d by c u r v e ( e ) i n F i g . 2 ; t h a t i s , c o n s t a n t i n t h e l i q u i d and l i n e a r l y a p p r o a c h i n g  the f i n a l  value i n t h e s o l i d i n a d i s t a n c e equal t o t h e diameter o f the p a r t i c l e .  143 The  c o n d i t i o n s proposed by Neuman and h i s co-workers a r e  t h a t when  A  G n  e  t  whereas i f A G  <  net  t r a p p i n g , o f the p a r t i c l e w i l l  0  >0 r e j e c t i o n w i l l 5-7  were checked e x p e r i m e n t a l l y and  occur.  be f a v o r e d ,  These c o n d i t i o n s  u s i n g m a t r i x e s of naphthalene  b i pheni1 and o r g a n i c p a r t i c l e s of a c e t a l , n y l o n , p o l y -  e s t r y r e n e and t e f l o n .  These m a t e r i a l s covered  h y d r o p h i l i c t o hydrophobic  particles.  p a r t i c l e s was q u a n t i f i e d through  t h e range from  The r e j e c t i o n of the  t h e i r m o b i l i t y a t an i n t e r -  face moving h o r i z o n t a l l y a t a v e l o c i t y w e l l below 8 ym s e c . - 1  The  r e s u l t s which appear i n Table  the thermodynamic p r e d i c t i o n s : engulfment and A G  net  >0  I were i n agreement w i t h  i . e . AG ^.<0 corresponds to ne  rejection.  In a d d i t i o n Omenyi and Neuman  7  confirmed  t h e importance  of t h e s u r f a c e thermodynamic e f f e c t s o b s e r v i n g the behavior of g l a s s spheres ( h y d r o p h i l i c ) and g l a s s spheres with s i l i c o n  (hydrophobic)  covered  which were pushed and trapped  respectively.  One important  aspect o f t h i s work which i s u n c l e a r i s  i n t h e procedure t o o b t a i n t h e s u r f a c e t e n s i o n o f t h e P-S interface,  Ps  To o b t a i n the s u r f a c e t e n s i o n s they used 59 60  the f o l l o w i n g s e m i - e m p i r i c a l r e l a t i o n s s o l i d s w i t h low i n t e r f a c i a l values f o r w a t e r ) .  energies  '  obtained f o r  ( o f the order of  144  cos  a'  CO.. 015 a  a = y  s v  a  =  sl  -2:C.0Va .a, •+ a , ) • ' ^— CQ.015/a a -1}  ] y  s v  s y  ;-  S V  l  y  .;.-3(b)  ~  lV  1 - 0.015/a  ...3Ca)  v  s v  a  l  y  and Young's; e q u a t i o n a  where:  sv  " s.l a  =  °lv  £ 5 §  e  ...3(c)  y  i s t h e Y o u n g ' s c o n t a c t a n g l e , a » a -| a n d - j a r e s v  the i n t e r f a c i a l  s  a  v  e n e r g i e s o f s o l i d / v a p o r , s o l i d / l i q u i d and  l i q u i d and l i q u i d / v a p o r i n t e r f a c e s r e s p e c t i v e l y . These e q u a t i o n s a r e used determined enabling  0 S V  i n t h e f o l l o w i n g way.  V a l u e s , o f -| a n d a  v  experimentally are put into Equation 3(a) t o be c a l c u l a t e d .  i n E q u a t i o n 3 ( b ) g i v e s a -| • s  Substituting . a  gv  Alternatively,  and may a l s o  be o b t a i n e d u s i n g E q u a t i o n 3 ( c ) r e p l a c i n g t h e v a l u e s o f .  ff  , a-,  o V  I  and e .  V  v  The c o n t a c t a n g l e s  Jr  room t e m p e r a t u r e  Q were measured a t y  using t h e s e s s i l e drop technique  the m a t r i x and p a r t i c l e m a t e r i a l as s u b s t r a t e s . of a  s  v  employing The v a l u e s  c a l c u l a t e d a r e then e x t r a p o l a t e d t o t h e m e l t i n g p o i n t  using estimated c o e f f i c i e n t s , da /dT, f o r the variation of s v  s u r f a c e e n e r g y with, What  temperature.  '.i s ~ u n c l e a r i n t h e procedure  to obtain a ps  i s t h e v a l i d i t y o f t h e a s s u m p t i o n made t o u s e E q u a t i o n 3 ( b ) . They assumed t h a t " . . . r e l a t i o n s between thermodynamic  145  F i g u r e 3. Changes i n f r e e energy d u r i n g the engulfement of a cube or a sphere of u n i t area by a s o l i d . T h e s u r f a c e e n e r g i e s f o r the b u l k media do not change due to the presence of a t h i r d i n t e r f a c e .  146 q u a n t i t i e s may he e x p e c t e d state of aggregation...."  n o t t o d e p e n d s t r o n g l y on t h e With, t h i s a s s u m p t i o n ,  then,.  E q u a t i o n 3 (b 1 h o l d s when i n s t e a d o f t h r e e p h a s e s different  states of aggregation  f o r which, i t w a s d e v e l o p e d ,  remaining  a gaseous  Then a^  phase.  in which  proposed  1  v  particle-interface  in  i s a^^.  a different  In a t y p i c a l  phases  i s r e p l a c e d by a  c o n d i t i o n f o r pushing  t h e i n t e r f a c i a l e n e r g i e s a .p a ^  involved.  three  ( p a r t i c l e and m a t r i x ) and t h e  E q u a t i o n 3(b) and t h e r e s u l t  Pikunov  Csolid, l i q u i d a n d g a s ) ,  there are s t i l l  b u t two o f them s o l i d p h a s e s  in three  pushing  v  and a  are  c o n f i g u r a t i o n the estimated  s e p a r a t i o n i s a s s u m e d t o be much  less  t h a n 1 ym, t h a t i s t h e p a r t i c l e a n d i n t e r f a c e a r e n e a r l y i n I f S i s the surface area of the p a r t i c l e i n  "contact".  " c o n t a c t " with the i n t e r f a c e the energy SXap-| + o )  while the energy  l s  f a c e i s So  .  of  of  the system i s  the p a r t i c l e - s o l i d  inter-  T h e d i f f e r e n c e b e t w e e n t h e two e n e r g i e s i s  ps a s s u m e d t o be t h e d r i v i n g  force f o r pushing, therefore  when cr  ps  the p a r t i c l e w i l l a  < a  pi  , + af - . IS.  ...  4(a)  b e t r a p p e d , a n d when  ps. > p l a  +  a  l s  --^(b)  147 These r e l a t i o n s , however,  the p a r t i c l e w i l l he r e j e c t e d , were not p r o y e d  T h i s and o t h e r c o n d i t i o n s  experimentally,  are i n c l u d e d i n formal  theories predicting c r i t i c a l  t i e s and t h e y w i l l be  p r e s e n t e d i n t h e next s ; e c t i o n .  .  b)  T h e o r y o f Uhlmann e t a l .  veloci-  4  In t h i s t h e o r y t h e d r i v i n g f o r c e f o r p u s h i n g i s c o n s i d e r e d t o be t h e d i f f e r e n c e i n s u r f a c e e n e r g y °ps ~ ^ p l a  +  a  ls^  P P  a s  r o  o s e c  ' by P i k u n o v .  & e F 0  However, t h i s  i n t e r f a c i a l energy i s not c o n s i d e r e d ' e f f e c t i v e u n t i l the p a r t i c l e i s very c l o s e to the i n t e r f a c e . account  To t a k e t h i s i n t o  an e x p r e s s i o n i s assumed f o r t h e dependence o f t h e  d r i v i n g e n e r g y w i t h d i s t a n c e between t h e p a r t i c l e and t h e i n t e r f a c e g i v e n by 'id' Aa  -  Aa  Q  L  1  n  0 '  d_  where d i s t h e p a r t i c l e / i n t e r f a c e d i s t a n c e and d i s an 0  assumed minimum d i s t a n c e between p a r t i c l e and i n t e r f a c e where t h e y a r e i n c o n t a c t .  A minimum d i s t a n c e d  Q  is  neceS'  s a r y i n t h e d e r i v a t i o n t o a v o i d i n f i n i t e v a l u e s o f A a. The v a l u e o f t h e c o n s t a n t n i s some p o s i t i v e number between 4 and 5. The v a r i a t i o n i n s u r f a c e f r e e e n e r g y g i v e n by 5 corresponds  to a chemical  Equation  p o t e n t i a l which, i s w r i t t e n as  148  F i g u r e 4. The e f f e c t i v e s u r f a c e energy of t h e PLS system as a f u n c t i o n of d i s t a n c e , d i s a minimum d i s t a n c e at which p a r t i c l e and s o l i d a r e c o n s i d e r e d to be i n c o n t a c t .  149 . . .6  where V  q  i s th.e a t o m i c yolame o f t h e l i q u i d .  Such, p o t e n -  t i a l p r o v i d e s ; the. d r i v i n g f o r c e t o c a r r y m a t e r i a l i n t o t h e l i q u i d f i l m between t h e p a r t i c l e and t h e i n t e r f a c e . Two  f o r m u l a t i o n s a r e c o n s i d e r e d : a) d i f f u s i o n i n t h e  r e g i o n o f c o n t a c t a l o n e and b) d i f f u s i o n w i t h v i s c o u s d r a g . a) In a p p r o a c h ( a ) two b a s i c e q u a t i o n s a r e p r o p o s e d . From mass c o n s e r v a t i o n , a s s u m i n g t r a n s p o r t by d i f f u s i o n i n the f l u i d f i l m , t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d : r dr  i r  a  dr  J  D  * • *'  From t h e e q u i l i b r i u m c o n d i t i o n a t t h e i n t e r f a c e , a change i n f r e e e n e r g y a s s o c i a t e d w i t h t h e c u r v a t u r e o f t h e i n t e r f a c e i n t h e c o n t a c t r e g i o n i s compensated by a change i n c h e m i c a l p o t e n t i a l i n t h e f i l m g i v e n by E q u a t i o n 5. The e q u i l i b r i u m c o n d i t i o n i s given as: L  at  H ri/R - ^  -V^r ^ 1  =0  •••  8  where L a,t i s t h e l a t e n t heat o f f u s i o n p e r atom a„ o is a l e n g t h o f t h e o r d e r o f a m o l e c u l a r d i a m e t e r and O/R - l / p +  s  .)  i s t h e n e t c u r v a t u r e o f t h e i n t e r f a c e ; 1/R i s t h e c u r v a t u r e o f t h e i r r e g u l a r i t i e s ; o f t h e p a r t i c l e s and 1 / p * is. t h e  150 c o n t r i b u t i o n o f tlie s e p a r a t i o n between p a r t i c l e and face,  d-j i s the s e p a r a t i o n a t the c e n t e r o f the  inter-  contact  r e g i o n (see F i g . 4 ) . 7 w i t h u g i v e n by f o r m u l a  The s o l u t i o n o f E q u a t i o n p e r m i t s the r e p l a c e m e n t  o f La  and 1 / P  Q  s  6  i n E q u a t i o n 8. T h i s  l e a d s to La. where  2d.  1 R  vk.T 4 n+1  nr.o-  1  V o Dd,1  rl  o  =  0  L  l a t e n t heat per u n i t volume  r.  e f f e c t i v e c o n t a c t r a d i u s , t h a t i s the  u  r a d i u s at w h i c h s e p a r a t i o n becomes. ihflh'itg  v  =  growth v e l o c i t y  Using Equation 9 the c r i t i c a l in the f o l l o w i n g way. branches of ( d ^ / d )  determined  T h i s e q u a t i o n g i v e s two d e f i n i t e v e r s u s v (see F i g . 5 ) .  n  Q  group i n which ( d / d ) 1  v e l o c i t y can be  Q  n  decreases with v  ji.Sj  The  upper  stable.  This  i s a s s o c i a t e d w i t h the c a s e i n w h i c h the i n t e r f a c e i s f l a t and the. p a r t i c l e i s f a r from the i n t e r f a c e .  The  lower  branches, a r e uns.tab.le a g a i n s t s m a l l s e p a r a t i o n f l u c t u a t i o n s . T h e r e f o r e the c r i t i c a l  v e l o c i t y i s the l a r g e s t growth r a t e  in the s t a b l e upper c o n f i g u r a t i o n . the c r i t i c a l given  by  Maximizing  E q u a t i o n 9.  v e l o c i t y , a t which, p a r t i c l e s a r e t r a p p e d , i s  151 \ (n + 1) C.La V D/kTR ) 2  0  b)  . . . IQ  o  When t h e d r a g f o r c e g i v e n by f o r m u l a (;1 ) i s i n -  cluded i n the analysts. Equation 9 transforms to La  2d-  1  n  y.kT  6n.R R v  nr.  4 n+1 V Jo) d 1,  . . .11  d sr o h 2  where t h e l a s t term i s t h e c o n t r i b u t i o n t o t h e f r e e e n e r g y due t o t h e d r a g f o r c e w h i c h i s t r a n s m i t t e d t o t h e i n t e r face through the i r r e g u l a r i t i e s only.  R  i s the p a r t i c l e  Q  r a d i u s and h i s t h e d e p t h o f t h e c o n t a c t r e g i o n i n t o t h e solid.  T h i s i s assumed t o be i n d e p e n d e n t  of the p a r t i c l e  radius. The c r i t i c a l  v e l o c i t y then  d„hLa„d, s o 1 6nR R n 2  1 +  1+  becomes  6 R n ( n + l )V D"0.5 n  0  d hd kT  Q  s  w h i c h r e d u c e s t o E q u a t i o n 10 when d ,h, s  Q  .. .12  ]  n  0.  In E q u a t i o n 12  d-| and n a r e unknown; n i s assumed t o be 5 and d i s -  t a n c e s a r e o f t h e o r d e r o f 1Q~ . 7  Replacing the remaining  parameters with these corresponding to the water-ice s y s t e m t h e c r i t i c a l y e l o c i t y i s one o r d e r o f m a g n i t u d e l a r g e r than e x p e r i m e n t a l y a l u e s o b t a i n e d by Uhlmann. R e d u c t i o n o f t h e d i f f u s i o n c o e f f i c i e n t s f o r b u l k l i q u i d by a f a c t o r o f 400 i s r e q u i r e d t o f i t t h e o r y w i t h  experiment.  152  i/d,  V F i g u r e 5 Schematic r e p r e s e n t a t i o n of s e p a r a t i o n v s . growth v e l o c i t y f o r v a r i o u s p a r t i c l e s i z e s . The upper branches are s t a b l e and the lower are u n s t a b l e a g a i n s t f l u c t u a t i o n i n growth v e l o c i t y . T h e c r i t i c a l i s the maximum v e l o c i t y .  F i g u r e 6. The B o i l i n g and C i s s e scheme of p u s h i n g . The used i n the theory are r e p r e s e n t e d .  symbols  153  On th.e o t h e r hand and c o n t r a d i c t i n g what may he. e x p e c t e d , t h e V a l u e s g i y e n by E q u a t i o n 11 a r e a l m o s t o r d e r s o f m a g n i t u d e l e s s than t h o s e o b t a i n e d from 12.  For a p a r t i c l e of r a d i u s R  Q  three  Equation  = 2 um i n w a t e r - i c e t h e  c r i t i c a l v e l o c i t i e s c a l c u l a t e d from E q u a t i o n 11 and 12 a r e 0.6 um s e c " ' and 200 ura s e c " ' r e s p e c t i v e l y . 3  c)  B o i l i n g and Cisse" T h eo r y The main d i f f e r e n c e between t h i s t h e o r y and t h e  p r e v i o u s t h e o r y o f Uhlmann  i s t h a t d i f f u s i o n i s c o n s i d e r e d to be  the - l i m i t i n g ; c a s e f o r f l u i d t o move i n t o t h e l i q u i d  film  between t h e p a r t i c l e and i n t e r f a c e . T h e r e f o r e d i f f u s i o n i s included i n the v i s c o u s drag.  The r e p u l s i v e f o r c e f o r  p u s h i n g o r i g i n a t e s from an i n t e r f a c e c u r v a t u r e e f f e c t o n l y . The d r a g f o r c e i s r e c a l c u l a t e d t a k i n g i n t o a c c o u n t t h e curvature of the i n t e r f a c e i n the region of contact. I t s maximum v a l u e c o r r e s p o n d s .to an i n t e r f a c e w i t h s p h e r i c a l shape b e h i n d t h e p a r t i c l e .  I f the radius of the [interface  i s R/a, where R i s t h e p a r t i c l e r a d i u s and  a  i s a coef-  f i c i e n t l e s s than o r e q u a l t o 1, t h e d r a g f o r c e i s : Fta]  =  6ffin.VR /d(.l - a ) 2  2  . . . 13  where V = growth, v e l o c i t y d = P-S. d i s t a n c e , a t t h e c e n t e r o f the. c o n t a c t r e g i o n  Mechanical  e q u i l i b r i u m i s f e a s i b l e when  154 F(al 2  a 2 cr, „ > i  ... 1 4  u  s  R  T h i s i n d i c a t e s t h a t the. maximum f o r c e t r a n s m i t t e d t o t h e 2 s o l i d t h r o u g h t h e s m a l l c o n t a c t a r e a •nr i s compensated by Q  2  t h e c u r v a t u r e e f f e c t (.see F i g . 6 ) . In t h e a r e a  Trr  Q  t h e .:  t r a n s p o r t m a t e r i a l may be d i f f u s i o n o r a f l o w p r o c e s s .  In  the l i m i t , a s s u m i n g a d i f f u s i o n mechanism, f o r t h e l e a s t a d v a n t a g e o u s s e q u e n c e g i v e n by a random walk a n a l y s i s rl  < 4bD/V  ...15 < 4kT/3^nV  Where i t was used t h e S t o k e s - E i n s t e i n r e l a t i o n D =  kT/3Tuy,a  and b, t h e l a y e r h e i g h t f o r g r o w t h , was t a k e n as t h e i n teratomic distance a . In o r d e r t o make t h e i n t e r f a c e r e s p o n s i v e t o t h e p a r t i c l e , a was assumed t o have t h e e x p r e s s i o n a  =  exp-g(.d- a ) / a 0  where g. i s a p a r a m e t e r ,  ...16  0  f o r g>> d / a t h e r e i s i n t i m a t e conQ  t a c t and f o r d >> g. a , a t e n d s t o z e r o , then t h e r e i s no Q  interaction. O p e r a t i n g w i t h E q u a t i o n s 13 t o 16 t h e f o l l o w i n g  155 e x p r e s s i o n f o r ttie y e . l o c i t y t s o b t a i n e d n ;V R = w ( a ] 4 k T a 2  2  3  ] s  aj^  ...17  where ¥ ( a ) i s a f u n c t i o n o f a a c c o r d i n g t o Y(CX) = •2 a (1-«•)'' (6-1 no;) with, a maximum-va 1 ue o f f ( a ) . , = 0.34 f o r max m  3=1.  Equation  v  17 i s t h e s i m p l e t h e o r y v a l i d f o r s m a l l .  smooth s p h e r i c a l p a r t i c l e s .  F o r l a r g e p a r t i c l e s g r a v i t y and  t h e number o f i r r e g u l a r i t i e s a t w h i c h t h e i n t e r a c t i o n i s c a r r i e d o u t s h o u l d be c o n s i d e r e d .  The number o f c o n t a c t  points  N i s d i r e c t l y i n t r o d u c e d as a f a c t o r i n t h e s e c o n d member of Equation  17>.  T h i s i s . : s u p p o r t e d by t h e e x p e r i m e n t a l  observation that the c r i t i c a l  v e l o c i t y f o r a small  particle  at a g r a i n , g r a i n b o u n d a r y and t r i p l e j u n c t i o n a r e r e l a t e d by V(g)  i  V(g.b.)//2~  V(g.) = V ( t . j . ) / / 3 ~ when t h e number o f c o n t a c t p o i n t s and a l s o t h e g r a v i t y a r e i n cluded the f o l l o w i n g formulas are obtained f o r the c r i t i c a l veloci ty: R < R  b  2 2 3 T_~ _4 ¥i (_ aJ) k ,-r-i Ta  a  2,.2 3 . 2 Y ( a ) • .  .._3 -. 4¥.(«) ,, R kT a  w  n  R>R  b  v  n  R  ] s  , .. ...18(a)  Q  t  n  V  R  D  +  - ^ g A p a ^ V R  n  b  u  a  .» ...18(b) l o 7  Q  156 R»R  b  nVR ~ N 2 ( x k T a / R ^ g A p  ...18(c)  3  ls  where R  fa  b  i s t h e bump r a d i u s , g i s t h e a c c e l e r a t i o n o f  g r a v i t y and Ap i s t h e d e n s i t y d i f f e r e n c e between p a r t i c l e and  melt. A comparison  o f t h e model p r e d i c t i o n s w i t h  experimental  measurements w i l l be made i n t h e n e x t s e c t i o n . N e v e r t h e l e s s , t h i s t h e o r y does n o t p r o v i d e enough i n f o r m a t i o n t o p r e d i c t when p u s h i n g w i l l be p r e s e n t f o r a g i v e n p a r t i c l e and s o l i d i f y i n g m a t e r i a l .  I f i t i s assumed  t h a t t h e i n t e r f a c e always c u r v e s b e h i n d a f o r e i g n p a r t i c l e l e a v i n g a t h i n f i l m o f l i q u i d i n between, t h i s t h e o r y p r e d i c t s p u s h i n g always o c c u r s . M o r e o v e r , parameter  as t h e o n l y p a r t i c l e  appearing i n t h i s theory i s the p a r t i c l e r a d i u s ,  p u s h i n g s h o u l d be i n d e p e n d e n t  o f t h e type o f p a r t i c l e f o r  a g i v e n moving i n t e r f a c e . T h i s i s c l e a r l y c o n t r a d i c t o r y to what i s o b s e r v e d , as d i s c u s s e d i n s e c t i o n 2. d)  The D i s j o i n i n g P r e s s u r e and Chernov  et a l .  Theory  So f a r i t has been seen t h a t t h e models p r e s e n t e d used t h e r m o d y n a m i c q u a n t i t i e s , l i k e i n t e r f a c i a l whose v a l u e s a r e supposed  energies  t o be u n a l t e r e d by t h e f a c t t h a t  157  \\ \\ \ \ \V h  (L)  //////// (S)  F i g u r e 7. A p a r t i c l e ( s o l i d , l i q u i d or gas) i n "contact" w i t h s o l i d . T h e d i s j o i n i n g p r e s s u r e i n a gas bubble as a f u n c t i o n s e p a r a t i o n h i s measured by p r e s s i n g the bubble a g a i n s t the  a of solid,  n(h)  \\  OC \  0  \  h  F i g u r e 8. Two t y p i c a l b e h a v i o u r of the d i s j o i n i n g p r e s s u r e f o r a bubble i n a p o l a r l i q u i d ( 1 ) , and i n a n o n - p o l a r l i q u i d (.2). Branches oc and fi g i v e s t a b l e f i l m s w h i l e branch # i s u n s t a b l e .  158 the f i l m between p a r t i c l e and s o l i d i s v e r y t h i n , o f the o r d e r o f 10"^,- cm- o r 10"^ cm . 8  In a d d i t i o n , none o f the  t h e o r i e s p r e d i c t i n g p u s h i n g ( e x c e p t the Uhlmann e t a l . t h e o r y ) i n v o l v e s p r o p e r t i e s o f a l l . t h r e e media a t the same t i m e as might be e x p e c t e d  in a theory.  theory p a r t i a l l y attempts  to i n c l u d e t h i s i n t h e i r  u s i n g the c o n c e p t o f " d i s j o i n i n g  The Chernov e t a l . theory  p r e s s u r e " i n t r o d u c e d many  y e a r s ago ( s e e < F r e n k e l . Y. I . , Ch.V,  r e f . 9).  Measurements  of t h i s p r e s s u r e made by p r e s s i n g a gas b u b b l e  ( F i g . 7)  against different  s o l i d s and a v a r i e t y o f l i q u i d s gave 3 4 -2 v a l u e s o f the o r d e r o f 10 t o 10 dyn cm for distances -5  between b u b b l e and s o l i d a f the o r d e r o f 10  cm = 1,000  °A.  They o b s e r v e d two m a r k e d l y  different pressure v a r i a t i o n s  with distance f o r d i f f e r e n t  l i q u i d s as shown i n F i g . 8.  The f u l l  l i n e corresponds  the dashed  to a p o l a r l i q u i d l i k e  l i n e i s for a non-polar l i q u i d l i k e a metal.  can be o b s e r v e d t h a t the d i s j o i n i n g steadily  water,  as the i n t e r f a c e  f i n i t y (g b r a n c h )  pressure increases  i s approached  s t a r t i n g from i n -  u n t i l a maximum i s i r e a c h e d .  After this  p o i n t the p r e s s u r e d r o p s g i v i n g an u n s t a b l e f i l m (y  branch)  then r i s e s s h a r p l y as the d i s t a n c e between b u b b l e and interface  becomes v e r y s m a l l (^ b r a n c h ) .  corresponds  It  The dashed  the line  t o a n o n - p o l a r f i l m and does not show an  u n s t a b l e r e g i o n where the p r e s s u r e i s n e g a t i v e .  This dis-  j o i n i n g p r e s s u r e c o n t r i b u t e s to the chemical p o t e n t i a l the f i l m ; t h e amount g i v e n by n(h.)°,, so t h a t  of  159 u-j = -  - ir(h.)n  M  where ^  00  w  ... 19  i s t h e c h e m i c a l p o t e n t i a l o f t h e b u l k l i q u i d and n  i s t h e a t o m i c volume.  The p r e s s u r e i s c o n s i d e r e d p o s i t i v e  when t h e f i l m t e n d s t o t h i c k e n and t h e r e f o r e i n t h i s c a s e a p a r t i c l e i s pushed away from t h e i n t e r f a c e . The m i g r a t i o n of s o l i d p a r t i c l e s observed during s o l i d i f i c a t i o n  suggests  t h a t t h e same p r e s s u r e i s p r e s e n t i n t h e f i l m between s o l i d or l i q u i d p a r t i c l e s and a s o l i d - l i q u i d i n t e r f a c e . T h i s p r e s s u r e i s b u i l t up from t h r e e main i n t e r a c t i o n s Van d e r Waals Debye  (dispersion) (electrostatic)  Structural In m e t a l s  only  Van d e r Waals and s t r u c t u r a l i n t e r a c t i o n s a r e  possible.  The f i r s t c o n t r i b u t i o n f o r a s m a l l d i s t a n c e h  between p a r t i c l e and t h e f r o n t i s g i v e n by t h e e x p r e s s i o n ' ' ^ 8  n(h) B  3  =  B /h  ...20  3  3  i s a c o n s t a n t f o r t h e s y s t e m P-L-S i n c o n s i d e r a t i o n . When  Bg>0 t h e r e i s p u s h i n g , w i t h B <;0 t h e r e i s t r a p p i n g . 3  11-14 In t h e r i g u r o u s L i f s h i ' t z - V a n d e r Waals  theory  what i s c a l l e d d i s j o i n i n g p r e s s u r e i s o r i g i n a l l y c a l c u l a t e d f o r a s y s t e m s i m i l a r t o t h a t shown i n F i g . 9 a s t h e f o r c e 5  per u n i t a r e a between t h e b o d i e s 1 and 2.  This force is  1 0  1  160 g i v e n by t h e e q u a t i o n n  =  F =  ...21(a)  8uh  co  wi t h  e,(i§) - e,(iO  e (iC)  e-| (i§) + e ( U )  £ (i5) +  3  9  z  - eo(ic)  ...21(b)  e (U) 3  0  where  i s t h e d i e l e c t r i c c o n s t a n t o f t h e i t h - m e d i u m as a  f u n c t i o n o f t h e i m a g i n a r y p a r t o f t h e complex f r e q u e n c y to. These d i e l e c t r i c c o n s t a n t s a r e r e l a t e d to the imaginary p a r t s e"(to) o f t h e complex d i e l e c t r i c c o n s t a n t s the  Kramer-Kronig  through  relations co  0  The p r e s e n c e o f e"(co), w h i c h d e s c r i b e s t h e a b s o r p t i o n p r o p e r t i e s o f a s u b s t a n c e , g i v e s t h e name o f d i s p e r s i o n interaction to this force,  to p l a y s t h e r o l e o f a c h a r -  a c t e r i s t i c f r e q u e n c y f o r t h e t h r e e media P-L-S, and depending  on i t s s i g n t h e f o r c e w i l l be e i t h e r a t t r a c t i v e  or r e p u l s i v e .  In t h i s t h e o r y to>o g i v e s Bg>o and t h e f o r c e  is a t t r a c t i v e .  Chernov's convention i s opposed to t h i s ,  f o r B >o 3  the d i s j o i n i n g pressure i s p o s i t i v e , the f i l m  t e n d s t o t h i c k e n and t h e r e f o r e t h e p a r t i c l e i s r e p e l l e d . In E q u a t i o n 2 1 ( a ) and ( b ) , f o r example, i f media 1  and 2 are the same, i . e . , s o l i d and p a r t i c l e , e-j =• the f o r c e i s always a t t r a c t i v e .  I f the media  separating  s o l i d and p a r t i c l e i s vacuum £3 =• 1 and the f o r c e i s a l w a y s a t t r a c t i v e as w e l l .  But when the f i l l i n g medium i s  a l i q u i d and the two o t h e r b o d i e s the f o r c e may  are d i f f e r e n t m a t e r i a l s  be a t t r a c t i v e o r r e p u l s i v e , d e p e n d i n g  on t h e v a l u e o f to.  It is necessary  .;  to m e n t i o n t h a t  Jo or  i s no c a l c u l a t e d or measured v a l u e o f the c o n s t a n t s Bg f o r a s o l i d - l i q u i d s y s t e m o f the same m a t e r i a l  there  and  at  the m e l t i n g p o i n t , t h a t i s , f o r an a d v a n c i n g s o l i d i f y i n g interface.  f o r a r e p u l s i v e c a s e ( i . e . , to<o).  constants values -1 4 10  M o r e o v e r , t h e r e i s no t a b u l a t e d  values of The  these  measured  f o r i d e n t i c a l 1 and 3 m a t e r i a l s a r e o f the o r d e r -1 5 - 10  e r g . f o r B^ i n r e a s o n a b l e  calculated v a l u e s ' " 1 0  1 7  o f co.  1 8  agreement w i t h  For d i s t a n c e P-S  of the  larger  than , the wave . l e n g t h ..at; which the i n t e r a c t i o n . i s p e r f o r m e d but. s h o r t e r ,  than 1 0 ~  4  cm, the L i s h i t z - V a n der Waal f o r c e  i s a f u n c t i o n o f the f o u r t h power o f the s e p a r a t i o n  P-S.  In the s p e c i a l c a s e i n which the l i q u i d f i l m i s a m e t a l the f o r c e i s a l w a y s a t t r a c t i v e and depends on the power o f the s e p a r a t i o n .  T h i s f o r c e i s g i v e n by  fifth the  expression fi c F  =  2  0.0034  ...23 a l 3  where a ^ i s the d.c. c o n d u c t i v i t y o f the m e t a l .  162  F i g u r e 9. Three media,solid(.1). - l i q u i d or gas (31 and s o l i d , l i q u i d or gas-(.2) l e a d i n g to an i m p o r t a n t i n t e r a c t i o n at v e r y c l o s e d i s t a n c e s (.10 ^) d e s c r i b e d by the L i f s h i t z ^ V a n der Waals t h e o r y . -  r  r —  F i g u r e 10. The Chernov scheme of p u s h i n g . thoery are represented.  The symbols used  in  the  163 The  s t r u c t u r a l component o f t h e d i s j o i n i n g p r e s s u r e  is a t t r i b u t e d to the order present i n the l i q u i d near the c r y s t a l .  When two s o l i d s w i t h t h e i r a s s o c i a t e d  l i q u i d l a y e r a r e b r o u g h t t o g e t h e r an " e n t r o p y  ordered  repulsion"  a r i s e s as a r e s u l t o f t h e i n t e r a c t i o n between t h e two layers.  In a p o l a r l i q u i d i t i s e a s y t o i m a g i n e t h i s as a  dipole-dipole  i n t e r a c t i o n at the surface  which i s t r a n s m i t t e d  through the l i q u i d .  s t r u c t u r a l i n t e r a c t i o n i s strong  of the c r y s t a l s In w a t e r , t h e  enough t o e x p l a i n , by  i t s e l f , t h e r e j e c t i o n o f p a r t i c l e s a t an a d v a n c i n g w a t e r 21 2 i c e i n t e r f a c e and a l s o t h e m o t i o n o f a l o a d e d w i r e i n i c e . ' This i n t e r a c t i o n a r i s e s from the f a c t that water i s p o l a r 19 20  r  c  and a s s y m e t r i c and i c e i s n o n - p o l a r . may be p r e s e n t i n any o t h e r m a t e r i a l M e t a l s , however, a r e n o t i n c l u d e d A l t h o u g h , as p o i n t e d  '  This  interaction  w i t h t h e same  properties.  i n t h i s group o f m a t e r i a l s .  o u t i n r e f . 10, any i n t e r f a c i a l  order-  i n g i n t e r a c t i o n s h o u l d be e f f e c t i v e t h r o u g h a Van d e r W a a l s ; force.  T h e r e f o r e i t may be e x p e c t e d t h a t t h i s f o r c e would  be i n c l u d e d  i n the general  Van. der Waals f o r c e s  expression  21.  Therefore i f the  can. account f o r t h e r e j e c t i o n o f p a r t i c l e s  a t a s o l i d - l i q u i d i n t e r f a c e t h e s i g n o f t h e c o n s t a n t ui may d e t e r m i n e when p u s h i n g o c c u r s i n a g i v e n which t h e r e  PLS s y s t e m i n  i s no 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 i m i l a r t o t h e two p r e v i o u s t h e o r i e s , Chernov e t a l .  164 proposed .two e q u a t i o n s  f o r the e q u i l i b r i u m .  For thermo-  dynamic e q u i l i b r i u m o f t h e i n t e r f a c e t h e f o l l o w i n g is  equation  proposed ls ]  fia  (R 1  +  2. ) AS(T -T)-ASm [ C ( r ) - r ]  R  ]  +  a,  -ji(h)fi = 0 ::.24  where  Q,  = atomic = S-L  R.j , R AS  volume o f the l i q u i d  i n t e r f a c i a l energy  = main c r y s t a l  2  surface curvature  radii  = entropy of f u s i o n  co  -T--1 = i n t e r f a c e t e m p e r a t u r e f o r p l a n e i n t e r f a c e T  = i n t e r f a c e t e m p e r a t u r e f o r t h e cave r e g i o n behind  m  the  particle  = liquidus slope  C(r) = c o n c e n t r a t i o n of s o l u t e at a d i s t a n c e r from a v e r t i c a l c e n t e r p l a n e o f the p a r t i c l e as shown i n F i g u r e C°°  10.  = c o n c e n t r a t i o n of s o l u t e f a r from the , p a r t i c l e where the i n t e r f a c e i s f l a t .  In E q u a t i o n  24 the c o n t r i b u t i o n s o f the Gibb-Thomson  e f f e c t , due to t e m p e r a t u r e ,  i m p u r i t y and d i s j o i n i n g  pressure  are i n c l u d e d in that order.  The s o l u t i o n o f E q u a t i o n  24 f o r t h e i n t e r f a c e shape  i n pure m a t e r i a l s i s a p p r o x i m a t e d by a p a r a b o l o i d and a p l a n e as f o l l o w s :  Z(r)  H  -R + h£ - v r /2R  r>r  where Z ( r ) i s t h e d i s t a n c e  of the contact  the p a r t i c l e - i n t e r f a c e d i s t a n c e = R/R .  plane,  passing  region,  at the center  ho i s  of the contact  R i s t h e p a r t i c l e r a d i u s and  2  is the distance  o  from a . h o r i z o n t a l  i n t e r f a c e , ro i s the radius k> = R/R|  25  o  of the p a r t i c l e , to the s o l i d - 1 i q u i d  through the center  region,  165  r<rg  outside  the contact  region.  These  symbols  a r e shown i n F i g . 10.  The cluding  drag force onto the p a r t i c l e i s c a l c u l a t e d i n the fact that the sink  p a r t i c l e i s curved. 25 i s u s e d .  (the i n t e r f a c e ) behind the  F o r t h e shape o f t h e i n t e r f a c e  W i t h t h e same e q u a t i o n  the d i s j o i n i n g  Equation pressure  over the whole p a r t i c l e i s c a l c u l a t e d ; t h i s gives the repulsive  force onto the p a r t i c l e .  Equating  to t h e d i s j o i n i n g f o r c e t h e e q u a t i o n equilibrium 67rnVR  i s obtained  (l-v) h  l-v(3-y)-  2  *  0  3  (l.-v)  + v  h(r )  h 2  < o>j  R  h|  w h e r e t h e f i r s t member  v  form:  ,2  Q  L-  B  , .  force  f o r the mechanical  i n the following ,  2  the drag  -  h^Cr")  r  26  i s t h e drag f o r c e and t h e second i s  166 the r e p u l s i v e f o r c e due t o t h e d i s j o i n i n g adequate  s i m p l i f i c a t i o n , the c r i t i c a l  pressure.vr'With  v e l o c i t i e s a r e un-  ambiguously  o b t a i n e d from E q u a t i o n 26 f o r s m a l l and l a r g e  particles.  The r e s p e c t i v e f o r m u l a s g i v e n f o r t h e " e x a c t "  solution  o f E q u a t i o n 26 a r e :  Small P a r t i c l e s V  c  2 P><A /I 0.14 a 1/3 = -r^— B~ nR BR  . . .27  3  2 R>A / l  Large P a r t i c l e s V  c  + 0.15  B /riR.) 3  (GAS/B fi)  where X and 1 a r e c h a r a c t e r i s t i c relations B  X  =  (aa^ /GAS)  = 10" erg,  ...28  1 / 4  3  1 / 2  S  l e n g t h s g i v e n by t h e  and 1 = ( B f t / G A S )  1 / 4  3  .  For  a = 3 x l 0 " c m , AS = 2 k ; 3 x l 0 " e r g d e g " , 2 1 2 2 G = 10 deg cm" , X '/I = 5 x 1 0 " cm. T h a t i s t h e t r a n s i t i o n from s m a l l t o l a r g e p a r t i c l e s o c c u r s a t R= 500 ym. -4/3 At t;h'is r a di us t h e c r i t i c a l v e l o c i t y changes from a R to R" d e p e n d e n c e on t h e p a r t i c l e r a d i u s . One i m p o r t a n t 1 5  3  2 3  3  1 6  1  1  f a c t o f t h i s t h e o r y i s t h a t E q u a t i o n s 27 and 28 do n o t c o n t a i n unknown p a r a m e t e r s ;  i e x c e p t - f o r the value o f B  e s t i m a t e d from t h o s e o b t a i n e d f o r systems force i s attractive  as i t was m e n t i o n e d  3  which i s  i n which t h e above.  In t h e n e x t s e c t i o n the t h e o r i e s a r e compared w i t h experiments.  Appendix-I I CALCULATION OF u  168 APPENDIX I I given by formula 19 i s : co  e - [ ( i - 6=3(12) -^ (iE) - ^ ( i ^ ) 2  0  3  d5 e (ic) + e (i5) e ( i C ) + 63(1?) T  3  19  2  substituting the d i e l e c t r i c constants f o r 2 CO  the integral i s : (O  =  1'2 4 A(0_p  T  2 / I p23 / ~  A(JO  1 3  0  <  :  ?  i  -  _d£  -  rrr  -pi ) 3  (e  2 +  ...A-l V3>  where 2  _  Aco„ • •  -  2  _  P1J  2 PT  co .  -  2 PJ  co  and 1 , 2 , 2 ,  Using the fact that the integrant of A-l i s an even function, we can evaluate the same integral within -«>,<» l i m i t s , i . e . : 2/  fU)-d£ = / f(?)dg  = . ZiriX Res f(0  0  where the residues of f(5) in the upper simi-space should be considered. The two p01 es with\pqs 11ive imaginary parts are:  Evaluating the residue by the formula: Res f U ) = a  = lim f U ) o 9 s  we obtain: Res f U ) = 1  •  Hl3  21  (  "  V3  <W  Similarly Res ?  f(s) =  ?  i f7  2i  2  2  w  '( o ) 2  p  2  3  p  1  -  3  2 U  p  2  3  )  Substituting in formula A-l and operating i t i s f i n a l l y obtained  w =  A t 0  .2 i3  A w  P  P  .2 23 a  2 ' "023  TT  CO  -1/2  ' "013  2 _ CO2/ 2o o ~ CO2 xJ p23 pin o3 I CO p23 pl3' 0  n o  To obtain the values of co . = n 4irNe p  2  with n = valence, N =  rri  no atoms/unit volume the following procedure i s followed: Liquid Pb 46 Hodgson  obtained the following relation between constants fly  A co 2  p  = 1.344xl0"  4  p  Q  elf  1  where y = c o l l i s i o n frequency P = e l e c t r i c a l r e s i s t i v i t y from optical constants, f o r lead q  the values g i v e n are: 47  fiy = 2.86 eV and p = 98.2 y ohm-cm. Q  Substituting values f l u , ^ ) ' = 14.7 eV 1  or ^ X  - 2.23xl0  16  sec"  1  = 845.3 A°  P b ( 1 )  P  Solid Pb The following values are used: n = 4 45 m* = e f f e c t i v e mass = 1.12 m e = 4.8GxlO" ues 10  oo  m = 9.10x10  Na.d _ —6.022xl0 x 11 _ = TT - 3.19x1022 Pb 207.19 23  nN  = N  Na  = Arogadro's number  d  = density = atomic weight  M  M n  M  r  Pb  gr Q  f  N  M  Therefore, substituting values f J  b  P  (  s  )  =  12,52 eV  or < /  b ( s )  P Pb(s)  *p  = 1 .goxlO^sec" = 992.1 A°  1  Q  V  L  N  171 Sol)id Fe 1/N = 0.01195 m at 400°C (Metals Handbook, Vol. 1, 9th Ed. ASM) 3  ri  m*  = 2.1  ) ) ref. = 12 m )  5Q  .tt  with these values *J*M  =4.5eV  or  ^ > s  =  ./e(s) P  =  6  2  .  8 2 x ] Q  7g4  A  15  s e c  -l  o  Substituting these values of co i n co P to = 1.904xl0 sec 15  and f o r  _1  from Equation 19 (a) B, = ^  =  2.54xl0" erg. 14  NOTE: tt This anomalous very high value of effective mass i s typical of the t r a n s i t i o n metals, i . e . Fe, Co, N i , Pd, Pt.  

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